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+{"text":"\\section{Introduction}\nLearning to learn is essential in human intelligence but is still a wide area of research in machine learning.\n\\textit{Meta-learning} has emerged as a popular approach to enable models to perform well on new tasks using limited data.\nIt involves first a \\textit{meta-training} process, when the model learns valuable features from a set of tasks.\nThen, at test time, using only few datapoints from a new, unseen task, the model (1) \\textit{adapts} to this new task (i.e., performs \\textit{few-shot learning} with \\textit{context data}), \nand then (2) \\textit{infers} by making predictions on new, unseen \\textit{query inputs} from the same task.\nA popular baseline for meta-learning, which has attracted a large amount of attention, is Model-Agnostic Meta-Learning (MAML) \\citep{maml}, in which the adaptation process consists of fine-tuning the parameters of the model via gradient descent.\n\nHowever, meta-learning methods can often struggle in several ways when deployed in challenging real-world scenarios. First, when context data is too limited to fully identify the test-time task, accurate prediction can be challenging. As these predictions can be untrustworthy, this necessitates the development of meta-learning methods that can express uncertainty during adaptation \\citep{bayesian_maml, alpaca}. In addition, meta-learning models may not successfully adapt to ``unusual'' tasks, i.e., when test-time context data is drawn from an \\textit{out-of-distribution} (OoD) task not well represented in the training dataset \\citep{ood_maml, meta_learning_ood}.\nFinally, special care has to be taken when learning tasks that have a large degree of heterogeneity.\nAn important example is the case of tasks with a \\textit{multimodal} distribution, i.e., when there are no common features shared across all the tasks, but the tasks can be broken down into subsets (modes) in a way that the ones from the same subset share common features \\citep{mmaml}.\n\n\\textbf{Our contributions.}~\\, We present \\textsc{UnLiMiTD}{} (\\textit{uncertainty-aware meta-learning for multimodal task distributions}), a novel meta-learning method that leverages probabilistic tools to address the aforementioned issues.\nSpecifically, \\textsc{UnLiMiTD}{} models the true distribution of tasks with a learnable distribution constructed over a linearized neural network and uses analytic Bayesian inference to perform uncertainty-aware adaption.\nWe present three variants (namely, \\approach-\\textsc{I}, \\approach-\\textsc{R}{}, and \\approach-\\textsc{F}) that reflect a trade-off between learning a rich prior distribution over the weights and maintaining the full expressivity of the network; we show that \\approach-\\textsc{F}{} strikes a balance between the two, making it the most appealing variant.\nFinally, we demonstrate that (1) our method allows for efficient probabilistic predictions on in-distribution tasks, that compare favorably to, and in most cases outperform, the existing baselines, (2) it is effective in detecting context data from OoD tasks at test time, and that (3) both these findings continue to hold in the multimodal task-distribution setting.\n\nThe rest of the paper is organized as follows. Section~\\ref{sec:problem_statement} formalizes the problem. Section~\\ref{sec:background} presents background information on the linearization of neural networks and Bayesian linear regression.\nWe detail our approach and its three variants in Section~\\ref{sec:approach}.\nWe discuss related work in detail in Section~\\ref{sec:related_work}. Finally, we present our experimental results concerning the performance of \\textsc{UnLiMiTD}{} in Section~\\ref{sec:results} and conclude in Section~\\ref{sec:conclusion}.\n\n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{images\/flowchart.pdf}\n    \\caption{ The true task distribution $p(f)$ can be multimodal, i.e., containing multiple clusters of tasks (e.g., lines and sines). Our approach \\textsc{UnLiMiTD}{} fits $p(f)$ with a parametric, tuneable distribution $\\Tilde{p}_\\xi(f)$ yielded by Bayesian linear regression on a linearized neural network.}\n    \\label{fig:flowchart}\n\\end{figure}\n\n\\section{Problem statement}\\label{sec:problem_statement}\nA task $\\mathcal{T}^i$ consists of a function $f_i$ from which data is drawn.\nAt test time, the prediction steps are broken down into (1) \\textit{adaptation}, that is identifying $f_i$ using $K$ context datapoints $(\\xinput^i, \\youtput^i)$ from the task, and (2) \\textit{inference}, that is making predictions for $f_i$ on the \\textit{query inputs} $\\xinput^i_*$.\nLater the predictions can be compared with the \\textit{query ground-truths} $\\youtput^i_*$ to estimate the quality of the prediction, for example in terms of mean squared error (MSE).\nThe meta-training consists in learning valuable features from a \\textit{cluster of tasks}, which is a set of similar tasks (e.g., sines with different phases and amplitudes but same frequency), so that at test time the predictions can be accurate on tasks from the same cluster.\nWe take a probabilistic, functional perspective and represent a cluster by $p(f)$, a theoretical distribution over the function space that describes the probability of a task belonging to the cluster.\nLearning $p(f)$ is appealing, as it allows for performing OoD detection in addition to making predictions. Adaptation amounts to computing the conditional distribution given test context data, and one can obtain an uncertainty metric by evaluating the negative log-likelihood (NLL) of the context data under $p(f)$.\n\nThus, our goal is to construct a parametric, learnable functional distribution $\\Tilde{p}_\\xi(f)$ that approaches the theoretical distribution $p(f)$, with a structure that allows tractable conditioning and likelihood computation, even in deep learning contexts.\nIn practice, however, we are not given $p(f)$, but only a meta-training dataset $\\mathcal{D}$ that we assume is sampled from $p(f)$: $\\mathcal{D}=\\{ (\\widetilde{\\xinput}^i, \\widetilde{\\youtput}^i)\\}_{i=1}^{N}$, where $N$ is the number of tasks available during training, and $(\\widetilde{\\xinput}^i, \\widetilde{\\youtput}^i) \\sim \\mathcal{T}^i$ is the entire pool of data from which we can draw subsets of context data $(\\xinput^i, \\youtput^i)$.\nConsequently, in the meta-training phase, we aim to optimize $\\Tilde{p}_\\xi(f)$ to capture properties of $p(f)$, using only the samples in $\\mathcal{D}$.\n\nOnce we have $\\Tilde{p}_\\xi(f)$, we can evaluate it both in terms of how it performs for few-shot learning (by comparing the predictions with the ground truths in terms of MSE), as well as for OoD detection \n(by measuring how well the NLL of context data serves to classify in-distribution tasks against OoD tasks, measured via the AUC-ROC score). \\nopagebreak[4]\n\n\n\n\n\\section{Background}\n\\label{sec:background}\n\\subsection{Bayesian linear regression and Gaussian Processes}\n\\label{sec:reglin}\nEfficient Bayesian meta-learning requires a tractable inference process at test time. In general, this is only possible analytically in a few cases.\nOne of them is the Bayesian linear regression with Gaussian noise and a Gaussian prior on the weights. \nViewing it from a nonparametric, functional approach, this model is equivalent to a Gaussian process (GP) \\citep{rasmussen}.\n\nLet ${\\bm{X}} = ({\\bm{x}}_1, \\dots, {\\bm{x}}_K) \\in \\mathbb{R}^{{N_x} \\times K}$ be a batch of $K$ ${N_x}$-dimensional inputs, and let ${\\bm{y}} = ({\\bm{y}}_1, \\dots, {\\bm{y}}_K) \\in \\mathbb{R}^{{N_y} K}$ be a vectorized batch of ${N_y}$-dimensional outputs. In the Bayesian linear regression model, these quantities are related according to\n$\n    {\\bm{y}} = \\phi({\\bm{X}})^\\top \\hat{\\param} + \\varepsilon \\in \\mathbb{R}^{{N_y} K}\n$\nwhere $\\hat{\\param} \\in \\mathbb{R}^P$ are the weights of the model, and the inputs are mapped via $\\phi:\\mathbb{R}^{{N_x} \\times K} \\rightarrow \\mathbb{R}^{P \\times {N_y} K}$. Notice how this is a generalization of the usual one-dimensional linear regression (${N_y}=1$).\n\n\nIf we assume a Gaussian prior on the weights $\\hat{\\param} \\sim \\mathcal{N}({\\bm{\\mu}}, {\\bm{\\Sigma}})$ and a Gaussian noise $\\varepsilon \\sim \\mathcal{N}(\\bm{0}, {\\bm{\\Sigma}}_\\varepsilon)$ with ${\\bm{\\Sigma}}_\\varepsilon = \\sigma_\\varepsilon^2 {\\bm{I}}$, then the model describes a multivariate Gaussian distribution on ${\\bm{y}}$ for any ${\\bm{X}}$. Equivalently, this means that this model describes a GP distribution over functions, with mean and covariance function (or kernel)\n\\begin{align}\n\\begin{split}\n\\label{eq:prior_pred_dist}\n{\\bm{\\mu}}_{\\text{prior}} ({\\bm{x}}_t) & = \\phi({\\bm{x}}_t)^\\top {\\bm{\\mu}}, \\\\\n\\text{cov}_{\\text{prior}}  ({\\bm{x}}_{t_1}, {\\bm{x}}_{t_2}) & =  \\phi({\\bm{x}}_{t_1})^\\top {\\bm{\\Sigma}} \\phi({\\bm{x}}_{t_2}) + {\\bm{\\Sigma}}_\\varepsilon  =: k_{\\bm{\\Sigma}}({\\bm{x}}_{t_1}, {\\bm{x}}_{t_2}) + {\\bm{\\Sigma}}_\\varepsilon .\n\\end{split}\n\\end{align}\nThis GP enables tractable computation of the likelihood of any batch of data $({\\bm{X}}, {\\bm{Y}})$ given this distribution over functions. The structure of this distribution is governed by the feature map $\\phi$ and the prior over the weights, specified by ${\\bm{\\mu}}$ and ${\\bm{\\Sigma}}$.\n\nThis distribution can also easily be conditioned to perform inference. Given a batch of data $({\\bm{X}}, {\\bm{Y}})$, the posterior predictive distribution is also a GP, with an updated mean and covariance function\n\\begin{align}\n    \\label{eq:post_pred_dist}\n    \\begin{split}\n        {\\bm{\\mu}}_{\\text{post}} ({\\bm{x}}_{t_*}) & =  k_{\\bm{\\Sigma}}({\\bm{x}}_{t_*}, {\\bm{X}}) \\left( k_{\\bm{\\Sigma}}({\\bm{X}}, {\\bm{X}}) + {\\bm{\\Sigma}}_\\varepsilon \\right)^{-1} {\\bm{Y}}, \\\\\n        \\text{cov}_{\\text{post}}  ({\\bm{x}}_{{t_1}_*}, {\\bm{x}}_{{t_2}_*}) & =  k_{\\bm{\\Sigma}}({\\bm{x}}_{{t_1}_*}, {\\bm{x}}_{{t_2}_*}) - k_{\\bm{\\Sigma}}({\\bm{x}}_{{t_1}_*}, {\\bm{X}}) \\left( k_{\\bm{\\Sigma}}({\\bm{X}}, {\\bm{X}}) + {\\bm{\\Sigma}}_\\varepsilon \\right)^{-1} k_{\\bm{\\Sigma}}({\\bm{X}}, {\\bm{x}}_{{t_2}_*}).\n    \\end{split}\n\\end{align}\nHere, ${\\bm{\\mu}}_{\\text{post}}({\\bm{X}}_*)$ represents our model's adapted predictions for the test data, which we can compare to ${\\bm{Y}}_*$ to evaluate the quality of our predictions, for example, via mean squared error (assuming that test data is clean, following \\citet{rasmussen}). \nThe diagonal of $\\text{cov}_{\\text{post}}({\\bm{X}}_*, {\\bm{X}}_*)$ can be interpreted as a per-input level of confidence that captures the ambiguity in making predictions with only a limited amount of context data.\n\n\\subsection{The linearization of a neural network yields an expressive linear regression model}\n\\label{sec:linearization}\nAs discussed, the choice of feature map $\\phi$ plays an important role in specifying a linear regression model.\nIn the deep learning context, recent work has demonstrated that the linear model obtained when linearizing a deep neural network with respect to its weights at initialization, wherein the Jacobian of the network operates as the feature map, can well approximate the training behavior of wide nonlinear deep neural networks \\citep{jacot,nonlinear,liu2020linearity,shallow_nns_infinite_width,dnns_infinite_wdith}.\n\n\nLet $f$ be a neural network $f: \\left({\\bm{\\theta}}, {\\bm{x}}_t \\right) \\mapsto {\\bm{y}}_t$, where ${\\bm{\\theta}} \\in \\mathbb{R}^{P}$ are the parameters of the model, ${\\bm{x}} \\in \\mathbb{R}^{{N_x}}$ is an input and ${\\bm{y}} \\in \\mathbb{R}^{{N_y}}$ an output.\nThe linearized network (w.r.t. the parameters) around $\\param_0$ is\n\\begin{displaymath}\n    f({\\bm{\\theta}}, {\\bm{x}}_t) - f(\\param_0, {\\bm{x}}_t) \\approx {\\bm{J}}_{\\bm{\\theta}}( f )(\\param_0, {\\bm{x}}_t) ({\\bm{\\theta}} - \\param_0),\n\\end{displaymath}\nwhere ${\\bm{J}}_{\\bm{\\theta}}(f)(\\cdot, \\cdot): \\mathbb{R}^P \\times \\mathbb{R}^{N_x} \\rightarrow \\mathbb{R}^{{N_y} \\times P}$ is the Jacobian of the network (w.r.t. the parameters).\n\nIn the case where the model accepts a batch of $K$ inputs ${\\bm{X}} = ({\\bm{x}}_1, \\dots, {\\bm{x}}_K)$ and returns ${\\bm{Y}} = ({\\bm{y}}_1, \\dots, {\\bm{y}}_K)$, we generalize $f$ to $g: \\mathbb{R}^P \\times \\mathbb{R}^{{N_x} \\times K}  \\rightarrow \\mathbb{R}^{{N_y} \\times K}$, with ${\\bm{Y}} = g({\\bm{\\theta}}, {\\bm{X}})$.\nConsequently, we generalize the linearization:\n\\begin{displaymath}\ng({\\bm{\\theta}}, {\\bm{X}}) - g(\\param_0, {\\bm{X}}) \\approx {\\bm{J}}(\\param_0, {\\bm{X}}) ({\\bm{\\theta}} - \\param_0),\n\\end{displaymath}\nwhere ${\\bm{J}}(\\cdot, \\cdot): \\mathbb{R}^P \\times \\mathbb{R}^{{N_x} \\times K} \\rightarrow \\mathbb{R}^{{N_y} K \\times P}$ is a shorthand for ${\\bm{J}}_{\\bm{\\theta}}(g)(\\cdot, \\cdot)$.\nNote that we have implicitly vectorized the outputs, and throughout the work, we will interchange the matrices $\\mathbb{R}^{{N_y} \\times K}$ and the vectorized matrices $\\mathbb{R}^{{N_y} K}$.\n\nThis linearization can be viewed as the ${N_y} K$-dimensional linear regression\n\\begin{equation}\n    \\label{eq:linearized_network}\n    {\\bm{z}} = \\phi_{\\param_0}({\\bm{X}})^\\top \\hat{\\param} \\in \\mathbb{R}^{{N_y} K},\n\\end{equation}\nwhere the feature map $\\phi_{\\param_0}(\\cdot): \\mathbb{R}^{{N_x} \\times K} \\rightarrow \\mathbb{R}^{P \\times {N_y} K}$ is the transposed Jacobian ${\\bm{J}}(\\param_0, \\cdot)^\\top$.\nThe parameters of this linear regression $\\hat{\\param} = \\left( {\\bm{\\theta}} - \\param_0 \\right)$ are the \\textit{correction} to the parameters chosen as the linearization point.\nEquivalently, this can be seen as a kernel regression with the kernel $ k_{\\param_0}({\\bm{X}}_1,{\\bm{X}}_2) = {\\bm{J}}(\\param_0, {\\bm{X}}_1) {\\bm{J}}(\\param_0, {\\bm{X}}_2)^\\top$, which is commonly referred to as the Neural Tangent Kernel (NTK) of the network. Note that the NTK depends on the linearization point $\\param_0$. \nBuilding on these ideas, \\citet{maddox} show that the NTK obtained via linearizing a DNN \\textit{after} it has been trained on a task yields a GP that is well-suited for adaptation and fine-tuning to new, similar tasks. Furthermore, they show that networks trained on similar tasks tend to have similar Jacobians, suggesting that neural network linearization can yield an effective model for multi-task contexts such as meta-learning. In this work, we leverage these insights to construct our parametric functional distribution $\\Tilde{p}_\\xi(f)$ via linearizing a neural network model.\n\n\\section{Our approach: \\textsc{UnLiMiTD}}\n\\label{sec:approach}\nIn this section, we describe our meta-learning algorithm \\textsc{UnLiMiTD}{} and the construction of a parametric functional distribution $\\Tilde{p}_\\xi(f)$ that can model the true underlying distribution over tasks $p(f)$.\nFirst, we focus on the single cluster case, where a Gaussian process structure on $\\Tilde{p}_\\xi(f)$ can effectively model the true distribution of tasks, and detail how we can leverage meta-training data $\\mathcal{D}$ from a single cluster of tasks to train the parameters $\\xi$ of our model.\nNext, we will generalize our approach to the multimodal setting, with more than one cluster of tasks. Here, we construct $\\Tilde{p}_\\xi(f)$ as a mixture of GPs and develop a training approach that can automatically identify the clusters present in the training dataset without requiring the meta-training dataset to contain any additional structure such as cluster labels.\n\n\\subsection{Tractably structuring the prior predictive distribution over functions via a Gaussian distribution over the weights}\nIn our approach, we choose $\\Tilde{p}_\\xi(f)$ to be the \nGP distribution over functions that arises from a Gaussian prior on the weights of the linearization of a neural network (\\eqref{eq:linearized_network}). Consider a particular task $\\mathcal{T}^i$ and a batch of $K$ context data $(\\xinput^i, \\youtput^i)$.\nThe resulting prior predictive distribution, derived from \\eqref{eq:prior_pred_dist} after evaluating on the context inputs, is ${\\bm{Y}} | \\xinput^i \\sim \\mathcal{N}( {\\bm{\\mu}}_{\\youtput \\mid \\xcontextinput}, {\\bm{\\Sigma}}_{\\youtput \\mid \\xcontextinput})$, where\n\\begin{equation}\n    \\label{eq:prior_pred_dist_ntk}\n    {\\bm{\\mu}}_{\\youtput \\mid \\xcontextinput} = {\\bm{J}}(\\param_0, \\xinput^i) {\\bm{\\mu}}, \\quad {\\bm{\\Sigma}}_{\\youtput \\mid \\xcontextinput} = {\\bm{J}}(\\param_0, \\xinput^i) {\\bm{\\Sigma}} {\\bm{J}}(\\param_0, \\xinput^i)^\\top + {\\bm{\\Sigma}}_\\varepsilon.\n\\end{equation}\nIn this setup, the parameters $\\xi$ of $\\Tilde{p}_\\xi(f)$ that we wish to optimize are the linearization point $\\param_0$, and the parameters of the prior over the weights $({\\bm{\\mu}}, {\\bm{\\Sigma}})$.\nGiven this Gaussian prior, it is straightforward to compute the joint NLL of the context labels $\\youtput^i$,\n\\begin{align}\n\\label{eq:single-nll}\n    \\mathrm{NLL}(\\xinput^i, \\youtput^i) = \\frac12\\left( \\left\\| \\youtput^i - {\\bm{\\mu}}_{\\youtput \\mid \\xcontextinput} \\right\\|^2_{{\\bm{\\Sigma}}_{\\youtput \\mid \\xcontextinput}^{-1}} + \\log\\det {\\bm{\\Sigma}}_{\\youtput \\mid \\xcontextinput} + {N_y} K \\log 2 \\pi \\right).\n\\end{align}\nThe NLL (a) serves as a loss function quantifying the quality of $\\xi$ during training and (b) serves as an uncertainty signal at test time to evaluate whether context data $(\\xinput^i, \\youtput^i)$ is OoD.\nGiven this model, \\textit{adaptation} is tractable as we can condition this GP on the context data analytically. In addition, we can efficiently make probabilistic predictions by evaluating the mean and covariance of the resulting posterior predictive distribution on the query inputs, using \\eqref{eq:post_pred_dist}.\n\n\\subsubsection{Parameterizing the prior covariance over the weights}\n\\label{sec:prior_covariance}\nWhen working with deep neural networks, the number of weights $P$ can surpass $10^6$. While it remains tractable to deal with $\\param_0$ and ${\\bm{\\mu}}$, whose memory footprint grows linearly with $P$, it can quickly become intractable to make computations with (let alone store) a dense prior covariance matrix over the weights ${\\bm{\\Sigma}} \\in \\mathbb{R}^{P \\times P}$. Thus, we must impose some structural assumptions on the prior covariance to scale to deep neural network models.\n\n\n\\textbf{Imposing a unit covariance.}~\\, One simple way to tackle this issue would be to remove ${\\bm{\\Sigma}}$ from the learnable parameters $\\xi$, i.e., fixing it to be the identity ${\\bm{\\Sigma}} = {\\bm{I}}_{P}$. In this case, $\\xi = (\\param_0, {\\bm{\\mu}})$. \nThis computational benefit comes at the cost of model expressivity, as we lose a degree of freedom in how we can optimize our learned prior distribution $\\Tilde{p}_\\xi(f)$. In particular, we are unable to choose a prior over the weights of our model that captures correlations between elements of the feature map.\n\n\\textbf{Learning a low-dimensional representation of the covariance.}~\\,\nAn alternative is to learn a low-rank representation of ${\\bm{\\Sigma}}$, allowing for a learnable weight-space prior covariance that can encode correlations. Specifically, we consider a covariance of the form ${\\bm{\\Sigma}} = {\\bm{Q}}^\\top \\diag{{\\bm{s}}^2} {\\bm{Q}}$, where ${\\bm{Q}}$ is a fixed projection matrix on an $s$-dimensional subspace of $\\mathbb{R}^{P}$, while ${\\bm{s}}^2$ is learnable.\nIn this case, the parameters that are learned are $\\xi = (\\param_0, {\\bm{\\mu}}, {\\bm{s}})$.\nWe define ${\\bm{S}} := \\diag{{\\bm{s}}^2}$.\nThe computation of the covariance of the prior predictive (\\eqref{eq:prior_pred_dist_ntk}) could then be broken down into two steps:\n\\begin{displaymath}\n\\left\\{\n    \\begin{array}{l}\n        A := {\\bm{J}}(\\param_0, \\xinput^i) {\\bm{Q}}^\\top \\\\\n        {\\bm{J}}(\\param_0, \\xinput^i) {\\bm{\\Sigma}} {\\bm{J}}(\\param_0, \\xinput^i)^\\top = A {\\bm{S}} A^\\top \n    \\end{array}\n\\right.\n\\end{displaymath}\nwhich requires a memory footprint of $O(P(s + {N_y} K) )$, if we include the storage of the Jacobian.\nBecause ${N_y} K \\ll P$ in typical deep learning contexts, it suffices that $s \\ll P$ so that it becomes tractable to deal with this new representation of the covariance.\n\n\\textbf{A trade-off between feature-map expressiveness and learning a rich prior over the weights.} Note that even if a low-dimensional representation of ${\\bm{\\Sigma}}$ enriches the prior distribution over the weights, it also restrains the expressiveness of the feature map in the kernel by projecting the $P$-dimensional features ${\\bm{J}}(\\param_0, {\\bm{X}})$ on a subspace of size $s \\ll P$ via ${\\bm{Q}}$.\nThis presents a trade-off: we can use the full feature map, but limit the weight-space prior covariance to be the identity matrix by keeping ${\\bm{\\Sigma}} = {\\bm{I}}$ (case \\approach-\\textsc{I}). Alternatively, we could learn a low-rank representation of ${\\bm{\\Sigma}}$ by randomly choosing $s$ orthogonal directions in $\\mathbb{R}^{P}$, with the risk that they could limit the expressiveness of the feature map if the directions are not relevant to the problem that is considered (case \\approach-\\textsc{R}).\nAs a compromise between these two cases, we can choose the projection matrix more intelligently and project to the most impactful subspace of the full feature map --- in this way, we can reap the benefits of a tuneable prior covariance while minimizing the useful features that the projection drops. To select this subspace, we construct this projection map by choosing the top $s$ eigenvectors of the Fisher information matrix (FIM) evaluated on the training dataset $\\mathcal{D}$ (case \\approach-\\textsc{F}). Recent work has shown that the FIM for deep neural networks tends to have rapid spectral decay \\citep{scod}, which suggests that keeping only a few of the top eigenvectors of the FIM is enough to encode an expressive task-tailored prior. See Appendix~\\ref{app:fim} for more details. \n\n\\subsubsection{Generalizing the structure to a mixture of Gaussians}\n\\label{sec:mixture}\n\nWhen learning on multiple clusters of tasks, $p(f)$ can become non-unimodal, and thus cannot be accurately described by a single GP.\nInstead, we can capture this multimodality by structuring $\\Tilde{p}_\\xi(f)$ as a \\textit{mixture} of Gaussian processes.\n\n\\textbf{Building a more general structure.}~\\, We assume that at train time, a task $\\mathcal{T}^i$ comes from any cluster $\\left\\{\\mathcal{C}_j \\right\\}_{j=1}^{j=\\alpha}$ with equal probability.\nThus, we choose to construct $\\Tilde{p}_\\xi(f)$ as an equal-weighted mixture of $\\alpha$ Gaussian processes.\n\nFor each element of the mixture, the structure is similar to the single cluster case, where the parameters of the cluster's weight-space prior are given by $({\\bm{\\mu}}_j, {\\bm{\\Sigma}}_j)$. We choose to have both the projection matrix ${\\bm{Q}}$ and the linearization point $\\param_0$ (and hence, the feature map $\\phi(\\cdot) = {\\bm{J}}(\\param_0,\\cdot)$) shared across the clusters. This yields improved computational efficiency, as we can compute the projected features once, simultaneously, for all clusters.\nThis yields the parameters $\\xi_\\alpha = (\\param_0, {\\bm{Q}}, ({\\bm{\\mu}}_1, {\\bm{s}}_1), \\ldots, ({\\bm{\\mu}}_\\alpha, {\\bm{s}}_\\alpha))$.\n\nThis can be viewed as a mixture of linear regression models, with a common feature map but \nseparate, independent prior distributions over the weights for each cluster. These separate distributions are encoded using the low-dimensional representations ${\\bm{S}}_j$ for each ${\\bm{\\Sigma}}_j$. \nNotice how this is a generalization of the single cluster case, for when $\\alpha=1$, $\\Tilde{p}_\\xi(f)$ becomes a Gaussian and $\\xi_\\alpha = \\xi$\\footnote{In theory, it is possible to drop ${\\bm{Q}}$ and extend the identity covariance case to the multi-cluster setting; however, this leads to each cluster having an identical covariance function, and thus is not effective at modeling heterogeneous behaviors among clusters.}.\n\n\n\\textbf{Prediction and likelihood computation.}~\\, The NLL of a batch of inputs under this mixture model can be computed as\n\\begin{equation}\n    \\label{eq:nll_mixture}\n    \\mathrm{NLL}_{\\text{mixt}}(\\xinput^i, \\youtput^i) = \\log \\alpha - \\log \\add \\exp (-\\mathrm{NLL}_1(\\xinput^i, \\youtput^i), \\ldots, -\\mathrm{NLL}_\\alpha(\\xinput^i, \\youtput^i)),\n\\end{equation}\nwhere $\\mathrm{NLL}_j(\\xinput^i, \\youtput^i)$ is the NLL with respect to each individual Gaussian, as computed in \\eqref{eq:single-nll}, and $\\log\\add\\exp$ computes the logarithm of the sum of the exponential of the arguments, taking care to avoid underflow issues.\n\nTo make exact predictions, we would require conditioning this mixture model. As this is not directly tractable, we propose to first \\textit{infer the cluster} from which a task comes from, by identifying the Gaussian $\\mathcal{G}_{j_0}$ that yields the highest likelihood for the context data $\\left( \\xinput^i, \\youtput^i \\right)$. Then, we can \\textit{adapt} by conditioning $\\mathcal{G}_{j_0}$ with the context data and finally \\textit{infer} by evaluating the resulting posterior distribution on the queried inputs $\\xinput^i_*$.\n\n\\subsection{Meta-training the Parametric Task Distribution}\nThe key to our meta-learning approach is to estimate the quality of $\\Tilde{p}_\\xi(f)$ via the NLL of context data from training tasks, and use its gradients to update the parameters of the distribution $\\xi$.\nOptimizing this loss over tasks in the dataset draws $\\Tilde{p}_\\xi(f)$ closer to the empirical distribution present in the dataset, and hence towards the true distribution $p(f)$.\n\n\nWe present three versions of \\textsc{UnLiMiTD}, depending on the choice of structure of the prior covariance over the weights (see Section~\\ref{sec:prior_covariance} for more details).\n\\approach-\\textsc{I}{} (Algorithm~\\ref{alg:meta_training_identity}) is the meta-training with the fixed identity prior covariance. \\approach-\\textsc{R}{} and \\approach-\\textsc{F}{} (Algorithm~\\ref{alg:meta_training_learnt_cov}) learn a low-dimensional representation of that prior covariance, either with random projections or with FIM-based projections.\n\n\\begin{algorithm}[t]\n\\caption{\\footnotesize \\approach-\\textsc{I}: meta-training with identity prior covariance}\n\\footnotesize\n\\label{alg:meta_training_identity}\n\\begin{algorithmic}[1]\n        \\State Initialize $\\param_0$, ${\\bm{\\mu}}$.\n        \\ForAll{epoch}\n            \\State Sample $n$ tasks $\\{ \\mathcal{T}^i, (\\xinput^i, \\youtput^i) \\}_{i=1}^{i=n}$\n            \\ForAll{$\\mathcal{T}^i, (\\xinput^i, \\youtput^i)$}\n                \\State $NLL_i \\gets \\Call{GaussNLL}{\\youtput^i; {\\bm{J}}{\\bm{\\mu}},~ {\\bm{J}}\\jac^\\top + {\\bm{\\Sigma}}_\\varepsilon}$ \\Comment{${\\bm{J}} = {\\bm{J}}(\\param_0, \\xinput^i)$}\n            \\EndFor\n            \\State Update $\\param_0$, ${\\bm{\\mu}}$ with $\\nabla_{\\param_0 \\cup {\\bm{\\mu}}} \\sum_i NLL_i$\n        \\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[t]\n\\caption{\\footnotesize \\approach-\\textsc{R}{} and \\approach-\\textsc{F}{}: meta-training with a learnt covariance}\n\\footnotesize\n\\label{alg:meta_training_learnt_cov}\n\\begin{algorithmic}[1]\n        \\If{using random projections}\n            \\State Find random projection ${\\bm{Q}}$\n            \\State Initialize $\\param_0$, ${\\bm{\\mu}}$, ${\\bm{s}}$\n        \\ElsIf{using FIM-based projections}\n            \\State Find intermediate $\\param_0$, ${\\bm{\\mu}}$ with \\approach-\\textsc{I}{} \\Comment{see Alg.~\\ref{alg:meta_training_identity}}\n            \\State Find ${\\bm{Q}}$ via \\Call{FIMProj}{s}; initialize ${\\bm{s}}$. \\Comment{see Alg.~\\ref{alg:fim_proj}}\n        \\EndIf\n        \\ForAll{epoch}\n            \\State Sample $n$ tasks $\\{ \\mathcal{T}^i, (\\xinput^i, \\youtput^i) \\}_{i=1}^{i=n}$\n            \\ForAll{$\\mathcal{T}^i, (\\xinput^i, \\youtput^i)$}\n                \\State $NLL_i \\gets \\Call{GaussNLL}{\\youtput^i; {\\bm{J}}{\\bm{\\mu}},~ {\\bm{J}} {\\bm{Q}}^\\top \\diag{{\\bm{s}}^2} {\\bm{Q}} {\\bm{J}}^\\top + {\\bm{\\Sigma}}_\\varepsilon}$ \\Comment{${\\bm{J}} = {\\bm{J}}(\\param_0, \\xinput^i)$}\n            \\EndFor\n            \\State Update $\\param_0$, ${\\bm{\\mu}}$, ${\\bm{s}}$ with $\\nabla_{\\param_0 \\cup {\\bm{\\mu}} \\cup {\\bm{s}}} \\sum_i NLL_i$\n        \\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\n\\textbf{Computing the likelihood.}~\\,\nIn the algorithms, the function \\Call{GaussNLL}{$\\youtput^i$; $m$, $K$} stands for NLL of $\\youtput^i$ under the Gaussian $\\mathcal{N}(m, K)$ (see \\eqref{eq:single-nll}).\nIn the mixture case, we instead use \\Call{MixtNLL}{}, which wraps \\eqref{eq:nll_mixture} and calls \\Call{GaussNLL}{} for the individual NLL computations (see discussion in Section~\\ref{sec:mixture}).\nIn this case, ${\\bm{\\mu}}$ becomes $\\{{\\bm{\\mu}}_j\\}_{j=1}^{j=\\alpha}$ and ${\\bm{s}}$ becomes $\\{{\\bm{s}}_j\\}_{j=1}^{j=\\alpha}$ when applicable.\n\n\\textbf{Finding the FIM-based projections.}~\\,\nThe FIM-based projection matrix aims to identify the elements of $\\phi = {\\bm{J}}(\\param_0, {\\bm{X}})$ that are most relevant for the problem (see Section~\\ref{sec:prior_covariance} and Appendix~\\ref{app:fim}).\nHowever, this feature map evolves during training, because it is $\\param_0$-dependent.\nHow do we ensure that the directions we choose for ${\\bm{Q}}$ remain relevant during training?\nWe leverage results from \\citet{ntk_evolution}, stating that the NTK (the kernel associated with the Jacobian feature map, see Section~\\ref{sec:linearization}) changes significantly at the beginning of training and that its evolution slows down as training goes on. This suggests that as a heuristic, we can compute the FIM-based directions after partial training, as they are unlikely to deviate much after the initial training. \nFor this reason, \\approach-\\textsc{F}{}\n(Algorithm~\\ref{alg:meta_training_learnt_cov}) first calls \\approach-\\textsc{I}{} (Algorithm~\\ref{alg:meta_training_identity}) before computing the FIM-based ${\\bm{Q}}$ that yields intermediate parameters $\\param_0$ and ${\\bm{\\mu}}$.\nThen the usual training takes place with the learning of ${\\bm{s}}$ in addition to $\\param_0$ and ${\\bm{\\mu}}$.\n\n\\section{Related work}\n\\label{sec:related_work}\n\\textbf{Bayesian inference with linearized DNNs.}~\\,\nBayesian inference with neural networks is often intractable because the posterior predictive has rarely a closed-form expression.\nWhereas \\textsc{UnLiMiTD}{} linearizes the network to allow for practical Bayesian inference, existing work has used other approximations to tractably express the posterior.\nFor example, it has been shown that in the infinite-width approximation, the posterior predictive of a Bayesian neural network behaves like a GP \\citep{shallow_nns_infinite_width, dnns_infinite_wdith}. This analysis can in some cases yield a good approximation to the Bayesian posterior of a DNN \\citep{cnns_infinite_width}.\nIt is also common to use Laplace's method to approximate the posterior predictive by a Gaussian distribution and allow practical use of the Bayesian framework for neural networks.\nThis approximation relies in particular on the computation of the Hessian of the network: this is in general intractable, and most approaches use the so-called Gauss-Newton approximation of the Hessian instead \\citep{laplace_scalable}.\nRecently, it has been shown that the Laplace method using the Gauss-Newton approximation is equivalent to working with a certain linearized version of the network and its resulting posterior GP \\citep{laplace_linearization}.\n\nBayesian inference is applied in a wide range of subjects. For example, recent advances in transfer learning have been possible thanks to Bayesian inference with linearized neural networks.\n\\citet{maddox} have linearized pre-trained networks and performed domain adaptation by conditioning the prior predictive with data from the new task: the posterior predictive is then used to make predictions. Our approach leverages a similar adaption method and demonstrates how the prior distribution can be learned in a meta-learning setup.\n\n\\textbf{Meta-learning.}~\\,\nMAML is a meta-learning algorithm that uses as adaptation a few steps of gradient descent \\citep{maml}.\nIt has the benefit of being model-agnostic (it can be used on any model for which we can compute gradients w.r.t. the weights), whereas \\textsc{UnLiMiTD}{} requires the model to be a differentiable regressor.\nMAML has been further generalized to probabilistic meta-learning models such as PLATIPUS or BaMAML \\citep{bayesian_maml, probabilistic_maml}, where the simple gradient descent step is augmented to perform approximate Bayesian inference. These approaches, like ours, learn (during meta-training) and make use of (at test-time) a prior distribution on the weights. In contrast, however, \\textsc{UnLiMiTD}{} uses exact Bayesian inference at test-time.\nMAML has also been improved for multimodal meta-learning via MMAML \\citep{mmaml, revisit_mmaml}. Similarly to our method, they add a step to identify the cluster from which the task comes from \\citep{mmaml}.\nOoD detection in meta-learning has been studied by \\citet{ood_maml}, who build upon MAML to perform OoD detection in the classification setting, to identify unseen classes during training.\n\\citet{meta_learning_ood} also implemented OoD detection for classification, by learning a Gaussian mixture model on a latent space. \n\\textsc{UnLiMiTD}{} extends these ideas to the regression task, aiming to identify when test data is drawn from an unfamiliar function.\n\nALPaCA is a Bayesian meta-learning algorithm for neural networks, where only the last layer is Bayesian \\citep{alpaca}.\nSuch framework yields an exact linear regression that uses as feature map the activations right before the last layer.\nOur work is a generalization of ALPaCA, in the sense that \\textsc{UnLiMiTD}{} restricted to the last layer matches ALPaCA's approach.\nMore on this link between the methods is discussed in Appendix~\\ref{app:link_with_alpaca}.\n\n\\section{Results and discussion}\n\\label{sec:results}\nWe wish to evaluate four key aspects of \\textsc{UnLiMiTD}.\n(1) At test time, how do the probabilistic predictions compare to baselines?\n(2) How well does the detection of context data from OoD tasks perform?\n(3) How do these results hold in the multimodal setting?\n(4) Which approach performs better between (a) the identity covariance (\\approach-\\textsc{I}), (b) the low-dimensional covariance with random directions (\\approach-\\textsc{R}) and the compromise (c) using FIM-based directions (\\approach-\\textsc{F}) (see trade-off in Section~\\ref{sec:prior_covariance})?\nThat is, what is best between learning a rich prior distribution over the weights, keeping a full feature map, and a compromise between the two?\n\nWe consider a cluster of sine tasks, one of linear tasks and one of quadratic tasks, regression problems inspired from \\citet{mmaml}.\nDetails on the problems can be found in Appendix~\\ref{app:problem-details}.\n\n\\textbf{Unimodal meta-learning: The meta-learned prior accurately fits the tasks.}~\\,\nFirst, we investigate the performance of \\textsc{UnLiMiTD}{} on a unimodal task distribution consisting of sinusoids of varying amplitude and phase, using the single GP structure for $\\Tilde{p}_\\xi(f)$.\nWe compare the performance between \\approach-\\textsc{I}, \\approach-\\textsc{R}{} and \\approach-\\textsc{F}.\nWe also compare the results between training with an infinite amount of available sine tasks (infinite task dataset), and with a finite amount of available tasks (finite task dataset).\nMore training details can be found in Appendix~\\ref{app:train-details-single}.\nExamples of predictions at the test time are available in Figure~\\ref{fig:single-predictions}, along with confidence levels.\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/predictions\/single_pred_fim_infinite_1.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/predictions\/single_pred_fim_infinite_5.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/predictions\/single_pred_fim_infinite_10.pdf}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/predictions\/single_pred_fim_finite_1.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/predictions\/single_pred_fim_finite_5.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/predictions\/single_pred_fim_finite_10.pdf}\n    \\end{subfigure}\n    \\caption{ Example of predictions for a varying number of context inputs $K$, after meta-training with \\approach-\\textsc{F}. Top: \\approach-\\textsc{F}, infinite task dataset. Bottom: \\approach-\\textsc{F}, finite task dataset. The standard deviation is from the posterior predictive distribution.\n    Note how the uncertainty levels are coherent with the actual prediction error.\n    Also, note how uncertainty decreases when there is more context data.\n    Notice how \\approach-\\textsc{F}{} recovers the shape of the sine even with a low number of context inputs.\n    Finally, note how \\approach-\\textsc{F}{} is able to reconstruct the sine even when trained on fewer tasks (bottom). More comprehensive plots available in Figure~\\ref{fig:single-predictions-full}.}\n    \\label{fig:single-predictions}\n\\end{figure}\n\nIn both OoD detection and quality of predictions, \\approach-\\textsc{R}{} and \\approach-\\textsc{F}{} \nperform better than \\approach-\\textsc{I}{} (Figure~\\ref{fig:single-performance}), and this is reflected in the quality of the learned prior $\\Tilde{p}_\\xi(f)$ in each case (see Appendix~\\ref{app:additional-single}).\nWith respect to the trade-off mentioned in Section~\\ref{sec:prior_covariance}, we find that for small networks, a rich prior over the weights matters more than the full expressiveness of the feature map, making both \\approach-\\textsc{R}{} and \\approach-\\textsc{F}{} appealing.\nHowever, after running further experiments on a deep-learning image-domain problem, this conclusion does not hold for deep networks (see Appendix~\\ref{app:deep}), where keeping an expressive feature map is important (\\approach-\\textsc{I}{} and \\approach-\\textsc{F}{} are appealing in that case).\nThus, \\approach-\\textsc{F}{} is the variant that we retain, for it allows similar or better performances than the other variants in all situations.\n\nNote how \\approach-\\textsc{F}{} outperforms MAML: it achieves much better generalization when decreasing the number of context samples $K$ (Figure~\\ref{fig:single-performance}).\nIndeed, \\approach-\\textsc{F}{} trained with a finite task dataset performs better than MAML with an infinite task dataset: it is able to capture better the common features of the tasks with a smaller task dataset.\n\n\\begin{figure}[t]\n     \\centering\n     \\begin{subfigure}[t]{0.3\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/single_data.pdf}\n         \\caption{Examples of context data from in-dist. and OoD tasks}\n         \\label{fig:single-data}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[t]{0.3\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{single_auc.pdf}\n         \\caption{AUC for OoD detection}\n         \\label{fig:single-auc}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[t]{0.3\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{single_mse.pdf}\n         \\caption{MSE on predictions}\n         \\label{fig:single-mse}\n     \\end{subfigure}\n    \\caption{Unimodal case: Performance of \\textsc{UnLiMiTD}{} for OoD detection and inference, as a function of the number of context datapoints $K$. The training dataset consists of sinusoids, while OoD tasks are lines and quadratic tasks. We compare different variants (\\approach-\\textsc{I}, \\approach-\\textsc{R}{} and \\approach-\\textsc{F}), and against MAML for predictions. We also compare training with a finite and infinite task dataset. Note how \\approach-\\textsc{R}{} and \\approach-\\textsc{F}{} have efficient OoD detection and outperform MAML in predictions. Also, note how MAML trained with an infinite task dataset performs worse than \\approach-\\textsc{R}{} and \\approach-\\textsc{F}{} trained on a finite task dataset.}\n    \\label{fig:single-performance}\n\\end{figure}\n\n\\textbf{Multimodal meta-learning: Comparing the mixture model against a single GP.}~\\,\nNext, we consider a multimodal task distribution with training data consisting of sinusoids as well as lines with varying slopes.  \nHere, we compare the performance between choosing the mixture structure or the single GP structure (see discussion in Section~\\ref{sec:mixture}): in both cases, we use \\approach-\\textsc{F}.\nMore training details can be found in Appendix~\\ref{app:train-details-multi}).\n\nBoth the OoD detection and the prediction performances are better with the mixture structure than with the single GP structure (Figure~\\ref{fig:multi-performance}), indicating that the mixture model is a useful structure for $\\Tilde{p}_\\xi(f)$.\nThis is reflected in the quality of the learned priors (see Appendix \\ref{app:additional-multi} for qualitative results including samples from the learned priors).\nNote how the single GP structure still performs better than both MAML and MMAML for prediction, especially in the low-data regime.\nThis demonstrates the strength of our probabilistic approach for multimodal meta-learning: even if the probabilistic assumptions are not optimal, the predictions are still accurate and can beat baselines.\n\n\\begin{figure}[t]\n     \\centering\n     \\begin{subfigure}[t]{0.3\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/multi_data.pdf}\n         \\caption{Examples of context data from in-dist. and OoD tasks}\n         \\label{fig:multi-data}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[t]{0.3\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{multi_auc.pdf}\n         \\caption{AUC for OoD detection}\n         \\label{fig:multi-auc}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[t]{0.3\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{multi_mse.pdf}\n         \\caption{MSE on predictions}\n         \\label{fig:multi-mse}\n     \\end{subfigure}\n    \\caption{Multimodal case: Performance of \\textsc{UnLiMiTD}{} for OoD detection and inference, as a function of the number of context datapoints $K$. The training dataset includes both sines and lines, while OoD tasks are quadratic functions. We compare the different variants (\\approach-\\textsc{F}{} with a single GP or a mixture model), and against MAML\/MMAML for predictions. Note how both versions of \\approach-\\textsc{F}{} yield better predictions than the baselines. In particular, even with a single GP, \\approach-\\textsc{F}{} outperforms the baselines.}\n    \\label{fig:multi-performance}\n\\end{figure}\n\n\\section{Conclusion}\\label{sec:conclusion}\n\nWe propose \\textsc{UnLiMiTD}{}, a novel meta-learning algorithm that models the underlying task distribution using a parametric and tuneable distribution, leveraging Bayesian inference with linearized neural networks.\nWe compare three variants, and show that among these, the Fisher-based parameterization, \\approach-\\textsc{F}{}, effectively balances scalability and expressivity, even for deep learning applications.\nWe have demonstrated that (1) our approach makes efficient probabilistic predictions on in-distribution tasks, which compare favorably to, and often outperform, baselines, (2) it allows for effective detection of context data from OoD tasks, and (3) that both these findings continue to hold in the multimodal task-distribution setting.\n\nThere are several avenues for future work. One direction entails understanding how the performance of \\approach-\\textsc{F}{} is impacted if the FIM-based directions are computed too early in the training and the NTK changes significantly afterwards.\nOne could also generalize our approach to non-Gaussian likelihoods, making \\textsc{UnLiMiTD}{} effective for classification tasks.\nFinally, further research can push the limits of multimodal meta-learning, e.g., by implementing non-parametric Bayesian methods to automatically infer an optimal number of clusters, thereby eliminating a hyperparameter of the current approach.\n\n\n\\subsubsection*{Acknowledgements}\nThe authors acknowledge the MIT SuperCloud \\citep{supercloud} and Lincoln Laboratory Supercomputing Center for providing HPC resources that have contributed to the research results reported within this paper.\nThe authors would like to thank MISTI MIT-France for supporting this research.\nC.A. further acknowledges support from Mines Paris Foundation. N.A. acknowledges support from the Edgerton Career Development Professorship.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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4}\\xspace}\n\\newcommand\\heplike{\\textsf{HEPLike}\\xspace}\n\\newcommand\\SFOLD{\\textsf{SFOLD}\\xspace}\n\\newcommand\\HFOLD{\\textsf{HFOLD}\\xspace}\n\\newcommand\\FeynHiggs{\\textsf{FeynHiggs}\\xspace}\n\\newcommand\\Mathematica{\\textsf{Mathematica}\\xspace}\n\\newcommand\\Kernel{\\textsf{Kernel}\\xspace}\n\\newcommand\\WSTP{\\textsf{WSTP}\\xspace}\n\\newcommand\\lilith{\\textsf{Lilith}\\xspace}\n\\newcommand\\nulike{\\textsf{nulike}\\xspace}\n\\newcommand\\gamLike{\\textsf{gamLike}\\xspace}\n\\newcommand\\gamlike{\\gamLike}\n\\newcommand\\daFunk{\\textsf{daFunk}\\xspace}\n\\newcommand\\pippi{\\textsf{pippi}\\xspace}\n\\newcommand\\MultiNest{\\textsf{MultiNest}\\xspace}\n\\newcommand\\multinest{\\MultiNest}\n\\newcommand\\great{\\textsf{GreAT}\\xspace}\n\\newcommand\\twalk{\\textsf{T-Walk}\\xspace}\n\\newcommand\\diver{\\textsf{Diver}\\xspace}\n\\newcommand\\ddcalc{\\textsf{DDCalc}\\xspace}\n\\newcommand\\tpcmc{\\textsf{TPCMC}\\xspace}\n\\newcommand\\nest{\\textsf{NEST}\\xspace}\n\\newcommand\\luxcalc{\\textsf{LUXCalc}\\xspace}\n\\newcommand\\xx{\\raisebox{0.2ex}{\\smaller ++}\\xspace}\n\\newcommand\\Cpp{\\textsf{C\\xx}\\xspace}\n\\newcommand\\Cppeleven{\\textsf{C\\raisebox{0.2ex}{\\smaller ++}11}\\xspace}\n\\newcommand\\plainC{\\textsf{C}\\xspace}\n\\newcommand\\Python{\\textsf{Python}\\xspace}\n\\newcommand\\python{\\Python}\n\\newcommand\\Fortran{\\textsf{Fortran}\\xspace}\n\\newcommand\\YAML{\\textsf{YAML}\\xspace}\n\\newcommand\\Yaml{\\YAML}\n\\newcommand{\\textsf{Capt'n General}\\xspace}{\\textsf{Capt'n General}\\xspace}\n\n\\newcommand\\beq{\\begin{equation}}\n\\newcommand\\eeq{\\end{equation}}\n\n\\newcommand{\\mail}[1]{\\href{mailto:#1}{#1}}\n\\renewcommand{\\url}[1]{\\href{#1}{#1}}\n\n\\newcommand{\\TODO}[1]{\\textbf{\\textcolor{red}{To do: #1}}}\n\\newcommand{\\Pat}[1]{\\textbf{\\color{teal}Pat: #1}}\n\\newcommand{\\Anders}[1]{\\textbf{\\color{brown}Anders: #1}}\n\\newcommand{\\Martin}[1]{\\textbf{\\color{orange}Martin: #1}}\n\n\n\\def\\alpha_t{\\alpha_t}\n\\def\\alpha_b{\\alpha_b}\n\\def\\alpha_s{\\alpha_s}\n\\def\\alpha_{\\tau}{\\alpha_{\\tau}}\n\\def\\mathcal{O}(\\at){\\mathcal{O}(\\alpha_t)}\n\\def\\mathcal{O}(\\ab){\\mathcal{O}(\\alpha_b)}\n\\def\\mathcal{O}(\\atau){\\mathcal{O}(\\alpha_{\\tau})}\n\\def\\mathcal{O}(\\at\\ab){\\mathcal{O}(\\alpha_t\\alpha_b)}\n\\def\\mathcal{O}(\\at\\as){\\mathcal{O}(\\alpha_t\\alpha_s)}\n\\def\\mathcal{O}(\\ab\\as){\\mathcal{O}(\\alpha_b\\alpha_s)}\n\\def\\mathcal{O}(\\at\\ab + \\ab^2){\\mathcal{O}(\\alpha_t\\alpha_b + \\alpha_b^2)}\n\\def\\mathcal{O}(\\at^2 + \\at\\ab + \\ab^2){\\mathcal{O}(\\alpha_t^2 + \\alpha_t\\alpha_b + \\alpha_b^2)}\n\\def\\mathcal{O}(\\at\\as + \\at^2){\\mathcal{O}(\\alpha_t\\alpha_s + \\alpha_t^2)}\n\\def\\mathcal{O}(\\at\\as +\\ab\\as){\\mathcal{O}(\\alpha_t\\alpha_s +\\alpha_b\\alpha_s)}\n\\def\\mathcal{O}(\\at\\as + \\at^2 +\\ab\\as){\\mathcal{O}(\\alpha_t\\alpha_s + \\alpha_t^2 +\\alpha_b\\alpha_s)}\n\\def\\mathcal{O}(\\at^2){\\mathcal{O}(\\alpha_t^2)}\n\\def\\mathcal{O}(\\ab^2){\\mathcal{O}(\\alpha_b^2)}\n\\def\\mathcal{O}(\\atau^2){\\mathcal{O}(\\alpha_{\\tau}^2)}\n\\def\\mathcal{O}(\\ab \\atau){\\mathcal{O}(\\alpha_b \\alpha_{\\tau})}\n\\def\\mathcal{O}((\\at+\\ab)^2){\\mathcal{O}((\\alpha_t+\\alpha_b)^2)}\n\\def\\mathcal{O}(\\as){\\mathcal{O}(\\alpha_s)}\n\\def\\mathcal{O}(\\atau^2 +\\ab \\atau ){\\mathcal{O}(\\alpha_{\\tau}^2 +\\alpha_b \\alpha_{\\tau} )}\n\n\n\\begin{document}\n\n\\begin{frontmatter}\n\n\n\n\\title{GAMBIT and its Application in the Search for Physics Beyond the Standard Model}\n\n\n\\author[imperial,oslo]{Anders Kvellestad}\n\\author[imperial,uq]{Pat Scott}\n\\author[adelaide]{Martin White}\n\\address[imperial]{Department of Physics, Imperial College London, Blackett Laboratory, Prince Consort Road, London SW7 2AZ, UK}\n\\address[oslo]{Department of Physics, University of Oslo, N-0316 Oslo, Norway}\n\\address[uq]{School of Mathematics and Physics, The University of Queensland, St.\\ Lucia, Brisbane, QLD 4072, Australia}\n\\address[adelaide]{Department of Physics, University of Adelaide, Adelaide, SA 5005, Australia}\n\n\\begin{abstract}\nThe Global and Modular Beyond-Standard Model Inference Tool (\\textsf{GAMBIT}\\xspace) is an open source software framework for performing global statistical fits of particle physics models, using a wide range of particle and astroparticle data. In this review, we describe the design principles of the package, the statistical and sampling frameworks, the experimental data included, and the first two years of physics results generated with it. This includes supersymmetric models, axion theories, Higgs portal dark matter scenarios and an extension of the Standard Model to include right-handed neutrinos. Owing to the broad spectrum of physics scenarios tackled by the \\textsf{GAMBIT}\\xspace community, this also serves as a convenient, self-contained review of the current experimental and theoretical status of the most popular models of dark matter.\n\\end{abstract}\n\n\n\\begin{keyword}\n  Supersymmetry \\sep Axions \\sep Global fits \\sep Right-handed neutrinos \\sep Higgs portal dark matter\n\n\n\n\\end{keyword}\n\n\\end{frontmatter}\n\n\n\\section{Introduction}\n\nThe core of the scientific method in the physical sciences is the identification of mathematical theories that describe some aspect of our Universe in terms of a number of free parameters. The use of global statistical fits, in either a Bayesian or frequentist framework, allows us to find the preferred parameter values of a candidate theory given experimental data, and to compare the abilities of different models to describe that data. Our current knowledge of particle physics is enshrined in the Standard Model (SM), and global fit techniques are routinely used to provide the most accurate estimates the parameters of the neutrino sector~\\cite{Bari13,Tortola14,NuFit15}, the CKM matrix~\\cite{CKMFitter}, and the electroweak sector~\\cite{ZFitter,GFitter11}.\n\nDespite its incredible successes in explaining experimental data, the SM still faces a number of experimental and theoretical challenges.  Many if not all of these can be explained by new physics Beyond the Standard Model (BSM).  Such physics could show up in a number of experiments, including direct searches for new particles at high energy particle colliders~\\cite{ATLAS_diphoton,ATLAS15,CMS_SMS}, measurements of rare Standard Model processes~\\cite{gm2exp,BelleII,CMSLHCb_Bs0mumu}, direct searches for dark matter~\\cite{XENON2013,PICO60,LUX2016}, indirect astroparticle searches for distant annihilation or decay of dark matter~\\cite{BringmannWeniger,LATdwarfP8,IC79_SUSY}, and cosmological observations~\\cite{Planck15cosmo,Slatyer15a,keVsterile_whitepaper}. Unfortunately, despite the existence of many candidate theories beyond the SM, there is no unambiguous prediction of what we expect to observe, or in which experimental field we expect to observe it.  It is therefore highly likely that the next theory of particle physics will have to be pieced together by combining clues from a number of disparate fields and experiments. In the process, it is essential to also consistently combine \\emph{null} results in experiments that had the potential to discover a given candidate theory, but failed to do so. Even in the complete absence of positive discoveries in the near future, it is essential to determine which candidate BSM theories are now comfortably excluded, and which regions of which candidate theories are now the most amenable to future discovery.\n\nGlobal fits of BSM theories have thus been a very active area of research for well over a decade \\cite{Baltz04,Allanach06,SFitter, Ruiz06}, with increases in computing power opening the option of exploring models with larger and larger parameter spaces. Nevertheless, it remains a considerable challenge to efficiently explore the high-dimensional parameter spaces of candidate theories whilst rigorously calculating likelihoods for a large range of experiments, each of which may require a costly simulation procedure. To further complicate matters, one must consistently handle systematic uncertainties that may be correlated across different datasets, resulting from either instrumental effects, or our imprecise knowledge of the nuclear, astro- or particle physics relevant to a given set of experiments. Prior to 2017, most global fits were focussed on supersymmetric theories, involving dedicated software that was built from the ground up with a knowledge of the supersymmetric parameters~\\cite{Baltz04,Allanach06,SFitter, Ruiz06,Strege15,Fittinocoverage,Catalan:2015cna,MasterCodeMSSM10,2007NewAR..51..316T,2007JHEP...07..075R,Roszkowski09a,Martinez09,Roszkowski09b,Roszkowski10,Scott09c,BertoneLHCDD,SBCoverage,Nightmare,BertoneLHCID,IC22Methods,SuperbayesXENON100,SuperBayesGC, Buchmueller08,Buchmueller09,MasterCodemSUGRA,MasterCode11,MastercodeXENON100,MastercodeHiggs,Buchmueller:2014yva,Bagnaschi:2016afc,Bagnaschi:2016xfg,Allanach:2007qk,Abdussalam09a,Abdussalam09b,Allanach11b,Allanach11a,Farmer13,arXiv:1212.4821,Fowlie13,Henrot14,Kim:2013uxa,arXiv:1503.08219,arXiv:1604.02102,Han:2016gvr, Bechtle:2014yna, arXiv:1405.4289, arXiv:1402.5419, MastercodeCMSSM, arXiv:1312.5233, arXiv:1310.3045, arXiv:1309.6958, arXiv:1307.3383, arXiv:1304.5526, arXiv:1212.2886, Strege13, Gladyshev:2012xq, Kowalska:2012gs, Mastercode12b, arXiv:1207.1839, arXiv:1207.4846, Roszkowski12, SuperbayesHiggs, Fittino12, Mastercode12, arXiv:1111.6098, Fittino, Trotta08, Fittino06,\narXiv:1608.02489, arXiv:1507.07008, Mastercode15, arXiv:1506.02499, arXiv:1504.03260, Mastercode17}. These results typically covered low-dimensional subsets of the minimal supersymmetric SM (MSSM) or, in some cases the next-to-minimal variant, with relatively few global studies of other theories completed~\\cite{Cheung:2012xb,Arhrib:2013ela,Sming14,Chowdhury15,Liem16,LikeDM,Banerjee:2016hsk,Matsumoto:2016hbs,Cuoco:2016jqt,Cacchio:2016qyh,BertoneUED,Chiang:2018cgb,hepfit,Matsumoto:2018acr}.\n\nIn 2017, the \\textsf{GAMBIT}\\xspace collaboration released the Global and Modular Beyond-Standard Model Inference Tool (\\textsf{GAMBIT}\\xspace) \\cite{gambit}, an open-source package able to produce results in both the Bayesian and frequentist statistical frameworks, and easily extendible to new BSM models and new experimental datasets. A fully modular design enables much of the code to be reused when changing the theoretical model of interest. \\textsf{GAMBIT}\\xspace includes a wide variety of efficient sampling algorithms for posterior evaluation and optimisation, and ensures computational efficiency through massive, multi-level parallelisation, both of the sampling algorithms and individual likelihood calculations.\n\nThe purpose of this article is to give a brief introduction to the \\textsf{GAMBIT}\\xspace software and science programme, reviewing the most important results obtained with \\textsf{GAMBIT}\\xspace in the first few years since its initial release. These serve to illustrate the versatility of the code in attacking completely different BSM models, and the constraining power of the highly detailed and rigorous simulations of different particle and astroparticle datasets in \\textsf{GAMBIT}\\xspace. Given the centrality of dark matter (DM) in the current search for BSM physics, this review also serves as a convenient summary of the status of the most widely-studied DM candidates.\n\nIn Section~\\ref{sec:gambit}, we describe the structure and design of the \\textsf{GAMBIT}\\xspace package, including the core framework, the means by which \\textsf{GAMBIT}\\xspace supports generic BSM models, the sampling and statistics module, and the various physics modules able to produce theoretical predictions and experimental likelihoods. In Section~\\ref{sec:physics}, we summarise the results of recent \\textsf{GAMBIT}\\xspace global fits of various supersymmetric theories, Higgs portal and axion DM models, and a right-handed neutrino extension of the SM.  We then conclude in Section~\\ref{sec:summary}.\n\n\\section{The \\textsf{GAMBIT}\\xspace software}\n\\label{sec:gambit}\n\nThe core \\textsf{GAMBIT}\\xspace software is written in \\Cppeleven, but interfaces seamlessly with extensions and existing physics codes written in \\python, \\Mathematica, \\Fortran and \\plainC.  Since the release of \\textsf{GAMBIT}\\xspace \\textsf{1.0.0} in 2017 \\cite{gambit}, the most notable updates have been versions \\textsf{1.1} (adding support for \\Mathematica) \\cite{gambit_addendum}, \\textsf{1.2} (adding support for \\python and higher-spin Higgs portal models) \\cite{HP}, \\textsf{1.3} (adding support for axion and axion-like particles) \\cite{Axions} and \\textsf{1.4} (adding support for right-handed neutrinos) \\cite{RHN}.  The current public release is \\textsf{v1.4.2}.  The source code is openly available from \\href{http:\/\/gambit.hepforge.org}{http:\/\/gambit.hepforge.org} under the 3-clause BSD license.\n\n\\subsection{Core design}\n\\label{sec:core}\n\nThe core principles of \\textsf{GAMBIT}\\xspace's software design are modularity and flexibility.  All theoretical predictions and experimental likelihood evaluations are separated into a series of smaller, self-contained sub-calculations, with each sub-calculation represented by a single function. Each function is assigned a metadata string that identifies the physical quantity that the function is able to calculate.  Examples might be the mass of the lightest Higgs boson, or the likelihood for the latest run of the LUX direct detection experiment. Functions are further tagged with additional metadata strings indicating any other physical inputs required for them to run.  In the case of the Higgs mass, one might require e.g. the SM electroweak vacuum expectation value and the masses and couplings of various other particles.  In the case of the LUX likelihood, one might require the number of events observed by LUX, its detector efficiency as a function of nuclear recoil energy, and the theoretically predicted event rate.\nAt runtime, \\textsf{GAMBIT}\\xspace identifies which functions are actually required for the analysis of a given theory, and connects them dynamically in order to enable the calculation in the most efficient manner possible.\n\nThe individual functions are grouped together according to physics theme, into seven different \\textbf{physics modules}.  We describe these specific modules in Section \\ref{sec:modules} below.  Individual functions are thus referred to as \\textbf{module functions}.  The module functions are the true building blocks of a \\textsf{GAMBIT}\\xspace analysis, allowing the code to automatically adapt itself to incorporate new observables, likelihoods, theories and experimental datasets.  The metadata string associated with the output of a module function is referred to as its \\textbf{capability}, and the metadata associated with the required inputs are referred to as \\textbf{dependencies}.  The process of dynamically connecting the outputs of module functions to the inputs of others at runtime thus consists of matching dependency strings to the capability strings of other functions (and ensuring that their \\Cpp types also match).  This process is known as \\textbf{dependency resolution}, and is performed by the \\textsf{GAMBIT}\\xspace dependency resolver.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width = 0.75\\textwidth]{figures\/dep_tree}\n\t\\caption{An example \\textsf{GAMBIT}\\xspace dependency tree for a simple fit of flavour Wilson coefficients to $b\\to s\\gamma$ and $B\\to ll$ data. Boxes (graph nodes) correspond to single module functions.  Function capabilities are marked in red, and return types of the functions, their actual function names and enveloping modules are indicated in black.  Arrows (graph edges) indicate the direction of information flow, from the capability (output) of one function to the dependencies (inputs) of others.  The input file used to instigate this fit (\\texttt{WC\\_lite.yaml}) is one of the example files distributed with \\textsf{GAMBIT}\\xspace.  This particular fit makes use of the \\textsf{GAMBIT}\\xspace modules \\textsf{FlavBit}\\xspace, \\textsf{SpecBit}\\xspace and \\textsf{DecayBit}\\xspace, as well as the backend (external package) \\superiso \\textsf{4.1} \\cite{Mahmoudi:2007vz,Mahmoudi:2008tp,Mahmoudi:2009zz}.}\n\t\\label{fig:dep_tree}\n\\end{figure}\n\nThe result is a (potentially extremely large) directed graph connecting the outputs and inputs of different module functions, known as a \\textbf{dependency tree}.  The module functions themselves constitute graph nodes, and their resolved dependencies graph edges.  An example of such a graph is shown in Fig.\\ \\ref{fig:dep_tree}.  For such a graph to constitute a viable computational pathway to all theoretical predictions and experimental likelihoods of interest, two things are required.  The first is for the actual numerical values of some dependencies to be known in advance.  These are the parameters of the theory that the user wishes to analyse, and must be chosen `from on high' before the dependency tree can be evaluated.  These are selected by the user's chosen statistical parameter sampling algorithm, discussed below in Section \\ref{sec:stats}.\n\nFrom the values of a model's parameters, all other intermediate quantities can be obtained, as long as the second criterion is also met. This condition is that no closed loops exist in the graph, i.e.\\ there are no dependencies of any module function upon things that can only be computed by knowing the result of the function. Many algorithms exist within graph theory for taking such a directed acyclic graph and obtaining a linear ordering of its nodes that respects the underlying structure of the graph.  \\textsf{GAMBIT}\\xspace uses the \\textsf{Boost::Graph} library to obtain such an ordering, and then employs that ordering to evaluate the module functions in turn.  This ensures that all module functions run before any other functions that depend upon their results.  Within topologically equivalent subsets of the ordering, \\textsf{GAMBIT}\\xspace also further dynamically optimises the module function evaluation order for speed, according to previous function evaluation times and likelihoods.\n\nModule functions may also make use of functions provided by external packages, or \\textbf{backends}.  These are also connected dynamically at runtime to module functions by the dependency resolver, in much the same way as it ensures that dependencies upon the results of other module functions are fulfilled.  This layer of abstraction allows \\textsf{GAMBIT}\\xspace to provide its module functions with seamless and interchangeable access to functions from external codes written in \\plainC, \\Cpp, \\python\\textsf{2}, \\python\\textsf{3}, \\Mathematica and all variants of \\Fortran.  The \\textsf{GAMBIT}\\xspace build system allows users to select and automatically download, configure and build whatever combination of backends they prefer to use, and the dependency resolver automatically adapts to the presence or absence of different backends when selecting which functions to connect to others.  Backends presently supported in version \\textsf{1.4.2} of \\textsf{GAMBIT}\\xspace are \\textsf{Capt'n General}\\xspace \\cite{HP}, \\ds \\cite{darksusy4,darksusy}, \\ddcalc \\cite{DarkBit,HP}, \\textsf{FeynHiggs}\\xspace \\cite{Heinemeyer:1998yj,Heinemeyer:1998np,Degrassi:2002fi,Frank:2006yh,Hahn:2013ria}, \\gamlike \\cite{DarkBit}, \\gmtwocalc \\cite{gm2calc}, \\textsf{HiggsBounds}\\xspace \\cite{Bechtle:2008jh,Bechtle:2011sb,Bechtle:2013wla}, \\textsf{HiggsSignals}\\xspace \\cite{HiggsSignals}, \\textsf{micrOMEGAs}\\xspace \\cite{Belanger:2001fz,Belanger:2004yn,Belanger:2006is,Belanger:2008sj,Belanger:2010gh,Belanger:2013oya,micromegas}, \\nulike \\cite{IC22Methods,IC79_SUSY}, \\textsf{Pythia}\\xspace \\cite{Sjostrand:2006za,Sjostrand:2014zea}, \\spheno \\cite{Porod:2003um,Porod:2011nf}, \\superiso \\cite{Mahmoudi:2007vz,Mahmoudi:2008tp,Mahmoudi:2009zz}, \\susyhd \\cite{Vega:2015fna} and \\susyhit \\cite{Djouadi:2006bz}.  Many more are also already supported in the current development version, which will be released in 2020.\n\n\\begin{figure}[tbp]\n\t\\centering\n\t\\includegraphics[width = \\textwidth]{figures\/BasicStructure}\n\t\\caption{The overall structure of a \\textsf{GAMBIT}\\xspace run, illustrating the roles of the input \\YAML file, modules, module functions, capabilities, dependencies, backends, backend functions, dependency resolver, hierarchical model database and the sampling machinery.  The user specifies one or more models to scan in the input \\YAML file, and chooses likelihoods and observables to compute in the scan, making their choice by capability rather than by choosing specific functions.  The dependency resolver automatically identifies and connects appropriate module and backend functions in order to facilitate the computation of the requested likelihoods and observables, and the scanning machinery (\\textsf{ScannerBit}\\xspace) selects parameter combinations to pass through the resulting dependency tree.  From \\cite{gambit}.}\n\t\\label{fig:corechain}\n\\end{figure}\n\nActual runs of \\textsf{GAMBIT}\\xspace are driven by a single input file, in \\YAML format.  In this file, the user selects the model(s) to analyse, gives details of which algorithms to use in order to sample the models' parameters, and provides a list of all likelihoods and physical observables that should be calculated in the scan.  The model parameter values constitute one boundary condition for dependency resolution (the dependency tree must begin from the parameters), and the target likelihoods and observables the other (the final outputs of the tree must be the required likelihoods and observables).  The dependency resolver is then responsible for identifying and filling in all the required steps in between.  To help direct this process and break degeneracies in the valid choices available to the dependency resolver at each step, the \\YAML input file may also set rules that the dependency resolver must respect. These may be e.g.\\ restrictions about which functions should be selected to fill which specific dependencies, or which version of a given backend should be used throughout the run.  These rules can be arbitrarily complicated, general or specific.  A rule can also contain explicit keyword options that will be passed to all module functions that fulfil the rule, allowing enormous control to be exercised over the details of the individual calculations from a single input file.\n\nThe core design elements of \\textsf{GAMBIT}\\xspace described so far in this section are module functions, backend abstraction, dependency resolution, and an input format that borders on its own programming language.  Together, these combine to provide an extremely flexible and extendible framework for performing global analyses of theories for BSM physics.  Fig.\\ \\ref{fig:corechain} illustrates how all of these features work together to enable a \\textsf{GAMBIT}\\xspace scan.  Further technical details can be found in Ref.\\ \\cite{gambit}.\n\n\n\\subsection{Model support}\n\\label{sec:models}\n\nAnother feature illustrated in Fig.\\ \\ref{fig:corechain} is the \\textsf{GAMBIT}\\xspace hierarchical model database.  Models are defined both in terms of their parameters, and in terms of their relationships to each other via parameter translation routines.  Models may descend from one another, meaning that a parameter combination in a child model can be translated `up' its family tree to a point in an appropriate subspace of its parent model, or in any other more distant ancestor model.  Cross-family `friend' translation pathways can also be defined.  These pathways allow module functions to be designed to work with one model, but to be used with another model without further alteration, so long as a translation pathway exists from one model to the other.\n\nModule functions, backend functions and all rules set in \\YAML files can be endowed with model-specific restrictions.  This allows the model-dependence of every sub-calculation to be tracked explicitly, and for the dependency resolver to explicitly ensure that the entire dependency tree of every scan is validated for use with the model under investigation.\n\nThe complete model database of \\textsf{GAMBIT}\\xspace \\textsf{1.4.2} is shown in Fig.\\ \\ref{fig:model_tree}.\n\n\\begin{sidewaysfigure}[tbp]\n\t\\centering\n\t\\includegraphics[width = \\textheight]{figures\/model_tree}\\vspace{2mm}\n\t\\caption{Hierarchical model database of \\textsf{GAMBIT}\\xspace \\textsf{1.4.2}. Models are shown as boxes, child-to-parent translations as black arrows, and friend translations as red arrows.}\n\t\\label{fig:model_tree}\n\\end{sidewaysfigure}\n\n\n\\subsection{Sampling and statistics}\n\\label{sec:stats}\n\nIn carrying out a global statistical analysis of a BSM theory, one may be interested in determining which parameter combinations are able to explain the totality of observed data within a given model, and to what extent -- or one may be more interested in using the experimental data to choose between entire theories.  The first of these tasks is parameter estimation, whereas the second is model comparison.  There are two philosophically distinct ways of posing both these questions:\n\\begin{enumerate}\n\\item How probable is it that we would have observed the data that we have, if a model and a specific combination of its parameters were true?\n\\item How probable is it that a model (or a specific combination of its parameters) is true, given the data that we have observed to date?\n\\end{enumerate}\nQuestion 1 concerns frequentist statistics, whereas Question 2 is fundamentally Bayesian.\n\nIn the context of parameter estimation, the choice of question dictates whether the appropriate quantity to consider is the frequentist profile likelihood, or the Bayesian posterior.  The profile likelihood for some parameters of interest $\\boldsymbol{\\theta}$ is the maximum value of the likelihood at each parameter combination $\\boldsymbol{\\theta}$, regardless of the values of any other parameters $\\boldsymbol{\\alpha}$:\n\\begin{equation}\n  \\hat{\\mathcal{L}}(\\boldsymbol{\\theta}) = \\mathrm{max}_{\\boldsymbol{\\alpha}}\\,\\mathcal{L}(\\boldsymbol{\\theta},\\boldsymbol{\\alpha}),\n\\end{equation}\nwhere the parameters $\\boldsymbol{\\alpha}$ are other `nuisance' parameters not of direct interest. Conversely, the Bayesian posterior probability distribution for the parameters $\\boldsymbol{\\theta}$ is given by Bayes' Theorem as the integral of the likelihood over $\\boldsymbol\\alpha$, weighted by ones prior belief $\\pi(\\boldsymbol{\\theta},\\boldsymbol{\\alpha})$ as to the plausibility of different values of $\\boldsymbol{\\theta}$ and $\\boldsymbol{\\alpha}$:\n\\begin{equation}\n\\mathcal{P}(\\boldsymbol\\theta) = \\int \\mathcal{P}(\\boldsymbol\\theta,\\boldsymbol{\\alpha})\\,\\mathrm{d}\\boldsymbol{\\alpha} = \\frac{1}{\\mathbb{Z}}\\int \\mathcal{L}(\\boldsymbol\\theta,\\boldsymbol{\\alpha})\\pi(\\boldsymbol\\theta,\\boldsymbol{\\alpha})\\,\\mathrm{d}\\boldsymbol{\\alpha}.\n\\end{equation}\nHere $\\mathbb{Z} \\equiv \\int \\mathcal{L}(\\boldsymbol\\theta,\\boldsymbol{\\alpha})\\pi(\\boldsymbol\\theta,\\boldsymbol{\\alpha})\\,\\mathrm{d}\\boldsymbol{\\alpha}\\,\\mathrm{d}\\boldsymbol{\\theta}$ is a normalisation factor referred to as the model evidence; taking ratios of evidences of different models is the most common method of Bayesian model comparison.  In contrast, frequentist model testing typically involves building up the distribution of the likelihood or other test statistic by simulation, in order to determine the precise probability of obtaining the observed (or worse-fitting) data if the model is assumed to be correct.\n\nThe choice of Bayesian posterior or profile likelihood has strong implications for the required design of the algorithm with which to sample the model parameters: efficiently obtaining converged estimates of profile likelihood and posterior distributions requires drastically different sampling distributions.  In neither case is random sampling at all sufficient nor correct, whether for accurate estimation of statistical properties nor for making statements about what is `typical' or `normal' within the parameter space of a given theory.  Efficient profile likelihood evaluation requires fast location of and convergence towards the maximum likelihood, whereas efficient posterior and evidence evaluation requires samples obtained with a density approximately proportional to the value of the posterior itself (as indeed is the case in most other numerical integration problems).\n\nSampling and statistical considerations in \\textsf{GAMBIT}\\xspace are handled mostly by the \\textsf{ScannerBit}\\xspace module \\cite{ScannerBit}.  It contains all the tools necessary to transform the likelihood function provided by the dependency resolver into converged profile likelihoods and Bayesian posteriors.  It also facilitates Bayesian model comparison by calculating evidences (see e.g.\\ Refs.\\ \\cite{HP,Axions} for recent examples), and frequentist model testing by providing information that can be used to perform statistical simulations (see Ref.\\ \\cite{EWMSSM} for a detailed example).\n\nFor Bayesian analyses, \\textsf{ScannerBit}\\xspace provides a series of different prior transformations, allowing the user to choose what assumptions to make about the probabilities of different model parameter at the beginning of a run, and to sample accordingly.\n\n\\textsf{ScannerBit}\\xspace contains interfaces to various built-in and external implementations of a number of leading sampling algorithms.  These include algorithms optimised for profile likelihood evaluation, and algorithms optimised for posterior and evidence calculations.  Amongst these are \\twalk \\cite{ScannerBit}, a built-in ensemble Markov Chain Monte Carlo (MCMC) well suited to posterior evaluation, \\great \\cite{great}, a regular MCMC, \\diver \\cite{ScannerBit}, a differential evolution optimiser able to efficiently map profile likelihoods, and \\multinest \\cite{Feroz:2007kg,Feroz:2008xx} and \\textsf{polychord} \\cite{Handley:2015}, nested samplers well suited to evidence and posterior evaluation.  Detailed performance comparisons between the different samplers can be found in Ref.\\ \\cite{ScannerBit}.\n\nFor consistency and the convenience of module function writers, \\textsf{GAMBIT}\\xspace also provides a series of relatively simple pre-profiled and pre-marginalised likelihood functions \\cite{gambit}.  These functions provide likelihoods where the influence of one or more Gaussian or log-normally distributed nuisance parameters is profiled or integrated out without the assistance of explicit sampling by \\textsf{ScannerBit}\\xspace.\n\n\\subsection{Physics modules}\n\\label{sec:modules}\nThe physics content of \\textsf{GAMBIT}\\xspace currently resides in seven modules, which contain the module and backend functions relevant for all necessary theoretical calculations, simulations of particle astrophysics experiments and likelihood calculations. Future \\textsf{GAMBIT}\\xspace updates will both refine the code in each module, and add new modules for new branches of physics (such as the forthcoming \\textsf{CosmoBit} module).\n\n\\subsubsection{SpecBit}\n\nBSM physics theories necessarily introduce new particles.  The first step in evaluating the likelihood of any parameter combination in a new theory is typically to calculate the masses and decay branching fractions of the new particles. These calculations get very complicated once one moves beyond tree level, as loop corrections can involve any number of new states in the theory, and loop corrections that shift the masses and decays of the existing SM particles must also be taken into account.\n\nParticle mass and coupling calculations are handled in the \\textsf{SpecBit}\\xspace module, which includes module functions for obtaining the pole masses and mixings of all new physical states in a model, scheme-dependent quantities such as those defined in the \\DR and $\\overline{MS}$\\xspace schemes, and SM masses and couplings (e.g. couplings of the SM-like Higgs). Generally, this information is obtained by running an appropriate spectrum generator but, in the simple case that the pole masses of a model are specified as input parameters, \\textsf{SpecBit}\\xspace simply formats the information to match that expected from a spectrum generator. In any case, it is important to realise that a spectrum cannot be stored simply as a set of numbers, since different experimental likelihoods may require predictions of running particle properties at different physical scales. Thus, \\textsf{SpecBit}\\xspace facilitates the passing of a spectrum object to module functions that contains knowledge of the renormalisation group equations of a model, allowing module functions in other parts of the \\textsf{GAMBIT}\\xspace code to locally run the \\DR or $\\overline{MS}$\\xspace parameters to the appropriate scale. Although \\textsf{SpecBit}\\xspace can be extended to include any model, its development has tended to proceed through updates that add functionality for the specific models targeted in \\textsf{GAMBIT}\\xspace physics papers. To date, this includes functions for GUT-\\cite{CMSSM} and weak-scale \\cite{MSSM,EWMSSM} parameterisations of the MSSM, singlet DM models with either a scalar, fermion or vector DM candidate \\cite{SSDM,SSDM2,HP}, minimal electroweak triplet and quintuplet DM \\cite{Piteration,McKay2}, and a low-energy object that holds SM-like particle information. A range of backends is used to supply the \\textsf{SpecBit}\\xspace calculations, including \\SPheno \\cite{Porod:2003um,Porod:2011nf} and \\FlexibleSUSY \\cite{Athron:2014yba} for BSM mass spectrum calculations. The latter is typically used for all spectrum generation requirements outside of the MSSM, including for the scalar singlet model examples described in this review. The Higgs and $W$ masses can also be calculated via the \\FeynHiggs \\cite{Heinemeyer:1998yj,Heinemeyer:1998np,Degrassi:2002fi,Frank:2006yh,Hahn:2013ria,Bahl:2016brp} backend.\n\n\\subsubsection{DecayBit}\n\nParticle decay calculations are handled by the \\textsf{DecayBit}\\xspace module, after accepting the masses and couplings of particles from \\textsf{SpecBit}\\xspace. These are used to calculate decay widths and branching fractions for each particle, which are stored in a single decay table entry for each particle. The collection of entries is then gathered into a full \\gambit decay table, which is passed on to other \\gambit modules.\n\n\\textsf{DecayBit}\\xspace includes known SM particle decays, modifications of SM particle decays through new physics effects, and the decays of BSM particles. For the SM, \\textsf{DecayBit}\\xspace contains the Particle Data Group results for the total widths for the $W$, $Z$, $t$, $b$, $\\tau$ and $\\mu$ (plus antiparticles), and for the most common mesons $\\pi^0$, $\\pi^\\pm$, $\\eta$, $\\rho^0, \\rho^\\pm$ and $\\omega$~\\cite{PDB}. In addition, partial widths to all distinct final states are provided for $W$, $Z$, $t$, $b$, $\\tau$, $\\mu$, $\\pi^0$ and $\\pi^\\pm$. These ``pure SM'' decays are used in \\textsf{GAMBIT}\\xspace whenever an SM decay acquires no BSM contribution in a model, or when the only effect of the BSM physics is to introduce a new decay channel, in which case the pure SM decays can be appended to the new list of decay channels. For the pure SM Higgs boson, the user can decide whether to calculate the partial and total decay widths at the predicted value of the Higgs mass with \\textsf{FeynHiggs}\\xspace, or to use pre-computed tables provided in \\textsf{DecayBit}\\xspace, sourced from Ref.\\ \\cite{YellowBook13}.\n\nBSM decays are handled on a model-by-model basis. For Higgs portal DM models, \\textsf{DecayBit}\\xspace contains analytic expressions for the partial width for a Higgs decay to two DM particles, and this decay is added to the list of SM Higgs partial widths, before rescaling the decay branching fractions and the total width. For MSSM variants, \\textsf{DecayBit}\\xspace calculates both the decays of all sparticles and additional Higgs bosons, and the SUSY corrections to the decays of the SM-like Higgs boson and the top quark. Higgs decay results may be sourced from either \\HDECAY via \\SUSYHIT, or \\FeynHiggs, whilst top quark decays are only available via \\FeynHiggs. Sparticle decays are obtained from \\SDECAY via \\SUSYHIT, but we note that a patch to the code is required to allow \\gambit to call functions from a shared library, and to solve problems with negative decay widths for some models due to large and negative 1-loop QCD corrections. Full details are given in~\\cite{SDPBit}; the patch is applied automatically when \\SUSYHIT is retrieved and built from within \\textsf{GAMBIT}\\xspace.\n\nA recent update of \\textsf{DecayBit}\\xspace has seen the addition of routines for observables relating to right-handed neutrino studies. This includes the invisible decay width of the $Z$ boson $\\Gamma_{\\rm{inv}}$, and the leptonic $W$ boson decay widths $\\Gamma_{W\\to e \\bar\\nu_e}$, $\\Gamma_{W\\to \\mu \\bar\\nu_\\mu}$ and $\\Gamma_{W\\to \\tau \\bar\\nu_\\tau}$. Measurements and uncertainties are taken from Ref.~\\cite{PDG17}, whilst theoretical results are taken from Refs.~\\cite{Drewes:2015iva,Antusch:2014woa,Antusch:2015mia,Ferroglia:2012ir,Antusch:2015mia,Fernandez-Martinez:2015hxa,Abada:2013aba,Dubovyk:2018rlg,Antusch:2014woa}.\n\n\\subsubsection{PrecisionBit}\n\nSome of the most severe constraints on BSM physics scenarios come from precision measurements of the electroweak sector, and other SM quantities. In \\textsf{GAMBIT}\\xspace, these are handled by the \\textsf{PrecisionBit}\\xspace module, which provides nuisance likelihoods for SM quantities such as the top quark mass and strong coupling constant, which have been measured with high precision. A related function is the calculation of the BSM corrections to SM observables such as the mass of the $W$ boson and the weak mixing angle, and the provision of likelihood functions that compare these predictions with experimental data.\n\nA schematic representation of \\textsf{PrecisionBit}\\xspace is shown in Fig.\\ \\ref{fig:precisionbit}, providing our first interesting example of the interaction between \\textsf{GAMBIT}\\xspace modules. Standard Model parameters that do not require BSM correction calculations are provided directly by the \\textsf{GAMBIT}\\xspace SM model, and are used in the calculation of SM nuisance likelihoods. The BSM parameters, meanwhile, are first used by \\textsf{SpecBit}\\xspace in the calculation of particle masses, couplings and precision Higgs properties. \\textsf{PrecisionBit}\\xspace then updates the results to form a precision-updated spectrum (including a dedicated calculation of the $W$ mass) which is used for calculating Higgs and $W$ mass likelihoods, in addition to a suite of electroweak precision observables (EWPO).\n\nLikelihoods exist for the Fermi coupling constant ($G_F$), the fine-structure constant ($\\alpha_{\\mathrm{em}}$), the $\\overline{MS}$\\xspace light quark ($u$,$d$,$s$) masses at $\\mu=2$~GeV, the charm ($m_c(m_c)$) and bottom ($m_b(m_b)$) masses, and the $W$, $Z$ and Higgs boson masses. There are also calculations and likelihoods for other precision observables such as the anomalous magnetic moment of the muon $a_\\mu=\\frac12(g-2)_\\mu$, the effective leptonic weak mixing angle sin$^2\\theta_{W,eff}$, and the departure from 1 of the ratio of the Fermi constants implied by the neutral and weak currents $\\Delta\\rho$. Note that, for the full suite of observables, calculations are currently only included for the MSSM; calculations for other models will be added as the corresponding models are implemented in \\textsf{GAMBIT}\\xspace.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.59\\textwidth]{figures\/PrecisionBit}\n\\caption{Schematic representation of the structure of \\textsf{PrecisionBit}\\xspace. From \\cite{SDPBit}.}\n\\label{fig:precisionbit}\n\\end{figure}\n\nLike \\textsf{DecayBit}\\xspace, \\textsf{PrecisionBit}\\xspace has also recently been updated to include observables for right-handed neutrino studies. These consist of right-handed neutrino contributions to $m_W$ and sin$^2\\theta_{W,eff}$.\n\n\\subsubsection{DarkBit}\nBSM physics models that include particle DM candidates can potentially give rise to observable consequences in a wide range of astrophysical DM experiments.\n\nIn gamma-ray indirect detection, the \\textsf{DarkBit}\\xspace module contains a dedicated signal yield calculator, along with an interface to \\gamlike, a likelihood calculator for current and future gamma-ray experiments. This combination can cope with signatures that result from an arbitrary mixture of final states, which significantly extends previous tools.\n\nFurther indirect detection constraints come from an interface to the \\nulike neutrino telescope likelihood package~\\cite{IC79_SUSY}.\n\nDirect DM search experiments are handled by the dedicated \\ddcalc package, which can be extended to include the effects of generic interactions between Weakly Interacting Massive Particles (WIMPs) and nucleons, as parameterised through effective operators. This includes both spin-dependent and spin-independent scattering.  The package models a wide range of direct search experiments including Xenon100, SuperCDMS, SIMPLE, LUX, PandaX, PICO-60 and PICO-2L.\n\nFinally, the relic density of dark matter can be computed via interfaces to \\textsf{DarkSUSY}\\xspace and \\textsf{micrOMEGAs}\\xspace~\\cite{darksusy,micromegas_nu}, and used to constrain models by computing a likelihood based on the value observed by \\textit{Planck} \\cite{Planck18cosmo}.\n\nThe basic structure of \\textsf{DarkBit}\\xspace applicable to WIMP theories is sketched in Fig.\\ \\ref{fig:flow}, providing a good example of \\textsf{GAMBIT}\\xspace's modular design principle. None of the likelihoods requires knowledge of the BSM physics parameters, instead only requiring knowledge of derived quantities that can be shared between likelihood calculations. The first step in \\textsf{DarkBit}\\xspace is to create a Process Catalogue containing information on particle annihilation processes, using the particle masses and couplings provided by \\textsf{SpecBit}\\xspace. For indirect detection calculations, this is used to create the gamma ray or neutrino spectrum of the annihilation products, via a weighted sum of indiviual contributions. For long decay chains, a native cascade decay Monte Carlo generator is used. This final annihilation spectrum is then passed to the likelihood calculators for gamma ray and neutrino telescope experiments. The Process Catalogue is also used to provide the effective annihilation rate for relic density calculations, which is then passed to a Boltzmann solver, followed by the relic density likelihood calculator. For direct dectection signatures, the model parameters are used to set the WIMP-nucleon couplings, which are then used in the calculation of the direct detection likelihood via the \\ddcalc package.\n\nA recent update of \\textsf{DarkBit}\\xspace has added various module functions required for the calculation of axion observables and likelihoods. The included observables are detailed in Section~\\ref{sec:axionL}.\n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=0.60\\linewidth]{figures\/DarkBit_flow}\n  \\caption{Schematic overview of the \\textsf{DarkBit}\\xspace module. The two-letter insets indicate what backend codes can\n    be used:  \\textsf{DarkSUSY}\\xspace (DS), \\textsf{micrOMEGAs}\\xspace (MO), \\gamLike (GL), \\nulike (NL) and\n    \\ddcalc (DC).  From \\cite{DarkBit}.\n    }\n  \\label{fig:flow}\n\\end{figure*}\n\nAn important additional function of \\textsf{DarkBit}\\xspace is to constrain nuisance parameters for various astrophysical unknowns that strongly affect direct and indirect searches for DM. \\textsf{DarkBit}\\xspace contains likelihoods for the parameters of the local DM spatial and velocity distributions, plus the nuclear matrix elements that enter direct search WIMP-nucleon scattering calculations.\n\n\\subsubsection{FlavBit}\n\nA very powerful indirect probe of BSM physics comes from the measurement of flavour physics processes, as theoretical predictions for these observables would be shifted by loop corrections from new particles. The excellent precision of flavour phyics measurements allows them to be sensitive to much higher energy scales than direct searches for new particles. Indeed, recent measurements from the LHCb experiment~\\cite{Aaij:2016flj,Aaij:2015esa,Aaij:2015yra,Aaij:2014ora,Aaij:2013qta,Aaij:2019wad,Aaij:2014pli} and from $B$ factories~\\cite{Lees:2012xj,Lees:2013uzd,Huschle:2015rga,Abdesselam:2016cgx,Abdesselam:2016llu,Aubert:2006vb,Lees:2015ymt,Wei:2009zv,Wehle:2016yoi} show tensions with the SM that are generating a considerable amount of theoretical interest.\n\n\\textsf{FlavBit}\\xspace implements flavour physics constraints from rare decay observables using the effective Hamiltonian approach, in which the cross-sections for transitions from initial states $i$ to final states $f$ are proportional to the squared matrix elements $|\\langle f |{\\cal H}_{\\rm eff}|i\\rangle|^2$.  For example, an effective Hamiltonian for $b \\rightarrow  s$ transitions given by\n\\begin{equation}\n\\mathcal{H}_{\\rm eff}  =  -\\frac{4G_{F}}{\\sqrt{2}} V_{tb} V_{ts}^{*} \\sum_{i=1}^{10} \\Bigl(C_{i}(\\mu) \\mathcal{O}_i(\\mu)+C'_{i}(\\mu) \\mathcal{O}'_i(\\mu)\\Bigr)\\;.\n\\end{equation}\nThe local operators $\\mathcal{O}_i$ represent long-distance interactions.  The Wilson coefficients $C_i$ can be calculated using perturbative methods, by requiring matching between the high-scale theory and the low-energy effective theory, at some scale $\\mu_W$ which is of the order of $m_W$. The Wilson coefficients can then be evolved to the characteristic scale for $B$ physics calculations ($\\mu_b$, of the order of the $b$ quark mass) using the renormalisation group equations of the \\emph{effective} field theory. A similar approach can be taken to $b\\rightarrow d$ transitions, using a different basis of low-energy operators. The original list of observables in \\textsf{FlavBit}\\xspace was divided into four categories:\n\n\\begin{itemize}\n\\item {\\bf Tree-level leptonic and semi-leptonic decays: }includes decays of $B$ and $D$ mesons to leptons, including $B^\\pm \\to \\tau \\nu_\\tau$, $B \\to D^{(*)} \\tau \\nu_\\tau$ and $B \\to D^{(*)} \\ell \\nu_\\ell$.\n\\item {\\bf Electroweak penguin transitions: }includes measurements of rare decays of the form $B \\to M \\ell^+\\ell^-$ (where $M$ is a meson with a smaller mass than the parent meson), such as angular observables of the decay $B^0 \\to K^{*0} \\mu^+\\mu^-$.\n\\item {\\bf Rare purely leptonic decays: }includes $B$ decays with only leptons in the final state, such as $B^0_{(s)} \\to \\mu^+ \\mu^-$.\n\\item {\\bf Other observables: }includes $b\\to s$ transitions in the radiative decays $B \\to X_s \\gamma$, the mass difference ($\\Delta M_s$ between the heavy $B_H$ and light $B_L$ eigenstates of the $B^0_s$ system, and kaon and pion decays (e.g. the leptonic decay ratio ${\\cal B}(K^\\pm\\to \\mu \\nu_\\mu)\/{\\cal B}(\\pi^\\pm\\to \\mu \\nu_\\mu)$).\n\\end{itemize}\n\nTheoretical calculations for these processes are handled via an interface to \\superiso~\\cite{Mahmoudi:2007vz,Mahmoudi:2008tp,Mahmoudi:2009zz}. Experimental results used in the calculation of likelihoods come from a variety of sources, including the PDG, the BaBar and Belle experiments, the HFAG collaboration and the LHCb experiment. Full details are given in~\\cite{FlavBit}.\n\nMore recently, \\textsf{FlavBit}\\xspace has been updated with observables relevant to right-handed neutrinos. These include:\n\\begin{itemize}\n\\item {\\bf Lepton-flavour violating (LFV) muon and tau decay searches} performed by the MEG, BaBar, Belle, ATLAS, LHCb and SINDRUM collaborations~\\cite{TheMEG:2016wtm,Aubert:2009ag,Hayasaka:2007vc,Lees:2010ez,Hayasaka:2010np,Aad:2016wce,Aaij:2014azz,Bellgardt:1987du}. LFV processes can also result in a neutrinoless $\\mu - e$ conversion inside a nucleus, and these are included in the form of three results using Ti, Pb and Au nuclei obtained by the SINDRUM II experiment~\\cite{Kaulard:1998rb,Honecker:1996zf,Bertl:2006up}.\n\\item {\\bf Tests of lepton universality violation} in the semileptonic decays of $B$ mesons $B^{0\/\\pm} \\to X^{0\/\\pm} l^+ l^-$, as performed by LHCb \\cite{Aaij:2014ora,Aaij:2017vbb}.\n\\end{itemize}\n\nA forthcoming major update to the \\textsf{FlavBit}\\xspace module will add an interface for \\superisofour, with added support for theory uncertainty covariance matrices. The experimental likelihoods will also receive an update, via a new interface to the \\heplike package \\cite{HEPLike}.\n\n\\begin{figure}[tp]\n\\centering\n\\includegraphics[width=0.60\\linewidth]{figures\/colliderbit-flow}\n\\caption{Schematic diagram of the \\textsf{ColliderBit}\\xspace processing chain for LHC likelihoods. From \\cite{ColliderBit}.}\n\\label{fig:lhcchain}\n\\end{figure}\n\n\\subsubsection{ColliderBit}\n\\label{sec:colliderbit}\nA leading source of constraints on BSM physics models comes from high-energy collider searches for new particles, plus the relatively recent measurements of the Higgs boson mass and decay branching fractions. The \\textsf{ColliderBit}\\xspace module includes the most comprehensive list of recent LHC particle searches of any public package, alongside a new interpolation of LEP results for supersymmetric particle searches. Higgs signal strength and mass measurements (including limits on possible signatures arising from new Higgs bosons) are handled via an interface to the \\textsf{HiggsSignals}\\xspace~\\cite{HiggsSignals} and \\textsf{HiggsBounds}\\xspace~\\cite{Bechtle:2008jh,Bechtle:2011sb} packages, which includes data from LEP, the Tevatron and the LHC.\n\nLHC constraints are particularly difficult to model rigorously for general models. Searches for new particles are often optimised on, and interpreted in terms of, so-called ``simplified models'', which feature only a few options from the much broader phenomenology of the parent model. For example, searches for supersymmetric particles might assume that only a particular pair of sparticles is ever produced, with decays fixed to a particular final state. The resulting exclusion limit will never apply directly to a more general model, although one can obtain approximate limits by scaling individual simplified model limits by the known cross-sections and branching ratios for each parameter point~~\\cite{Kraml:2013mwa,Papucci:2014rja}.\n\nIn \\textsf{ColliderBit}\\xspace, we provide more rigorous limits by performing an actual reproduction of the ATLAS and CMS limit-setting procedures, as shown in Fig.\\ \\ref{fig:lhcchain}. This includes a cross-section calculation for new particle production processes, followed by Monte Carlo simulation of LHC events for each parameter point using a custom parallelised version of the \\textsf{Pythia\\,8}\\xspace generator~\\cite{Sjostrand:2006za,Sjostrand:2014zea}. The results can either be fed at the truth level into code that reproduces the kinematic selections of a wide range of LHC analyses, or passed through a custom detector simulation based on four-vector smearing before analysis. Cross-sections are currently taken at leading order (plus leading log) from \\textsf{Pythia\\,8}\\xspace, but a forthcoming update will allow user-specified cross-sections. The final step of the process is to calculate a combined likelihood by either taking the signal region in a given final state for each experiment that is expected to have the highest sensitivity to the model in question, or by using a covariance matrix for analyses in cases where this is published by the relevant experimental collaboration. The list of \\textsf{ColliderBit}\\xspace analyses is continually updated, and currently includes a broad selection of searches for supersymmetric particles, plus monojet searches for DM particles.\n\n\\subsubsection{NeutrinoBit}\n\\label{sec:neutrinobit}\nThe \\textsf{NeutrinoBit}\\xspace module contains a variety of module functions for calculating observables and likelihoods in the neutrino sector, both for SM(-like) neutrinos and for right-handed neutrinos (RHNs). RHNs could cause observable consequences in a number of experiments, although it should be noted that the recent \\textsf{GAMBIT}\\xspace study focussed on models that are capable of explaining the light neutrino oscillation data, which excludes most sterile neutrino dark matter models. This is because long-lived RHNs would require very small couplings with SM matter, in which case their contribution to light neutrino mass generation is negligible.\n\n\\textsf{NeutrinoBit}\\xspace currently contains likelihoods dealing with the following classes of experimental data:\n\n\\begin{itemize}\n\\item \\textbf{Active neutrino mixing: }\\textsf{NeutrinoBit}\\xspace includes likelihoods for the 3-flavour SM-like active neutrino mixing observables $\\theta_{12}$, $\\theta_{13}$, $\\theta_{23}$ (mixing angles), $\\delta_{\\mathrm{CP}}$ (CP-phase) and $\\Delta m^2_{21}$ and $\\Delta m^2_{3\\ell}$ (mass splittings) with $\\ell = 1$ for normal mass ordering and $\\ell = 2$ for inverted mass ordering. The likelihoods use the one-dimensional $\\Delta\\chi^2$ tables provided by the NuFIT collaboration~\\cite{Esteban:2016qun,NuFit}.  These in turn include results from the solar neutrino experiments Homestake (chlorine)~\\cite{Cleveland:1998nv}, Gallex\/GNO~\\cite{Kaether:2010ag}, SAGE~\\cite{Abdurashitov:2009tn}, SNO~\\cite{Aharmim:2011vm}, the four phases of Super-Kamiokande~\\cite{Hosaka:2005um,Cravens:2008aa,Abe:2010hy} and two phases of Borexino~\\cite{Bellini:2011rx,Bellini:2008mr,Bellini:2014uqa}.  They also include results from the atmospheric experiments IceCube\/DeepCore~\\cite{Aartsen:2014yll}, the reactor experiments KamLAND~\\cite{Gando:2013nba}, Double-Chooz~\\cite{An:2016srz}, Daya-Bay~\\cite{An:2016ses} and Reno~\\cite{reno}, the accelerator experiments MINOS~\\cite{Adamson:2013whj,Adamson:2013ue}, T2K~\\cite{t2k} and NO$\\nu$A~\\cite{nova}, and the cosmic microwave background results from Planck~\\cite{Ade:2015xua}.\n\\item \\textbf{Lepton universality: }\\textsf{NeutrinoBit}\\xspace contains likelihoods for lepton universality violation in fully leptonic decays of charged mesons, $X^+ \\to l^+ \\nu$.\n\\item \\textbf{CKM unitarity: }The determination of the CKM matrix elements usually relies on the assumption that the active-sterile neutrino mixing matrix is zero. The presence of non-trivial mixing thus modifies the CKM matrix elements, and the experimentally-observed values can be used to simultaneously constrain the true CKM element values, and the active-sterile mixing matrix $\\Theta$. \\textsf{NeutrinoBit}\\xspace constructs a likelihood based on the deviations of the true values of $(V_{CKM})_{us}$ and $(V_{CKM})_{ud}$ from their experimentally-measured values.\n\\item \\textbf{Neutrinoless double-beta decay: }In a double-beta decay process, two neutrons decay into two protons, with the emission of two electrons and two anti-neutrinos. Majorana neutrinos would give rise to lepton number violation, resulting in neutrinoless double-beta decay ($0\\nu\\beta\\beta$). In addition, the exchange of RHNs can modify the effective neutrino mass $m_{\\beta\\beta}$, which is constrained by half-life measurements of $0\\nu\\beta\\beta$ decay. The best upper limits currently come from the GERDA experiment (Germanium)~\\cite{Agostini:2017iyd} with $m_{\\beta\\beta}<0.15-0.33\\;\\text{eV}$ (90\\% CL), and KamLAND-Zen (Xenon)~\\cite{KamLAND-Zen:2016pfg}, $m_{\\beta\\beta}<0.061-0.165\\;\\text{eV}$ (90\\% CL). \\textsf{NeutrinoBit}\\xspace uses these values to define one-sided Gaussian likelihoods, with theoretical calculations for RHN models taken from Refs.~\\cite{Drewes:2016lqo,Faessler:2014kka}\n\\item \\textbf{Big Bang Nucleosynthesis: }RHNs can affect the abundances of the primordial elements if they decay shortly before BBN, as the typical energy of the decay products is significantly higher than the plasma energy at that time. This can lead to the dissociation of formed nuclei, or the creation of deviations from thermal equilibrium.  The requirement that RHNs decay before BBN implies an upper limit on their lifetime which, in turn, results in a constraint on the total mixing with the active neutrinos. \\textsf{NeutrinoBit}\\xspace currently includes a basic BBN likelihood that uses decay expressions from Refs.~\\cite{Gorbunov:2007ak,Canetti:2012kh}, and requires the lifetime of each RHN to be less than 0.1s~\\cite{Ruchayskiy:2012si}. A more comprehensive update will be released in future, associated with the new \\textsf{CosmoBit} module.\n\\item \\textbf{Direct RHN searches: }Direct searches for RHNs can be performed by looking for peaks in the lepton energy spectrum of a meson decay, looking for evidence of production in beam dump experiments, and by studying the decay of vector bosons or mesons in $e^+e^-$ or $pp$ colliders. \\textsf{NeutrinoBit}\\xspace contains likelihoods for RHN searches at the PIENU~\\cite{PIENU:2011aa}, PS-191~\\cite{Bernardi:1987ek}, CHARM~\\cite{Bergsma:1985is}, E949~\\cite{Shaykhiev:2011zz,Artamonov:2014urb}, NuTeV~\\cite{Vaitaitis:1999wq}, DELPHI~\\cite{Abreu:1997uq}, ATLAS~\\cite{Aad:2015xaa} and CMS~\\cite{Sirunyan:2018mtv} experiments.\n\\end{itemize}\n\n\\section{Applications to new physics}\n\\label{sec:physics}\n\\subsection{Supersymmetry}\n\\label{sec:SUSY}\n\nSupersymmetry (SUSY) has long been one of the leading candidates for BSM physics, owing to its potential for simultaneously answering several of the questions left open by the SM. In particular, the hierarchy problem and the dark matter ``WIMP miracle'' suggest the possible existence of SUSY states around the weak scale.\n\nMost phenomenological explorations of SUSY take the MSSM as their starting point. On top of its minimal supersymmetrisation of the SM, the MSSM effectively parameterises our ignorance about the high-scale mechanism responsible for breaking SUSY. This is done by including in the Lagrangian all gauge-invariant and renormalisable terms that break SUSY ``softly'', that is, without re-introducing the quadratic Higgs mass divergences that gave rise to the hierarchy problem. In this way the MSSM provides a unified framework for exploring a wide range of possible manifestations of SUSY, but at the price of a vast parameter space: if no further assumptions are made the soft SUSY-breaking terms introduce more than one hundred free parameters.\n\nMany different assumptions have been employed in the literature to reduce this parametric freedom and improve predictability.  The resulting models broadly fit in two categories.\n\nThe first category consists of high-scale models that take inspiration from the fact that SUSY can provide gauge coupling unification at some high Grand Unified Theory (GUT) scale, typically around $10^{16}$\\,GeV. In these models a small number of unified mass and coupling parameters are defined at the GUT scale and then run down to the electroweak scale where phenomenological predictions are calculated. Thus, the assumption of high-scale unification constrains the model to a low-dimensional subspace of the full MSSM space, effectively imposing a set of characteristic correlations among the many MSSM parameters at the weak scale.\n\nProbably the most studied SUSY model in this category is the Constrained MSSM (\\textsf{CMSSM}\\xspace)~\\cite{Nilles:1983ge}. Here the parameter space is reduced to only four continuous parameters and a sign choice: the unified soft-breaking scalar mass, $m_0$; the unified soft-breaking gaugino mass, $m_{1\/2}$; the unified trilinear coupling, $A_0$; the ratio of the vacuum expectation values for the two Higgs doublets, $\\tan\\beta\\equiv v_\\mathrm{u}\/v_\\mathrm{d}$; and the sign of the supersymmetric Higgsino mass parameter $\\mu$. The \\textsf{CMSSM}\\xspace has been studied in global fits for over a decade~\\cite{Han:2016gvr, Bechtle:2014yna, arXiv:1405.4289, arXiv:1402.5419, MastercodeCMSSM, arXiv:1312.5233, arXiv:1310.3045, arXiv:1309.6958, arXiv:1307.3383, arXiv:1304.5526, arXiv:1212.2886, Strege13, Gladyshev:2012xq, Kowalska:2012gs, Mastercode12b, arXiv:1207.1839, arXiv:1207.4846, Roszkowski12, SuperbayesHiggs, Fittino12, Mastercode12, arXiv:1111.6098, Fittino, Trotta08, Ruiz06, Allanach06, Fittino06, Baltz04, SFitter}, most recently in the \\textsf{GAMBIT}\\xspace analysis in~\\cite{CMSSM}.\n\n\nTwo much-studied generalisations of the \\textsf{CMSSM}\\xspace are the Non-Universal Higgs Mass models 1 and 2 (\\textsf{NUHM1}\\xspace and \\textsf{NUHM2}\\xspace)~\\cite{Matalliotakis:1994ft,Olechowski:1994gm,Berezinsky:1995cj,Drees:1996pk,Nath:1997qm}. These models loosen the tight link in the \\textsf{CMSSM}\\xspace between the Higgs sector and the sfermions by separating out the soft-breaking mass parameters of the Higgs sector from the common scalar mass parameter $m_0$. This is achieved by introducing either one (\\textsf{NUHM1}\\xspace) or two (\\textsf{NUHM2}\\xspace) additional parameters at the GUT scale. In recent years the \\textsf{NUHM1}\\xspace and \\textsf{NUHM2}\\xspace have been studied in several global fit analyses~\\cite{MastercodeCMSSM, arXiv:1312.5233, Strege13, Fittino12, Mastercode15, Buchmueller:2014yva, arXiv:1405.4289}. \\textsf{GAMBIT}\\xspace global fits of the \\textsf{NUHM1}\\xspace and \\textsf{NUHM2}\\xspace were performed along with the fit of the \\textsf{CMSSM}\\xspace in~\\cite{CMSSM}. In Section~\\ref{sec:GUT-SUSY} we summarise the \\textsf{GAMBIT}\\xspace results for these GUT-scale SUSY models.\n\nThe second category of MSSM sub-models are the weak-scale models. Here the focus is on exploring a broad range of weak-scale phenomenological scenarios in an economical manner, by varying only the MSSM parameters that most directly impact the observables under study. With all MSSM parameters defined near the weak scale, these models are mostly agnostic to questions concerning physics at very high scales, such as grand unification. The models are often labeled as \\textsf{MSSM}$n$ (or as \\textsf{pMSSM}$n$ for the \\textit{phenomenological} \\textsf{MSSM}$n$), with $n$ specifying the number of weak-scale MSSM parameters that are treated as free parameters.\n\nVarious such weak-scale models have been subjected to global fit analyses in the past few years~\\cite{arXiv:1608.02489, arXiv:1507.07008, Mastercode15, arXiv:1506.02499, arXiv:1504.03260, Mastercode17}. The \\textsf{GAMBIT}\\xspace analyses in this category are~\\cite{MSSM}, which looks at a seven-dimensional MSSM parameterisation (\\textsf{MSSM7}\\xspace), and~\\cite{EWMSSM}, in which the fast LHC simulation capabilities of \\textsf{ColliderBit}\\xspace are used for a collider-focused fit of the four-dimensional MSSM ``electroweakino'' (chargino and neutralino) sector (\\textsf{EWMSSM}\\xspace). We summarise the \\textsf{GAMBIT}\\xspace results for the \\textsf{MSSM7}\\xspace in Section~\\ref{sec:MSSM7} and for the \\textsf{EWMSSM}\\xspace in Section~\\ref{sec:EWMSSM}.\n\nThe phenomenological richness of the MSSM means that a wide range experimental results are relevant for constraining the parameter space. The mass and signal strength measurements for the $125$\\,GeV Higgs boson and the measurement of the relic density of dark matter are of particular importance. We note that the impact of the relic density measurement depends strongly on whether the SUSY model is assumed to account for the full relic density, or, more conservatively, some arbitrary fraction of it. The \\textsf{GAMBIT}\\xspace studies reviewed here all take the latter approach.\n\nMeasurements of electroweak precision observables such as $m_W$ and the muon $g-2$, and of flavour observables like $BR(B \\rightarrow X_s \\gamma)$ and $BR(B_{(s)} \\rightarrow \\mu^+ \\mu^-)$, introduce further important requirements on the SUSY parameter space. Finally, the null results from direct and indirect dark matter searches, and from collider searches for sparticles and additional Higgs bosons, essentially rule out some parts of SUSY parameter space. Though, to determine the exact implications of such null-result collider searches is far from trivial, as will be illustrated by the \\textsf{EWMSSM}\\xspace results discussed in Section~\\ref{sec:EWMSSM}.\n\n\n\\subsubsection{Results for the \\textsf{CMSSM}\\xspace, \\textsf{NUHM1}\\xspace and \\textsf{NUHM2}\\xspace}\n\\label{sec:GUT-SUSY}\n\nThe \\textsf{GAMBIT}\\xspace global fits of the \\textsf{CMSSM}\\xspace, \\textsf{NUHM1}\\xspace and \\textsf{NUHM2}\\xspace in~\\cite{CMSSM} are interpreted in terms of frequentist profile likelihood maps, identifying the best-fit point and the $1\\sigma$ and $2\\sigma$ preferred regions relative to this point. As the results for \\textsf{NUHM2}\\xspace are qualitatively similar to those for \\textsf{NUHM1}\\xspace, we here focus on the \\textsf{CMSSM}\\xspace and \\textsf{NUHM1}\\xspace results.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width = 0.49\\textwidth]{figures\/susy\/CMSSM_2_3_like2D_post}\n  \\includegraphics[width = 0.49\\textwidth]{figures\/susy\/CMSSM_2_3_ColourMechanism_post}\\\\\n  \\includegraphics[width = 0.49\\textwidth]{figures\/susy\/NUHM1_2_3_like2D_post}\n  \\includegraphics[width = 0.49\\textwidth]{figures\/susy\/NUHM1_2_3_ColourMechanism_post}\\\\\n  \\includegraphics[height=3.1mm]{figures\/susy\/rdcolours4.pdf}\n  \\caption{\n  Profile likelihood in the $(m_0,m_{1\/2})$ plane in the \\textsf{CMSSM}\\xspace (\\textit{top left}) and the \\textsf{NUHM1}\\xspace (\\textit{bottom left}). The right-hand panels show the mechanisms that contribute to bringing the predicted DM relic density close to or below the observed value. The white contours show the $1\\sigma$ and $2\\sigma$ preferred regions relative to the best-fit point (white star). From~\\cite{CMSSM}.\n  }\n  \\label{fig:CMSSM_NUHM1_m0_m12}\n\\end{figure}\n\nThe profile likelihood maps for the $(m_0,m_{1\/2})$ planes of the \\textsf{CMSSM}\\xspace and \\textsf{NUHM1}\\xspace are shown in the left panels of Fig.\\ \\ref{fig:CMSSM_NUHM1_m0_m12}. The \\textsf{NUHM1}\\xspace plane is clearly less constrained compared to the \\textsf{CMSSM}\\xspace. The underlying reason is the additional parametric freedom in the Higgs sector of the \\textsf{NUHM1}\\xspace, where the MSSM Higgs parameters $m_{H_u}^2$ and $m_{H_d}^2$ are not unified with $m_0^2$ at the GUT scale, but are rather set by an independent parameter $m_H$ through the GUT-scale requirement $m_{H_u} = m_{H_d} \\equiv m_H$. (In the \\textsf{NUHM2}\\xspace, $m_{H_u}$ and $m_{H_d}$ are taken as independent parameters at the GUT scale.) We note that as $m_H$ is taken to be a real parameter, we have $m_{H_u}^2 = m_{H_d}^2 > 0$ at the GUT scale. The correct shape of the Higgs potential at the weak scale must therefore be generated through radiative corrections, as is the case for the \\textsf{CMSSM}\\xspace.\n\nThe right-hand panels in Fig.\\ \\ref{fig:CMSSM_NUHM1_m0_m12} help us understand the preferred parameter space in more detail. In these panels different sub-regions of the $2\\sigma$ region are coloured according to which mechanism(s) contribute to keeping the DM relic density close to or below the observed value. The following criteria are used to define the DM mechanism regions in~\\cite{CMSSM} and in the \\textsf{MSSM7}\\xspace study in~\\cite{MSSM}:\n\\begin{itemize}\n\\item stop co-annihilation: $m_{\\tilde{t}_1} \\leq 1.2\\,m_{\\tilde{\\chi}^0_1}$,\n\\item sbottom co-annihilation: $m_{\\tilde{b}_1} \\leq 1.2\\,m_{\\tilde{\\chi}^0_1}$,\n\\item stau co-annihilation: $m_{\\tilde{\\tau}_1} \\leq 1.2\\,m_{\\tilde{\\chi}^0_1}$,\n\\item chargino co-annihilation: $\\tilde{\\chi}^0_1$ $\\ge50\\%$ Higgsino,\\footnote{For brevity we refer to this mechanism simply as ``chargino co-annihilation'', though it also includes co-annihilations with the next-to-lightest neutralino. Further, for many points in this region the most important effect is simply enhanced $\\tilde{\\chi}^0_1$--$\\tilde{\\chi}^0_1$ annihilations, owing to the dominantly Higgsino $\\tilde{\\chi}^0_1$ composition.}\n\\item $A\/H$ funnel: $1.6\\,m_{\\tilde{\\chi}^0_1} \\leq \\textrm{($m_A$ or $m_H$)} \\leq 2.4\\,m_{\\tilde{\\chi}^0_1}$,\n\\item $h\/Z$ funnel: $1.6\\,m_{\\tilde{\\chi}^0_1} \\leq \\textrm{($m_Z$ or $m_h$)} \\leq 2.4\\,m_{\\tilde{\\chi}^0_1}$.\n\\end{itemize}\nThe coloured regions overlap for parameter points where more than one mechanism contributes.\n\nIn the \\textsf{CMSSM}\\xspace the overall highest-likelihood point is found in the stop co-annihilation region, at $m_0 \\lesssim 4.5$\\,TeV. This region is associated with large, negative values for the trilinear coupling, $A_0 \\lesssim -5$\\,TeV, and $\\tan\\beta \\lesssim 16$. Only two other DM mechanisms are active within the best-fit parameter space, namely the $A\/H$ funnel and chargino co-annihilation. Thus, in contrast with earlier \\textsf{CMSSM}\\xspace fits, these results show that the stau co-annihilation region has fallen out of the preferred parameter space. This is mainly driven by the likelihood contribution from the LHC Higgs measurements, which penalise the lower-$m_0$ region where the lightest stau get sufficiently close in mass to the lightest neutralino.\n\nAs discussed above, the link between $m_0$ and the Higgs sector is relaxed in the \\textsf{NUHM1}\\xspace. This opens up the parameter space at lower $m_0$, allowing the stau co-annihilation region back within the $2\\sigma$ preferred region, as seen in the lower right panel of Fig.\\ \\ref{fig:CMSSM_NUHM1_m0_m12}. A second consequence of $m_0$ being decoupled from the Higgs sector in the \\textsf{NUHM1}\\xspace is that the allowed chargino co-annihilation region is extended to arbitrarily small $m_0$ values, compared to in the \\textsf{CMSSM}\\xspace. We can understand this by investigating the \\textsf{CMSSM}\\xspace case: the chargino co-annihilation DM mechanism is important when the MSSM Higgsino mass parameter $\\mu$ is smaller than the bino mass parameter $M_1$ at the weak scale, as in that case the $\\tilde{\\chi}^0_1$ will be the lightest state in a triplet of near mass-degenerate Higgsinos (two neutralinos and one chargino).\\footnote{In the general MSSM it is also possible to have chargino co-annihilation between a pair of wino-dominated $\\tilde{\\chi}^0_1$ and $\\tilde{\\chi}^\\pm_1$, when $|M_2| < |M_1|, |\\mu|$. However, this mechanism is not available in the models discussed here, as the GUT-scale relation $M_1 = M_2 = M_3 \\equiv m_{1\/2}$ leads to $M_2 \\sim 2 M_1$ at the weak scale.}\nIn the \\textsf{CMSSM}\\xspace, the MSSM Higgsino mass parameter $\\mu$ is strongly linked to $m_0$ via the conditions for EWSB; reducing $m_0$ effectively increases $\\mu$. The bino mass parameter $M_1$ is on the other hand controlled by $m_{1\/2}$ via the GUT-scale relation $M_1 = M_2 = M_3 \\equiv m_{1\/2}$. For a fixed value of $m_{1\/2}$, lowering $m_0$ therefore eventually leads to $M_1 \\ll |\\mu|$, resulting in a bino-dominated $\\tilde{\\chi}^0_1$ significantly lower in mass than the Higgsino-dominated neutralinos\/chargino. In the \\textsf{NUHM1}\\xspace, on the other hand, the $\\mu$ parameter is mostly controlled by $m_H$. This allows for $|\\mu| < M_1$, and thus chargino co-annihilation, also in the low-$m_0$ region.\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width = 0.49\\textwidth]{figures\/susy\/CMSSM_105_201_like2D_post}\n  \\includegraphics[width = 0.49\\textwidth]{figures\/susy\/CMSSM_105_201_ColourMechanism_post}\\\\\n  \\includegraphics[height=3.1mm]{figures\/susy\/rdcolours3.pdf}\n  \\caption{\n  Profile likelihood in the $(m_{\\tilde{\\chi}^0_1},\\Omega_\\chi h^2)$ plane of the \\textsf{CMSSM}\\xspace (\\textit{left}), and the mechanisms that bring the predicted relic density close to or below the measured value (\\textit{right}). The stars show the best-fit points, while the white contours outline the $1\\sigma$ and $2\\sigma$ regions. From~\\cite{CMSSM}.\n  }\n  \\label{fig:CMSSM_mN1_oh2}\n\\end{figure}\n\nAs mentioned above, in these fits the observed DM relic density is only imposed as an upper bound, to leave open the possibility for non-MSSM contributions in the observed DM density. While this choice broadens the allowed parameter space, it is worth noting that the parameter regions that fully explain the relic density can have equally high likelihoods as those with a lower predicted relic density. This can be seen in the left panel of Fig.~\\ref{fig:CMSSM_mN1_oh2}, which shows the profile likelihood in the \\textsf{CMSSM}\\xspace plane of the neutralino mass $m_{\\tilde{\\chi}^0_1}$ and the predicted relic density $\\Omega_{\\chi} h^2$. For most $m_{\\tilde{\\chi}^0_1}$ values there is little variation in the profile likelihood when moving up to a point where the prediction saturates the observed value (dashed purple line).\n\nThe right-hand panel in Fig.~\\ref{fig:CMSSM_mN1_oh2} shows that, in the \\textsf{CMSSM}\\xspace, the lowest predicted neutralino masses are found within the stop and chargino co-annihilation regions, extending down to $m_{\\tilde{\\chi}^0_1} \\sim 250$\\,GeV. In the \\textsf{NUHM1}\\xspace and \\textsf{NUHM2}\\xspace, the chargino co-annihilation and stau co-annihilation regions extend further down, to $m_{\\tilde{\\chi}^0_1} \\sim 150$\\,GeV. The chargino co-coannihilation region in Fig.~\\ref{fig:CMSSM_mN1_oh2} also illustrates the well-known result that a dominantly Higgsino $\\tilde{\\chi}^0_1$ produces the entire observed relic density when $m_{\\tilde{\\chi}^0_1} \\sim 1$\\,TeV. Moving along the observed relic density towards higher neutralino masses, additional contributions from resonant $A\/H$-funnel annihilations become more and more important.\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.49\\textwidth]{figures\/susy\/CMSSM_105_217_ColourMechanism_post_wExps.pdf}\n  \\includegraphics[width=0.49\\textwidth]{figures\/susy\/NUHM1_105_217_ColourMechanism_post_wExps.pdf}\\\\\n  \\includegraphics[height=3.1mm]{figures\/susy\/rdcolours4.pdf}\n  \\caption{\n  The $2\\sigma$ preferred regions in the plane of the spin-independent neutralino-proton cross-section versus the neutralino mass for the \\textsf{CMSSM}\\xspace (\\textit{left}) and the \\textsf{NUHM1}\\xspace (\\textit{right}), coloured according to the mechanism(s) that limit the predicted DM relic density. The pink lines show the observed 90\\% CL exclusion limit from LUX~\\cite{LUXrun2} and projected limits for XENON1T (two tonne-years of exposure), XENONnT\/LZ (20 tonne-years of exposure)~\\cite{XENONnTLZ} and DARWIN (200 tonne-years of exposure)~\\cite{DARWIN}. The $1\\sigma$ and $2\\sigma$ regions are shown as white contours; best-fit points are marked by stars. From~\\cite{CMSSM}.\n  }\n  \\label{fig:CMSSM_NUHM1_direct_detection}\n\\end{figure}\n\nDirect detection DM searches seem the most promising experimental probe for the SUSY scenarios preferred in these fits. In Fig.\\ \\ref{fig:CMSSM_NUHM1_direct_detection} the preferred \\textsf{CMSSM}\\xspace (left) and \\textsf{NUHM1}\\xspace regions are shown in the plane of the lightest neutralino mass versus the spin-independent neutralino-proton cross-section. The predicted cross-section is scaled by the fraction $f$ of the full DM relic density that the given parameter point attributes to neutralinos. The solid pink line shows the $90\\%$\\,CL exclusion limit from the LUX 2016 result~\\cite{LUXrun2}, which was included as a likelihood component in these fits. The dashed and dotted lines show projected $90\\%$\\,CL limits for the XENON and DARWIN experiments~\\cite{XENONnTLZ, DARWIN}. While the stop co-annihilation regions will largely remain out of reach, as will much of the stau co-annihilation region in the \\textsf{NUHM1}\\xspace, both the chargino co-annihilation and the $A\/H$ funnel regions can be fully probed in future direct detection searches.\n\nFinally, we note that the \\textsf{CMSSM}\\xspace, \\textsf{NUHM1}\\xspace and \\textsf{NUHM2}\\xspace fit results in~\\cite{CMSSM} indicate that these models no longer hold much promise for resolving the observed discrepancy in the muon anomalous magnetic moment. The strong constraints on the low-mass parameter space -- in particular from LHC sparticle searches, DM direct detection and the LHC Higgs measurements -- push the fits towards heavier sfermion and electroweakino spectra, thus diminishing the possible SUSY contribution to the muon $(g-2)$.\n\n\n\n\\subsubsection{Results for the \\textsf{MSSM7}\\xspace}\n\\label{sec:MSSM7}\n\nWe now move on to the weak-scale parameterisations of the MSSM, starting with the \\textsf{GAMBIT}\\xspace analysis of the \\textsf{MSSM7}\\xspace in~\\cite{MSSM}. Here the free parameters are the wino mass parameter, $M_2$; the $(3,3)$ elements of the $\\mathbf{A}_u$ and $\\mathbf{A}_d$ MSSM trilinear coupling matrices, $(\\mathbf{A}_u)_{33} \\equiv A_{u_3}$ and $(\\mathbf{A}_d)_{33} \\equiv A_{d_3}$ (the other trilinear couplings are set to 0); the soft-breaking Higgs mass parameters, $m_{H_u}^2$ and $m_{H_u}^2$; a common parameter $m_{\\tilde{f}}^2$ for the sfermion soft-breaking mass parameters; and the ratio of the Higgs vacuum expectation values, $v_u\/v_d \\equiv \\tan\\beta$. All the parameters are defined at the scale $Q = 1$\\,TeV, except $\\tan\\beta$, which is defined at $Q = m_Z$.\n\nWhile this model is a weak-scale MSSM parameterisation, the GUT-inspired relation\n\\begin{align}\n\\frac{3}{5}\\cos^2\\theta_\\mathrm{W}M_1 = \\sin^2\\theta_\\mathrm{W}M_2 = \\frac{\\alpha}{\\alpha_\\mathrm{s}}M_3,\n\\label{eq:GUT_relation}\n\\end{align}\nis imposed to limit the dimensionality of the parameter space. Equation~\\ref{eq:GUT_relation} represents an expected weak-scale relation between $M_1$, $M_2$ and $M_3$ if they originate from a common GUT-scale parameter, like $m_{1\/2}$ in the \\textsf{CMSSM}\\xspace.\n\nAs in the GUT-scale models, the Higgsino mass parameter $\\mu$ is determined from the input parameters -- most importantly $m_{H_u}^2$ and $m_{H_u}^2$ -- and the requirements for EWSB. Since Eq.\\ \\ref{eq:GUT_relation} implies that $|M_1| < |M_2|$, we again have three \\textit{a priori} possibilities for the composition of the neutralino state: dominantly bino ($|M_1| < |\\mu|$), dominantly Higgsino ($|\\mu| < |M_1|$), or a bino-Higgsino mixture ($|M_1| \\sim |\\mu|$).\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.49\\textwidth]{figures\/susy\/MSSM7_111_112_like2D.pdf}\n  \\includegraphics[width=0.49\\textwidth]{figures\/susy\/MSSM7_111_112_ColourMechanism_post.pdf}\\\\\n  \\includegraphics[width=0.49\\textwidth]{figures\/susy\/MSSM7_105_201_like2D.pdf}\n  \\includegraphics[width=0.49\\textwidth]{figures\/susy\/MSSM7_105_201_ColourMechanism_post.pdf}\\\\\n  \\includegraphics[height=3.1mm]{figures\/susy\/MSSM7_rdcolours5.pdf}\n  \\caption{\n  Profile likelihoods in the $(\\mu,M_1)$ plane (\\textit{top left}) and the $(m_{\\tilde{\\chi}^0_1},\\Omega_\\chi h^2)$ plane of the \\textsf{MSSM7}\\xspace. The right-hand panels show the $2\\sigma$ preferred parameter regions coloured according to which mechanism(s) contribute to limit the relic density. The stars mark the best-fit points, while the white contours show the $1\\sigma$ and $2\\sigma$ preferred regions. From~\\cite{MSSM}.\n  }\n  \\label{fig:MSSM7}\n\\end{figure}\n\nThe global fit analysis in~\\cite{MSSM} finds that all these three neutralino scenarios are allowed within the $2\\sigma$ preferred parameter space of the \\textsf{MSSM7}\\xspace. This can be seen in the top panels of Fig.\\ \\ref{fig:MSSM7}, showing the profile likelihood in the $(\\mu, M_1)$ plane (left) and the active mechanisms that bring the relic density close to or below the observed value (right). In the $\\mu < |M_1|$ regions of the plane, corresponding to a mostly Higgsino $\\tilde{\\chi}^0_1$, the chargino co-annihilation and $A\/H$ funnel mechanisms dominate. Moving towards larger $\\mu$ we enter the bino-Higgsino mixture scenario at $\\mu \\sim |M_1|$, before reaching the bino-$\\tilde{\\chi}^0_1$ scenario at $\\mu > |M_1|$. Here the chargino co-annihilation mechanism is no longer relevant, so an acceptable relic density must be achieved either through efficient $A\/H$ funnel annihilations, co-annihilations with the lightest stop or sbottom, or a combination of these mechanisms.\\footnote{The lack of a stau co-annihilation region in the \\textsf{MSSM7}\\xspace is related to the assumption of a common sfermion mass parameter defined at the low scale of $Q=1$\\,TeV. The differences in sfermion masses are then mostly determined by the amount of L\/R mixing in the sfermion mass matrices, rather than RGE running of mass parameters. Since the L\/R mixing terms for both up-type and down-type sfermions are proportional to the corresponding Yukawa couplings, the light stop ends up being the lightest sfermion across much of parameter space, and the light sbottom is always lighter than then light stau.}\n\n\nThe overall best-fit point in the \\textsf{MSSM7}\\xspace is found in the chargino co-annihilation region, with $m_{\\tilde{\\chi}^0_2} \\approx m_{\\tilde{\\chi}^\\pm_1} \\approx m_{\\tilde{\\chi}^0_1} \\approx 260$\\,GeV. As can be seen in the lower panels of Fig.\\ \\ref{fig:MSSM7}, the predicted neutralino relic density at this point can only explain around 10\\% of the observed DM relic density. However, with only slightly heavier neutralino masses there are \\textsf{MSSM7}\\xspace scenarios that achieve close to the same likelihood values -- well within the $1\\sigma$ region -- and account for the full relic density. These are scenarios with a mostly bino $\\tilde{\\chi}^0_1$ and efficient $\\tilde{\\chi}^0_1$--$\\tilde{\\chi}^0_1$ annihilations through the $A\/H$ funnel.\n\nThe cutoff of this $A\/H$ funnel region at $m_{\\tilde{\\chi}^0_1} \\sim 250$\\,GeV, corresponding to $m_{A\/H} \\sim 500$\\,GeV, is due to several independent likelihood contributions that penalize the lower-mass scenarios. In particular, the constraint on BSM contributions to $BR(B \\rightarrow X_s \\gamma)$ plays an important role here, as the $A^0$ mass is closely related to the $H^\\pm$ mass, and a light charged Higgs will induce sizable SUSY contributions to this decay. Further important constraints on this region come from the LHC Higgs measurements, and also from LHC gluino searches, as the gluino mass parameter $M_3$ is connected to $M_1$ via Eq.\\ \\ref{eq:GUT_relation}, giving $M_3 \\sim 5 M_1$.\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.49\\textwidth]{figures\/susy\/MSSM7_105_217_like2D.pdf}%\n  \\includegraphics[width=0.49\\textwidth]{figures\/susy\/MSSM7_105_217_ColourMechanism_post.pdf}\\\\\n  \\includegraphics[height=3.1mm]{figures\/susy\/MSSM7_rdcolours5.pdf}\n  \\caption{\n\tProfile likelihood in the plane of the neutralino mass versus the spin-independent neutralino-proton cross-section in the \\textsf{MSSM7}\\xspace (left), and the relic density mechanisms that are active in different parts of the $2\\sigma$ region (right). The predicted neutralino-proton cross-section is rescaled at each point by the fraction $f$ of the observed DM relic density that the neutralino relic prediction accounts for. 90\\% CL exclusion limits are shown for the full LUX exposure~\\cite{LUXrun2} and the projected reach for for XENON1T (two tonne-years of exposure), XENONnT\/LZ (20 tonne-years of exposure)~\\cite{XENONnTLZ} and DARWIN (200 tonne-years of exposure)~\\cite{DARWIN}. The $1\\sigma$ and $2\\sigma$ regions are outlined by white contours. The stars mark the best-fit points. From~\\cite{MSSM}.\n   }\n  \\label{fig:MSSM7_direct_detection}\n\\end{figure}\n\nWe note that even the $h\/Z$ funnel mechanisms are present within the $2\\sigma$ parameter regions, for $m_{\\tilde{\\chi}^0_1} \\approx 45$\\,GeV and $m_{\\tilde{\\chi}^0_1} \\approx 62$\\,GeV. However, the allowed scenarios in this low-$m_{\\tilde{\\chi}^0_1}$ region have an almost pure Higgsino $\\tilde{\\chi}^0_1$ anyway, so this alone ensures a predicted relic density far below the observed value, also explaining how the otherwise strong constraints from DM direct detection are avoided.\n\nAs for the GUT-scale models discussed in the previous section, direct DM searches seem the most promising probe of the \\textsf{MSSM7}\\xspace scenarios preferred by this fit. Figure \\ref{fig:MSSM7_direct_detection} shows the profile likelihood (left) and the active DM mechanisms (right) across the plane of the neutralino mass and the spin-independent neutralino-proton cross-section. We see that future direct detection experiments will explore not only the full chargino co-annihilation region, but almost the entire $1\\sigma$ region preferred in the \\textsf{GAMBIT}\\xspace fit.\n\nConcerning the muon $(g-2)$ discrepancy, the fit in~\\cite{MSSM} shows that there is little hope that the \\textsf{MSSM7}\\xspace can provide an explanation. This is not particularly surprising: because the model dimensionality is kept low, relating all sfermion mass parameters to the common $m_{\\tilde{f}}^2$ parameter at the weak scale, it is impossible to get sufficiently light smuons and muon sneutrinos without simultaneously causing significant tension with other observables such as LHC squark searches.\n\n\n\n\\subsubsection{Results for the \\textsf{EWMSSM}\\xspace}\n\\label{sec:EWMSSM}\n\nCurrent SUSY searches by the ATLAS and CMS experiments at the LHC are usually optimised and interpreted assuming a simplified model. These models typically include only two or three different sparticles and assume 100\\% of decays occur to the signal processes. Such theory simplifications are a necessary compromise given the level of detail and complexity in experimental searches. Nevertheless it leaves open an important question: what impact do the results from ATLAS and CMS SUSY searches have on the parameter space of more realistic models like the MSSM?\n\nThe \\textsf{GAMBIT}\\xspace analysis in~\\cite{EWMSSM} takes on this question in the context of LHC searches for neutralinos and charginos. The canonical simplified model for these searches is one that assumes production of a purely wino $\\tilde{\\chi}^0_2 \\tilde{\\chi}^\\pm_1$ pair, with subsequent decays to a purely bino $\\tilde{\\chi}^0_1$ via $\\tilde{\\chi}^0_2 \\rightarrow Z \\tilde{\\chi}^0_1$ and $\\tilde{\\chi}^\\pm_1 \\rightarrow W^\\pm \\tilde{\\chi}^0_1$. This gives motivation for a search for events with leptons, jets and missing energy (see e.g.\\ \\cite{Aaboud:2018jiw, Sirunyan:2017lae}). The \\textsf{GAMBIT}\\xspace study assumes a phenomenologically far richer model, referred to as the \\textsf{EWMSSM}\\xspace.  This is the effective theory obtained when assuming that all sparticles except the MSSM electroweakinos are too heavy to affect current collider searches. The \\textsf{EWMSSM}\\xspace is thus a model with six sparticles -- four neutralinos and two charginos -- controlled by only four free MSSM parameters: $M_1$, $M_2$, $\\mu$ and $\\tan\\beta$. Loosely speaking, the bino soft-mass $M_1$ controls the mass of one neutralino, the wino soft-mass $M_2$ controls the masses of one neutralino and one chargino, and the Higgsino mass parameter $\\mu$ sets the masses of two neutralinos and one chargino.\n\nIn contrast to the global fits discussed in the previous two sections, the fit in \\cite{EWMSSM} focuses exclusively on collider constraints. This choice allows the fit to explore the full range of possible collider scenarios in the \\textsf{EWMSSM}\\xspace without further enlarging the model parameter space. Keeping the dimensionality of the parameter space fairly low is of critical importance, due the large computational expense of this fit: for each sampled \\textsf{EWMSSM}\\xspace parameter point, \\textsf{ColliderBit}\\xspace is used to run full Monte Carlo simulations of the relevant ATLAS and CMS searches. While running full simulations at each point in a global fit is always computationally challenging, it is particularly so when simulating electroweakino searches due to the low signal acceptance rates in these searches.\\footnote{For most of the included LHC searches there is no public information on how background estimates are correlated across signal regions. In these cases the single signal region with the best expected sensitivity must be identified at each \\textsf{EWMSSM}\\xspace parameter point in the fit. This adds to the already substantial computational cost, as distinguishing between ``competing'' signal regions often requires higher Monte Carlo statistics than what is needed to get reasonable signal estimates for each individual signal region alone.}\n\nThe analysis in \\cite{EWMSSM} includes \\textsf{ColliderBit}\\xspace simulations of most of the $13$\\,TeV electroweakino searches that were available at the time of the study~\\cite{Aaboud:2018jiw,Aaboud:2018sua,Aaboud:2018htj,Aaboud:2018zeb,CMS:2017fth,Sirunyan:2018iwl,Sirunyan:2017qaj,CMS-PAS-SUS-16-039}. The combined likelihood obtained from these simulations is the main component in the fit likelihood function. The other collider observables going into the total likelihood are a collection of SUSY cross-section limits from LEP and the invisible decay widths of the $Z$ and the $125$\\,GeV Higgs.\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.49\\textwidth]{figures\/susy\/MSSMEW_155_151_like2D.pdf}\n  \\includegraphics[width=0.49\\textwidth]{figures\/susy\/MSSMEW_152_155_like2D.pdf}\\\\\n  \\includegraphics[width=0.49\\textwidth]{figures\/susy\/MSSMEW_152_153_like2D.pdf}\n  \\includegraphics[width=0.49\\textwidth]{figures\/susy\/MSSMEW_154_153_like2D.pdf}\n  \\caption{\n  Profile likelihood in four different \\textsf{EWMSSM}\\xspace mass planes: the $(m_{\\tilde{\\chi}^\\pm_1},m_{\\tilde{\\chi}^0_1})$ plane (top left), the $(m_{\\tilde{\\chi}^0_2},m_{\\tilde{\\chi}^\\pm_1})$ plane (top right), the $(m_{\\tilde{\\chi}^0_2},m_{\\tilde{\\chi}^0_3})$ plane (bottom left), and the $(m_{\\tilde{\\chi}^0_4},m_{\\tilde{\\chi}^0_3})$ plane (bottom right). The white contours show the $1\\sigma$ and $2\\sigma$ preferred regions. The star marks the best-fit point. From~\\cite{EWMSSM}.\n  }\n  \\label{fig:EWMSSM_mass_planes}\n\\end{figure}\n\nThe main result from \\cite{EWMSSM} is that, when combined, the ATLAS and CMS electroweakino results prefer \\textsf{EWMSSM} scenarios with a distinct pattern of relatively light neutralino and chargino masses (Fig.\\ \\ref{fig:EWMSSM_mass_planes}). The preferred $2\\sigma$ parameter region has all six neutralinos and charginos below $\\sim$$700$\\,GeV, with the lightest neutralino below $\\sim$$200$\\,GeV. The lightest neutralino is always dominantly bino, but it also has a non-negligible wino or Higgsino component. Further, the best-fit parameter region predicts two characteristic $\\gtrsim m_Z$ gaps in the mass spectrum: the first between the mostly bino $\\tilde{\\chi}^0_1$ and the mostly wino (Higgsino) $\\tilde{\\chi}^0_2$\/$\\tilde{\\chi}^\\pm_1$, and the second between $\\tilde{\\chi}^0_2$\/$\\tilde{\\chi}^\\pm_1$ and the mostly Higgsino (wino) $\\tilde{\\chi}^0_4$\/$\\tilde{\\chi}^\\pm_2$.\\footnote{In the preferred scenarios, $\\tilde{\\chi}^0_3$ is always mostly Higgsino and thus fairly close in mass to the other Higgsino-dominated states, i.e.\\ either $\\tilde{\\chi}^0_2$\/$\\tilde{\\chi}^\\pm_1$ or $\\tilde{\\chi}^0_4$\/$\\tilde{\\chi}^\\pm_2$.}\n\nAt first sight this result may seem surprising. None of the included ATLAS and CMS searches have seen a convincing SUSY signal, yet when combined they prefer the low-mass region over the decoupling region, where all \\textsf{EWMSSM}\\xspace collider predictions would align with SM expectations. The reason is that the \\textsf{EWMSSM}\\xspace is able to simultaneously fit a pattern of small excesses across several of the simulated LHC searches, while at the same time avoiding generating too much tension with the other searches. The excesses that mostly drive this result come from searches for 2-, 3-, and 4-lepton final states in ATLAS \\cite{Aaboud:2018jiw,Aaboud:2018sua,Aaboud:2018zeb}, specifically in signal regions that target leptons from on-shell $Z$ and $W$ decays. This explains the preference in the fit for electroweakino mass spectra with two $\\gtrsim m_Z$ mass gaps.\n\n\\begin{figure}\n  \\centering\n  %\n  \\includegraphics[width=0.32\\textwidth]{figures\/susy\/MSSMEW_155_151_obs2D_605.pdf}\n  \\includegraphics[width=0.32\\textwidth]{figures\/susy\/MSSMEW_152_153_obs2D_605.pdf}\n  \\includegraphics[width=0.32\\textwidth]{figures\/susy\/MSSMEW_154_153_obs2D_605.pdf}\n  %\n  \\includegraphics[width=0.32\\textwidth]{figures\/susy\/MSSMEW_155_151_obs2D_607.pdf}\n  \\includegraphics[width=0.32\\textwidth]{figures\/susy\/MSSMEW_152_153_obs2D_607.pdf}\n  \\includegraphics[width=0.32\\textwidth]{figures\/susy\/MSSMEW_154_153_obs2D_607.pdf}\n  %\n  \\includegraphics[width=0.32\\textwidth]{figures\/susy\/MSSMEW_155_151_obs2D_608.pdf}\n  \\includegraphics[width=0.32\\textwidth]{figures\/susy\/MSSMEW_152_153_obs2D_608.pdf}\n  \\includegraphics[width=0.32\\textwidth]{figures\/susy\/MSSMEW_154_153_obs2D_608.pdf}\n  %\n  \\includegraphics[width=0.32\\textwidth]{figures\/susy\/MSSMEW_155_151_obs2D_610.pdf}\n  \\includegraphics[width=0.32\\textwidth]{figures\/susy\/MSSMEW_152_153_obs2D_610.pdf}\n  \\includegraphics[width=0.32\\textwidth]{figures\/susy\/MSSMEW_154_153_obs2D_610.pdf}\n  %\n  \\caption{\n  Contributions to the total fit likelihood from the ATLAS searches in Ref.\\ \\cite{Aaboud:2018zeb} (top), Ref.\\ \\cite{Aaboud:2018jiw} (second and third rows), and Ref.\\ \\cite{Aaboud:2018sua} (bottom), shown across the full $3\\sigma$ regions in the $(m_{\\tilde{\\chi}^\\pm_1}, m_{\\tilde{\\chi}^0_1})$ plane (left), the $(m_{\\tilde{\\chi}^0_2}, m_{\\tilde{\\chi}^0_3})$ plane (middle), and the $(m_{\\tilde{\\chi}^0_4}, m_{\\tilde{\\chi}^0_3})$ plane (right). In the blue regions a non-zero signal prediction in the given search improves the overall fit, while in red regions the signal prediction worsens the fit. In the white regions the given search is not sensitive. The orange contours outline the $1\\sigma$, $2\\sigma$ and $3\\sigma$ regions preferred in the fit. The white star marks the best-fit point. From~\\cite{EWMSSM}.\n  }\n  \\label{fig:EWMSSM_mass_planes_per_analysis}\n\\end{figure}\n\nTo understand the interplay between the analyses contributing to the excess, we can look at their individual likelihood contributions across the combined best-fit surface. This is done in Fig.\\ \\ref{fig:EWMSSM_mass_planes_per_analysis}, where the contributions from four ATLAS results are displayed across the preferred $3\\sigma$ regions in three different mass planes. When reading these plots it is important to keep in mind that the plotted points are those parameter samples picked out by profiling the \\textit{total} likelihood. The sharp changes in analysis likelihood seen in some plots are due to abrupt changes in what scenarios are picked out by this profiling, which again changes which signal region is selected to set the analysis likelihood value.\n\nOne example of the interplay between analyses is seen by comparing the middle panels on the first and third rows. The first of these show the likelihood contribution from an ATLAS search for 4-lepton final states, with the leptons coming from two $Z$ bosons~\\cite{Aaboud:2018zeb}. We see that fitting a 4-lepton excess in the \\textsf{EWMSSM}\\xspace relies on having non-negligible production of $\\tilde{\\chi}^0_3$, as this allows for signal leptons from the decays $\\tilde{\\chi}^0_3 \\rightarrow Z \\tilde{\\chi}^0_{1,2}$. The second of these panels is for an ATLAS search for 3-lepton final states~\\cite{Aaboud:2018jiw}, designed to target $\\tilde{\\chi}^0_2 \\tilde{\\chi}^\\pm_1$ production. For a given $m_{\\tilde{\\chi}^0_2}$, reducing $m_{\\tilde{\\chi}^0_3}$ to $\\lesssim 600$\\,GeV (as preferred by the 4-lepton search) also improves the fit to this 3-lepton search, which for high $m_{\\tilde{\\chi}^0_3}$ sees some tension with the data. At lower $m_{\\tilde{\\chi}^0_3}$ production processes with $\\tilde{\\chi}^0_3$ come into play, involving more complicated event topologies. At the same time the production cross-section for the $\\tilde{\\chi}^0_2 \\tilde{\\chi}^\\pm_1$ pair is reduced somewhat, due to a higher Higgsino component. The combined effect is a change in which 3-lepton signal region is identified as having the best expected sensitivity.\n\nThe combined excess in the $13$\\,TeV searches is estimated in~\\cite{EWMSSM} to have a local significance of $3.3\\sigma$. The impact of $8$\\,TeV LHC results on the preferred low-mass scenarios is investigated by post-processing all parameter samples in the $1\\sigma$ region with simulations of relevant ATLAS and CMS electroweakino searches at $8$\\,TeV \\cite{Aad:2015jqa,ATLAS:2LEPEW_20invfb,ATLAS:3LEPEW_20invfb,CMS:3LEPEW_20invfb}. The result is an upwards shift in the best-fit mass spectrum, by $\\sim20$\\,GeV in all masses, and a small reduction of the estimated significance of the excess, to $2.9\\sigma$.\n\nWe also note that even though the \\textsf{EWMSSM}\\xspace fit did not include DM constraints, parts of the preferred parameter space do give acceptable relic density predictions while avoiding exclusion from current direct and indirect DM searches. This is possible for scenarios with $m_{\\tilde{\\chi}^0_1}$ close to $m_Z\/2$ or $m_h\/2$, where resonant annihilations via the $Z\/h$ funnel can bring the predicted relic density close to or below the observed value.\n\nWhile the small excess seen in the \\textsf{EWMSSM}\\xspace fit is quite possibly due to background fluctuations, the fit demonstrates two important points. First, that LHC constraints on light SUSY can be significantly weaker in realistic SUSY such as the MSSM than in simplified models.\\footnote{While not discussed here, the analysis in~\\cite{EWMSSM} shows that for every mass hypothesis in the $(m_{\\tilde{\\chi}^0_2}, m_{\\tilde{\\chi}^0_1})$ plane -- not just for points in the best-fit region -- there is a point in the \\textsf{EWMSSM}\\xspace parameter space that fits the \\textit{combined} collider results at least as well as the SM expectation.} Second, that proper statistical combinations of collider searches can be a powerful tool to uncover suggestive patterns in BSM parameter spaces.\n\n\n\n\\subsection{Higgs Portal models for dark matter}\n\nNo definitive evidence has yet been uncovered for non-gravitational interactions of DM with the SM.  At some level however, such interactions must be inevitably generated by effective operators connecting Lorentz-invariant, gauge singlet combinations of SM particles to equivalently symmetric combinations of DM fields.  The lowest-dimension such operator in the SM is the Higgs bilinear $H^\\dagger H$.  Depending on the spin and gauge representation of a DM candidate $X$, the lowest-order Lorentz- and gauge-invariant DM operator may be either the bilinear $X^\\dagger X$, or a lone DM field. Operators linear in $X$ are only consistent if $X$ is itself a Lorentz invariant (i.e.\\ a scalar), and a gauge singlet.  If it is to be a viable DM candidate however, $X$ must be stable on cosmological timescales. The most straightforward way to achieve this is for $X$ to hold a different charge to SM particles under some new unbroken (typically discrete) symmetry.  This has the effect of forbidding terms linear in $X$, preventing the field from decaying.\n\nThe lowest-order operator connecting $X$ to the SM guaranteed to exist at some level is therefore the so-called `Higgs portal' operator $X^\\dagger X H^\\dagger H$.  Following electroweak symmetry breaking, this operator gives rise to a mass term for $X$ proportional to $v_0^2$ (with $v_0$ the vacuum expectation value of the Higgs field), a Higgs-DM-DM vertex proportional to $v_0$, and a direct four-particle vertex between two Higgses and two DM particles.  The new 3-particle and 4-particle interactions of $X$ with the Higgs boson lead to DM annihilation (enabling thermal production and possible indirect detection), spin-independent DM-nucleon scattering (leading to possible direct detection), DM production at colliders (with the possibility for signals in e.g.\\ monojet searches), and invisible decays of the Higgs to two DM particles when $m_X < m_h\/2$.\n\nDepending on the Lorentz representation of DM, $X^2 H^2$ may be a fully renormalisable dimension 4 operator (if $X$ is a scalar), an effective dimension 4 operator (if $X$ is a vector), or an effective dimension 5 operator (if $X$ is a fermion).  All three of these cases have been considered in detail in the literature, with a particular focus on models where $X$ is itself a gauge singlet and the $X^2H^2$ term is therefore the sole link between DM and the SM.  The most commonly studied cases have been the $\\mathbb{Z}_2$-symmetric scalar [\\citenum{SilveiraZee,McDonald94,Burgess01,Davoudiasl:2004be,Goudelis09,Yaguna09,Profumo2010a,Andreas:2010dz,Arina11,Mambrini11, Raidal:2011xk,Mambrini:2011ik,He:2011de,Drozd:2011aa,Okada:2012cc,Cheung:2012xb,Okada:2013bna,Cline:2013gha,Chacko:2013lna, Endo:2014cca,Craig:2014lda, Feng15,Duerr15,arXiv:1510.06165,Duerr16,He:2016mls,Han:2016gyy,Dupuis:2016fda,Cuoco:2016jqt,Binder:2017rgn,Ghorbani:2018yfr,Chiang:2018gsn,Stocker:2018avm,Hardy:2018bph,Bernal:2018kcw,Glioti:2018roy,Urbano:2014hda,Escudero:2016gzx,Kanemura:2011nm,Djouadi:2011aa,Djouadi:2012zc,Bishara:2015cha,Ko:2016xwd,Beniwal:2015sdl,Kamon:2017yfx,Dutta:2017sod,Dick:2018lqx}; \\textsf{GAMBIT}\\xspace analyses \\citenum{SSDM,SSDM2}], vector [\\citenum{Djouadi:2011aa,Kanemura:2011nm,Djouadi:2012zc,Bishara:2015cha,Chen:2015dea,DiFranzo:2015nli, Beniwal:2015sdl,Ko:2016xwd,Kamon:2017yfx,Dutta:2017sod,arXiv:1704.05359,Dick:2018lqx,Baek:2014jga}; \\textsf{GAMBIT}\\xspace analysis \\citenum{HP}] and fermionic [\\citenum{Djouadi:2011aa,Kanemura:2011nm,LopezHonorez:2012kv,Djouadi:2012zc,Urbano:2014hda, Baek:2014jga,Bishara:2015cha,Beniwal:2015sdl,Ko:2016xwd,Fedderke:2014wda,Matsumoto:2014rxa,arXiv:1506.04149, arXiv:1506.08805,arXiv:1506.06556, Escudero:2016gzx,Kamon:2017yfx,Dutta:2017sod,Dick:2018lqx,Matsumoto:2018acr}; \\textsf{GAMBIT}\\xspace analysis \\citenum{HP}] variants, along with the $\\mathbb{Z}_3$-symmetric scalar [\\citenum{Belanger2013a,Kang:2017mkl,2017JHEP...10..088B,Hektor:2019ote,Kannike:2019mzk}; \\textsf{GAMBIT}\\xspace analysis \\citenum{SSDM2}].\n\n\n\\subsubsection{$\\mathbb{Z}_2$-symmetric scalar singlet}\n\nThe simplest Higgs portal model for DM, and indeed probably the most minimal of all models for particle DM, is a single, real, gauge-singlet scalar field $S$, protected from decay by a $\\mathbb{Z}_2$ symmetry.  The only new renormalisable Lagrangian terms allowed by gauge, Lozentz and $\\mathbb{Z}_2$ symmetry are\n\\begin{equation}\n\\mathcal{L}_{\\mathbb{Z}_2} = \\frac12 \\mu_{\\scriptscriptstyle S}^2 S^2 + \\frac14\\lambda_{\\sss S} S^4 + \\frac12\\lambda_{h\\sss S} S^2|H|^2.\n\\label{L_S}\n\\end{equation}\nThe model is fully specified by the $S$ bare mass $\\mu_{\\scriptscriptstyle S}$, the dimensionless $S$ quartic self-coupling $\\lambda_{\\sss S}$, and the dimensionless Higgs portal coupling $\\lambda_{h\\sss S}$.  For the most part, the $S$ quartic coupling has little impact on the phenomenology of the model, as it leads only to DM self-interactions, which are not sufficiently constrained by existing data to place strong limits on $\\lambda_{\\sss S}$.  A key exception, however, is the impact of $\\lambda_{\\sss S}$ on the running of gauge couplings under renormalisation group flow, which can have important implications for stability of the electroweak vacuum.\n\nDenoting the physical SM Higgs field by $h$, following electroweak symmetry breaking $H \\rightarrow \\left[0, (v_0+h)\/\\sqrt{2}\\right]^\\text{T}$.  This generates new vertices of the form $v_0hS^2$ and $h^2S^2$, and induces a shift to the $S$ bare mass, such that at tree level\n\\begin{equation}\nm_{\\sss S} = \\sqrt{\\mu_{\\scriptscriptstyle S}^2 + \\frac12{\\lambda_{h\\sss S} v_0^2}}.\n\\label{ms}\n\\end{equation}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{figures\/figure_1.pdf}\n\\caption{Feynman diagrams for annihilation, semi-annihilation, nuclear scattering and Higgs decays in scalar singlet Higgs portal models.  $N, f$ and $V$ refer to nucleons, fermions and SM electroweak vector bosons ($Z$ and $W$), respectively. Diagrams are shown for the $\\mathbb{Z}_3$-symmetric case, where DM exists in $S$ and anti-$S$ (i.e. $S^*$) states, but the same diagrams apply in the $\\mathbb{Z}_2$-symmetric case with $S=S^*$, except for semi-annihilation (which is absent in the $\\mathbb{Z}_2$ model).  The same diagrams also apply to $\\mathbb{Z}_2$-symmetric vector and fermionic Higgs portal models (with $S$ replaced by the relevant DM particle and semi-annihilation also forbidden by the $\\mathbb{Z}_2$ symmetry). From \\cite{SSDM2}.}\n\\label{fig:diagrams}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_singlet_19_17_like2D_SingletDM_Z2_low_X}\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_singlet_19_17_like2D_SingletDM_Z2_full_X}\n  \\caption{Profile likelihoods of parameters in the $\\mathbb{Z}_2$-symmetric scalar singlet Higgs portal dark matter model, including constraints from direct and indirect detection, the relic density of dark matter and LHC searches for invisible decays of the Higgs boson, along with various Standard Model, dark matter halo and nuclear uncertainties.  \\textit{Left}: the low-mass resonance region.  \\textit{Right}: the full mass range.  Contours show 1 and 2$\\sigma$ confidence regions, with white corresponding to the main scan (including the 2018 XENON1T direct search \\cite{Aprile:2018dbl}) and grey to a secondary scan using the 2017 XENON1T result \\cite{Aprile:2017iyp}.  White stars indicate the location of the best-fit point.  From \\protect\\cite{SSDM2}.}\n\t\\label{fig:z2scalar1}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_singlet_19_200_like2D_SingletDM_Z2_full_X}\n  \\caption{Results from the same analysis of the $\\mathbb{Z}_2$-symmetric scalar singlet Higgs portal dark matter model as shown in Fig.\\ \\ref{fig:z2scalar1}, but plotted in the plane of the effective spin-independent nuclear scattering cross-section and the scalar mass, in order to compare directly to the sensitivity of direct detection experiments.  All models have their effective cross-section defined as $f\\sigma_\\mathrm{SI}$, where $f\\equiv \\Omega_S \/ \\Omega_\\mathrm{DM}$ is the fraction of the relic density constituted by the scalar singlet.  Experiments assume $f=1$ when publishing their results. Contours show 1 and 2$\\sigma$ confidence regions, and stars best fits. From \\protect\\cite{SSDM2}.}\n\t\\label{fig:z2scalar2}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_singlet_19_17_like2D_SingletDM_Z2_full_vs_X_cut}\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_singlet_19_200_like2D_SingletDM_Z2_full_vs_X_cut}\n  \\caption{Regions in the $\\mathbb{Z}_2$-symmetric scalar singlet model that satisfy all experimental constraints, stabilise the electroweak vacuum and remain perturbative up to scales of $10^{15}$\\,GeV. Contours show 1 and 2$\\sigma$ confidence regions, and stars best fits. Grey contours show the allowed regions without the requirements of vacuum stability and perturbativity. From \\protect\\cite{SSDM2}.}\n\t\\label{fig:z2scalar3}\n\\end{figure}\n\nThe interaction with the physical Higgs endows $S$ with essentially all of the classic phenomenology of WIMP DM, via the diagrams shown in Fig.\\ \\ref{fig:diagrams} -- along with the added possibility of Higgs decays $h\\to SS$ where $m_{\\sss S} \\le m_h\/2$.  The leading constraints on the model come from searches for gamma rays from dark matter annihilation in dwarf spheroidal galaxies \\cite{LATdwarfP8}, the observed relic density of dark matter \\cite{Planck18cosmo}, direct searches performed by the XENON1T \\cite{Aprile:2018dbl} and PandaX \\cite{Cui:2017nnn} experiments, and searches for invisible Higgs decays at the LHC \\cite{Belanger:2013xza,CMS-PAS-HIG-17-023}.\n\nThe resulting preferred regions of parameter space are shown in Fig.\\ \\ref{fig:z2scalar1}.  These results explicitly allow models where $S$ is only a fraction of the observed DM, and include a fully self-consistent rescaling of the predicted signals at direct and indirect searches according to the fraction $f \\le 1$ of DM constituted by $S$ at each point in the parameter space.  The allowed parameter space splits into three regions: one at high masses where direct detection loses sensitivity, a second at intermediate mass where the 4-boson vertex boosts the annihilation cross-section and depletes the relic density, and another at and immediately below $m_{\\sss S} = m_h\/2$, where $S$ annihilates highly efficiently via an $s$-channel resonance mediated by the Higgs, depleting the relic density to below the observed value even for very small values of $\\lambda_{h\\sss S}$.\n\nThe Higgs invisible width constraint rules out large couplings $\\lambda_{h\\sss S}$ at singlet masses below the resonance.  The thermal relic density of $S$ provides the lower limit of the low-mass and high-mass allowed regions.  Indirect detection plays the leading role only on the high-mass edge of the resonance, where thermal effects in the early Universe push annihilation slightly off resonance but late-time annihilation remains strongly boosted.  Direct detection plays a significant role throughout the parameter space, as can be seen in Fig.\\ \\ref{fig:z2scalar2}.  Except for the very bottom of the resonance region, the entirety of the model will soon be probed by direct detection.\n\nGamma-ray lines do not provide any meaningful constraint, as the partial annihilation cross-section for $SS\\to\\gamma\\gamma$ is only appreciable in parts of the parameter space where the relic density is significantly suppressed.  Likewise, monojet searches only constrain very large values of $\\lambda_{h\\sss S}$ already excluded by other constraints or expected to lead to new strong dynamics.  Indeed, both these points also apply to all other Higgs portal models that we discuss in this review.\n\nGiven that the Higgs portal operator is not just an effective interaction, but a fully renormalisable operator in this model, it is also important to consider the UV behaviour of the theory.  Due to the observed values of the top and Higgs masses, the SM posesses a second minimum in its scalar potential at $\\gtrsim \\mathcal{O}(10^{15})$\\,GeV, causing the low-scale vacuum in which we reside to be metastable.  Adding an additional scalar to the SM impacts the running of the Higgs quartic coupling, raising its value at high scales.  This can prevent the quartic coupling from running negative, and make the low-scale minimum a global rather than a local one.  The catch is that $\\lambda_{\\sss S}$ must be relatively large in order to achieve this effect.  Fig.\\ \\ref{fig:z2scalar3} shows the parts of the parameter space, consistent with all experimenal constraints, where $\\lambda_{\\sss S}$ can be pushed high enough to stabilise the SM vacuum, but without pushing any of the couplings non-perturbative below a scale of $10^{15}$\\,GeV. Clearly, the $\\mathbb{Z}_2$-symmetric scalar singlet can solve the vacuum stability problem without introducing new strong dynamics, and satistfy all experimental constraints, but only in a region around \\mbox{$m_{\\sss S} = 1$--2\\,TeV} and $\\sigma_\\mathrm{SI} \\sim 10^{-45}$\\,cm$^2$.  Curiously, this is also in the region consistent with the (admittedly very small) excess seen in the most recent XENON1T results \\cite{Aprile:2018dbl}.  In any case, this hypothessis will clearly be tested very quickly in the upcoming runs of the LZ and XENONnT \\cite{Akerib:2015cja,XENONnTLZ} experiments.\n\nThe results in Figs.\\ \\ref{fig:z2scalar1}--\\ref{fig:z2scalar3} are based on profile likleihood analyses, and illustrate what is possible in each parameter plane, were one able to freely vary the other parameters of the theory (including nuisance parameters) in order to achieve the best possible fit to all available data.  If one instead carries out a Bayesian analysis, looking instead at the posterior probability density for these parameters, a different picture emerges.  In this case, parameter combinations become more likely if they can provide a good fit for a broader range of values of the other parameters of the theory, i.e.\\ if they can fit the data with less fine tuning.  In this case, the low-mass resonance region is strongly disfavoured, as `hitting' the resonance and avoiding the relic density constraint for a given value of $m_{\\sss S}$ requires some fine-tuning of various SM nuisance parameters such as $m_h$; the same is true to a lesser extent for the intermediate-mass region as well.  We therefore see that from a Bayesian perspective, the region where the singlet model stabilises the SM vacuum is in fact favoured over the other regions of the theory, even before considering the implications for vacuum stability.\n\n\\begin{figure}[t]\n\t\\centering\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_singlet_20_17_like2D_SingletDM_Z3_full_X}\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_singlet_20_19_like2D_SingletDM_Z3_full_X}\n  \\caption{Profile likelihoods of parameters in the $\\mathbb{Z}_3$-symmetric scalar singlet Higgs portal dark matter model, including constraints from direct and indirect detection, the relic density of dark matter and LHC searches for invisible decays of the Higgs boson, along with various Standard Model, dark matter halo and nuclear uncertainties.  Contours show 1 and 2$\\sigma$ confidence regions, with white corresponding to the main scan (including the 2018 XENON1T direct search \\cite{Aprile:2018dbl}) and grey to a secondary scan using the 2017 XENON1T result \\cite{Aprile:2017iyp}.  White stars indicate the location of the best-fit point.  From \\protect\\cite{SSDM2}.}\n\t\\label{fig:z3scalar1}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_singlet_20_17_obs2D_60_SingletDM_Z3_full_X}\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_singlet_20_19_obs2D_60_SingletDM_Z3_full_X}\n  \\caption{Results from the same analysis of the $\\mathbb{Z}_3$-symmetric scalar singlet Higgs portal dark matter model as shown in Fig.\\ \\ref{fig:z3scalar1}, but shaded according to the semi-annihilation fraction $\\alpha$ (Eq.\\ \\protect\\ref{eqn:sa_fraction}). From \\protect\\cite{SSDM2}.}\n\t\\label{fig:z3scalar2}\n\\end{figure}\n\n\\subsubsection{$\\mathbb{Z}_3$-symmetric scalar singlet}\n\nIn contrast to the self-adjoint $\\mathbb{Z}_2$-symmetric scalar singlet, a $\\mathbb{Z}_3$ symmetry leads to a complex scaler DM candidate, with both DM ($S$) and anti-DM ($S^*$) states contributing to the relic density.  This symmetry also allows an additional cubic term in the Lagrangian,\n\\begin{equation}\n\\mathcal{L}_{\\mathbb{Z}_3} = \\mu_{\\scriptscriptstyle S}^2 S^\\dagger S + \\lambda_{\\sss S} (S^\\dagger S)^2 + \\frac{\\mu_3}{2}(S^{\\dagger 3}+S^3) + \\lambda_{h\\sss S} S^\\dagger S|H|^2,\n\\end{equation}\nwhere we have introduced the new dimension-1 $S$ cubic coupling $\\mu_3$.  This new coupling allows for so-called semi-annihilation processes $SS\\to S^*h$ and $S^*S^*\\to Sh$, shown in Fig.\\ \\ref{fig:diagrams}.\n\nCompared to the $\\mathbb{Z}_2$-symmetric model, semi-annihilation is able to deplete the relic density of DM at intermediate masses and open up an entirely new region of viable parameter space.  This is shown in terms of the profile likelihood in Fig.\\ \\ref{fig:z3scalar1}, and highlighted in terms of the semi-annihilation fraction\n\\begin{equation}\n\\alpha=\\frac{1}{2}\\frac{\\langle\\sigma v_\\mathrm{rel}\\rangle_{SS\\rightarrow hS}}{\\langle\\sigma v_\\mathrm{rel}\\rangle+\\frac{1}{2}\\langle\\sigma v_\\mathrm{rel}\\rangle_{SS\\rightarrow hS}},\\label{eqn:sa_fraction}\n\\end{equation}\nin Fig.\\ \\ref{fig:z3scalar2}.  Here $\\langle\\sigma v_\\mathrm{rel}\\rangle$ is the thermally averaged (semi-)annihilation cross-section weighted by the relative velocity between annihilating particles.\n\nThe vacuum structure of the theory is also more complicated than that of the $\\mathbb{Z}_2$-symmetric model, as regions where $\\mu_3 \\geq 2\\sqrt{\\lambda_{\\sss S}}m_{\\sss S}$ or $\\mu_{\\scriptscriptstyle S}^2<0$ and $\\lambda_{h\\sss S}$ is large can possess a second, $\\mathbb{Z}_3$-breaking minimum.  The results shown in Figs.\\ \\ref{fig:z3scalar1} and \\ref{fig:z3scalar2} avoid these regions, demanding that $S$ does not itself obtain a VEV, and that the potential remains bounded from below.\n\nLike the $\\mathbb{Z}_2$-symmetric variant, the $\\mathbb{Z}_3$-symmetric model can in principle completely stabilise the SM vacuum.  However, because of the various factors of 2 introduced relative to the $\\mathbb{Z}_2$ case, by virtue of DM not being self-adjoint, the region where this is possible is in fact in strong tension with the results from both XENON1T \\cite{Aprile:2018dbl} and PandaX \\cite{Cui:2017nnn}.  $\\mathbb{Z}_3$-symmetric models that stabilise the SM vacuum and produce the entire observed DM relic density are ruled out at 99\\% confidence; those constituting only a fraction of DM are ruled out at 98\\% confidence.  The same is expected of other $\\mathbb{Z}_N$-symmetric models with $N>3$, which also feature non-self-adjoint DM.\n\nAs in the $\\mathbb{Z}_2$-symmetric case, a Bayesian analysis prefers the higher-mass part of the parameter space, due to the fine-tuning needed to achieve agreement with all experimental data in both the resonance and semi-annihilation (intermediate mass) regions.  In this case, the additional tuning in $\\mu_3$ required to satisfy the condition $\\mu_3 \\leq 2\\sqrt{\\lambda_{\\sss S}}m_{\\sss S}$ -- and to achieve sufficient semi-annihilation in the intermediate-mass region -- further penalises these regions.\n\n\\begin{figure}[t]\n\t\\centering\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_VDM_2_1_like2D_combined_low_mass}\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_VDM_2_1_like2D_combined_high_mass}\n  \\caption{Profile likelihoods of parameters in the $\\mathbb{Z}_2$-symmetric vector singlet Higgs portal dark matter model, including constraints from direct and indirect detection, the relic density of dark matter and LHC searches for invisible decays of the Higgs boson, along with various Standard Model, dark matter halo and nuclear uncertainties.  \\textit{Left}: the low-mass resonance region.  \\textit{Right}: the full mass range.  Grey shading indicates the area that fails the unitarity cut (Eq.\\ \\ref{eq:vec_unitarity}). Orange annotations indicate the edge of the allowed parameter space along which the model reproduces the entire cosmological abundance of dark matter. Contours show 1 and 2$\\sigma$ confidence regions. White stars indicate the location of the best-fit point. From \\protect\\cite{HP}.}\n\t\\label{fig:vector}\n\\end{figure}\n\n\n\\subsubsection{$\\mathbb{Z}_2$-symmetric vector singlet}\n\nIf DM is a $\\mathbb{Z}_2$-symmetric vector singlet $V_\\mu$ interacting with the SM via the Higgs portal, its effective Lagrangian takes the form\n\\begin{equation}\n\\mathcal{L_V} = -\\frac{1}{4} W_{\\mu\\nu} W^{\\mu\\nu} + \\frac{1}{2} \\mu_V^2 V_\\mu V^\\mu - \\frac{1}{4!} \\lambda_{V} (V_\\mu V^\\mu)^2 + \\frac{1}{2} \\lambda_{hV} V_\\mu V^\\mu H^\\dagger H.\n\\label{eq:Lag_V}\n\\end{equation}\nHere $W_{\\mu\\nu} \\equiv \\partial_\\mu V_\\nu - \\partial_\\nu V_\\mu$ is the field strength tensor for the new vector.  The tree-level DM mass has exactly the same form as Eq.\\ \\ref{ms}.  Although all terms here are dimension 4, the theory is not renormalisable, as it possesses an explicit mass term for $V_\\mu$.  Perturbative unitarity is violated at energies above this mass.  In the \\textsf{GAMBIT}\\xspace analysis \\cite{HP}, this issue was avoided by excluding the region of parameter space\n\\begin{equation}\n0 \\le \\lambda_{hV} \\le 2m_V^2\/v_0^2\n\\label{eq:vec_unitarity}\n\\end{equation}\nfrom the analysis.\n\nThe phenomenology of the vector model is very similar to that of the $\\mathbb{Z}_2$-symmetric scalar variant, with the only major difference being the absence of the intermediate-mass solution due to the unitarity requirement (Fig.\\ \\ref{fig:vector}; the region excluded from the analysis due to the unitarity condition is shown in grey).  The Bayesian analysis once again prefers the high-mass region due to the fine-tuning of nuisance parameters required in the resonance region.\n\n\\subsubsection{$\\mathbb{Z}_2$-symmetric Dirac \\& Majorana fermionic singlets}\n\nThe Lagrangians of the fermionic singlet Higgs portal models are\n\\begin{align}\n  \\mathcal{L}_{\\chi} &= \\frac{1}{2} \\overline{\\chi} (i\\slashed{\\partial} - \\mu_\\chi) \\chi - \\frac{1}{2}\\frac{\\lambda_{h\\chi}}{\\Lambda_\\chi} \\Big(\\cos\\theta \\, \\overline{\\chi}\\chi + \\sin\\theta \\, \\overline{\\chi}i\\gamma_5 \\chi \\Big) H^\\dagger H,\n  \\label{eq:Lag_chi}\\\\\n  \\mathcal{L}_{\\psi} &= \\overline{\\psi} (i \\slashed{\\partial} - \\mu_\\psi) \\psi \\nonumber - \\frac{\\lambda_{h\\psi}}{\\Lambda_\\psi} \\Big(\\cos\\theta \\, \\overline{\\psi}\\psi + \\sin\\theta \\, \\overline{\\psi}i\\gamma_5 \\psi \\Big) H^\\dagger H,\n  \\label{eq:Lag_psi}\n\\end{align}\nwith the Majorana variant denoted $\\chi$ and the Dirac variant $\\psi$.  These noticeably possess dimension-5 effective portal operators suppressed by the scale of new physics $\\Lambda$, with both scalar ($CP$-even) and pseudoscalar ($CP$-odd) couplings.  The degree to which the portal interaction violates $CP$ is dictated by the mixing angle $\\theta$, where $\\theta = 0$ corresponds to pure $CP$ conservation and $\\theta = \\frac\\pi2$ to maximal $CP$ violation.\n\nAs in the scalar and vector models, the portal interaction produces terms quadratic in the DM field following electroweak symmetry breaking.  The pseudoscalar coupling leads to an imaginary mass term, which must be rotated away with the field transformation $X \\rightarrow e^{i\\gamma_5 \\alpha\/2} X$ for $X \\in \\{\\chi, \\psi\\}$, in order to arrive at the physical (real) mass.  This introduces a new parameter $\\alpha$.  The physical masses are then\n\\begin{equation}\nm_{X}^2 = \\left(\\mu_{X} + \\frac{1}{2}\\frac{\\lambda_{hX}}{\\Lambda_{X}} v_0^2 \\cos\\theta \\right)^2 + \\left(\\frac{1}{2}\\frac{\\lambda_{X}}{\\Lambda_{X}}v_0^2 \\sin\\theta \\right)^2.\n\\end{equation}\nThe rotation parameter $\\alpha$ is fixed by the requirement that the mass be real, so all phenomenology can be described by three parameters: $m_X$, $\\lambda_X\/\\Lambda_X$ and $\\xi \\equiv \\theta + \\alpha$.  Notably, the pure $CP$-conserving theory ($\\theta = 0$) remains $CP$-conserving after electroweak symmetry breaking ($\\xi = 0$), but maximal $CP$ violation before electroweak symmetry breaking does not correspond to maximal violation after the symmetry is broken (i.e. $\\theta = \\frac\\pi2 \\notimplies \\xi = \\frac\\pi2$).\n\nWhilst the $CP$-even Higgs portal coupling leads to the familiar velocity and momentum-independent nuclear scattering cross-section, the $CP$-odd coupling gives rise to an interaction suppressed by $q^2$, the square of the momentum exchanged in the scattering event.  This leads to an overall suppression of direct detection signals and corresponding constraints for $\\xi \\rightarrow \\frac\\pi2$.  Conversely, the $CP$-odd coupling produces a velocity and momentum-independent annihilation cross-section, whereas the $CP$-even coupling gives rise to a velocity-suppressed annihilation cross-section.\n\n\\begin{figure}[tbp]\n\t\\centering\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_MDM_2_1_like2D_combined_low_mass}\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_MDM_2_1_like2D_combined_high_mass}\\\\\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_MDM_2_3_like2D_combined}\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_MDM_3_1_like2D_combined}\n  \\caption{Profile likelihoods of parameters in the $\\mathbb{Z}_2$-symmetric Majorana fermion singlet Higgs portal dark matter model, including constraints from direct and indirect detection, the relic density of dark matter and LHC searches for invisible decays of the Higgs boson, along with various Standard Model, dark matter halo and nuclear uncertainties.  The upper-left panel shows a zoomed-in view of the low-mass resonance region.  Grey shading indicates the area that fails the unitarity cut (Eq.\\ \\ref{eq:fermion_unitarity}). Orange annotations indicate the edge of the allowed parameter space along which the model reproduces the entire cosmological abundance of dark matter. Contours show 1 and 2$\\sigma$ confidence regions.  White stars indicate the location of the best-fit point. From \\protect\\cite{HP}.}\n\t\\label{fig:fermion1}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\t\\centering\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_MDM_2_1_post2D_xi_free_TWalk_low_mass}\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_MDM_2_1_post2D_xi_free_TWalk_high_mass}\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_MDM_2_3_post2D_xi_free_TWalk}\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_MDM_3_1_post2D_xi_free_TWalk}\n  \\caption{Posterior probability densities from a Bayesian analysis of the $\\mathbb{Z}_2$-symmetric Majorana fermion singlet Higgs portal dark matter model, using the same likelihood functions as Fig.\\ \\protect\\ref{fig:fermion1}.  White bullets indicate posterior means; other annotations are as in Fig.\\ \\protect\\ref{fig:fermion1}. From \\protect\\cite{HP}.}\n\t\\label{fig:fermion2}\n\\end{figure}\n\nProfile likelihoods from the global fit to the Majorana fermion model are shown in Fig.\\ \\ref{fig:fermion1}.  Results for Dirac fermion dark matter are broadly very similar, and differ from the Majorana case only in the exact location of the border of the allowed parameter space, reflecting the essentially inconsequential nature of the relative factors of 2 between the two Lagrangians.  Grey regions correspond to the regime\n\\begin{equation}\n\\lambda_{hX}\/\\Lambda_{X} \\geq 2\\pi\/m_X,\n\\label{eq:fermion_unitarity}\n\\end{equation}\nwhere the validity of the EFT becomes questionable.  Further discussion on this issue can be found in Ref.\\ \\cite{HP}; it would also be possible to unitarise the theory, and draw further constraints in this region, using the $K$-matrix formalism \\cite{Bell:2016obu,Balaji:2018qyo}.\n\nThe preferred regions in the mass-coupling plane (upper panels of Fig.\\ \\ref{fig:fermion1}) include the now-familiar resonance and high-mass regions.  However, unlike the vector and scalar models, these are fully connected by valid models at all masses, with the preferred region bounded from below mostly by the relic density constraint, supported by indirect detection.  This is because profiling over $\\xi$ allows for the selection of $CP$-violating couplings in order to avoid constraints from direct detection.  The degree of tuning in $\\xi$ required to achieve this is apparent in the lower panels of Fig.\\ \\ref{fig:fermion1}, where it is clear that good fits can be found for any value of $\\xi$ in the resonance region, but that higher masses require some degree of $CP$ violation in order to avoid direct detection.  This becomes even clearer in the equivalent Bayesian results shown in Fig.\\ \\ref{fig:fermion2}, where intermediate masses and couplings are disfavoured relative to other regions, due to the need to make $CP$ violation nearly maximal in order to avoid direct detection.\n\n\\begin{figure}[t]\n\t\\centering\n  \\includegraphics[width=0.495\\columnwidth]{figures\/plot_MDM_3_post1D_xi_free_TWalk}\n  \\caption{Marginalised one-dimensional posterior probability density for the $CP$-mixing parameter $\\xi$ in the $\\mathbb{Z}_2$-symmetric Majorana fermion singlet Higgs portal dark matter model.  This result has been extracted from the same analysis as that shown in Fig.\\ \\protect\\ref{fig:fermion2}.  The value $\\xi = 0 = \\pi$ corresponds to $CP$ conservation; a clear preference for violation of $CP$ symmetry is evident. The blue bullet indicates the posterior mean value of $\\xi$, and the red star the value of $\\xi$ at the best-fit sample.  From \\cite{HP}.}\n\t\\label{fig:fermion3}\n\\end{figure}\n\nIntegrating the posterior over all parameters other than $\\xi$ (Fig.\\ \\ref{fig:fermion3}), there is a clear preference for $CP$ violation.  This reflects the fact that the more $CP$ violation permitted, the broader the range of other parameters able to give good fits to the combined data of all experiments.  Performing Bayesian model comparison between the full model and its pure $CP$-conserving subspace (i.e.\\ $\\xi = 0$) results in Bayes factors of between 70:1 and 140:1, depending on the adopted priors.  This indicates a strong preference for $CP$ violation in fermionic Higgs portal models.  Bayesian model comparison between the scalar, vector and fermionic variants of the Higgs portal DM model reveals essentially equal odds for each of the scalar and fermionic models, but a 6:1 preference for all of these models over the vector variant.\n\n\\subsection{Axions}\n\n\\subsubsection{Axion models and their implementation in \\textsf{GAMBIT}\\xspace}\n\nAxions are an intriguing theoretical possibility due to their ability to solve the strong-$CP$ problem of the SM whilst providing a credible DM candidate~\\cite{Preskill:1982cy,Abbott:1982af,Dine:1982ah,1986_turner_axiondensity}. One can also use axion-like particles to reconcile various tensions between astrophysical observations and theory, including the cooling of white dwarfs~\\cite{Isern:1992gia,1205.6180,1211.3389,1512.08108,1605.06458,1605.07668,1708.02111}, and the transparency of the Universe to gamma rays~\\cite{0707.4312,0712.2825,1001.0972,1106.1132,1201.4711,1302.1208}.\n\nThe strong-$CP$ problem is ultimately a fine-tuning problem, arising from the fact that the SM symmetries permit a $CP$-odd term in the SM Lagrangian density of the form:\n\n\\begin{equation}\n\\mrm{\\pazocal{L}}{QCD} \\supset - \\frac{\\mrm{\\alpha}{S}}{8\\pi} \\mrm{\\theta}{QCD} G_{\\mu \\nu}^a \\widetilde{G}^{\\mu \\nu, a} \\, , \\label{eq:QCDLagrangian}\n\\end{equation}\nwhere $G_{\\mu \\nu}^a$ is the gluon field strength tensor, $\\widetilde{G}^{\\mu \\nu, a}$ is its dual (both of which have the $SU(3)$ gauge index $a$ explicitly shown), and $\\mrm{\\alpha}{S}$ is the strong coupling constant. The angle $\\mrm{\\theta}{QCD} \\in [-\\pi, \\pi]$ is a free parameter. In the SM, the term also receives a contribution from the chiral anomaly which, for down- and up-type Yukawa matrices $Y_d$ and $Y_u$, replaces $\\mrm{\\theta}{QCD}$ by the effective angle\n\\begin{equation}\n\\mrm{\\theta}{eff} \\equiv \\mrm{\\theta}{QCD} - \\arg \\left [ \\det (Y_dY_u) \\right ] \\, . \\label{eq:thetaeff}\n\\end{equation}\nA non-zero $\\mrm{\\theta}{eff}$ would result in $CP$-violating effects in strong interactions, which are severely constrained by observed upper limits on the electric dipole moment of the neutron, demanding $|\\mrm{\\theta}{eff}|\\mathrel{\\rlap{\\lower4pt\\hbox{$\\sim$}}\\raise1pt\\hbox{$<$}} \\num{e-10}$ \\cite{1509.04411}. Naively, this can only be avoided in the SM by fine-tuning the value of $\\mrm{\\theta}{QCD}$ to cancel the contribution from the chiral anomaly.\n\nAn alternative solution, first proposed by Peccei and Quinn~\\cite{1977_pq_axion1,1977_pq_axion2}, is to add a new global, axial $U(1)$ symmetry spontaneously broken by the vacuum expectation value $v$ of a complex scalar field. This breaking has an associated pseudoscalar Nambu-Goldstone boson, $a(x)$, which supplements $\\mrm{\\theta}{eff}$ by a new term $Na(x)\/v$, where the non-zero integer $N$ is the colour anomaly of the added symmetry. The Vafa-Witten theorem~\\cite{1984_vafa_vafawitten1,1984_vafa_vafawitten2} can then be used to show that  $\\mrm{\\theta}{eff}+Na(x)\/v$ is dynamically driven to zero, solving the strong $CP$ problem.\n\nIn the resulting theory of the QCD axion, the axion is practically massless until the time of the QCD phase transition, due to a shift symmetry of the $U(1)$ phase, which prevents a mass term in the Lagrangian. After this, however, it picks up a small, temperature-dependent mass due to breaking of the continuous shift symmetry by fluctuations of the gluon fields. This gives rise to an effective axion potential\n\\begin{equation}\n\tV(a) = f_a^2 \\, m_a^2 \\, \\left[1 - \\cos (a\/f_a) \\right] \\, , \\label{eq:axion_eff_pot}\n\\end{equation}\nwhere $m_a$ is the temperature-dependent axion mass and $f_a \\equiv v\/N$. The zero-temperature axion mass, $m_{a,0}$, can be calculated using next-to-leading order chiral perturbation theory, and it turns out to be inversely proportional to $f_a$ for the QCD axion. At higher temperatures, numerical estimates of the mass are available from lattice QCD results, which can be described to a good approximation by\n\\begin{equation}\nm_a(T) =  m_{a,0}\n\\begin{cases}\n\\hfil 1 \\hfil & \\mathrm{if \\; } T \\leq T_\\chi \\\\\n\\left ( \\frac{T_\\chi}{T} \\right )^{\\beta\/2} & \\mathrm{otherwise}\n\\end{cases} \\, . \\label{eq:axionmass}\n\\end{equation}\n$T_\\chi$ and $\\beta$ are in principle calculable, but can be left as nuisance parameters in order to account for systematic uncertainties in the calculations.\n\nIn fact, QCD axions are only one instance of a general class of \\emph{axion-like particles} (ALPs), which could generally result from the breaking of a $U(1)$ symmetry at some scale $f_a$, with mass generation occurring via the explicit breaking of the residual symmetry at some lower scale $\\Lambda$~\\cite{1987_kim_lightpseudoscalars,1002.0329,1801.08127}. It can be shown that in a Friedmann-Robertson-Walker-{Lema\\^itre} universe, a QCD axion or ALP field $\\theta(t)=a(t)\/f_a$ satisfies the equation of motion\n\\begin{equation}\n  \\ddot{\\theta} + 3H(t) \\, \\dot{\\theta} + m_a^2(t) \\, \\sin (\\theta) = 0,\n\\label{eq:AxionFieldEq}\n\\end{equation}\nwhere we have assumed the canonical axion potential of\n\\begin{equation}\n  V(\\theta) = f_a^2m_a^2\\left[1-\\cos (\\theta)\\right]. \\label{eq:potential}\n\\end{equation}\nThis is subject to the boundary condition $\\theta(\\mrm{t}{i}) = \\mrm{\\theta}{i}$ and $\\dot{\\theta}(\\mrm{t}{i}) = 0$, where $\\mrm{\\theta}{i}$ is called the \\textit{initial misalignment angle}.\n\nThe \\textsf{GAMBIT}\\xspace collaboration completed a comprehensive study of axion and broader ALP theories in 2018~\\cite{Axions}, using an extensive list of experimental constraints. These rely on the interactions of ALPs with SM matter, which can be studied in an effective field theory framework~\\cite{Kaplan:1985dv,1985_srednicki_axioneft,1986_georgi_axioneft}.\n\nThe most general axion\/ALP model in \\textsf{GAMBIT}\\xspace assumes the effective Lagrangian density to take the form\n\\begin{equation}\n\t\\pazocal{L}_a^\\mathrm{int} = -\\frac{f_ag_{a\\gamma\\gamma}}{4} \\theta F_{\\mu\\nu}\\widetilde{F}^{\\mu\\nu} - \\frac{f_ag_{aee}}{2m_e} \\bar{e}\\gamma^\\mu\\gamma_5e\\partial_\\mu \\theta \\, .\\label{eq:ax:lagrange}\n\\end{equation}\nNote that this provides for possible axion-photon and axion-electron interactions, whilst ignoring terms for other interactions that do not currently give rise to interesting experimental observables. The complete family tree of \\textsf{GAMBIT}\\xspace axion\/ALP models is shown in Fig.\\ \\ref{fig:AxionModelTree}, headed by the \\textsf{GeneralALP}\\xspace model, whose parameters have all now been defined. This provides a phenomenological description of axion physics that is not constrained to give physical solutions, as the couplings are not inversely proportional to $f_a$.\n\nThe \\textsf{QCDAxion}\\xspace model appears as a child model, and differs from the more general case by having tight constraints on some parameters, arising from the known relationships with the QCD scale. The axion-electron coupling is traded for the model-dependent form factor~$C_{aee}$\n\\begin{equation}\n\tg_{aee} = \\frac{m_e}{f_a} \\; C_{aee}\\, , \\label{eq:qcdaxioncouplings1}\n\\end{equation}\nwhilst the axion-photon coupling is replaced by the model-dependent ratio of the electromagnetic and colour anomalies~$E\/N$\n\\begin{equation}\n\tg_{a\\gamma\\gamma} = \\frac{\\mrm{\\alpha}{EM}}{2\\pi f_a}\\left(\\frac{E}{N} - \\widetilde{C}_{a\\gamma\\gamma}\\right) \\, .\\label{eq:qcdaxioncouplings2}\n\\end{equation}\n$\\widetilde{C}_{a\\gamma\\gamma}$ is a model-independent contribution from axion-pion mixing, which is taken from Ref.~\\cite{1511.02867}, and assigned a nuisance likelihood with a relevant uncertainty. Note that the ratio $E\/N$ should in principle take discrete values, but it is sampled as a continuous parameter for convenience, seeing as the possible rational values that it can take are close together. The final nuisance parameter of the \\textsf{QCDAxion}\\xspace model is $\\Lambda_\\chi$, which results from replacing the parameter  $m_{a,0}$ of the \\textsf{GeneralALP}\\xspace model by an energy scale such that\n\\begin{equation}\n\tm_{a,0} \\equiv \\frac{\\Lambda_\\chi^2}{f_a} \\, .\n\\end{equation}\nThe value of $\\Lambda_\\chi$ is taken from first-principle calculations of the zero-temperature axion mass provided in Ref.~\\cite{1511.02867},\\footnote{This value was later updated in \\cite{Gorghetto:2018ocs}, after the appearance of Ref.\\ \\cite{Axions}.} and it is subject to a Gaussian nuisance likelihood.\n\nThe other models of interest for this review are the \\textsf{KSVZAxion}\\xspace and \\textsf{DFSZAxion}\\xspace model variants, which involve further field content being added to the SM. In \\textsf{KSVZAxion}\\xspace models~\\cite{1979_kim_ksvz,1980_shifman_ksvz}, the SM is supplemented by one or more electrically neutral, heavy quarks, and there are no tree-level interactions between the axion and SM fermions. There is, however, still an axion-photon interaction, which generates an axion-electron interaction at one loop. The \\textsf{GAMBIT}\\xspace study investigated four different \\textsf{KSVZAxion}\\xspace models, distinguished only by the choice of $E\/N$ from the set 0, 2\/3, 5\/3 and 8\/3.\n\n\\textsf{DFSZAxion}\\xspace models supplement the SM by an additional Higgs doublet~\\cite{1980_zhitnitsky_dfsz,1981_dine_dfsz}, which results in direct axion-electron interactions. Defining the ratio of the two Higgs vacuum expectation values to be  $\\tan (\\beta^\\prime)$, one can write two variants of the \\textsf{DFSZAxion}\\xspace scenario as\n\\begin{align}\n\t\\begin{array}{lll}\n\t\tC_{aee} = \\sin^2 (\\beta^\\prime)\\left\/3\\right., \\quad \\phantom{.} & E\/N = 8\/3  \\quad \\phantom{.}& (\\textsf{DFSZAxion-I}\\xspace) \\, \\\\\n\t\tC_{aee} = \\left[1-\\sin^2 (\\beta^\\prime) \\right]\\left\/3\\right., \\quad \\phantom{.} & E\/N = 2\/3 \\quad \\phantom{.} & (\\textsf{DFSZAxion-II}\\xspace) \\,\n\t\\end{array} \\label{eq:dfsz:caee}.\n\\end{align}\nIt is thus convenient to replace the parameter~$C_{aee}$ in the \\textsf{QCDAxion}\\xspace model by $\\tan (\\beta^\\prime)$.\n\n\\begin{figure}[bt]\n\t\\centering\n\t\\input{include\/tree.tex}\n\t\\caption{Family tree of axion models in \\textsf{GAMBIT}\\xspace. The numbers in brackets refer to the number of model parameters; $(n+m)$ indicates $n$ (largely unconstrained) fundamental parameters of the model and $m$ (typically well-constrained) nuisance parameters. From \\cite{Axions}.}\n\t\\label{fig:AxionModelTree}\n\\end{figure}\n\n\\subsubsection{Experimental constraints on axions}\n\\label{sec:axionL}\nMany experiments are sensitive to the axion theories described here, and current null results place tight constraints on axions for specific combinations of masses and coupling strengths. Here we provide a brief review of those constraints, referring the reader to Ref.~\\cite{Axions} for a detailed description of the experimental likelihoods.\n\n\\begin{itemize}\n\\item \\textbf{Light-shining-through-wall (LSW) experiments: }Photon-axion interactions would allow photons to pass through a wall by becoming an axion, only to convert back to a photon on the other side. LSW experiments attempt to observe this by shining laser light onto an opaque material in the presence of a strong magnetic field. The \\textsf{GAMBIT}\\xspace LSW likelihood uses the results from the ALPS-I experiment, using data for both evacuated and gas-filled magnets~\\cite{1004.1313}.\n\\item \\textbf{Helioscopes: }Axion production in the Sun can be probed by observing the solar disc with a long magnet contained in an opaque casing. Any axions produced in the Sun that made it to Earth would pass through the exterior, and potentially convert to photons within the magnetic field in the interior. The details of solar axion production depend on the solar model, in addition to the axion-photon and axion-electron couplings. The \\textsf{GAMBIT}\\xspace axion studies utilise the AGSS09met solar model~\\cite{Serenelli09,AGSS} and its more recent iteration~\\cite{1611.09867}, and utilise two separate likelihoods for the 2007 and 2017 results of the CAST experiment~\\cite{hep-ex\/0702006,1705.02290}.\n\\item \\textbf{Haloscopes (cavity experiments): }Axion haloscopes aim to detect resonant axion-photon conversion inside a tunable cavity~\\cite{1983_Sikivie, 1985_Sikivie}, with microwave cavities providing the greatest current sensitivity to axions. Unfortunately, the resonant nature of the experiment means that one obtains highly sensitive constraints only within a very narrow mass range. The ability of haloscope experiments to detect axions depends on their cosmological abundance, as well as the galactic DM velocity distribution~\\cite{2011_Hoskins}.  The \\textsf{GAMBIT}\\xspace study combines separate likelihood terms for the Rochester-Brookhaven-Fermi (RBF)~\\cite{DePanfilis:1987dk,Wuensch:1989sa}, University of Florida (UF)~\\cite{Hagmann:1990tj}, ADMX 1998-2009~\\cite{astro-ph\/9801286,Asztalos:2001tf,astro-ph\/0310042,astro-ph\/0603108,0910.5914} and ADMX 2018~\\cite{1804.05750} datasets.\n\\item \\textbf{Dark matter relic density: }Although axions are not a thermal relic such as those encountered in WIMP models, the relic abundance of axion DM is calculable numerically via the details of the realignment mechanism that follow from the equation of motion given in Eq~\\ref{eq:AxionFieldEq}. This can be compared with the observed value from the most recent \\emph{Planck} analysis~\\cite{Planck15cosmo}. The \\textsf{GAMBIT}\\xspace axion study applied this as both an upper limit (in which case axions are allowed to provide only a component of DM) and, in separate analyses of each model, a measurement. In the former case, anticipated yields in experiments that rely on the local DM density were scaled accordingly.\n\\item \\textbf{Distortions of gamma-ray spectra: }Axion-photon conversions could occur in strong galactic or inter-galactic magnetic fields, resulting in a distortion of the spectra of distant sources~\\cite{Raffelt:1987im,hep-ph\/0111311,hep-ph\/0204216,0704.3044}. There is a critical energy scale $E_{crit}$ at which photons will efficiently convert into axions, and it can be shown that spectral distortions only occur in real measurements when the critical energy lies within the spectral window of the instrument~\\cite{1205.6428,1305.2114}. This has the effect of localising constraints from spectral distortion measurements to specific ranges of the axion mass. The \\textsf{GAMBIT}\\xspace axion study utilises a likelihood based on H.E.S.S studies of the active galactic nucleus PKS 2155-304~\\cite{1311.3148}.\n\\item \\textbf{Supernova 1987A: }If axions had been produced in the SN1987A supernova explosion, they could have been converted to photons in the Galactic magnetic field, and detected as a coincident gamma ray burst by the Solar Maximum Mission~\\cite{Chupp:1989kx}. The absence of this observation has been used to constrain axion properties. The \\textsf{GAMBIT}\\xspace study uses a likelihood based on Ref.~\\cite{1410.3747}.\n\\item \\textbf{Horizontal Branch stars and the R parameter: }The existence of axions would provide an extra mechanism of energy loss for stars, causing them to cool faster~\\cite{Sato:1975vy,Raffelt:1990yz,book_raffelt_laboratories}. This would affect the relative time that stars spend on the Horizontal Branch (HB) and upper Red Giant Branch (RGB), which in turn sets the observed ratio of the numbers of stars on these branches ($R = \\mrm{N}{HB}\/\\mrm{N}{RGB}$). Theory suggests that axions would have the most significant impact on the lifetimes of HB stars, leading to a reduction in $R$. The \\textsf{GAMBIT}\\xspace $R$ parameter likelihood is based on the comparison of a calculation of the $R$ parameter for axion theories~\\cite{1512.08108,1983A&A...128...94B,Raffelt:1989xu,1311.1669,1406.6053} with the observed value of $\\mrm{R}{obs}=1.39 \\pm 0.03$~\\cite{1406.6053}, which is based on a weighted average of cluster count obervations \\cite{astro-ph\/0403600}.\n\\item \\textbf{White Dwarf cooling hints: }White dwarfs (WDs) are intriguing axion laboratories for several reasons. The first is that energy loss via axion production in WDs can be probed experimentally by using measurements of the oscillations of their radii and luminosities. These can be related to their internal structure via astroseismology, and measurements of the decrease in the oscillation periods can be related to energy loss. The second reason is that WDs have electron-degenerate cores, allowing us to probe the axion-electron coupling rather than the electron-photon coupling. A number of previous studies have calculated the expected period decrease in the presence of axions.  The \\textsf{GAMBIT}\\xspace WD cooling likelihood is based on interpolation of the results and uncertainties found in Refs.~\\cite{1205.6180,1211.3389,1605.06458,1605.07668}. Current evidence suggests that WDs actually require an additional cooling mechanism relative to standard models, but this remains controversial due to a number of experimental and theoretical issues. The \\textsf{GAMBIT}\\xspace axion paper thus contains studies generated both with and without WD cooling hints added to the combined likelihood.\n\\end{itemize}\n\n\\subsubsection{\\textsf{GAMBIT}\\xspace results for the \\textsf{QCDAxion}\\xspace model}\n\nAlthough the \\textsf{GAMBIT}\\xspace axion paper contained results for all of the models described above, we will here concentrate on the \\textsf{QCDAxion}\\xspace results in the interests of brevity. The various parameters (including nuisance parameters) are shown in Table~\\ref{tab:priors:QCDAxion}, along with the chosen priors and prior ranges. For each of the nuisance parameters, the prior range is chosen to cover a range of approximately $-5\\sigma$ to $+5\\sigma$ around the known central value, where $\\sigma$ is the known uncertainty. The range of $E\/N$ values is selected to encompass those encountered in a broad range of previous axion model studies, whilst the range on $f_a$ is driven by the requirement that the range of possible axion masses reaches from very small masses to the the largest mass allowed by bounds on hot DM. Our choice of a log prior for $f_a$ is motivated by the fact that the scale is unknown. $C_{aee}$ is explored in a generous range around 1, whilst the causal structure of the early Universe motivates our use of a flat prior on the initial misalignment angle $\\mrm{\\theta}{i}$. The local DM density $\\rho_0$ is given the same treatment as in previous \\textsf{GAMBIT}\\xspace studies.\n\n\\begin{table}[tbp]\n\t\\caption{Prior choices for \\textsf{QCDAxion}\\xspace models in \\cite{Axions}.\\label{tab:priors:QCDAxion}}\n\t\\footnotesize\n\t\\centering\n\t\\begin{tabular}{@{}lccc}\n\t\t\\toprule\n\t\t\\textbf{Model} & \\multicolumn{2}{l}{\\textbf{Parameter range\/value}} & \\textbf{Prior type} \\\\\n\t\t\\midrule\n\t\t\\textsf{QCDAxion}\\xspace & \\iuo{f_a}{\\text{G\\eV}\\xspace} & \\prrange{e6}{e16} & log \\\\\n\t\t& \\iuo{\\Lambda_\\chi}{\\text{M\\eV}\\xspace}& \\prrange{73}{78} & flat \\\\\n\t\t& $\\widetilde{C}_{a\\gamma\\gamma}$ & \\prrange{1.72}{2.12} & flat \\\\\n\t\t& $E\/N$ & \\prrange{-1.33333}{174.667} & flat \\\\\n\t\t& $C_{aee}$ & \\prrange{e-4}{e4} & log \\\\\n\t\t& $\\mrm{\\theta}{i}$ & \\prrange{-3.14159}{3.14159} & flat \\\\\n\t\t& $\\beta$ & \\prrange{7.7}{8.2} & flat \\\\\n\t\t& \\iuo{T_\\chi}{\\text{M\\eV}\\xspace}& \\prrange{143}{151} & flat \\\\\n\t\t\\midrule\n\t\tLocal DM density & \\iuo{\\rho_0}{\\text{G\\eV}\\xspace\\per\\centi\\metre^3} & \\prrange{0.2}{0.8} & flat \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\\end{table}\n\n\\begin{figure}[tbp]\n\t\\centering\n\t{\n\t\t\\includegraphics[width=0.49\\linewidth]{figures\/diver\/plot_2_10M1_QCDAxion_100_6_like2D}\n\t\t\\hfill\n\t\t\\includegraphics[width=0.49\\linewidth]{figures\/diver\/plot_2_10M2_QCDAxion_100_6_like2D}\n\t}\n\t{\n\t\t\\includegraphics[width=0.49\\linewidth]{figures\/diver\/plot_2_10M1_QCDAxion_100_108_like2D}\n\t\t\\hfill\n\t\t\\includegraphics[width=0.49\\linewidth]{figures\/diver\/plot_2_10M2_QCDAxion_100_108_like2D}\n\t}\n\t\\caption{Profile likelihoods~(from \\diver) for \\textsf{QCDAxion}\\xspace models with upper limits~(\\textit{left}) and matching condition~(\\textit{right}) for the observed DM relic density. The upper and lower panels show the constraints on the anomaly ratio, $E\/N$, and the absolute value of the initial misalignment angle, $|\\mrm{\\theta}{i}|$, respectively. From \\cite{Axions}. \\label{fig:QCDAxion:frequentist}}\n\\end{figure}\n\nFig.\\ \\ref{fig:QCDAxion:frequentist} shows profile likelihood distributions in various planes of the \\textsf{QCDAxion}\\xspace parameters, obtained without the presence of WD cooling hints in the combined likelihood. The left panels show the result of imposing the relic density constraint as an upper limit. The exclusion of the low-$f_a$ (high mass) region, except at very low values of the axion-photon coupling (which is related to $E\/N$), arises from the $R$ parameter and CAST results. The slight reduction in the profile likelihood for masses lower than approximately \\SI{0.1}{\\micro\\ensuremath{\\text{e}\\mspace{-0.8mu}\\text{V}}\\xspace} also comes from the $R$ parameter likelihood; for such masses, the maximum allowed value for the axion-electron coupling ($C_{aee} \\leq \\num{e4}$) is not large enough to perfectly satisfy the $R$-parameter constraint. If axions are assumed to saturate the relic abundance of DM (right top panel), the high-mass region is excluded entirely due to the fact that the realignment mechanism cannot produce enough DM. The bottom row of Fig.\\ \\ref{fig:QCDAxion:frequentist} shows the allowed values for the initial misalignment angle. In the case that axions supply all of DM, we recover the familiar result that $\\left|\\mrm{\\theta}{i}\\right| \\ll 1$, for QCD axion masses of $m_{a,0} \\mathrel{\\rlap{\\lower4pt\\hbox{$\\sim$}}\\raise1pt\\hbox{$<$}} \\SI{0.1}{\\micro\\ensuremath{\\text{e}\\mspace{-0.8mu}\\text{V}}\\xspace}$, a fine-tuning that we will discuss further in the context of a Bayesian analysis.\n\nIn the left panel of Fig.\\ \\ref{fig:cooling:QCDAxion:RelicDens:twoD}, we show the marginalised Bayesian posterior in the $\\Omega_a h^2-m_{a,0}$ plane without WD cooling hints, demonstrating that the scan can find viable parts of the parameter space where axions consistent with all current experimental observations can account for a sizable fraction of dark matter. The situation is similar even when WD cooling hints are included (not shown). One can also observe an interesting bound on the axion mass. If the DM relic density constraint is applied as an upper limit, we find $\\SI{0.73}{\\micro\\ensuremath{\\text{e}\\mspace{-0.8mu}\\text{V}}\\xspace} \\le m_{a,0} \\le \\SI{6.1}{\\milli\\ensuremath{\\text{e}\\mspace{-0.8mu}\\text{V}}\\xspace}$ at 95\\% credibility (equal-tailed interval). Meanwhile, if axions must provide all of the observed dark matter, this changes to $\\SI{0.53}{\\micro\\ensuremath{\\text{e}\\mspace{-0.8mu}\\text{V}}\\xspace} \\le m_{a,0} \\le \\SI{0.13}{\\milli\\ensuremath{\\text{e}\\mspace{-0.8mu}\\text{V}}\\xspace}$.\n\n\\begin{figure}\n        \\centering\n        {\n                \\includegraphics[width=0.49\\linewidth]{figures\/twalk\/plot_2_3042_QCDAxion_100_10_post2D}\n                \\includegraphics[width=0.49\\linewidth]{figures\/twalk\/plot_2_3141_QCDAxion_100_107_post2D}\n        }\n        \\caption{Marginalised posterior for \\textsf{QCDAxion}\\xspace models with the DM relic density constraint treated as an upper limit. \\textit{Left}: constraints on the energy density in axions today, $\\Omega_a h^2$, without the inclusion of WD cooling hints.  \\textit{Right}: constraints on the absolute value of the axion-photon coupling, $|g_{a\\gamma\\gamma}|$.  This panel also includes WD cooling hints, but they have little impact on the result.  For comparison, the right panel also shows the region for which QCD axions are not theoretically possible (red line and shading), as well as the frequentist $2\\sigma$ C.L. constraints on more general ALP models (dashed lines). From \\cite{Axions}. \\label{fig:cooling:QCDAxion:RelicDens:twoD}}\n\\end{figure}\n\nIn the right panel of Fig.\\ \\ref{fig:cooling:QCDAxion:RelicDens:twoD}, we show the marginalised posterior in the $|g_{a\\gamma\\gamma}|-m_{a,0}$ plane with the DM relic density constraint applied as an upper limit, and with WD cooling hints included. Also shown are the {na\\\"ive} bounds on the parameter space that result from phenomenological constraints on \\textsf{GeneralALP}\\xspace models and the maximum value of $E\/N$. The shape of the preferred region is partly formed by the effect of fine-tuning. At low axion masses, this is required to avoid dark matter overproduction, whilst at large axion masses it is required to achieve low values of $|g_{a\\gamma\\gamma}|$ through cancellations between $E\/N$ and $\\tilde{C}_{a\\gamma\\gamma}$. The preferred parameter region is localised within a few orders of magnitude in mass around $m_{a,0} \\sim \\SI{100}{\\micro\\ensuremath{\\text{e}\\mspace{-0.8mu}\\text{V}}\\xspace}$ and $g_{a\\gamma\\gamma} \\sim \\SI{e-12}{\\text{G\\eV}\\xspace^{-1}}$.\n\n\\begin{table}[tbp]\n\t\\caption{Prior choices for \\textsf{DFSZAxion-I}\\xspace, \\textsf{DFSZAxion-II}\\xspace and \\textsf{KSVZAxion}\\xspace models in \\cite{Axions}. Note that the priors listed in the first section of the table apply to all three models.\\label{tab:priors:DFSZvsKSVZ}}\n\t\\footnotesize\n\t\\centering\n\t\\begin{tabular}{lcccl}\n\t\t\\toprule\n\t\t\\textbf{Model} & \\multicolumn{2}{l}{\\textbf{Parameter range\/value}} & \\textbf{Prior type} & \\textbf{Comments}\\\\\n\t\t\\midrule\n\t\t& \\iuo{f_a}{\\text{G\\eV}\\xspace} & \\prrange{e6}{e16} & log & Applies to all\\\\\n\t\t& \\iuo{\\Lambda_\\chi}{\\text{M\\eV}\\xspace}& \\prrange{73}{78} & flat & Applies to all\\\\\n\t\t& $\\widetilde{C}_{a\\gamma\\gamma}$ & \\prrange{1.72}{2.12} & flat & Applies to all\\\\\n\t\t& $\\mrm{\\theta}{i}$ & \\prrange{-3.14159}{3.14159} & flat & Applies to all\\\\\n\t\t& $\\beta$ & \\prrange{7.7}{8.2} & flat & Applies to all\\\\\n\t\t& \\iuo{T_\\chi}{\\text{M\\eV}\\xspace}& \\prrange{143}{151} & flat & Applies to all\\\\\n\t\t\\midrule\n\t\t\\textsf{DFSZAxion-I}\\xspace & $E\/N$ & $8\/3$ & delta & \\\\\n\t\t& $\\tan(\\beta')$ & \\prrange{0.28}{140.0} & log & \\\\\n\t\t\\textsf{DFSZAxion-II}\\xspace & $E\/N$ & $2\/3$ & delta & \\\\\n\t\t& $\\tan(\\beta^\\prime)$ & \\prrange{0.28}{140.0} & log & \\\\\n\t\t\\textsf{KSVZAxion}\\xspace & $E\/N$ & $0$, $2\/3$, $5\/3$, $8\/3$ & delta & Discrete\\\\\n\t\t\\midrule\n\t\tLocal DM density & \\iuo{\\rho_0}{\\text{G\\eV}\\xspace\\per\\centi\\metre^3} & \\prrange{0.2}{0.8} & flat \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\\end{table}\n\nThe \\textsf{GAMBIT}\\xspace Bayesian analysis of axion models also includes a model comparison of the \\textsf{QCDAxion}\\xspace, \\textsf{DFSZAxion-I}\\xspace, \\textsf{DFSZAxion-II}\\xspace and \\textsf{KSVZAxion}\\xspace, based on scans of the latter models that use the priors defined in Table~\\ref{tab:priors:DFSZvsKSVZ}. Bayesian evidence values $\\mathbb{Z}(\\mathcal{M})$ for each model $\\mathcal{M}$ were calculated using the \\MultiNest nested sampling package, before constructing the Bayes factor \\cite{Jeffreys:1939xee,10.2307\/2291091,10.2307\/4356165}\n\\begin{equation}\n\t\\mathcal{B} \\equiv \\frac{\\mathbb{Z}(\\mathcal{M}_1)}{\\mathbb{Z}(\\mathcal{M}_2)} \\equiv \\frac{\\int \\!  \\mathcal{L}\\left(\\text{data}\\left| \\right. \\boldsymbol{\\theta_1} \\right) \\pi_1(\\boldsymbol{\\theta_1}) \\, \\mathrm{d} \\boldsymbol{\\theta_1}}{\\int \\!  \\mathcal{L}\\left(\\text{data}\\left| \\right. \\boldsymbol{\\theta_2} \\right) \\pi_2(\\boldsymbol{\\theta_1}) \\, \\mathrm{d} \\boldsymbol{\\theta_2}} \\, ,\n\\end{equation}\nwhich relates two models $\\mathcal{M}_1$ and $\\mathcal{M}_2$ with parameters $\\boldsymbol{\\theta_1}$ and~$\\boldsymbol{\\theta_2}$. $\\pi_1$ and $\\pi_2$ are the priors on the parameters of the two models, and $\\mathcal{L}$ is the likelihood. The Bayes factor is connected to the ratio of posterior probabilities of the models being correct\n\\begin{equation}\n\t\\frac{\\mathcal{P}\\left(\\mathcal{M}_1 \\left| \\text{data} \\right.\\right)}{\\mathcal{P}\\left(\\mathcal{M}_2 \\left| \\text{data} \\right.\\right)} = \\mathcal{B} \\; \\frac{\\pi(\\mathcal{M}_1)}{\\pi(\\mathcal{M}_2)},\n\\end{equation}\nwhere the prior probabilities of the models themselves being correct are given by $\\pi(\\mathcal{M}_1)$ and $\\pi(\\mathcal{M}_2)$.  In the following, it is assumed that $\\pi(\\mathcal{M}_1)=\\pi(\\mathcal{M}_2)$, causing the the posterior odds ratio to be equal to the Bayes factor.\n\nWithout cooling hints, the odds ratios for pairs of models provide insufficient evidence to favour any particular scenario. However, if it is demanded that axions solve the DM and WD cooling problems simultaneously, there is a positive preference for the \\textsf{QCDAxion}\\xspace model over the DFSZ- and KSVZ-type models, at a level of about 5:1. This results from the larger $C_{aee}$ values allowed in the \\textsf{QCDAxion}\\xspace model, which peaks at $C_{aee}\\approx 100$ in the one-dimensional marginalised posterior. Such a large coupling may cause a problem for model building. A frequentist analysis of the same scenario allows both the \\textsf{DFSZAxion}\\xspace and \\textsf{KSVZAxion}\\xspace models to be rejected with respect to the \\textsf{QCDAxion}\\xspace model with better than 99\\% confidence; if DM is instead allowed to consist only partially of axions, only \\textsf{KSVZAxion}\\xspace models can be rejected in this way.\n\n\\subsection{Right-handed neutrinos}\n\\subsubsection{Model definition}\nThe addition of right-handed ``sterile'' neutrinos to the SM has been proposed to explain the existence of neutrino flavour oscillations, which imply a non-zero neutrino mass.  They also serve an aesthetic theoretical purpose, as neutrinos are the only elementary fermions in the SM to not have both left- and right-handed incarnations. Moreover, sterile neutrinos are permitted to have both a Dirac mass term $\\bar{\\nu_L}M_D\\nu_R$ and a Majorana mass term $\\bar{\\nu_R}M_M\\nu_R^c$, and specific choices of the latter allow sterile neutrinos to solve cosmological problems such as the baryon asymmetry of the Universe~\\cite{Fukugita:1986hr,Akhmedov:1998qx,Asaka:2005pn}, and the DM problem~\\cite{Dodelson:1993je, Shi:1998km}.\n\nA convenient parameterisation of a right-handed neutrino sector is the Casas-Ibarra parametrisation, amended to include 1-loop corrections to the left-handed neutrino mass matrix~\\cite{Casas:2001sr,Lopez-Pavon:2015cga}. This involves writing a matrix that encodes the mixing among left-handed neutrinos (LHNs) and right-handed neutrinos (RHNs) as\n\\begin{align}\n\\Theta = iU_{\\nu}\\sqrt{m_{\\nu}^\\text{diag}}\\mathcal{R}\\sqrt{\\tilde{M}^\\text{diag}}^{-1},\n\\label{CItheta}\n\\end{align}\nwhere $U_{\\nu}$ is the PMNS matrix, $m_{\\nu}^\\text{diag}$ is a diagonalised, one-loop-corrected LHN mass matrix and $\\tilde{M}^\\text{diag}$ is the analogous RHN mass matrix. $\\mathcal{R}$ is a complex, orthogonal matrix written as the product\n\\begin{align}\n\\mathcal{R} = \\mathcal{R}^{23}\\mathcal{R}^{13}\\mathcal{R}^{12}\\;,\n\\label{Rorder}\n\\end{align}\nwhere the $\\mathcal{R}^{ij}$ can, in turn, be parameterised by complex angles $\\omega_{ij}$ with\n\\begin{align}\n\\mathcal{R}^{ij}_{ii} &= \\mathcal{R}^{ij}_{jj} = \\cos\\omega_{ij}, \\\\\n\\mathcal{R}^{ij}_{ij} &= -\\mathcal{R}^{ij}_{ji} = \\sin\\omega_{ij}, \\\\\n\\mathcal{R}^{ij}_{kk} &= 1; k \\neq i,j\\;.\n\\end{align}\n\nWorking in the flavour basis in which the Yukawa couplings of the charged leptons are diagonal by construction, the PMNS matrix $U_\\nu$ can be written as\n\\begin{align}\\label{UnuParameterisation}\n  U_{\\nu} = V^{23}U_{\\delta}V^{13}U_{-\\delta}V^{12}\\mathrm{diag}(e^{i\\alpha_1\/2},e^{i\\alpha_2\/2},1)\\;,\n\\end{align}\nwhere $U_{\\pm\\delta} = \\mathrm{diag}(e^{{\\mp}i\\delta\/2},1,e^{{\\pm}i\\delta\/2})$\nand $V^{ij}$ is parameterised by the LHN mixing angles $\\theta_{ij}$. The non-zero elements of $V^{ij}$ are analogous to those of $\\mathcal{R}$.  $\\alpha_1$, $\\alpha_2$ and $\\delta$ are $CP$-violating phases that are not excluded \\emph{a priori}.\n\n\\begin{table}[t]\n  \\centering\n  \\begin{tabular}{l}\n    \\toprule\n    \\textbf{Parameter} \\\\\n    \\midrule\n    Active neutrino parameters\\\\\n    \\quad$\\theta_{12}$ [rad] \\\\\n    \\quad$\\theta_{23}$ [rad] \\\\\n    \\quad$\\theta_{13}$ [rad] \\\\\n    \\quad$m_{\\nu_0}$ [eV] \\\\\n    \\quad$\\Delta m^2_{21}$ $[10^{-5}\\,\\text{eV}^2]$ \\\\\n    \\quad$\\Delta m^2_{3l}$ $[10^{-3}\\,\\text{eV}^2]$ \\\\\n    \\quad$\\alpha_1$, $\\alpha_2$ [rad] \\\\\n    Sterile neutrino parameters\\\\\n    \\quad$\\delta$ [rad] \\\\\n    \\quad Re $\\omega_{ij}$ [rad] \\\\\n    \\quad Im $\\omega_{ij}$ \\\\\n    \\quad$M_I$ [GeV] \\\\\n    \\quad$R_{\\rm{order}}$ \\\\\n    Nuisance parameters \\\\\n    \\quad$m_h$ [GeV]\\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{The full list of scanned parameters for the \\textsf{GAMBIT}\\xspace right-handed neutrino study \\cite{RHN}.}\n  \\label{tab:scanpars}\n\\end{table}\n\n\\begin{figure}[tbp]\n  \\centering\n  \\includegraphics[width=0.45\\linewidth]{figures\/NH_M_Ue_capped.pdf}\n  \\includegraphics[width=0.45\\linewidth]{figures\/IH_M_Ue_capped.pdf}\\\\\n  \\includegraphics[width=0.45\\linewidth]{figures\/NH_M_Umu_capped.pdf}\n  \\includegraphics[width=0.45\\linewidth]{figures\/IH_M_Umu_capped.pdf}\\\\\n  \\includegraphics[width=0.45\\linewidth]{figures\/NH_M_Utau_capped.pdf}\n  \\includegraphics[width=0.45\\linewidth]{figures\/IH_M_Utau_capped.pdf}\n  \\caption{Profile likelihoods of right-handed neutrino models in the $M_I$ vs $U_{eI}^2$ (top), $M_I$ vs $U_{\\mu I}^2$ (middle) and $M_I$ vs $U_{\\tau I}^2$ (bottom) planes.  Results are shown for normal (left) and inverted (right) neutrino mass ordering. From \\cite{RHN}.}\n  \\label{fig:M_Ue_capped}\n\\end{figure}\n\n\\begin{figure}[tbp]\n  \\centering\n  \\includegraphics[width=0.45\\linewidth]{figures\/NH_M_U_capped.pdf}\n  \\includegraphics[width=0.45\\linewidth]{figures\/IH_M_U_capped.pdf}\n  \\caption{Profile likelihood of right-handed neutrino models in the $M_I$ vs $U_I^2$ plane for normal (left) and inverted (right) neutrino mass ordering. From \\cite{RHN}.}\n  \\label{fig:M_U_capped}\n\\end{figure}\n\nA comprehensive, frequentist \\textsf{GAMBIT}\\xspace study of this scenario has recently been completed \\cite{RHN}.  The full list of parameters considered in the \\textsf{GAMBIT}\\xspace RHN study is given in Table~\\ref{tab:scanpars}.  Separate scans were done for the cases of a normal and an inverted mass hierarchy, and the scanning strategy used a number of carefully targeted scans to ensure that the regions near the various experimental bounds were convergently sampled. The scans used the full range of likelihoods implemented in \\textsf{NeutrinoBit}\\xspace, documented in Section~\\ref{sec:neutrinobit}, in addition to the extra routines described in \\textsf{FlavBit}\\xspace, \\textsf{DecayBit}\\xspace and \\textsf{PrecisionBit}\\xspace. The main analysis used a capped likelihood, in which each point is forced to have a likelihood which is equal to or worse than the SM ($\\mathcal{L} = \\min[\\mathcal{L}_{\\rm{SM}}, \\mathcal{L}_{\\rm{RHN}}]$). This is due to a number of excesses in individual experimental observations that --- although combining to give a small overall significance --- would bias the presentation of exclusion limits on RHN parameters. In this review, we concentrate on the resulting limits on RHNs, and direct the reader to the original study~\\cite{RHN} for a detailed discussion of the excesses.\n\nThe 1-loop Casas-Ibarra parameterisation used in the GAMBIT analysis \\cite{RHN} is valid for seesaw scenarios where the active-sterile mixing, $|\\Theta|^2$, is small. In principle, additional $|\\Theta|^4$ corrections could be expected in low-scale seesaw scenarios, and could only be captured by an exact expansion (e.g.\\ Schechter-Valle \\cite{Schechter:1980gr}). Nevertheless, the loss of generality in the Casa-Ibarra approximation is outweighed by its numerical and computational benefits. This parameterisation allows one to explicitly choose the masses of both active and sterile neutrinos, and is automatically consistent with oscillation data, allowing oscillation parameters to be easily treated as Gaussian nuisances.  Alternative parameterisations would constitute a different effective prior on both the sterile and active neutrino parameters. Because the GAMBIT analysis is based on profile likelihoods, which are by construction prior-independent, switching parameterisation would only have the impact of making sampling less efficient, rather than causing any physical or statistical effect.\n\n\\subsubsection{RHN results}\n\nIn Fig.\\ \\ref{fig:M_Ue_capped}, we show, as functions of the heavy neutrino masses $M_I$, the constraints on the couplings $U^2_{\\alpha I}$ to the active neutrino flavours $\\alpha = (e,\\mu,\\tau)$; in Fig.\\ \\ref{fig:M_U_capped} we also show the overall constraints on their combination $U^2_I=\\sum_\\alpha U^2_{\\alpha I}$. The index $I$ can refer to any of the heavy neutrino flavours $I=(1,2,3)$, as their labelling has no physical significance. The profile likelihood is mostly flat at low values of the couplings, but exhibits characteristic drop-offs at higher values that result from specific experimental observations. The most dominant constraint varies with the RHN mass.\n\nAbove the masses of the weak gauge bosons, direct searches at colliders are not relevant, and the leading constraints on the RHN properties come from electroweak precision observables, CKM measurements and searches for lepton flavour violation (LFV). The upper limits on the $\\tau$ couplings are much larger than on the $e$ and $\\mu$ couplings, due to the fact that the EWPO and LFV limits are stronger for the $e$ and $\\mu$ flavours.\n\nWhen $M_I$ is between the $D$ meson and $W$ boson masses, direct search experiments dominate, as RHNs are efficiently produced via the $s$-channel exchange of on-shell $W$ bosons. The DELPHI and CMS results compete to impose the strongest bound in this region.\n\nBelow the $D$ meson mass, the dominant constraints come from direct searches at beam-dump experiments, in particular CHARM and NuTeV (above the kaon mass), PS-191 and E949 (between the pion and the kaon mass), and pion decay experiments at even lower mass. In the case of the $\\tau$ couplings, the direct search constraints are much weaker, and the most significant constraint instead comes from DELPHI searches for long-lived particles.\n\nFor $M_I$ values below 0.3\\,GeV, the global constraints are stronger than the sum of the individual contributions, due to an interplay between the lower bound from BBN, the upper bounds from direct searches and the constraints on RHN mixing from neutrino oscillation data (which disfavour large hierarchies amongst the couplings to individual SM flavours). The BBN lifetime constraint does not have an observable effect on the individual couplings, but it does force their combination to be greater than a certain value (as seen in Fig.\\ \\ref{fig:M_U_capped}).\n\nFinally, we point out that although this analysis included the active neutrino oscillation likelihoods contained in \\textsf{NeutrinoBit}, based on the results of \\textsf{NuFit} \\cite{NuFit15}, it would not change the results even slightly if one were to replace these with nuisance likelihoods from either of the other main 3-flavour neutrino fitting groups \\cite{deSalas:2017kay,Capozzi:2018ubv}.  This is because the results of all three groups are highly consistent, and the preferred parameter region is where the approximate $B-L$ symmetry holds and the oscillation constraints are essentially irrelevant.  As the fits allow $m_{\\nu_0}\\to 0$, there is no lower limit implied on $M_I$ from oscillation data, but rather only from BBN (the effects of which were modelled under the massless neutrino approximation).\n\n\n\\section{Summary}\n\\label{sec:summary}\n\n\\textsf{GAMBIT}\\xspace is an open-source software framework for combining all relevant experimental constraints on theories for physics Beyond the Standard Model.  It includes extensive libraries of theory and likelihood routines for dark matter, flavour, collider, neutrino and precision observables, along with spectrum and decay calculations, all for a range of popular and highly plausible theories of new physics.  It features an array of different statistical samplers, a hierarchical model database, an automated engine for building calculations based on graph theory, and the ability to connect to external physics calculators with ease.\n\nIn the two years since its release, \\textsf{GAMBIT}\\xspace has produced seven global analyses of leading theories for physics beyond the Standard Model.  In supersymmetry, this includes analyses of the CMSSM, NUHM1, NUHM2, a 7-parameter weak-scale MSSM, and an electroweakino effective field theory known as the EWMSSSM.  Results indicate a $3.3\\sigma$ combined preference in LHC searches for weak production of light charginos and neutralinos.  In Higgs portal dark matter, \\textsf{GAMBIT}\\xspace results cover $\\mathbb{Z}_2$ and $\\mathbb{Z}_3$-symmetric scalar singlet models, as well as $\\mathbb{Z}_2$-symmetric vector and fermion models.  All can provide good fits to experimental constraints, but fermionic models strongly prefer to violate $CP$.  Scalar models can not only solve the dark matter problem, but can also stabilise the vacuum of the standard model if and only if they possess TeV-scale masses and respect a $\\mathbb{Z}_2$ symmetry.  GAMBIT studies indicate that QCD axions are most likely to constitute a fraction of dark matter rather than the entire amount, and to reside in a mass window between about $10^{-7}$ and $10^{-3}$\\,eV. Right-handed neutrinos are constrained by a wide array of different searches at different masses; interactions with electrons and muons are the most strongly constrained, with constraints on couplings to tau leptons somewhat weaker.\n\n\\textsf{GAMBIT}\\xspace is a powerful tool for testing theories of new physics.  The code can be obtained from \\url{https:\/\/gambit.hepforge.org}.  All samples, input files and benchmark points resulting from the \\textsf{GAMBIT}\\xspace physics studies discussed in this review can also be obtained from \\textsf{Zenodo}, by visiting \\url{https:\/\/zenodo.org\/communities\/gambit-official}.\n\n\\section*{Acknowledgements}\nWe thank our collaborators within the \\textsf{GAMBIT}\\xspace Community for their essential and extensive contributions to the work reviewed in this article, and Tom\\'as Gonzalo in particular for comments on right-handed neutrinos.  The majority of plots presented in the review were produced with \\pippi \\cite{pippi}.  We acknowledge PRACE for awarding us access to Marconi at CINECA, Italy.\n\n\\bibliographystyle{JHEP_pat}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\nVisual place recognition in changing environments is the problem of finding matchings between two sets of observations, database and query, despite severe appearance changes.\nIt is required for loop closure detection in SLAM (Simultaneous Localization and Mapping) and for candidate selection in image-based 6D localization systems.\nThe common pipeline for visual place recognition in changing environments is shown in Fig.~\\ref{fig:intro} (top):\nGiven the two sets of images, database (the ``map'') and query (later or current run), a descriptor is computed for each image.\nSubsequently, each descriptor of the database is compared pairwise to each descriptor of the query to get a similarity matrix $\\hat{S}^{DB\\times Q}$ that can finally be used to determine potential place matches.\nAs illustrated in Fig.~\\ref{fig:intro} (bottom), there are various additional approaches to improve the place recognition pipeline either by preprocessing the descriptors, e.g., with feature standardization, or by postprocessing the similarity matrix, e.g., with sequence search in the similarity matrix.\nIn addition to these pre- and postprocessing steps, additional information can be exploited like database image poses or the so far rarely used intra-database and intra-query similarities.\nHowever, there are only few methods that exploit such additional knowledge in order to further improve the place recognition performance.\n\nIn this paper, we address the problem of systematically exploiting additional information by proposing a versatile, expandable framework for different kinds of prior knowledge from or about database and query.\nSpecifically, we\n\\begin{itemize}\n \\item propose a versatile, expandable graph-based framework that formulates place recognition as a sparse non-linear least squares optimization problem in a factor graph (Sec.~\\ref{sec:algo})\n \\item discuss different sources of information in place recognition problems and propose a loop-rule and an exclusion-rule to exploit inherent structural properties of the place recognition problem (Sec.~\\ref{sec:binary_factor})\n \\item demonstrate how this framework can be used to integrate the different sources of information, e.g., we provide implementations of the loop- and exclusion-rules in terms of cost functions for factors. These either exploit intra-set descriptor similarities within database and\/or query, or, if available, additional knowledge about database image poses (Sec.~\\ref{sec:binary_factor})\n \\item demonstrate the versatility of the framework by proposing an n-ary factor that mimics the sequence processing approach of SeqSLAM \\cite{seqSLAM} (Sec.~\\ref{sec:nary_factor})\n \\item present the optimization using standard non-linear least squares optimization tools (Sec.~\\ref{sec:optim})\n \\item provide comprehensive experimental evaluation on a variety of datasets, configurations and existing methods from the literature (Sec.~\\ref{sec:results})\n\\end{itemize}\n\nThe paper closes with a discussion of the current implementation and potential extensions in Sec.~\\ref{sec:discussion}.\n\n\\begin{figure}[tb]\n \\centering \n \\includegraphics[width=1\\linewidth]{images\/pr_pipeline.pdf}\n \\vspace{-0.7cm}\n \\caption{The basic place recognition pipeline (above the horizontal dashed line) can be extended with additional information (below this line). Established approaches are standardization of descriptors and sequence processing. We propose a graph-based framework to integrate various sources of additional information in a common optimization formulation. An example for so far rarely used information are intra-set similarities.} \n \\vspace{-0.4cm}\n \\label{fig:intro}\n\\end{figure}\n\n\n\\section{RELATED WORK}\n\\label{sec:related_work}\nVisual place recognition in changing environments is a subject of active research.\nAn overview of existing methods is given in \\cite{Lowry2016}.\nIn the present paper, the basic source of information to match places are image descriptors.\nThe authors of \\cite{Suenderhauf2015} demonstrated that intermediate CNN-layer outputs like the \\textit{conv3}-layer from AlexNet \\cite{alexnet} trained for image classification can serve as holistic image descriptors to match places across condition changes between database and query.\nMoreover, there are CNNs especially trained for place recognition that return either holistic image descriptors like NetVLAD \\cite{netvlad} or local features like DELF \\cite{delf}.\nThe performance of holistic descriptors can be further improved by additional pre- and postprocessing steps.\nTo improve the performance of holistic descriptors, feature standardization and unsupervised learning techniques like PCA- and clustering-based methods can be used \\cite{Schubert2020}.\nIn \\cite{Neubert2019} it is shown how a neurologically-inspired neural network can be used to combine several descriptors along a sequence to a new descriptor for each place.\n\\cite{Schubert2019} extended this approach to encode additional odometry information in the new descriptors.\nGiven the final pairwise similarities between descriptors from database and query, sequence-based methods \\cite{seqSLAM}\\cite{hmm} for postprocessing can be used to improve the place recognition performance further.\n\nIn this paper, we propose a graph-based approach to optimize the descriptor similarities by incorporating prior knowledge.\nIn \\cite{hmm} Hansen and Browning used a hidden Markov model (HMM) to formulate a graph-based sequence search method in the similarity matrix.\nNaseer et al. \\cite{Naseer14} used a flow network with edges defined for sequence search to improve place recognition results.\nIn \\cite{Zhang2019} a graph is used to prevent place matches between adjacent places and to distribute high matching scores to neighboring places.\nIn contrast to these approaches, our approach exploits not only sequence information but also integrates other additional knowledge, e.g., about intra-set similarities in the database set or the query set.\nThe literature provides several approaches for localization where the database is known in advance.\nFor example, given the images from the database (or another representative training set), FabMap \\cite{fabmap} learns statistics about feature occurrences in order to determine the most descriptive features.\nMcManus et al. \\cite{McManus2014} train offline condition-invariant broad-region detectors from beforehand collected images with a variety of appearances at particular locations.\nIn \\cite{Neubert2015} intra-database descriptor comparisons are used to reduce the number of required image comparisons during the query run.\nVysotska and Stachniss \\cite{Vysotska2017} use binary intra-database place matches to perform jumps within the similarity matrix during sequence search.\nIn contrast, in our presented approach we use potentially continuous intra-database similarities to optimize the place recognition result instead of just accelerating it.\nMoreover, we do not only use this information to find loops but in addition to potentially inhibit wrong loop closures.\n\nGraphical models, and in particular factor-graphs, are a well established tool in mobile robotics \\cite{Dellaert2017}, e.g., in the form of robust pose graph SLAM \\cite{Sunderhauf2012}.\nSimilar to the here proposed approach, pose graph SLAM represents each place as a node and the edges (factors) model constraints between these places. \nHowever, a significant difference is that pose graph SLAM deals with spatial information, i.e., the places are represented by pose coordinates in the world and the factors are spatial transformations between these poses (e.g., odometry or detected loop closures). \nIn contrast, our presented approach is intended to be used \\textit{before} SLAM to establish loop closures. \nIn particular, we do not optimize metric errors between spatial constraints but errors in the mutual consistency of descriptor similarities.\n\n\n\\section{ARCHITECTURE OF THE GRAPHICAL MODEL}\\label{sec:algo}\nA graphical model serves as a structured representation of prior knowledge in terms of dependencies, rules, and available information.\nGiven the variable nodes in the graph with their initial values together with dependencies between nodes based on additional knowledge, optimization algorithms can be used to modify the variables in order to resolve inconsistencies between nodes.\nWe are going to exploit this mechanism and present a graph-based framework for visual place recognition that optimizes the initial similarity values $\\hat{S}^{DB\\times Q}$ from pairwise image descriptor comparisons.\n\nThe graph-based framework is expressed as factor graph.\nFactor graphs are graphical representations of least squares problems -- for a detailed introduction to factor graphs please refer to \\cite{Dellaert2017}.\nThe graph's basic architecture consists of nodes with unary factors that penalize deviations from the initial similarity values.\nDepending on additional knowledge, we can add different factors in the graph to introduce connections (i.e. dependencies) between nodes.\nTwo architectures with nodes, unary factors, and additional factors are shown in Fig.~\\ref{fig:graph_intra_set}. \nEach factor defines a quadratic cost function based on the variables it connects. \nThe resulting combined optimization problem is defined as a weighted sum of the individual cost functions.\nThe optimization is subject of Sec.~\\ref{sec:optim}.\nHere, we first define the basic architecture of the graph with corresponding nodes and unary factors.\nNext, we propose two ways to exploit prior knowledge with binary and n-ary factors.\nWe will structure the explanation of each factor by the \\textit{prior knowledge} that can be exploited, a corresponding \\textit{rule} that expresses the resulting dependency between nodes, a proposed related \\textit{cost function} $f(\\cdot)$ that punishes a violation of this rule, and the factor's \\textit{usage} in the graph.\nExcept for the unary factor, each factor is optional and depends on the available knowledge.\nAccordingly, the proposed framework can be extended in future work with additional factors.\n\n\\subsection{Graph nodes}\\label{sec:algo:nodes}\nThe graph-based framework is designed to optimize the initial pairwise descriptor similaritiy matrix $\\hat{S}^{DB\\times Q} \\in \\mathbb{R}^{M\\times N}$.\n$M$ is the number of database images and $N$ the number of query images.\nAccordingly, we define $M\\times N$ nodes where each node $s_{ij} \\in S^{DB\\times Q}$ is a variable version of its initial value $\\hat{s}_{ij} \\in \\hat{S}^{DB\\times Q}$ with $s_{ij},\\hat{s}_{ij} \\in [0,1]$.\n\n\\subsection{Unary factor} \\label{sec:unary_factor}\n\\subsubsection{Prior Knowledge}\nDescriptor similarities $\\hat{s}_{ij}$ are the primary source of information for place recognition.\nBeyond initializing the variables to these similarities, we have to prevent too large deviations of $s_{ij}$ from $\\hat{s}_{ij}$, in particular to prevent trivial solutions during optimization.\n\n\\subsubsection{Rule ''prior``}\n\\begin{align}\\label{eq:rule_unary}\n s_{ij} \\approx \\hat{s}_{ij}\n\\end{align}\n\n\\subsubsection{Cost function ''prior``}\n\\begin{align}\\label{eq:cost_unary}\n f_\\text{prior}(\\cdot) = (s_{ij} - \\hat{s}_{ij})^2\n\\end{align}\n\n\\subsubsection{Usage}\nEach node $s_{ij}$ is connected with a single unary factor to its initial value $\\hat{s}_{ij}$ as shown in Fig.~\\ref{fig:graph_intra_set}.\nThus, $M\\times N$ unary factors are used in a graph.\n\n\\subsection{Binary factors for the exploitation of intra-database or intra-query similarities from poses or descriptor-comparisons}\\label{sec:binary_factor}\n\n\\subsubsection{Prior Knowledge -- intra-database similarities from poses}\nIn some applications, the database is created with a more advanced sensor setup than the query.\nDue to the missing position information for the query images, direct position-based matching cannot be conducted.\nNevertheless, available poses for database images can be used to create binary intra-database similarities $\\hat{s}^{DB}_{ij} \\in \\hat{S}^{DB}$ that encode whether two database images $i$ and $j$ show the same place ($\\hat{s}^{DB}_{ij}:=1$) or different places ($\\hat{s}^{DB}_{ij}:=0$).\n\n\\subsubsection{Prior Knowledge -- intra-set similarities from image descriptors}\nEven if position data is not available, we can acquire similar information directly from image comparisons within the database and also within the query to get intra-database similarities $\\hat{s}^{DB}_{ij} \\in \\hat{S}^{DB}$ and intra-query similarities $\\hat{s}^Q_{ij} \\in \\hat{S}^Q$ with $\\hat{s}^{DB}_{ij}, \\hat{s}^Q_{ij} \\in [0,1]$.\nThese intra-set similarities are an inherent and almost always available source of additional information, which has not been used often yet.\nThe computation of intra-set similarities can usually be done more reliably than the computation of inter-set similarities, because the condition \\textit{within} database or query is potentially more constant than between both. \nThe intra-set image comparisons could be done with methods like FabMap~\\cite{fabmap} that are more suited for place recognition under constant condition.\nFig.~\\ref{fig:s_matrices} shows how the intra-set similarities $\\hat{S}^{DB}$, $\\hat{S}^Q$ are related to the inter-set similarities $\\hat{S}^{DB\\times Q}$ and how they can reveal loops and zero-velocity stages.\n\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=0.8\\linewidth]{images\/S_matrices.pdf}\n \\vspace{-0.2cm}\n \\caption{\n The information of the similarity matrix between database and query (bottom right) can be extended with similarity information of images \\textit{within} the query set (top right) and\/or \\textit{within} the database set (bottom left).}\n \\vspace{-0.2cm}\n \\label{fig:s_matrices}\n\\end{figure}\n\nWe define separate binary factors for both intra-set similarities, either from poses or from image descriptors.\nEach binary factor is a combination of two complementing rules and cost functions $f^Q_\\text{loop}(\\cdot)+f^Q_\\text{exclusion}(\\cdot)$ (Eq.~(\\ref{eq:cost_sim_q}), (\\ref{eq:cost_dissim_q})) and $f^{DB}_\\text{loop}(\\cdot)+f^{DB}_\\text{exclusion}(\\cdot)$ (Eq.~(\\ref{eq:cost_sim_db}), (\\ref{eq:cost_dissim_db})), respectively.\n\n\\subsubsection{Rule ''loop``}\nIf $\\hat{s}^{Q}_{jl}$ is high (indicated by ''$\\hat{s}^{Q}_{jl}\\uparrow$``), the $j$-th and $l$-th query image likely show the same place.\nIf the $i$-th database image is compared to these two query images, the following ternary relation can be derived:\n\\begin{align}\\label{eq:rule_sim2}\n    s_{ij} \\approx s_{il} &\\text{, iff } \\hat{s}^Q_{jl}\\uparrow\n\\end{align}\n\nThe rule expresses that ''$s_{ij}$ should be equal to $s_{il}$ iff the query images $j$, $l$ show the same place``, because if both query images $j$, $l$ show the same place, database image $i$ can either be equal to both or to none of both.\nAccordingly, this rule exploits loops within the intra-query similarities.\nThis rule is inherent to the place recognition problem and valid as well for intra-database similarities:\n\\begin{align}\\label{eq:rule_sim1}\n    s_{ij} \\approx s_{kj} &\\text{, iff } \\hat{s}^{DB}_{ik}\\uparrow\n\\end{align}\n\n\\subsubsection{Cost function ''loop``}\nFor equation (\\ref{eq:rule_sim2}) and (\\ref{eq:rule_sim1}), cost functions similar to (\\ref{eq:cost_unary}) can be formulated:\n\\begin{align}\n f^Q_\\text{loop}(\\cdot) &= \\hat{s}^Q_{jl} \\cdot (s_{ij} - s_{il})^2 \\label{eq:cost_sim_q} \\\\\n f^{DB}_\\text{loop}(\\cdot) &= \\hat{s}^{DB}_{ik} \\cdot (s_{ij} - s_{kj})^2 \\label{eq:cost_sim_db} \n\\end{align}\nThe quadratic error term is multiplied by the intra-set similarity $\\hat{s}^Q_{jl}$ or $\\hat{s}^{DB}_{ik}$ to apply weighting in case of non-binary intra-set similarities from image descriptors or to turn it on and off for binary intra-set similarities.\n\n\\subsubsection{Rule ''exclusion``}\nIf $\\hat{s}^{Q}_{jl}$ is low (indicated by ''$\\hat{s}^{Q}_{jl}\\downarrow$``), the $j$-th and $l$-th query image probably show different places.\nIf the $i$-th database image is compared to these two query images, the following ternary relation can be derived:\n\\begin{align}\\label{eq:rule_dissim2}\n    \\neg(s_{ij}\\uparrow \\land s_{il}\\uparrow) \n    &\\text{, iff } \\hat{s}^Q_{jl}\\downarrow\n\\end{align}\nThis rule expresses that ''not both similarity measurements $s_{ij}$ AND $s_{il}$ can be high iff the query images $j$, $l$ show different places``, i.e., the rule excludes one or both similarities $s_{ij}$, $s_{il}$ from being high; otherwise a single database image $i$ would show two different places concurrently.\nAgain, this rule is inherent to the place recognition problem and is supposed to add valuable information that can be exploited.\nAs before, the same rule is valid for intra-database similarities:\n\\begin{align}\\label{eq:rule_dissim1}\n    \\neg(s_{ij}\\uparrow \\land s_{kj}\\uparrow) \n    &\\text{, iff } \\hat{s}^{DB}_{ik}\\downarrow\n\\end{align}\n\n\\subsubsection{Cost function ''exclusion``}\nIt is less natural how to express the \\textit{rule ''exclusion``} in a cost function. This is part of the discussion in Sec.~\\ref{sec:discussion}.\nIn this work, we define the following cost functions for equation (\\ref{eq:rule_dissim2}) and (\\ref{eq:rule_dissim1}):\n\\begin{align}\n f^Q_\\text{exclusion}(\\cdot) &= (1-\\hat{s}^Q_{jl}) \\cdot (s_{ij} \\cdot s_{il})^2 \\label{eq:cost_dissim_q} \\\\\n f^{DB}_\\text{exclusion}(\\cdot) &= (1-\\hat{s}^{DB}_{ik}) \\cdot (s_{ij} \\cdot s_{kj})^2 \\label{eq:cost_dissim_db}\n\\end{align}\nThe quadratic error term is multiplied by the negated intra-set similarity $(1-\\hat{s}^Q_{jl})$ or $(1-\\hat{s}^{DB}_{ik})$ to weight it in case of non-binary intra-set similarities from image descriptors or to turn it on and off for binary intra-set correspondences.\n\n\\subsubsection{Usage}\nFig.~\\ref{fig:graph_intra_set} (left) shows a graphical model with nodes $s_{ij}$, unary factors $f_\\text{prior}(\\cdot)$, and the proposed binary factors.\nTo apply the binary factors for existing intra-database similarities $\\hat{S}^{DB}$, each node $s_{ij}$ has to be connected to every node $s_{kj}$ for all $k = 1,\\ldots,M$ with $k\\neq i$ within each column in $S^{DB\\times Q}$; i.e., $\\binom{M}{2}$ factors per column.\nFor existing intra-query similarities $\\hat{S}^Q$, each node $s_{ij}$ has to be connected to every node $s_{il}$ for all $l = 1,\\ldots,N$ with $l\\neq j$ within each row in $S^{DB\\times Q}$; i.e., $\\binom{N}{2}$ factors per row.\nThe potentially high number of connections is part of the discussion in Sec.~\\ref{sec:discussion}.\n\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=\\linewidth]{images\/graph.pdf}\n \\caption{Illustration of the graph structure. \\textit{(left)} Unary and binary factors. \\textit{(right)} Unary and n-nary factors, which connect structured local blocks.}\n \\label{fig:graph_intra_set}\n\\end{figure}\n\n\\subsection{N-ary factors for the exploitation of sequences}\\label{sec:nary_factor}\n\n\\subsubsection{Prior Knowledge}\nIf both database and query are recorded as spatio-temporal sequence, sequences also appear in the inter-set similarities $S^{DB\\times Q}$ (see Fig.~\\ref{fig:s_matrices}).\nIn this case, SeqSLAM \\cite{seqSLAM} showed the benefit from a simple combination of similarities of neighboring images.\nOriginally, it was implemented as postprocessing of the similarity matrix $S^{DB\\times Q}$. \nWe can integrate such sequential information within the graph in a similar fashion using n-ary factors.\n\n\\subsubsection{Rule ''sequence``}\n\\begin{align}\\label{eq:rule_seq}\n s_{ij} \\approx \\frac{1}{L}\\sum_{\\forall \\{k,l\\} \\in \\text{Seq}({i,j};L,v_p)} s_{kl}\n\\end{align}\n$\\text{Seq}({i,j};L,v_p)$ is a function that returns similarities that are part of a sequence along a line segment with slope $v_p$ and sequence length $L$ within $S^{DB\\times Q}$ around $s_{ij}$.\nFor a full explanation how SeqSLAM works, please refer to \\cite{seqSLAM}.\n\n\\subsubsection{Cost function ''sequence``}\n\n\\begin{align}\n  f_\\text{seq}(\\cdot) = (s_{ij} - \\frac{1}{L}\\max_{v_p \\in V}{(\\sum_{\\{k,l\\} \\in \\text{Seq}(\\{i,j\\};L,v_p)} s_{kl})})^2\n\\end{align}\nwith $V$ being the set of allowed velocities within $S^{DB\\times Q}$.\n\n\\subsubsection{Usage}\nFig.~\\ref{fig:graph_intra_set} (right) shows a graphical model with nodes $s_{ij}$, unary factors $f_\\text{prior}(\\cdot)$, and the proposed n-ary factors for sequences.\nAs for the unary factors in Sec.~\\ref{sec:unary_factor}, each node in the graph is equipped with one n-ary factor.\nEach n-ary factor connects its node to all nodes that are part of any sequence with length $L$ and slope $v_p\\in V$.\nAccordingly, $M\\times N$ n-ary factors are introduced into a graph if sequences are exploited.\n\n\\section{OPTIMIZATION OF THE GRAPHICAL MODEL}\\label{sec:optim}\nThe objective of the graph-based framework is the formulation of dependencies from prior knowledge for a subsequent optimization of the similarities $s_{ij}$ to incorporate the prior knowledge into the final similarity values, and to potentially resolve contradictory dependencies.\n\nWe use factor graphs as graphical representation of least squares problems and defined every cost function for each type of factor as (weighted) quadratic error function.\nAccordingly, the global error $E$ is defined as a sum over the (weighted) cost functions of all factors in the graph.\n\n\\subsection{Weighting of factor's costs for global error computation in the graph}\nAs usually done in error computation for a graph, costs from different factor-types have to be balanced by weighting them separately.\nTherefore, we normalize the cost of each factor by the number of factors per factor-type, and introduce user-specified weights $w$ for all factors except for the basic unary factor (which gets weight 1).\n\nFor the unary factor of our basic architecture with cost function $f_\\text{prior}(\\cdot)$, we define the partial global error $E_\\text{prior}$ with\n\\begin{align}\n E_\\text{prior} = \\frac{1}{MN} \\sum_{i=1}^M \\sum_{j=1}^N f_\\text{prior}(s_{ij})\n\\end{align}\n\nIn case of dependencies in the graph that are introduced by available intra-database or intra-query similarities, we define a partial global error $E^{DB}_{loop,exclusion}$ or $E^Q_{loop,exclusion}$ for the weighted summation over all binary factors (Sec.~\\ref{sec:binary_factor}): \n\\begin{align}\n E^{DB}_\\text{loop,exclusion} = &\\frac{1}{N\\binom{M}{2}} \\sum_{j=1}^N \\sum_{i=1}^{M-1} \\sum_{k=i+1}^M w^{DB}_\\text{loop} \\cdot f^{DB}_\\text{loop}(s_{ij}, s_{kj}) \\nonumber\\\\\n &+ w^{DB}_\\text{exclusion} \\cdot f^{DB}_\\text{exclusion}(s_{ij}, s_{kj})\n \\\\\n E^Q_\\text{loop,exclusion} = &\\frac{1}{M\\binom{N}{2}} \\sum_{i=1}^M \\sum_{j=1}^{N-1} \\sum_{l=j+1}^N w^Q_\\text{loop} \\cdot f^Q_\\text{loop}(s_{ij}, s_{il}) \\nonumber\\\\\n &+ w^Q_\\text{exclusion} \\cdot f^Q_\\text{exclusion}(s_{ij}, s_{il})\n\\end{align}\n\nFor the n-ary factors (Sec.~\\ref{sec:nary_factor}) in case of sequence exploitation, the partial global error $E_\\text{seq}$ is defined with\n\\begin{align}\n E_\\text{seq} = \\frac{1}{MN} \\sum_{i=1}^M \\sum_{j=1}^N w_\\text{seq} \\cdot f_\\text{seq}(s_{ij}, \\text{Seq}(\\{i,j\\};L,v^*_{ij})\n\\end{align}\nwith $v^*_{ij}$ being the optimal velocity (slope) with the highest mean of connected similarities around $s_{ij}$.\n\nFinally, summation over all partial global errors $E_i$ that occur in the graph yields the global error $E$:\n\\begin{align}\n E = \\sum_{\\forall i} E_i\n\\end{align}\n\n\n\\subsection{Implementation of the optimization}\nError $E$ is a sum solely over quadratic cost functions.\nThus, many tools for non-linear least squares (NLSQ) optimization can be used (e.g., scipy's \\textit{least\\_squares}-function in Python).\nFor an easier formulation of the optimization problem, factor graph tools like g2o \\cite{g2o} can be used.\nHowever, one should be aware that depending on the introduced factors, the optimization problem can get quite huge.\nThus, efficient implementations should be preferred that perform a quick and memory efficient optimization.\nSec.~\\ref{sec:seq} reports our achieved runtimes that are presumably sufficient for many applications.\nSec.~\\ref{sec:discussion} provides some more discussion on alternative optimization methods and approximation techniques for runtime improvements.\n\n\\section{EXPERIMENTAL RESULTS}\\label{sec:results}\nWe present experiments that investigate the performance gains achieved by the graph-based framework with three different extensions that exploit 1)~intra-database similarities; 2)~intra-database and intra-query similarities; 3)~intra-database similarities, intra-query similarities and sequences.\nIn order to evaluate the potential benefit beyond available pre- and postprocessing methods, we repeat these experiments with the descriptor standardization approach from \\cite{Schubert2020} for preprocessing and SeqSLAM for postprocessing.\nIn a final experiment, we compare our graph-based method with sequence-based approaches from the literature.\n\n\\subsection{Experimental Setup}\n\\subsubsection{Image descriptor}\nNetVLAD \\cite{netvlad} is used as CNN-image descriptor in all experiments.\nWe use the author's implementation trained on the Pitts30k dataset with VGG-16 and whitening.\n\n\\subsubsection{Metric}\nThe performance is measured with average precision which is the area under the precision-recall curve.\n\n\\subsubsection{Datasets}\nAll experiments are based on the five different datasets Nordland~\\cite{nordland}, StLucia (Various Times of the Day)~\\cite{stlucia}, CMU (Visual Localization)~\\cite{cmu}, GardensPoint Walking~\\cite{gardenspoint} and Oxford RobotCar~\\cite{robotcar} with different characteristics regarding the environment, appearance changes, in-sequence loops, stops, or viewpoint changes.\nWe use the datasets as described in our previous work~\\cite{Neubert2019}.\nImages from StLucia, CMU and Oxford were sampled with one frame per second, which preserves varying camera speeds, stops and loops in the datasets, and leads to translation and orientation changes during revisits. GardensPoint contains lateral viewpoint changes.\n\nFor the binary intra-database similarities $\\hat{S}^{DB}_{pose}$ from poses, we use the GPS from the datasets or a main diagonal for \\textit{GardensPoint Walking} and \\textit{Nordland}.\nFor the query sequence, we assume that no GPS is available.\n\n\\subsubsection{Implementation}\\label{sec:implementation}\nGraph creation and optimization were implemented in Python with scipy's \\textit{least\\_squares}-optimization function; the \\textit{Trust Region Reflective} algorithm was used for minimization.\nDue to the huge number of factors within a graph in case of the usage of intra-database and intra-query similarities, we divided $S^{DB\\times Q}$ into equally sized patches with height and width $\\leq 500$, and optimized each patch separately.\nNo information is shared between patches, and the n-ary factors are truncated at borders.\nA full optimization on $S^{DB\\times Q}$ without patches was performed if only intra-database similarities $\\hat{S}^{DB}$ were used.\nThe variables in the optimization are initialized with $\\hat{S}^{DB\\times Q}$ from the pairwise descriptor comparisons.\n\n\\subsubsection{Parameters}\nIn all experiments, we used a fixed parameter set that was determined from a toy-example and a small real-world dataset.\nWe used $w^{DB}_\\text{loop}=4, w^{DB}_\\text{exclusion}=40$ for intra-set similarities from GPS, $w^{DB}_\\text{loop}=w^Q_\\text{loop}=1, w^{DB}_\\text{exclusion}=w^Q_\\text{exclusion}=20$ for intra-set similarities from descriptors, and $w_\\text{seq}=10$, $L=11$ for sequences.\n\n\\subsection{Contributions of information sources and rules}\\label{sec:exp1}\nIn Sec.~\\ref{sec:algo} we identified four rules: \\textit{''prior``}, \\textit{''loop``}, \\textit{''exclusion``} and \\textit{''sequence``}.\nIn the following, we are going to evaluate the influence of the rules when they are successively added and exploited in the graph.\nNote that rule \\textit{''prior``} alone would merely return the initial values $\\hat{S}^{DB\\times Q}$.\nInput to the graph are the pairwise descriptor similarities from the raw NetVLAD descriptors that serve as baseline as well (termed ''pairwise``).\nAll results are summarized in Table~\\ref{tab:no_std_seq}.\n\\input{tables\/res.tex}\n\n\n\\subsubsection{Exploitation of intra-database similarities from poses or descriptors (rules \\textit{''loop``} and \\textit{''exclusion``})}\\label{sec:exp2}\nIntra-database similarities $\\hat{S}^{DB}$ can be used in most place recognition setups as they can be acquired either from the pairwise image comparisons within the database or from poses, e.g., from GPS or SLAM.\n\nTable~\\ref{tab:no_std_seq} shows the results (indicated by $\\hat{S}^{DB}_{pose}$ and $\\hat{S}^{DB}_{desc}$) when intra-database similarities are used either from poses or descriptor comparisons.\nIn most cases, the pairwise performance is significantly improved and never gets worse.\nIf intra-database similarities from poses (here: GPS) are used, the performance gain is slightly better, presumably because place matchings and distinctions from poses are binary and less error prone.\nHowever, even when the intra-database similarities from descriptor comparisons are used, the performance can be improved significantly.\n\nThe results support that most existing place recognition pipelines could be improved with the proposed graph-based framework together with intra-database similarities.\nMoreover, since it is not necessary to know the query sequence in advance, the proposed graph-based framework can be employed in an online fashion.\n\n\n\\subsubsection{Exploitation of intra-database and intra-query similarities from poses or descriptors (rules \\textit{''loop``} and \\textit{''exclusion``})}\nIn addition to the previous experiment, supplementary intra-query similarities could be used to model dependencies within the graph not only between database images but also between query images.\nWe used intra-query similarities solely from descriptor comparisons since we do not assume global pose information during the query run; otherwise, place matchings could be received directly from pose-comparisons.\n\nTable~\\ref{tab:no_std_seq} shows the results (indicated by $\\hat{S}^{DB}_{pose}{+}\\hat{S}^Q_{desc}$ and $\\hat{S}^{DB}_{desc}{+}\\hat{S}^Q_{desc}$).\nFor intra-database similarities from poses, the performance is improved only for a few sequence-combinations compared to the performance with intra-database but without intra-query similarities.\nWhen using intra-query similarities in addition to intra-database similarities from descriptors, the performance could be improved at least by additional $5\\%$ for $50\\%$ of all datasets.\n\nThe results indicate that additional data from intra-query similarities within the proposed graph-based framework can be used to improve the place recognition performance further.\nIt is important to note that the system again never performs worse in comparison to a graph with intra-database but without intra-query similarity exploitation.\nThe result is interesting, since intra-query similarities from descriptors could always be collected for a subsequent postprocessing.\n\n\\subsubsection{Additional exploitation of sequences within the graph (rules \\textit{''loop``}, \\textit{''exclusion``} and \\textit{''sequence``})}\nIn this experiment, we used sequence information within the graph in addition to the intra-database and intra-query similarities.\nTable~\\ref{tab:no_std_seq} shows the results (indicated by $\\hat{S}^{DB}_{pose}{+}\\hat{S}^Q_{desc}{+}\\text{Seq}$ and $\\hat{S}^{DB}_{desc}{+}\\hat{S}^Q_{desc}{+}\\text{Seq}$).\nWith sequence information the graph can again significantly improve the place recognition performance compared to the previous experiments without this additional assumption no matter if the intra-database similarities come from poses or descriptor comparisons.\nMoreover, the full setup of the graph with all proposed factors clearly outperforms the baseline.\n\n\\subsection{Combination with preprocessed descriptors}\\label{sec:exp2_std}\nThe place recognition performance can be improved if the descriptors are preprocessed \\cite{Schubert2020}.\nTo investigate the influence of preprocessing on all configurations of the graph from Sec.~\\ref{sec:exp1}, we repeated all previous experiments with standardized image descriptors \\cite{Schubert2020}.\nResults are shown in Table~\\ref{tab:std_seq} (left).\nSince the preprocessing requires complete knowledge of the query descriptors, we do not provide results for the online configuration of the graph.\nThe full configuration of the graph with intra-set similarities and sequences can again show significant performance improvements for the majority of the sequence-combinations.\nThe results demonstrate that the graph-based framework benefits from better performing descriptors.\n\\input{tables\/res_cat_STD_seq.tex}\n\n\\subsection{Combination with sequence-based postprocessing}\\label{sec:exp_postproc}\nIn the next experiment, a modified version of SeqSLAM~\\cite{seqSLAM} without local contrast normalization and the single matching constraint is used; otherwise SeqSLAM would fail on datasets with in-sequence loops.\nIt postprocesses the inter-set similarities $S^{DB\\times Q}$ either from the pairwise descriptor comparison (baseline) or from all configurations of the graph from Sec.~\\ref{sec:exp1}.\n\nResults are shown in Table~\\ref{tab:std_seq} (right);\nnote that descriptor comparisons from the raw NetVLAD descriptors \\textit{without} sequence-based postprocessing are used as inputs to the graphs.\nThe baseline performance after postprocessing compared to the baseline performance without postprocessing (Table~\\ref{tab:no_std_seq}) is already comparatively high.\nAccordingly, it is hard to achieve high performance improvements.\nNonetheless, the graph-based framework could improve the results in almost $50\\%$ of the cases often by more than $10\\%$ -- however, intra-database similarities from both poses and descriptors seem to be sufficient, since more information could not be used for further performance improvements on most datasets.\n\n\\subsection{Combination with pre- and postprocessing}\nWe also conducted experiments with preprocessing from Sec.~\\ref{sec:exp2_std} \\textit{and} postprocessing from Sec.~\\ref{sec:exp_postproc}.\nThe baseline performance got already almost perfect for most of the datasets, so that performances could only be improved slightly by less than $5\\%$ with the graph-based approach.\nAgain, the performance was never worse than the baseline.\n\n\\subsection{Comparison with state-of-the-art sequence-based methods}\\label{sec:seq}\nTo compare performance and runtime of our method with approaches from the literature, we conduced additional experiments with the sequence-based methods SeqSLAM~\\cite{seqSLAM}, MCN~\\cite{Neubert2019}, VPR~\\cite{Vysotska2017} and ABLE~\\cite{able}.\nThe sequence length was $L=11$, if required.\nThe experiments were performed twice without and with feature standardization~\\cite{Schubert2020} for descriptor preprocessing.\n\nTable~\\ref{tab:seq} shows the achieved performances.\nOur method clearly outperformed the compared approaches and achieved the best performance on most datasets.\nOnly on Nordland SeqSLAM and ABLE achieved better performance, since they benefit from constant camera speed in database and query.\nWith feature standardization, ABLE could additionally achieve best performance on GardensPoint, and MCN performed best on three Oxford datasets.\n\nRuntimes were measured on an Intel i7-7700K CPU with 64GB RAM.\nThe maximum runtimes per query for all methods are shown in Table~\\ref{tab:seq} (bottom).\nOur method required approx. 5.7sec per query on Oxford\\#2 with 3413 database images, while all other approaches needed less than 500msec.\nAll reported runtimes are presumably sufficient for applications like loop closure detection in SLAM, since loops need not to be detected with the full frame rate of a camera.\nSec.~\\ref{sec:discussion} provides some more discussion on the computational efficiency and potential improvements.\n\n\\input{tables\/res_seq.tex}\n\n\\section{DISCUSSION AND CONCLUSION}\\label{sec:discussion}\nThe previous sections presented our approach to use a graphical model as a flexible framework to model different kinds of additional information available in place recognition.\nThe experiments demonstrated that the presented method can considerably improve place recognition results in various configurations in terms of available data (e.g., poses of database images), the subset of applied rules (e.g., using sequences or not), or restriction to online place recognition.\nThe representation and optimization using graphical models offers a high degree of flexibility.\nIn the remainder of this last section, we want to discuss aspects of the proposed system and some particularly interesting possible extensions.\n\nThe graph-based optimization is performed on the pairwise descriptor similarities $S^{DB\\times Q}$.\nThis makes it relatively independent of the actually chosen image descriptor, which may require only a slight parameter adjustment.\nThis property even allows an optimization of place descriptor similarities from different sensor modalities like LiDAR.\n\nIn this paper, we defined several factor-types to express prior knowledge (``rules'') about place recognition problems.\nEach factor implements a cost function for a rule. \nOften, there are alternative formulations of a cost function for a rule.\nFor example, the cost function in case of no loop in query ($\\hat{s}^Q_{ij}\\downarrow$; Eq.~(\\ref{eq:cost_dissim_q})) or database ($\\hat{s}^{DB}_{ij}\\downarrow$; Eq.~(\\ref{eq:cost_dissim_db})) is defined as multiplication $(s_1\\cdot s_2)^2$.\nEspecially for this rule, alternative formulations may apply like $\\min(s_1, s_2)^2$ or $\\max(s_1+s_2-1, 0)^2$; both are piecewise linear which could be beneficial.\n\nFactor graphs are often (but not exclusively) used in combination with probabilities. Presumably, a more probabilistic view on the proposed graph-based framework could provide additional insights.\nFor instance, the chosen cost functions (\\ref{eq:cost_unary}), (\\ref{eq:cost_sim_db}) and (\\ref{eq:cost_dissim_q}) with structure $(s_1-s_2)^2$ can be considered as the negative log-likelihood of a single Gaussian:\n\\vspace{-0.02cm}\n\\begin{align}\n (s_1-s_2)^2 \\Leftrightarrow -\\ln(e^{-(s_1-s_2)^2})\n\\end{align}\nMoreover, a piecewise linear formulation of the discussed factor above could help to formulate the proposed graph-based framework in a more probabilistic way; for instance a cost function $\\min(s_1, s_2)^2$ corresponds to the negative log-likelihood of a maximum of two Gaussians:\n\\begin{align}\n \\min(s_1, s_2)^2 \\Leftrightarrow -\\ln(\\max(e^{-(s_1-0)^2}, e^{-(s_2-0)^2})\n\\end{align}\n\nThese cost functions, however, solely work on the descriptor similarities.\nIn the related work (Sec.~\\ref{sec:related_work}), we already mentioned the important difference to pose graph SLAM, which solely works on spatial poses (and not their similarities).\nAn interesting question for future work is whether the proposed approach can be extended to also directly work on descriptors instead of their similarities (i.e., the variables would be descriptor vectors, not their scalar similarities).\nThis significantly increases the complexity of the optimization problem, but could allow the simultaneous optimization of spatial poses and descriptors for a potentially tightly coupled loop closure detection and SLAM.\n\nEven without such an extension, as indicated in Sec.~\\ref{sec:algo} and Sec.~\\ref{sec:optim}, the number of factors or connections between nodes can get quite huge, and grows cubically if intra-set similarities are used.\nIn our experiments, we addressed the problem by dividing $S^{DB\\times Q}$ into patches if both intra-set similarities were used (Sec.~\\ref{sec:implementation}).\nPresumably, using more efficient implementations, e.g. C++ implementations in Ceres \\cite{ceres}, can improve memory consumption and computational efficiency.\nAnother promising direction are approximation techniques like a systematic removal of low-relevance connections in the graph.\nFinally, different optimization techniques could be used: In earlier work on graph optimization, minimization techniques based on hill-climbing algorithms like ICM (iterated conditional modes) were used \\cite[p.599]{Koller2009}; these may allow a different and more compact representation and optimization of the graph.\n\n\\input{root.bbl}\n\n\\end{document}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe ubiquity of opportunity offered by the Internet of Things (IoT) is providing new ways to embed computation and storage into everyday devices. Reductions in the cost of hardware, coupled with technological advancements, are enabling a constant stream of innovative new products and services that rely upon IoT architectures. Such developments are particularly evident in the `wearable' and healthcare industries, with parallel innovations occurring in the Industrial Internet of Things (IIoT) \\cite{bessis2013,hill2017a}.\n\nAlong with the ability to sense environments, conditions and processes, comes the requirement to measure, monitor, analyse and evaluate performance. As such, Business Intelligence (BI) is undergoing a resurgence as consumer demand moves beyond traditional dashboards through forecasting, towards prescriptive analytics. Of late, vendors have delivered BI often as a flexible, on-demand service, making use of the underlying elastic cloud platforms that enterprise software applications are typically deployed upon \\cite{alaqrabi2012,alaqrabi2013,alaqrabi2015,alaqrabi2018}.\n\nAdvancements in technology have increased the ability to connect such a variety of embedded devices to larger pools of resources such as clouds. Integrating embedded devices and cloud servers raises an important discussion regarding the nature of data generated or transmitted by IoT devices. These approaches must be secure and provide the necessary privacy controls for users. At present, the security and privacy concerns created by these devices play a central role in the successful integration of these two technologies \\cite{kalra2015}.\n\nThe heterogeneous nature of IoT environments makes it much harder to detect the insider and outsider attacks in such universal platforms \\cite{Alrawais2017}.\n\nExperiences with clouds, especially those that have public or hybrid architectures, illustrates that cloud services and applications require multi-layered approaches to external and internal threats to security. This is most pertinent in the IIoT environment, where business systems contain valuable Intellectual Property (IP) that can only be protected by retaining tight security over working practices.\n\nThis article describes a model for delivering analytics services to and between components such as those encountered in IoT architectures. A number of adversarial attacks are simulated to demonstrate the effectiveness of a cloud-inspired multi-layer security model in the IoT domain.\n\\section{Cloud Computing Model}\nThe core principles of cloud computing: on-demand self-service, broad network access, resource pooling, rapid elasticity and service metering (NIST Special Publication 800-145) \\cite{hill2013}, make utility computing an attractive proposition for environments that contain distributed resources.\nThe NIST definition \\cite{nist2011} describes five options for the deployment of cloud computing as follows: public clouds, private clouds hosted onsite, private clouds hosted off-site, onsite community clouds and offsite community clouds.\n\nAbstraction is one of the most compelling motivations for system architects; this enables myriad heterogeneous resources to be viewed as a homogeneous whole. Cloud architectures of any of the five types defined by NIST are modelled using a framework that consists of seven layers. The physical infrastructure foundation is layer one, upon which a resource abstraction (virtualisation) second layer resides. Layer three is the resource composition layer, and layer four refers to the Infrastructure as a Service model (IaaS). Platform as a Service (PaaS) exists on layer five, Software as a Service (SaaS) on layer seven and finally the cloud tenants' applications sit within layer seven \\cite{demchenko,hill2013}.\n\nWe have considered the application of analytics services in the context of an enterprise Business Intelligence application that is likely to be delivered across a heterogeneous network of devices. In a cloud environment this is typified by off-site private or community clouds, whereby multiple disparate business organisations can host their own enterprise software services and data repositories in what is, to all intents and purposes, a set of hardware resources that is dedicated to each tenant \\cite{demchenko2012,hill2013}.\nThe tenants access their dedicated resources via a VPN, thus maintaining clear separation between corporate services and their respective analytics implementations. However, should any of the businesses wish to share services to promote operational efficiencies, this can be enabled via community-based agreements \\cite{demchenko2012}.\n\nWe have used the cloud model as inspiration to propose a multi-layer service-oriented framework that can deliver secure services (such as BI analytics) across distributed infrastructure. Security and privacy are key concerns for tenants of shared cloud infrastructure such as that found in off-site private and community clouds. The sharing of services such as malware detection and inoculation has significant benefits for organisations who may not otherwise have strategies or the resources to maintain their own defences. We shall now briefly review services dedicated to security and privacy within clouds in the following section.\n\n\n\\section{Secure Service Delivery}\nIn keeping with the service orientation models described so far, security as a service can be considered as a multi-layer framework in itself \\cite{panian2008}, protecting each layer of the cloud computing model \\cite{carvakho2011}. Such a service lends itself to a utility offering in the same way that clouds are rapidly provisioned and expanded on demand, for a cost that is quantifiable and chargeable to the consumer.\n\nAs described above, resource abstraction through virtualisation is a key principle for a cloud-inspired architecture, and therefore any security and privacy services need to be made available to all relevant Virtual Machines (VM) for each client \\cite{kumar2011,luo2011}. \n\nTherefore, security and privacy control services must be orchestrated via appropriate service interfaces between each cloud and its respective tenants that are accessing the services \\cite{kumar2011}. This control will be managed by the relevant virtualisation security manager for each VM, and as a consequence the control must be located within the virtualisation layer.\n\nPrivacy is managed via policy resources that need to be retained securely in a protected space such as a digital vault for encryption keys or digital certificates and such like \\cite{pearson2009,diaz2013}, all contained within a tier above other layers such as authentication and client metadata layers \\cite{pervez2012,chadwick2012}.\n\nAny instances whereby session packets do not have the requisite authority to invoke a particular service interaction will result in the session being terminated, and is a key security feature that is embedded within the framework.\n\n\\section{Multi-layered Security Model}\nOur proposed model is illustrated in Figure \\ref{fig:multi} and describes the cloud-inspired architecture containing multiple layers. Each layer incorporates security and privacy services that make use of firewalls as gateways for session traffic from each of the respective clients.\n\nWe have implemented the model within Opnet, the description of which is as follows.\nEach firewall is represented by a Cisco PIX 535, which operates across the various layers including network, transport and application. Access control lists within the firewalls enable the governance of traffic based on IP addresses, protocols and  ports for common and bespoke applications.\n\nThe firewalls are configured at the application layer to filter traffic from different URLs, encrypted sessions (HTTPS) and various clients such as Java, and are able to utilise Internetwork Key Exchange (IKE) to encrypt all sessions via DES, 3DES or AES.\n\nAs an example we have defined four separate LANs to represent different clients\/cloud tenants, and they each access the cloud through separate firewalls.\n\nTo illustrate an adversarial scenario, we have incorporated a simulated distributed attack from three malicious parties, who each are attempting to access the network through independent firewalls. Such an attack is a key concern for early adopters of IoT technologies, as the pervasive use of wireless communications presents many opportunities for business vulnerabilities to be exposed.\n\n\\begin{figure}[tb]\n \\includegraphics[width=\\linewidth]{.\/figures\/Multilayer_hierarchical_Inter_Cloud_architecture.JPG}\n \\caption{Multilayer hierarchical inter-cloud architecture.}\n \\label{fig:multi}\n\\end{figure}\nAll of the multiple layers of the cloud are illustrated in Figure \\ref{fig:cloudlay}. We have embraced the cloud concept of resource abstraction and exploited this in the multi-layer model by representing each of the layers as separate clouds. Within each cloud layer exists an array of computing hardware to maximise performance and therefore minimise response times to service requests. It is an imperative that the model must not introduce excessive overheads into the normal functionality of the system.\n\nThe cloud layers are arranged such that inbound traffic from clients are filtered by the firewalls and then passed through successive layers until the system is satisfied that the requests can be delivered to the analytics functionality residing on the cloud applications layer. We now describe each of the layers and the role that they play within the model in turn:\n\n\\begin{itemize}\n\\item {\\bf Tenant firewalls}. This cloud layer consists of a number of databases that hold authentication data for each of the tenants, in order for them to be permitted access to the VM that have been assigned to them as part of their subscription. This is the gateway for each session, $S$ to be invoked.\n\n\\item {\\bf Tenant metadata}. Beyond the authentication criteria that is marshalled by the tenant firewalls layer, there exists further metadata for each client tenant. This metadata describes the credentials to authorise access to specific instances of repositories, applications and services. The detail is embedded within the session packets and is used to verify whether the session can continue or not.\n\\item {\\bf Digital vaults}. The vaults are a secure place to retain digital signatures\/certificates and decryption keys so that only the requisite authorities can access their own content and services. Public key encryption ensures that legitimate session requests are honoured and bogus requests are terminated.\n\\item {\\bf Intrusion Prevention System}. The occurrence of adversarial attacks is not limited to the correct authentication at the start of a session. Intrusion detection prevents attacks on sessions that are in progress, such as SQL injection in web forms. This might be considered a route into the DB\\textsubscript{META} or DB\\textsubscript{VAULT} repositories from the perspective of an adversary. This cloud layer prevents such activity from continuing.\n\n\\item {\\bf Malware protection}. Similar to the IPS cloud layer, an anti-malware layer contiuously montitors for trojan activity, where malicious exploits are embedded and concealed within session packets, only to be executed at the application layer. Records of the monitoring and detection history are retained within DB\\textsubscript{ANTIMAL}.\n\\item {\\bf Tenant applications}. This layer hosts the tenant's applications themselves, in this case the analytics functionality of enterprise BI. The preceding layers ensure that only a marshalled session, $S$ can access this layer, which in the case of BI potentially provides access to confidential business operations and performance data. For some functionality, there will be a requirement for further authentication from a user, such as an account and password, etc.\n\n\\end{itemize}\n\\subsection{Tenant repositories}\nThe final cloud layer hosts the back-end repositories that serve myriad tenant applications. For an analytics application, this would be the databases\/data warehouses\/data lakes that retain the underlying business data, together with any processed data objects for reporting and analysis. These objects are typically accessed by tenant users through reporting dashboards and data visualisation suites, abstracting users away from the complexities of database organisation.\n\\begin{figure}[h]\n \\includegraphics[width=\\linewidth]{.\/figures\/Cloud_layers_modeled_in_this_research.jpg}\n \\caption{Cloud layers modelled in this work.}\n \\label{fig:cloudlay}\n\\end{figure}\n\\subsection{Simulation design}\nWe have elected to model and simulate the proposed system (Figure \\ref{fig:cloudlay}) so that we can examine the operational characteristics in terms of performance, and its ability to provide a collection of services that are resilient towards malicious attacks.\n\n\n\nFigure \\ref{fig:virmac} illustrates how the cloud layers have been mapped to individual profiles. Each profile consists of a collection of VMs, that host the contents of the model as described in the previous section, namely: security and privacy services, tenant applications and repositories.\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=0.85\\linewidth]{.\/figures\/Virtual_Machines_with_applications_packaged.jpg}\n\\caption{Virtual Machines with applications packaged.}\n \\label{fig:virmac}\n\\end{figure}\n\\section{Modelling adversarial attacks}\nIf we now consider an adversarial attack upon the model, we can see how the system protects against such a scenario.\nFigure \\ref{fig:attack} shows the situation where a malicious agent attempts to infiltrate the system to obtain access to an authorised tenant's VM. Whilst the cloud service requires verification to be able to subscribe and access the remote resources, what appears to be a legitimate tenant could actually be an adversary that is masquerading as a valid client, who has the objective of entering the cloud and then attacking other tenant VMs from within the same cloud.\n\nTools such as Metasploit can be employed to automate the delivery of exploits in a rapid fashion. This could enable a bogus tenant to create surreptitious means of exposing sensitive data, unbeknown to any other party.\n\n\\begin{figure}[tb]\n\\centering\n \\includegraphics[width=0.9\\linewidth]{.\/figures\/An_attack_scenario_showing_the_problem.jpg}\n \\caption{An attack scenario showing the problem.}\n \\label{fig:attack}\n\\end{figure}\n\nPosing as a legitimate tenant, the adversarial agent would in this case be attacking from a VM that is authorised and hosted within the cloud. As such, the conventional cloud security processes would not be able to detect such activity.\nIn effect, the activity is obscured by the sheer volume of VMs that exist within a cloud environment.\nThis is a significant challenge for cloud service providers, particularly as service orientation through Microservice Architectures becomes more prevalent \\cite{hill2017}. If we extend this to the IoT domain, there is a stronger desire to package functionality into services, to be hosted on distributed, connected hardware. Therefore, the ability to successfully address this issue is a key feature of this work.\nWe propose a solution whereby the collection of VMs are organised into a hierarchy, as Figure \\ref{fig:ahier} shows.\nIf an adversarial agent enters the cloud via a subscription, and is assigned VM2, it has the potential to employ cross-channel attacks against VM1 and VM3, thereby exploiting the presence of virtual links between differing VMs. Typically, cloud security controls are deployed to prevent external attacks rather than insider attacking.\n\\begin{figure}[tb]\n \\includegraphics[width=\\linewidth]{.\/figures\/A_hierarchical_framework_presented_as_a_solution_to_the_attack_scenario_in_Figure_4.jpg}\n \\caption{A hierarchical framework presented as a solution to the attack scenario in Figure \\ref{fig:attack}.}\n \\label{fig:ahier}\n\\end{figure}\nThis architecture limits the opportunity to commit further exploits since the attacker is prevented from moving to the next control using virtual links. The only option is to proceed using a real network link by requesting a session in Control A, to communicate with VMs 4,5 and 6. Since there now exists Control A, the attacker has to successfully satisfy {\\tt Tenant Metadata Inspection} in order to proceed with the penetration.\nOf course, an orchestrated and sustained attempt to commit an attack will mean that we should anticipate an adversarial agent will also have obtained valid credentials, either by posing as a legitimate tenant or otherwise.\nIn this case, Control B will have required that the malicious agent would need to navigate the entire stack of cloud layers before access could be gained to the analytics interface.\n\nIf the attacker has satisfied the cloud validation and metadata inspection layers, the only way forward now is to plant exploits in the hope that these will lie undetected. However, the Intrusion Protection layer, and the Anti-Malware layer both offer protection for subversive, covert attacks from the inside.\nThe result is that our proposed model prevents data breaches, even when adversarial attacks are launched from what appears to be genuine service subscribers.\nWe can see in Figure \\ref{fig:control} a sequence in which various security controls might be instantiated. The security policies of the host system (or systems) will inform the order in which controls are implemented, to suit the goals desired by the infrastructure provider. It is also evident how a tenant's session is routed through the various VMs in order to access the relevant analytics services.\n\\begin{figure}[tb\n \\includegraphics[width=\\linewidth]{.\/figures\/The_controls_positioned_on_each_layer_of_the_hierarchy.jpg}\n \\caption{The position of the controls at each layer of the hierarchy.}\n \\label{fig:control}\n\\end{figure}\n\\section{Session packet inspection}\nIn this section we shall describe a detailed walkthrough of the security model and explain the use of session packet inspection.\n\nReferring back to Figure \\ref{fig:ahier}, VM1 will host a client that ultimately will access the enterprise analytics application hosted in VM7. A malicious agent that has access to either VM2 or VM3 (or both) can only attack the client of the analytics application, rather than the application itself.\nVM2 will be used to test the validity of the VM1 client using a VM identification number, and assuming that all is well, will launch a form via VM3 to request details from the tenant.\nThe details requested from the tenant will vary each time that VM3 is executed, but they will always refer to some personal details that can be used to help identify the correct tenant.\nOnce the form on VM3 has been completed, VM2 will use the responses to verify the details against those held in the {\\tt MetaDB} repository. Once the metadata has been verified, the session can continue. If the malicious tenant has the intention of acquiring metadata about other tenants, it would need to deliver an exploit into {\\tt MetaDB}. Whilst it might be expected that the system is fully patched, it is still conceivable that a vulnerability exists, enabling an adversary to progress to the next cloud layer.\nSince a session cannot be interrupted, any evidence left by malicious exploits will still remain as the agent will not be able to remove the incriminating evidence of the exploit. As such, the attacker can only proceed to the next layer by exposing that an exploit has been used to obtain entry. If the malicious agent has obtained private keys, they still cannot progress without exposing details of the exploit within the session.\nLayers 5 and 6 both have the capability to detect malware, which of course is reliant upon adequate, proactive security maintenance to ensure that all exploit databases are current.\n\nIt is feasible that an attacker could compromise VMs 3 and 4 with a fresh exploit that has yet to be discovered and documented, in which case the anti-malware cloud layer will be unaware of this exploit as well. Whilst this situation may foil malware detection, there still exists Intrusion Prevention within the cloud layer, which by its nature monitors and reports upon anomalies.\n\nOur model enables system architects the ability to add or subtract security controls as required for a given set of policies, merely by specifying additional cloud layers for the hierarchy. When a session is authenticated as satisfying the requirements of each layer, it can progress to subsequent layers until the destination application layer is reached.\nOne advantage of the use of multiple VMs is that much of the computation can be performed in parallel, and therefore the majority of packet inspection incurs a minimal overhead in VMs 2,3 and 4. However, session packet inspection across cloud layers, especially anything that involves Intrusion Prevention or anti-malware detection will result in an additional overhead. \n\\subsection{Mapping controls to the seven layer model}\nWith reference back to the NIST seven layer model \\cite{nist2011}, the controls of our proposed model can be mapped to the PaaS layer as per Figure \\ref{fig:mapp}, with the exception of the firewalls which naturally reside in the IaaS layer \\cite{hill2013,carvakho2011}.\nTenant VMs are hosted in layers 1-3 of the NIST model; users interact directly with clients that do not store data.\n\nSessions commence within tenant VMs, before moving to the application in layer 7 via several layers of verification and authentication in layers 4 and 5. If the tenant has a SaaS platform, this will be made available through layer 6, otherwise all bespoke applications are resident in layer 7.\n\\subsection{Session flow}\nThe proposed architecture is located on layers 4 (IaaS) and 5 (PaaS) of the NIST seven-layer model. All firewalls are categorised as IaaS due to the functionality of verifying VM instance IDs, and also relating these checks to the authorisation data provided by a tenant to secure access to the cloud.\nWhilst VM IDs are assigned in layers 2 and 3, access controls are assigned at layer 4. Subsequent controls are assigned outside of the VM layer, and are primarily concerned with session packet inspection, for a given session $S$.\nInformation is requested directly from the tenant to satisfy the DB\\textsubscript{META} and DB\\textsubscript{VAULT} checkpoint controls, and is supplemented by DB\\textsubscript{IPS} and DB\\textsubscript{ANTIMAL} controls that perform the session packet inspection function. DB\\textsubscript{IPS} and DB\\textsubscript{META} are therefore categorised as PaaS controls.\nIt is likely that controls will also exist at the application layer. For instance, a SaaS instance will require user authentication, as will a custom enterprise application\\cite{ouf2011}. User role profiles are useful in such scenarios to manage different levels of system access within an enterprise application, to reflect the role, responsibilities and authority of a particular stakeholder.\n\n\n\n\\begin{figure*}[htb]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{.\/figures\/mapping.jpg}\n \\caption{Mapping of the proposed architecture with the NIST seven layer cloud model\\cite{nist2011}.}\n \\label{fig:mapp}\n\\end{figure*}\n\\section{Algorithm design}\nAlgorithm 1 describes the sequence of inspections that are performed within the proposed multi-layer security model. This algorithm represents the core functionality and can be augmented as additional cloud layers are defined in response to the security needs of an organisation. In the case of a single enterprise the model may tend towards fewer augmentations.\nHowever, enterprises that collaborate, or who choose to adopt shared services across distributed platforms, will no doubt adopt additional layers in order to securely manage service access. \n\\begin{algorithm}\n\\SetAlgoLined\nInput: $S$, $P_{CT}$, ($DB_{FW}$, $DB_{META}$, $DB_{VAULT}$, $DB_{IPS}$, $DB_{ANTIMAL}$), 1=permit, 0=deny\\\\\nTenant session: $S$\\\\\nContents of session packets:$P_{CT}$\\\\\nContents of FW: $DB_{FW}$\\\\\nContents of $TENANT_{META}: DB_{META}$\\\\\nContents of $TENANT_{VAULT}: DB_{VAULT}$\\\\\nContents of $IPS: DB_{IPS}$\\\\\nContents of $ANTIMALWARE: DB_{ANTIMAL}$\\\\\nFlags: 1=permit, 0=deny\\\\\n \n Initialise $S$\\;\n Set $S=1$, Match$(P_{CT})$;\\\\\n \\ForEach{\\{$DB_{FW}$, $DB_{META}$, $DB_{VAULT}$, $DB_{IPS}$, $DB_{ANTIMAL}$\\}} {%\n  \\eIf{$P_{CT}\\in$ \\{$DB_{FW}$,$DB_{META}$, $DB_{VAULT}$\\} AND $P_{CT}\\notin$ \\{$DB_{IPS}$,$DB_{ANTIMAL}$\\}}{\n  set $S=1$; AuthoriseTenantAccess()\\; \/\/tenant access authorised\\;}{\n  set $S=0$; DenyTenantAccess()\\; \/\/tenant access denied\\;\n  }\n}\n Output: $S$\n \\caption{Multi-layer hierarchical packet inspection}\n\\end{algorithm}\n\\subsection{Security model logic}\nOur proposed model seeks to address security concerns in distributed service applications across heterogeneous hardware resources by enabling session packet inspection to take place at a number of checkpoints. Each session packet is scrutinised and compared with a number of repositories such as DBMETA, DBVAULT, etc. Each inspection stage shall now be described in turn.\n\\begin{enumerate}\n\\item {\\bf DB\\textsubscript{FW}:} For each instance, a client initiates a session, which is inspected as a second stage. This session will possess a VM ID together with authentication and authorisation verification data.\nIn addition, the DB\\textsubscript{FW} must have an entry that relates to the VM ID, otherwise the session is terminated.\n\\item {\\bf DB\\textsubscript{META}:} A further inspection is then performed to confirm tenant metadata as requested by the cloud host administrators.\nOnce the metadata has been provided, this will be added to session packets. The session can only continue if the session metadata matches that which exists in the DB\\textsubscript{META} repository.\n\\item {\\bf DB\\textsubscript{VAULT}:} At this stage the VM ID within the session will be inspected to verify that a private key exists within the DB\\textsubscript{VAULT} so that encrypted databases can be accessed.\n\\item {\\bf DB\\textsubscript{IPS}:} On reaching this stage, the session itself is now regarded as being fully authenticated and is authorised to progress to subsequent layers. The next stage is to inspect the sessions for evidence of potential exploits that match those held in the DB\\textsubscript{IPS} repository. If a match occurs, the session is terminated.\n\\item {\\bf DB\\textsubscript{ANTIMAL}:} This stage performs an anti-malware check, which in conjunction with the DB\\textsubscript{IPS} inspection, prevents fully authenticated VM sessions being able to penetrate the upper layers of the security model by masquerading as legitimate cloud tenants. \n\\end{enumerate}\nThe algorithm ensures that after a session is initiated, it is inspected at each layer, where each layer is represented by a separate cloud. A key principle is that session packet data must match the firewall data, tenant metadata and vault data before a session can be authorised. Furthermore, the session cannot be granted access to application layers until it has been successfully screened against IPS and anti-malware repositories.\n\\subsection{Implementing the model}\nFor our example scenario, the LAN contains 500 clients. Each of the clients is assigned three VMs, with an assigned destination being the tenant client's metadata repositories rather than the eventual analytics application. This was a conscious decision to prohibit any tenant sessions from attempting to subvert the security and privacy controls of the cloud.\nFor a physical network, this control would most likely be enacted by a firewall in an application layer, or as part of a setting in a virtual network controller. For consistency we have replicated this by ensuring that the destinations of the metadata servers are directed towards the tenant vaults; this is a faithful representation of the intentions of the algorithm, in that it is governing the marshalling of each session packet by enforcing checkpoints in a particular sequence. In the case of a session packet not containing a tenant key that matches a corresponding entry in DB\\textsubscript{VAULT}, the encryption key is not assigned and the packets are dropped \\cite{diaz2013}. We have represented the adversarial agents as clients who have the authorisation of a valid tenant, in order to simulate an attack from within.\n\\section{Results}\nFigure \\ref{fig:dbsess} shows the sessions that have been hosted by the tenant LAN in the simulation. We observed that sessions only existed within the tenant LAN for tenants that had matching metadata in DB\\textsubscript{META}. It follows that the sessions that were invoked within the tenant LAN, had consistency between tenant metadata (DB\\textsubscript{META}), tenant vault (DB\\textsubscript{VAULT}), and the associated VM ID from the initial authentication.\nThe simulation demonstrated that these validations were maintained throughout the experiment for all network hops, illustrating that VMs were complying with their specified destination configurations, and there was no evidence of VMs by-passing authentication layers. As such, all security and privacy control rules are enforced by the model.\n\n\\begin{figure}[tb\n\\centering\n  \\includegraphics[width=0.9\n  \\linewidth]{.\/figures\/Client_DB_sessions_on_Tenants__LAN.jpg}\n\\caption{Client DB sessions on Tenants' LAN.}\n  \\label{fig:dbsess}\n\\end{figure}\nVMs initiated by malicious agents have separate profiles from those of legitimate tenants, even though a malicious agent may appear legitimate at the outset. This means that adversaries had distinct metadata and decryption keys in their vault repositories.\nWe can see in Figure \\ref{fig:ipdropped} the instances where a malicious agent's packets have been dropped as a result of packet inspection through either the IPS or anti-malware layers, preventing deeper penetration into the system. As discussed earlier, an internal attack (an agent with legitimate DB\\textsubscript{META} data) would need to compromise DB\\textsubscript{VAULT}, DB\\textsubscript{IPS} and DB\\textsubscript{ANTIMAL} in sequence if it is to successfully reach the analytics application layer.\n\\begin{figure}[tb\n \\includegraphics[width=\\linewidth]{.\/figures\/IP_packets_from_the_hackers__machines_are_dropped.jpg}\n \\caption{After initial attempts to penetrate the system, IP packets from the attackers' machines are dropped.}\n \\label{fig:ipdropped}\n\\end{figure}\nConversely, an authorised tenant may elect to execute DB sessions on their own LAN, since the application profiles can include references to their VMs, as per Figure \\ref{fig:auth}. In such cases, the sessions are authorised in the sense that they fulfil the relevant rule in Algorithm 1. Since we have chosen to model each of the layers as clouds, each of the multiple layers can in fact be serviced by different cloud providers. This architecture thus demonstrates significant flexibility and is attractive to system architects who are proactively designing systems that will rely upon heterogeneous hardware and distributed resources, such as the IoT and IIoT environments.\nAn enterprise may adopt this security model so that it can take the opportunity to employ software applications and services that are themselves hosted on offsite private or community clouds. However, the use of these services may increase operational costs, although this is off-set by a reduction in capital expenditure. Operational costs may also increase indirectly through the maintenance charges associated with managing the database updates of five security control layers.\n\\begin{figure}[tb\n \\includegraphics[width=\\linewidth]{.\/figures\/Authorized_tenant_LANs_established_and_ran_DB_sessions_with_the_TENANT_METADATA}\n \\caption{Authorised tenant LANs established, DB sessions initiated with the TENANT METADATA.}\n \\label{fig:auth}\n\\end{figure}\n\\section {Conclusions\nThis article proposes a multi-layer hierarchical inter-cloud security model, that is inspired by the NIST seven-layer model of cloud computing. By using sequential session packet inspection techniques, we have demonstrated an architecture that exhibits considerable resilience towards both external attack as well as more surreptitious internal adversarial behaviour. Whilst VM vulnerabilities are well documented in multi-tenant shared environments, our five layers for packet inspection enables the architecture to identify and compartmentalise malicious activity. Thus, penetration of a firewall is in itself insufficient as a means of attempting to access the application layer, as it is then necessary to create an evidence trail of exploits that cannot be hidden from the IPS and anti-malware packet inspection layers. The proposed model is particularly suited to architectures that have a requirement to remain flexible for future scaling (which is often a driver for the adoption of cloud infrastructure), such as those built upon microservices. \nOur solution is mapped to the NIST model in order to assist cloud and IoT system architects to incorporate this work into their own designs.\n                                 \n                                 \n                                 \n                                 \n                                 \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCompact groups of galaxies are associations of three to seven galaxies, where the projected distances between them is of the order of their diameters, and where the group\nshows a low velocity dispersion, making compact groups an ideal place to study\ngalaxy interaction and intergalactic star formation (e.g. \\cite[Torres-Flores et\nal. 2010]{tor09b}, \\cite[de Mello et al. 2008]{dem08b}, \\cite[de Mello, Torres--Flores \\& Mendes de Oliveira 2008]{dem08a}, \\cite[Mendes de Oliveira et al. 2004]{mdeo04}). The main goal of this work is to search for a link between the evolutionary stage of a group and the presence of young intergalactic objects which may have formed during galaxy interactions. For this, we analyze a subsample of seven compact groups (HCG 2, 7, 22, 23, 92, 100 and NGC 92) which span a wide range of evolutionary stages, from HI rich groups to strongly interacting groups, where the galaxies show tidal tail features and a deficiency in neutral HI gas. In order to analyze the evolutionary stage of each group, we used new Fabry-Perot velocity maps, GALEX\/UV data and optical R-band images. The velocity fields and rotation curves help constraining the evolutionary stage of each compact group while ultraviolet light contains important information regarding the age of the young stellar population that may be present in the intragroup medium.\n\n\\section{UV analysis}\n\nWe searched for ultraviolet emitting regions in the vicinity of all seven targets, using the SExtractor software (SE, \\cite[Bertin \\& Arnouts 1996]{ber96}) in the FUV, NUV and R sky-subtracted images of our compact group sample. We compare the field density of regions detected in the compact group with a control sample outside the group. HCG 92 and HCG 22 have the highest field density in this study. No excess was found in HCG 2, HCG 7, HCG 23, HCG 100 and NGC 92  (\\cite[Torres-Flores et al. 2009]{tor09}).\n\n\\section{Fabry-Perot analysis}\n\nIn order to constrain the evolutionary stage of each compact group, we\ninspected the velocity field and rotation curve of each galaxy to search for\ninteraction indicators, in a similar way to that done by \\cite[Plana et al. (2003)]{pla03} and \\cite[Amram et al. (2003)]{amr03}. In the case of NGC 92, it shows a prominent tidal tail in its velocity field. At the tip of this tail, there is a tidal dwarf galaxy candidate, having an age of about 40 Myrs (\\cite[Torres-Flores et al. 2009]{tor09}).\n\n\\section{Conclusions}\n\nWe used multiwavelength data to study the evolutionary stages of the compact groups of\ngalaxies HCG 2, 7, 22, 23, 92, 100 and NGC 92. New Fabry-Perot velocity fields, rotation curves and GALEX NUV\/FUV images were analyzed for four and seven of these groups respectively. Groups HCG 7 and 23 are in an early stage of interaction whereas\nHCG 2 and 22 show limited interaction features and HCG 92, 100 and NGC 92 are in a\nlate stage of evolution, having HI gas in the intragroup medium, galaxies with peculiar velocity fields and several young star-forming regions in the intergalactic medium.\n\n\\acknowledgments\n\nS. T--F. acknowledges the financial support of FAPESP through the Doctoral position, under contract 2007\/07973-3 and Eiffel scholarship.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nChemical signalling that exploits molecule or ion for communication has been widely found in nature with diverse propagation distances. In a relatively long range, animals or insects utilize pheromone to communicate with the members of their species for mate selection, identity recognition, alarm inform, etc. \\cite{Shorey.Pheromone.13,Akyildiz13}. At micro-scale environment, hormones or other chemical substances are prevalently transmitted or received in tiny organisms such as cells. This process is the so-called cell signalling that is crucial to cells' survival~\\cite{Han.Signalling.17}.\n\nMolecular communication (MC) is an emerging technique inspired from the aforementioned communication schemes in vivo, whose history dates back to 2005 \\cite{Tadashi.MC.05}. MC is available at both macro-scale and micro-scale~\\cite{Farsad.MCSurvey.16}. At micro-scale, MC is suitable for connection among nanomachines, whose communication distance ranges from a nanometer to a hundred nanometers. Nanomachine is one of the most remarkable progress of nanotechnology that has the potential to revolutionize many aspects of technology, and ultimately benefit our life. However, a single nanomachine can only perform very simple tasks due to its size constraint, whereas complex applications including biopsy, targeted drug delivery, environmental sensing, cell sorting, etc., require coordination among a swarm of nanomachines~\\cite{Wang.Nanomachine.13}. MC is regarded as a prominent candidate for nanonetworking because of its bio-compatible and energy-efficient characteristics. It has been acknowledged as the most important communication paradigm in IEEE 1906.1 standard~\\cite{Tadashi.10year.17}.\n\nModulation plays a significant role in MC as it determines the system's achievable performance~\\cite{Suzuki.MCmodelling.17}. \nThere are a few of modulation schemes for MC proposed in the literature, including the concentration shift keying (CSK)~\\cite{Kuran.ModuMC.11}, molecular shift keying (MoSK)~\\cite{Kuran.ModuMC.11}, isomer-based ratio shift Keying (IRSK)~\\cite{Kim.Isomodu.13} and pulse position modulation (PPM)~\\cite{Llatser.PPM.13}, in which messages are encoded as the concentration, type, ratio and release time of transmitted molecules, respectively. Advanced modulation schemes have also been considered in the context of the single-input single-output~(SISO) based MC (SISO-MC)~\\cite{Arjmandi.MCSK.13,Kabir.DMoSK.15,Mosayebi.typesignmodu.18}. However, SISO-MC is not always suitable for the scenarios of high-speed transmission and some of the applications where reliability is indispensable. To solve these problems, some well-known techniques in the conventional wireless communications, e.g., multiple-input multiple-output (MIMO), have been redesigned for MC \\cite{KC.frontierWC.12}. Specifically, MIMO based MC (MIMO-MC) is a recent trend in MC research, dated back to 2012 when it was first proposed in \\cite{KC.MIMOMC.12}. It is shown that in MIMO-MC, spatial diversity enhances the bit error rate (BER) performance, while spatial multiplexing may increase transmission rate significantly~\\cite{KC.MIMOMC.12}. In 2013, a micro-scale MIMO-MC system was introduced~\\cite{Tadashi.MCbook.13}, where a group of sender nanomachines simultaneously transmit messages to a group of receiver nanomachines through the medium where they reside. In the same year, the first SISO-MC prototype at macro-scale was implemented \\cite{Farsad.Prototype.13}, which mentioned that MIMO principle may be introduced to improve the transmission rate. In 2016, a $2 \\times 2$ MIMO-MC prototype achieving spatial multiplexing was implemented, which achieves a 1.7 times higher data rate than its SISO counterpart~\\cite{Chae.MCMIMOproto.16}. These research and practice demonstrate that spatial multiplexing in MIMO-MC is feasible for rate increase, although the data rate is not doubled due to the existence of interference and overhead. In~2017, MIMO-MC technique gained more attention than ever before, due to the appearance of the training-based channel estimation~\\cite{Rou.CEMIMOMC.17}, spatial diversity coding techniques~\\cite{Damrath.Spatialcoding.17} and the introduction of machine learning based channel modeling methods~\\cite{Lee.MLMIMOMC.17}. Moreover, in \\cite{WG.MCSIMOsyn.18}, synchronization was investigating in the context of the single-input multiple-output (SIMO) based MC (SIMO-MC). Now, we have no doubt that MIMO-MC constitutes a promising technique for performance improvement in MC. However, a typical challenge in MIMO-MC is the inter-link interference (ILI) in addition to the inter-symbol interference (ISI) existing in all MC systems. Due to the ILI, first, it increases the detection complexity at the receiver side. Second, the study in \\cite{Chae.MCMIMOproto.16}~manifests that the BER of $2 \\times 2$ MIMO-MC prototype is 2.2 times higher than that of its SISO-MC counterpart, meaning that high-reliability MC is challenging. Third, the energy consumption in MIMO-MC is significant in comparison to SISO-MC, due to more devices being simultaneously activated at both transmitter and receiver sides. This problem can be serious in micro-scale MC, since the power supply of nanomachine is limited and their computing capability is low. Consequently, the development of energy-efficient and low-complexity transmission schemes for MIMO-MC is demanding.\n\nIn this paper, we propose a spatial modulation based MC (SM-MC) for MIMO-MC implementation, which combines a space-dependent modulation with a concentration-relied modulation. By allowing only the space modulation, we also propose the special case of SM-MC, namely, the space shift keying based MC (SSK-MC). It can be shown that both SM-MC and SSK-MC are able to combat the aforementioned problems of the MIMO-MC. It is well known that SM~\\cite{Mesleh.SM.08} and SSK~\\cite{Jeganathan.SSK.09} in radio-based wireless communications exhibit low complexity and high energy efficiency~\\cite{Wen.IMsurvey.17}. By exploiting the spatial domain for message encoding via activating an active transmit antenna in each time slot, SM and SSK are capable of eliminating the inter-antenna interference (IAI). Note that, the major distinction between SM and SSK is that SM uses both spatial and signal constellation to transmit information, while SSK only exploits the spatial constellation.\n\nIn our design for SM-MC and SSK-MC, the spatial domain is reflected by the concentration difference at the receiver side, when different transmitters emit molecules. By contrast, the signal constellation domain is contributed by the different constellation level in CSK. Hence, the SM-MC\/SSK-MC receivers can collaborate to identify the activated transmitter nanomachine for detecting the space symbol, while the SM-MC receiver can decode the concentration symbol according to the rule of CSK. Furthermore, in order to enhance the reliability of detection, detector assisted by either \\emph{equal gain combining} (EGC) or \\emph{selection combining} (SC) is developed for detecting the concentration symbol. In this paper, we derive the SER expressions for both the SM-MC and SSK-MC, and conduct computer simulations to examine their performance as well as to validate our theoretical analysis. Our studies and results show that both the SM-MC and SSK-MC outperform the MIMO-MC in terms of the SER performance, and our analytical results are accurate enough for predicting the achieving SER performance of the SM-MC and SSK-MC systems.\n \nThe remainder of this paper is organized as follows. Section~II reviews the system model of MIMO-MC over diffusion channels. In Section~III, we present the principles of SM-MC and SSK-MC based on the architecture of MIMO-MC. In this section, we also derive the resulting SER of the SM-MC and SSK-MC. Section~IV evaluates the SER performance of SM-MC and SSK-MC by invoking the MIMO-MC with on-off keying (OOK) modulation and the SISO-MC with quadruple CSK (QCSK) modulation as the benchmarks. Finally, the research is concluded in Section~V.\n\n\\textit{Notation}: Boldface uppercase and lowercase letters indicate matrices and vectors, respectively. $\\mathbb{R}^{n\\times m}$ indicates a real number matrix with $n\\times m$ dimensions. $\\mathbb{E}[\\cdot]$, $\\left \\lVert \\cdot \\right \\rVert$, $|\\cdot|$, $Q(\\cdot)$ and $\\text{Pr}[\\cdot]$ represent expectation, Euclidean norm, absolute value, Q-function and probability of an event, respectively.\n\n\\section{Review of MIMO-MC}\nWe consider a MIMO-MC system in a 3-D unbounded environment with point sources and spherical receivers, which are assumed to be memoryless as \\cite{Yilmaz.14,Zamiri.CDMAMC.16,Tepekule.Memoryless.15}, meaning that they have no information of the previously detected symbols. Perfect synchronization is also assumed in our MIMO-MC. In this section, we propose the channel and communication models for MIMO-MC, which constitutes the fundamentals of our subsequent SM-MC scheme and its special case of the SSK-MC scheme.\n\\subsection{Channel Model of MIMO-MC}\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics[width=3.25in]{Fig1MIMOMC.eps}\n\t\\caption{System model of MIMO-MC.}\n\t\\label{fig_1}\n\\end{figure}\n\nWe consider an $N \\times N$ diffusion based MIMO-MC system operated at micro-scale in this paper \\cite{Tadashi.MCbook.13}, where $N$ transmit and receive nanomachines are attached to the cell membrane at both transmitter and receiver sides, respectively. Furthermore, we assume that there are a transmission control center and a reception decision center, respectively, located at the centers of the transmitter and receiver cells. The function of transmission control center is to coordinate the transmitter nanomachines to emit molecular pulses according to the information to be transmitted, while the reception decision center connecting all the receiver nanomachines decodes the information based on the received signals from $N$ receiver nanomachines. To clearly demonstrate the communication system model of the MIMO-MC, we exemplify a $4 \\times 4$ MIMO-MC system with $4$ pairs of transceivers as shown in Fig. 1, where the lipid bilayer of cell membrane are shown in green color. As shown in Fig. 1, the pair of transmitter and receiver of a link and the molecules transmitted between them are marked using the same color for the sake of easy distinction. Under the assumption that perfect alignment is achieved in the MIMO-MC architecture, the spacing between adjacent transmitters or adjacent receivers is equally set as $r$, and the distance between a pair transmitter and receiver is expressed as $d$. Thus, the distance from the $j$-th transmitter to the $i$-th receiver is given by\n\\begin{align}\nd_{ji} =\n\\begin{cases}\n\\hspace{1cm}d,  & \\text{for $i=j$}, \\\\\n\\sqrt{d^{2}+|j-i|^{2}r^{2}}, & \\text{for $i\\neq j$.}\n\\end{cases}\n\\end{align}\n\nIn this paper, we assume $M$-ary CSK modulation and denote the concentration of molecules in the $i$-th receiver at time $t$ in response to the $j$-th transmitter as $c_{m,ij}(t)$ when a pulse of $S_{m}$ molecules with $m\\in\\{0,1,\\ldots, M-1\\}$ is emitted at $t=0$. Consequently, the concentration at receiver can be formulated according to Fick's second law of diffusion as \\cite{Dk.receiverMC.13}\n\\begin{align}\nc_{m,ij}(t)=S_{m}\\frac{1}{{(4\\pi Dt)}^\\frac{3}{2}}\\exp \\left(-\\frac{d_{ji}^2}{4Dt}\\right),\n\\begin{array}{c}\n\\hspace{-0.3cm}j\\in\\{1,2,\\ldots, N\\},\\\\\n\\hspace{-0.3cm}i\\in\\{1,2,\\ldots, N\\},\\\\\n\\hspace{-0.4cm}m\\in\\{0,1,\\ldots, M-1\\},\\\\\n\\end{array}\n\\end{align}\nwhere $D$ is the diffusion coefficient of the propagation medium that is assumed to be homogeneous in this paper. We assume that the spherical receiver nanomachine has a volume of $V_{\\sss \\text{RX}}=\\frac{4}{3}\\pi{\\rho}^{3}$ with $\\rho$ being the radius of the receiver. Molecular concentration is assumed to be uniform inside a passive receiver when $d\\gg\\rho$. Based on these assumptions, the expected number of molecules inside the $i$-th receiver at time $t$ can be formulated by \\cite{MM.studyofCEMC.14,ANoel.Passivereceiver.16}\n\\begin{align}\nN_{m,ij}(t)=V_{\\sss\\rm{RX}}c_{m,ij}(t)=V_{\\sss\\rm{RX}}S_{m}h_{ij}(t),&\\text{\\quad\\quad for $t>0,$}\n\\end{align}\nwhere $h_{ij}(t)$ indicates the probability that an information molecule released at $t=0$ is sensed by the passive receiver at time~$t$ \\cite{GeC.adadetec.18,GD.AmpliDetect.17}, which is given by\n\\begin{align}\nh_{ij}(t)=\\frac{1}{{(4\\pi Dt)}^\\frac{3}{2}}\\exp\\left(-\\frac{d_{ji}^2}{4Dt}\\right).\n\\end{align}\nIn MC, $h_{ij}(t)$ represents the channel impulse response (CIR) between the $j$-th transmitter and the $i$-th receiver. Therefore, the concentration vector $\\mathbf{c}_{m,j}(t)$$\\in \\mathbb{R}^{N \\times 1}$ collecting the expected concentration of all receivers in response to the $j$-th transmitter can be written as\n\\begin{align}\n\\mathbf{c}_{m,j}(t)&\\triangleq\\big[c_{m,1j}(t), \\ldots, c_{m,ij}(t), \\ldots, c_{m,Nj}(t)\\big]^{T}\\nonumber\\\\\n&=S_{m}\\big[h_{1j}(t), \\ldots, h_{ij}(t), \\ldots, h_{Nj}(t)\\big]^{T}\\nonumber\\\\\n&=S_{m}\\mathbf{h}_{j}(t),\n\\end{align}\nwhere $\\mathbf{h}_{j}(t)$ is the CIR vector from the $j$-th transmitter to all the $N$ receivers at time $t$. Let us define $\\mathbf{H}(t)$$\\in \\mathbb{R}^{N \\times N}$ as the channel matrix of an $N \\times N$ MIMO-MC system at time $t$, which can be represented as\n\\begin{align}\n\\mathbf{H}(t)&=\\Big[\\mathbf{h}_{1}(t); \\ldots ;\\mathbf{h}_{j}(t); \\ldots ; \\mathbf{h}_{N}(t)\\Big],\n\\end{align}\nwhose entries are given by (4). Based on our assumptions, we can know that the diagonal elements of $\\mathbf{H}(t)$, such as $h_{jj}(t)$, have the same value, given by (4) associated with setting $i=j$ and $d_{jj}=d$. By contrast, a non-diagonal element $h_{ij}(t)$ gives the probability of the ILI from the transmitter $j$ to the receiver~$i$.\n\nIn this paper, we consider the amplitude detection \\cite{Llatser.PPM.13}. To achieve this, we assume that all the receivers sense the concentration at a certain time, e.g., at the time when the concentration at a receiver generated by its paired transmitter reaches the peak value, which can be obtained by solving the partial derivative equation $\\frac{\\partial{c_{m,jj}(t)}}{\\partial{t}}=0$. Specifically, when an impulse of molecules is emitted by a transmitter at $t=0$, the peak concentration presenting at its paired receiver can be found to be at the time of\n\\begin{align}\nt_{p}=\\frac{d^{2}}{6D}.\n\\end{align}\nExplicitly, the peak time is irrelevant to $S_{m}$. In this paper, without further explanation, it is assumed that each receiver of the MIMO-MC samples for concentration after a time interval $t_{p}$ seconds from the emission of the chemical impulse by its paired transmitter. Therefore, upon substituting (1) and (7) into (2), we can derive the maximum concentration of a receiver in response to its paired transmitter as\n\\begin{align}\nc_{m,jj}(t_{p})=S_{m}\\bigg(\\frac{3}{2\\pi ed^{2}}\\bigg)^{\\frac{3}{2}},\n\\end{align}\nwhen an impulse of $S_{m}$ molecules is released. Similarly, $c_{m,ij}(t_{p})$ can be derived, given by\n\\begin{align}\nc_{m,ij}(t_{p})=S_{m}\\bigg(\\frac{3}{2\\pi d^{2}}\\bigg)^{\\frac{3}{2}}\\exp \\left(-\\frac{3d_{ji}^2}{2d^2}\\right).\n\\end{align}\n\n\\subsection{Communication Model of MIMO-MC}\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=3.25in]{Fig2ILIISIinMIMO_MC.eps}\n\t\\caption{Concentration expected at one receiver of a $2\\times2$ MIMO-MC affected by its ISI and the ILI from one other link, when $\\text{SNR}=10\\ \\rm{dB}$, $D=2.2\\times10^{-9}$$m^{2}\/s$, $d=20um$, $r=15um$ and $T_{s}=0.2s$.}\n\t\\label{fig_2}\n\\end{figure}\n\nBinary CSK (BCSK) is the simplest CSK with $M=2$, which emits a chemical pulse containing $S_{1}$ molecules towards its paired receiver for transmitting bit ``1'', or a pulse of $S_{0}$ molecules for transmitting bit ``0''. As a special case of BCSK, the OOK modulation, which is prevalently adopted in the existing MIMO-MC \\cite{KC.MIMOMC.12,Chae.MCMIMOproto.16,Lee.MLMIMOMC.17,Damrath.Spatialcoding.17}, keeps silent without any molecule emission for bit ``0'', i.e., $S_{0}=0$. For the general CSK, it has been revealed in literature that the CSK with $M\\geq4$ is usually unable to attain satisfactory error performance in SISO-MC \\cite{Suzuki.MCmodelling.17,Kabir.DMoSK.15}. Note that in Fig. 1, we considered OOK as the modulation scheme for MIMO-MC, where the red, yellow, and purple colored links represent transmitting $S_{1}$ molecules to their corresponding receivers, while the blue colored one keeps silent without any emission of molecules. Furthermore, in Fig. 1, there are still a few of blue-colored molecules, which are the residual molecules of the previous transmissions. \n\nLet us denote the transmit signal vector at the sampling time $t$ as\n\\begin{align}\n\\mathbf{x}(t)=\\Big[x_{1}(t),\\ldots,x_{i}(t),\\ldots, x_{N}(t)\\Big]^{T},\n\\end{align}\nwhere $x_{i}(t)$$\\in \\{S_{0},S_{1}\\}$ denotes the number of molecules emitted by the $i$-th transmitter. Then, based on (6) and (9), the concentration vector corrupted by noise sensed at time $t$ can be expressed as\n\\begin{align}\n\\mathbf{y}_{\\sss \\text{MIMO}}(t)&=\\Big[{y}_{1,\\sss \\text{ MIMO}}(t),\\ldots,{y}_{i,\\sss \\text{ MIMO}}(t),\\ldots, {y}_{N,\\sss \\text{ MIMO}}(t)\\Big]^{T}\\nonumber\\\\\n&=\\sum_{l=0}^{L}\\mathbf{H}(t+lT_{s})\\mathbf{x}(t-lT_{s})+{\\mathbf{n}_{\\sss \\text{MIMO}}}(t),\n\\end{align}\nwhere $y_{i,\\sss \\text{ MIMO}}(t)$ represents the concentration sensed by the $i$-th receiver at time $t$, $T_{s}$ is the symbol duration. We assume that the ISI and ILI last for $L$ and $L+1$ symbol durations, respectively, where ISI is generated by the paired transmitter, while the ILI is the interference resulted from the other links. Practically, there is also noise in MC, which is given by the form of the noise vector ${\\mathbf{n}_{\\sss \\text{MIMO}}}(t)$$\\in \\mathbb{R}^{N \\times 1}$, having\n\\begin{align}\n\\mathbf{n}_{\\sss \\text{MIMO}}(t)=[n_{1,\\sss \\text{ MIMO}}(t),\\ldots,n_{i,\\sss \\text{ MIMO}}(t),\\ldots, n_{N,\\sss \\text{ MIMO}}(t)]^{T},\n\\end{align}\nwhere $n_{i,\\sss \\text{ MIMO}}(t)$ is a signal dependent noise sensed by the $i$-th receiver. Similar to \\cite{Damrath.Spatialcoding.17}, $y_{i,\\sss \\text{ MIMO}}(t)$ is the concentration sensed by the $i$-th receiver at time $t$:\n\\begin{align}\ny_{i,\\sss \\text{ MIMO}}(t)= \\sum_{j=1}^{N}\\sum_{l=0}^{L}{h}_{ij}(t+lT_{s}){x}_{j}(t-lT_{s})+n_{i,\\sss \\text{ MIMO}}(t).\n\\end{align}\n\nThe impact of ISI and ILI on a desired signal is illustrated in Fig. 2 for a $(2\\times2)$ MIMO-MC system, where the ILI is not only from the previous molecular symbols, but also from the current one released by the unpaired transmitter. However, it is worth noting that the current ILI plays a significant role in confusing the detection of a desired molecular signal, since the current ILI, as shown in Fig. 2, generates the strongest interference. The existing studies on SISO-MC often ignore the effect of ISI or only take one previous symbol into account \\cite{AN.Enzyme.14}, due to the fact that this previous symbol generates the highest ISI. However, when MIMO-MC is considered, as shown in Fig.~2, the effect of the current ILI may significantly surpass that of the ISI. In other words, the interference in MIMO-MC system may be dominated by the current ILI generated by the other unpaired transmitters, and the ISI generated by one previous symbol transmitted by the paired transmitter. Therefore, if we express (13) as\n\\begin{align}\ny_{i,\\sss \\text{ MIMO}}(t)=\n\\underbrace{{h}_{ii}(t){x}_{i}(t)}_{\\text{desired signal}}+\\underbrace{I_{i,\\sss \\text{ MIMO}}(t)}_{\\text{sum of interference}}+\\underbrace{n_{i,\\sss \\text{ MIMO}}(t)}_{\\text{noise}},\n\\end{align}\nthe noise component can be assumed to follow the Gaussian distribution, depending on the current symbol at time $t$ as \\cite{MengLS.receiverdesign.14}:\n\\begin{align}\nn_{i,\\sss \\text{ MIMO}}(t)\\sim \\mathcal{N}\\big(\\mu_{ni,\\sss \\text{ MIMO}}(t),\\sigma^{2}_{ni,\\sss \\text{ SM}}(t)\\big),\n\\end{align}\nwith\n\\begin{align}\n\\mu_{ni,\\sss \\text{ MIMO}}(t)=0,\\quad \\sigma^{2}_{ni,\\sss \\text{ SM}}(t)=\\frac{{h}_{ii}(t){x}_{i}(t)+I_{i,\\sss \\text{ MIMO}}(t)}{V_{\\sss \\text{RX}}}.\\nonumber\n\\end{align}\n\nThe interference component in (14) can be approximated as\n\\begin{align}\nI_{i,\\sss \\text{ MIMO}}(t)\\approx\\underbrace{\\sum\\nolimits_{j\\neq i}{h}_{ij}(t){x}_{j}(t)}_{\\text{current ILI}}+\\underbrace{{h}_{ii}(t+T_{s}){x}_{i}(t-T_{s})}_{\\text{last ISI}},\n\\end{align}\nwhen considering that the ILI is only from the current unpaired transmitter, while the ISI is only from one previous symbol sent by the paired transmitter.\n\\section{Spatial modulation based molecular communication}\n\nFrom the above description, we can know that MIMO-MC experiences both ILI and ISI, which may significantly degrade the detection performance. In order to combat these problems, in this section, we propose the spatial modulation based MC (SM-MC) as one of the implementation of the MIMO-MC. In our proposed scheme, both spatial and concentration domains are exploited for conveying information simultaneously, but only one type of molecules is used for transmission.\n\\subsection{Transmitter of SM-MC}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=3.2in]{Fig3DiagramSM_MC.eps}\n\t\\caption{Transmitter diagram of SM-MC.}\n\t\\label{fig_3}\n\\end{figure}\n\nThe ideology of SM-MC is inspired by the SSK technique having been widely studied in MIMO communications. Typically, in SSK modulation, only one transmit antenna is activated during each symbol period. At the receiver, the index of the activated transmit antenna can be detected, when the channel state information is available. The SSK modulation can be implemented in conjunction with a conventional amplitude-phase modulation, forming the SM \\cite{LLY.PSM.11}. Similarly, the SM-MC proposed in this paper is the combination of a SSK modulation and a CSK modulation. Specifically, given a symbol transmitted, one of the transmitters releases a pulse of molecules, with the number of molecules determined also by the data symbol being transmitted. In detail, the transmit schematic diagram for our SM-MC system is depicted in Fig.~3. Hence, when the $j$-th space symbol is transmitted, with $j\\in\\{1,2,\\ldots, N\\}$, the transmit signal vector has a form of\n\\begin{align}\n\\mathbf{x}(t)=\\Big[0,0,\\ldots,\\underbrace{S_{m}}_{\\text{j-th transmitter}},\\ldots, 0,0\\Big]^{T},\n\\end{align}\nwhere only the $j$-th transmitter is activated to emit $S_{m}>0$ molecules, when the $m$-th concentration symbol $S_{m}$ is selected. The space symbol is denoted as $S^{j}$, when the $j$-th transmitter is activated. \nWe assume that the space symbols $S^{j}$ and the concentration symbols $S_{m}$ are independent of each other, solely depended on the input data stream. Then, we have\n\\begin{align}\n\\text{Pr}[S_{m}]=\\frac{1}{M},\\quad m\\in\\{0,1,\\ldots, M-1\\},\n\\end{align}\nand\n\\begin{align}\n\\text{Pr}[S^{j}]=\\frac{1}{N},\\quad j\\in\\{1,2,\\ldots, N\\}.\n\\end{align}\n\nFurthermore, if we express the SM symbols representing the combinations of the space and concentration symbols as $S^{j}_{m}$, we have\n\\begin{align}\n\\text{Pr}[S^{j}_{m}]=\\frac{1}{MN},\n\\begin{array}{c}\nj\\in\\{1,2,\\ldots, N\\},\\\\\nm\\in\\{0,1,\\ldots, M-1\\}.\\\\\n\\end{array}\n\\end{align}\n\nTherefore, the data rate of the SM-MC measured in bits per molecular symbol is given as\n\\begin{align}\nR_{\\sss \\text{SM}}=\\log_{2}N+\\log_{2}M,\n\\end{align}\nwhere both the values of $N$ and $M$ are assumed to be an integer power of $2$.\n\nBased on (17), the concentration signal observed at the $i$-th receiver in SM-MC systems is similar to (14), expressed as\n\\begin{align}\ny_{i,\\sss \\text{ SM}}(t)=S_{m}{h}_{ij}(t)+{I}_{i, \\sss \\text{ SM}}(t)+n_{i,\\sss \\text{ SM}}(t),\n\\end{align}\nwhere both $S_{m}{h}_{ij}(t)$ and $n_{i,\\sss \\text{ SM}}(t)$ are dependent on the current molecular symbol being received, representing the expected number of molecules received and the noise component, respectively, at time $t$, when the $j$-th transmitter is activated to emit a chemical impulse with $S_{m}$ molecules. However, unlike the case in MIMO-MC \\cite{Chae.MCMIMOproto.16}, the ${I}_{i, \\sss \\text{ SM}}(t)$ in (22) consists of only the ISI component that is resulted from a previous molecular symbol emitted by the $i$-th transmitter, since we assume that ILI only occurs with the current transmission, while ISI is only experienced from one previous emission by a paired transmitter. Let us denote the previous molecular symbol as $S^{\\bar{j}}_{\\bar{m}}$, then we have\n\\begin{align}\n{I}_{i, \\sss \\text{ SM}}(t)=\n\\begin{cases}\n\\ 0, & \\text{for $i\\neq j$}, \\\\\n\\ S_{\\bar{m}}{h}_{ii}(t+T_{s}),& \\text{for $i= \\bar{j}$.}\n\\end{cases}\n\\end{align}\n\nSimilar to (15), the distribution of the noise component in (22) can be expressed as\n\\begin{align}\nn_{i,\\sss \\text{ SM}}(t)\\sim \\mathcal{N} \\Big(\\mu_{ni,\\sss \\text{ SM}}(t),\\sigma^{2}_{ni,\\sss \\text{ SM}}(t)\\Big),\n\\end{align}\nassociated with\n\\begin{align}\n\\mu_{ni,\\sss \\text{ SM}}(t)=0,\\quad \\sigma^{2}_{ni,\\sss \\text{ SM}}(t)=\\frac{S_{m}{h}_{ij}(t)+{I}_{i, \\sss \\text{ SM}}(t)}{V_{\\sss \\text{RX}}}.\\nonumber\n\\end{align}\n\nFurthermore, based on (22), (23) and (24), the distribution of $y_{i,\\sss \\text{ SM}}(t)$ can be found, and expressed as\n\\begin{align}\ny_{i,\\sss \\text{ SM}}(t)\\hspace{-0.1cm}\\sim\\hspace{-0.05cm}\\mathcal{N}\\Big(\\hspace{-0.1cm}S_{m}{h}_{ij}(t)+{I}_{i, \\sss \\text{ SM}}(t),\\sigma^{2}_{ni,\\sss \\text{ SM}}(t)\\hspace{-0.05cm}\\Big).\n\\end{align}\n\n\\begin{table}[t]\n\t\\caption{Bits per pulse for different modulation schemes in MC}  \n\t\\begin{tabular*}{9cm}{lclc}  \n\t\t\\toprule \n\t\tScheme & Bits per pulse&Scheme & Bits per pulse   \\\\  \n\t\t\\toprule  \n\t\tBMoSK& 1&$N\\times N$ MIMO (OOK)  & 2\\\\\n\t\tBCSK & 1&$2\\times2$ SM (BCSK)& 2 \\\\\n\t\tBSSK & 1&$8$SSK & 3\\\\\n\t\tOOK  & 2&$2\\times2$ SM (QCSK)& 3\\\\\n\t\tQCSK & 2&$4\\times4$ SM (BCSK)& 3\\\\\n\t\tQSSK & 2&$16$SSK & 4\\\\ \n\t\t\\bottomrule  \n\t\\end{tabular*}  \n\\end{table}  \nFrom the above analysis, we can readily realize that our proposed SM-MC transmission scheme is capable of mitigating the ILI experienced by the general MIMO-MC \\cite{Chae.MCMIMOproto.16}. Furthermore, the SM-MC scheme is more energy-efficient than the MIMO-MC. In MIMO-MC, when fixing a modulation scheme, increasing $N$ may transmit more bits per symbol but at the expense of higher energy consumption. By contrast, SM-MC and its special scheme of SSK-MC can obtain a logarithmic increase in bits per pulse transmission, which is contributed by the space modulation, with the increase of $N$, but without demanding any extra transmission energy. In order to highlight the energy efficiency achieved by the SM-MC and SSK-MC schemes, we compare the different modulation techniques of MC in Table~I. Note that in Table~I, we assume that the input data are independently uniformly distributed binary data. Hence, for example, when the OOK is used, transmitting $M$ bits requires about $M\/2$ pulses. Therefore, we have the average bits per pulse equaling 2. As another example, when the $N\\times N$ MIMO-MC using OOK is considered, it also has the average bits per pulse equaling 2.\n\n\\subsection{Signal Detection in SM-MC Systems}\nIn theory, the maximum-likelihood (ML) detection can be implemented by the SM-MC to jointly decode the index of the activated transmitter and the concentration symbol. When the memoryless receivers are taken into account, the ML detection based on (22) can be formulated as\n\\begin{align}\n<\\hat{j},\\hat{m}>=\\mathop{\\arg\\min}_{j\\in\\{1,2,\\dots,N\\},m\\in\\{0,1,\\dots,M-1\\}}\\left \\lVert\\mathbf{y}_{\\sss \\text{SM}}(t)-S_{m}\\mathbf{h}_{j}(t)\\right \\rVert^{2},\n\\end{align}\nwhere $\\hat{j}$ and $\\hat{m}$ are the estimated indices of the space and concentration symbols, respectively. Note that $S_{m}$ in SM-MC is always greater than zero, i.e., $S_{m}>0$, otherwise the space symbol is unable to be detected. However, (26) imposes a search complexity of $\\mathcal{O}(NM)$, when $NM$ is relatively large, hence it is not practical to deploy in MC systems, due to the constraint of the size and computing power of MC receivers. To this end, we suggest a low-complexity successive detection based on a scheme that has been commonly used in SM for wireless communications \\cite{JJ.optdetectSM.08,RZ.GPSM.13}. In this scheme, the CSK symbol is detected after the detection of the space symbol. To be more specific, we detect the index of the activated transmitter via the concentration comparison of the $N$ receivers based on the fact that the receiver paired with the activated transmitter is most likely to have the maximum concentration at the sampling time, as it is located with the minimum distance from its paired transmitter. Accordingly, the detection of the space symbol can be formulated as\n\\begin{align}\n\\hat{j}= \\mathop{\\arg\\max}_{j\\in\\{1,2,\\dots,N\\}}\\;{y_{j,\\sss \\text{ SM}}(t)}.\n\\end{align}\nAfter the detection of the space symbol, the $\\hat{j}$-th receiver's concentration can be used to detect the concentration symbol, which can be described as\n\\begin{align}\n\\hat{m}=\\mathop{\\arg\\min}_{m\\in\\{0,1,\\dots,M-1\\}}|{y}_{\\hat{j},\\sss \\text{ SM}}(t)-S_{m}h_{{\\hat{j}\\hat{j}}}(t)|^{2},\n\\end{align}\n\nFrom (27) and (28), we can readily know that the complexity of our proposed detector is $\\mathcal{O}(N+M)$. Provided that $N\\geqslant2$, $M\\geqslant2$, the complexity of the proposed detection scheme is lower than that of the ML detection of (26). It is worth noting that, once there is a transmitter activated, all the $N$ receivers can collect the molecules released. Hence, all the receivers can collaborate to make more reliable detection of the CSK symbol at the second stage. We refer to this detector as the EGC, described as\n\\begin{align}\n\\hat{m}_{\\sss \\text{EGC}}&=\\mathop{\\arg\\min}_{m\\in\\{0,1,\\dots,M-1\\}}\\left \\lVert\\mathbf{y}_{\\sss \\text{SM}}(t)-S_{m}\\mathbf{h}_{\\hat{j}}(t)\\right \\lVert^{2}.\n\\end{align}\nIt can be shown that the complexity of (29) combined with (27) is still $\\mathcal{O}(N+M)$, although the total number of computation of (29) is significantly higher than that of (28). However, the performance of the EGC-assisted detection can significantly outperform that of the detector of (28), which will be validated by the simulation results in Section IV.\n\\subsection{Error Performance Analysis of SM-MC}\nIn this subsection, we focus on the SER analysis of the successive detection, given by (27) and (29). Since the detection is dependent on the current molecular symbol $S^{j}_{m}$ and the previous one $S^{\\bar{j}}_{\\bar{m}}$, let us denote the correct detection probability of both space and concentration symbols as $P_{c,\\sss \\text{ SSK}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}})$ and $P_{c,\\sss \\text{ CSK}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}})$, respectively. Remembering that the $S^{j}_{m}$ and the $S^{\\bar{j}}_{\\bar{m}}$ are mutually independent, we can express the correct detection probability of space symbol as\n\\begin{align}\nP_{c,\\sss \\text{ SSK}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}})=\\prod_{i\\neq j}Q\\left(-\\frac{\\mu_{ji|S^{\\bar{j}}_{\\bar{m}}}}{\\sigma_{ji|S^{\\bar{j}}_{\\bar{m}}}}\\right),\n\\end{align}\nwhere $Q(\\cdot)$ is the Q-function. When given $i\\neq j$, $\\mu_{ji|S^{\\bar{j}}_{\\bar{m}}}$ and $\\sigma^{2}_{ji|S^{\\bar{j}}_{\\bar{m}}}$ are shown as\n\\begin{align}\n\\mu_{ji|S^{\\bar{j}}_{\\bar{m}}}\\hspace{-0.15cm}=\\hspace{-0.15cm}\n\\begin{cases}\n\\vspace{0.15cm} \n\\\n\\hspace{-0.225cm}S_{m}\\hspace{-0.075cm}\\big(h_{jj}\\hspace{-0.05cm}(t)\\hspace{-0.1cm}-\\hspace{-0.1cm}h_{ij}\\hspace{-0.05cm}(t)\\hspace{-0.06cm}\\big)\\hspace{-0.1cm}-\\hspace{-0.1cm}S_{\\bar{m}}h_{ii}\\hspace{-0.05cm}(t\\hspace{-0.1cm}+\\hspace{-0.1cm}T_{s}\\hspace{-0.05cm}),\\hspace{-0.3cm}&\\text{for $i= \\bar{j}$ and $j\\neq \\bar{j}$,}\\\\\n\\\n\\vspace{0.15cm}  \\hspace{-0.225cm}S_{m}\\hspace{-0.075cm}\\big(h_{jj}\\hspace{-0.05cm}(t)\\hspace{-0.1cm}-\\hspace{-0.1cm}h_{ij}\\hspace{-0.05cm}(t)\\hspace{-0.06cm}\\big)\\hspace{-0.1cm}+\\hspace{-0.1cm}S_{\\bar{m}}h_{jj}\\hspace{-0.05cm}(t\\hspace{-0.1cm}+\\hspace{-0.1cm}T_{s}\\hspace{-0.05cm}),\\hspace{-0.3cm}&\\text{for $i\\neq \\bar{j}$ and $j= \\bar{j}$,}\\\\\n\\ \\hspace{-0.225cm}S_{m}\\hspace{-0.075cm}\\big(h_{jj}\\hspace{-0.05cm}(t)\\hspace{-0.1cm}-\\hspace{-0.1cm}h_{ij}\\hspace{-0.05cm}(t)\\hspace{-0.06cm}\\big), \\hspace{-0.3cm}&\\text{for $i\\neq \\bar{j}$ and $j\\neq \\bar{j}$.}\\nonumber\n\\end{cases}\n\\end{align}\n\\begin{align}\n\\sigma^{2}_{ji|S^{\\bar{j}}_{\\bar{m}}}\\hspace{-0.15cm}=\\hspace{-0.15cm}\n\\begin{cases}\n\\vspace{0.15cm}\n\\ \\hspace{-0.25cm}\\frac{S_{m}\\big(h_{jj}(t)+h_{ij}(t)\\big)+S_{\\bar{m}}h_{ii}(t+T_{s})}{V_{\\sss\\text{RX}}},&\\text{for $i= \\bar{j}$ and $j\\neq \\bar{j}$,}\\\\\n\\vspace{0.15cm}\n\\ \\hspace{-0.25cm}\\frac{S_{m}\\big(h_{jj}(t)+h_{ij}(t)\\big)+S_{\\bar{m}}h_{jj}(t+T_{s})}{V_{\\sss\\text{RX}}},&\\text{for $i\\neq \\bar{j}$ and $j= \\bar{j}$,}\\\\\n\\ \\hspace{-0.25cm}\\frac{S_{m}\\big(h_{jj}(t)+h_{ij}(t)\\big)}{V_{\\sss\\text{RX}}},&\\text{for $i\\neq \\bar{j}$ and $j\\neq \\bar{j}$.}\\nonumber\n\\end{cases}\n\\end{align}\n\n\\begin{IEEEproof}\n\tSee Appendix A.\n\\end{IEEEproof} \n\nSpecifically, when only the space symbol $S^{j}$ is transmitted, the SM-MC with $M=1$ degenerates to the SSK-MC and the detection in (27) is non-coherent. The detection performance of the SSK-MC is solely described by the expectation of $P_{c,\\sss \\text{ SSK}}(S^{j}|S^{\\bar{j}})$, where $S^{j}$ and $S^{\\bar{j}}$ are also mutually independent. Note that in SSK-MC, both $S^{j}$ and $S^{\\bar{j}}$ are integers. Correspondingly, considering that the transmitted symbols are uniform distribution, we have\n\\begin{align}\nP_{c,\\sss \\text{ SSK}}&=\\mathbb{E}\\Big[\\mathbb{E}\\big[P_{c,\\sss \\text{ SSK}}(S^{j}|S^{\\bar{j}})\\big]\\Big]\\nonumber\\\\\n&=\\sum_{j=1}^{N}\\sum_{\\bar{j}=1}^{N}P_{c,\\sss \\text{ SSK}}(S^{j}|S^{\\bar{j}})\\text{Pr}[S^{\\bar{j}}]\\text{Pr}[S^{j}]\\nonumber\\\\\n&=\\frac{1}{N^{2}}\\sum_{j=1}^{N}\\sum_{\\bar{j}=1}^{N}\\prod_{i\\neq j}Q\\left(\\frac{-\\mu_{ji|S^{\\bar{j}}}}{\\sigma_{ji|S^{\\bar{j}}}}\\right).\n\\end{align}\nNote that in (31), the subscript $m$ is omitted, since there is no concentration symbol transmitted in SSK-MC. Furthermore, the error probability can be easily derived as\n\\begin{align}\nP_{e,\\sss \\text{ SSK}}=1-P_{c,\\sss \\text{ SSK}}=1-\\frac{1}{N^{2}}\\sum_{j=1}^{N}\\sum_{\\bar{j}=1}^{N}\\prod_{i\\neq j}Q\\left(\\frac{-\\mu_{ji|S^{\\bar{j}}}}{\\sigma_{ji|S^{\\bar{j}}}}\\right).\n\\end{align}\n\nHaving considered the space symbol, the error rate of the CSK detection can be analyzed based on the estimate $\\hat{j}$. Here, we analyze the more general EGC-assisted detector of (29), which utilizes all receivers' observations to detect the CSK symbol. Given that a pulse of $S_{m}$ molecules is transmitted by the $j$-th transmitter and that the previously emitted molecular symbol is $S^{\\bar{j}}_{\\bar{m}}$, the probability of erroneous detection of the CSK symbol can be upper bounded by\n\\begin{align}\nP_{e,\\sss \\text{ CSK}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}})&\\leqslant \\sum_{m\\neq n}\\text{Pr}[S_{m}\\rightarrow S_{n}|S^{\\bar{j}}_{\\bar{m}}],\n\\end{align}\nwhere $\\text{Pr}[S_{m}\\rightarrow S_{n}|S^{\\bar{j}}_{\\bar{m}}]$ is also dependent on the detection of the space symbol. Hence, it can be expressed as\n\\begin{align}\n\\text{Pr}[S_{m}\\rightarrow S_{n}|S^{\\bar{j}}_{\\bar{m}}]=&\\sum_{\\hat{j}\\neq j}\\text{Pr}_{\\sss \\text{ SSK}}[\\hat{j}|S^{\\bar{j}}_{\\bar{m}}]\\ \\text{Pr}[S_{m}\\rightarrow S_{n}|S^{\\bar{j}}_{\\bar{m}},\\hat{j}]\\nonumber\\\\\n&+\\text{Pr}_{\\sss \\text{ SSK}}[j|S^{\\bar{j}}_{\\bar{m}}]\\ \\text{Pr}[S_{m}\\rightarrow S_{n}|S^{\\bar{j}}_{\\bar{m}},j],\n\\end{align}\nwhere $\\text{Pr}_{\\sss \\text{ SSK}}[\\hat{j}|S^{\\bar{j}}_{\\bar{m}}]$ and $\\text{Pr}_{\\sss \\text{ SSK}}[j|S^{\\bar{j}}_{\\bar{m}}]$ have been derived in (43) of Appendix~A for the detection of the SSK symbol. Hence, below we only focus on the unknown components, i.e., $\\text{Pr}[S_{m}\\rightarrow S_{n}|S^{\\bar{j}}_{\\bar{m}},\\hat{j}]$ and $\\text{Pr}[S_{m}\\rightarrow S_{n}|S^{\\bar{j}}_{\\bar{m}},j]$, in (34), which are given by (35) and (36), respectively, shown on the top of the next page. In these formulas, the $\\sigma^{2}_{ni,\\sss \\text{ SM}}(t)$ component has been defined in (24).\n\\begin{figure*}[t]\n\t\\begin{align}\n\t&\\text{Pr}[S_{m}\\rightarrow S_{n}|S^{\\bar{j}}_{\\bar{m}},\\hat{j}]=\\text{Pr}\\bigg[\\sum_{i=1}^{N}|{y}_{i,\\sss \\text{ SM}}(t)-S_{m}{h}_{i\\hat{j}}(t)|^{2}>\\sum_{i=1}^{N}|{y}_{i,\\sss \\text{ SM}}(t)-S_{n}{h}_{i\\hat{j}}(t)|^{2}\\Big|S^{\\bar{j}}_{\\bar{m}}\\bigg]\\nonumber\\\\\n\t&=\n\t\\begin{cases}\n\t\\ 1-Q\\left(\\frac{-S_{\\bar{m}}{h}_{\\bar{j}\\hat{j}}(t){h}_{\\bar{j}j}(t+T_{s})-S_{m}\\sum_{i=1}^{N}{h}_{i\\hat{j}}(t){h}_{ij}(t)+\\frac{(S_{m}+S_{n})}{2}\\sum_{i=1}^{N}{h}^{2}_{i\\hat{j}}(t)}{\\sqrt{\\sum_{i=1}^{N}{h}^{2}_{i\\hat{j}}(t)\\sigma^{2}_{ni,\\sss \\text{ SM}}(t)}}\\right),&\\text{for $S_{m}>S_{n}$}, \\\\\n\t\\ Q\\left(\\frac{-S_{\\bar{m}}{h}_{\\bar{j}\\hat{j}}(t){h}_{\\bar{j}j}(t+T_{s})-S_{m}\\sum_{i=1}^{N}{h}_{i\\hat{j}}(t){h}_{ij}(t)+\\frac{(S_{m}+S_{n})}{2}\\sum_{i=1}^{N}{h}^{2}_{i\\hat{j}}(t)}{\\sqrt{\\sum_{i=1}^{N}{h}^{2}_{i\\hat{j}}(t)\\sigma^{2}_{ni,\\sss \\text{ SM}}(t)}}\\right),&\\text{for $S_{m}<S_{n}$}.\n\t\\end{cases}\n\t\\end{align}\n\t\\begin{align} \n\t&\\text{Pr}[S_{m}\\rightarrow S_{n}|S^{\\bar{j}}_{\\bar{m}},j]\n\t=\\text{Pr}\\bigg[\\sum_{i=1}^{N}|{y}_{i,\\sss \\text{ SM}}(t)-S_{m}{h}_{ij}(t)|^{2}>\\sum_{i=1}^{N}|{y}_{i,\\sss \\text{ SM}}(t)-S_{n}{h}_{ij}(t)|^{2}\\Big|S^{\\bar{j}}_{\\bar{m}}\\bigg]\\nonumber\\\\\n\t&=\n\t\\begin{cases}\n\t\\ 1-Q\\left(\\frac{-S_{\\bar{m}}{h}_{\\bar{j}\\hat{j}}(t){h}_{\\bar{j}j}(t+T_{s})+\\frac{S_{n}-S_{m}}{2}\\sum_{i=1}^{N}{h}^{2}_{ij}(t)}{\\sqrt{\\sum_{i=1}^{N}{h}^{2}_{ij}(t)\\sigma^{2}_{ni,\\sss \\text{ SM}}(t)}}\\right),  & \\text{for $S_{m}>S_{n}$}, \\\\\n\t\\ Q\\left(\\frac{-S_{\\bar{m}}{h}_{\\bar{j}\\hat{j}}(t){h}_{\\bar{j}j}(t+T_{s})+\\frac{S_{n}-S_{m}}{2}\\sum_{i=1}^{N}{h}^{2}_{ij}(t)}{\\sqrt{\\sum_{i=1}^{N}{h}^{2}_{ij}(t)\\sigma^{2}_{ni,\\sss \\text{ SM}}(t)}}\\right), & \\text{for $S_{m}<S_{n}$}.\n\t\\end{cases}\n\t\\end{align}\n\t\\rule{\\textwidth}{0.3mm}\n\\end{figure*}\n\nConsequently, when both the SSK and the CSK symbols are considered, the error probability of the SM-MC system conditioned on that $S^{j}_{m}$ and $S^{\\bar{j}}_{\\bar{m}}$ are transmitted as the current and previous symbol, is given by \n\\begin{align}\nP_{e,\\sss \\text{ SM}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}})=&1\\hspace{-0.1cm}-\\hspace{-0.1cm}\\Big(\\hspace{-0.1cm}1-P_{e,\\sss \\text{ SSK}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}})\\hspace{-0.1cm}\\Big)\\Big(1-P_{e,\\sss \\text{ CSK}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}})\\hspace{-0.1cm}\\Big)\\nonumber\\\\\n\\leqslant&P_{e,\\sss \\text{ SSK}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}})+\\sum_{m\\neq n}\\text{Pr}[S_{m}\\rightarrow S_{n}|S^{\\bar{j}}_{\\bar{m}}]\\nonumber\\\\\n&-P_{e,\\sss \\text{ SSK}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}})\\sum_{m\\neq n}\\text{Pr}[S_{m}\\rightarrow S_{n}|S^{\\bar{j}}_{\\bar{m}}].\n\\end{align}\nFurthermore, if we ignore the negative term in (37), we have\n\\begin{align}\nP_{e,\\sss \\text{ SM}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}})\n\\leqslant&P_{e,\\sss \\text{ SSK}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}})+\\sum_{m\\neq n}\\text{Pr}[S_{m}\\rightarrow S_{n}|S^{\\bar{j}}_{\\bar{m}}].\n\\end{align}\n\nWhen comparing (37) and (38), we can see that when the signal-to-noise ratio (SNR) is sufficiently high, making the detection of both the SSK and CSK symbols sufficiently reliable, then we can ignore the production term in (37), and directly use (38) to obtain the approximate conditional error probability, as shown in Section IV.\n\nFinally, the average SER of the SM-MC systems can be approximately evaluated as\n\\begin{align}\nP_{e,\\sss \\text{ SM}}=\\frac{1}{M^{2}N^{2}}\\sum_{j=1}^{N}\\sum_{m=0}^{M-1}\\sum_{\\bar{j}=1}^{N}\\sum_{\\bar{m}=0}^{M-1}P_{e,\\sss \\text{ SM}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}}).\n\\end{align}\n\n\\begin{table}[b]\n\t\\caption{System parameters}  \n\t\\begin{tabular*}{9cm}{lccc}\\toprule \n\t\tParameter & Variable & Value &Unit   \\\\ \n\t\t\\toprule \n\t\tDiffusion Coefficient &$D$   & $2.2\\times10^{-9}$&$m^{2}\/s$  \\\\  \n\t\tLink Distance &$d$  & $20$ &$\\rm{\\mu m}$ \\\\\n\t\tReceiver Radius &$\\rho$  & $0.1$&$\\rm{\\mu m}$  \\\\ \n\t\tSNR Range &  & [$0$, $20$]&$\\rm{dB}$\\\\   \n\t\tSNR Interval &  & $2$&$\\rm{dB}$\\\\ \n\t\tOrder of Concentration Symbol& $M$& $[2,4]$\\\\  \n\t\tOrder of Space Symbol& $N$& $[2,4]$\\\\  \n\t\tSymbol Sequence Length &  & $10^{6}$\\\\   \n\t\tReplication Times&  & $20$ \\\\ \n\t\t\\bottomrule\n\t\\end{tabular*}  \n\\end{table}\n\\section{Numerical Results}\nIn this section, we present the analytical and simulation results for the SERs of the SM-MC and SSK-MC systems. Performance comparison among various modulation schemes is also provided.\n\nThe definition of SNR in SISO-MC has been proposed in \\cite{LLYang.MCanalysis.16} when OOK modulation schemes are considered. Here, we generalize the SNR definition to the $N\\times N$ SM-MC and MIMO-MC with $M$-ary CSK modulation. Specifically, in SM-MC, the SNR can be defined similar to SISO-MC, as the ratio between the average received power of the desired link from a single transmitted impulse of molecules and the noise power, expressed as\n\\begin{align}\n{\\text{SNR}}&=\\frac{P_{s}}{P_{n}}=\\frac{1}{MN}\\sum_{j=1}^{N}\\sum_{m=0}^{M-1}S_{m}h_{jj}(t_{p})V_{\\sss\\rm{RX}}\\nonumber\\\\\n&=\\bigg(\\frac{3}{2\\pi e}\\bigg)^{\\frac{3}{2}}\\frac{V_{\\sss\\rm{RX}}}{Md^{3}}\\sum_{m=0}^{M-1}S_{m},\n\\end{align}\nwhere $S_{m}>0$ should be satisfied. Eq. (40) implies that SNR is merely dependent on the molecule number of molecules released in average by a chemical impulse, the transceiver distance $d$ and the volume of receiver $V_{\\sss {\\text{RX}}}$. Similarly, the SNR of each link in the $N\\times N$ MIMO-MC with $M$-ary CSK can be defined as\n\\begin{align}\n{\\text{SNR}}=\\bigg(\\frac{3}{2\\pi e}\\bigg)^{\\frac{3}{2}}\\frac{V_{\\sss {\\text{RX}}}}{NMd^{3}}\\sum_{m=0}^{M-1}S_{m}.\n\\end{align}\nNote that (41) is $1\/N$ of the (40), indicating that all the transmitters are activated to emitted chemical pulses in each symbol duration, when $S_{m}>0$. Specifically, when the OOK modulation is used in the $N\\times N$ MIMO-MC, only $N\/2$ transmitters are activated in average to emitted chemical pulses with $S_{1}$ molecules during each symbol period. Hence, the SNR of each link in the $N\\times N$ MIMO-MC is then defined as\n\\begin{align}\n{\\text{SNR}}=\\bigg(\\frac{3}{2\\pi e}\\bigg)^{\\frac{3}{2}}\\frac{2V_{\\sss {\\text{RX}}}}{Nd^{3}}S_{1},\n\\end{align}\nwhere $S_{1}$ is the number of molecules for transmitting bit ``1''.\n\n=When the transceiver distance $d$ is fixed, the same SNR implies the same molecular energy consumption. We can make relatively fair SER comparison between different modulation schemes of MC with the same transmission rate, under the same energy consumption. In order to achieve this, in our simulation for SM-MC with BCSK, we set $S_{1}=2S_{0}$. By contrast, for SISO-MC with QCSK, we have $S_{0}=0$ and $S_{3}=\\frac{3}{2}S_{2}=3S_{1}$. The other system parameters are presented in Table~II. Note that in this section, the SSK-MC with the $2\\times2$ and $4\\times4$ MIMO architectures, which for brevity are referred to as the BSSK-MC and QSSK-MC.\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics[width=3.5in]{Fig_perform1SSK.eps}\n\t\\caption{SER comparison between BSSK-MC and QSSK-MC, when receiver separation distance $r=12.5\\ \\rm{\\mu m}$ and different symbol durations $T_{s}=0.1\\ \\rm{s},0.2\\ \\rm{s},0.8\\ \\rm{s}$ are assumed.}\n\t\\label{fig_4}\n\\end{figure}\n\nFig. 4 shows the analytical and simulation results for the SER of the BSSK-MC and QSSK-MC, when different symbol durations are considered. Clearly, the analytical SER agrees well with the corresponding simulated SER. In the low SNR region, the SER of QSSK-MC is much higher than that of the BSSK-MC, while in the high SNR region, the SER performance of QSSK-MC is becomes better than that of the BSSK-MC, when $T_{s}=0.1\\ \\rm{s}$ and $T_{s}=0.2\\ \\rm{s}$. By contrast, when $T_{s}=0.8\\ \\rm{s}$, the SER of BSSK-MC is always lower than that of QSSK-MC in the whole SNR region considered. However, the gap between their SER performance decreases with the increase of SNR. As shown in Fig. 4, the SER performance of BSSK-MC and QSSK-MC is highly dependent on the symbol duration $T_{s}$ . When $T_{s}$ increases, the SER reduces due to the fact that the ISI reduces, as $T_{s}$ increases.\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics[width=3.5in]{Fig_perform3SSK.eps}\n\t\\caption{SER comparison between BSSK-MC and QSSK-MC, when $T_{s}=0.5\\ \\rm{s}$, and different receiver separation distances are considered.}\n\t\\label{fig_5}\n\\end{figure}\n\nFig. 5 manifests the analytical and simulation SER of BSSK-MC and QSSK-MC, when different transceiver separation distances are considered. Again, the simulation and analytical results match well, which hence validate our theoretical analysis. As shown in the figure, for a given transceiver separation distance, there is a gap between the SER performance of BSSK-MC and that of QSSK-MC in the low SNR region, with BSSK-MC always outperforming the QSSK-MC. However, as the SNR increases, the SER curves of BSSK-MC and QSSK-MC converge and become nearly the same at high SNR. Therefore, when sufficient source of information molecules, i.e., SNR, is available, higher throughput can be attained by utilizing higher order modulation schemes in the SSK-MC, while achieving the required error performance.\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics[width=3.5in]{Fig_perform2SM.eps}\n\t\\caption{SER comparison between the $2\\times2$ SM-MC and $4\\times4$ SM-MC, when receiver separation distance $r=10\\ \\rm{\\mu m}$ and different symbol durations $T_{s}=0.15\\ \\rm{s},0.3\\ \\rm{s},1\\ \\rm{s}$ are assumed.}\n\t\\label{fig_6}\n\\end{figure}\n\n\nFig. 6 depicts the theoretical SER upper bound and the simulated SER of the SM-MC, when the $2\\times2$ and the $4\\times4$ MIMO architectures are respectively considered. In this figure, the transceiver separation distance is fixed to $r=10\\ \\rm{\\mu m}$, whilst the symbol durations $T_{s}$ is set to $0.15\\ \\rm{s}$, $0.3\\ \\rm{s}$ or $1\\ \\rm{s}$. The results show that the upper bound is tight, which becomes tighter as the SNR increases. As shown in Fig. 6, in the low SNR regime, the SER of the $4\\times4$ SM-MC is much higher than that of the $2\\times2$ SM-MC. By contrast, in the high SNR regime, the SER of the $4\\times4$ SM-MC is better than that of the $2\\times2$ SM-MC, when $T_{s}=0.15\\ \\rm{s}$ or $T_{s}=0.3\\ \\rm{s}$. Furthermore, when $T_{s}=1\\ \\rm{s}$, the SER of the BSSK-MC is always lower than that of the QSSK-MC in the SNR regime considered. The explanation for these results are similar to that for the results shown in Fig. 4.\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics[width=3.5in]{Fig_perform4SM.eps}\n\t\\caption{SER comparison between the $2\\times2$ SM-MC and $4\\times4$ SM-MC, when $T_{s}=1\\ \\rm{s}$, and $r=8\\ \\rm{\\mu m},10\\ \\rm{\\mu m},12\\ \\rm{\\mu m}$ are assumed.}\n\t\\label{fig_7}\n\\end{figure}\n\nFig. 7 also demonstrates the theoretical SER upper bound and the simulation SER results of the SM-MC, where the symbol duration is set to $T_{s}=1\\ \\rm{s}$, while various transceiver separation distances are considered. Similar to Fig. 6, the SER upper bounds match well with their corresponding results obtained from simulations. There is also an SER gap between the $2\\times2$ SM-MC and its $4\\times4$ counterpart. However, when the transceiver separation distance increases, the SER difference evaluated in $\\rm{dB}$ reduces. We should note that when the link distance $d$ and the symbol duration $T_{s}$ are fixed, the SER performance of SM-MC is mainly determined by the separation distance $r$ between the receivers.\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics[width=3.5in]{Fig_perform5SM.eps}\n\t\\caption{SER comparison of the SM-MC and SSK-MC with that of the MIMO-MC using OOK modulation and SISO-MC using QCSK modulation, when $T_{s}=0.2\\ \\rm{s}$, and $r=10\\ \\rm{\\mu m}$ or $15\\ \\rm{\\mu m}$.}\n\t\\label{fig_8}\n\\end{figure}\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics[width=3.5in]{Fig_perform6SM.eps}\n\t\\caption{SER performance of the SM-MC with EGC-assisted and SC-assisted detection schemes, when $T_{s}=1\\ \\rm{s}$, and $r=10\\ \\rm{\\mu m},12.5\\ \\rm{\\mu m}$ or $15\\ \\rm{\\mu m}$.}\n\t\\label{fig_9}\n\\end{figure}\n\nTo verify the effectiveness of our proposed SSK-MC and SM-MC, we compare the simulated SER performance of the SISO-MC with QCSK, $2 \\times 2$ MIMO-MC with OOK, $4 \\times 4$ SSK-MC and the $2 \\times 2$ SM-MC in Fig. 8. All of them have the same transmission rate of 2 bits per symbol. As shown in Fig. 8, when $r=15\\ \\rm{\\mu m}$, the QSSK-MC outperforms all the other schemes and achieves the best SER performance. Both Both the SSK-MC and SM-MC benefit from the employment of space modulation, making them significantly surpass the $2 \\times 2$ MIMO-MC in the high SNR regime. The SISO-MC with QCSK attains the worst SER performance among the schemes considered. However, the situation is completely different when the separation distance is reduced to $r=10\\ \\rm{\\mu m}$. In this case, the SISO-MC with QCSK achieves the lowest SER, and outperforms all the other MIMO schemes. As shown in Fig. 8, the $2 \\times 2$ MIMO-MC with OOK has similar SER performance as the QSSK-MC and the $2 \\times 2$ SM-MC when the SNR is between $0\\ \\rm{dB}$ and $12\\ \\rm{dB}$. When for the increasing SNR, its SER performance becomes poorer than that of the other schemes.\n\nThe results in Fig. 8 implies that the separation distance $r$ is important for the achievable SER performance, especially when comparing the modulation schemes in SISO-MC. When $r$ is appropriately selected, the MIMO-MC schemes are capable of outperforming SISO-MC with QCSK, owning to the significant ILI reduction in the MIMO-MC schemes, which is beneficial to transmitting information in the space domain. By contrast, if $r$ is not sufficiently large, the advantage of using space symbol vanishes due to the high ILI.\n\nIn Fig. 9, we compare the SER performance of the SC-assisted SM-MC of (28) with that of the EGC-assisted SM-MC of (29). From Fig. 9, we observe that the SER performance of EGC-assisted SM-MC is better than that of SC-assisted SM-MC, when there is only a single receiver utilized to decode the concentration symbol. For the $4 \\times 4$ SM-MC with $r=10\\ \\rm{\\mu m}$, EGC-assisted SM-MC slightly outperforms SC-assisted SM-MC, when the SNR is lower than about $16 \\ \\rm{dB}$, while their performance converges at high SNR. When $r=12.5\\ \\rm{\\mu m}$, and the $2 \\times 2$ and $4 \\times 4$ SM-MC are considered, we observe that the EGC-assisted SM-MC always outperforms its SC-assisted counterpart. Additionally, when the same combining scheme is employed at the receivers, a small-scale SM-MC with fewer links outperforms a large-scale SM-MC with more links. Fig.9 also shows, when $2 \\times 2$ SM-MC with $r=15\\ \\rm{\\mu m}$ is considered, the SER performance of EGC-assisted SM-MC becomes increasingly better than that of SC-assisted SM-MC, and approximately $1.5\\ \\rm{dB}$ gain can be obtained at the SER of $10^{-3}$.\n\nBased on the aforementioned results, in order to improve the SER performance of the SSK-MC and that of SM-MC, the following approaches may be applied. Firstly, the symbol duration $T_{s}$ may be increased to reduce ISI, which hence improves the SER of the SSK-MC and SM-MC systems, but at the cost of the transmission rate. Secondly, the separation distance between adjacent transmitters and that between adjacent receivers may be increased for mitigating the ILI. This is an effective method to improve the SER performance provided that tehre are spaces for transmitter and receiver deployment. Furthermore, the number of molecules per pulse maybe increased, which increases SNR, and therefore enhances the reliability of the information transmission in SSK-MC and the SM-MC system, but at the cost of energy consumption.\n\n\\section{Conclusions and Future Work}\nIn this paper, we have proposed a SM-MC scheme and its special case of SSK-MC by introducing new degrees of freedom from the spatial domain to transmit more information in MC. Our studies showed that when the separation distance between transmit\/receive nanomachines is appropriate, additional information conveyed in the space domain is achievable, and our proposed SM-MC and SSK-MC schemes are capable of outperforming the conventional SISO-MC and MIMO-MC scheme. The reason behind is that our proposed SM-MC can effectively eliminate the ILI, which usually severely affects the performance of the conventional MIMO-MC. Moreover, it is an energy-efficient system where only one nanomachine transmitter is activated during each symbol period, befitting the twofold goal in small scale communication systems of low energy consumption and low-complexity.\n\nIn this paper, we have also presented theoretical analysis and some simulation results under the assumption of the homogeneous propagation medium as done in the literature. This may not accord with a realistic situation where various medium exist in the communication channel. In our future work, we will consider the diffusion based MC with more realistic channel modelling, as well as the design of corresponding MIMO-MC transceivers.\n\n\\begin{appendices}\n\\section{  }\nLet us denote $\\text{Pr}_{\\sss \\text{ SSK}}[i|S^{\\bar{j}}_{\\bar{m}}]$ as the probability that the $i$-th receiver senses the maximum molecular concentration, which can be expressed as\n\t\\begin{align}\n\t\\text{Pr}_{\\sss \\text{ SSK}}[i|S^{\\bar{j}}_{\\bar{m}}]=\\text{Pr}\\Big[\\hspace{-0.05cm}\\big(y_{i,\\sss \\text{ SM}}(t)\\hspace{-0.1cm}>\\hspace{-0.05cm}y_{1,\\sss \\text{ SM}}(t)\\hspace{-0.05cm}\\big)\\hspace{-0.1cm}\\cap\\hspace{-0.1cm}\\cdots\\hspace{-0.1cm}\\cap\\hspace{-0.1cm}\\big(y_{i,\\sss \\text{ SM}}(t)\\hspace{-0.1cm}>\\hspace{-0.05cm}y_{N,\\sss \\text{ SM}}(t)\\hspace{-0.05cm}\\big)\\hspace{-0.05cm}\\Big],\n\t\\end{align}\nwhen given that the previously sent molecular symbol is $S^{\\bar{j}}_{\\bar{m}}$. In order to derive the error detection probability of the space symbol, let us first derive the correct detection probability of the space symbol, when $S_{m}$ molecules are released from the $j$-th transmitter. Based on (43), this probability can be expressed as\n\\begin{align}\nP_{c,\\sss \\text{ SSK}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}})&=\\text{Pr}_{\\sss \\text{ SSK}}[j|S^{\\bar{j}}_{\\bar{m}}]\\nonumber\\\\\n&=\\text{Pr}\\hspace{-0.05cm}\\Big[\\hspace{-0.075cm}\\big(y_{j,\\sss \\text{ SM}}(t)\\hspace{-0.1cm}>\\hspace{-0.1cm}y_{1,\\sss \\text{ SM}}(t)\\hspace{-0.075cm}\\big)\\hspace{-0.1cm}\\cap\\hspace{-0.1cm}\\cdots\\hspace{-0.1cm}\\cap\\hspace{-0.1cm}\\big(y_{j,\\sss \\text{ SM}}(t)\\hspace{-0.1cm}>\\hspace{-0.1cm}y_{N,\\sss \\text{ SM}}(t)\\hspace{-0.075cm}\\big)\\hspace{-0.075cm}\\Big],\n\\end{align}\nwhich is the probability that the $j$-th receiver senses the maximum concentration. When the reception processes of the receiver nanomachines are assumed to be independent, (44) can be expressed as\n\\begin{align}\nP_{c,\\sss \\text{ SSK}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}})=\\prod_{i\\neq j}\\text{Pr}[y_{j,\\sss \\text{ SM}}(t)-y_{i,\\sss \\text{ SM}}(t)>0|S^{\\bar{j}}_{\\bar{m}}].\n\\end{align}\n\nFurthermore, when given $S^{\\bar{j}}_{\\bar{m}}$, $y_{j,\\sss \\text{ SM}}(t)-y_{i,\\sss \\text{ SM}}(t)$ can be written as\n\\begin{align}\ny_{ji|S^{\\bar{j}}_{\\bar{m}}}=&S_{m}\\big(h_{jj}(t)-h_{ij}(t)\\big)+I_{j,\\sss \\text{ SM}}(t)-I_{i,\\sss \\text{ SM}}(t)\\nonumber\\\\&+n_{j,\\sss \\text{ SM}}(t)-n_{i,\\sss \\text{ SM}}(t).\n\\end{align}\nAccording to (25), $y_{ji|S^{\\bar{j}}_{\\bar{m}}}$ follows the Normal distribution of\n\\begin{align}\ny_{ji|S^{\\bar{j}}_{\\bar{m}}}\\sim \\mathcal{N} \\left(\\mu_{ji|S^{\\bar{j}}_{\\bar{m}}},\\sigma^{2}_{ji|S^{\\bar{j}}_{\\bar{m}}}\\right).\n\\end{align}\nWhen given $i\\neq j$, and according to \\cite{Pat.NormalDist.96}, $\\mu_{ji|S^{\\bar{j}}_{\\bar{m}}}$ and $\\sigma^{2}_{ji|S^{\\bar{j}}_{\\bar{m}}}\\hspace{-0.15cm}$ are given by\n\\begin{align}\n\\mu_{ji|S^{\\bar{j}}_{\\bar{m}}}\\hspace{-0.15cm}=\\hspace{-0.15cm}\n\\begin{cases}\n\\vspace{0.15cm} \n\\\n\\hspace{-0.225cm}S_{m}\\hspace{-0.075cm}\\big(h_{jj}\\hspace{-0.05cm}(t)\\hspace{-0.1cm}-\\hspace{-0.1cm}h_{ij}\\hspace{-0.05cm}(t)\\hspace{-0.06cm}\\big)\\hspace{-0.1cm}-\\hspace{-0.1cm}S_{\\bar{m}}h_{ii}\\hspace{-0.05cm}(t\\hspace{-0.1cm}+\\hspace{-0.1cm}T_{s}\\hspace{-0.05cm}),\\hspace{-0.3cm}&\\text{for $i= \\bar{j}$ and $j\\neq \\bar{j}$,}\\\\\n\\\n\\vspace{0.15cm}  \\hspace{-0.225cm}S_{m}\\hspace{-0.075cm}\\big(h_{jj}\\hspace{-0.05cm}(t)\\hspace{-0.1cm}-\\hspace{-0.1cm}h_{ij}\\hspace{-0.05cm}(t)\\hspace{-0.06cm}\\big)\\hspace{-0.1cm}+\\hspace{-0.1cm}S_{\\bar{m}}h_{jj}\\hspace{-0.05cm}(t\\hspace{-0.1cm}+\\hspace{-0.1cm}T_{s}\\hspace{-0.05cm}),\\hspace{-0.3cm}&\\text{for $i\\neq \\bar{j}$ and $j= \\bar{j}$,}\\\\\n\\ \\hspace{-0.225cm}S_{m}\\hspace{-0.075cm}\\big(h_{jj}\\hspace{-0.05cm}(t)\\hspace{-0.1cm}-\\hspace{-0.1cm}h_{ij}\\hspace{-0.05cm}(t)\\hspace{-0.06cm}\\big), \\hspace{-0.3cm}&\\text{for $i\\neq \\bar{j}$ and $j\\neq \\bar{j}$,}\n\\end{cases}\n\\end{align}\n\\begin{align}\n\\sigma^{2}_{ji|S^{\\bar{j}}_{\\bar{m}}}\\hspace{-0.15cm}=\\hspace{-0.15cm}\n\\begin{cases}\n\\vspace{0.15cm}\n\\ \\hspace{-0.25cm}\\frac{S_{m}\\big(h_{jj}(t)+h_{ij}(t)\\big)+S_{\\bar{m}}h_{ii}(t+T_{s})}{V_{\\sss\\text{RX}}},&\\text{for $i= \\bar{j}$ and $j\\neq \\bar{j}$,}\\\\\n\\vspace{0.15cm}\n\\ \\hspace{-0.25cm}\\frac{S_{m}\\big(h_{jj}(t)+h_{ij}(t)\\big)+S_{\\bar{m}}h_{jj}(t+T_{s})}{V_{\\sss\\text{RX}}},&\\text{for $i\\neq \\bar{j}$ and $j= \\bar{j}$,}\\\\\n\\ \\hspace{-0.25cm}\\frac{S_{m}\\big(h_{jj}(t)+h_{ij}(t)\\big)}{V_{\\sss\\text{RX}}},&\\text{for $i\\neq \\bar{j}$ and $j\\neq \\bar{j}$,}\n\\end{cases}\n\\end{align}\nrespectively. Therefore, we have\n\\begin{align}\n\\text{Pr}[y_{ji|S^{\\bar{j}}_{\\bar{m}}}\\hspace{-0.1cm}>\\hspace{-0.1cm}0]\\hspace{-0.1cm}&=\\hspace{-0.2cm}\\int_{0}^{+\\infty}\\hspace{-0.5cm}\\frac{1}{\\sqrt{2\\pi}\\sigma_{ji|S^{\\bar{j}}_{\\bar{m}}}}\\exp\\hspace{-0.1cm}\\left(\\hspace{-0.15cm}-\\frac{(y_{ji|S^{\\bar{j}}_{\\bar{m}}}-\\mu_{ji|S^{\\bar{j}}_{\\bar{m}}})^{2}}{2\\sigma^{2}_{ji|S^{\\bar{j}}_{\\bar{m}}}}\\hspace{-0.125cm}\\right)\\hspace{-0.125cm}dy_{ji|S^{\\bar{j}}_{\\bar{m}}}\\nonumber\\\\\n&=Q\\left(-\\frac{\\mu_{ji|S^{\\bar{j}}_{\\bar{m}}}}{\\sigma_{ji|S^{\\bar{j}}_{\\bar{m}}}}\\right).\n\\end{align}\nFinally, when substituting (50) into (45), the probability of correct detection of the space symbol $j$ can be formulated as\n\\begin{align}\nP_{c,\\sss \\text{ SSK}}(S^{j}_{m}|S^{\\bar{j}}_{\\bar{m}})=\\prod_{i\\neq j}Q\\left(-\\frac{\\mu_{ji|S^{\\bar{j}}_{\\bar{m}}}}{\\sigma_{ji|S^{\\bar{j}}_{\\bar{m}}}}\\right).\n\\end{align}\n\n\n\\end{appendices}\n\\bibliographystyle{IEEEtran} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n The classification problem of finite dimensional algebras is one of the important problems of modern algebra. So far two approaches are known to the solution of the problem. One of them is structural (basis free, invariant) approach. For instance, the classification of finite dimensional simple and semi-simple associative algebras by Wedderburn and simple and semi-simple Lie algebras by Cartan are examples of such approach. But it is observed that this approach becomes more difficult when one considers more general types of algebras. Another approach to the solution of the problem is coordinate based approach (see \\cite{B2015,D2003,E2004,G2011,P2011} for the latest results). These two approaches somehow are complementary to each other.\n \n In two-dimensional case a complete classification, by basis free approach, is stated in \\cite{P2000} over any basic field. In this paper we follow the coordinate (basis, structure constants) based approach, we consider such problem for two-dimensional algebras over any algebraically closed field $\\mathbf{F}$. We provide a list of algebras, given by their matrices of structure constants (MSC), such that any 2-dimensional algebra is isomorphic to only one algebra from the list. A similar result has been stated in \\cite{G2011}. The main difference between these two results is as follows. In \\cite{G2011} the authors can state the existence only whereas the uniqueness can not be guaranteed. Moreover, the approach to the problem followed in the present paper is totally different that in \\cite{G2011}. Some details of our approach for arbitrary dimensional case have been given in \\cite{B2015}.\n \n The paper is organized as follows. First we study the problem over algebraically closed fields of characteristics not 2 and 3, then the solution is given for algebras over algebraically closed fields of characteristics 2 and the last subsection of paper contains the result over algebraically closed fields of characteristics 3. In each of these cases we present the list of algebras via their matrices of structure constants.\n \n\\section{Classification of two-dimensional algebras}\n\nTo classify the main part of two-dimensional algebras we use a particular case of the following result from \\cite{B2015}.  Let $n$, $m$  be any natural numbers, $\\tau: (G,V)\\rightarrow V$ be a fixed linear algebraic representation of an algebraic subgroup $G$ of $GL(m,\\mathbf{F})$ on the $n$-dimensional vector space $V$ over $\\mathbf{F}$.  Assume that there exists a nonempty $G$-invariant subset $V_0$ of $V$ and an algebraic map $P: V_0\\rightarrow G$ such that\n\\begin{center}\n\t$P(\\tau( g,\\mathbf{v}))=P(\\mathbf{v}) g^{-1}$,\n\\end{center}\nwhenever  $\\mathbf{v}\\in V_0$ and $g\\in G$. The following result holds true \\cite{B2015}.\n\n{\\bf Theorem.} \\textit{ Elements $\\mathbf{u},\\mathbf{v}\\in V_0$ are $G$-equivalent, that is $\\mathbf{u}=\\tau( g,\\mathbf{v})$ for some $g\\in G$, if and only if \\[\\tau(\n\tP(\\mathbf{u}),\\mathbf{u})=\\tau( P(\\mathbf{v}),\\mathbf{v}).\\]}\n\nLet $\\mathbf{A}$ be any 2 dimensional algebra over $\\mathbf{F}$ with multiplication $\\cdot$ given by a bilinear map $(\\mathbf{u},\\mathbf{v})\\mapsto \\mathbf{u}\\cdot \\mathbf{v}$ whenever $\\mathbf{u}, \\mathbf{v}\\in \\mathbf{A}$. If $e=(e^1,e^2)$ is \nbasis for $\\mathbf{A}$ as a vector space over $\\mathbf{F}$ then one can represent this bilinear map by a matrix \\[A=\\left(\n\\begin{array}{cccc}\nA^{1}_{1,1} & A^{1}_{1,2}& A^{1}_{2,1}& A^{1}_{2,2} \\\\\nA^{2}_{1,1} & A^{2}_{1,2}& A^{2}_{2,1}& A^{2}_{2,2} \\\\\n\\end{array}\n\\right) \\in Mat(2\\times 4;\\mathbf{F})\\] such that \\[\\mathbf{u}\\cdot \\mathbf{v}=eA(u\\otimes v)\\] for any $\\mathbf{u}=eu,\\mathbf{v}=ev,$\nwhere $u=(u_1, u_2),$ and  $v=(v_1, v_2)$ are column coordinate vectors of $\\mathbf{u}$ and $\\mathbf{v},$ respectively, $(u\\otimes v)=(u_1v_1,u_1v_2,u_2v_1,u_2v_2)$, $e^i\\cdot e^j=A^{1}_{i,j}e^1+A^{2}_{i,j}e^2$ whenever $i,j=1,2$.\nSo the algebra $\\mathbf{A}$ is presented by the matrix $A\\in Mat(2\\times 4;\\mathbf{F})$ (called the matrix of MSC of $\\mathbf{A}$ with respect to the basis $e$).\n\nIf $e'=(e'^1,e'^2)$ is also another basis for $\\mathbf{A}$, $g\\in G=GL(2,\\mathbf{F})$, $e'g=e$ and  $\\mathbf{u}\\cdot \\mathbf{v}=e'B(u'\\otimes v')$, where $\\mathbf{u}=e'u',\\mathbf{v}=e'v'$, then\n\\[\\mathbf{u}\\cdot \\mathbf{v}=eA(u\\otimes v)=e'B(u'\\otimes v')=eg^{-1}B(gu\\otimes gv)=eg^{-1}B(g\\otimes g)(u\\otimes v)\\] as far as $\\mathbf{u}=eu=e'u'=eg^{-1}u',\\mathbf{v}=ev=e'v'=eg^{-1}v'$.\nTherefore the equality\n\\[B=gA(g^{-1})^{\\otimes 2}\\] is valid, where for $g^{-1}=\\left(\\begin{array}{cccc} \\xi_1& \\eta_1\\\\ \\xi_2& \\eta_2\\end{array}\\right)$ one has\n\\[(g^{-1})^{\\otimes 2}=g^{-1}\\otimes g^{-1}=\\left(\\begin{array}{cccc} \\xi _1^2 & \\xi _1 \\eta _1 & \\xi _1 \\eta _1 & \\eta _1^2 \\\\\n\\xi _1 \\xi _2 & \\xi _1 \\eta _2 & \\xi _2 \\eta _1 & \\eta _1 \\eta _2 \\\\\n\\xi _1 \\xi _2 & \\xi _2 \\eta _1 & \\xi _1 \\eta _2 & \\eta _1 \\eta _2 \\\\\n\\xi _2^2 & \\xi _2 \\eta _2 & \\xi _2 \\eta _2 & \\eta _2^2\n\\end{array}\n\\right).\\]\n\n{\\bf Definition.} Two-dimensional algebras $\\mathbf{A}$, $\\mathbf{B}$, given by\n\ttheir matrices of structural constants $A$, $B$, are said to be isomorphic if $B=gA(g^{-1})^{\\otimes 2}$ holds true for some $g\\in GL(2,\\mathbf{F})$.\n\nNote that the following identities \\begin{equation} \\label{2.1} Tr_1(gA(g^{-1})^{\\otimes 2})=Tr_1(A)g^{-1},\\ \\ Tr_2(gA(g^{-1})^{\\otimes 2})=Tr_2(A)g^{-1}\\end{equation} hold true whenever $A\\in Mat(2\\times 4,\\mathbf{F}), g\\in GL(2,\\mathbf{F})$, where \\[Tr_1(A)=(A^1_{1,1}+A^2_{2,1}, A^1_{1,2}+A^2_{2,2}),\\ Tr_2(A)=(A^1_{1,1}+A^2_{1,2}, A^1_{2,1}+A^2_{2,2})\\] are row vectors.\n\nWe divide $Mat(2\\times 4,\\mathbf{F})$ into the following five disjoint subsets:\n\\begin{itemize}\n\t\\item[1.] All $A$ for which the system $\\{Tr_1(A),Tr_2(A)\\}$ is linear independent.\n\t\\item[2.] All $A$ for which the system $\\{Tr_1(A),Tr_2(A)\\}$ is linear dependent and $Tr_1(A),Tr_2(A)$ are nonzero vectors.\n\t\\item[3.] All $A$ for which $Tr_1(A)$ is nonzero vector and $Tr_2(A)=(0,0)$.\n\t\\item[4.] All $A$ for which $Tr_1(A)=(0,0)$ and $Tr_2(A)$ is nonzero vector.\n\t\\item[5.] All $A$ for which$Tr_1(A)=Tr_2(A)=(0,0)$.\n\\end{itemize}\n\nDue to (1) it is clear that algebras with matrices from these different classes can not be isomorphic. We deal with each of these subsets separately.\nFurther, for the simplicity, we use the notation \\[A=\\left(\\begin{array}{cccc} \\alpha_1 & \\alpha_2 & \\alpha_3 &\\alpha_4\\\\ \\beta_1 & \\beta_2 & \\beta_3 &\\beta_4\\end{array}\\right),\\] where\n$\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, \\beta_1, \\beta_2, \\beta_3, \\beta_4$ stand for any elements of $\\mathbf{F}$.\n\nAs it has been stated above we consider three cases\ncorresponding to $Char(\\mathbf{F})\\neq 2,3,$ $Char(\\mathbf{F})=2$ and $Char(\\mathbf{F})=3,$ respectively.\n\n\\subsection{Characteristic not $2$, $3$ case}\n\n{\\bf Theorem 1.} \\textit{Over an algebraically closed field $\\mathbf{F}$ with the characteristic not 2 and 3, any non-trivial 2-dimensional algebra is isomorphic to only one of the following listed, by their matrices of structure constants,  algebras:\n\t\\[A_{1}(\\mathbf{c})=\\left(\n\t\\begin{array}{cccc}\n\t\\alpha_1 & \\alpha_2 &\\alpha_2+1 & \\alpha_4 \\\\\n\t\\beta_1 & -\\alpha_1 & -\\alpha_1+1 & -\\alpha_2\n\t\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\alpha_2, \\alpha_4, \\beta_1)\\in \\mathbf{F}^4,\\]\n\t\\[A_{2}(\\mathbf{c})=\\left(\n\t\\begin{array}{cccc}\n\t\\alpha_1 & 0 & 0 & 1 \\\\\n\t\\beta _1& \\beta _2& 1-\\alpha_1&0\n\t\\end{array}\\right)\\simeq \\left(\n\t\\begin{array}{cccc}\n\t\\alpha_1 & 0 & 0 & 1 \\\\\n\t-\\beta _1& \\beta _2& 1-\\alpha_1&0\n\t\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_1, \\beta_2)\\in \\mathbf{F}^3,\\]\n\t\\[A_{3}(\\mathbf{c})=\\left(\n\t\\begin{array}{cccc}\n\t0 & 1 & 1 & 0 \\\\\n\t\\beta _1& \\beta _2 & 1&-1\n\t\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\beta_1, \\beta_2)\\in \\mathbf{F}^2,\\]\n\t\\[A_{4}(\\mathbf{c})=\\left(\n\t\\begin{array}{cccc}\n\t\\alpha _1 & 0 & 0 & 0 \\\\\n\t0 & \\beta _2& 1-\\alpha _1&0\n\t\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_2)\\in \\mathbf{F}^2,\\]\n\t\\[A_{5}(\\mathbf{c})=\\left(\n\t\\begin{array}{cccc}\n\t\\alpha_1& 0 & 0 & 0 \\\\\n\t1 & 2\\alpha_1-1 & 1-\\alpha_1&0\n\t\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=\\alpha_1\\in \\mathbf{F},\\]\n\t\\[A_{6}(\\mathbf{c})=\\left(\n\t\\begin{array}{cccc}\n\t\\alpha_1 & 0 & 0 & 1 \\\\\n\t\\beta _1& 1-\\alpha_1 & -\\alpha_1&0\n\t\\end{array}\\right)\\simeq \\left(\n\t\\begin{array}{cccc}\n\t\\alpha_1 & 0 & 0 & 1 \\\\\n\t-\\beta _1& 1-\\alpha_1 & -\\alpha_1&0\n\t\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_1)\\in \\mathbf{F}^2,\\]\n\t\\[A_{7}(\\mathbf{c})=\\left(\n\t\\begin{array}{cccc}\n\t0 & 1 & 1 & 0 \\\\\n\t\\beta_1& 1& 0&-1\n\t\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=\\beta_1\\in \\mathbf{F},\\]\n\t\\[A_{8}(\\mathbf{c})=\\left(\n\t\\begin{array}{cccc}\n\t\\alpha_1 & 0 & 0 & 0 \\\\\n\t0 & 1-\\alpha_1 & -\\alpha_1&0\n\t\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=\\alpha_1\\in \\mathbf{F},\\]\n\t\\[A_{9}=\\left(\n\t\\begin{array}{cccc}\n\t\\frac{1}{3}& 0 & 0 & 0 \\\\\n\t1 & \\frac{2}{3} & -\\frac{1}{3}&0\n\t\\end{array}\\right),\\\n\t\\ A_{10}=\\left(\n\t\\begin{array}{cccc}\n\t0 & 1 & 1 & 0 \\\\\n\t0 &0& 0 &-1\n\t\\end{array}\n\t\\right),\\]\n\t\\[ A_{11}=\\left(\n\t\\begin{array}{cccc}\n\t0 & 1 & 1 & 0 \\\\\n\t1 &0& 0 &-1\n\t\\end{array}\n\t\\right),\\\n\t\\ A_{12}=\\left(\n\t\\begin{array}{cccc}\n\t0 & 0 & 0 & 0 \\\\\n\t1 &0&0 &0\\end{array}\n\t\\right).\\]}\n\n{\\bf Proof.}\n{\\bf The first subset case.} In this case let $P(A)$ stand for a nonsingular matrix $\\left(\\begin{array}{cc} \\alpha_1+\\beta_3& \\alpha_2+\\beta_4\\\\ \\alpha_1+\\beta_2 & \\alpha_3+\\beta_4\\end{array}\\right)$  with rows\n\\[Tr_1(A)=(\\alpha _1+\\beta _3, \\alpha _2+\\beta _4),\\ Tr_2(A)=(\\alpha _1+\\beta _2, \\alpha _3+\\beta _4).\\] Due to (\\ref{2.1}) we have the identity \\[P(g A(g^{-1})^{\\otimes 2})= P(A)g^{-1},\\] which means that one can apply the  Theorem as $V=\\mathbf{F}^8$ and $\\tau(g,A)= g A(g^{-1})^{\\otimes 2}.$ In this case for $V_0$ one can take \\[ V_0=\\{A\\in Mat(2\\times 4,\\mathbf{F}): \\det(P(A))=\\alpha_1(\\alpha_3-\\alpha_2)+\\beta_4(\\beta_3-\\beta_2)+\\alpha_3\\beta_3-\\alpha_2\\beta_2\\neq 0.\\}\\] Therefore,\ntwo-dimensional algebras $\\mathbf{A}$, $\\mathbf{B}$, given by\ntheir matrices of structure constants $A$, $B \\in V_0$, are isomorphic if and only if the equality \\[P(B)B(P(B)^{-1}\\otimes P(B)^{-1})=P(A)A(P(A)^{-1}\\otimes P(A)^{-1})\\] holds true.\n\nFor $\\xi_1=\\frac{\\alpha_3+\\beta_4}{\\Delta}$, $\\xi_2=-\\frac{\\alpha_1+\\beta_2}{\\Delta}$, $\\eta_1=-\\frac{\\alpha_2+\\beta_4}{\\Delta}$, $\\eta_2=\\frac{\\alpha_1+\\beta_3}{\\Delta}$, where $\\Delta=\\Delta(A)=\\det(P(A))$, one has\n\\[P(A)^{-1}=\\left(\\begin{array}{cc} \\xi_1& \\eta_1\\\\ \\xi_2& \\eta_2\\end{array}\\right)\\] and\n$A'=\\left(\\begin{array}{cccc} \\alpha'_1 & \\alpha'_2 & \\alpha'_3 &\\alpha'_4\\\\ \\beta'_1 & \\beta'_2 & \\beta'_3 &\\beta'_4\\end{array}\\right)=P(A)A(P(A)^{-1}\\otimes P(A)^{-1})$ is matrix consisting of columns\n\\[\\left(\\begin{array}{c}\n-\\frac{\\beta _1 \\eta _1 \\xi _1^2}{\\Delta }+\\frac{\\alpha _1 \\eta _2 \\xi _1^2}{\\Delta }+\\frac{2 \\alpha _1 \\eta _1 \\xi _1 \\xi _2}{\\Delta }-\\frac{2 \\beta _4 \\eta _2 \\xi _1 \\xi _2}{\\Delta }-\\frac{2 \\eta _1 \\eta _2 \\xi _1 \\xi _2}{\\Delta ^2}+\\frac{\\eta _2 \\xi _1^2 \\xi _2}{\\Delta ^2}-\\frac{\\beta _4 \\eta _1 \\xi _2^2}{\\Delta }+\\frac{\\alpha _4 \\eta _2 \\xi _2^2}{\\Delta }+\\frac{\\eta _1 \\xi _1 \\xi _2^2}{\\Delta ^2}  \\\\\n\\frac{\\beta _1 \\xi _1^3}{\\Delta }-\\frac{3 \\alpha _1 \\xi _1^2 \\xi _2}{\\Delta }+\\frac{\\eta _2 \\xi _1^2 \\xi _2}{\\Delta ^2}+\\frac{3 \\beta _4 \\xi _1 \\xi _2^2}{\\Delta }+\\frac{\\eta _1 \\xi _1 \\xi _2^2}{\\Delta ^2}-\\frac{2 \\xi _1^2 \\xi _2^2}{\\Delta ^2}-\\frac{\\alpha _4 \\xi _2^3}{\\Delta }\n\\end{array}\n\\right),\\]\n\\[\\left(\\begin{array}{c}-\\frac{\\beta _1 \\eta _1^2 \\xi _1}{\\Delta }+\\frac{2 \\alpha _1 \\eta _1 \\eta _2 \\xi _1}{\\Delta }-\\frac{\\beta _4 \\eta _2^2 \\xi _1}{\\Delta }-\\frac{\\eta _1 \\eta _2^2 \\xi _1}{\\Delta ^2}+\\frac{\\alpha _1 \\eta _1^2 \\xi _2}{\\Delta }-\\frac{2 \\beta _4 \\eta _1 \\eta _2 \\xi _2}{\\Delta }-\\frac{\\eta _1^2 \\eta _2 \\xi _2}{\\Delta ^2}+\\frac{\\alpha _4 \\eta _2^2 \\xi _2}{\\Delta }+\\frac{2 \\eta _1 \\eta _2 \\xi _1 \\xi _2}{\\Delta ^2}\\\\\n\\frac{\\beta _1 \\eta _1 \\xi _1^2}{\\Delta }-\\frac{\\alpha _1 \\eta _2 \\xi _1^2}{\\Delta }-\\frac{2 \\alpha _1 \\eta _1 \\xi _1 \\xi _2}{\\Delta }+\\frac{2 \\beta _4 \\eta _2 \\xi _1 \\xi _2}{\\Delta }+\\frac{2 \\eta _1 \\eta _2 \\xi _1 \\xi _2}{\\Delta ^2}-\\frac{\\eta _2 \\xi _1^2 \\xi _2}{\\Delta ^2}+\\frac{\\beta _4 \\eta _1 \\xi _2^2}{\\Delta }-\\frac{\\alpha _4 \\eta _2 \\xi _2^2}{\\Delta }-\\frac{\\eta _1 \\xi _1 \\xi _2^2}{\\Delta ^2}\n\\end{array}\n\\right),\\]\n\n\\[\\left(\\begin{array}{c} -\\frac{\\beta _1 \\eta _1^2 \\xi _1}{\\Delta }+\\frac{2 \\alpha _1 \\eta _1 \\eta _2 \\xi _1}{\\Delta }-\\frac{\\beta _4 \\eta _2^2 \\xi _1}{\\Delta }-\\frac{\\eta _1 \\eta _2^2 \\xi _1}{\\Delta ^2}+\\frac{\\eta _2^2 \\xi _1^2}{\\Delta ^2}+\\frac{\\alpha _1 \\eta _1^2 \\xi _2}{\\Delta }-\\frac{2 \\beta _4 \\eta _1 \\eta _2 \\xi _2}{\\Delta }-\\frac{\\eta _1^2 \\eta _2 \\xi _2}{\\Delta ^2}+\\frac{\\alpha _4 \\eta _2^2 \\xi _2}{\\Delta }+\\frac{\\eta _1^2 \\xi _2^2}{\\Delta ^2}\\\\\n\\frac{\\beta _1 \\eta _1 \\xi _1^2}{\\Delta }-\\frac{\\alpha _1 \\eta _2 \\xi _1^2}{\\Delta }+\\frac{\\eta _2^2 \\xi _1^2}{\\Delta ^2}-\\frac{2 \\alpha _1 \\eta _1 \\xi _1 \\xi _2}{\\Delta }+\\frac{2 \\beta _4 \\eta _2 \\xi _1 \\xi _2}{\\Delta }-\\frac{\\eta _2 \\xi _1^2 \\xi _2}{\\Delta ^2}+\\frac{\\beta _4 \\eta _1 \\xi _2^2}{\\Delta }+\\frac{\\eta _1^2 \\xi _2^2}{\\Delta ^2}-\\frac{\\alpha _4 \\eta _2 \\xi _2^2}{\\Delta }-\\frac{\\eta _1 \\xi _1 \\xi _2^2}{\\Delta ^2}\\end{array}\n\\right),\\]\n\n\n\\[\\left(\\begin{array}{c} -\\frac{\\beta _1 \\eta _1^3}{\\Delta }+\\frac{3 \\alpha _1 \\eta _1^2 \\eta _2}{\\Delta }-\\frac{3 \\beta _4 \\eta _1 \\eta _2^2}{\\Delta }-\\frac{2 \\eta _1^2 \\eta _2^2}{\\Delta ^2}+\\frac{\\alpha _4 \\eta _2^3}{\\Delta }+\\frac{\\eta _1 \\eta _2^2 \\xi _1}{\\Delta ^2}+\\frac{\\eta _1^2 \\eta _2 \\xi _2}{\\Delta ^2}\\\\\n\\frac{\\beta _1 \\eta _1^2 \\xi _1}{\\Delta }-\\frac{2 \\alpha _1 \\eta _1 \\eta _2 \\xi _1}{\\Delta }+\\frac{\\beta _4 \\eta _2^2 \\xi _1}{\\Delta }+\\frac{\\eta _1 \\eta _2^2 \\xi _1}{\\Delta ^2}-\\frac{\\alpha _1 \\eta _1^2 \\xi _2}{\\Delta }+\\frac{2 \\beta _4 \\eta _1 \\eta _2 \\xi _2}{\\Delta }+\\frac{\\eta _1^2 \\eta _2 \\xi _2}{\\Delta ^2}-\\frac{\\alpha _4 \\eta _2^2 \\xi _2}{\\Delta }-\\frac{2 \\eta _1 \\eta _2 \\xi _1 \\xi _2}{\\Delta ^2}\\end{array}\n\\right).\\]\n\nFrom the above it can be easily seen that $\\alpha'_1=-\\beta'_2$, $\\alpha'_3=\\alpha'_2+1$, $\\beta'_3=\\beta'_2+1$ and $\\beta'_4=-\\alpha'_2,$ i.e.\n\\[A'=\\left(\n\\begin{array}{cccc}\n\\alpha'_1 & \\alpha'_2 &\\alpha'_2+1 & \\alpha'_4 \\\\\n\\beta'_1 & -\\alpha'_1 & -\\alpha'_1+1 & -\\alpha'_2\n\\end{array}\\right).\\]\nTherefore, the main role in the finding of $A'$ is played by the functions\n\n$\\alpha'_1=-\\frac{\\beta _1 \\eta _1 \\xi _1^2}{\\Delta }+\\frac{\\alpha _1 \\eta _2 \\xi _1^2}{\\Delta }+\\frac{2 \\alpha _1 \\eta _1 \\xi _1 \\xi _2}{\\Delta }-\\frac{2 \\beta _4 \\eta _2 \\xi _1 \\xi _2}{\\Delta }-\\frac{2 \\eta _1 \\eta _2 \\xi _1 \\xi _2}{\\Delta ^2}+\\frac{\\eta _2 \\xi _1^2 \\xi _2}{\\Delta ^2}-\\frac{\\beta _4 \\eta _1 \\xi _2^2}{\\Delta }+\\frac{\\alpha _4 \\eta _2 \\xi _2^2}{\\Delta }+\\frac{\\eta _1 \\xi _1 \\xi _2^2}{\\Delta ^2},$\n\n$\\beta'_1=\\frac{\\beta _1 \\xi _1^3}{\\Delta }-\\frac{3 \\alpha _1 \\xi _1^2 \\xi _2}{\\Delta }+\\frac{\\eta _2 \\xi _1^2 \\xi _2}{\\Delta ^2}+\\frac{3 \\beta _4 \\xi _1 \\xi _2^2}{\\Delta }+\\frac{\\eta _1 \\xi _1 \\xi _2^2}{\\Delta ^2}-\\frac{2 \\xi _1^2 \\xi _2^2}{\\Delta ^2}-\\frac{\\alpha _4 \\xi _2^3}{\\Delta }\n,$\n\n$\\alpha'_2=-\\frac{\\beta _1 \\eta _1^2 \\xi _1}{\\Delta }+\\frac{2 \\alpha _1 \\eta _1 \\eta _2 \\xi _1}{\\Delta }-\\frac{\\beta _4 \\eta _2^2 \\xi _1}{\\Delta }-\\frac{\\eta _1 \\eta _2^2 \\xi _1}{\\Delta ^2}+\\frac{\\alpha _1 \\eta _1^2 \\xi _2}{\\Delta }-\\frac{2 \\beta _4 \\eta _1 \\eta _2 \\xi _2}{\\Delta }-\\frac{\\eta _1^2 \\eta _2 \\xi _2}{\\Delta ^2}+\\frac{\\alpha _4 \\eta _2^2 \\xi _2}{\\Delta }+\\frac{2 \\eta _1 \\eta _2 \\xi _1 \\xi _2}{\\Delta ^2},$\n\n$\\alpha'_4=-\\frac{\\beta _1 \\eta _1^3}{\\Delta }+\\frac{3 \\alpha _1 \\eta _1^2 \\eta _2}{\\Delta }-\\frac{3 \\beta _4 \\eta _1 \\eta _2^2}{\\Delta }-\\frac{2 \\eta _1^2 \\eta _2^2}{\\Delta ^2}+\\frac{\\alpha _4 \\eta _2^3}{\\Delta }+\\frac{\\eta _1 \\eta _2^2 \\xi _1}{\\Delta ^2}+\\frac{\\eta _1^2 \\eta _2 \\xi _2}{\\Delta ^2}.$\n\nIt can be easily verified that these functions can take any values in $\\mathbf{F}.$ Therefore, the values of $\\alpha'_1,\\beta'_1, \\alpha'_2, \\alpha'_4$ also can be any elements in $\\mathbf{F}$. Using obvious redenotions one can list the following non-isomorphic ``canonical'' algebras from the first subset, given by their MSCs as follows:\n\\[ A_{1}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & \\alpha_2 &\\alpha_2+1 & \\alpha_4 \\\\\n\\beta_1 & -\\alpha_1 & -\\alpha_1+1 & -\\alpha_2\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\alpha_2, \\alpha_4, \\beta_1) \\in \\mathbf{F}^4.\\]\n\nFor any algebra $\\mathbf{A}$ from the first subset there exists a unique algebra from  the class $A_{1}(\\mathbf{c})$ isomorphic to $\\mathbf{A}$. Note that the appearance of $A_{1}(\\mathbf{c})$ for the first subset does not depend on either algebraically closeness of $\\mathbf{F}$ or its characteristic.\n\n{\\bf The second and third subset cases.} In these cases one can make $Tr_1(A)g=(1,0)$. It implies that $Tr_2(A)g=(\\lambda,0)$. Therefore, we can assume that\n\\[Tr_1(A)=(\\alpha _1+\\beta _3,\\alpha _2+\\beta _4)=(1,0)\\ \\mbox{and}\\ Tr_2(A)=(\\lambda(\\alpha _1+\\beta _3),\\lambda(\\alpha _2+\\beta _4))=(\\lambda,0).\\]\nHere if $\\lambda$ is zero we can also cover the third subset. So let us consider\n\n\\[A=\\left(\n\\begin{array}{cccc}\n\\alpha _1 & \\alpha _2 & \\lambda  \\alpha _2+(\\lambda -1)\\beta _4 & \\alpha _4 \\\\\n\\beta _1 & (\\lambda -1)\\alpha _1+\\lambda  \\beta _3 & \\beta _3 & \\beta _4\n\\end{array}\n\\right)=\\left(\n\\begin{array}{cccc}\n\\alpha _1 & \\alpha _2 & \\alpha _2 & \\alpha _4 \\\\\n\\beta _1 & \\lambda-\\alpha _1 & 1-\\alpha _1& -\\alpha _2\n\\end{array}\\right),\\] with respect to $g\\in GL(2,\\mathbf{F})$ of the form \\[g^{-1}=\\left(\\begin{array}{cc} 1& 0\\\\ \\xi_2& \\eta_2\\end{array}\\right),\\] as far as\n$(1,0)g^{-1}=(1,0)$ if and only if $g^{-1}$ is of the above form. In this case for the entries of $A'$ we have $A'=\\left(\n\\begin{array}{cccc}\n\\alpha'_1 & \\alpha'_2 & \\alpha'_2 & \\alpha'_4 \\\\\n\\beta'_1 & \\lambda-\\alpha'_1 & 1-\\alpha'_1& -\\alpha'_2\n\\end{array}\\right)=gA(g^{-1})^{\\otimes 2}$, where $\\lambda$ is same for the $A$ and $A'$,\none has\n\n$\\alpha '_1=\\frac{1}{\\Delta }(\\alpha _1 \\eta _2 +2\\alpha _2\\eta _2\\xi _2+\\alpha _4 \\eta _2 \\xi _2^2)=\\alpha _1+2\\alpha _2\\xi _2+\\alpha _4\\xi _2^2.$\n\n$\\alpha '_2=\\frac{-1}{\\Delta }(-\\alpha _2\\eta^2_2 -\\alpha _4 \\eta _2^2 \\xi _2)=(\\alpha _2+\\alpha _4\\xi _2)\\eta _2.$\n\n$\\alpha '_4=\\frac{-1}{\\Delta }(\\beta _1 \\eta _1^3+[(\\lambda-2)\\alpha _1+(1+\\lambda)\\beta _3]\\eta _1^2 \\eta _2-[(1+\\lambda)\\alpha _2+(\\lambda-2)\\beta _4]\\eta _1 \\eta _2^2-\\alpha _4 \\eta _2^3)=\\alpha _4 \\eta_2^2.$\n\n$\\beta '_1=\n\\frac{1}{\\Delta }(\\beta _1 \\xi _1^3+[(\\lambda-2)\\alpha _1 +(\\lambda+1)\\beta _3]\\xi _1^2 \\xi _2-[(1+\\lambda)\\alpha _2+(\\lambda-2)\\beta _4] \\xi _1 \\xi _2^2-\\alpha _4 \\xi _2^3)$\n\n\\qquad $=\\frac{\\beta _1+(1+\\lambda-3 \\alpha _1)\\xi _2-3 \\alpha _2 \\xi _2^2-\\alpha _4 \\xi _2^3}{\\eta_2}.$\n\nSo it is enough to consider the system\n\n$\\alpha '_1=\\alpha _1+2 \\alpha _2 \\xi _2+\\alpha _4 \\xi _2^2,$\n\n$\\alpha '_2=(\\alpha _2+\\alpha_4\\xi_2) \\eta _2,$\n\n$\\alpha '_4=\\alpha _4 \\eta _2^2,$\n\n$\\beta '_1=\\frac{\\beta _1+(1+\\lambda-3\\alpha _1)\\xi _2-3 \\alpha _2 \\xi _2^2-\\alpha _4 \\xi _2^3}{\\eta _2}.$\n\nNow we have to consider a few cases again.\n\nCase 1: $\\alpha_4\\neq 0$. In this case one can make $$\\alpha '_2=0, \\alpha '_4=1, \\alpha '_1=\\alpha_1-\\frac{\\alpha^2_2}{\\alpha_4}\\ \\mbox{and}\\ \\beta '_1=\\sqrt{\\alpha_4}\\left(\\beta _1-\\left(1+\\lambda-3\\alpha _1\\right)\\frac{\\alpha_2}{\\alpha_4}-2\\frac{\\alpha^3_2}{\\alpha^2_4}\\right)$$\nand\nonce again using redenotion one can represent the corresponding ``canonical'' MSCs as follows\n\\[ A_{2}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 1 \\\\\n\\beta _1& \\beta_2 & 1-\\alpha_1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_1, \\beta_2)\\in \\mathbf{F}^3,\\] as far as the expressions $\\alpha_1-\\frac{\\alpha^2_2}{\\alpha_4}$, $\\sqrt{\\alpha_4}(\\beta _1-(1+\\lambda-3\\alpha _1)\\frac{\\alpha_2}{\\alpha_4}-2\\frac{\\alpha^3_2}{\\alpha^2_4})$, $\\lambda-(\\alpha_1-\\frac{\\alpha^2_2}{\\alpha_4} ) $ may have any values in $\\mathbf{F}$. It can be checked that algebras $\\left(\n\\begin{array}{cccc}\n\t\\alpha_1 & 0 & 0 & 1 \\\\\n\t\\beta _1& \\beta_2 & 1-\\alpha_1&0\n\\end{array}\\right)$ and $\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 1 \\\\\n-\\beta _1& \\beta_2 & 1-\\alpha_1&0\n\\end{array}\\right)$ are isomorphic.\nNote that the result $A_{2}(\\mathbf{c})$, does not depend on the characteristic of $\\mathbf{F}$.\n\nCase 2: $\\alpha _4=0.$\n\nSubcase 2 - a): $\\alpha _2\\neq 0.$  If $\\alpha _2\\neq 0$ then one can make\n$\\alpha '_1=0$, $\\alpha '_2=1$, $\\beta '_1=\\alpha_2\\beta _1-\\frac{2+2\\lambda-3\\alpha _1}{4}\\alpha_1$ to get the following set of canonical matrices of structural constants\n\\[ A_{3}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n0 & 1 & 1 & 0 \\\\\n\\beta _1& \\beta_2 & 1&-1\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\beta_1, \\beta_2)\\in \\mathbf{F}^2.\\]\n\nSubcase 2 - b): $\\alpha _2= 0.$  If $\\alpha _2=0$ then $\\alpha '_1=\\alpha _1,$ $\\alpha '_2=0,$ $\\alpha '_4=0,$ $\\beta '_1=\\frac{\\beta _1+(1+\\lambda-3\\alpha _1)\\xi _2}{\\eta _2}.$\n\n\\underline{Subsubcase: 2 - b) - 1: $1+\\lambda-3\\alpha _1\\neq 0.$} If $1+\\lambda-3\\alpha _1\\neq 0$, that is $\\lambda-\\alpha _1\\neq 2\\alpha_1-1$,  one can make $\\beta'_1=0$ to get\n\\[A_{4}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha _1 & 0 & 0 & 0 \\\\\n0 & \\beta_2 & 1-\\alpha _1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_2) \\in \\mathbf{F}^2, \\ \\mbox{with}\\  \\beta_2\\neq 2\\alpha_1-1.\\]\n\n\\underline{Subsubcase: 2 - b) - 2: $1+\\lambda-3\\alpha _1= 0.$} If $1+\\lambda-3\\alpha _1=0$ and $\\beta_1\\neq 0$ one can make $\\beta'_1=1$ to get\n\\[A_{5}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha _1 & 0 & 0 & 0 \\\\\n1 & 2\\alpha _1-1 & 1-\\alpha _1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=\\alpha_1\\in F.\\] If $\\beta_1=0$ one has $\\lambda-\\alpha _1= 2\\alpha_1-1$ and therefore\n$A'=\\left(\n\\begin{array}{cccc}\n\\alpha _1 & 0 & 0 & 0 \\\\\n0 & 2\\alpha _1-1 & 1-\\alpha _1&0\n\\end{array}\\right)$\\\\ which is $A_{4}(\\mathbf{c})$ with $\\beta_2=2\\alpha_1-1$.\n\n{\\bf The fourth subset case.}\nBy the similar justification as in the second and the third subsets case it is enough to consider  \\[ A=\\left(\n\\begin{array}{cccc}\n\\alpha _1 & \\alpha _2 & \\alpha _2 & \\alpha _4 \\\\\n\\beta _1 & 1-\\alpha _1 & -\\alpha _1 & -\\alpha _2\n\\end{array}\n\\right)\\ \\mbox{and}\\ \\ g^{-1}=\\left(\\begin{array}{cc} 1& 0\\\\ \\xi_2& \\eta_2\\end{array}\\right),\\\n\\mbox{where one has}\\]\n\n$\\alpha'_1= \\alpha _1+2 \\alpha _2 \\xi _2+\\alpha _4 \\xi _2^2 ,$\n\n$\\alpha'_2= (\\alpha _2+\\alpha _4\\xi _2)\\eta_2,$\n\n$\\alpha'_4=\\alpha _4 \\eta _2^2,$\n\n$\\beta'_1=\\frac{\\beta _1+\\xi _2-3 \\alpha _1 \\xi _2-3 \\alpha _2 \\xi _2^2-\\alpha _4 \\xi _2^3}{\\eta _2}.$\n\nTherefore we get the canonical MSCs as follows\n\\[A_{6}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 1 \\\\\n\\beta _1& 1-\\alpha_1 & -\\alpha_1&0\n\\end{array}\\right)\\simeq \\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 1 \\\\\n-\\beta _1& 1-\\alpha_1 & -\\alpha_1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_1) \\in \\mathbf{F}^2,\\]\n(it is $A_{2}(\\mathbf{c})$, where the $2^{nd}$ and $3^{rd}$ columns are interchanged and $\\alpha_1+\\beta_2=0$) or\n\\[ A_{7}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n0 & 1 & 1 & 0 \\\\\n\\beta _1& 1& 0&-1\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=\\beta_1\\in \\mathbf{F},\\]\n(it is $A_{3}(\\mathbf{\\mathbf{c}})$, where the $2^{nd}$ and $3^{rd}$ columns are interchanged and $\\beta_2=0$) or\n\\[A_{8}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha _1 & 0 & 0 & 0 \\\\\n0 & 1-\\alpha _1 & -\\alpha _1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=\\alpha_1\\in \\mathbf{F},\\]\n(it is $A_{4}(\\mathbf{c})$, where the $2^{nd}$ and $3^{rd}$ columns are interchanged and $\\alpha_1+\\beta_2=0$) or\n\\[A_{9}=\\left(\n\\begin{array}{cccc}\n\\frac{1}{3} & 0 & 0 & 0 \\\\\n1 & \\frac{2}{3} & -\\frac{1}{3} &0\n\\end{array}\\right)\\] (it is $A_{5}(\\mathbf{c})$, where the $2^{nd}$ and $3^{rd}$ columns are interchanged and $3\\alpha_1-1=0$).\n\n{\\bf The fifth subset case.} In this case\n\\[A=\\left(\n\\begin{array}{cccc}\n\\alpha _1 & \\alpha _2 & \\alpha _2 & \\alpha _4 \\\\\n\\beta _1 & -\\alpha _1 & -\\alpha _1 & -\\alpha _2\n\\end{array}\n\\right),\\ g^{-1}=\\left(\\begin{array}{cc} \\xi_1& \\eta_1\\\\ \\xi_2& \\eta_2\\end{array}\\right)\\mbox{and}\\\nA'=\\left(\n\\begin{array}{cccc}\n\\alpha' _1 & \\alpha' _2 & \\alpha' _2 & \\alpha' _4 \\\\\n\\beta' _1 & -\\alpha' _1 & -\\alpha' _1 & -\\alpha' _2\n\\end{array}\n\\right)\\]\nsuch that:\n\n$\\alpha' _1=\\frac{1}{\\Delta }\\left(-\\beta _1 \\eta _1 \\xi _1^2+\\alpha _1 \\eta _2 \\xi _1^2+2 \\alpha _1 \\eta _1 \\xi _1 \\xi _2+2 \\alpha _2 \\eta _2 \\xi _1 \\xi _2+\\alpha _2 \\eta _1 \\xi _2^2+\\alpha _4 \\eta _2 \\xi _2^2\\right),$\n\n$\\alpha' _2=\\frac{-1}{\\Delta }\\left(\\beta _1 \\eta _1^2 \\xi _1-2 \\alpha _1 \\eta _1 \\eta _2 \\xi _1-\\alpha _2 \\eta _2^2 \\xi _1-\\alpha _1 \\eta _1^2 \\xi _2-2 \\alpha _2 \\eta _1 \\eta _2 \\xi _2-\\alpha _4 \\eta _2^2 \\xi _2\\right),$\n\n$\\alpha' _4=\\frac{-1}{\\Delta }\\left(\\beta _1 \\eta _1^3-3 \\alpha _1 \\eta _1^2 \\eta _2-3 \\alpha _2 \\eta _1 \\eta _2^2-\\alpha _4 \\eta _2^3\\right),$\n\n$\\beta' _1=\\frac{1}{\\Delta }\\left(\\beta _1 \\xi _1^3-3 \\alpha _1 \\xi _1^2 \\xi _2-3 \\alpha _2 \\xi _1 \\xi _2^2-\\alpha _4 \\xi _2^3\\right).$\n\nIf $\\alpha _4\\neq 0$ by making $\\frac{\\eta_2}{\\eta_1}$ equal to any root of the polynomial $p(t)=\\beta _1-3 \\alpha _1t-3 \\alpha _2t^2-\\alpha _4t^3$ one can make $\\alpha'_4=0$. Therefore, further it is assumed that $\\alpha _4= 0$.\n\nLet us consider $g$ with $\\eta _1 = 0$ to have $\\alpha'_4=0.$ In this case $\\Delta=\\xi_1\\eta_2$ and \\\\\n$\\alpha' _1=\\xi _1\\left(\\alpha _1+2\\alpha _2\\frac{\\xi_2}{\\xi_1}\\right),$\\\\\n$\\alpha' _2=\\alpha _2\\eta_2,$\\\\\n$\\beta' _1=\\frac{ \\xi _1^2}{\\eta_2 }\\left(\\beta _1-3 \\alpha _1 \\frac{\\xi_2}{\\xi_1}-3 \\alpha _2(\\frac{\\xi_2}{\\xi_1})^2\\right).$\\\\\nCase a: $\\alpha_2\\neq 0$. One can consider $\\frac{\\xi_2}{\\xi_1}=\\frac{-\\alpha _1}{2\\alpha _2}$ to get $\\alpha' _1=0, \\alpha' _2=1$ and $\\beta' _1=\\xi^2_1\\frac{3\\alpha^2 _1+4\\alpha _2\\beta _1}{4}.$ Therefore one can make $\\beta' _1$ equal to $0$ or $1,$ depending on $3\\alpha^2 _1+4\\alpha _2\\beta _1$ to have\n\\[A_{10}=\\left(\n\\begin{array}{cccc}\n0 & 1 & 1 & 0 \\\\\n0 &0& 0 &-1\n\\end{array}\n\\right)\\ \\mbox{or}\\ A_{11}=\\left(\n\\begin{array}{cccc}\n0 & 1 & 1 & 0 \\\\\n1 &0& 0 &-1\n\\end{array}\n\\right).\\]\nCase b: $\\alpha_2=0$. Then $\\alpha'_2=\\alpha'_4=0$ and $\\alpha'_1=\\xi_1\\alpha_1, \\beta' _1=\\frac{\\xi^2_1}{\\eta_2}\\left(\\beta _1-3 \\alpha _1 \\frac{\\xi_2}{\\xi_1}\\right).$ Therefore if $\\alpha_1\\neq 0$ one can make $\\alpha'_1=1, \\beta' _1=0$ to get $A'=\\left(\n\\begin{array}{cccc}\n1 & 0 & 0 & 0 \\\\\n0 &-1& -1 &0\n\\end{array}\n\\right)$ , which isomorphic to $A_{10},$  if $\\alpha_1= 0$ then $\\alpha'_1=0$ and one can make $\\beta' _1=1$ to come to\n\\[A_{12}=\\left(\n\\begin{array}{cccc}\n0 & 0 & 0 & 0 \\\\\n1 &0&0 &0\\end{array}\n\\right)\\]\nA routine check in each subset case shows that the corresponding algebras presented above\nare not isomorphic. \n\n\\subsection{Characteristic $2$ case}\nIn this case the result can be summarized in the following form.\n\n{\\bf Theorem 2.}\\textit{ Over an algebraically closed field $\\mathbf{F}$ characteristic 2 any non-trivial 2-dimensional algebra is isomorphic to only one of the following listed, by their matrices of structure constants,  algebras:\n\\[A_{1,2}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & \\alpha_2 &\\alpha_2+1 & \\alpha_4 \\\\\n\\beta_1 & -\\alpha_1 & -\\alpha_1+1 & -\\alpha_2\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\alpha_2, \\alpha_4, \\beta_1)\\in \\mathbf{F}^4,\\]\n\\[A_{2,2}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 1 \\\\\n\\beta _1& \\beta_2 & 1-\\alpha_1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_1, \\beta_2)\\in \\mathbf{F}^3,\\]\n\\[A_{3,2}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 1 & 1 & 0 \\\\\n0& \\beta_2 & 1-\\alpha_1&1\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_2)\\in \\mathbf{F}^2,\\]\n\\[A_{4,2}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha _1 & 0 & 0 & 0 \\\\\n0 & \\beta_2 & 1-\\alpha _1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1,\\beta_2)\\in \\mathbf{F}^2,\\]\n\\[A_{5,2}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 0 \\\\\n1 & 1 & 1-\\alpha_1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=\\alpha_1\\in \\mathbf{F},\\]\n\\[A_{6,2}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 1 \\\\\n\\beta _1& 1-\\alpha_1 & -\\alpha_1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_1)\\in \\mathbf{F}^2,\\]\n\\[A_{7,2}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 1 & 1 & 0 \\\\\n0& 1-\\alpha_1& -\\alpha_1&-1\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=\\alpha_1\\in \\mathbf{F},\\]\n\\[A_{8,2}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 0 \\\\\n0 & 1-\\alpha_1 & -\\alpha_1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=\\alpha_1\\in \\mathbf{F},\\]\n\\[A_{9,2}=\\left(\n\\begin{array}{cccc}\n1 & 0 & 0 & 0 \\\\\n1 & 0 & 1&0\n\\end{array}\\right),\\ \\ A_{10,2}=\\left(\n\\begin{array}{cccc}\n0 & 1 & 1 & 0 \\\\\n0 &0& 0 &-1\n\\end{array}\n\\right),\\]\n\\[ A_{11,2}=\\left(\n\\begin{array}{cccc}\n1 & 1 & 1 & 0 \\\\\n0 &-1& -1 &-1\n\\end{array}\n\\right),\\\n\\ A_{12,2}=\\left(\n\\begin{array}{cccc}\n0 & 0 & 0 & 0 \\\\\n1 &0&0 &0\\end{array}\n\\right).\\]}\n\n{\\bf Proof.}\nIn this case $\\alpha=-\\alpha$ for any $\\alpha\\in \\mathbf{F}$, but sometimes we use $-\\alpha$ also to keep similarity with the previous case.\n\n{\\bf The first subset case.} It has been noted earlier that in this case the result doesn't depend either on algebraically closeness of $\\mathbf{F}$ or its characteristics. Therefore, one has \\[A_{1,2}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & \\alpha_2 &\\alpha_2+1 & \\alpha_4 \\\\\n\\beta_1 & -\\alpha_1 & -\\alpha_1+1 & -\\alpha_2\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\alpha_2, \\alpha_4, \\beta_1)\\in \\mathbf{F}^4.\\]\n\n{\\bf The second and third subset cases.}\nObviously in this case one can make $Tr_1(A)g=(1,0)$. This implies $Tr_2(A)g=(\\lambda,0)$. Therefore, it can be assumed that  $Tr_1(A)=(\\alpha _1+\\beta _3,\\alpha _2+\\beta _4)=(1,0)$ and $Tr_2(A)=( \\lambda(\\alpha _1+\\beta _3),\\lambda(\\alpha _2+\\beta _4)=(\\lambda,0)$. Here also we allow $\\lambda$ to be zero to include the third subset's case. Therefore, we consider\n\\[A=\\left(\n\\begin{array}{cccc}\n\\alpha _1 & \\alpha _2 & \\alpha_2 & \\alpha _4 \\\\\n\\beta _1 & \\lambda-\\alpha _1 & 1-\\alpha _1& -\\alpha _2\n\\end{array}\\right)\\ \\mbox{with respect to}\\ g^{-1}=\\left(\\begin{array}{cc} 1& 0\\\\ \\xi_2& \\eta_2\\end{array}\\right) \\ \\mbox{and}\\]\n\n$\\alpha '_1=\\alpha _1+\\alpha _4\\xi _2^2,$\n\n$\\alpha '_2=(\\alpha _2+\\alpha _4\\xi _2)\\eta _2,$\n\n$\\alpha '_4=\\alpha _4 \\eta_2^2,$\n\n$\\beta '_1=\\frac{\\beta _1+(1+\\lambda-3 \\alpha _1)\\xi _2-3 \\alpha _2 \\xi _2^2-\\alpha _4 \\xi _2^3}{\\eta_2}=\\frac{\\beta _1+(1+\\lambda+\\alpha _1)\\xi _2+ \\alpha _2 \\xi _2^2+\\alpha _4 \\xi _2^3}{\\eta_2}.$\n\\begin{itemize}\n\t\\item[1.] As we have observed earlier in $\\alpha_4\\neq 0$ case one gets\n\t\\[A_{2,2}(\\mathbf{c})=\\left(\n\t\\begin{array}{cccc}\n\t\\alpha_1 & 0 & 0 & 1 \\\\\n\t\\beta _1& \\beta_2 & 1-\\alpha_1&0\n\t\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_1, \\beta_2)\\in \\mathbf{F}^3.\\]\n\t\\item[2.] Let now assume that $\\alpha _4=0.$\n\t\\begin{itemize}\n\t\t\\item[2-a).] If $\\alpha _2\\neq0$ then one can make\n\t\t$\\alpha '_1=\\alpha_1$, $\\alpha '_2=1$, $\\alpha'_4=0$ and $\\beta'_1=0$ to get \\[A_{3,2}(\\mathbf{c})=\\left(\n\t\t\\begin{array}{cccc}\n\t\t\\alpha_1 & 1 & 1 & 0 \\\\\n\t\t0& \\beta_2 & 1-\\alpha_1&-1\n\t\t\\end{array}\\right), \\ \\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_2)\\in \\mathbf{F}^2.\\]\n\t\t\n\t\t\\item[2-b).] If $\\alpha _2=0$ then $\\alpha '_1=\\alpha _1,$ $\\alpha '_2=0,$ $\\alpha '_4=0,$ and $\\beta '_1=\\frac{\\beta _1+(1+\\lambda+\\alpha _1)\\xi _2}{\\eta _2}.$\n\t\t\\begin{itemize}\n\t\t\t\\item[2-b)-1.] If $1+\\lambda+\\alpha _1\\neq 0$, that is $\\lambda-\\alpha _1\\neq 1$, one can make $\\beta'_1=0$ to get\n\t\t\t\\[A_{4,2}(\\mathbf{c})=\\left(\n\t\t\t\\begin{array}{cccc}\n\t\t\t\\alpha _1 & 0 & 0 & 0 \\\\\n\t\t\t0 & \\beta_2 & 1-\\alpha _1&0\n\t\t\t\\end{array}\\right),\\  \\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_2)\\in \\mathbf{F}\\ \\mbox{with}\\ \\beta_2\\neq 1.\\]\n\t\t\t\n\t\t\t\\item[2-b)-2.] Let $1+\\lambda+\\alpha _1=0$. In this case\n\t\t\t\\begin{itemize}\n\t\t\t\t\\item if $\\beta_1\\neq 0$ one can make $\\beta'_1=1$ to have\n\t\t\t\t\\[A_{5,2}(\\mathbf{c})=\\left(\n\t\t\t\t\\begin{array}{cccc}\n\t\t\t\t\\alpha _1 & 0 & 0 & 0 \\\\\n\t\t\t\t1 & 1 & 1-\\alpha _1&0\n\t\t\t\t\\end{array}\\right), \\ \\mbox{where}\\ \\mathbf{c}= \\alpha _1 \\in \\mathbf{F}.\\]\n\t\t\t\t\\item if $\\beta_1=0$ then\n\t\t\t\t\\[A'=\\left(\n\t\t\t\t\\begin{array}{cccc}\n\t\t\t\t\\alpha _1 & 0 & 0 & 0 \\\\\n\t\t\t\t0 & 1 & 1-\\alpha _1&0\n\t\t\t\t\\end{array}\\right), \\ \\mbox{which is}\\ A_{4,2}(\\mathbf{c}) \\ \\mbox{with}\\  \\beta_2=1.\\]\n\t\t\t\\end{itemize}\n\t\t\\end{itemize}\n\t\\end{itemize}\n\\end{itemize}\n{\\bf The fourth subset case.} For this case we have \\[ A=\\left(\n\\begin{array}{cccc}\n\\alpha _1 & \\alpha _2 & \\alpha _2 & \\alpha _4 \\\\\n\\beta _1 & 1-\\alpha _1 & -\\alpha _1 & -\\alpha _2\n\\end{array}\n\\right)\\ \\mbox{and}\\ g^{-1}=\\left(\\begin{array}{cc} 1& 0\\\\ \\xi_2& \\eta_2\\end{array}\\right)\\]\ntherefore\n\n$\\alpha'_1= \\alpha _1+\\alpha _4 \\xi _2^2 ,$\n\n$\\alpha'_2= (\\alpha _2+\\alpha _4\\xi _2)\\eta_2,$\n\n$\\alpha'_4=\\alpha _4 \\eta _2^2,$\n\n$\\beta'_1=\\frac{\\beta _1+(1+\\alpha _1)\\xi _2+\\alpha _2 \\xi _2^2+\\alpha _4 \\xi _2^3}{\\eta _2}.$\n\nHence, one gets\n$A_{6,2}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 1 \\\\\n\\beta _1& 1-\\alpha_1 & -\\alpha_1&0\n\\end{array}\\right) \\ \\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_1)\\in \\mathbf{F}^2$\\\\ (it is $A_{2,2}(\\mathbf{c})$, where the $2^{nd}$ and $3^{rd}$ columns are interchanged and $\\alpha_1+\\beta_2=0$),\n\\[A_{7,2}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 1 & 1 & 0 \\\\\n0&1-\\alpha_1&-\\alpha_1&-1\n\\end{array}\\right),\\ \\ \\mbox{where}\\ \\mathbf{c}=\\alpha_1\\in \\mathbf{F}\\] (it is $A_{3,2}(\\mathbf{c})$, where the $2^{nd}$ and $3^{rd}$ columns are interchanged and $\\alpha_1+\\beta_2=0$),\n\\[A_{8,2}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 0 \\\\\n0 & 1-\\alpha_1 & -\\alpha_1&0\n\\end{array}\\right)\\ \\ \\mbox{where}\\ \\mathbf{c}=\\alpha_1\\in \\mathbf{F}\\] (it is $A_{4,2}(\\mathbf{c})$, where the $2^{nd}$ and $3^{rd}$ columns are interchanged and $\\alpha_1+\\beta_2=0$)\nand\n\\[A_{9,2}=\\left(\n\\begin{array}{cccc}\n1 & 0 & 0 & 0 \\\\\n1 &0 & 1&0\n\\end{array}\\right)\\] (it is $A_{5,2}(\\mathbf{c})$, where the $2^{nd}$ and $3^{rd}$ columns are interchanged and $\\alpha_1+1=0$).\n\n{\\bf The fifth subset case.} In this case\n\\[A=\\left(\n\\begin{array}{cccc}\n\\alpha _1 & \\alpha _2 & \\alpha _2 & \\alpha _4 \\\\\n\\beta _1 & -\\alpha _1 & -\\alpha _1 & -\\alpha _2\n\\end{array}\\right),\\ \\ g^{-1}=\\left(\\begin{array}{cc} \\xi_1& \\eta_1\\\\ \\xi_2& \\eta_2\\end{array}\\right)\\]\nand for the entries of $A'=\\left(\n\\begin{array}{cccc}\n\\alpha' _1 & \\alpha' _2 & \\alpha' _2 & \\alpha' _4 \\\\\n\\beta' _1 & -\\alpha' _1 & -\\alpha' _1 & -\\alpha' _2\n\\end{array}\n\\right)$ one has\n\n$\\alpha' _1=\\frac{1}{\\Delta }\\left(\\beta _1 \\eta _1 \\xi _1^2+\\alpha _1 \\eta _2 \\xi _1^2+ \\alpha_2 \\eta _1\\xi^2_2+\\alpha _4 \\eta _2 \\xi _2^2\\right),$\n\n$\\alpha' _2=\\frac{1}{\\Delta }\\left(\\beta _1 \\eta _1^2 \\xi _1+\\alpha _1 \\eta _1^2 \\xi _2+\\alpha _2 \\eta _2^2 \\xi _1+\\alpha _4 \\eta _2^2 \\xi _2\\right),$\n\n$\\alpha' _4=\\frac{1}{\\Delta }\\left(\\beta _1 \\eta _1^3+ \\alpha _1 \\eta _1^2 \\eta _2+\\alpha _2 \\eta _1 \\eta _2^2+\\alpha _4 \\eta _2^3\\right),$\n\n$\\beta' _1=\\frac{1}{\\Delta }\\left(\\beta _1 \\xi _1^3+\\alpha _1 \\xi _1^2 \\xi _2+\\alpha _2 \\xi _1 \\xi _2^2+\\alpha _4 \\xi _2^3\\right).$\n\nAs it was observed one can assume that $\\alpha_4=0$.\n\nLet us consider $g$ with $\\eta _1 = 0$ to have $\\alpha'_4=0.$ In this case $\\Delta=\\xi_1\\eta_2$ and \\\\\n$\\alpha' _1=\\xi _1\\alpha _1,$\\\\\n$\\alpha' _2=\\alpha _2\\eta_2,$\\\\\n$\\beta' _1=\\frac{ \\xi _1^2}{\\eta_2 }\\left(\\beta _1+ \\alpha _1 \\frac{\\xi_2}{\\xi_1}+ \\alpha _2(\\frac{\\xi_2}{\\xi_1})^2\\right).$\\\\\nIf one of $\\alpha _1, \\alpha _2$ is not zero one can make $\\beta' _1=0$ and depending on $\\alpha _1, \\alpha _2$ one can have $\\alpha' _1=0, \\alpha' _2=1$ or $\\alpha' _1=1, \\alpha' _2=1$ or $\\alpha' _1=1, \\alpha' _2=0$ to get\n$A_{10,2}=\\left(\n\\begin{array}{cccc}\n0 & 1 & 1 & 0 \\\\\n0 &0& 0 &-1\n\\end{array}\n\\right)$ or $A_{11,2}=\\left(\n\\begin{array}{cccc}\n1 & 1 & 1 & 0 \\\\\n0 &-1& -1 &-1\n\\end{array}\n\\right)$ or $A'=\\left(\n\\begin{array}{cccc}\n1 & 0 & 0 & 0 \\\\\n0 &-1& -1 &0\n\\end{array}\n\\right)$ which is isomorphic to $A_{10,2}$\\\\\nIf $\\alpha _1=\\alpha _2=0$ then one can make $\\beta' _1=1$ to get\n\\[A_{12,2}=\\left(\n\\begin{array}{cccc}\n0 & 0 & 0 & 0 \\\\\n1 &0& 0 &0\n\\end{array}\n\\right).\\]\n\n\\subsection{Characteristic $3$ case}\nIn this case we summarize the result as follows.\n\n{\\bf Theorem 3.}\\textit{ Over an algebraically closed field $\\mathbf{F}$ characteristics 3 any non-trivial 2-dimensional algebra is isomorphic to only one of the following listed, by their matrices of structure constant matrices,  algebras:\n\\[A_{1,3}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & \\alpha_2 &\\alpha_2+1 & \\alpha_4 \\\\\n\\beta_1 & -\\alpha_1 & -\\alpha_1+1 & -\\alpha_2\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\alpha_2, \\alpha_4, \\beta_1)\\in \\mathbf{F}^4,\\]\n\\[A_{2,3}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 1 \\\\\n\\beta _1& \\beta _2& 1-\\alpha_1&0\n\\end{array}\\right)\\simeq\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 1 \\\\\n-\\beta _1& \\beta _2& 1-\\alpha_1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_1, \\beta_2)\\in \\mathbf{F}^3,\\]\n\\[A_{3,3}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n0 & 1 & 1 & 0 \\\\\n\\beta _1& \\beta _2 & 1&-1\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\beta_1, \\beta_2)\\in \\mathbf{F}^2,\\]\n\\[A_{4,3}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha _1 & 0 & 0 & 0 \\\\\n0 & \\beta _2& 1-\\alpha _1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_2)\\in \\mathbf{F}^2,\\]\n\\[A_{5,3}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1& 0 & 0 & 0 \\\\\n1 & -1-\\alpha_1 & 1-\\alpha_1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=\\alpha_1\\in \\mathbf{F},\\]\n\\[A_{6,3}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 1 \\\\\n\\beta _1& 1-\\alpha_1 & -\\alpha_1&0\n\\end{array}\\right)\\simeq \\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 1 \\\\\n-\\beta _1& 1-\\alpha_1 & -\\alpha_1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_1)\\in \\mathbf{F}^2,\\]\n\\[A_{7,3}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n0 & 1 & 1 & 0 \\\\\n\\beta_1& 1& 0&-1\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=\\beta_1\\in \\mathbf{F},\\]\n\\[A_{8,3}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 0 \\\\\n0 & 1-\\alpha_1 & -\\alpha_1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=\\alpha_1\\in \\mathbf{F},\\]\n\\[A_{9,3}=\\left(\n\\begin{array}{cccc}\n0 & 1& 1& 0 \\\\\n1 &0&0 &-1\\end{array}\n\\right),\\\n\\ A_{10,3}=\\left(\n\\begin{array}{cccc}\n0 & 1 & 1 & 0 \\\\\n0 &0&0 &-1\\end{array}\n\\right),\\]\n\\[ A_{11,3}=\\left(\n\\begin{array}{cccc}\n1 & 0 & 0 & 0 \\\\\n1 &-1&-1 &0\\end{array}\n\\right),\\ \\ A_{12,3}=\\left(\n\\begin{array}{cccc}\n0 &0 &0 & 0 \\\\\n1 &0& 0 &0\n\\end{array}\n\\right).\\]}\n\n{\\bf Proof.}\n\\quad {\\bf The first subset case.} One has\n$$A_{1,3}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & \\alpha_2 &\\alpha_2+1 & \\alpha_4 \\\\\n\\beta_1 & -\\alpha_1 & -\\alpha_1+1 & -\\alpha_2\n\\end{array}\\right), \\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\alpha_2, \\alpha_4, \\beta_1)\\in \\mathbf{F}^4.$$\n\n{\\bf The second and third subset cases.} One can consider\n\\[A=\\left(\n\\begin{array}{cccc}\n\\alpha _1 & \\alpha _2 & \\alpha _2 & \\alpha _4 \\\\\n\\beta _1 & \\lambda-\\alpha _1 & 1-\\alpha _1& -\\alpha _2\n\\end{array}\\right),\\] where $\\lambda$ may be zero as well, with respect to\n$g^{-1}=\\left(\\begin{array}{cc} 1& 0\\\\ \\xi_2& \\eta_2\\end{array}\\right)$ in that case for the entries of $A'$ we get\n\n$\\alpha '_1=\\alpha _1+2 \\alpha _2 \\xi _2+\\alpha _4 \\xi _2^2,$\n\n$\\alpha '_2=(\\alpha _2+\\alpha_4\\xi_2) \\eta _2,$\n\n$\\alpha '_4=\\alpha _4 \\eta _2^2,$\n\n$\\beta '_1=\\frac{\\beta _1+(1+\\lambda)\\xi _2-\\alpha _4 \\xi _2^3}{\\eta _2}.$\n\\begin{itemize}\n\t\\item[1.] $\\alpha_4\\neq 0$. In this case we obtain\n\t$$ A_{2,3}(\\mathbf{c})=\\left(\n\t\\begin{array}{cccc}\n\t\\alpha_1 & 0 & 0 & 1 \\\\\n\t\\beta _1& \\beta_2 & 1-\\alpha_1&0\n\t\\end{array}\\right)\\simeq \\left(\n\t\\begin{array}{cccc}\n\t\\alpha_1 & 0 & 0 & 1 \\\\\n\t-\\beta _1& \\beta_2 & 1-\\alpha_1&0\n\t\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1, \\beta_1, \\beta_2)\\in \\mathbf{F}^3.$$\n\t\n\t\\item[2.] $\\alpha _4=0.$\n\t\\begin{itemize}\n\t\t\\item[2-a)] If $\\alpha _2\\neq0$ one can make $\\alpha '_1=0$, $\\alpha'_2=1$ to get \\[ A_{3,3}(\\mathbf{c})=\\left(\n\t\t\\begin{array}{cccc}\n\t\t0 & 1 & 1 & 0 \\\\\n\t\t\\beta _1& \\beta_2 & 1&-1\n\t\t\\end{array}\\right)\\ \\mbox{with}\\ \\mathbf{c}=(\\beta_1,\\beta_2)\\in \\mathbf{F}^2.\\]\n\t\t\n\t\t\\item[2-b)] If $\\alpha _2=0$ then $\\alpha '_1=\\alpha _1,$ $\\alpha '_2=0,$ $\\alpha '_4=0,$ and $\\beta '_1=\\frac{\\beta _1+(1+\\lambda)\\xi _2}{\\eta _2}.$\n\t\t\\begin{itemize}\n\t\t\t\\item[2-b)-1.] Let $1+\\lambda\\neq 0$, that is $\\lambda\\neq -1$, one can make $\\beta'_1=0$ to get\n\t\t\t\\[A_{4,3}(\\mathbf{c})=\\left(\n\t\t\t\\begin{array}{cccc}\n\t\t\t\\alpha _1 & 0 & 0 & 0 \\\\\n\t\t\t0 & \\beta_2 & 1-\\alpha _1&0\n\t\t\t\\end{array}\\right), \\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1,\\beta_2)\\in \\mathbf{F}^2 \\ \\mbox{with}\\  \\beta_2\\neq -1-\\alpha_1.\\]\n\t\t\t\n\t\t\t\\item[2-b)-2.] Let $1+\\lambda=0.$\n\t\t\t\\begin{itemize}\n\t\t\t\t\\item If $\\beta_1\\neq 0$ one can make $\\beta'_1=1$ to get\n\t\t\t\t\\[A_{5,3}(\\mathbf{c})=\\left(\n\t\t\t\t\\begin{array}{cccc}\n\t\t\t\t\\alpha_1 & 0 & 0 & 0 \\\\\n\t\t\t\t1 & -1-\\alpha_1 & 1-\\alpha_1&0\n\t\t\t\t\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=\\alpha_1\\in \\mathbf{F}.\\]\n\t\t\t\t\\item If $\\beta_1=0$ one has $\\lambda=-1$ and therefore\n\t\t\t\t$A'=\\left(\n\t\t\t\t\\begin{array}{cccc}\n\t\t\t\t\\alpha _1 & 0 & 0 & 0 \\\\\n\t\t\t\t0 & -1-\\alpha _1 & 1-\\alpha _1&0\n\t\t\t\t\\end{array}\\right)$ which is $A_{4,3}(\\mathbf{c})$ with $\\beta_2=-1-\\alpha_1$.\n\t\t\t\\end{itemize}\n\t\t\\end{itemize}\n\t\\end{itemize}\n\\end{itemize}\n\n{\\bf The fourth subset case.} It is easy to see that in this case the result can be derived from the second and third subsets case. So we get\n\\[A_{6,3}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 1 \\\\\n\\beta _1& 1-\\alpha_1 & -\\alpha_1&0\n\\end{array}\\right)\\simeq \\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 1 \\\\\n-\\beta _1& 1-\\alpha_1 & -\\alpha_1&0\n\\end{array}\\right),\\ \\mbox{where}\\ \\mathbf{c}=(\\alpha_1,\\beta_1)\\in \\mathbf{F}^2\\]\\\\ (it is $A_{2,3}(\\mathbf{c})$, where the $2^{nd}$ and $3^{rd}$ columns are interchanged and $\\alpha_1+\\beta_2=0$),\n\\[ A_{7,3}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n0 & 1 & 1 & 0 \\\\\n\\beta_1& 1& 0&-1\n\\end{array}\\right), \\ \\mbox{where}\\ \\mathbf{c}=\\beta_1\\in \\mathbf{F}\\]\\\\ (it is $A_{3,3}(\\mathbf{c})$, where the $2^{nd}$ and $3^{rd}$ columns are interchanged and $\\beta_2=0$),\n\\[A_{8,3}(\\mathbf{c})=\\left(\n\\begin{array}{cccc}\n\\alpha_1 & 0 & 0 & 0 \\\\\n0 & 1-\\alpha_1 & -\\alpha_1&0\n\\end{array}\\right), \\mbox{where}\\ \\mathbf{c}=\\alpha_1\\in \\mathbf{F}\\]\\\\ (it is $A_{4,3}(\\mathbf{c})$, where the $2^{nd}$ and $3^{rd}$ columns are interchanged and $\\alpha_1+\\beta_2=0$).\n\n{\\bf The fifth subset case.}\n\nHere it is enough to consider $\\alpha_4=0$ case. Therefore we have\n\n$\\alpha' _1=\\frac{1}{\\Delta }\\left(-\\beta _1 \\eta _1 \\xi _1^2+\\alpha _1 \\eta _2 \\xi _1^2-\\alpha _1 \\eta _1 \\xi _1 \\xi _2-\\alpha _2 \\eta _2 \\xi _1 \\xi _2+\\alpha _2 \\eta _1 \\xi _2^2\\right),$\n\n$\\alpha' _2=\\frac{-1}{\\Delta }\\left(\\beta _1 \\eta _1^2 \\xi _1+\\alpha _1 \\eta _1 \\eta _2 \\xi _1-\\alpha _2 \\eta _2^2 \\xi _1-\\alpha _1 \\eta _1^2 \\xi _2+\\alpha _2 \\eta _1 \\eta _2 \\xi _2\\right),$\n\n$\\alpha' _4=\\frac{-1}{\\Delta }\\beta _1 \\eta _1^3,$\n\n$\\beta' _1=\\frac{1}{\\Delta }\\beta _1 \\xi _1^3.$\n\nMaking $\\eta_1=0$ results in $\\alpha'_1=\\alpha_1 \\xi_1-\\alpha _2\\xi_2,$ $\\alpha'_4=0$, $\\alpha'_2=\\alpha _2\\eta_2$,  $\\beta'_1=\\frac{\\xi^2_1}{\\eta_2}\\beta _1.$\n\\begin{itemize}\n\t\\item[a)] If $\\alpha_2\\neq 0$ one can make $\\alpha'_1= 0$ and $\\alpha'_2=1$.\n\t\\begin{itemize}\n\t\t\\item[a)-1.] If $\\beta_1\\neq 0$ one can reduce $\\beta'_1=1$ to get\n\t\t$A_{9,3}=\\left(\n\t\t\\begin{array}{cccc}\n\t\t0 & 1 & 1 & 0 \\\\\n\t\t1 &0&0 &-1\\end{array}\n\t\t\\right).$\n\t\t\n\t\t\\item[a)-2.] If $\\beta_1= 0$ then $\\beta'_1=0$ and one gets\n\t\t$A_{10,3}=\\left(\n\t\t\\begin{array}{cccc}\n\t\t0 &1 & 1 & 0 \\\\\n\t\t0 &0& 0 &-1\n\t\t\\end{array}\n\t\t\\right).$\n\t\\end{itemize}\n\t\\item[b)] If $\\alpha_2= 0$ then $\\alpha'_2= 0$.\n\t\\begin{itemize}\n\t\t\\item[b)-1.] If $\\beta_1\\neq 0$  we reduce $\\beta'_1=1$ and $\\alpha'_1$ may take one or zero depending on $\\alpha_1$. Therefore one has\n\t\t\\[A_{11,3}=\\left(\n\t\t\\begin{array}{cccc}\n\t\t1 &0 & 0 & 0 \\\\\n\t\t1 &-1&-1 &0\n\t\t\\end{array}\n\t\t\\right)\\ \\mbox{or}\\ A_{12,3}=\\left(\n\t\t\\begin{array}{cccc}\n\t\t0 &0 &0 & 0 \\\\\n\t\t1 &0& 0 &0\n\t\t\\end{array}\n\t\t\\right).\\]\n\t\t\n\t\t\\item[b)-2.] If $\\beta_1=0$  then $\\beta'_1=0$, $\\alpha'_4=0$ and one can make , $\\alpha'_1=1$ to get\n\t\t\\[A'=\\left(\n\t\t\\begin{array}{cccc}\n\t\t1 &0& 0 & 0 \\\\\n\t\t0 &-1& -1 &0\n\t\t\\end{array}\n\t\t\\right)\\ \\mbox{isomorphic to}\\ A_{10,3}.\\]\n\t\\end{itemize}\n\\end{itemize}\n\n{\\bf Remark 1.} From the proof of Theorems it is clear that the same results remain be true if one assumes the existence of a root in $\\mathbf{F}$ of every second and third order polynomial over $\\mathbf{F}$ (instead of assumption that $\\mathbf{F}$ to be algebraically closed). The results can be used for getting complete classification of different classes, for example,  associative, Jordan, alternative and etc., 2-dimensional algebras. \n\n{\\bf Remark 2.} In \\cite{A} the class $A_{3,2}(\\mathbf{c})$ should be understood as it is in this paper.\n \n{\\bf Acknowledgment.}\n  The first author thanks M.A.A. Ahmed for the help in computer computations, the second author's research is supported by FRGS14-153-0394, MOHE and the third author acknowledges MOHE for a support by grant 01-02-14-1591FR.\n\n\n\\vspace{0,2cm}\n\n\\bibliographystyle{aipproc}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nIn this paper, we prove the conjecture by Gyenge, N\\'emethi and\nSzendr\\H{o}i \\cite{MR3671513,MR3799153} giving a formula of the\ngenerating function of Euler numbers of Hilbert schemes of points\n$\\operatorname{Hilb}^n({\\mathbb C}^2\/\\Gamma)$ on a simple singularity\n${\\mathbb C}^2\/\\Gamma$, where $\\Gamma$ is a finite subgroup of $\\operatorname{\\rm SL}(2)$. When\n$\\Gamma$ is of type $A$, Euler numbers were computed by Dijkgraaf and\nSu\\l kowski \\cite{MR2391107}, and Toda \\cite{MR3394393}. The formula in\n\\cite{MR3671513,MR3799153} is given in a different form, and makes\nsense for arbitrary $\\Gamma$. \nThe formula was proved for type $D$, as well as type $A$, in\n\\cite{MR3799153}.\nOur proof of the formula in type $E$ is new.\n\nThe formula \\cite{MR3671513,MR3799153} is written as a specialization\nof the character for the `Fock space', which is the tensor product of\nthe basic representation of the affine Lie algebra ${\\mathfrak{g}}_{\\mathrm{aff}}$\ncorresponding to $\\Gamma$, and the usual Fock space representation of\nthe Heisenberg algebra. The Fock space was realized in a geometric way\nas the direct sum of homology groups of various connected components\nof $\\operatorname{Hilb}^n({\\mathbb C}^2)^\\Gamma$ for all $n$. See\n\\cite{MR1989196}. The character is well-known (see e.g.,\n\\cite[\\S12]{Kac})\n\\begin{equation*}\n    Z =\n    \\prod_{m=1}^\\infty\n    (1 - e^{-m\\delta})^{-(n+1)}\n    \\sum_{\\vec{m}\\in{\\mathbb Z}^n} \n    e^{-\\frac{(\\vec{m}, \\mathbf C\\vec{m})}2\\delta}\n    \\prod_{i=1}^n e^{-m_i\\alpha_i},\n\\end{equation*}\nwhere $\\alpha_i$ is the $i$th simple root of ${\\mathfrak{g}}_{\\mathrm{fin}}$, the\nunderlying finite dimensional complex simple Lie algebra, $\\delta$ is\nthe positive primitive imaginary root, $n$ is the rank of\n${\\mathfrak{g}}_{\\mathrm{fin}}$, and $\\mathbf C$ is the finite Cartan matrix for\n${\\mathfrak{g}}_{\\mathrm{fin}}$. Here we omit the factor $e^{\\Lambda_0}$ for the $0$th\nfundamental weight $\\Lambda_0$. The well-known formula for the basic\nrepresentation gives $(1 - e^{-m\\delta})^{-n}$ instead of the power\n$-(n+1)$. The additional $-1$ comes from the Heisenberg algebra.\n\\begin{NB}\n  Suppose the dimension vector $\\bv$ for ${\\mathfrak{M}}(\\bv,\\Lambda_0)$ is\n  $m\\delta + \\bv_{\\mathrm{fin}}$ with\n  $\\bv_{\\mathrm{fin}} = \\sum_{i\\neq 0} m_i \\alpha_i$. By the Frenkel-Kac\n  construction, we have the `bottom' case with\n  $\\mathop{\\text{\\rm dim}}\\nolimits {\\mathfrak{M}}(\\bv,\\Lambda_0) = 0$, and raise its vacuum by the\n  Heisenberg algebra. For the bottom, we have\n  \\begin{equation*}\n    0 = \\mathop{\\text{\\rm dim}}\\nolimits {\\mathfrak{M}}(m\\delta+\\bv_{\\mathrm{fin}},\\Lambda_0)\n    = 2m - (m\\delta+\\bv_{\\mathrm{fin}}, \\mathbf C(m\\delta+\\bv_{\\mathrm{fin}}))\n    = 2m - (\\bv_{\\mathrm{fin}}, \\mathbf C \\bv_{\\mathrm{fin}})\n  \\end{equation*}\n  as $\\mathbf C\\delta = 0$. Thus\n  $m = \\frac12 (\\bv_{\\mathrm{fin}}, \\mathbf C \\bv_{\\mathrm{fin}})$.\n\\end{NB}%\n\nThe conjectural formula in \\cite{MR3671513}, which we will prove, is\nthe following:\n\\begin{Theorem*}\\label{thm:1}\n  The generating function of Euler numbers of\n  $\\operatorname{Hilb}^n({\\mathbb C}^2\/\\Gamma)$\n  \\begin{equation*}\n    \\sum_{n=0}^\\infty e^{-n\\delta} \\chi(\\operatorname{Hilb}^n({\\mathbb C}^2\/\\Gamma))\n  \\end{equation*}\n  is obtained from $Z$ by the substitution\n  \\begin{equation*}\n    e^{-\\alpha_i} = \\exp\\left( \\frac{2\\pi i}{h^\\vee+1}\\right)\n    \\quad (i\\neq 0),\n  \\end{equation*}\n  where $h^\\vee$ is the dual Coxeter number.\n\\end{Theorem*}\n\nLet us sketch the strategy of our proof.\nIn order to compute Euler numbers of\n$\\operatorname{Hilb}^n({\\mathbb C}^2\/\\Gamma)$, we use a recent result of Craw,\nGammelgaard, Gyenge and Szendr\\H{o}i \\cite{2019arXiv191013420C}:\n$\\operatorname{Hilb}^n({\\mathbb C}^2\/\\Gamma)$ with the reduced scheme\nstructure is isomorphic to a quiver variety\n${\\mathfrak{M}}_{\\zeta^\\bullet}(n\\delta,\\Lambda_0)$, where $\\zeta^\\bullet$ is a\nstability parameter\n\\begin{equation*}\n  \\zeta_i^\\bullet = 0 \\quad\\text{for all $i\\neq 0$},\\quad\n  \\zeta_0^\\bullet < 0.\n\\end{equation*}\nThis stability parameter is different from the standard one $\\zeta$\n\\begin{equation*}\n    \\zeta_i < 0\\quad\\text{for all $i\\in I$},\n \n \n\\end{equation*}\nwhich gives connected components ${\\mathfrak{M}}_{\\zeta}(\\bv, \\Lambda_0)$ of\n$\\operatorname{Hilb}^{n}({\\mathbb C}^2)^\\Gamma$. Here $\\bv$ is an isomorphism\nclass of ${\\mathbb C}[x,y]\/I$ as a $\\Gamma$-module for an ideal\n$I \\subset {\\mathbb C}[x,y]$ of the coordinate ring ${\\mathbb C}[x,y]$ of ${\\mathbb C}^2$,\ninvariant under the $\\Gamma$-action.\n\nAs $\\zeta^\\bullet$ lies in walls given by roots in the space of\nstability parameters, ${\\mathfrak{M}}_{\\zeta^\\bullet}(n\\delta,\\Lambda_0)$ is\nsingular in general. Nevertheless it has a representation theoretic\nmeaning, as shown in \\cite{Na-branching}: the quiver variety\n${\\mathfrak{M}}_{\\zeta^\\bullet}(n\\delta,\\Lambda_0)$ is responsible for the\nrestriction of ${\\mathfrak{g}}_{\\mathrm{aff}}$ to ${\\mathfrak{g}}_{\\mathrm{fin}}$.\nThe essential geometric ingredient for this relation is a projective\nmorphism\n\\begin{equation*}\n    \\pi_{\\zeta^\\bullet,\\zeta}\\colon\n    {\\mathfrak{M}}_{\\zeta}(\\bv,\\Lambda_0)\n    \\to {\\mathfrak{M}}_{\\zeta^\\bullet}(n\\delta,\\Lambda_0)\n    = \\operatorname{Hilb}^n({\\mathbb C}^2\/\\Gamma),\n    \\qquad n = \\bv_0.\n\\end{equation*}\nBy a local description of singularities of quiver varieties\n(\\cite[\\S2.7]{Na-branching}, which went back to\n\\cite[\\S6]{Na-quiver}), fibers of $\\pi_{\\zeta^\\bullet,\\zeta}$ are\nlagrangian subvarieties in quiver varieties ${\\mathfrak{M}}_{\\zeta}(\\bv^s,\\bw^s)$\nassociated with the finite $ADE$ quiver for ${\\mathfrak{g}}_{\\mathrm{fin}}$, the finite\ndimensional complex simple Lie algebra underlying ${\\mathfrak{g}}_{\\mathrm{aff}}$.\nAs a simple application of this result, we observe that\n\\begin{itemize}\n\\item Euler numbers of $\\operatorname{Hilb}^n({\\mathbb C}^2\/\\Gamma)$ can be\n  written in terms of Euler numbers of ${\\mathfrak{M}}_\\zeta(\\bv,\\Lambda_0)$ and\n  quiver varieties ${\\mathfrak{M}}_\\zeta(\\bv^s,\\bw^s)$ of the finite $ADE$ type.\n\\end{itemize}\nIn fact, we get \\emph{more} data than we need: we have a\nstratification of $\\operatorname{Hilb}^n({\\mathbb C}^2\/\\Gamma)$ so that\n$\\pi_{\\zeta^\\bullet,\\zeta}$ is a fiber bundle over each stratum. We\ncan compute Euler numbers of all strata from Euler numbers of\n${\\mathfrak{M}}_\\zeta(\\bv,\\Lambda_0)$ and quiver varieties\n${\\mathfrak{M}}_\\zeta(\\bv^s,\\bw^s)$ of the finite $ADE$ type. See \\eqref{eq:5}\nfor the precise formula.\n\nThere are several algorithms\\footnote{We make a distinction between an\n  algorithm for a computation and an explicit computation as in\n  \\cite{Na:E8}. Namely a computation results finitely many $\\pm$,\n  $\\times$, integers and variables. We do not require that the final\n  expression is readable by a human. The expression like\n  $\\sum_{i=1}^{2^{(2^{100})}} a_i$ with explicit $a_i$ is an\n  algorithm.} to compute Euler numbers of quiver varieties. For\nexample, \\cite{MR2144973} and \\cite{MR2651380}. They are complicated,\nand hard to use in practice.\nOn the other hand, the conjectural formula in \\cite{MR3671513} above\ndoes \\emph{not} contain such a complicated algorithm. It is just given\nby a simple substitution.\nIt means that we should have a drastic simplification if we take a\n\\emph{linear combination} of complicated Euler numbers of quiver\nvarieties of finite $ADE$ type. We do not need to compute Euler\nnumbers of individual strata, as we only need their sum.\n\nThis simplification has a representation theoretic origin. Let us\nexplain it.\n\nThe above specialized character is called the \\emph{quantum dimension}:\n\\begin{equation*}\n  \\mathop{\\text{\\rm dim}}\\nolimits_q V = \\left.\\operatorname{ch} V\\right|_{e^{-\\alpha_i}=\\exp(\\frac{2\\pi i}{h^\\vee+1})}.\n\\end{equation*}\nHere $V$ is a finite dimensional representation $V$ of ${\\mathfrak{g}}_{\\mathrm{fin}}$.\nIt was introduced by Andersen~\\cite[Def.~3.1]{MR1182414} (see also\nParshall-Wang~\\cite{MR1247547}). We choose a specific root of unity\n$\\exp(\\frac{2\\pi i}{2(h^\\vee+1)})$, where these papers study more\ngeneral roots of unity $\\zeta$.\n\nTherefore the conjectural formula in \\cite{MR3671513} states that the\ngenerating function of Euler numbers of\n$\\operatorname{Hilb}^n({\\mathbb C}^2\/\\Gamma)$ is given by the quantum\ndimension of the Fock space at $\\exp(\\frac{2\\pi i}{2(h^\\vee+1)})$,\nrestricted from ${\\mathfrak{g}}_{\\mathrm{aff}}$ to ${\\mathfrak{g}}_{\\mathrm{fin}}$, just keeping track\n$e^{-\\delta}$.\n\nA direct sum of homology groups (which are isomorphic to complexified\n$K$-group) of Lagrangian subvarieties of quiver varieties of the\nfinite $ADE$ type carries a structure of a finite dimensional\nrepresentation of the quantum loop algebra\n$\\mathbf U_{q}(\\mathbf L{\\mathfrak{g}}_{\\mathrm{fin}})$. See \\cite{Na-qaff}. It is called a\n\\emph{standard module} of $\\mathbf U_{q}(\\mathbf L{\\mathfrak{g}}_{\\mathrm{fin}})$.\n\nIt turns out that the above simplification is a consequence of the\nfollowing representation theoretic result of an independent interest:\n\\begin{Theorem*}\\label{thm:2}\n  The quantum dimention of arbitrary standard module of\n  $\\mathbf U_q(\\mathbf L{\\mathfrak{g}}_{\\mathrm{fin}})$ of type $ADE$ at\n  $\\zeta=\\exp(\\frac{2\\pi i}{2(h^\\vee+1)})$ is equal to $1$.\n\\end{Theorem*}\n\nSince standard modules are tensor products of $l$-fundamental modules\n\\cite{VV-std}, it is enough to prove this result for $l$-fundamental\nmodules.\nThe author is told by Naoi that this result is a special case of more\ngeneral conjecture posed by Kuniba \\cite[Conj.~2 (A.6a)]{MR1202213}\n(see also Kuniba-Nakanishi-Suzuki \\cite[Conjecture~14.2]{MR2773889})\nformulated for Kirillov-Reshetikhin modules.\n(Recall $l$-fundamental modules are the simplest examples of\nKirillov-Reshetikhin modules.)\nIt is not difficult to check the general conjecture for type $A$. Type\n$D$ case was shown in \\cite{MR3043892}, while type $E_6$ case was\nshown in \\cite{MR3282650}.\n\nAlthough we only need the \\emph{simplest special} case of more general\nconjecture, we could not find a proof of the relevant result for type\n$E_7$, $E_8$ in the literature. Therefore we give its proof.\nFortunately the necessary explicit computation for type $E$ from the\nalgorithm in \\cite{MR2144973} was already done in \\cite{Na:E8} by\nusing a supercomputer. Alternatively we could quote the computation by\nKleber \\cite{MR1436775}, which assumed fermionic formula conjectured\nat that time. The fermionic formula was proved later by\nDi~Francesco and Kedem~\\cite{MR2428305}.\n\\begin{NB}\n  based on many earlier works including \\cite{MR1993360}, as well as\n  results from cluster algebras.\n\\end{NB}%\n\nWe also give a proof of the known cases $A$, $D$, $E_6$ for the\ncompleteness. We encounter a new feature in $E_7$, $E_8$, which did\nnot arise in other cases. Hence our check in the simplest case is yet\nnontrivial.\n\nBecause it is about Kirillov-Reshetikhin modules, it suggests that a\ngeneral conjecture should be studied in the framework of cluster\nalgebras, as in \\cite{MR2428305}. Note also that Euler numbers of\n(graded) quiver varieties are understood in the context of cluster\nalgebras in a recent work of Bittmann \\cite{2019arXiv191113110B}. Thus\nthe suggestion is compatible with the approach in this paper, though\ncluster algebras play no role in this paper.\n\nThe paper is organized as follows. In \\cref{sec:Euler} we deduce\n\\cref{thm:1} from \\cref{thm:2} after recalling results from\n\\cite{Na-branching}. In \\cref{sec:2} we prove \\cref{thm:2}. It is\nproved by the case by case analysis. In \\cref{sec:rationally} we\ndiscuss an additional topic, the rationally smoothness of\n$\\operatorname{Hilb}^n({\\mathbb C}^2\/\\Gamma)$. It is rationally smooth if\n$n=2$, but not so in general.\n\n\\subsection*{Acknowledgments}\n\nThe author thanks\nA.~Gyenge\nand\nB.~Szendr\\H{o}i\nfor discussion on \\cite{MR3671513,MR3799153,2019arXiv191013420C}, and\nA.~Craw\non \\cite{2019arXiv191013420C} and \\cite{MR1202213,MR2773889}.\nHe also thanks\nA.~Kuniba,\nS.~Naito,\nT.~Nakanishi\nand\nK.~Naoi\nfor discussion on the conjecture in \\cite{MR1202213,MR2773889},\nD.~Muthiah\nfor discussion on rational smoothness,\nand\nR.~Yamagishi\nfor discussion on \\cite{2017arXiv170905886Y}.\n\nThe research is supported in part by the World Premier International\nResearch Center Initiative (WPI Initiative), MEXT, Japan, and by JSPS\nGrants Numbers 16H06335, 19K21828.\n\n\\section{Euler numbers of quiver varieties}\\label{sec:Euler}\n\n\\subsection{Quiver varieties}\n\nLet $\\Gamma$ be a nontrivial finite subgroup of $\\operatorname{\\rm SL}(2)$.\nWe define the affine Dynkin diagram via the McKay correspondence (see\ne.g., \\cite[Ch.~8]{MR3526103} for detail): let\n$\\{ \\rho_i\\}_{i\\in I}$ be the set of isomorphism classes of\nirreducible representations of $\\Gamma$ with the trivial\nrepresentation $\\rho_0$. We identify $\\rho_i$ with a vertex of a\ngraph. We draw $a_{ij}$ edges between $\\rho_i$ and $\\rho_j$ where\n$a_{ij} = \\mathop{\\text{\\rm dim}}\\nolimits \\operatorname{Hom}_\\Gamma(\\rho_i, \\rho\\otimes\\rho_j) =\n\\mathop{\\text{\\rm dim}}\\nolimits\\operatorname{Hom}_\\Gamma(\\rho_j,\\rho\\otimes\\rho_i)$, where $\\rho$ is the\n$2$-dimensional representation of $\\Gamma$ given by the inclusion\n$\\Gamma\\subset\\operatorname{\\rm SL}(2)$. Then the graph is an affine Dynkin diagram of\ntype $ADE$. Let ${\\mathfrak{g}}_{\\mathrm{aff}}$ denote the corresponding affine Lie\nalgebra, and ${\\mathfrak{g}}_{\\mathrm{fin}}$ the underlying finite dimensional complex\nsimple Lie algebra corresponding to the Dynkin diagram obtained from\nthe affine one by removing $\\rho_0$. Let $n$ be the rank of\n${\\mathfrak{g}}_{\\mathrm{fin}}$, which is the number of vertices in $I$ minus $1$.\n\nWe use the convention of the root system for ${\\mathfrak{g}}_{\\mathrm{aff}}$ as in\n\\cite[Ch.~6 and \\S12.4]{Kac}.\nLet $\\alpha_i$ be the $i$th simple root of ${\\mathfrak{g}}_{\\mathrm{aff}}$ corresponding\nto $\\rho_i$, $\\delta$ be the primitive positive imaginary root of\n${\\mathfrak{g}}_{\\mathrm{aff}}$. We have $\\delta = \\sum a_i \\alpha_i$, and $a_i$ is equal\nto the dimension of $\\rho_i$.\nLet $\\alpha_i^\\vee$ be the $i$th simple coroot. We take the scaling\nelement $d$ satisfying\n\\begin{equation*}\n  \\langle\\alpha_i, d\\rangle = \\delta_{i0}.\n\\end{equation*}\nThen $\\langle \\alpha_i^\\vee, d\\rangle_{i\\in I}$ is a base of\n$\\mathfrak h_{\\mathrm{aff}}$, the Cartan subalgebra of ${\\mathfrak{g}}_{\\mathrm{aff}}$.\nWe define the fundamental weights $\\Lambda_i$ ($i\\in I$) by\n\\begin{equation*}\n  \\langle \\Lambda_i, \\alpha_j^\\vee\\rangle = \\delta_{ij},\\quad\n  \\langle \\Lambda_i, d\\rangle = 0.\n\\end{equation*}\nThen $\\langle \\alpha_i, \\Lambda_0\\rangle_{i\\in I}$ forms a base of\n$\\mathfrak h_{\\mathrm{aff}}^*$.\n\nWe choose an orientation of edges in the affine Dynkin diagram and\nconsider the corresponding affine quiver $Q = (I,\\Omega)$.\nWe take dimension vectors $\\bw = (\\bw_i)$, and\n$\\bv = (\\bv_i)\\in {\\mathbb Z}_{\\ge 0}^{I}$, and\nconsider quiver varieties ${\\mathfrak{M}}_\\zeta(\\bv,\\bw)$,\n${\\mathfrak{M}}_{\\zeta^\\bullet}(\\bv,\\bw)$, where $\\zeta$, $\\zeta^\\bullet$ are\nstability parameters such that\n\\begin{equation*}\n  \\zeta_i < 0\\quad\\text{for all $i\\in I$},\\qquad\n  \\zeta_i^\\bullet = 0 \\quad\\text{for all $i\\neq 0$},\\quad\n  \\zeta_0^\\bullet < 0.\n\\end{equation*}\nSee \\cite[\\S2]{Na-branching} for the definition of quiver varieties\nfor these stability conditions.\nSince $\\zeta^\\bullet$ lives in the boundary of a chamber containing\n$\\zeta$, we have a projective morphism\n\\begin{equation*}\n  \\pi_{\\zeta^\\bullet,\\zeta}\\colon\n  {\\mathfrak{M}}_\\zeta(\\bv,\\bw)\\to {\\mathfrak{M}}_{\\zeta^\\bullet}(\\bv,\\bw).\n\\end{equation*}\nSee \\cite[\\S2]{Na-branching}.\nIn fact, this is an example studied in \\cite[\\S2.8]{Na-branching}\nassociated with a division $I = I^0\\sqcup I^+$ with\n$I^0 = I\\setminus \\{0\\}$, $I^+ = \\{0\\}$.\n\nWe identify $\\bv$, $\\bw$ with weights of ${\\mathfrak{g}}_{\\mathrm{aff}}$ by\n\\begin{equation*}\n  \\bv = \\sum \\bv_i \\alpha_i, \\quad \\bw = \\sum \\bw_i \\Lambda_i.\n\\end{equation*}\nIt is convenient to use a different convention for the dimension\nvector $\\bv$:\n\\begin{equation}\\label{eq:3}\n  \\bv = m\\delta + \\sum_{i\\in I^0} m_i \\alpha_i, \\qquad \\text{i.e., }\n  m = \\bv_0, \\quad m_i = \\bv_i - \\bv_0 a_i.\n\\end{equation}\n\nLet us take $\\bw = \\Lambda_0$.\nIt is well-known that ${\\mathfrak{M}}_\\zeta(\\bv,\\Lambda_0)$ is the $\\Gamma$-fixed\npoint component of Hilbert schemes $I$ of points in the affine plane\n${\\mathbb C}^2$ such that ${\\mathbb C}[x,y]\/I$ is isomorphic to\n$\\bigoplus \\rho_i^{\\oplus \\bv_i}$, as a $\\Gamma$-module.\nEuler number for ${\\mathfrak{M}}_\\zeta(\\bv,\\Lambda_0)$ is also known. It was\ngiven in \\cite{MR1989196}. Since it was stated without an explanation,\nlet us explain how it is derived. We change the stability condition\n$\\zeta$ to see that ${\\mathfrak{M}}_\\zeta(\\bv,\\bw)$ is diffeomorphic to a moduli\nspace of framed rank $1$ torsion free sheaves on the minimal\nresolution of ${\\mathbb C}^2\/\\Gamma$. Then rank $1$ torsion free sheaves are\nideal sheaves twisted by line bundles, hence Euler numbers are given\nby G\\\"ottsche formula for Hilbert schemes of points \\cite{MR1032930}.\nMoreover this latter picture gives the Frenkel-Kac construction of the\nbasic representation of ${\\mathfrak{g}}_{\\mathrm{aff}}$ (see e.g., \\cite[\\S14.8]{Kac}),\nhence\nwe get the formula of $Z$ in Introduction. We have used the convention\nin \\eqref{eq:3}.\n\nWe can define a structure of a representation of the affine Lie\nalgebra ${\\mathfrak{g}}_{\\mathrm{aff}}$ on the direct sum of homology groups\n$\\bigoplus_\\bv H_*({\\mathfrak{M}}_\\zeta(\\bv,\\Lambda_0))$. See \\cite{MR1989196}\nand references therein. We can also construct a structure of a\nrepresentation of the Heisenberg algebra commuting with ${\\mathfrak{g}}_{\\mathrm{aff}}$\nby \\cite{MR1441880,Lecture}.\n\nOn the other hand, Craw, Gammelgaard, Gyenge and Szendr\\H{o}i\n\\cite{2019arXiv191013420C} recently proved that the Hilbert scheme\n$\\operatorname{Hilb}^n({\\mathbb C}^2\/\\Gamma)$ of $n$ points in ${\\mathbb C}^2\/\\Gamma$\nwith the reduced scheme structure is isomorphic to\n${\\mathfrak{M}}_{\\zeta^\\bullet}(\\bv,\\Lambda_0)$ with $\\bv = n\\delta$.\n\n\\subsection{Stratification}\\label{subsec:strat}\n\nAs is explained in \\cite[\\S2]{Na-branching},\n${\\mathfrak{M}}_{\\zeta^\\bullet}(\\bv,\\bw)$ parametrizes $S$-equivalence classes of\n$\\zeta^\\bullet$-semistable framed representations of the preprojective\nalgebra. Therefore its points are represented by direct sum of\n$\\zeta^\\bullet$-stable representations. Under the above choice of\n$\\zeta^\\bullet$, we have one distinguished summand, giving a point in\n${\\mathfrak{M}}_{\\zeta^\\bullet}^{\\mathrm{s}}(\\bv',\\bw)$ for some $\\bv'\\le\\bv$\n(component-wise) such that $\\bv'_0 = \\bv_0$, and other summands are\nsimple representations $S_i$ with $i\\in I^0$. See\n\\cite[\\S2.6]{Na-branching}.\nSince multiplicities of $S_i$ can be read off from the difference\n$\\bv - \\bv'$, we can regard\n${\\mathfrak{M}}_{\\zeta^\\bullet}^{\\mathrm{s}}(\\bv',\\bw)$ as a subset of\n${\\mathfrak{M}}_{\\zeta^\\bullet}(\\bv,\\bw)$.\nBy \\cite[Prop.~2.30]{Na-branching}\n${\\mathfrak{M}}^{\\mathrm{s}}_{\\zeta^\\bullet}(\\bv',\\bw)\\neq\\emptyset$ if and only\nif $\\bw - \\bv'$ is an $I^0$-dominant weight appearing in the basic\nrepresentation $V(\\Lambda_0)$ of the affine Lie algebra ${\\mathfrak{g}}$\nassociated with $\\Gamma$.\nHere $I^0$-dominant means that\n$\\langle \\bw - \\bv', \\alpha_i^\\vee\\rangle \\ge 0$ for $i\\in I^0$.\nWe thus have the stratification\n\\begin{equation*}\n  {\\mathfrak{M}}_{\\zeta^\\bullet}(\\bv,\\bw)\n  = \\bigsqcup_{\\text{$\\bv'$ as above}} {\\mathfrak{M}}_{\\zeta^\\bullet}^{\\mathrm{s}}(\\bv',\\bw). \n\\end{equation*}\n\nMoreover the transversal slice to the stratum\n${\\mathfrak{M}}^{\\mathrm{s}}_{\\zeta^\\bullet}(\\bv',\\bw)$ is locally isomorphic to\na quiver variety ${\\mathfrak{M}}_0(\\bv^s,\\bw^s)$ around $0$, associated with the\nfinite $ADE$ quiver $Q\\setminus \\{\\rho_0\\}$ such that dimension\nvectors are given by\n\\begin{equation*}\n  \\bv^s = \\bv - \\bv', \\qquad\n  \\bw^s_i = \\langle \\bw - \\bv', \\alpha_i^\\vee\\rangle \\quad i\\in I^0.\n\\end{equation*}\nNote that $\\bv-\\bv'$ has no $0$th component as $\\bv'_0 = \\bv_0$.\nNote also that $\\bw^s_i \\ge 0$, as $\\bw - \\bv'$ is $I^0$-dominant.\nIt is also known that the inverse image of the slice in\n${\\mathfrak{M}}_\\zeta(\\bv,\\bw)$ under $\\pi_{\\zeta^\\bullet,\\zeta}$ is locally\nisomorphic to ${\\mathfrak{M}}_{\\zeta}(\\bv^s,\\bw^s)$ around\n$\\mathfrak L(\\bv^s,\\bw^s)$, the inverse image of the origin $0$ of\n${\\mathfrak{M}}_0(\\bv^s,\\bw^s)$ under the projective morphism\n${\\mathfrak{M}}_\\zeta(\\bv^s,\\bw^s)\\to {\\mathfrak{M}}_0(\\bv^s,\\bw^s)$.\nSee \\cite[\\S2.7]{Na-branching} and the references therein for these\nclaims on transversal slices. There is also an algebraic approach in\n\\cite{CB:normal}.\n\nFor $\\bw = \\Lambda_0$, we have\n\\begin{equation*}\n  \\bw^s_i = - \\sum_{j\\in I^0} m'_j \\langle \\alpha_j, \\alpha_i^\\vee\\rangle,\n  \\qquad \\text{where $\n    \\bv' = m\\delta + \\sum_{j\\in I^0} m'_j \\alpha_j$\n    in the convention \\eqref{eq:3}}.\n\\end{equation*}\nNote that $\\bv = m\\delta + \\sum m_i \\alpha_i$ and $\\bv'$ share the\nsame $m$ for the coefficient of $\\delta$, as $\\bv_0 = \\bv'_0$.\nIn particular,\n\\begin{equation}\\label{eq:6}\n    \\sum_{i\\in I^0} \\bw^s_i \\Lambda_i - \\bv^s_i \\alpha_i\n    \\begin{NB}\n        = - \\sum_{i\\in I^0} \\sum_{j\\in I^0} m'_j \\langle\\alpha_j,\n        \\alpha_i^\\vee \\rangle \\Lambda_i - (m_i - m'_i)\\alpha_i\n    \\end{NB}\n    = - \\sum_{i\\in I^0} m_i \\alpha_i.\n\\end{equation}\nWe also note\n\\begin{equation}\\label{eq:7}\n    m'_i \\le 0 \\qquad (i\\in I^0),\n\\end{equation}\nas $(m'_i)_{i\\in I^0} = - (\\bw^s)_{i\\in I^0} \\mathbf C^{-1}$, and the\ninverse of the Cartan matrix $\\mathbf C$ has positive entries. This\nwas proved in \\cite[Prop.~A.1]{2019arXiv191013420C} by a different\nmethod.\n\n\\subsection{Examples}\n\n\\begin{Example}\n  Let\n  ${\\mathfrak{M}}_{\\zeta^\\bullet}(n\\delta, \\Lambda_0)\\cong\n  \\operatorname{Hilb}^n({\\mathbb C}^2\/\\Gamma)$.\n  Consider a stratum containing $(n-1)$-distinct points in\n  ${\\mathbb C}^2\\setminus\\{0\\}\/\\Gamma$ together with trivial\n  representations. We have\n  $\\bv' = (n-1)\\delta + \\alpha_0 = n\\delta - \\sum_{i\\in I^0} a_i\n  \\alpha_i$, and hence\n  \\begin{equation*}\n    \\bv^s = \\sum_{i\\in I^0} a_i \\alpha_i, \\quad\n    \\bw^s_i = \\sum_{j\\in I^0} C_{ij} a_j.\n\\end{equation*}\nNote that $\\bw^s$ has entries $1$ at vertices in the finite quiver\n$I^0$ which are connected to the $0$-vertex in the affine quiver. In\nparticular, ${\\mathfrak{M}}_0(\\bv^s,\\bw^s)$ is ${\\mathbb C}^2\/\\Gamma$,\n${\\mathfrak{M}}_{\\zeta}(\\bv^s,\\bw^s)$ is its minimal resolution, the very\nfirst example of a quiver variety considered by Kronheimer \\cite{Kr},\nbefore the definition of quiver varieties was introduced.\n\nThis is obvious, as the transversal slice is\n${\\mathfrak{M}}_{\\zeta^\\bullet}(\\delta,\\Lambda_0) \\cong {\\mathbb C}^2\/\\Gamma$, as we can\nignore $(n-1)$-distinct points.\n\\end{Example}\n\n\\begin{Example}[Yamagishi \\cite{2017arXiv170905886Y}]\\label{ex:Yam}\n  Consider\n  ${\\mathfrak{M}}_{\\zeta^\\bullet}(2\\delta, \\Lambda_0) =\n  \\operatorname{Hilb}^2({\\mathbb C}^2\/\\Gamma)$. The formal neighborhoods of\n  fibers of $\\pi_{\\zeta^\\bullet,\\zeta}$ over $0$-dimensional strata in\n  ${\\mathfrak{M}}_{\\zeta^\\bullet}(2\\delta,\\Lambda_0)$ were determined by Yamagishi\n  \\cite{2017arXiv170905886Y}. He identified the formal neighborhood\n  with that of the intersection of the nilpotent cone for the complex\n  simple Lie algebra ${\\mathfrak{g}}_{\\mathrm{fin}}$ and the Slodowy slice to a\n  `sub-subregular' orbit. There is only one $0$-dimensional stratum\n  except type $A_1$, $A_2$, $D_n$, while there are none for $A_1$,\n  $A_2$, two for $D_n$ ($n > 4$) and three for $D_4$.\n  We can give corresponding vector $\\bv'$ as follows:\n  \\begin{itemize}\n  \\item If $Q$ is of type $A_1^{(1)}$ or $A_2^{(1)}$, there is no such\n    $\\bv'$.\n  \\item If $Q$ is of type $A_{n}^{(1)}$ with $n > 2$,\n    $\\bv' = 2\\alpha_0 + \\alpha_1 + \\alpha_{n}$.\n  \\item If $Q$ is of type $D_n^{(1)}$ with $n > 4$,\n    $\\bv' = \\dynkin[edge\n    length=.6cm,labels={2,0,2,2,2,2,1,1}]{D}[1]{***.****} = 2\\alpha_0\n    + 2\\alpha_2 + \\dots + 2\\alpha_{n-2} + \\alpha_{n-1} + \\alpha_n$ or\n    $\\bv' = \\dynkin[edge\n    length=.6cm,labels={2,1,2,1,0,0,0,0,0}]{D}[1]{****.****} =\n    2\\alpha_0 + \\alpha_1 + 2\\alpha_2 + \\alpha_3$.\n  \\item If $Q$ is of type $D_4^{(1)}$, we have three possibilities: in\n    addition to the second example in $D_n^{(1)}$ above, we have\n    $(\\alpha_1,\\alpha_3) \\mapsto (\\alpha_1,\\alpha_4)$,\n    $(\\alpha_3,\\alpha_4)$, as its cyclic permutation.\n\n  \\item If $Q$ is of type $E_8^{(1)}$, we have\n    $\\bv' = \\dynkin[edge\n    length=.6cm,labels={2,0,1,1,2,2,2,2,2}]{E}[1]{8}$. The cases\n    $E_6^{(1)}$, $E_7^{(1)}$ are similar.\n  \\end{itemize}\n  Here $0$th vertex is $\\circ$, and other vertices are $\\bullet$.\n\n  Transversal slices are\n  \\begin{itemize}\n  \\item If $Q$ is of type $A_{n}^{(1)}$ with $n>2$,\n    $\\bv^s = \\dynkin[edge\n    length=.6cm,labels={1,2,2,2,2,1}]{A}{***.***}$,\n    $\\bw^s = \\dynkin[edge\n    length=.6cm,labels={0,1,0,0,1,0}]{A}{***.***}$.\n\n  \\item If $Q$ is of type $D_n^{(1)}$,\n    $\\bv^s = \\dynkin[edge\n    length=.6cm,labels={2,2,2,2,1,1}]{D}{***.***}$,\n    $\\bw^s = \\dynkin[edge\n    length=.6cm,labels={2,0,0,0,0,0}]{D}{***.***}$ in the first case,\n    and\n    $\\bv^s = \\dynkin[edge\n    length=.6cm,labels={1,2,3,4,4,4,4,2,2}]{D}{*****.****}$,\n    $\\bw^s = \\dynkin[edge\n    length=.6cm,labels={0,0,0,1,0,0,0,0,0}]{D}{*****.****}$ in the\n    second case.\n\n  \\item If $Q$ is of type $D_4^{(1)}$,\n    $\\bv^s = \\dynkin[edge length=.6cm,labels={1,2,1,2}]{D}{4}$,\n    $\\bw^s = \\dynkin[edge length=.6cm,labels={0,0,0,2}]{D}{4}$, and\n    its cyclic permutation.\n\n  \\item If $Q$ is of type $E_8^{(1)}$,\n    $\\bv^s = \\dynkin[edge\n    length=.6cm,labels={4,5,7,10,8,6,4,2}]{E}{8}$,\n    $\\bw^s = \\dynkin[edge\n    length=.6cm,labels={1,0,0,0,0,0,0,0}]{E}{8}$.\n\\end{itemize}\n\nIn particular, it means that the quiver variety ${\\mathfrak{M}}_0(\\bv^s,\\bw^s)$\nis isomorphic to the intersection of Slodowy slice and the nilpotent\ncone in the formal neighborhood of the origin.\n\nThis result was known before for type $A_n$ and the second case of\n$D_n$. The type $A_n$ case was proved in \\cite[\\S8]{Na-quiver}. See\nalso \\cite{MR2130242}. For the second case of type $D_n$, it was shown\nin \\cite{MR3269179}. In these cases, ${\\mathfrak{M}}_0(\\bv^s,\\bw^s)$ itself is\nisomorphic to the intersection of Slodowy slice and the nilpotent\ncone, not only in the formal neighborhood.\nThe exceptional cases are conjectured, but not shown as far as the\nauthor knows.\n\\end{Example}\n\n\\begin{Remark}\n  In cases for the above Example, it is known that corresponding\n  Coulomb branches of the quiver gauge theories (which are affine\n  Grassmannian slices by \\cite{2016arXiv160403625B}) are\n  `next-to-minimal' nilpotent orbits closures in ${\\mathfrak{g}}_{\\mathrm{fin}}$ by\n  \\cite{MR3131493}.\n\\end{Remark}\n\n\\begin{Example}[Yamagishi \\cite{2017arXiv170905886Y}]\n  Let us consider $\\bv = \\delta+\\alpha_0$, $\\bw = \\Lambda_0$, hence\n  ${\\mathfrak{M}}_{\\zeta^\\bullet}(\\delta+\\alpha_0,\\Lambda_0)$. Note that this is\n  different from\n  ${\\mathfrak{M}}_{\\zeta^\\bullet}(\\delta,\\Lambda_0) =\n  \\operatorname{Hilb}^1({\\mathbb C}^2\/\\Gamma) = {\\mathbb C}^2\/\\Gamma$. Let us show\n  that ${\\mathfrak{M}}_{\\zeta^\\bullet}(\\delta+\\alpha_0,\\Lambda_0)$ is \\emph{not}\n  isomorphic to ${\\mathbb C}^2\/\\Gamma$.\n\n  This ${\\mathfrak{M}}_{\\zeta^\\bullet}(\\delta+\\alpha_0,\\Lambda_0)$ appears as the\n  union of strata in\n  ${\\mathfrak{M}}_{\\zeta^\\bullet}(2\\delta,\\Lambda_0) =\n  \\operatorname{Hilb}^2({\\mathbb C}^2\/\\Gamma)$. Thus all possible strata are\n  determined already as above: besides the open stratum for\n  $\\bv' = \\bv$, we have a single $0$-dimensional stratum except for\n  $A_1$, $A_2$, $D_n$, none for $A_1$, $A_2$, three for $D_4$, two for\n  $D_n$ ($n > 4$).\n  The transversal slice ${\\mathfrak{M}}_0(\\bv^s,\\bw^s)$ is given by the same\n  $\\bw^s$ as above, $\\bv^s$ is subtracting\n  $\\sum_{i\\in I^0} a_i\\alpha_i$ from the above example. Concretely it\n  is\n  \\begin{itemize}\n  \\item For type $A_n^{(1)}$ with $n > 2$,\n    $\\bv^s = \\dynkin[edge\n    length=.6cm,labels={0,1,1,1,1,0}]{A}{***.***}$,\n    $\\bw^s = \\dynkin[edge\n    length=.6cm,labels={0,1,0,0,1,0}]{A}{***.***}$.\n\n  \\item For type $D_n^{(1)}$ with $n > 4$,\n    $\\bv^s = \\dynkin[edge\n    length=.6cm,labels={1,0,0,0,0,0}]{D}{***.***}$,\n    $\\bw^s = \\dynkin[edge\n    length=.6cm,labels={2,0,0,0,0,0}]{D}{***.***}$ in the first case,\n    and\n    $\\bv^s = \\dynkin[edge\n    length=.6cm,labels={0,0,1,2,2,2,2,1,1}]{D}{*****.****}$,\n    $\\bw^s = \\dynkin[edge\n    length=.6cm,labels={0,0,0,1,0,0,0,0,0}]{D}{*****.****}$ in the\n    second case.\n\n  \\item For type $D_4^{(1)}$,\n    $\\bv^s = \\dynkin[edge length=.6cm,labels={0,0,0,1}]{D}{4}$,\n    $\\bw^s = \\dynkin[edge length=.6cm,labels={0,0,0,2}]{D}{4}$, and\n    its cyclic permutation.\n\n  \\item For type $E_8^{(1)}$,\n    $\\bv^s = \\dynkin[edge\n    length=.6cm,labels={2,2,3,4,3,2,1,0}]{E}{8}$,\n    $\\bw^s = \\dynkin[edge\n    length=.6cm,labels={1,0,0,0,0,0,0,0}]{E}{8}$.\n  \\end{itemize}\n\n  Let us consider\n  \\(\n    \\pi_{\\zeta^\\bullet,\\zeta}\\colon {\\mathfrak{M}}_{\\zeta}(\\bv,\\bw)\n    \\to {\\mathfrak{M}}_{\\zeta^\\bullet}(\\bv,\\bw).\n  \\)\n  The domain ${\\mathfrak{M}}_{\\zeta}(\\bv,\\bw)$ is the minimal resolution of\n  ${\\mathbb C}^2\/\\Gamma$. The above description of the transversal slice\n  immediately conclude the following:\n  \\begin{itemize}\n  \\item ${\\mathfrak{M}}_{\\zeta^\\bullet}(\\bv,\\bw)$ is obtained from the minimal\n    resolution of ${\\mathbb C}^2\/\\Gamma$ by collapsing $\\mathbb P^1$'s whose\n    vertices are not connected to the $0$-vertex in the affine Dynkin\n    diagram.\n  \\end{itemize}\n\n  There are no such vertices for $A_1$, $A_2$. For type $A_n$\n  ($n > 2$), we collapse $(n-2)$ $\\mathbb P^1$'s corresponding to\n  vertices except the leftmost and rightmost. For type $D_n$, we\n  collapse $(n-1)$ $\\mathbb P^1$'s except the second one from the\n  left, and producing singularities of type $A_1$ and $D_{n-2}$, where\n  we understand $D_2 = A_1\\times A_1$, $D_3 = A_3$.\n  For type $E_8$, the $7$ $\\mathbb P^1$'s except the rightmost one are\n  collapsed.\n\\end{Example}\n\n\\subsection{Euler numbers}\n\nBy \\cref{subsec:strat} we can relate Euler numbers of\n${\\mathfrak{M}}_{\\zeta}(\\bv,\\bw)$, ${\\mathfrak{M}}_{\\zeta^\\bullet}(\\bv',\\bw)$ and\n${\\mathfrak{M}}_{\\zeta}(\\bv^s,\\bw^s)$. Let $\\chi(\\ )$ denote the Euler number of\na space. We have\n\\begin{equation*}\n  \\begin{split}\n  \\chi({\\mathfrak{M}}_{\\zeta}(\\bv,\\bw))\n  & = \\sum_{\\bv'} \\chi({\\mathfrak{M}}_{\\zeta^\\bullet}^{\\mathrm{s}}(\\bv',\\bw))\n  \\chi(\\mathfrak L_{\\zeta}(\\bv^s,\\bw^s)) \\\\\n  & = \\sum_{\\bv'} \\chi({\\mathfrak{M}}_{\\zeta^\\bullet}^{\\mathrm{s}}(\\bv',\\bw))\n  \\chi({\\mathfrak{M}}_{\\zeta}(\\bv^s,\\bw^s)).\n  \\end{split}\n\\end{equation*}\nIt is known \\cite[Cor.~5.5]{Na-quiver} that the central fiber\n$\\mathfrak L_{\\zeta}(\\bv^s,\\bw^s)$ of\n${\\mathfrak{M}}_{\\zeta}(\\bv^s,\\bw^s)\\to {\\mathfrak{M}}_0(\\bv^s,\\bw^s)$ is homotopic to\n${\\mathfrak{M}}_{\\zeta}(\\bv^s,\\bw^s)$, hence the second equality follows.\n\nFrom $\\chi({\\mathfrak{M}}_{\\zeta}(\\bv,\\bw))$,\n$\\chi({\\mathfrak{M}}_{\\zeta}(\\bv^s,\\bw^s))$, we compute\n$\\chi({\\mathfrak{M}}_{\\zeta^\\bullet}^{\\mathrm{s}}(\\bv',\\bw))$ recursively as\n\\begin{equation}\\label{eq:5}\n  \\chi({\\mathfrak{M}}_{\\zeta^\\bullet}^{\\mathrm{s}}(\\bv,\\bw))\n  = \\chi({\\mathfrak{M}}_\\zeta(\\bv,\\bw)) - \\sum_{\\bv'\\neq \\bv}\n  \\chi({\\mathfrak{M}}_{\\zeta^\\bullet}^{\\mathrm{s}}(\\bv',\\bw))\n  \\chi({\\mathfrak{M}}_{\\zeta}(\\bv^s,\\bw^s)).\n\\end{equation}\nHere we use $\\bv^s = 0$, hence ${\\mathfrak{M}}_{\\zeta}(\\bv^s,\\bw^s)$ is a point\nfor $\\bv'=\\bv$.\n\nWe take the generating function of Euler numbers as\n\\begin{equation}\\label{eq:8}\n  \\sum_{\\bv} \\chi({\\mathfrak{M}}_{\\zeta}(\\bv,\\bw)) e^{-\\bv}\n  = \\sum_{\\bv'} \\chi({\\mathfrak{M}}_{\\zeta^\\bullet}^{\\mathrm{s}}(\\bv',\\bw)) e^{-\\bv'}\n  \\sum_{\\bv^s}\n  \\chi({\\mathfrak{M}}_{\\zeta}(\\bv^s,\\bw-\\bv'|_{I^0})) e^{-\\bv^s},\n\\end{equation}\nwhere\n$\\bw - \\bv'|_{I^0} = \\bw^s = \\sum_{i\\in I^0} \\langle \\bw -\n\\bv',\\alpha_i^\\vee\\rangle \\Lambda_i$.\n\nWe claim\n\\begin{equation}\\label{eq:1}\n  \\left.\\sum_{\\bv^s}\n    \\chi({\\mathfrak{M}}_{\\zeta}(\\bv^s,\\bw^s)) \n    \\prod_{i\\in I^0} e^{\\bw^s_i \\Lambda_i\n      - \\bv^s_i \\alpha_i}\n  \\right|_{\n    e^{-\\alpha_i} = \\exp(\\frac{2\\pi i}{h^\\vee+1})\n  }\n  = 1.\n\\end{equation}\n\\begin{NB}\n  $\\mathop{\\text{\\rm dim}}\\nolimits {\\mathfrak{M}}_{\\zeta}(\\bv^s,\\bw^s) = \\langle 2\\bw^s-\\bv^s, \\bv^s\\rangle$.\n\\end{NB}%\n\nWe take $\\bw = \\Lambda_0$ and switch to the convention\n\\eqref{eq:3}. Then the left hand side of \\eqref{eq:8} is $Z$ in\nIntroduction. By \\eqref{eq:6}, \\eqref{eq:1} implies that\n\\begin{NB}\n\\begin{multline*}\n    \\sum_{m, \\vec{m}} \\chi({\\mathfrak{M}}_{\\zeta}(m\\delta+\\sum_{i\\in I^0} m_i\\alpha_i,\n  \\Lambda_0)) \n  e^{-m\\delta} \\prod_{i\\in I^0} e^{-m_i\\alpha_i}\n\\\\\n  = \\sum_{m,\\vec{m}'} \\chi({\\mathfrak{M}}_{\\zeta^\\bullet}^{\\mathrm{s}}(\n  m\\delta+\\sum_{i\\in I^0} m'_i\\alpha_i,\\Lambda_0)) \n  e^{-m\\delta} \\prod_{i\\in I^0} e^{-m_i\\alpha_i}\n  \\sum_{\\vec{m''}}\n  \\chi({\\mathfrak{M}}_{\\zeta}(\\sum_{i\\in I^0} m''_i\\alpha_i,\\bw^s)) \n  \\prod_{i\\in I^0}e^{-m''_i\\alpha_i}.\n\\end{multline*}\n\\end{NB}\n\\begin{equation*}\n    \\left.Z\\right|_{\n      e^{-\\alpha_i} = \\exp(\\frac{2\\pi i}{h^\\vee+1})\n    }\n    = \\sum_{m,\\vec{m}'} \\chi({\\mathfrak{M}}_{\\zeta^\\bullet}^{\\mathrm{s}}(\n  m\\delta+\\sum_{i\\in I^0} m_i'\\alpha_i,\\Lambda_0)) \n  e^{-m\\delta}.\n\\end{equation*}\nBy \\eqref{eq:7} we consider\n${\\mathfrak{M}}_{\\zeta^\\bullet}^{\\mathrm{s}}( m\\delta+\\sum_{i\\in I^0}\nm_i'\\alpha_i,\\Lambda_0)$ as a stratum of\n${\\mathfrak{M}}_{\\zeta^\\bullet}(m\\delta,\\Lambda_0) =\n\\operatorname{Hilb}^m({\\mathbb C}^2\/\\Gamma)$. Therefore the right hand side\nof the above is the generating function of Euler numbers of\n$\\operatorname{Hilb}^m({\\mathbb C}^2\/\\Gamma)$.\nThus we have proved \\cref{thm:1}.\n\nRecall that\n$\\bigoplus_{\\bv^s} H_*(\\mathfrak L_{\\zeta}(\\bv^s,\\bw^s)) \\cong\n\\bigoplus_{\\bv^s} K(\\mathfrak L_{\\zeta}(\\bv^s,\\bw^s))\\otimes_{{\\mathbb Z}}{\\mathbb C}$\nis the so-called standard module of the quantum loop algebra\n$\\mathbf U_q(\\mathbf L{\\mathfrak{g}}_{\\mathrm{fin}})$, specialized at $q=1$\n\\cite{Na-qaff}.\nThe above \\eqref{eq:1} means that its character (as a\n${\\mathfrak{g}}_{\\mathrm{fin}}$-module), specialized at\n$e^{-\\alpha_i} = \\exp(\\frac{2\\pi i}{h^\\vee+1})$ is equal to $1$.\nAs we mentioned in Introduction, this specialization is the quantum\ndimension. Thus \\eqref{eq:1} follows from \\cref{thm:2}.\n\nIn order to prove \\eqref{eq:1}, we may assume $\\bw^s$ is a fundamental\nweight: there is a torus action on framing vector spaces, and the\ninduced action on ${\\mathfrak{M}}_{\\zeta}(\\bv^s,\\bw^s)$. Let us suppose\n$\\bw^s = \\Lambda_i+\\Lambda_j$ for $i,j\\in I^0$ for simplicity. Then\nthe torus fixed point set is\n\\begin{equation*}\n  \\bigsqcup_{\\bv^1+\\bv^2=\\bv^s} {\\mathfrak{M}}_{\\zeta}(\\bv^1,\\Lambda_i)\n  \\times {\\mathfrak{M}}_{\\zeta}(\\bv^2,\\Lambda_j).\n\\end{equation*}\nAs Euler number is equal to the sum of Euler numbers of fixed points\nwith respect to a torus action, \\eqref{eq:1} for $\\bw^s$ follows from\n\\eqref{eq:1} for $\\Lambda_i$ and $\\Lambda_j$.\nThis result is compatible with what we mentioned in Introduction:\nstandard modules are tensor products of $l$-fundamental modules\n\\cite{VV-std}, as $l$-fundamental modules correspond to the case\n$\\bw^s$ is a fundamental weight.\n\nA standard module depends also on spectral parameter, which is\nspecialization homomorphism\n$K_{\\prod_{i\\in I^0} \\operatorname{GL}(\\bw^s_i)}(\\mathrm{pt})\\to {\\mathbb C}$. But the\nrestriction of a standard module to $\\mathbf U_q({\\mathfrak{g}}_{\\mathrm{fin}})$ is\nindependent of the spectral parameter. Hence the spectral parameter is\nnot relevant for \\cref{thm:2}.\n\n\\begin{Remark}\n  In order to prove \\cref{thm:1}, we need to check \\eqref{eq:1} only\n  when $\\bw^s$ is contained in the root lattice. But $\\Lambda_i$ does\n  not satisfy this condition in general, hence the above reduction\n  cannot be performed among $\\bw^s$ in the root lattice.\n\\end{Remark}\n\n\\section{Quantum dimensions of standard modules}\\label{sec:2}\n\n\\subsection{Quantum dimension}\n\nLet $V$ be a finite dimensional representation of ${\\mathfrak{g}}_{\\mathrm{fin}}$. The\nspecialized character\n\\begin{equation*}\n  \\left.\\operatorname{ch} V\\right|_{e^{-\\alpha_i}=\\exp(\\frac{2\\pi i}{h^\\vee+1})}\n\\end{equation*}\nis called the \\emph{quantum dimension} of $V$, denoted by $\\mathop{\\text{\\rm dim}}\\nolimits_q V$,\nand was introduced by Andersen~\\cite[Def.~3.1]{MR1182414} (see also\nParshall-Wang~\\cite{MR1247547}), where\n$\\zeta = \\exp(\\frac{2\\pi i}{2(h^\\vee+1)})$. Note $2\\rho$ in\n$K_{2\\rho} = q^{2\\rho}$ in \\cite{MR1182414} should be understood as an\nelement in the dual of the weight lattice by\n$\\langle 2\\rho,\\lambda\\rangle = (2\\rho,\\lambda)\/(\\alpha_0,\\alpha_0)$\nfor a weight $\\lambda$. Here $\\alpha_0$ is a short root. See\n\\cite[Lemma~1.1]{MR1247547}.\nIt was assumed that $\\zeta$ is a primitive $\\ell$-th root of unity\nwith odd $\\ell$ in \\cite{MR1182414}, the definition still makes sense\nfor our choice. By the Weyl character formula, we have (see\n\\cite[(3.2)]{MR1182414}, \\cite[Th.~1.3]{MR1247547})\n\\begin{equation*}\n  \\left.\\operatorname{ch} V\\right|_{e^{-\\alpha_i}=\\exp(\\frac{2\\pi i}{h^\\vee+1})}\n  = \\mathop{\\text{\\rm dim}}\\nolimits_q V\n  = \\prod_{\\alpha\\in\\Delta^+_{\\mathrm{fin}}} \\frac{\n    \\zeta^{d_\\alpha\\langle\\lambda+\\rho,\\alpha^\\vee\\rangle} -\n    \\zeta^{-d_\\alpha\\langle\\lambda+\\rho,\\alpha^\\vee\\rangle}}{\n    \\zeta^{d_\\alpha\\langle\\rho,\\alpha^\\vee\\rangle} -\n    \\zeta^{-d_\\alpha\\langle\\rho,\\alpha^\\vee\\rangle}},\n\\end{equation*}\nwhere $\\lambda$ is the highest weight of an irreducible representation\n$V = V(\\lambda)$, $\\Delta^+_{\\mathrm{fin}}$ is the set of positive roots of\n${\\mathfrak{g}}_{\\mathrm{fin}}$, $d_\\alpha\\in \\{1,2,3\\}$ is the square length of $\\alpha$\ndivided by the length of a short root, and $\\rho$ is the half sum of\npositive roots. (Since we are only considering type $ADE$, we have\n$d_\\alpha = 1$ for any $\\alpha$.)\nWe have\n$d_\\alpha\\langle\\rho,\\alpha^\\vee\\rangle\\le \\langle\\rho,\\theta\\rangle =\nh^\\vee$, hence the denominator does not vanish.\n\nLet us introduce $\\zeta$-integers by\n\\begin{equation*}\n  [n]_{\\zeta} \\overset{\\operatorname{\\scriptstyle def.}}{=} \\frac{\\zeta^{n} - \\zeta^{-n}}\n  {\\zeta - \\zeta^{-1}},\n\\end{equation*}\nso that\n\\begin{equation}\\label{eq:4}\n  \\left.\\operatorname{ch} V\\right|_{e^{-\\alpha_i}=\\exp(\\frac{2\\pi i}{h^\\vee+1})}\n  = \\prod_{\\alpha\\in\\Delta^+}\n  \\frac{[d_\\alpha\\langle\\lambda+\\rho,\\alpha^\\vee\\rangle]_{\\zeta}}\n  {[d_\\alpha\\langle\\rho,\\alpha^\\vee\\rangle]_{\\zeta}}.\n\\end{equation}\nWe have\n\\begin{equation*}\n  [2(h^\\vee+1) + k]_{\\zeta} = [k]_{\\zeta}\n\\end{equation*}\nas $\\zeta^{2(h^\\vee+1)} = 1$, as well as\n\\begin{equation}\\label{eq:2}\n  [h^\\vee+1-k]_{\\zeta} = [k]_{\\zeta}\n\\end{equation}\nas $\\zeta^{h^\\vee+1-k} \\zeta^{k} = \\zeta^{h^\\vee+1} = -1$. In\nparticular, we have $[h^\\vee+1]_{\\zeta} = 0$. The former is usual\nanalogy between roots of unity and characteristic $2(h^\\vee+1)$. The\nlatter is a new feature at an even root of unity.\n\n\\subsection{type $A$}\n\nConsider type $A_{n-1}$, i.e., ${\\mathfrak{g}}_{\\mathrm{fin}} = \\algsl(n,{\\mathbb C})$. We have\n$h^\\vee = n$.\n\nIn type $A_{n-1}$, it is known that the $k$th $l$-fundamental module\nis the $k$th fundamental representation of $\\operatorname{\\rm SL}(n)$.\n\nThe quantum dimension of the $k$th fundamental representation\n$V(\\Lambda_k)$ is\n\\begin{equation*}\n  \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_k)\n  = \\begin{bmatrix}\n    n \\\\ k\n  \\end{bmatrix}_{\\zeta}\n  = \\frac{[n]_{\\zeta}[n-1]_{\\zeta}\\dots [n-k+1]_{\\zeta}}\n  {[k]_{\\zeta} [k-1]_{\\zeta} \\dots [1]_{\\zeta}}.\n\\end{equation*}\n\\begin{NB}\nIn fact, the denominator of \\eqref{eq:4} is equal to\n\\begin{equation*}\n  \\prod_{1\\le i < j\\le n} [j-i]_\\zeta,\n\\end{equation*}\nas positive roots $\\alpha$ are\n$\\alpha_{i+1} + \\dots + \\alpha_j$.\n\nOn the other hand, from the numerator, we have\n\\begin{equation*}\n  \\prod_{1\\le i < j\\le n} [\\lambda_i - \\lambda_j + j-i]_\\zeta,\n\\end{equation*}\nwhere $\\lambda$ is regarded as a partition $(1^k)$.\nSince $\\lambda_i - \\lambda_j$ is $1$ if $i\\le k$, $j>k$ and $0$\notherwise, we have\n\\begin{equation*}\n  \\prod_{1\\le i<j\\le n} \\frac{[\\lambda_i-\\lambda_j+j-i]_\\zeta}\n  {[j-i]_\\zeta} =\n  \\prod_{i=1}^k \\prod_{j=k+1}^n \\frac{[j-i+1]_\\zeta}{[j-i]_\\zeta}\n  = \\prod_{i=1}^k \\frac{[n+1-i]_\\zeta}{[k+1-i]_\\zeta},\n\\end{equation*}\nas required.\n\\begin{NB2}\n  Older version: the denominator is\n  \\begin{equation*}\n    [n-1]_{\\zeta}[n-2]^2_{\\zeta}\\dots [1]_{\\zeta}^{n-1}.\n  \\end{equation*}\n  The numerator is\n  \\begin{equation*}\n    [n]_{\\zeta} [n-1]_{\\zeta}^2 \\dots [n-k+1]_{\\zeta}^{k}\n    [n-k]_{\\zeta}^{k} \\dots [k+1]_{\\zeta}^{n-k-1} [k]_{\\zeta}^{n-k-1}\n    \\dots [1]_{\\zeta}^{n-2},\n  \\end{equation*}\n  where we assume $k\\le n-k$. This is checked by considering whether\n  $\\langle\\lambda,\\alpha^\\vee\\rangle$ is $0$ or $1$. In fact, among\n  $(n-1)$ coroots $\\alpha^\\vee$ with\n  $\\langle\\rho,\\alpha^\\vee\\rangle = 1$, we have one $\\alpha^\\vee$ with\n  $\\langle\\Lambda_k,\\alpha^\\vee\\rangle = 1$. All others have\n  $\\langle\\Lambda_k,\\alpha^\\vee\\rangle = 0$. Thus we get\n  $[2]_{\\zeta}[1]^{n-2}_{\\zeta}$.\n  Among $(n-2)$ coroots $\\alpha^\\vee$ with\n  $\\langle\\rho,\\alpha^\\vee\\rangle = 2$, we have two $\\alpha^\\vee$ with\n  $\\langle\\Lambda_k,\\alpha^\\vee\\rangle = 1$. All others have\n  $\\langle\\Lambda_k,\\alpha^\\vee\\rangle = 0$. Thus we get\n  $[3]_{\\zeta}^2 [2]^{n-4}_{\\zeta}$. This continues until\n  $\\langle\\rho,\\alpha^\\vee\\rangle = k$. At this last stage we get\n  $[k+1]_{\\zeta}^k[k]_{\\zeta}^{n-2k}$. In total we get\n  \\begin{equation*}\n    [k+1]^k_{\\zeta} [k]_{\\zeta}^{n-k-1}\\cdots\n    [2]_{\\zeta}^{n-3} [1]^{n-2}_{\\zeta}\n  \\end{equation*}\n\n  Next at $(n-k-1)$ $\\alpha^\\vee$ with\n  $\\langle\\rho,\\alpha^\\vee\\rangle = k+1$, we get\n  $[k+2]_{\\zeta}^k [k+1]_{\\zeta}^{n-2k-1}$. This will\n  continue until $k$ $\\alpha^\\vee$ with\n  $\\langle\\rho,\\alpha^\\vee\\rangle = n-k$, where we get\n  $[n-k]_{\\zeta}^{k}$. In this stage we get\n  \\begin{equation*}\n    [n-k+1]_{\\zeta} [n-k]_{\\zeta}^{k-1} \\cdots\n    [k+1]_{\\zeta}^{n-2k-1}.\n  \\end{equation*}\n\n  After that, we get\n  \\begin{equation*}\n    [n-k+1]_{\\zeta}^{k-1} \\dots\n    [n-1]^2_{\\zeta} [n]_{\\zeta},\n  \\end{equation*}\n  as all roots get $1$ by $\\langle\\Lambda_k,\\alpha^\\vee\\rangle$. In\n  total, we get the above.\n\\end{NB2}%\n\\end{NB}%\nBy \\eqref{eq:2}, this is equal to $1$, as $[n-i]_\\zeta$ cancels with\n$[i+1]_\\zeta$.\n\nHence \\cref{thm:2} is proved for type $A$.\n\n\\begin{NB}\nMore generally we have\n\\begin{equation*}\n  \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\lambda) = \\prod_{1\\le i<j\\le n}\n  \\frac{[\\lambda_i-\\lambda_j+j-i]_\\zeta}{[j-i]_\\zeta}\n\\end{equation*}\ncan be expressed by a specialized Schur polynomial.\n\\end{NB}%\n\n\\subsection{type $D$}\nConsider type $D_n$, i.e., ${\\mathfrak{g}}_{\\mathrm{fin}} = \\operatorname{\\mathfrak{so}}(2n,{\\mathbb C})$. We have\n$h^\\vee = 2n-2$.\n\nIt is known that the $k$th $l$-fundametal module for $1\\le k\\le n-2$\nis isomorphic as an $\\operatorname{\\mathfrak{so}}(2n,{\\mathbb C})$-module to\n\\begin{equation*}\n  {\\textstyle \\bigwedge}^k({\\mathbb C}^{2n}) \\oplus {\\textstyle \\bigwedge}^{k-2}({\\mathbb C}^{2n})\\oplus \\cdots,\n\\end{equation*}\nwhere $\\cdots$ ends as ${\\textstyle \\bigwedge}^1({\\mathbb C}^{2n}) = {\\mathbb C}^{2n}$ if $k$ is odd,\nand ${\\textstyle \\bigwedge}^0({\\mathbb C}^{2n}) = {\\mathbb C}$ if $k$ is even. See\n\\cite[Remark~5.9]{MR1988990}. For $k=n-1$, $n$, the $l$-fundamental\nmodule is isomorphic to the $k$th fundamental representation of\n$\\operatorname{\\mathfrak{so}}(2n,{\\mathbb C})$.\n\nAs in \\cite{MR1153249}, positive roots are\n\\begin{equation*}\n  \\{ L_i + L_j \\mid i < j\\} \\sqcup \\{ L_i - L_j \\mid i < j\\},\n\\end{equation*}\nand simple roots are\n\\begin{equation*}\n  \\alpha_i = L_i - L_{i+1} \\quad (i=1,\\dots, n-1),\\qquad\n  \\alpha_n = L_{n-1} + L_{n}\n\\end{equation*}\nwith the standard inner product $(L_i,L_j) = \\delta_{ij}$.\nFundamental weights are\n\\begin{equation*}\n  \\begin{split}\n    & \\Lambda_i = L_1+ L_2 + \\dots + L_i\\quad (i=1,\\dots, n-2), \\\\\n    & \\Lambda_{n-1} = \\frac12(L_1 + L_2 + \\dots + L_{n-2} + L_{n-1} - L_n),\\\\\n    & \\Lambda_n = \\frac12(L_1 + L_2 + \\dots + L_{n-2} + L_{n-1} + L_n).\n  \\end{split}\n\\end{equation*}\nWeyl vector $\\rho$ is\n\\begin{equation*}\n  \\rho = \\sum_{i=1}^n (n-i) L_i.\n\\end{equation*}\n\nWe have\n\\(\n   (\\Lambda_k, L_1+L_2) = 2\n\\)\nunless $k=1$, $n-1$, $n$. If this holds, we have\n\\begin{equation*}\n  [(\\Lambda_k + \\rho, L_1+L_2)]_\\zeta = [2n-1]_\\zeta = 0.\n\\end{equation*}\nThus\n\\begin{equation*}\n  \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_k) = 0 \\quad \\text{if $k\\neq 1$, $n-1$, $n$}.\n\\end{equation*}\n\nWe have\n\\begin{equation*}\n  \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_1) = \\frac{[n]_\\zeta [2n-2]_\\zeta}{[1]_\\zeta [n-1]_{\\zeta}} = 1,\n\\end{equation*}\nas $[2n-2]_\\zeta = [1]_\\zeta$, $[n]_\\zeta = [n-1]_\\zeta$ by \\eqref{eq:2}.\n \nNext we consider the case $\\lambda = \\Lambda_{n-1}$. We have\n\\begin{equation*}\n  (\\Lambda_{n-1},L_i - L_j) =\n  \\begin{cases}\n    0 & \\text{if $j\\le n-1$},\\\\\n    1 & \\text{if $j=n$},\n  \\end{cases}\n  \\qquad\n  (\\Lambda_{n-1},L_i + L_j) =\n  \\begin{cases}\n    1 & \\text{if $j\\le n-1$},\\\\\n    0 & \\text{if $j=n$}.\n  \\end{cases}\n\\end{equation*}\nThese imply\n\\begin{equation*}\n  \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_{n-1}) =\n  \\frac{[2n-2]_\\zeta[2n-4]_\\zeta \\dots [4]_\\zeta}\n  {[n-1]_\\zeta[n-2]_\\zeta\\dots [3]_\\zeta}\n  =\n  \\begin{cases}\n    \\frac{[2n-2]_\\zeta[2n-4]_\\zeta \\dots [n]_\\zeta}\n    {[n-1]_\\zeta [n-3]_\\zeta \\dots [3]_\\zeta} & \\text{if $n$ is even},\\\\\n    \\frac{[2n-2]_\\zeta[2n-4]_\\zeta \\dots [n+1]_\\zeta}\n    {[n-2]_\\zeta [n-4]_\\zeta \\dots [3]_\\zeta} & \\text{if $n$ is odd}.\n  \\end{cases}\n\\end{equation*}\nThis is $1$ by \\eqref{eq:2}. Note $[2n-2]_\\zeta = [1]_\\zeta = 1$. We\nalso have $\\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_n) = 1$ as we have a diagram automorphism\n$n\\leftrightarrow n-1$, $i\\leftrightarrow i$ ($i\\neq n-1,n$).\n\nIn summary,\n\\begin{equation*}\n  \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_k) =\n  \\begin{cases}\n    1 & \\text{if $k=1,n-1, n$},\\\\\n    0 & \\text{otherwise}.\n  \\end{cases}\n\\end{equation*}\nHence \\cref{thm:2} is proved.\n\n\\subsection{type $E_6$}\nWe have $h^\\vee = 12$.\nThe numbering of vertices is\n$\n\\dynkin[edge\n  length=.6cm,label]{E}{6}.$ \n  We need to compute quantum dimensions of $V(\\Lambda_k)$, as well as\n  of $V(\\Lambda_1+\\Lambda_6)$ since it appears in standard\n  representations when $\\bw^s$ is the $4$th fundamental\n  weight.\n\nPositive roots are\n\\begin{gather*}\n  \\alpha_1, \\dots, \\alpha_6, \\alpha_1 + \\alpha_3, \\alpha_3 + \\alpha_4,\n  \\alpha_4 + \\alpha_5, \\alpha_5 + \\alpha_6, \\alpha_2+ \\alpha_4,\\\\\n  \\alpha_1 + \\alpha_3 + \\alpha_4, \\alpha_2+\\alpha_3+\\alpha_4,\n  \\alpha_2+\\alpha_4+\\alpha_5, \\alpha_3+\\alpha_4+\\alpha_5,\n  \\alpha_4+\\alpha_5+\\alpha_6,\\\\\n  \\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4,\n  \\alpha_1+\\alpha_3+\\alpha_4+\\alpha_5,\n  \\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5,\\\\\n  \\alpha_2+\\alpha_4+\\alpha_5+\\alpha_6,\n  \\alpha_3+\\alpha_4+\\alpha_5+\\alpha_6,\\\\\n  \\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5,\n  \\alpha_1+\\alpha_3+\\alpha_4+\\alpha_5+\\alpha_6,\\\\\n  \\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5+\\alpha_6,\n  \\alpha_2+\\alpha_3 + 2\\alpha_4 + \\alpha_5,\\\\\n  \\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5+\\alpha_6,\n  \\alpha_1+\\alpha_2+\\alpha_3+2\\alpha_4+\\alpha_5,\n  \\alpha_2+\\alpha_3+2\\alpha_4+\\alpha_5+\\alpha_6,\n  \\\\\n  \\alpha_1+\\alpha_2+2\\alpha_3+2\\alpha_4+\\alpha_5,\n  \\alpha_1+\\alpha_2+\\alpha_3+2\\alpha_4+\\alpha_5+\\alpha_6,\n  \\alpha_2+\\alpha_3+2\\alpha_4+2\\alpha_5+\\alpha_6,\n  \\\\\n  \\alpha_1+\\alpha_2+2\\alpha_3+2\\alpha_4\n  +\\alpha_5+\\alpha_6,\n  \\alpha_1+\\alpha_2+\\alpha_3+2\\alpha_4\n  +2\\alpha_5+\\alpha_6,\n  \\\\\n  \\alpha_1+\\alpha_2+2\\alpha_3+2\\alpha_4\n  +2\\alpha_5+\\alpha_6,\n  \\\\\n  \\alpha_1+\\alpha_2+2\\alpha_3+3\\alpha_4\n  +2\\alpha_5+\\alpha_6,\\\\\n \n  \\alpha_1+2\\alpha_2+2\\alpha_3+3\\alpha_4+2\\alpha_5+\\alpha_6.\n\\end{gather*}\nHence the denominator of \\eqref{eq:4} is\n\\begin{equation*}\n  [1]_\\zeta^6 [2]_\\zeta^5 [3]_\\zeta^5 [4]_\\zeta^5 [5]_\\zeta^4\n  [6]_\\zeta^3 [7]_\\zeta^3 [8]_\\zeta^2 [9]_\\zeta [10]_\\zeta [11]_\\zeta.\n\\end{equation*}\n\nWe compute the quantum dimension of the first fundamental representation as\n\\begin{equation*}\n  \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_1) =\n  \\frac{[2]_\\zeta[3]_\\zeta[4]_\\zeta[5]_\\zeta^2[6]_\\zeta^2[7]_\\zeta^2 [8]_\\zeta^2\n  [9]_\\zeta^2[10]_\\zeta[11]_\\zeta[12]_\\zeta}\n{[1]_\\zeta[2]_\\zeta[3]_\\zeta[4]_\\zeta^2[5]_\\zeta^2[6]_\\zeta^2[7]_\\zeta^2\n  [8]_\\zeta^2[9]_\\zeta[10]_\\zeta [11]_\\zeta}\n= \\frac{[9]_\\zeta[12]_\\zeta}{[1]_\\zeta[4]_\\zeta}.\n\\end{equation*}\n\\begin{NB}\n  the first fundamental representation is $27$-dimensional.\n\\end{NB}%\nThis is $1$ by \\eqref{eq:2}. We also have $\\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_6)=1$ by\nthe diagram automorphism.\n\nWe have\n\\begin{equation*}\n  \\begin{split}\n    & (\\Lambda_k+\\rho,\\alpha_1+2\\alpha_2+2\\alpha_3+3\\alpha_4+2\\alpha_5+\\alpha_6) = 13 \\quad\\text{for $k=2,3,5$},\\\\\n    & (\\Lambda_1+\\Lambda_6+\\rho,\\alpha_1+2\\alpha_2+2\\alpha_3+3\\alpha_4+2\\alpha_5+\\alpha_6) = 13.\n  \\end{split}\n\\end{equation*}\nTherefore $\\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_k) = 0$ for $k=2,3,5$,\n$\\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_1+\\Lambda_6) = 0$.\nWe also have\n\\begin{equation*}\n  (\\Lambda_4+\\rho,\\alpha_1+\\alpha_2+2\\alpha_3+3\\alpha_4\n  +2\\alpha_5+\\alpha_6) = 13,\n\\end{equation*}\nhence $\\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_4)=0$.\n\\begin{NB}\n    In fact, we have\n    \\begin{equation*}\n        \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_4) = \\frac{\n        [5]_\\zeta[9]_\\zeta^2[10]_\\zeta[13]_\\zeta[14]_\\zeta}\n      {[1]_\\zeta[2]_\\zeta^2[3]_\\zeta^2[7]_\\zeta}\n    \\end{equation*}\n\\end{NB}%\n\n\nIt is known that the $k$th fundamental module is isomorphic as a\n${\\mathfrak g}_{\\mathrm{fin}}$-module to\n\\begin{equation*}\n  \\begin{cases}\n    V(\\Lambda_k) & \\text{if $k=1,6$},\\\\\n    V(\\Lambda_2)\\oplus V(0) & \\text{if $k=2$},\\\\\n    V(\\Lambda_3)\\oplus V(\\Lambda_6) & \\text{if $k=3$},\\\\\n    V(\\Lambda_5)\\oplus V(\\Lambda_1) & \\text{if $k=5$},\\\\\n    V(\\Lambda_4)\\oplus V(\\Lambda_2)^{\\oplus 2}\n    \\oplus V(\\Lambda_1+\\Lambda_6) \\oplus V(0) & \\text{if $k=4$}.\n  \\end{cases}\n\\end{equation*}\nThis can be given by using the algorithm \\cite{MR2144973}, as we did\nfor $E_8$ in \\cite{Na:E8} with much smaller efforts. Instead, the list\ncan be found in \\cite{MR1436775}, which assumed fermionic formula\nconjectured at that time, and proved later in \\cite{MR2428305}\n\\begin{NB}\n, based\non many earlier works including \\cite{MR1993360}    \n\\end{NB}%\n.\n\nWe substitute the above computation of quantum dimensions of various\nmodules to the above combination.\nWe find that the combination has the quantum dimension always\n$1$. Hence \\cref{thm:2} is proved.\n\n\\subsection{type $E_7$}\nWe have $h^\\vee = 18$.\nThe numbering of vertices is\n$\n\\dynkin[edge\n  length=.6cm,label]{E}{7}.$ \n\nWe calculate as above, using Sage:\n\\begin{equation*}\n    \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_7) = \\frac{[10]_\\zeta[14]_\\zeta[18]_\\zeta}\n    {[1]_\\zeta[5]_\\zeta[9]_\\zeta} = 1\n\\end{equation*}\nand\n\\begin{equation*}\n    \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_k) = 0\\quad \\text{if $k\\neq 7$}.\n\\end{equation*}\n\nWe have a new pattern:\n\\begin{equation*}\n    \\mathop{\\text{\\rm dim}}\\nolimits_q V(2\\Lambda_1) =\n    \\frac{[12]_\\zeta[13]_\\zeta[14]_\\zeta[15]_\\zeta[18]_\\zeta[21]_\\zeta}{\n    [1]_\\zeta[2]_\\zeta[4]_\\zeta[5]_\\zeta[6]_\\zeta[7]_\\zeta}\n  = \\frac{[21]_\\zeta}{[2]_\\zeta}.\n\\end{equation*}\nWe use \\eqref{eq:2} for $k=-2$ this time:\n$[21]_\\zeta = [-2]_\\zeta$. This is $-[2]_\\zeta$. Hence\n\\begin{equation*}\n    \\mathop{\\text{\\rm dim}}\\nolimits_q V(2\\Lambda_1) = -1.\n\\end{equation*}\nThis is the first example of a representation whose quantum dimension\nis $-1$. Similarly we have\n\\begin{equation*}\n    \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_1+\\Lambda_7)\n    = \\frac{[12]_\\zeta[14]_\\zeta[15]_\\zeta[18]_\\zeta[20]_\\zeta}\n    {[1]_\\zeta^2 [4]_\\zeta [5]_\\zeta [7]_\\zeta}\n    = -1.\n\\end{equation*}\n\nOn the other hand, we get\n\\begin{equation*}\n    \\mathop{\\text{\\rm dim}}\\nolimits_q V(2\\Lambda_7) = \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_1+\\Lambda_6) \n    = \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_2+\\Lambda_7) = 0.\n\\end{equation*}\n\nIt is known that the $k$th $l$-fundamental module is isomorphic as a\n${\\mathfrak g}_{\\mathrm{fin}}$-module to\n\\begin{equation*}\n  \\begin{cases}\n    V(\\Lambda_k) & \\text{if $k=7$},\\\\\n    V(\\Lambda_1)\\oplus V(0) & \\text{if $k=1$},\\\\\n    V(\\Lambda_2)\\oplus V(\\Lambda_7) & \\text{if $k=2$},\\\\\n    V(\\Lambda_6)\\oplus V(\\Lambda_1)\\oplus V(0) & \\text{if $k=6$},\\\\\n    V(\\Lambda_3)\\oplus V(\\Lambda_6)\\oplus V(\\Lambda_1)^{\\oplus 2}\n    \\oplus V(0) & \\text{if $k=3$},\\\\\n    V(\\Lambda_5)\\oplus V(\\Lambda_2)^{\\oplus 2}\n    \\oplus V(\\Lambda_1+\\Lambda_7) \\oplus V(\\Lambda_7)^{\\oplus 2}\n    & \\text{if $k=5$},\\\\\n    \\begin{aligned}[t]\n    & V(\\Lambda_4)\\oplus V(\\Lambda_3)^{\\oplus 3}\n    \\oplus V(\\Lambda_2+\\Lambda_7)^{\\oplus 2}\n    \\oplus V(\\Lambda_1+\\Lambda_6) \\oplus V(2\\Lambda_1)\\\\\n    & \\quad \\oplus\n    V(\\Lambda_6)^{\\oplus 4}\\oplus V(\\Lambda_1)^{\\oplus 4}\n    \\oplus V(2\\Lambda_7)\\oplus V(0)^{\\oplus 2} \n    \\end{aligned}\n    & \\text{if $k=4$}.\n  \\end{cases}\n\\end{equation*}\nSubstituting the above computation, we find that these combinations\nhave its quantum dimension always $1$: for example, we have\n$\\mathop{\\text{\\rm dim}}\\nolimits_q V(2\\Lambda_1) + 2\\mathop{\\text{\\rm dim}}\\nolimits_q V(0) = (-1)+2=1$ in the last case.\nHence \\cref{thm:2} is proved.\n\n\\begin{NB}\n  $W(\\Lambda_2) = V(\\Lambda_2) \\oplus q V(\\Lambda_7)$,\n  \n  $W(\\Lambda_6) = V(\\Lambda_6) \\oplus q V(\\Lambda_1) \\oplus q^2 V(0)$,\n\n  $W(\\Lambda_3) = V(\\Lambda_2) \\oplus q V(\\Lambda_6) \\oplus (q+q^2)V(\\Lambda_1) \\oplus q^3 V(0)$\n\n  $W(\\Lambda_5) = V(\\Lambda_5) \\oplus (q+q^2) V(\\Lambda_2) \\oplus q V(\\Lambda_1+\\Lambda_7) \\oplus (q^2+q^3) V(\\Lambda_7)$\n\\end{NB}%\n\n\n\\subsection{type $E_8$}\nWe have $h^\\vee = 30$.\nThe numbering of vertices is\n$\n\\dynkin[edge\n  length=.6cm,label]{E}{8}.$ \n\nWe have similar patterns:\n\\begin{equation*}\n    \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_k) = 0 \\quad\\text{for any $k=1,\\dots,8$},\n\\end{equation*}\n\\begin{equation*}\n    \\mathop{\\text{\\rm dim}}\\nolimits_q V(2\\Lambda_8) =\n    \\frac{[20]_\\zeta[21]_\\zeta[24]_\\zeta[25]_\\zeta[30]_\\zeta[33]_\\zeta}\n    {[1]_\\zeta[2]_\\zeta[6]_\\zeta[7]_\\zeta[10]_\\zeta[11]_\\zeta} = -1,\n\\end{equation*}\n\\begin{equation*}\n    \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_7+\\Lambda_8) =\n    \\frac{[14]_\\zeta[18]_\\zeta[20]_\\zeta[22]_\\zeta[24]_\\zeta[25]_\\zeta[26]_\\zeta\n      [30]_\\zeta[32]_\\zeta[34]_\\zeta}\n    {[1]_\\zeta^2[3]_\\zeta[5]_\\zeta[6]_\\zeta[7]_\\zeta[9]_\\zeta[11]_\\zeta[13]_\\zeta\n    [17]_\\zeta} = 1,\n\\end{equation*}\n\\begin{equation*}\n    \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_6+\\Lambda_8) =\n    \\frac{[15]_\\zeta[18]_\\zeta[20]_\\zeta[21]_\\zeta[24]_\\zeta[25]_\\zeta[26]_\\zeta\n    [27]_\\zeta[30]_\\zeta[32]_\\zeta[33]_\\zeta[35]_\\zeta}\n  {[1]_\\zeta^2[2]_\\zeta[4]_\\zeta^{2}[5]_\\zeta[6]_\\zeta[7]_\\zeta[10]_\\zeta\n  [11]_\\zeta[13]_\\zeta[16]_\\zeta} = -1,\n\\end{equation*}\nand\n\\begin{gather*}\n    \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_1+\\Lambda_8) = \\mathop{\\text{\\rm dim}}\\nolimits_q V(2\\Lambda_1) =\n    \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_2+\\Lambda_8) = \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_1+\\Lambda_7) \\\\\n    = \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_1+2\\Lambda_8) = \\mathop{\\text{\\rm dim}}\\nolimits_q V(2\\Lambda_7) = \\mathop{\\text{\\rm dim}}\\nolimits_q\n    V(3\\Lambda_8) = \\mathop{\\text{\\rm dim}}\\nolimits_q V(2\\Lambda_1+\\Lambda_8)\\\\\n    = \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_3+\\Lambda_8) = \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_2+\\Lambda_7) = \n    \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_1+\\Lambda_6) = \\mathop{\\text{\\rm dim}}\\nolimits_q V(\\Lambda_1+\\Lambda_2) = 0.\n\\end{gather*}\n\nThe ${\\mathfrak g}_{\\mathrm{fin}}$-module structure of the $k$th $l$-fundamental module\nis known. Let us omit \\emph{negligible} modules (i.e., those with\n$\\mathop{\\text{\\rm dim}}\\nolimits_q = 0$) for brevity. We have\n\\begin{equation*}\n  \\begin{cases}\n    V(0) & \\text{if $k=1,2,7,8$},\\\\\n    V(2\\Lambda_8)\\oplus V(0)^{\\oplus 2} & \\text{if $k=3,6$},\\\\\n    V(2\\Lambda_8)^{\\oplus 5}\\oplus V(\\Lambda_7+\\Lambda_8)^{\\oplus 2}\n    \\oplus V(0)^{\\oplus 4} & \\text{if $k=5$},\\\\\n    V(\\Lambda_6+\\Lambda_8)^{\\oplus 4}\\oplus V(\\Lambda_7+\\Lambda_8)^{\\oplus 18}\n    \\oplus V(2\\Lambda_8)^{\\oplus 23} \\oplus V(0)^{\\oplus 10} \n    & \\text{if $k=4$}.\n  \\end{cases}\n\\end{equation*}\nRather miraculously we find that all have quantum dimensions $1$. For\nexample, we calculate the result as $(-4)+18+(-23)+10 = 1$ in the last\ncase. Hence \\cref{thm:2} is proved.\n\n\\section{Rationally smoothness}\\label{sec:rationally}\n\nAs mentioned in Introduction, the original motivation of study of\n${\\mathfrak{M}}_{\\zeta^\\bullet}(\\bv,\\bw)$ in \\cite{Na-branching} was its relation\nbetween the restriction from ${\\mathfrak{g}}_{\\mathrm{aff}}$ to ${\\mathfrak{g}}_{\\mathrm{fin}}$.\nHowever the Euler numbers of ${\\mathfrak{M}}_{\\zeta^\\bullet}(\\bv,\\bw)$ did not\nplay any role in \\cite{Na-branching}. Intersection cohomology groups of\n${\\mathfrak{M}}_{\\zeta^\\bullet}(\\bv,\\bw)$ appeared instead.\n\nTherefore we ask whether ${\\mathfrak{M}}_{\\zeta^\\bullet}(\\bv,\\bw)$ is a rational\nhomology manifold, i.e., its intersection cohomology complex is\nquasi-isomorphic to the constant sheaf. See \\cite[1.4]{BM} for the\noriginal definition of a rational homology manifold, and its\nequivalence to the present one.\nThis sounds optimistic, but Borho-MacPherson \\cite[\\S2.3]{BM} showed\nthat the nilpotent variety is a rational homology manifold.\nAlso ${\\mathfrak{M}}_0(\\bv,\\Lambda)$ is a symmetric power of ${\\mathbb C}^2\/\\Gamma$,\nwhich has only finite quotient singularity, hence is a rational\nhomology manifold.\n\n\\subsection{Example: Hilbert schemes of two points}\n\nConsider\n${\\mathfrak{M}}_{\\zeta^\\bullet}(2\\delta,\\Lambda_0) =\n\\operatorname{Hilb}^2({\\mathbb C}^2\/\\Gamma)$. \nAs we already mentioned in \\cref{ex:Yam}, Yamagishi showed that\n${\\mathfrak{M}}_{\\zeta^\\bullet}(2\\delta,\\Lambda_0)$ is locally isomorphic to the\nintersection of the nilpotent cone of ${\\mathfrak{g}}_{\\mathrm{fin}}$ and the Slodowy\nslice to a sub-subregular orbit around $0$-dimensional\nstrata.\nThe transversal slice to the bigger stratum is ${\\mathbb C}^2\/\\Gamma$. It is\nalso a rational homology manifold.\nCombined with the rational smoothness of the nilpotent variety\nmentioned above, we obtain the following.\n\n\\begin{Corollary}\n    ${\\mathfrak{M}}_{\\zeta^\\bullet}(2\\delta,\\Lambda_0) = \\mathrm{Hilb}^2({\\mathbb C}^2\/\\Gamma)$ is a rational homology manifold.\n\\end{Corollary}\n\nThis result can be also proved from \\cite{Na-branching}, together with\n\\cite{Na-qaff} and some Euler number computation. See the argument\nbelow in a simpler situation.\n\n\\begin{NB}\n  We look at \\cref{ex:Yam} for $\\bv^s$, $\\bw^s$, and compute the\n  multiplicities of the trivial representation in standard\n  modules.\n\n  Consider type $A_n$ with $n > 2$. We have\n  $\\bw^s = \\Lambda_2 + \\Lambda_{n-1}$. The standard module is\n  ${\\textstyle \\bigwedge}^2 {\\mathbb C}^{n+1}\\otimes {\\textstyle \\bigwedge}^{n-1} {\\mathbb C}^{n+1}$. It contains the\n  trivial representation with multiplicity $1$.\n\n  Consider type $D_n$. For $\\bw^s = 2\\Lambda_1$, the standard module\n  is the tensor square of the vector representation, which is\n  $\\operatorname{Sym}^2{\\mathbb C}^{2n}\\oplus {\\textstyle \\bigwedge}^2{\\mathbb C}^{2n}$, and the\n  former is the sum of the irreducible representation of the highest\n  weight $2\\Lambda_1$ and the trivial representation. (See e.g.,\n  \\cite[Exercise~19.21]{MR1153249}. Therefore the multiplicity of the\n  trivial representation is $1$.\n\\end{NB}%\n\n\\subsection{Counter-example}\n\nConsider $A_1^{(1)}$ with $n=4$, i.e.,\n$\\operatorname{Hilb}^4({\\mathbb C}^2\/({\\mathbb Z}\/2)) =\n{\\mathfrak{M}}_{\\zeta^\\bullet}(4\\delta,\\Lambda_0)$. We take\n$\\bv' = \\dynkin[arrows=false,edge length=.6cm,labels={4,2}]{B}{o*}$.\nThen $\\bv^s = 2$, $\\bw^s = 4$. Hence the transversal slice is\n${\\mathfrak{M}}_0(\\bv^s,\\bw^s)$, which is the nilpotent orbit in\n$\\mathfrak{sl}(4)$ of type $(2^2)$. It is well known to be a\nnon-rational homology manifold. For example, we can argue as\nfollows. By \\cite[Th.~15.1.1]{Na-qaff}\n$\\mathop{\\text{\\rm dim}}\\nolimits i_0^! \\mathrm{IC}({\\mathfrak{M}}_0(\\bv^s,\\bw^s))$ gives the multiplicities\nof the trivial representation in the standard module for $\\bw^s$. The\nlatter is $4$th tensor power of the natural $2$-dimensional\nrepresentation of $\\algsl(2)$. The multiplicity of the trivial\nrepresentation is $2$.\n\\begin{NB}\n  $(y+y^{-1})^4 = y^4 + 4y^2 + 6 + 4y^{-2} + y^{-4}\n  = (y^4 + y^2 + 1 + y^{-2} + y^{-4}) + 3(y^2 + 1 + y^{-2}) + 2$.\n\\end{NB}%\nHence ${\\mathfrak{M}}_0(\\bv^s,\\bw^s)$ is \\emph{not} a rational homology manifold,\nand ${\\mathfrak{M}}_{\\zeta^\\bullet}(4\\delta,\\Lambda_0)$ neither.\n\nAlternatively we argue as follows.\nWe realize this variety as the Coulomb branch of a quiver\ngauge gauge theory of type $A_3$ with\n$\\bv'' = \\dynkin[edge length=.6cm,labels={1,2,1}]{A}{3}$,\n$\\bw'' = \\dynkin[edge length=.6cm,labels={0,2,0}]{A}{3}$. It is the\nclosure of a $\\operatorname{\\rm SL}(4)[[z]]$-orbit in the affine Grassmannian for\n$\\operatorname{\\rm SL}(4)$, and hence the intersection cohomology is known by geometric\nSatake correspondence. We have\n\\begin{equation*}\n  \\mathop{\\text{\\rm dim}}\\nolimits i^!_0\\mathrm{IC}(\\mathcal M_C(\\bv'',\\bw'')) = \\mathop{\\text{\\rm dim}}\\nolimits V_0(2\\Lambda_2),\n\\end{equation*}\nwhere $i_0$ is the embedding of the identity element to the affine\nGrassmannian. The right hand side is the $0$-weight space in the\nrepresentation of $\\operatorname{\\rm SL}(4)$ with the highest weight $2\\Lambda_2$. This\nweight space is $2$-dimensional.\n\nIn general, if ${\\mathfrak{M}}_0(\\bv^s,\\bw^s)$ is happen to be the closure of a\n$G[[z]]$-orbit in an affine Grassmannian $\\mathrm{Gr}_G = G((z))\/G[[z]]$\n(e.g., it is true in type $A$ by \\cite{MR1968260} or the combination\nof \\cite{2016arXiv160602002N} and \\cite{2016arXiv160403625B}), it is a\nrational homology manifold if and only if weight spaces of the\ncorresponding representation are all $1$-dimensional.\\footnote{The\n  author thanks Dinakar Muthiah for this remark.}\n\n\\bibliographystyle{myamsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn any information system which requires user interaction, being able to understand the user is a crucial requirement, which is often tackled by limiting the user input (e.g., presenting predefined forms with fixed options). The more freedom you provide the user with, the more difficult interpretation the computer has to do to achieve a useful interaction. In such a context, being capable of disambiguating the input words (i.e., associating each word with its proper meaning in a given context) is the starting point of any interpretation process done by the computer. \n\nUsually, such a disambiguation process is tackled from a Natural Language Processing (NLP) perspective~\\cite{Navigli2009}, assuming rich linguistic information, such as Part of Speech (POS), dependencies between words, etc.,  which is very useful to perform the task. However, due to the world wide use of Web search engines, users are very used to keyword interfaces and they still express their needs in such terms. In this scenario, although there are some studies which point out that \\emph{keyword search queries} (aka., \\emph{Web search queries}) exhibit their own language structure~\\cite{barr2008linguistic,pinter2016syntactic,roy2016syntactic}, we still need methods to disambiguate the meanings of the words which do not need  such an information as it might not be available. \n\n\nRecent advances in NLP have focused on the development of different embedding models~\\cite{bengio2003neural,mikolov2013distributed,le2014distributed,mancini2017embedding,camacho2016nasari,pennington2014glove}, which are a set of language modeling and feature learning techniques where elements from a vocabulary are mapped to a vectorial space capturing their \\emph{distributional semantics}~\\cite{sahlgren2008distributional}.\nWhile there are different methods to build word embeddings, the latest (and most successful) word embedding techniques rely on neural network architectures~\\cite{bengio2003neural,mikolov2013distributed,le2014distributed,pennington2014glove}.\nTheir usage as the underlying input representation has boosted the performance of different NLP tasks~\\cite{socher2013parsing,socher2013recursive}. However, in the context of disambiguation tasks, one of the main limitations of word embeddings is that the possible meanings of a word are combined into a single representation, i.e., a single vector in the semantic space. Such a limitation can be avoided by representing individual meanings of words as distinct vectors in the space (e.g., sense2vec~\\cite{mancini2017embedding}). However, there are scenarios where we do not know all the different senses at training time (e.g., open domain scenarios where we cannot find all the possible meanings in a sense catalog), and, even if we would know them, we would require to have annotated data (which might be unavailable or expensive to obtain). Besides, we would need to train a model for each new scenario or new meaning that would be added to our catalog. Thus, we need a disambiguation method able to relate words and their senses in a flexible and general way (i.e., independently of the domain we are working in) by exploiting the available resources.\n\nIn this paper, we propose a keyword disambiguation method which is based on the semantic relatedness (the degree in which two objects are related by any kind of semantic relationship~\\cite{budan2006evaluating}) between words, taking advantage of the semantic information captured by word embeddings. Our proposal makes possible to measure the relatedness not only among plain words but also among senses of words (which, in a Semantic Web context, can be expressed as ontological terms), and it is able to work independently of the resources used, i.e., the sense inventory whose meanings we want to map to and the word embedding model used as input.\n\nFor this purpose, we build on the work by Gracia~and~Mena on semantic relatedness~\\cite{wise08} and disambiguation~\\cite{gracia2009multiontology}. These works exploited the information about word co-occurrence frequencies provided by existing Web search engines. We evolve and adapt them to improve their performance using different kinds of embeddings (both at word and sentence level).\nThe main benefit of such an adaptation is two-fold:~1) we exploit the semantics captured by embeddings which goes further than co-occurrence of terms, and~2) we decouple the proposal from any Web search engine, being able to use off-the-shelf models trained by third parties for our purposes. This has an important side-effect: our measure can be easily adapted to any domain which we have a document corpus from. Indeed, this adaptation would require a training step, but it would be unsupervised and the only data required would be the corpus of documents itself. \n\nTo evaluate our approach, we have carried out a thorough experimentation in the context of Word Sense Disambiguation (WSD), where we have used different pre-trained word embeddings publicly available on the Web, and WordNet\\footnote{\\label{wordnetfootnote}\\url{https:\/\/wordnet.princeton.edu\/}} as sense repository. Our measure improves the performance obtained in~\\cite{wise08}, and achieves state of the art WSD values without the need of specific training for a specific sense inventory. This is especially relevant, for example, for systems based on keyword input and\/or which have to work with dynamically selected ontologies~\\cite{bobed2016querygen} or even with ontologies extracted directly from the Web~\\cite{Halevy15}. All the experimental data and evaluation results are available online\\footnote{\\url{https:\/\/bit.ly\/2lqCzop}}.\n\nThe rest of the paper is structured as follows. Section~\\ref{sec:relatedwork} discusses related works. In Section~\\ref{sec:approachdescription} we describe our semantic relatedness measure approximation, in  Section~\\ref{sec:disambalgorithm} we present the disambiguation algorithm that we use, and Section~\\ref{sec:experimentaleval} summarizes our experimental results. Finally, our conclusions and future work appear in Section~\\ref{sec:conclusionsandfuturework}.\n\n\\section{Related Work}\n\\label{sec:relatedwork}\nSemantic relatedness is the degree in which two objects are related by any kind of semantic relationship~\\cite{budan2006evaluating} and lies at the core of many applications in NLP (such as WSD, Information Retrieval, Natural Language Understanding, or Entity Recognition). The term is often confused with semantic similarity, which measures the degree in which two objects are similar or equivalent. For example, \"car\" is similar to \"bus\", but is also related to \"road\" and \"driving\". It has received a great research interest and different types of methods have been developed: it can be statistically estimated (e.g. co-occurrence-based methods~\\cite{landauer1998introduction}) and learned (e.g., distributional measures that estimate semantic relatedness between terms using a multidimensional space model to correlate words and textual contexts~\\cite{mohammad2012distributional}); or it can be computed using a taxonomy or a graph (e.g., ontologies) to define the distance between terms or concepts~\\cite{pirro2012reword}. Indeed, most methods rely on particular lexical resources (dictionaries, thesauri, or well structured taxonomies such as WordNet\\textsuperscript{\\ref{wordnetfootnote}}).\n\nRegarding disambiguation, WSD methods can be classified into four conventional approaches: supervised~\\cite{vial2018improving}, unsupervised~\\cite{correa2018word}, semi-supervised~\\cite{yuan2016semi}, and knowledge-based methods~\\cite{chaplot2018knowledge}. For example, in a way similar to us, in the SemEval 2015 All-Words Sense Disambiguation and Entity Linking task\\footnote{\\label{semeval2015footnote}\\url{http:\/\/alt.qcri.org\/semeval2015\/task13\/}}, the majority of the approaches (LIMSI, SUDOKU, EBL-Hope, etc.) that best performed in WSD relied on the combination of unsupervised learning of semantic information from the content of a corpus (such as SemCor) and\/or on lexical resources as sense inventories (such as WordNet or BabelNet) to disambiguate the sense of words in natural language sentences. However, to our knowledge, no previous works (excepting those of Gracia~and~Mena~\\cite{wise08,gracia2009multiontology}) have studied specific methods for the disambiguation of words in keyword-based inputs, where the linguistic information is scarce.\n\nRegarding the resources we use in our approach, word embeddings represent words in a low-dimensional continuous space and they are used to capture syntactic and semantic information from massive amounts of textual content. In recent years, they have gained great popularity due to this ability and many NLP applications have taken advantage of the potential of these distributional models. Bengio et al.~\\cite{bengio2003neural} preceded a wide number of current language model techniques and several authors have proposed their own approaches~\\cite{le2014distributed,pennington2014glove,camacho2016nasari} to construct word embeddings vectors where \\emph{word2vec}~\\cite{mikolov2013distributed} is the most widely used.\n\nDespite their advantages, one of the main limitations of word embeddings is that possible meanings of a word are conflated into a single representation. Sense embeddings (e.g., sense2vec~\\cite{mancini2017embedding}) are proposed as a solution to this problem: individual meanings of words are represented as distinct vectors in the space. These approaches are classified in two categories by how they model meaning and where they obtain it from~\\cite{camacho2018word}: 1) unsupervised models which learn word senses from text corpora (by inducing different senses of a word, analyzing its contextual semantics in a text corpus and representing each sense based on the statistical knowledge derived from the corpus), and 2) knowledge-based methods which exploit sense inventories of lexical resources for representing meanings (such as WordNet\\textsuperscript{\\ref{wordnetfootnote}}, Wikipedia\\footnote{\\label{wikifootnote}\\url{https:\/\/www.wikipedia.org\/}}, BabelNet\\footnote{\\label{bnfootnote}\\url{https:\/\/babelnet.org\/}}, etc.). We can also find models that not only provide representations of words, but also the senses of the words in a joint embedded space. This is the case of NASARI vectors~\\cite{camacho2016nasari} which  not only provide accurate representation of word senses in different languages, but they also include both concepts and named entities, all included in a single unified semantic space. \nHowever, in the first case (i.e., unsupervised models), we cannot target a particular sense inventory or ontology to perform the disambiguation, not having control for example about the concept detail\/granularity, and, besides, the detected senses might not be aligned to any particular human-readable structure; in the second case, we need to know all the senses at training time, not being able to adapt to new scenarios (e.g., addition\/deletion of senses in the directory, evolving ontologies, etc.). Thus, the sweet spot would be neither to require re-training nor newly labelled data, while being capable to perform the disambiguation against any sense repository.\n\n\n\n\nAlthough sense embeddings capture and represent information about meanings and can be used to calculate the sense that a word has in a specific context, word embeddings have also been shown to have good performance in disambiguation tasks~\\cite{iacobacci2016embeddings}. Therefore, we wanted to explore how we could push further the usage of word embeddings for keyword disambiguation. Working at word level (as starting point) allows us to use a semantic relatedness measure between terms and reuse available resources, without needing to train explicitly new word embeddings neither for a specific task nor for newly added possible senses (i.e., adapting to any given sense dictionary or ontology). We have taken as baseline the works presented in~\\cite{wise08,gracia2009multiontology}. In~\\cite{gracia2009multiontology}, the authors provide a keyword disambiguation algorithm that uses the semantic relatedness measure defined in~\\cite{wise08} to find the appropriate sense for keywords. The authors focused on a method that exploits the Web as a source of knowledge, and a transformation of the Normalized Google Distance (NGD)~\\cite{cilibrasi2007google} into a mixed way of relatedness measure (between ontology terms and plain words). \nWe propose to, on the one hand, substitute this distance with a word embedding based one to take advantage of the semantics captured by embeddings, improving the performance regarding using just co-occurrence of terms; and, on the other hand, explore modifications of their algorithm to improve their disambiguation capabilities.\n\nFinally, as pointed out by Lastra-D\\'iaz~et~al.~\\cite{lastra2019reproducible}, the embeddings that behave the best for disambiguation purposes are those which capture not only distributional semantics of texts, but also structural information about the possible meanings. We aim at achieving this disambiguation performance in a more flexible way, decoupling linguistic surface from the actual sense catalog (i.e., ontology) in order to adapt to new (i.e., unknown at training time) possible meanings, and being capable to apply it to keyword inputs, where the linguistic information is scarce.\n\n\\section{Relatedness Measure based on Word Embeddings}\n\\label{sec:approachdescription}\nWord embeddings can be used out-of-the-box to compute relatedness between words. However, they do not suffice in situations in which a relatedness has to be computed between senses (e.g., ontology terms in a Semantic Web context) or between senses and words. To that end, we ground on a previously defined relatedness measure between senses proposed by Gracia~and~Mena~\\cite{wise08}.\nThe authors proposed a method to compute the semantic relatedness between ontology terms (which we can see as individual senses), and an extension to calculate it between plain words and terms.\nTheir proposal was built on the notion of the \\emph{ontological context} of a term, which is constructed  combining the synonyms and the hypernyms of the ontological term (or sense). Given an ontological term ${t}$, they defined its \\emph{ontological context} (denoted by ${OC(t)}$) as the minimum set of other ontological terms that belong to its semantic description, locating the term in the ontology and characterizing its meaning. For example, in the WordNet taxonomy, the class ``Java'' (in the sense of ``an Indonesian island''), is well characterized and distinguished from other senses by considering its direct hypernym ``Island'' (see Figure~\\ref{fig:Java}).\n\n\\begin{figure}\n\\includegraphics[width=8cm]{sense_example.png}\n\\caption{Example of the semantic description of the term \"Java\" in WordNet.}\n\\label{fig:Java}\n\\end{figure}\n\n\nThen, given two ontological terms $a$ and $b$, their relatedness measure is computed as: \n\n\\begin{align}\n\\label{eq:completeeq}\n\\begin{split}\n  rel(a,b)=w_0rel_0(a,b)+w_1rel_1(a,b), \\\\\n  (w_0 \\geq 0, w1 \\geq 0, w_0+w_1=1) \n\\end{split}\n\\end{align}\n\n\\noindent\nwith ${rel_0(a,b)}$ and ${rel_1(a,b)}$ computed as follows: \n\\begin{align}\n\\label{eq:level0}\n\\begin{split}\nrel_0(a,b)=\\frac{\\sum_{i, j}rel_{w}({syn_a}_i,{syn_b}_j)}{|Syn(a)||Syn(b)|}, \\\\\n(i=1..|Syn(a)|, j=1..|Syn(b)|)\n\\end{split}\n\\end{align}\n\\begin{align}\n\\label{eq:level1}\n\\begin{split}\nrel_1(a,b)=\\frac{\\sum_{i, j}rel_0({oc_a}_i,{oc_b}_j)}{|OC(a)||OC(b)|}, \\\\\n(i=1..|OC(a)|, j=1..|OC(b)|)\n\\end{split} \n\\end{align}\n\n\\noindent where ${rel_{w}}$ refers to the relatedness between words (as it will be defined later on in equations 7 and 9); ${Syn(a)=\\{{syn_a}_1, {syn_a}_2, ...\\}}$ and ${Syn(b)=\\{{syn_b}_1, {syn_b}_2, ...\\}}$ are the set of synonyms (equivalent labels, including the term label) of ontological terms \\textit{a} and \\textit{b}; ${OC(a)=}\\\\{\\{{oc_a}_1, {oc_a}_2, ...\\}}$ and ${OC(b)=\\{{oc_b}_1, {oc_b}_2, ...\\}}$ are the terms of their ontological context\\footnote{Notice that ${|Syn(x)| \\geq 1}$ and ${|OC(x)| \\geq 0}$.}. Each \\textit{a} and \\textit{b} is characterized by taking into account two levels of their semantic description: \\textit{Level 0)} the term label and its synonyms (Equation~\\ref{eq:level0}), and \\textit{Level 1)} its ontological context (Equation~\\ref{eq:level1}). $w_0$ and $w_1$ are used to weight these levels\\footnote{We set these values as $w_0 = w_1 = 0.5$ as indicated in~\\cite{wise08}.}.\n\nThis measure can be also applied between an ontology term ${t}$ and a plain word ${w}$ which provides us with a value which indicates the relatedness degree between a sense and a word. So, in that case, the previous equations are computed as follows:\n\\begin{align}\n\\begin{split}\nrel(t,w)=w_0rel_0(t,w)+w_1rel_1(t,w), \\\\ \n(w_0 \\geq 0, w1 \\geq 0, w_0+w_1=1)\n\\end{split} \\label{eq:termWord}\n\\end{align}\n\\begin{equation}\nrel_0(t,w)=\\frac{\\sum_{i,j}rel_{w}({syn_t}_i,w)}{|Syn(t)|}, \\\\\n(i=1..|Syn(t)|)\n\\end{equation}\n\\begin{equation}\nrel_1(t,w)=\\frac{\\sum_{i,j}rel_0({oc_t}_i,w)}{|OC(t)|}, \\\\\n(i=1..|OC(t)|)\n\\end{equation}\n\nHere, ${rel_{w}}$ is the distance that the authors used in~\\cite{wise08} to measure how two plain words are related. They proposed a generalization of the Cilibrasi and Vit\\'anyi's Normalized Google Distance \\textit{NGD(x,y)}~\\cite{cilibrasi2007google} to use any Web search engine as source of frequencies. This generalization is called Normalized Web Distance \\textit{NWD(x,y)}, whose smaller values represent greater semantic relation between words. Although most of NWD values fall between 0 and 1, it ranges from 0 to ${\\infty}$. Therefore, to obtain a proper relatedness measure in the range [0, 1] that increases inversely to distance, they proposed the following transformation:\n\\begin{equation}\n\\label{eq:relweb}\nrel_w(x, y)=relWeb(x, y)=e^{-2NWD(x,y)}\n\\end{equation}\n\n\n\nTo explore the use of emerging word-embedding techniques in this context and compare them with those based on search engines, we propose to exploit the semantic capabilities of word embeddings in this formulation and substitute the ${relWeb(x,y)}$ measure. We could use the cosine similarity distance between the embedding vectors of the words, i.e., using the following equation:\n\\begin{equation}\nsim(x,y)=cos(\\theta)=\\frac{X1 \\bullet X2}{||X1||\\cdot||X2||}\n\\end{equation}\n\\noindent where ${x}$ and ${y}$ are plain words, ${X1}$ and ${X2}$ their correspondent word embedding vectors, and ${\\theta}$ the angle between them. However, ${sim(x,y)}$ ranges in [-1, 1], so, in order to obtain a \ndistance in the range [0, 1] (so that we can substitute Equation~\\ref{eq:relweb} directly in Equation~\\ref{eq:level0}), we propose to use the \\textit{angular distance} instead, which is computed as follows:\n\\begin{equation}\n\\label{eq:arccos}\nrel_w(x, y)=ang. distance(x,y)=1-\\frac{arccos(sim(x,y))}{\\pi}\n\\end{equation}\n\n\\noindent So, in Equation~\\ref{eq:level0}, we use Equation~\\ref{eq:arccos} as ${rel_{w}}$ distance instead of Equation~\\ref{eq:relweb}.\nWe use this distance to compute the semantic relatedness between words, between ontology terms (or senses), or between ontology terms and words, obtaining a value between 0 and 1. For those cases in which the label of the ontological term is multi-word, we just compute the centroid of the set of words that form the label. While, at first, it might seem that we limit the coverage of the measure proposed in~\\cite{wise08} (it built on the results of Web search engines, which potentially cover any domain), we have to bear in mind the plethora of word embedding models directly available in the Web, as well as the possibility of using our own corpus of documents to fine tuning our measure for a particular domain (which is easier to have, rather than crawling the whole WWW).\n\n\\section{Disambiguation Algorithm}\n\\label{sec:disambalgorithm}\nWe ground our keyword disambiguation proposal on the disambiguation algorithm defined in~\\cite{gracia2009multiontology}, using the adapted semantic relatedness measure proposed in the previous section. This algorithm is based on the hypothesis that the most significant words in the disambiguation context are the most highly related to the word to disambiguate; such words conform the \\emph{active context} of the word being disambiguated.\n\nAs an overview, once the \\emph{active context} of each input keyword has been calculated, the algorithm performs three main steps:~1)~obtaining the semantic relatedness between the active context of a keyword and its possible senses,~2) calculating the overlap between the words in the active context and the semantic descriptions (i.e., \\emph{ontological context}) of the\npossible senses of the keyword to disambiguate, and 3) re-ranking the possible senses according to their frequency of use (only when such information is available for the sense inventory selected\\footnote{If we do not have such information, we assume that all senses are equally likely.}).\nApart from using the updated $rel_w$ measure to select the \\emph{active contexts}, we propose to modify the second step of this algorithm in order to study the influence of different approaches which exploit the semantic information captured by different word embeddings.\nIn the following subsections, we first detail the original algorithm which we base our proposal on, and, then, we describe the modifications that we propose to improve its performance using word embeddings.\n\n\\subsection{Background: Algorithm Description}\n\\label{sec:algorithmdescription}\nFirst of all, let us formally introduce the notion of \\emph{active context}. Let~${k}$ be an element of an input sequence of words~${\\mathbb{S}}$ with an intended meaning, ${\\mathbb{K} \\subseteq \\mathbb{S}}$ be the set of all possible keywords in the input, ${C \\subseteq \\mathbb{K}}$ the set of keywords of the disambiguation context (i.e., the complete disambiguation window considered, e.g., the sentence where the keyword appear), and ${k_d \\in \\mathbb{K}}$ the target keyword to disambiguate. \nThus:\n\\begin{definition}\nGiven a context $\\textit{C} \\in \\mathbb{K}$, and a word to disambiguate ${k_d} \\in \\mathbb{K}$, the \\textit{active context} ${C_a}$ of $k_d$ is a subset ${C_a \\subseteq C}$ such that \n$\\forall k_i \\in C_a, \\nexists k_j \\in (C - C_a)~\\ni rel(k_j,k_d) > rel(k_i,k_d)$.\n\\end{definition}\nIn other words, $C_a$ contains the words in the input that are the most related ones to $k_d$. \nTo obtain such a context, we stick to the method proposed in~\\cite{gracia2009multiontology}:~1)~removing repeated words and stopwords from \\textit{C}, ~2) applying a semantic relatedness (${rel_{w}}$ in our case) between each context word ${k_i \\in C}$ and the keyword to disambiguate ${k_d}$, and ~3) constructing ${C_a}$ with the context words whose relatedness scores above a certain threshold. The output of this process is the \\textit{active context} ${C_a \\subseteq C}$. The maximum cardinality of ${C_a}$ is set to a fixed  value(${|C_a| = 4}$) following Kaplan's experiments~\\cite{kaplan1955experiment}.\n\nOnce we have obtained $C_a$ for $k_d$, we\ncan apply the main algorithm, which takes as input $k_d$, $C_a$, and a set of possible senses for $k_d$, ${S_{k_d}}$.\nThe main steps are presented in Algorithm~\\ref{alg:disambiguation}\\footnote{We refer the interested reader to~\\cite{gracia2009multiontology} for the complete details.}: \n\n\n\\begin{algorithm}\n\n\\caption{Keyword disambiguation algorithm}\\label{alg:disambiguation}\n    \\SetKwInOut{Input}{Input}\n    \\SetKwInOut{keyword}{$K_d$}\n    \\SetKwInOut{senses}{$S_{k_d}$}\n    \\SetKwInOut{context}{$C_a$}\n    \\SetKwInOut{Output}{Output}\n    \n    \\Input{}\n    \\keyword{The keyword to disambiguate.}\n    \\senses{The set of possible senses for $K_d$.}\n    \\context{The active context selected for $K_d$.}\n    \\Output{A weight for each sense ${s_i \\in S_{k_d}}$.}\n    \\textbf{function} disambiguate $(K_d, S_{k_d}, C_a)$:\\\\\n    \\ForEach{sense $s_i \\in S_{k_d}$}{\\label{line:semrelfrom}\n        \\ForEach{keyword $k_j \\in C_a$}{\n        $r_j=rel(s_i,k_j)$\n        }\n        $score_{s_i}=\\sum_{j}r_j\/|C_a|$\n    }\\label{line:semrelto}\n    $maxScore=max(score_{s_1},\\ldots,score_{s_n})$ \\label{line:overlapfrom}\\\\\n    \\ForEach{sense $s_i \\in S_{k_d}$}{\n       \n        $newScore=score_{s_i}+(1-maxScore)*overlap(C_a,OC(s_i))$\n        $score_{s_i}=newScore$\\label{line:overlap}\n    }\\label{line:overlapto}\n    $maxScore=max(score_{s_1},\\ldots,score_{s_n})$\\label{line:freqfrom} \\\\\n    \\ForEach{sense $s_i \\in S_{k_d}$}{\n        \\If{$score_{s_i} > proximityFactor*maxScore$}{\n            $newScore=score_{s_i}+(1-maxScore)*normFreq(s_i)$\n            $score_{s_i}=newScore$\n        }\n    }\\label{line:freqto}\n\\end{algorithm}\n\n\\begin{enumerate}\n    \\item \\textbf{Applying the semantic relatedness:} First, the algorithm computes an initial disambiguation between the senses in ${S_{k_d}}$ and the active context $C_a$ (Lines~\\ref{line:semrelfrom}-\\ref{line:semrelto}). For this, we use the updated relatedness measure\n    presented in the previous section (Equations~\\ref{eq:termWord} and~\\ref{eq:arccos}). The score assigned to each sense (${score_{s_i}}$) is the mean of $rel(s_i,k_j)$ where ${s_i \\in S_{k_d}}$ is a candidate sense of the keyword being disambiguated, and ${k_j \\in C_a}$ is a keyword in the active context.\n    \\item \\textbf{Calculating the context overlap:} The disambiguation algorithm weights the scores taking into account the overlap between $C_a$ and the\n    ontological context of each sense, ${OC(s_i)}$ (Lines~\\ref{line:overlapfrom}-\\ref{line:overlapto}).\n    Note that ${OC(s_i)}$ includes its synonyms, glosses, and labels, as well as labels of other related terms, such as hypernyms, hyponyms, meronyms, holonyms, etc. The overlap is calculated (ignoring stopwords) as:\n    $$overlap(C_a,OC(s_i)) =\\frac{|OC(s_i)\\cap C_a|}{min(|{OC(s_i)}|,|C_a|)}$$\n   \n   \n   \n   \n    \\item \\textbf{Frequency of usage:} Finally,\n    the frequency of use of the highest scored senses is taken into account (Lines~\\ref{line:freqfrom}-~\\ref{line:freqto}), if such information is available. The proximity decision is handled by a ${proximityFactor\\in[0,1]}$, which is combined with the maximum of the scores of the senses ($proximityFactor * maxScore$) to obtain a threshold. The scores of the senses $s_i$ which are above such a threshold are then updated using:\n   \n   \n    $$normFreq({s_i})=\\sqrt{a*\\frac{frequency_{s_i}}{\\sum_{j}frequency_{s_j}}+b}$$ \n    \n    where $\\sum_{j}frequency_{s_j}$ is equal to the sum of the frequency of all senses of $k_d$, $a$ and $b$ are constrained\\footnote{We set $a = b = 0.5$ and $proximityFactor = 0.75$ as indicated in~\\cite{gracia2009multiontology}.} by $a,b \\in [0,1]$ and $a + b = 1$.\n\\end{enumerate}\n\nThe output of the disambiguation algorithm is a\nscore for each possible sense ${s_i \\in S_{k_d}}$ that represents the confidence level of being the right sense according to the active context $C_a$.\nNote that, in our approach, ${S_{k_d}}$ is not restricted to any particular dictionary, as it could be dynamically built from, e.g., different ontological resources.\n\n\\subsection{Proposed Modifications}\n\\label{sec:proposedmodifs}\nAs our aim is to study the best way to exploit word embeddings, we have analyzed their characteristics and explored different approaches to use them in the adopted disambiguation process. \nIn particular, in this section, we present a list of possible modifications to the Step~2 of the algorithm (Lines~\\ref{line:overlapfrom}-\\ref{line:overlapto}) to include and take advantage of the properties of word embeddings\nalong with the rationale behind them. For the rest of the section, let ${maxScore}$ be the maximum score among all senses in ${S_{k_d}}$, ${centroid}$ a function to calculate an average vector by the arithmetic mean of a set of vectors, and ${rel_{w}}$ the angular distance in Equation~\\ref{eq:arccos}. \nThus, the different approaches are described below:\n\n\\begin{itemize}\n    \\item \\textbf{Average:} The straightforward way to include the embeddings is to calculate directly the average vector of the different bag of words involved in the disambiguation, under the assumption that the semantically coherent groups of words should outstand from the others. Thus, instead of computing the overlap between the semantic descriptions $OC(s_i)$ of each sense $s_i$ and the current active context $C_a$, we propose to compute the average between the word vectors from ${C_a}$ and ${OC(s_i)}$ to obtain a new score. Line~\\ref{line:overlap} in Algorithm~\\ref{alg:disambiguation} changes to:\n    $$\\hspace{0.80cm}newScore(s_i)=score_{s_i}+(1-maxScore)*average({C_a},{OC(s_i)})$$\n    where ${average({C_a},{OC(s_i)})}$ is:\n    \\begin{align*}\n    \\begin{split}\n    average({C_a},{OC(s_i)})=\\frac{\\sum_{i, j}rel_{w}(k_i,w_j)}{|{C_a}||{OC(s_i)}|}, \\\\\n    (k_i \\in {C_a}, w_j \\in {OC(s_i)},  i=1..|{C_a}|, j=1..|{OC(s_i)}|)\n    \\end{split}\n    \\end{align*}\n    That is, we consider each set of words as a cluster in the vector space, and we represent them by their centroid. If there are elements that do not contribute to the semantic cohesion of the clusters, they will contribute negatively (they will increase the semantic distance) to select a particular sense for the target keyword\\footnote{We also studied other cluster-based distances measures (e.g., single linkage), but the results did not improve using the centroid-based measure, so we focused on the average vector which is broadly used in the literature.}.\n    \n    \\item \\textbf{Sense centroid without most frequent component:} As an evolution of the previous method, we studied the method described by Arora et al.~\\cite{arora2016simple}, called \\textit{Smooth Inverse Frequency (SIF)}. They propose to represent a sentence by a weighted average vector of its word vectors which the most frequent component using PCA\/SVD is substracted from. Thus, we propose to consider the semantic description ${OC(s_i)}$ of all senses of the sense inventory as sentences, and to calculate the SIF embedding of them. Then, during the disambiguation, we compute a new score (Line~\\ref{line:overlap}) for the sense being considered ${s_i}$ by measuring the distance between the centroid of the active context ${C_a}$ and the SIF vector of each $OC(s_i)$, following this computation:\\vspace{0.10cm}\n    \n    $\\hspace{0.25cm}newScore(s_i)=score_{si}+(1-maxScore)*$\n    \n    $\\vspace{0.15cm}\\hspace{2.30cm}rel_{w}(centroid({C_a}), SIF({OC(s_i)}))$\n    \n    Note that we do not substract the SIF vector from $C_a$ as all its words are already deemed as important.\n    The most frequent component vector we are removing may encompass those words that occur most frequently in a corpus and lack semantic content (e.g., stop-words),\n    thus not contributing to the actual disambiguation.\n    \n    \\item \\textbf{Top-K nearest words:} As a variant of the two previous methods, in this method, we select the top \\emph{k} nearest words from the semantic description ${OC(s_i)}$ of a sense to ${C_a} \\cup k_d$. After that, we compute the distance between centroids of the active context and the top K nearest words selected to obtain its new score: \\vspace{0.1cm}\n    \n    $\\hspace{0.25cm}newScore(s_i)=score_{s_i}+(1-maxScore) *$\n    \n    $\\hspace{2.30cm}rel_{w}(centroid({C_a}),centroid($\n    \n    $\\vspace{0.15cm}\\hspace{2.8cm}topKNearest(centroid({C_a}),{OC(s_i)})))$\n    \n   \n    In this case, we work under the same hypothesis as for the selecing an \\emph{active context}: the words that belong to the semantic description of the sense that are the closest ones to the active context and the keyword that is being disambiguated,\n    should be the most significant to contribute in making a correct disambiguation. \n    \n    \\item \\textbf{Doc2vec:} Finally, instead of treating the ontological descriptions as bag of words in this method, we consider them as proper documents and apply \\textit{doc2vec}~\\cite{le2014distributed}. In particular, each semantic description ${OC(s_i)}$ of the senses becomes a document, and \\textit{doc2vec} allows to calculate an embedding space for all of them. Then, we compute the distance between the centroid of the active context ${C_a}$ and the embedding calculated for the sense. Note that \\textit{doc2vec} learns as well a word embeddings model that it uses during training. We use those word vectors to create the centroid of the active context. Therefore, in a similar way, the new score is computed as:\\vspace{0.10cm}\n    \n    $\\hspace{0.25cm}newScore(s_i)=score_{s_i}+(1-maxScore)*$\n    \n    $\\vspace{0.15cm}\\hspace{2.30cm}rel_{w}(centroid({C_a}), doc2vec({OC(s_i)}))$\n    \n    We consider the semantic descriptions as documents\n    to capture the distributional semantics both at local (window) and global (document) scope.\n    \n\\end{itemize}\n\nWe report the best results that we obtained by applying these different approaches in the following section.\n\n\\section{Experimental Evaluation}\n\\label{sec:experimentaleval}\nIn this section, we discuss the results obtained in the experiments that we have carried out to evaluate our proposal.\nFirstly, we evaluated different available embedding models using the distance proposed in Equation~\\ref{eq:arccos}. We performed several tests comparing to human judgment in order to check how the \\emph{angular distance} behaved. Secondly, we evaluated the potential of our keyword disambiguation algorithm and the relatedness measure among ontology terms and words in the context of WSD, including all algorithm variations that we have proposed in Section~\\ref{sec:proposedmodifs}.\n\nFor the experiments, we used the following pre-trained vectors: \\textit{word2vec} trained on Google News corpus\\footnote{\\url{https:\/\/code.google.com\/archive\/p\/word2vec\/}}, \n\\textit{word2vec} trained on Wikipedia\\footnote{\\label{laugithub}\\url{https:\/\/github.com\/jhlau\/doc2vec}}, \\textit{doc2vec} trained also on Wikipedia\\footnote{Dump dated in 2015-12-01.}, \\textit{GloVe} trained on Wikipedia 2014 and Gigaword 5 corpus\\footnote{\\url{https:\/\/nlp.stanford.edu\/projects\/glove\/}}, and ${NASARI_{embed}}$ along with the \\textit{word2vec} word embeddings trained on the UMBC corpus\\footnote{\\url{http:\/\/lcl.uniroma1.it\/nasari\/\\#two}}. We used WordNet\\textsuperscript{\\ref{wordnetfootnote}} as the sense inventory.\n\n\\subsection{Correlation with Human Judgment}\n\\label{sec:correlation}\nIn order to validate the hypothesis of the suitability of using word embeddings along with the use of the \\emph{angular distance} to compute semantic relatedness, we first analysed the correlation of such a technique with human judgment in a basic word-to-word comparison. For this purpose, we used\ndifferent datasets available in the literature which contain pairs of words whose relatedness was manually assessed by different people. The datasets and their details can be seen in Table~\\ref{tab:datasets}.\n\n\n\\begin{table}[htb!]\n    \\caption{Correlation with human judgment benchmarks.}\n    \\centering\n    \\begin{tabular}{|c|c|c|c|}\n    \\hline\n        Dataset & \\#Word Pairs & \\#Human Judges \\\\\n        \\hline\n        MC-30~\\cite{miller1991contextual} & 30 & 38 \\\\\n        WordS353-Rel~\\cite{finkelstein2001placing} & 252 & 13 \\\\\n        WordS353-Sim~\\cite{finkelstein2001placing} & 203  & 16 \\\\\n        RG-65~\\cite{rubenstein1965contextual} & 65 & 51 \\\\\n        MEN dataset (train\/dev)~\\cite{bruni2014multimodal} & 2000\/1000 & crowdsourced\\footnotemark \\\\\n        GM dataset~\\cite{wise08} & 30 & 30 \\\\\n        \\hline \n    \\end{tabular}\n    \\label{tab:datasets}\n\\end{table}\n\\footnotetext{They used Amazon Mechanical Turk:  \\url{https:\/\/www.mturk.com\/}}\n\nThe results obtained for Spearman correlation are presented in Table~\\ref{tab:spearmancorrel}. Reported values, where available, were calculated using the widespread used cosine similarity. We can see that using the angular distance (Equation~\\ref{eq:arccos}) to calculate relatedness between pairs of words also offers a semantic correlation with the human judgment.\nIn particular, regarding the GM dataset~\\cite{wise08}, the authors reported a 78\\% using the previous \\emph{relWeb} measure (Equation~\\ref{eq:relweb}). We can see a strong improvement in this dataset by using word embeddings: we achieve up to a 87.3\\% using \\textit{word2vec} trained on Google News (taking into account the average of all models, we achieve an average of 81.2\\% for this dataset). These results enable us to use the angular distance as the core relatedness measure in Equations~\\ref{eq:completeeq} to~\\ref{eq:level1}. Note that for \\textit{word2vec} and \\textit{doc2vec} trained on Wikipedia we can not provide a comparison, because Lau \\& Baldwin~\\cite{lau-baldwin-2016-empirical} did not evaluate the correlation with human judgment.\n\n\\begin{table*}[!htb]\n  \\caption{Spearman correlation coefficients between the angular distance applied on word pairs and human judgment in different datasets. Upper values are our evaluations, lower ones are the reported values in the original papers using the cosine distance. Highlighted values equal or outperform the best result (78\\%) for the same dataset in~\\cite{wise08}.}\n  \\label{tab:spearmancorrel}\n  \\begin{tabular}{|c|c|c|c|c|c|c||c|}\n    \\hline\n    Vectors\\textbackslash Datasets & MC-30& WS353-Sim& WS353-Rel& RG-65& MEN& GM & Average\\\\\n    \\hline\n    GloVe & 70.4& 66.5& 56.1& 76.9& 74.2& \\textbf{84.5} & 71.4\\\\\n    \\hdashline\n    Reported at~Pennington et al.~\\cite{pennington2014glove} & 72.7 & 65.8 & - & 77.8 & - & - & 72.1\\\\\n    \\hline\n    Google News word2vec& 80.0 & 77.2& 63.5& 76.0& 77.0& \\textbf{87.3} & 76.8\\\\\n    \\hdashline\n    Reported at Camacho-Collados et al.~\\cite{camacho2016nasari} & 80.0 & 77.0 & - & - & - & - & 78.5\\\\\n    \\hline\n    ${NASARI_{embed}+UMBCw2v}$& 70.3& 72.7& 56.8& 70.7& 74.5& 74.7 & 70.0\\\\\n    \\hdashline\n    Reported at Camacho-Collados et al.~\\cite{camacho2016nasari} & 83.0 & 68.0 & - & 80.0 & - & - & 75.5\\\\\n    \\hline\n    Wikipedia word2vec& 80.9& 77.9& 62.2& 78.3& 76.9& \\textbf{81.8} & 76.3\\\\\n    \\hdashline\n    Reported at Lau \\& Baldwin~\\cite{lau-baldwin-2016-empirical} & - & - & - & - & - & - & -\\\\\n    \\hline\n    Wikipedia doc2vec& 73.3& 69.0& 52.3& 71.6& 72.0& \\textbf{77.8} & 69.3\\\\\n    \\hdashline\n    Reported at Lau \\& Baldwin~\\cite{lau-baldwin-2016-empirical} & - & - & - & - & - & - & -\\\\\n    \\hline\n  \\end{tabular}\n\\end{table*}\n\n\\subsection{Word Sense Disambiguation Evaluation}\nTo evaluate our proposal, we used three datasets oriented to WSD: SemCor 2.0 dataset\\footnote{\\url{http:\/\/web.eecs.umich.edu\/\\~mihalcea\/downloads.html\\#semcor}}, SemEval2013 all-words WSD dataset\\footnote{\\label{semeval2013footnote}\\url{https:\/\/www.cs.york.ac.uk\/semeval-2013\/task12.html}}, and SemEval2015 All-Words Sense Disambiguation and Entity Linking dataset\\textsuperscript{\\ref{semeval2015footnote}}.\nWe used WordNet\\textsuperscript{\\ref{wordnetfootnote}} as sense inventory. For SemCor~2.0, we specifically used WordNet~2.0 as such a dataset is annotated with this version, and for the rest of datasets we used WordNet 3.0.\n\nWe tested all the options proposed in Section~\\ref{sec:proposedmodifs} for the disambiguation algorithm, and we obtained that the \\textit{Top-K nearest words} option achieves the best results\\footnote{Achieving the best performance for ${K=15}$.}. Thus, due to space restrictions, we focus on \\textit{Top-K nearest words} option in this section\\footnote{\\label{expsurl}The interested reader can find all the details of the experiments at \\url{https:\/\/bit.ly\/2lqCzop}}.  \nRegarding the models, we selected \\textit{word2vec} trained in Google News and \\textit{word2vec} trained in Wikipedia \nbecause they showed better average correlation with human judgment in different datasets (see Table~\\ref{tab:spearmancorrel}); and ${NASARI_{embed}+UMBCw2v}$ word embeddings because, although they do not excelled in correlation with human judgment, they showed the best performance in all test datasets for WSD. Finally,\nin order to compare the results to~\\cite{wise08}, we report the precision results for SemCor 2.0; while we report the F-score results for the rest of datasets.\n\n\\subsubsection*{SemCor2.0 Experiments:}\nFollowing~\\cite{wise08}, in this set of experiments, for each of three selected highly ambiguous nouns (\\textit{plant}, \\textit{glass}, and \\textit{earth}), we took~10 random sentences from the corpus. Table~\\ref{tab:semcor2} presents the results: all cases outperform the results achieved in~\\cite{wise08}, which reported an average precision of 57\\%. Our best performance is an average precision of 63.15\\% with ${NASARI_{embed}+UMBCw2v}$ vectors.\nIn fact, SIF method shows equal or even slightly better performance in this particular dataset using ${NASARI_{embed}+UMBCw2v}$ vectors and \\textit{word2vec} trained in Google News vectors. However, in the rest of cases it is \\textit{Top-K nearest words} method that obtains the best results. \nIn addition, SIF method requires to preprocess the target sense inventory to calculate the sentence embeddings, introducing a mild dependence to it.\nOur selected method shows a good performance (it improves the results of the original algorithm), while allowing to be more decoupled from the actual sense inventory used.\n\n\\begin{table*}[!htb] \n  \\caption{Precision results for SemCor 2.0 dataset (10 random sentences) adopting Top-K nearest words. The two rightmost columns show the results using the \\textit{relWeb} based relatedness measure and Most Frequent Sense methods as reported in~\\cite{wise08}.}\n  \\label{tab:semcor2}\n  \\begin{tabular}{|c|c|c|c||c|c|}\n    \\hline\n     Experiment\\textbackslash Approach& Wikipedia word2vec & Google News word2vec & ${NASARI_{embed}+UMBCw2v}$ & \\textit{relWeb}* & Most Freq. Sense*\\\\\n    \\hline\n    10 sent. with PLANT& 58.44\\%& 63.03\\%& 66.20\\%& 80\\% & 40\\%\\\\\n    10 sent. with GLASS& 57.47\\%& 63.78\\%& 60.15\\%& 30\\% & 30\\%\\\\\n    10 sent. with EARTH& 59.21\\%& 56.38\\%& 62.33\\%& 60\\% & 60\\%\\\\\n    \\hline\n    AVERAGE& 58.41\\%& 61.13\\%& 63.15\\%& 57\\%& 43\\%\\\\ \n    \\hline\n  \\end{tabular}\n\\end{table*}\n\n\\subsubsection*{SemEval Results:} In Table~\\ref{tab:fscore}, we present the results obtained for SemEval 2013 and SemEval 2015. In this case, ${NASARI_{embed}+UMBC}\\\\{w2v}$ vectors achieved the best results, with an F-score of 64.39\\%. In SemEval 2013, UMCC-DLSI reported the best results, with a F-score of 64.7\\%, similar to ours. Besides, our results are similar to other state-of-the-art systems using sense embeddings\\KILL{vectors}: Camacho et al.~\\cite{camacho2016nasari} reported an F-Score value of 66.7\\%  with their ${NASARI_{lexical}}$ vectors evaluated in SemEval 2013. Unfortunately, they do not provide results for their ${NASARI_{embed}}$ vectors using WordNet.\nRegarding SemEval 2015, the best reported result in our task reached an F-score of 65.8\\%, while the baseline used to compare systems (BabelNet First Sense (BFS)) was an F-score of 67.5\\%. We reach an F-score value of 61.61\\%, which, while does not beat previous values, is meritory given that our approach is focused on situations where linguistic information might be scarce (e.g., keyword-based input).\n\n\\begin{table*}[!htb]\n  \\caption{F-Score results for SemEval 2013 and 2015 datasets  adopting Top-K nearest words\\KILL{and using WordNet 3.0 as sense inventory}. Values in column highlighted by ${\\diamond}$ correspond to the best system in each SemEval dataset. Values in column highlighted by ${^\\pm}$ are the baselines reported in SemEval 2013~and SemEval 2015.}\n  \\label{tab:fscore}\n  \\begin{tabular}{|c|c|c|c||c|c|}\n    \\hline\n     & Wikipedia word2vec & Google News word2vec & ${NASARI_{embed}+UMBCw2v}$ & Best system${^\\diamond}$ & Baseline${^\\pm}$ \\\\\n    \\hline\n    SemEval 2013 & 59.59\\% & 62.81\\% & 64.39\\% & 66.7\\% & 63.0\\%\\\\\n    SemEval 2015 & 61.32\\% & 60.37\\% & 61.61\\% & 65.8\\% & 67.5\\%\\\\\n    \\hline\n  \\end{tabular}\n\\end{table*}\n\nTo sum up, our proposal improves the results presented in~\\cite{wise08} by substituting their Web search engine-based measure to one that uses word embeddings. We also improve the disambiguation results reported in~\\cite{gracia2009multiontology} by adapting their algorithm to exploit the properties of the word embeddings. Our proposal achieves similar performance levels to the SOTA, while providing the flexibility to work independently of the resources used (i.e., word embeddings, sense inventory), and reducing the barriers to its application to any domain.\n\n\\section{Conclusions and future work}\n\\label{sec:conclusionsandfuturework}\nIn this paper, we have presented a keyword disambiguation approach based on a semantic relatedness measure which exploits the semantic information captured by word embeddings and tries to map the meanings from a sense inventory. We have visited the semantic relatedness measure proposed in~\\cite{wise08} to adapt it to work with word embeddings instead of relying on Web search engines, and we have improved a disambiguation algorithm~\\cite{gracia2009multiontology} by exploring different uses and types of embeddings (both at word and sentence level).\n\nTo validate our proposal, we have performed several experiments around Word Sense Disambiguation (WSD) where we have used different pre-trained word embeddings and WordNet as the resource to obtain the target senses of words. With our proposal:\n\\begin{itemize}\n    \\item We are able to relate words and meanings from a sense inventory (e.g., ontology terms) in a flexible way, by exploiting available resources and regardless the domain in which we are working. This makes our measure adaptive and general enough to be used for different contexts. \n    \\item We provide a method which can be adapted to any domain in a dictionary-decoupled way, provided that we have a document corpus which would allow us to capture the distributional semantics. This lowers the requirements of data in order to build more specific models for particular domains. \n    \\item We have tested the capabilities of different word embedding models, improving the results presented in~\\cite{wise08}. We evaluated our measure in the same SemCor 2.0 dataset described in this work, and we have obtained in the best case an average increase of 6\\% in precision (a relative improvement of about~11\\%).\n    \\item Being decoupled from a fixed pool of senses does not come at the expense of performance. We achieve similar quality of results than having an ad hoc and more expensive trained model capturing the possible senses. In particular, we have tested our measure in SemEval 2013 and SemEval 2015 datasets reaching an F-score of 64.39\\% and 61.61\\% respectively. These results are similar to the state of the art~\\cite{camacho2016nasari} using \\textit{sense2vec} approaches.\n\\end{itemize}\n\nAs future work, we want to further extend our approach to the field of concept discovery (similar to entity search~\\cite{Balog:2018:ER}, but focused on concepts rather than on instances). We also want to explore newer contextualized word embeddings, such as BERT or XLNet (based in ELMo~\\cite{peters2018deep}), and how they could be used in this context.\nFinally, we would like to propose a specific dataset exclusively for keyword disambiguation taking QALD\\footnote{QALD is a series of evaluation campaigns on Question Answering over Linked Data: (\\url{http:\/\/qald.aksw.org}).} datasets as baseline; we want to develop it in order to test our relatedness measure in a more appropriate dataset for the context in which we focus: the disambiguation of keyword-based inputs.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n  Finding a new chaotic system began from the last six decades to understand different natural and man made systems. Still scientific communities are investigating new chaotic system due to its practical applications in engineering \\cite{szemplinska1991}, finance \\cite{guegan2009}, plasma \\cite{ray2008, ray2009}, time series analysis \\cite{kodba2005, nazarimehr2018a} and natural observations like, climatology \\cite{selvam2007}, biology \\cite{may1987, ghosh2007}, geology \\cite{turcotte1997} etc. After discovery of Lorenz model \\cite{lorenz1963} in 1963, there are several chaotic models were investigated like,  R\\\"{o}ssler system \\cite{rossler1976}, Chen system \\cite{chen1999}, Chua's attractor \\cite{chua1986}, Sprott system \\cite{sprott1994, sprott2000}, logistic model \\cite{may1976}, predator-prey based model \\cite{hastings1991, ginoux2019} and many more. At first, theoretical background behind generation of chaotic solution in deterministic system was not developed. But, now different ways are brought to light to understand and realize the chaotic attractors and its dynamics, both from analytical as well as numerical approaches \\cite{eckmann1985, perc2005, silva2018}. Initially, three well-known routes were reported for generating chaos in dynamical systems, $\\it i.e.$, period-doubling, quasi-periodic and intermittent routes \\cite{ott2002}. Nowadays, researchers are  fascinated for finding chaotic attractor in a different manner.  One interesting topic is to find multistable chaotic attractors \\cite{bayani2019} which has important application and impact to explore a broad field of research \\cite{dudkowski2016}. Besides a new concept of attractor arises connected with multistability like attractors with no equilibria \\cite{wei2011, jafari2013b, pham2014}, line equilibria \\cite{jafari2013a, nazarimehr2018b}, plane of equilibrium \\cite{jafari2016} or coexistence of more than one chaotic attractors \\cite{dudkowski2016, ray2009} etc. Basin of attraction of chaotic attractor does not touch with small neighborhood of the unstable fixed point, which is quite different from the traditional chaotic attractors (called {\\it self-excited chaotic attractors}). This kind of attractor is familiar as {\\it hidden attractor} as finding of hidden attractor is difficult task usually \\cite{dudkowski2016}. Researches are also eager to control of the multistability \\cite{pisarchik2014}. But finding a simple chaotic system is still a challenging task with a great impact for basic research \\cite{liu2004, lu2002, wang2012}. We have also knowledge about chaotic systems in which different timescales are separated which have huge impact in various fields,  like chemical oscillatory model \\cite{epstein1983}, Hindmarsh-Rose neuronal model \\cite{hindmarsh1984}, prey-predator model \\cite {kuwamura2009} and so on. Well known features of chaotic attractor are sensitive dependence of initial conditions, boundedness, aperiodic and appearance of it is confirmed by existence of positive Lyapunov exponent in the parameter region \\cite{strogatz2014}.  Researchers are also very much interested to study the chaotic dynamics in delay systems due to its impactful applicability \\cite{wei2019, khajanchi2018, banerjee2019, ghosh2017}. With studying about generation of chaotic system,  parallel researchers have been focused about control of chaos in non-delay chaotic systems \\cite{ott1990, perc2006, jun2008} as well as delay chaotic systems \\cite{ghosh2008, bhowmick2014}. Its application includes chemical reaction oscillations, turbulent fluids etc. \\cite{boccaletti2000}. From the above discussion, the finding of a new chaotic system and exploring different dynamical states deserve a special attention.\n\n\\par In this paper, we propose a simple nonlinear system which exhibits single-scroll chaotic attractor in a certain parameter region. At first, we calculate the fixed points of the system and analyze their linear stability analyses. From these analyses, we observe that the two fixed points out of three always unstable, whereas one fixed point changes its stability via Hopf bifurcation and finally goes to chaotic state through period-doubling route. We also analytically derive the first Lyapunov coefficient to check the stability of the limit cycle. Then, we introduce slow-fast ratio parameter in the proposed model and investigate the previously mentioned system parameter region by varying this slow-fast parameter. We analytically calculate the slow manifold with the help of geometric singular perturbation theory. The chaotic behavior completely annihilate if the time scale ratio parameter is low.\n\\par The rest of the paper is organized as follows: Section-2 devotes the introduction of a new chaotic system and linear stability analysis of the fixed points. We also discuss the bifurcation diagram, Lyapunov exponent and two-dimensional parameter regions by varying the system parameters. In Sec-3, we discuss the effect of slow-fast parameter and controlling of chaos. Finally, conclusions are drawn in Sec-4.     \n\n\\section{Model and system dynamics}\nWe propose a three-dimensional model with two nonlinear turns in the following form,\n\\begin{equation}\n\\begin{array}{l}\\label{eq.1}\t\n\\dot{x} =-x+y+z,\\\\\n\\dot{y} =xy-z,\\\\\n\\dot{z} =-axz+y+b,\\\\\n\\end{array}\n\\end{equation}\nwhere $a,~b$ are two positive parameters and $x, y$ and $z$ are state variables. Here dot denotes time derivative with respect to time $t$. We first study the linear stability analysis of the equilibrium points of the above system. The equilibrium points are in the form $(x^*, f(x^*), g(x^*))$, where $f(x)$ and $g(x)$ are given by\n\\[\nf(x)=\\frac{x}{1+x},~~~g(x)=\\frac{x^2}{1+x}.\n\\]\nHere $x^*$ satisfies the following equation\n\\begin{equation}\n\\begin{array}{l}\\label{eq.2}\t\nax^*{^3}-(b+1)x^*-b=0.\n\\end{array}\n\\end{equation}\nSolving Eqn. (\\ref{eq.2}), we have three values of $x^*$ corresponding to three roots $x_1^*$, $x_2^*$ and $x_3^*$,  \n\\begin{equation*}\n\\begin{array}{l}\nx_1^*=2\\sqrt{\\frac{b+1}{3a}}cos(\\frac{\\theta}{3}),\\\\\nx_2^*=-2\\sqrt{\\frac{b+1}{3a}}cos(\\frac{\\pi-\\theta}{3}),\\\\\nx_3^*=-2\\sqrt{\\frac{b+1}{3a}}cos(\\frac{\\pi+\\theta}{3}),\n\\end{array}\n\\end{equation*}\n\\begin{figure}[ht]\n\t\\centerline{\n\t\t\\includegraphics[scale=0.6]{figure_1.png} \t}\n\t\\caption{(a) $a$-$b$ parameter space where region A denotes the existence of three fixed points, whereas one fixed point exists in the region B. (b) Variation of maximum real part of the eigenvalues corresponding  to the three fix points by varying the parameter $a$. Here maximum eigenvalues corresponding to two points $x_1^*$ and $x_2^*$ are always positive (red and magenta curves) and $x_3^*$ is negative up to $a<1.53$ and change the sign after that which is shown in inset figure. Here one fixed point changes its stability and other two are always unstable. Here $b=0.25$.} \n\t\\label{fig1} \n\\end{figure}\nwhere\n\\[\n\\theta=\\tan^{-1}\\Big(\\frac{1}{3b}\\sqrt{\\dfrac{4(b+1)^3-27ab^2}{3a}}\\Big).\n\\]\nThe above three roots of the Eqn. (\\ref{eq.2}) are real if  \n\\[\n0<a<\\frac{4(b+1)^3}{27b^2}.\n\\] \n  \nIn Fig.\\ \\ref{fig1}(a), we draw a curve $a=\\frac{4(b+1)^3}{27b^2}$. Left side of the curve in $a$-$b$ parameter space (denoted by A), system possesses three fixed points and only one fixed point exists at the right side of the curve (denoted by B). Now, we check the qualitative nature of equilibrium point $(x^*, f(x^*), g(x^*))$ of the system (\\ref{eq.1}) by linear stability analysis. The Jacobian matrix $J$ of the system at the equilibrium point $(x^*, f(x^*), g(x^*))$ is as follows,\n\\[\nJ=\n\\left[ {\\begin{array}{ccc}\n\t-1   &   1   &   1 \\\\\n\tf(x^*)   &   x^*   &   -1 \\\\\n\t-ag(x^*)   &   1   &   -ax^*\\\\\n\t\\end{array} } \\right].\n\\]\t\n\nSo, the characteristic equation of the matrix $J$ is\\\\\n\n\\begin{equation}\n\\begin{array}{l}\\label{eq.3}\n\\lambda^3+a_2\\lambda^2+a_1\\lambda+a_0=0,\n\\end{array}\n\\end{equation}\nwhere\\\\\n\\begin{equation*}\n\t\\begin{array}{l}\n\t\ta_2=1-x^*+ax^*,\\\\\n\t\ta_1=\\dfrac{1-ax^*{^3}+(a-1)x^*{^2}+(a-1)x^*}{1+x^*},\\\\\n\t\ta_0=\\dfrac{1-3ax^*{^2}-2ax^*{^3}}{1+x^*}.\n\t\\end{array}\n\\end{equation*}\nSolving Eqn. (\\ref{eq.3}), the three eigenvalues are\n\\begin{equation}\n\\begin{array}{l}\\label{eq.4}\n\\lambda_1=(z_1+z_2)-\\frac{a_2}{3},\\\\\n\\lambda_2=-\\frac{z_1+z_2}{2}-\\frac{a_2}{3}+i\\frac{\\sqrt3(z_1-z_2)}{2},\\\\\n\\lambda_3=-\\frac{z_1+z_2}{2}-\\frac{a_2}{3}-i\\frac{\\sqrt3(z_1-z_2)}{2},\\\\\n\\end{array}\n\\end{equation}\nwhere\n$\nz_1=(q+(p^3+q^2)^\\frac{1}{2})^\\frac{1}{2}$, $ z_2=(q-(p^3+q^2)^\\frac{1}{2})^\\frac{1}{2}$, $p=\\frac{a_1}{3}-\\frac{a_2^2}{9}$ and $ q=\\frac{a_1a_2-3a_0}{6}-\\frac{a_2^3}{27}$.\nIf $p^3+q^2>0$, Eqn.\\ (\\ref{eq.3}) gives one real root with a pair of complex conjugate eigenvalues and another case, $i.e.$,  $p^3+q^2<0$, the equation gives three  real roots.\n\n\\par Now, we fix the system parameter $b$ at $b=0.25$ and consider $a$ as a bifurcation parameter. In Fig.\\ \\ref{fig1}(b), we plot the  maximum real part of the eigenvalues of $J$ at $x_1^*, x_2^*,$ and $x_3^*$ with respect to the system parameter $a$. From this figure, the maximum real part of the eigenvalue (Re(${\\lambda_{3}}$)) of $J$ at third root, $i.e.,~x_3^*$ shows negative in the range $a \\in$ (0, 1.53) (blue line) and after that turns to positive (clearly shown in the inset figure of Fig.\\ \\ref{fig1}(b)), but for remaining two roots, $i.e.,~x_1^*,~x_2^*$, their associated maximum eigenvalues (Re(${\\lambda_{1,2}}$)) are always positive (magenta and red lines). Interestingly, we see that at $a=4.63$, there are one maximum real part of the eigenvalues corresponding to $x_1^*, x_2^*,$ and $x_3^*$ exists. This means for $a>4.63$, the Eqn. (\\ref{eq.2}) possesses only one real root ($B$ region in Fig.\\ \\ref{fig1}(a)). So it can be concluded that the Eqn. (\\ref{eq.1}) has only one stable fixed point corresponding to $x_3^*$ in the range $a \\in (0, 1.53)$. Now, for a specific value of $a=0.5$ and $b=0.25$, we calculate $x^*$ and associated eigenvalues  $\\lambda_{1, 2, 3}$ as follows:\n\\begin{equation*}\n\t\\begin{array}{l}\n\t\tx_1^*: x^*\\approx1.6731,\\\\ (\\lambda_{1},\\lambda_{2},\\lambda_{3})\\approx(1.6791,-0.9213\\pm0.9524i);\\\\\n\t\tx_2^*: x^*\\approx-1.4698,\\\\ (\\lambda_{1},\\lambda_{2},\\lambda_{3})\\approx(-3.0764,1.718,-0.3765);\\\\  \n\t\tx_3^*: x^*\\approx-0.2034,\\\\  (\\lambda_{1},\\lambda_{2},\\lambda_{3})\\approx(-0.9653,-0.0682\\pm1.1072i).      \n\t\\end{array}\n\\end{equation*}\n\n\\begin{figure*}[ht]\n\t\\centerline{\n\t\t\\includegraphics[scale=0.31]{figure_2_a.png}\n\t\t\\includegraphics[scale=0.31]{figure_2_b.png}\n\t\t\\includegraphics[scale=0.31]{figure_2_c.png}}\n\t\\caption{(a) Bifurcation diagram of the new system Eqn.\\ (\\ref{eq.1}) for $a\\in[2.4,3.037]$ showing period-doubling route to chaos.\n\t\t(b)\tZoom version of a periodic window shown as shaded region in figure (a). Here LPC, PD, and IC denote the limit point bifurcation of cycles, period-doubling, and interior crisis, respectively. (c) Corresponding maximum Lyapunov exponent with respect to $a\\in[2.4,3.037]$ to verify different states (periodic state and chaotic state). Here, we fix $b=0.25$. Initial condition is taken as $(x(0), y(0), z(0))=(0.3, 0.2, 0.1).$}\n\t\\label{fig2} \n\\end{figure*}\n\\begin{figure}[ht]\n\t\\centerline{\n\t\t\\includegraphics[scale=0.55]{figure_3_r.png}}\n\t\\caption{Phase portraits for different values of $a$: $a=2.4$, period-1 (magenta); $a=2.6$, period-2 (black); $a=2.75$, period-4 (blue) and $a=2.9$, chaotic state (red). From the phase diagrams, period-doubling route to chaos is easily observed.  Here $b=0.25$. }\n\t\\label{fig3} \n\\end{figure}\nUsing the sign of eigenvalues, the fixed points corresponding to $x_1^*, x_2^*,$ and $x_3^*$ of the system (\\ref{eq.1}) are unstable saddle focus, unstable saddle and stable focus, respectively at $a=0.5$ and $b=0.25$. So we focus on the fixed point corresponding to $x_3^*$ and study Hopf bifurcation analysis. The characteristic equation at the point $x_3^*$ becomes,\n\\begin{equation}\n\\begin{array}{l}\\label{eq.5}\n\\lambda^3+r_2\\lambda^2+r_1\\lambda+r_0=0,\n\\end{array}\n\\end{equation}\nwhere\n\\begin{equation*}\n\t\\begin{array}{l}\n\t\tr_2=1+(1-a)(\\frac{\\sqrt{15a}}{3a})cos(\\theta_1),\\\\\n\t\tr_1=\\dfrac{1-\\frac{(a-1)\\sqrt{15a}}{3a}cos(\\theta_1)+\\frac{5(a-1)}{3a}cos^2\\theta_1+\\frac{5\\sqrt{15a}}{9a}cos^3\\theta_1}{1-(\\frac{\\sqrt{15a}}{3a})cos(\\theta_1)},\\\\\n\t\tr_0=\\dfrac{1-5cos^2(\\theta_1)+\\frac{10\\sqrt{15a}}{9a}cos^3(\\theta_1)}{1-(\\frac{\\sqrt{15a}}{3a})cos(\\theta_1)},\n\t\\end{array}\n\\end{equation*}\nand,\n\\[\n\\theta_1=\\frac{\\sqrt{3a(125-27a)}}{9a}.\n\\]\nAccording to the Routh-Hurwitz stability criterion, the real parts of all the eigenvalues $\\lambda$ become negative if\n$r_2>0, r_0 >0,$ and $r_1r_2 >r_0$. Numerically, we calculate the Hopf bifurcation point at $a^* \\approx 1.524053$ where the real parts of the complex conjugate roots of Eqn. (\\ref{eq.5}) becomes zero \\cite{kuznetsov2004}.\n\\begin{figure}[ht]\n\t\\centerline{\n\t\t\\includegraphics[scale=0.65]{figure_4_r.png} \t}\n\t\\caption{ Basin of attraction of chaotic attractor at (a) $z=0.1$ plane, (b) $y=0.1$ plane, and (c) $x=0.1$ plane. White region represents the unbounded state. (d) First return map for the chaotic attractor. This map shows unimodal behavior which signify period-doubling route to chaos. Initial condition is chosen at $x(0)=0.3, y(0)=0.2,$ and $z(0)=0.1$. Parameter values are fixed at $a=2.9$ and $b=0.25$}\n\t\\label{fig4} \n\\end{figure} \n\nNow, we discuss about the nature of Hopf bifurcation \\cite{kuznetsov2004}. For this purpose, first we check the transversality condition and then calculate the value of first Lyapunov coefficient \\cite{kuznetsov2004, mello2009, wei2014}. The transversality condition, $\\ i.e.$, real part of $\\frac{d\\lambda(a^*)}{da}|_{\\lambda=1.0843i} \\approx 0.06839 (\\ne 0)$ is satisfied here. The first Lyapunov coefficient is $l_1\\approx0.04196~(> 0)$ (detail calculations are in appendix). So the fixed point (corresponding to $x_3^*$) deals with subcritical Hopf bifurcation at $a^* \\approx 1.524053$ and it indicates that there is an unstable limit cycle exists in the neighborhood of the fixed point.\n\n\\par  Next, we study the dynamical behavior of the equilibrium point corresponding to $x_3^*$ with respect to the parameter $a$ after crossing the value $a=a^*$. After crossing $a^*$, the fixed point loses its stability and a stable limit cycle of period-1 appears and persists up to $a\\approx2.493$. Then we achieve chaotic dynamics via period-doubling (PD) route \\cite{strogatz2014}. This scenario is depicted in Fig.\\ \\ref{fig2}(a). At $a\\approx3.037$, this chaotic attractor is destructed due to boundary crisis \\cite{grebogi1983} as the chaotic attractor collides with unstable fixed point (corresponding to $x_1^*$). We take a zoom version of periodic window  at the neighborhood of $a=2.845$ (shaded region in Fig.\\ \\ref{fig2}(a)) in Fig.\\ \\ref{fig2}(b). System changes from chaotic oscillation to periodic oscillation through limit point bifurcation of cycles (LPC). Chaotic oscillation again revives via PD bifurcation and amplifies via interior crisis (IC). Then we plot maximum Lyapunov exponent ($\\rm LE_{max}$) in Fig.\\ \\ref{fig2}(c) for verifying our result in Fig.\\ \\ref{fig2}(a). Positive value of $\\rm LE_{max}$ after a certain value of $a$ signifies the presence of chaos. For four different values of $a$, (i.e., $a=2.4, 2.6, 2.75$ and $2.9$), we display four different kind of attractors, $\\ i.e.$, limit cycle of period-1 (magenta), period-2 (black), period-4 (blue), and chaos (red), respectively. The occurrence of period-doubling cascade for generating chaos is verified by calculating Feigenbaum universal constant \\cite{strogatz2014}. The ratio of distances between successive intervals between the bifurcate values of the system parameter, represented by a constant factor\n\\[\n\\delta=\\lim_{k\\to\\infty} \\frac{a_{k+1}-a_{k}}{a_{k+2}-a_{k+1}}\n\\] \napproaches to 4.66992.... The value of Feigenbaum constant for our case are given in the following table: \n\\vspace{0.1cm}\n\\begin{table}[h!]\n\t\\centering\n\t\\begin{tabular}{||c c c c||} \n\t\t\\hline\n\t\tk & Period & Bifurcation parameter $a_k$ & Ratio $\\frac{a_{k+1}-a_{k}}{a_{k+2}-a_{k+1}}$ \\\\ [1ex] \n\t\t\\hline\\hline\n\t\t1 & $2^1$ & 2.491 & Not applicable \\\\ \n\t\t2 & $2^2$ & 2.7234 & Not applicable \\\\\n\t\t3 & $2^3$ & 2.7822 & 3.952 \\\\\n\t\t4 & $2^4$ & 2.7966 & 4.0833 \\\\\n\t\t5 & $2^5$ & 2.79969 & 4.6602 \\\\\n\t\t6 & $2^6$ & 2.8003517 & 4.6696 \\\\ [1ex] \n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\\vspace{1.0cm}\n\\begin{figure}[ht]\n\t\\centerline{\n\t\t\\includegraphics[scale=0.45]{figure_5.png} \t}\n\t\\caption{ Different dynamical behaviors in the $a$-$b$ parameter space. Here Cyan:  stable steady state, magenta: period-1, black: period-2, blue: period-4, and red: greater than period-4 or chaotic state. White region is for unbounded solution. The initial condition is $x(0)=0.3, y(0)=0.2$, and $z(0)=0.1$.}\n\t\\label{fig5} \n\\end{figure}\n\n Next, we identify the region of initial conditions for the chaotic state. The basin of attractions at $z=0.1$ plane, $y=0.1$ plane, and $x=0.1$ plane  for the chaotic attractor at $(a, b)=(2.9, 0.25)$ are depicted in Figs.\\ \\ref{fig4}(a-c). We get fully connected basin of attractions for respective three cases.  At the same set of parameter value, we also plot first return map, $\\ i.e.$, $x_n$ vs. $x_{n+1}$ in Fig.\\ \\ref{fig4}(d), where $x_{n}$ denotes the $n$-th local maxima of the time series. Unimodal nature of this first return map indicates the PD bifurcation route to generate chaos. \n\\par To explore the complete study by varying the two parameters $a$ and $b$ simultaneously, we plot a two parameter $(a$-$b)$ bifurcation diagram \\cite{ghosh2015} for the range $a \\in [0,5]$ and $b \\in [0.01,0.5]$ in Fig.\\ \\ref{fig5}. This demonstrates the whole dynamics of the system (\\ref{eq.1}). In this phase diagram, cyan, magenta, black, blue, and red regions represent steady state, limit cycle of period-1, period-2, period-4, and greater than period-4 (or chaotic),  respectively. White portion represents unbounded solution for higher values of $a$ and $b$. Interestingly, well-separated parameter space is observed in $(a$-$b)$ bifurcation diagram. \n   \n\\section{Slow-fast dynamical system and controlling chaos}\n In this section, we introduce a slow-fast time scale ratio parameter to create slow-fast dynamics in the proposed system. We try to understand the effect of this time scale parameter. A general framework of a slow-fast dynamical system \\cite{guckenheimer2008} can be written as,\n\t\\begin{equation*}\n\t\\begin{array}{lll}\n\t\t\t\t\n\t\\epsilon \\dfrac{dx}{dt}=f(x,y,\\eta); \n\t~\\dfrac{dy}{dt}= g(x,y,\\eta),\n\t\\end{array}\n\t\\end{equation*}\n\twhere $x\\in \\mathbb{R}^{m} \\;\\mbox{and}\\; y\\in \\mathbb{R}^{n}$ are fast and slow variables, respectively, $\\eta\\in \\mathbb{R}^{p}$ is a model parameter, $0<\\epsilon<1$ represents the ratio of time-scales. \tSetting $\\epsilon$=0, the trajectory of above equation converges to the solution of differential algebraic equation $f(x,y,\\eta) =0$ and $\\dot {y}=g(x,y,\\eta)$, where \t$S_{0}= \\{(x,y) \\in \\mathbb{R}^{m} \\times \\mathbb{R}^{n}| f(x,y,\\eta) =0\\}$ is a critical manifold.\n\n\n\\par Now, in our model we introduce a time-scale ratio parameter $\\epsilon ~(0<\\epsilon<1)$ in the Eqn. (\\ref{eq.1}) as follows \n\\begin{equation}\n\\begin{array}{l}\\label{eq.6}\t\n\\epsilon \\dot{x} =-x+y+z,\\\\\n\\dot{y} =xy-z,\\\\\n\\dot{z} =-axz+y+b,\\\\\n\\end{array}\n\\end{equation}\nso that $x$ is fast variable and $y, z$ are slow variables. We rescale the above Eqn. (\\ref{eq.6}) by setting $\\tau=t\/\\epsilon$ as \n\\begin{equation}\n\\begin{array}{l}\\label{eq.7}\t\nx^{'} =-x+y+z,\\\\\ny^{'} =\\epsilon(xy-z),\\\\\nz^{'} =\\epsilon(-axz+y+b),\\\\\n\\end{array}\n\\end{equation}\nwhere ${x^{'}}\\equiv\\frac{dx}{d\\tau}$. By setting $\\epsilon=0$, the trajectory of Eqn. (\\ref{eq.6}) converges to the solution of the differential algebraic equation  \n\\begin{equation}\n\\begin{array}{l}\\label{eq.8}\t\n0 =-x+y+z,\\\\\n\\dot{y} =xy-z,\\\\\n\\dot{z} =-axz+y+b.\\\\\n\\end{array}\n\\end{equation}\n\n\\begin{figure}[ht]\n\t\\centerline{\n\t\t\\includegraphics[scale=0.55]{figure_7.png} }\n\t\\caption{Plot of  the critical manifold $S_0$ and slow manifold $S_{\\epsilon}$ at $\\epsilon=0.85$ are illustrated by green and red colors, respectively. Chaotic attractor (blue line) is drawn for $a=3.17$ and $b=0.26$. The position of the unstable fixed point (solid dot): $(-0.2420, -0.2286, 0.0553)$.}\n\t\\label{fig6} \n\\end{figure}\n\nSo,  as per above definition, $S_0=\\{(x,y,z)\\in \\mathbb R^3 : x=y+z\\}$ is called a critical manifold. From Fenichel theory \\cite{fenichel1979}, we know that a subset $S_{\\epsilon}$ of $S_{0}$ exists for $\\epsilon~(>0)$ if $S_{\\epsilon}$ is normally hyperbolic. Our system is normally hyperbolic because of real part of $\\frac{\\partial (-x+y+z)}{\\partial x}=-1 (\\ne 0), (x,y,z)\\in S_{\\epsilon}$. So there exists a local invariant manifold (slow manifold) $S_{\\epsilon}$ and we derive this equation by using geometric singular perturbation method \\cite{fenichel1979, guckenheimer2008, desroches2012}.   \n\\par In $S_0$, relation among variables $x,y$ and $z$ is $x=y+z$. Let in $S_{\\epsilon}$, this relation becomes\n\\begin{equation}\n\\begin{array}{l}\\label{eq.9}\t\nx=y+z+O(\\epsilon).\n\\end{array}\n\\end{equation}\nBy Taylor series expansion in $\\epsilon$, we get  \n\\begin{equation}\n\\begin{array}{l}\\label{eq.10}\t\nx=y+z+\\epsilon H(y,z)+O(\\epsilon^2).\n\\end{array}\n\\end{equation}\nDifferentiate Eqn. (\\ref{eq.9}) and then multiply both sides by $\\epsilon$, we have,\n\\begin{equation}\n\\begin{array}{l}\\label{eq.11}\n\\epsilon \\dot{x}=\\epsilon \\dot{y}+\\epsilon \\dot{z}+O(\\epsilon^2).\n\\end{array}\n\\end{equation}\nUsing Eqns. (\\ref{eq.6}) and (\\ref{eq.9}) in Eqn. (\\ref{eq.11}), we get \n\\begin{equation}\n\\begin{array}{l}\\label{eq.12}\n\\epsilon \\dot{x}=\\epsilon\\Big (y^2-az^2+(1-a)yz+y-z+b\\Big )+O(\\epsilon^2).\n\\end{array}\n\\end{equation}\nAgain from first equation of Eqn. (\\ref{eq.6}), \n\\[\n\\epsilon \\dot{x} =-x+y+z.\n\\]\nNow using Eqn. (\\ref{eq.10}) in above equation, we get \n\\begin{equation}\n\\begin{array}{l}\\label{eq.13}\n\\epsilon \\dot{x}=-(\\epsilon H(y, z)+O(\\epsilon^2)).\n\\end{array}\n\\end{equation}\nComparing Eqns. (\\ref{eq.12}) and (\\ref{eq.13}), finally we obtain\\\\\n\\begin{equation}\n\\begin{array}{l}\\label{eq.14}\nH(y,z)=-\\Big (y^2-az^2+(1-a)yz+y-z+b\\Big ).\n\\end{array}\n\\end{equation}\n\\begin{figure*}[ht]\n\t\\centerline{\\includegraphics[scale=0.05]{figure_8_a_1.png}\n\t\t\\includegraphics[scale=0.5]{figure_8_a_2.png}}\n\t\\centerline{\\includegraphics[scale=0.6]{figure_8_b.png}}\n\t\\caption{Upper panel:  different dynamical states in $a$-$b$ parameter spaces for (a) $\\epsilon=0.9$, and (b) $\\epsilon=0.1$. Here, Cyan: stable steady state, magenta: period-1, black: period-2, blue: period-4, and red: greater than period-4 or chaotic state. White region is for unbounded solution. Lower panel: bifurcation diagrams with respect to the time-scale ratio parameter $\\epsilon$ for (c) $\\{a, b\\}=\\{3.17, 0.26\\}$  and (d) $\\{a, b\\}=\\{2.68, 0.304\\}$. The initial condition is $x(0)=0.3, y(0)=0.2$, and $z(0)=0.1$.  From this figure, it is noticed that chaotic state is annihilated by decreasing the value of time-scale parameter $\\epsilon$ and stable steady state and periodic solutions dominate the whole parameter region.}\n\t\\label{fig7} \n\\end{figure*}  \n\\par We use the above value of $H(y, z)$ in Eqn. (\\ref{eq.10}) to calculate the approximated slow manifold $S_{\\epsilon}$. We plot the critical manifold $S_0$ (in green color), an approximated slow manifold $S_{\\epsilon}$ (in red color) in Fig.\\ \\ref{fig6}. The chaotic attractor for the set of values $a=3.17, b=0.26$ and $\\epsilon=0.85$ is shown by blue line. This attractor lies on the both sides of the manifold $S_0$ and $S_{\\epsilon}$. We take the value $\\epsilon=0.85$ for $S_{\\epsilon}$ (though this value is not so small enough, but we choose this value due to the appearance of chaos).     \n\\par Next we study the effect of time-scale parameter $\\epsilon$ of the system (\\ref{eq.6}) in ($a$-$b$) parameter space for the previously mentioned range. For this, we first set $\\epsilon=0.9$ and different dynamical states like, stable fixed point, period-1, period-2, period-4 and greater than period-4 or chaos are shown in Fig.\\ \\ref{fig7}(a). The stable fixed point dominates in the ($a$-$b$) parameter space compare to Fig.\\ \\ref{fig5} for $\\epsilon=1.0$ case, i.e., without any time-scale.   Also the region of chaotic state  decreases compare to the system (\\ref{eq.1}) (in Fig.\\ \\ref{fig5}). If we decrease the value of time-scale parameter $\\epsilon$, then the chaotic region is completely annihilated in the ($a$-$b$) parameter space. Figure \\ref{fig7}(b) shows that the region of chaotic state is completely annihilated whereas region of stable fixed point is expanded and rest parts of the region are filled by period-1 limit cycle for $\\epsilon=0.1$. So by introducing the slow-fast parameter $\\epsilon$ in our original system (\\ref{eq.1}), we can control chaos. Now we take two sets of parameter values of $\\{a, b\\}, \\ i.e.,$ $\\{3.17, 0.26\\}$ and $\\{2.68, 0.304\\}$ from the Figs.\\ \\ref{fig7}(a, b), we have illustrated two bifurcation diagrams by varying slow-fast ratio parameter $\\epsilon \\in[0,1]$ for understanding the various change of dynamics in slow-fast system in Figs.\\ \\ref{fig7}(c, d), respectively. For the first set of parameter values,  the period-doubling route to chaos is observed, whereas for second case, the chaotic behavior in the system is terminated and two states (stable fixed point and period-1 limit cycle) are dominated.            \n\\section{Conclusions}\n\\par To summarize, in this paper, we have proposed a new simple chaotic system with two nonlinear terms. We have elaborately discussed the nature of fixed points through linear stability analysis. We have also analytically derived the first Lyapunov coefficient to show the nature of Hopf bifurcation by which fixed point changes its stability. Using bifurcation diagram, we have investigated the period-doubling route to chaos and also verified the results using Lyapunov exponent and first return map. We also separate the region of different dynamical behavior in the parameter space. Finally, we introduce slow-fastness in our model and try to understand the regarding changes of dynamics. Variation of dynamics has been captured in ($a$-$b$) parameter space for different values of the slow-fast parameter $\\epsilon$. Most interesting fact is that we can control the chaotic and higher periodic orbits by adjusting the value of $\\epsilon$\n\\\\\n\\par {\\bf Acknowledgments:}\nThe authors would like to thank Syamal K. Dana, Chittaranjan Hens, Gourab Kumar Sar, Sayeed Anwar, Soumen Majhi, Srilena Kundu, Sayantan Nag Chowdhury, Subrata Ghosh and Sarbendu Rakshit for helpful discussions and comments. \n\\section*{Appendix: Calculation of first Lyapunov coefficient} \n\\par Let us consider the differential equation \n\\begin{equation}\n\\begin{array}{l}\\label{eq.15}\n\\dot{x}=f(x,\\alpha),\n\\end{array}\n\\end{equation}\nwhere $x\\in \\mathbb{R}^3$ and $\\alpha\\in \\mathbb{R}^1$ represent the state variables and system parameter respectively, and $f:\\mathbb{R}^3\\times \\mathbb{R}^1 \\rightarrow \\mathbb{R}^1$ is a smooth function. Let the Eqn. (\\ref{eq.15}) has a fixed point $x=x_0$ at $\\alpha=\\alpha_0$. Now, replacing $x-x_0$ by $X$ and Eqn. (\\ref{eq.15}) becomes\n\\begin{equation}\n\\begin{array}{l}\\label{eq.16}\n\\dot{X}=F(X)=f(X,\\alpha_0).\n\\end{array}\n\\end{equation}\nHere, $F(X)$ is also a smooth function and we can expand it in Taylor series in terms of symmetric functions as follows,\n\\begin{equation}\n\\begin{array}{l}\\label{eq.17}\nF(X)=AX+\\frac{1}{2!}B(X,X)+\\frac{1}{3!}C(X,X,X)+O(||X||^4),\n\\end{array}\n\\end{equation}     \nwhere $A=f_x(0,\\alpha_0)$ is the Jacobian matrix and for i=1,2,3, \n\\begin{equation}\n\\begin{array}{l}\\label{eq.18}\nB_{i}(X,Y)=\\sum_{j,k=1}^{3} \\frac{\\partial^2{F_i(\\xi)}}{\\partial \\xi_{j} \\partial\\xi_{k}}|_{\\xi=0} X_j X_k,\\\\\nC_{i}(X,Y,Z)=\\sum_{j,k,l=1}^{3} \\frac{\\partial^3{F_i(\\xi)}}{\\partial \\xi_{j} \\partial\\xi_{k} \\partial\\xi_{l}}|_{\\xi=0} X_j X_k X_l.\n\\end{array}\n\\end{equation} \nIn our case, multilinear symmetric functions are\\\\\n\\begin{equation}\n\\begin{array}{l}\\label{eq.19}\nB(X,Y)=(0,x_1y_2+x_2y_1,-a^*(x_1y_3+x_3y_1)),\\\\\nC(X,Y,Z)=(0,0,0).\n\\end{array}\n\\end{equation}\n\n\\par We assume that A has a pair of complex eigenvalues  $\\lambda_{2,3}=\\pm iw_0(w_0>0)$ on the imaginary axis at $(x_0,\\alpha_0)$. Let generalized eigen space of $A$ be $T^C$ which is the largest invariant subspace spanned by eigenvectors corresponding to $\\lambda_{2,3}$. Let $p,q\\in$ C$^3$ be such eigen vectors then  \n\\begin{equation}\n\\begin{array}{l}\\label{eq.20}\nAq=iw_0q,\\\\\nA^Tp=iw_0p,\\\\\n<p,q>=\\sum_{i=1}^{3}\\bar{p}_iq_i=1,\n\\end{array}\n\\end{equation}\nwhere $A^T$ denotes the transpose of the matrix $A$ and $<...>$ is the standard scalar product over $\\mathbb{C}^3$. For our system (\\ref{eq.1}), we obtain\\\\\n$\nq\\approx(-0.0404 -0.5130i,~ 0.6402,~ -0.1244 - 0.5568i),\\\\\np \\approx(-0.1629 - 0.0591i,~ 0.78 - 0.1677i,~ 0.1501 - 0.8436i),\\\\\n$\nwhere $w_0\\approx 1.0843$.\n\\par  Now we choose any $y\\in T^C$ which is represented as $y=wq+\\bar w\\bar q$, where $w=<p,q>\\in \\mathbb{C}$. The two-dimensional center manifold associated with the eigenvalues $\\lambda_{2,3}= \\pm iw_{0}$ can be parameterized by $w$ and $\\bar w$ in the form $X=H(w,\\bar w)$. Here $H: \\mathbb{C}^2\\rightarrow\\mathbb{R}^3$ can be expanded as follows\n\\begin{equation}\n\\begin{array}{l}\\label{eq.21}\nH(w,\\bar w)=wq+\\bar w\\bar q\t+\\sum_{2\\le j+k\\le 3}\\frac{1}{j!k!}h_{jk}w^j\\bar w^k+O(|w|^4),\n\\end{array}\n\\end{equation}\nwhere $h_{jk}\\in \\mathbb{C}^3$ and $h_{jk}=\\bar h_{kj}$. Differentiating $H(w,\\bar w)$ with respect to $t$ and substituting this into Eqn. (\\ref{eq.15}) and using Eqn. (\\ref{eq.16}), we get the following differential equation\n\\begin{equation}\n\\begin{array}{l}\\label{eq.22}\nH_{w}\\dot{w}+H_{\\bar w}\\dot{\\bar w}=F(H(w,\\bar w)).\n\\end{array}\n\\end{equation}  \nUsing Eqns. (\\ref{eq.16}), (\\ref{eq.21}) and (\\ref{eq.22}), we determine the complex coefficients $h_{ij}$ and from the Eqn. (\\ref{eq.22}), the $w$  evolves on the center manifold,\\\\\n\\begin{equation}\n\\begin{array}{l}\\label{eq.23}\n\\dot{w}=iw_0w+\\frac{1}{2}G_{21}w|w|^2+O(|w|^4),\n\\end{array}\n\\end{equation}\nwhere $G_{21}\\in \\mathbb{C}$.\nSubstituting Eqn. (\\ref{eq.23}) into Eqn. (\\ref{eq.22}) and using Eqns.\\~ (\\ref{eq.17}) and (\\ref{eq.18}), we get the following coefficients,\n\\begin{equation}\n\\begin{array}{l}\\label{eq.24}\nh_{11}=-A^{-1}B(q,\\bar q),\\\\\nh_{20}=(2iw_0I_3-A)^{-1}B(q,q),\n\\end{array}\n\\end{equation}\nwhere $I_3$ is the 3 $\\times$ 3 identity matrix. Finally $G_{21}$ is obtained from the condition of existence of solution of the equation for $h_{21}$. So, \n\\begin{equation*}\n\t\\begin{array}{l}\n\t\tG_{21}=<p,~C(q,q,\\bar q)+B(\\bar q,h_{20})+2B(q,h_{11})>,\n\t\\end{array}\n\\end{equation*}\nwhere we use $<p,q>=1$. So the first Lyapunov coefficient is defined as \n\\[\nl_1=\\frac{1}{2}Re(G_{21}),\n\\]\nwhich determines the nonlinear stability of a nondegenerate codimension one Hopf bifurcation.\nNow, for calculating $l_1$, we first find  \\\\\n$ h11\\approx(   0.6348, 1.0854, -0.4507),$\\\\\n$h20\\approx( -0.2273 + 0.0747i, -0.1643 - 0.1240i, -0.2251 - 0.2942i).$\nusing Eqns. (\\ref{eq.18}), (\\ref{eq.24}) and (\\ref{eq.24}).\nFinally, we get $l_1\\approx0.04196~(> 0)$. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe European Union (EU) is now composed from 27 countries \nand is considered as a major world leading power \\cite{eusite}.\nJanuary 2021 has seen Brexit officially taking place, triggering the  \nwithdrawal of the United Kingdom (UK) from EU \\cite{wikibrexit}.\nThis event has important political, economical and social effects.\nHere we project and study its consequences from the view point\nof international trade between world countries.\nOur analysis is based on the UN COMTRADE database \\cite{comtrade} for\nthe multiproduct trade between world countries in \nrecent years. From this database we construct the world trade\nnetwork (WTN) and evaluate the influence and trade power \nof specific countries using the  Google matrix analysis\nof the WTN. We consider 27 EU countries as  a single\ntrade player having the trade exchange between EU and other countries.\nOur approach uses the Google matrix tools and algorithms developed for the WTN \n\\cite{wtn1,wtn2,wtn3,keu9} and other complex directed networks\n\\cite{rmp2015,politwiki}. The efficiency of the Google matrix and PageRank algorithms\nis well known from the World Wide Web network analysis \\cite{brin,meyer}.\n\nOur study shows that the Google matrix approach (GMA) allows to characterize \nin a more profound manner the trade power of countries \ncompared to the usual method relying on import and export analysis (IEA) between countries.\nGMA's deeper analysis power originates in the fact that it accounts for the multiplicity\nof transactions between countries while IEA only takes into account the \neffect of one step (direct link or relation) transactions. In this paper, we show that the \nworld trade network analysis with GMA identifies EU as the first trade player in the world, \nwell ahead of USA and China.\n\nThis paper is structured in the following way. Section~\\ref{sec:dataset} introduces first the UN COMTRADE dataset, \nand then gives a primer on the tools related to Google matrix analysis such as the trade balance metric and the REGOMAX algorithm.\nIn Section~\\ref{sec:results}, the central results of this papers are presented, which are discussed in \\ref{sec:discussion}. \n\n\\section{Data sets, algorithms and methods}\\label{sec:dataset}\n\nWe use the UN COMTRADE data \\cite{comtrade} for years 2012, 2014, 2016 and 2018 to \nconstruct the trade flows of the multiproduct WTN following the procedure\ndetailed in \\cite{wtn2,wtn3}. This paper gives the results for year 2018 only, others are \nto be found at \\cite{ourwebpage}. Each year is presented by a money matrix, \n$M^{p}_{cc^{\\prime}}$, giving the export flow of product $p$\nfrom country $c^{\\prime}$ to country $c$ (transactions are expressed in USD of current year).\nThe data set is given by $N_{c} = 168$ countries and territories \n(27 EU countries are considered as one country) and $N_{p} = 10$ principal type  of products\n(see the lists in \\cite{wtn1,wtn3}). These 10 products are: Food and live animals (0);\nBeverages and tobacco (1); Crude materials, inedible, except fuels (2);  Mineral fuels etc (3);\nAnimal and vegetable oils and fats (4); Chemicals and related products, n.e.s. (5);\nBasic manufactures (6); Machinery, transport equipment (7); \nMiscellaneous manufactured articles (8);\nGoods not classified elsewhere (9) (product index $p$ is given in brackets). They belong to \nthe Standard International Trade Classification (SITC Rev. 1)\nThus the total Google matrix $G$ size is given by all system nodes $N=N_c N_p= 1680$\nincluding countries and products.\n\nThe Google matrix $G_{ij}$ of direct trade flows\nis constructed in a standard way described in detail at \\cite{wtn2,wtn3}: monetary trade flows \nfrom a node $j$ to node $i$  are normalized to unity \nfor each column $j$ thus given the matrix $S$ of \nMarkov transitions for trade, the columns of dangling nodes with\nzero transactions are replaced by a column with all elements being $1\/N$. \nThe weight of each product is taken into account  via a\ncertain personalized vector taking into account \nthe weight of each product in the global trade volume.\nWe use the damping factor $\\alpha=0.5$. \nThe Google matrix is $G_{ij}=\\alpha S_{ij} + (1-\\alpha) v_i$\nwhere $v_i$ are components of positive column vectors called personalization vectors\nwhich take into account the weight of each product in the global trade \n($\\sum_i v_i=1$).\nWe also construct the matrix $G^*$ for the inverted trade flows.\n\nThe stationary probability distribution \ndescribed by $G$ is given by the PageRank vector\n$P$ with maximal eigenvalue $\\lambda=1$:\n$GP=\\lambda P =P$ \\cite{rmp2015,brin,meyer}. \nIn a similar way, for   the inverted flow, described by $G^*$, \nwe have the CheiRank vector $P^*$, \nbeing the eigenvector of $G^* P^* = P^*$. \n PageRank $K$ and CheiRank $K^*$ indexes are obtained from\nmonotonic ordering of probabilities of PageRank vector $P$ and \nof CheiRank vector $P^*$ as\n$P(K)\\ge P(K+1)$ and $P^*(K^*)\\ge P^*(K^*+1)$ with $K,K^*=1,\\ldots,N$.\nThe sums over all products $p$ gives the PageRank and CheiRank \nprobabilities of a given country as $P_c =\\sum_p P_{cp}$ and \n${P^*}_c =\\sum_p {P^*}_{cp}$ (and in a similar way\nproduct probabilities $P_p, {P^*}_p$) \\cite{wtn2,wtn3}.\nThus with these probabilities we obtain the related indexes $K_c, {K^*}_c$.\nWe also define \nfrom import and export trade volume the\n probabilities\n$\\hat{P}_p$, $\\hat{P}^*_p$, $\\hat{P}_c$, $\\hat{P}^*_c$,\n$\\hat{P}_{pc}$, $\\hat{P}^*_{pc}$ and \ncorresponding indexes\n$\\hat{K}_p$, $\\hat{K}^*_p$, $\\hat{K}_c$, $\\hat{K}^*_c$, $\\hat{K}$, $\\hat{K}^*$\n(these import and export probabilities\nare normalized to  unity\nby the total import and export volumes, see details in\n\\cite{wtn2,wtn3}). It is useful to note that qualitatively\nPageRank probability is proportional to the volume of ingoing\ntrade flow and CheiRank respectively to outgoing flow.\nThus, we can approximately consider that the\nhigh import gives a high  PageRank $P$ probability\nand a high export a high CheiRank $P^*$ probability.\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=0.99\\textwidth]{fig1.pdf}\n\t\\caption{Circles with country flags show country positions on\n  the plane of PageRank-CheiRank indexes $(K,K^*)$\n  (summation is done over all products) (left panel)\n  and on the plane of ImportRank-ExportRank  $\\hat{K}$, $\\hat{K}^*$ \n  from trade volume (right panel);\n  data is shown only for index values less than $61$ in year 2018\n \n  .} \\label{fig1}\n\\end{figure}\n\n\nAs in \\cite{wtn2,wtn3},\nwe define the trade balance of a given country\nwith PageRank and CheiRank probabilities given by\n$B_c = (P^*_c - P_c)\/(P^*_c + P_c)$.\nAlso we have from ImportRank and ExportRank probabilities\nas $\\hat{B}_c=  ({\\hat{P}^*}_c - \\hat{P}_c)\/({\\hat{P}^*}_c + \\hat{P}_c)$.\nThe sensitivity of \ntrade balance $B_c$ to the price of energy or machinery  can be obtained \nfrom the change of corresponding money volume\nflow related to SITC Rev.1 code $p=3$ (mineral fuels) or $p=7$ (machinery)\nby multiplying it by $(1+\\delta)$, renormalizing column to unity and \ncomputing all rank probabilities and\nthe derivatives $dB_c\/d\\delta$. \n\nWe also use the REGOMAX algorithm \\cite{politwiki,wtn3}\nto construct the reduced Google matrix $G_R$ \nfor a selected subset of WTN nodes $N_r \\ll N$.\nThis algorithm  takes into accounts all transitions of direct and\nindirect pathways happening in the full Google matrix $G$ \nbetween $N_r$ nodes of interest. We use this $G_R$ matrix\nto construct a reduced network of most strong transitions\n(``network of friends'')\nbetween a selection of nodes representing countries and products.\n\nEven if Brexit enter into play in 2021, we use UN COMTRADE data of previous years\nto make a projecting analysis of present and future power of EU composed of 27 countries.\n\nFinally we note that GMA allows to obtain interesting results for \nvarious types of directed networks including Wikipedia  \n\\cite{geopolitics,celestin} and protein-protein interaction \n\\cite{zinprotein,signor} networks.\n\n\\begin{table}[]\n    \\centering\n    \\caption{Top 20 countries of PageRank ($K$), CheiRank ($K^*$), \n    ImportRank and ExportRank in 2018.}\n    \\begin{tabular}{|c|l|l|l|l|}\n    \\hline\n    Rank & PageRank                   & CheiRank                   & ImportRank           & ExportRank           \\\\ \\hline\n1    & EU                       & EU                       & EU                 & China                \\\\\n2  & USA & China                                  & USA & EU                                  \\\\\n3  & China                                  & USA & China                                  & USA \\\\\n4    & United Kingdom             & Japan                      & Japan                & Japan                \\\\\n5    & India                      & Repub Korea          & United Kingdom       & Repub Korea    \\\\\n6    & U Arab Emirates       & India                      & Repub Korea    & Russia   \\\\\n7    & Japan                      & Russia         & India                & United Kingdom       \\\\\n8    & Mexico                     & U Arab Emirates       & Canada               & Mexico               \\\\\n9    & Repub Korea          & Singapore                  & Mexico               & Canada               \\\\\n10   & Canada                     & United Kingdom             & Singapore            & Singapore            \\\\\n11 & Singapore                              & South Africa                           & Switzerland             & Switzerland             \\\\\n12   & Switzerland & Thailand                   & U Arab Emirates & India                \\\\\n13   & Turkey                     & Malaysia                   & Russia   & Malaysia             \\\\\n14   & Russia         & Canada                     & Thailand             & Australia            \\\\\n15   & Australia                  & Mexico                     & Viet Nam             & U Arab Emirates \\\\\n16   & South Africa               & Turkey                     & Australia            & Thailand             \\\\\n17   & Thailand                   & Australia                  & Turkey               & Saudi Arabia         \\\\\n18   & Brazil                     & Switzerland & Malaysia             & Viet Nam             \\\\\n19   & Saudi Arabia               & Brazil                     & Indonesia            & Brazil               \\\\\n20   & Malaysia                   & Saudi Arabia               & Brazil               & Indonesia            \\\\ \\hline\n    \\end{tabular}\n\\label{tab1}\n\\end{table}\n\n\\section{Results}\\label{sec:results}\n\\subsection{CheiRank and PageRank of countries}\n\nThe positions of countries on the PageRank-CheiRank $(K,K^*)$ and\nImportRank-ExportRank $(\\hat{K},\\hat{K}^*)$ planes are shown in Fig.~\\ref{fig1}\nand in Table~\\ref{tab1}.\nThese results show a significant difference between these two types of ranking.\nIndeed, EU takes the top PageRank-CheiRank position $K=K^*=1$\nwhile with Export-Import Ranking it has \nonly $\\hat{K}=1; \\hat{K}^*=2$, with USA at $\\hat{K}=2, \\hat{K}^*=3$\nand China at $\\hat{K}=3, \\hat{K}^*=1$. Thus EU takes the leading positions\nin the GMA frame which takes into account  the muliplicity of trade transactions\nand characterizes the robust features of EU trade relations.\nAlso GMA shows that UK position is significantly weakened compared to IEA description\n(thus UK moves from $K^*=7$ in IEA to $K^*=10$ in GMA). From this data, we see also examples\nof other countries that significantly improve there rank positions in GMA\nframe compared to IEA: India ($K=5$, $K^*=6$, $\\hat{K}=7$, $\\hat{K}^*=12$), \nUnited Arab Emirates ($K=6$, $K^*=8$, $\\hat{K}=12$, $\\hat{K}^*=15$), \nSouth Africa ($K=16$, $K^*=11$, $\\hat{K}=23$, $\\hat{K}^*=23$). \nWe attribute this to well developed, deep \nand broad trade network of these countries which are well captured by GMA in contrast\nto IEA. Indeed, IEA only measures the volume of direct trade exchanges, while GMA characterises \nthe multiplicity of trade chains in the world.\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{fig2.pdf}\n\t\\caption{World map of trade balance of countries  \n$B_c = ({P_c}^* - P_c)\/({P_c}^* + P_c)$.\nTop: trade balance values are computed from the trade volume of Export-Import;\nbottom: trade balance values are computed from PageRank\nand CheiRank vectors; $B_c$ values are marked by color with the corresponding \ncolor bar marked by $j$; \ncountries absent in the UN COMTRADE report are marked by black color \n(here and in other Figs).}\\label{fig2}\n\\end{figure}\n\n\\subsection{Trade balance and its sensitivity to product prices}\n\nThe trade balance of countries in IEA and GMA frames is shown in Fig.~\\ref{fig2}.\nThe countries with 3 strongest positive balance are: \nEquatorial Guinea ($B_c=0.732$), Congo ($B_c=0.645$), Turkmenistan ($B_c=0.623$) in IEA and\nChina ($B_c=0.307$), Japan ($B_c= 0.244$), Russia ($B_c= 0.188$) in GMA.\nWe see that IEA marks top countries which have no significant world power\nwhile GMA marks countries with real significant world influence.\nFor EU and UK we have  respectively $B_c=-0.015; 0.020$ (EU) and  \n$B_c=-0.178; -0.187$ (UK) in IEA; GMA.\nThus the UK trade balance is significantly reduced in GMA\ncorresponding to a loss of network trade influence of UK\nin agreement with data of Fig.~\\ref{fig1} and Table~\\ref{tab1}.\n(We note that the balance variation bounds in GMA are smaller compared to IEA;\nwe attribute this to the fact of multiplicity of transactions \nin GMA that smooth various fluctuations which are  more typical for IEA).\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\textwidth]{fig3.pdf}\n\t\\caption{Sensitivity of country balance \n$dB_c\/d\\delta_s$ for product $s=3$ (mineral fuels).\nTop: probabilities are from the trade volume of Export-Import;\nbottom: probabilities are from PageRank\nand CheiRank vectors. Color bar marked by $j$ \ngives sensitivity.} \\label{fig3}\n\\end{figure}\n\nThe balance sensitivity $d B_c\/d \\delta_s$ to product $s=3$\n(mineral fuels (with strong petroleum and gas contribution))\nis shown in Fig.~\\ref{fig3}. The top 3 strongest \npositive sensitivities $d B_c\/d \\delta_s$ \nare found for Algeria (0.431), Brunei (0.415), South Sudan (0.411) in IEA\nand  Saudi Arabia (0.174), Russia (0.161), Kazakhstan (0.126)  in GMA.\nThe results of GMA are rather natural since Saudi Arabia, Russia and Kazakhstan are central petroleum producers. \nIt is worth noting that GMA ranks Iraq at the 4th position. \nThe 3 strongest negative sensitivities are Zimbabwe (-0.137), Nauru (-0.131), \nJapan (-0.106),  in IEA\nand Japan (-0.066), Korea (-0.062), Zimbabwe (-0.058), in GMA.\nFor China, India we have $d B_c\/d \\delta_s$ values being respectively:\n-0.073, -0.086 in IEA and -0.056, 0.010 in GMA. This shows that the trade network\nof India is more stable to price variations of product $s=3$.\nThese results  demonstrate that GMA selects more globally influential countries.\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\textwidth]{fig4.pdf}\n\t\\caption{Same as in Fig.~\\ref{fig3} but\nfor product $s=7$ (machinery).} \\label{fig4}\n\\end{figure}\n\nThe balance sensitivity $d B_c\/d \\delta_s$ to product $s=7$\n(machinery) is shown in Fig.~\\ref{fig4}. \nHere the top 3 strongest \npositive sensitivities $d B_c\/d \\delta_s$ are found\nin both  IEA and GMA for Japan (respectively 0.167, 0.151), \nRepub. Korea (0.143, 0.097), Philippines (0.130, 0.091).\nThe 3 strongest negative sensitivities are Brunei (-0.210), Iran (-0.202), \nUzbekistan (-0.190)  in IEA\nand Russia (-0.138), Kazakhstan (-0.102), Argentina (-0.097) in GMA.\nThus we again see that GMA selects more globally influential countries. \nThe sensitivity $d B_c\/d \\delta_s$ values for EU, UK, China, Russia, USA are:\n EU (0.048), UK (0.006), China (0.065), Russia (-0.170), USA (-0.027) in IEA;\n EU (0.000), UK (0.024), China (0.077), Russia (-0.138), USA (-0.018) in GMA.\nLatter GMA results show that even if machinery product ($s=7$) is very important for EU \nthe network power of trade with this product becomes dominated by Asian countries\nJapan, Repub. Korea, China, Philippines; \nin this aspect the position of UK is slightly better than EU. \n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\textwidth]{fig5.pdf}\n\t\\caption{Sensitivity of country balance \n$dB_c\/d\\delta_{cs}$ for product price $s=7$ (machinery) \nof EU (top left), USA (top right), China (bottom left)\nand $s=3$ (mineral fuel) of Russia (bottom right);\n$B_c$ is computed from PageRank and CheiRank vectors;\nsensitivity values are marked by color with the corresponding \ncolor bar marked by $j$. \nFor EU, USA, China, Russia we have  $dB_c\/d\\delta_{cs} = 0.11, 0.11, 0.14, 0.12$\nrespectively, these values are \nmarked by separate magenta color to highlight sensitivity \nof other countries in a better way.} \\label{fig5}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\textwidth]{fig6.pdf}\n\t\\caption{Network trade structure between \nEU, USA (US), China (CN), Russia (RU) \nwith 10 products. \nNetwork is obtained from the reduced\nGoogle matrix $G_R$ (left) and ${G^*}_R$ (right)\nby tracing four strongest outgoing links\n(similar to 4 ``best friends'').\nCountries are shown by circles with two letters of country\nand product index listed in Section 2. The arrow direction \nfrom node $A$ to node $B$ means that $B$ imports from $A$ (left)\nand  $B$ exports to $A$ (right).\nAll $40$ nodes are shown.\n} \\label{fig6}\n\\end{figure}\n\nIn Figs.~\\ref{fig3} and \\ref{fig4}, we have considered the sensitivity of \ncountry balance to a global price of a specific product (mineral fuel $s=3$ or machinery $s=7$). \nIn contrast, with GMA, we can also obtain the sensitivity of country balance to the price of\nproducts originating from a specific country. Such results are shown in Fig.~\\ref{fig5}.\nThey show that machinery ($s=7$) of EU gives a significant positive balance sensitivity \nfor UK and negative for Russia. This indicates a strong dependence of Russia \nfrom EU machinery.   Machinery of USA gives strong positive effect for Mexico and Canada\nwith a negative effect for EU, Russia, Brazil, Argentina.\nMachinery of China gives positive sensitivity for Asian countries\n(Repub. Korea, Japan, Philippines) and significant negative effect for Mexico.\nMineral fuels ($s=3$) of Russia gives positive effect for Kazakhstan, Uzbekistan, Ukraine\n(former USSR republics) and negative effect for competing petroleum and gas producers\nNorway and Algeria.\n\n\\subsection{Network structure of trade from reduced Google matrix}\n\nThe network structure for 40 nodes of 10 products of EU, USA, China and Russia is \nshown in Fig.~\\ref{fig6}. It is obtained from the reduced Google  matrix of $N_r=40$ nodes\nof global WTN network with $N=1680$ nodes on the basis of REGOMAX algorithm\nwhich takes into account all pathways between $N_r$ nodes via the global network of $N$ nodes.\nThe networks are shown for the direct  ($G$ matrix) and\ninverted  ($G^*$ matrix) trade flows. For each node we show only 4 strongest outgoing \nlinks (trade matrix elements) that heuristically can be considered as the four ``best friends''. \nThe resulting network structure clearly shows\nthe central dominant role of machinery product. For ingoing flows (import direction) of $G_R$ \nthe central dominance of machinery for  USA and EU is directly visible while  \nfor outgoing flows (export direction), machinery of EU and China dominate exports.\n\nIt is interesting to note that the network influence of EU with 27 countries\nis somewhat similar to the one constituted by a kernel of 9 dominant EU countries (KEU9)\n(being Austria, Belgium, France, Germany, Italy,\nLuxembourg, Netherlands, Portugal, Spain) discussed in \\cite{keu9}.  This shows\nthe leading role played by these KEU9 countries in the world trade influence of EU.\n\nFinally we note that additional data with figures and tables is available at \\cite{ourwebpage}.\n\n\\section{Discussion}\\label{sec:discussion}\n\nWe presented the Google matrix analysis of multiproduct WTN obtained from UN COMTRADE database\nin recent years. In contrast to the legacy Import-Export characterization of trade,\nthis new approach captures multiplicity of trade transactions between world countries\nand highlights in a better way the \nglobal significance and influence of trade relations between specific countries and products.\nThe Google matrix analysis clearly shows that the dominant position\nin WTN is taken by the EU of 27 countries despite the leave of UK after Brexit.\nThis result demonstrates the robust structure of worldwide EU trade.\nIt is in contrast with the usual Import-Export analysis in which  USA and China\nare considered as main players. We also see that machinery and mineral fuels\nproducts play a dominant role in the international trade.\nThe Google matrix analysis stresses the growing dominance of \nmachinery products of Asian countries (China, Japan, Republic of Korea).\n\nWe hope that the further development of Google matrix analysis \nof world trade will bring new insights in this complex system of world economy.\n\n{\\it Acknowledgments:}\nWe thank Leonardo Ermann for useful discussions.\nThis research has been partially supported through the grant\nNANOX $N^\\circ$ ANR-17-EURE-0009 (project MTDINA) in the frame \nof the {\\it Programme des Investissements d'Avenir, France} and\nin part by APR 2019 call of University of Toulouse and by \nRegion Occitanie (project GoIA).\nWe thank UN COMTRADE for providing us a friendly access\nto their detailed database.\n\n\n\n\n\n\\section{#1}\n    \\addcontentsline{lof}{section}{%\n        Section \\thesection: #1 \\vspace{10pt}\n    }\n}\n\n\n\n\n\\renewcommand{\\figurename}{SupFig.}\n\n\n\\date{February 2021}\n\n\\usepackage{graphicx}\n\\usepackage{hyperref}\n\n\\begin{document}\n\n\\title{Supplementary Material for: \\\\ Post-Brexit power of European Union \\\\ from the world trade network analysis}\n\n\\author{Justin Loye\\inst{1,2} \\and\nKatia Jaffr\\`es-Runser\\inst{3}\n\\and\nDima L. Shepelyansky\\inst{2}\n}\n\\authorrunning{J.Loye et al.}\n\\titlerunning{Supplementary Material for: Post-Brexit power of European Union from the WTN}\n\n\\institute{Institut de Recherche en Informatique de Toulouse, \\\\\nUniversit\\'e de Toulouse, UPS, 31062 Toulouse, France\\\\\n\\email{justin.loye@irit.fr}\\\\\n\\and\nLaboratoire de Physique Th\\'eorique, \nUniversit\\'e de Toulouse, CNRS, UPS, 31062 Toulouse, France\\\\\n\\email{dima@irsamc.ups-tlse.fr}\\\\\n\\and\nInstitut de Recherche en Informatique de Toulouse, \\\\\nUniversit\\'e de Toulouse, Toulouse INP, 31071 Toulouse, France\\\\\n\\email{katia.jaffresrunser@irit.fr}\\\\\n}\n\\maketitle  \n\n\\vspace{1.0cm}\n\nWe present additional results SupFig.~\\ref{fig:KKstar_2012_2016}, \\ref{fig:balance2012_2016}, \\ref{fig:importations},\n\\ref{fig:exportations}, \\ref{fig:dbalance_petroleum_2012_2016} and \\ref{fig:dbalance_machinery_2012_2016}.\n\nSupFig.~\\ref{fig:KKstar_2012_2016}, \\ref{fig:balance2012_2016}, \\ref{fig:dbalance_petroleum_2012_2016} and \\ref{fig:dbalance_machinery_2012_2016}\nreproduce for years 2012, 2014, 2016 respectively Fig.1, Fig.2, Fig.3 and Fig.4 of the main article. \nIn SupFig.~\\ref{fig:importations} and \\ref{fig:exportations}, we respectively show the volume of importation and exportation of countries in the world trade network.\n\n\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.85\\textwidth]{figS1.pdf}\n    \\caption{Circles with country flags show country positions on\n    the plane of PageRank-CheiRank indexes $(K,K^*)$\n    (summation is done over all products) (left panel)\n    and on the plane of ImportRank-ExportRank  $\\hat{K}$, $\\hat{K}^*$ \n    from trade volume (right panel);\n    data is shown only for index values less than $61$.\n   \n    First, second and third row correspond to years 2012, 2014, 2016.}\n    \\label{fig:KKstar_2012_2016}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.96\\textwidth]{figS2.pdf}\n    \\caption{World map of trade balance of countries  \n    $B_c = ({P_c}^* - P_c)\/({P_c}^* + P_c)$.\n    Left: probabilities are from PageRank and CheiRank vectors;\n    Right: probabilities are from the trade volume of Export-Import;\n    $B_c$ values are marked by color with the corresponding \n    color bar marked by $j$; \n   \n   \n    countries absent in the UN COMTRADE report are marked by black color \n    (here and in other SupFigs).\n    First, second and third row correspond to year 2012, 2014, 2016.}\n    \\label{fig:balance2012_2016}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{figS3.pdf}\n    \\caption{World map of countries importation volume in dollar.\n    Values lower than $10^8$ are clipped for better visibility.\n    First, second, third and fourth row correspond to years 2012, 2014, 2016, 2018.\n    }\n    \\label{fig:importations}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{figS4.pdf}\n    \\caption{World map of countries exportation volume in dollar.\n    Values lower than $10^8$ are clipped for better visibility.\n    First, second, third and fourth row correspond to years 2012, 2014, 2016, 2018.\n    }\n    \\label{fig:exportations}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.96\\textwidth]{figS5.pdf}\n    \\caption{Sensitivity of country balance \n    $dB_c\/d\\delta_s$ for product $s=3$ (mineral fuels).\n    Left: probabilities are from PageRank and CheiRank vectors;\n    Right: probabilities are from the trade volume of Export-Import.\n    Color bar marked by $j$ \n   \n    gives sensitivity..\n    First, second and third row correspond to year 2012, 2014, 2016.}\n    \\label{fig:dbalance_petroleum_2012_2016}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.96\\textwidth]{figS6.pdf}\n    \\caption{Same as in Fig.~\\ref{fig:dbalance_petroleum_2012_2016} but\n    for product $s=7$ (machinery).}\n    \\label{fig:dbalance_machinery_2012_2016}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe spectral heat content represents the total heat \nin a domain $D$ with Dirichlet boundary condition when the initial temperature is 1.\nThe spectral heat content for Brownian motions has been studied extensively.\nThe spectral heat content for isotropic stable processes was first studied in \\cite{Val2017}. \nSince then, considerable progress has been made toward understanding the asymptotic behavior of the spectral heat content for other L\\'evy processes (see \\cite{Val2016,  GPS19, P20, PS19}).\n\n\nThe following conjecture about the spectral heat content for isotropic $\\alpha$-stable\nprocesses, $\\alpha\\in (0,2)$, over bounded $C^{1,1}$ open sets (see Section \\ref{section:preliminaries} for the definition of $C^{1,1}$ open sets) was made in \\cite{Val2017}: \nAs $t\\downarrow 0$,\n\\begin{equation}\\label{eqn:Val conj}\nQ_{D}^{(\\alpha)}(t)=\n\\begin{cases}\n|D|-c_{1}|\\partial D|t^{1\/\\alpha} +O(t), \\quad \\alpha\\in (1,2),\\\\\n|D|-c_{2}|\\partial D|t\\ln(1\/t) +O(t), \\quad \\alpha=1,\\\\\n|D|-c_{3}\\text{Per}_{\\alpha}(D)t +o(t), \\quad \\alpha\\in (0,1),\n\\end{cases}\n\\end{equation}\nwhere $c_i, i=1, 2, 3,$ are constants and \n\\begin{equation}\\label{eqn:fractional perimeter}\n\\text{Per}_{\\alpha}(D):=\\int_{D}\\int_{D^{c}}\\frac{A_{d,\\alpha}}{|x-y|^{d+\\alpha}}dydx,\n\\quad \\mbox{ with }A_{d,\\alpha}=\\frac{\\alpha\\Gamma(\\frac{d+\\alpha}{2})}{2^{1-\\alpha}\\pi^{d\/2}\\Gamma(1-\\frac{\\alpha}{2})},\n\\end{equation}\nis the $\\alpha$-fractional perimeter of $D$.\nThis conjecture was resolved in dimension 1 in \\cite{Val2016} (actually, a slightly weaker version, with the error term being $o(t^{1\/\\alpha})$ in the case $\\alpha\\in (1, 2)$ and $o(t\\ln(1\/t))$ in the case $\\alpha=1$, of the conjecture was proved there).\nWe note that in \\cite{Val2017} the author \nconjectured, and also provided strong evidence, that the spectral heat content for isotropic $\\alpha$-stable processes  with $\\alpha\\in (0,2)$ must have an asymptotic expansion of the form as \\eqref{eqn:Val conj} for all dimensions $d\\geq2$, \nbut exact expressions for the coefficients $c_{i}$ were not provided.\nThen a two-term asymptotic expansion of the spectral heat content for L\\'evy processes of bounded variation in ${\\mathbb R}^{d}$ was established in \\cite{GPS19}.\nSince $\\alpha$-stable processes are of bounded variation if and only if $\\alpha\\in (0,1)$, \nthe result in \\cite{GPS19} proves \\eqref{eqn:Val conj} for $\\alpha\\in (0,1)$. \nThe purpose of this paper is to resolve the conjecture above for $\\alpha \\in [1,2)$ and $d\\geq 2$. \nIn fact, our result is slightly weaker than \\eqref{eqn:Val conj} since the error term is $o(t^{1\/\\alpha})$ for $\\alpha\\in (1, 2)$ and $o(t\\ln(1\/t))$ for $\\alpha=1$.\nWe also find explicit expressions for the constants $c_{1}$ and $c_{2}$.\nThe main results of this paper are Theorem \\ref{thm:main12} for $\\alpha\\in (1,2)$ and Theorem \\ref{thm:main1} for $\\alpha=1$. \nCombining Theorems \\ref{thm:main12} and \\ref{thm:main1} with \\cite[Corollary 3.5]{GPS19}, the asymptotic behavior of the spectral heat content for isotropic $\\alpha$-stable processes in bounded $C^{1,1}$ open sets $D$ can be stated as follows: \n\n\\begin{thm}\\label{thm:main}\nLet $D$ be a bounded $C^{1,1}$ open set in ${\\mathbb R}^{d}$, $d\\geq 2$ and \nlet\n$$\nf_{\\alpha}(t)=\n\\begin{cases}\nt^{1\/\\alpha} &\\text{ if } \\alpha\\in (1,2),\\\\\nt\\ln(1\/t) &\\text{ if } \\alpha=1,\\\\\nt &\\text{ if } \\alpha\\in (0,1).\n\\end{cases}\n$$\nThen, we have\n\\begin{equation}\\label{eqn:main result}\n\\lim_{t\\to 0}\\frac{|D|-Q_{D}^{(\\alpha)}(t)}{f_{\\alpha}(t)}=\n\\begin{cases}\n{\\mathbb E}[\\overline{Y}_{1}^{(\\alpha)}]|\\partial D| &\\text{if } \\alpha\\in (1,2),\\\\\n\\frac{|\\partial D|}{\\pi} &\\text{if }\\alpha=1,\\\\\n\\rm{Per}_{\\alpha}(D), &\\text{if } \n\\alpha\\in (0,1),\n\\end{cases}\n\\end{equation}\nwhere $\\overline{Y}_{t}^{(\\alpha)}=\\sup_{s\\le t}Y^{(\\alpha)}_s$ stands for the running supremum of a 1 dimensional symmetric $\\alpha$-stable process $Y^{(\\alpha)}_t$ and \n$\\rm{Per}_{\\alpha}(D)$ as defined in \\eqref{eqn:fractional perimeter}.\n\\end{thm}\nWe note that \\eqref{eqn:main result} is exactly the same form as \\cite[Theorem 1.1]{Val2016} if one interprets $|\\partial D|=2$ when $D$ is a bounded open interval in ${\\mathbb R}$, but the proof for $d\\geq 2$ is very different from the one dimensional case and much more challenging.\n\nThe two-term asymptotic expansion of the spectral heat content for Brownian motion was proved in \\cite{BD89}. \nThe crucial ingredient in \\cite{BD89} is the fact that \nindividual components of Brownian motion are independent.\nFor isotropic $\\alpha$-stable processes, $\\alpha\\in (0, 2)$, individual components  are not independent and the technique in \\cite{BD89} no longer works.\n When $\\alpha\\in (1,2)$,\nwe establish the lower bound for the heat loss $|D|-Q_{D}^{(\\alpha)}(t)$ by considering the most efficient way of exiting $D$ (see Lemma \\ref{lemma:lb}).\nIn order to establish the upper bound, we  approximate the heat loss \n$|D|-Q_{D}^{(\\alpha)}(t)$ by the heat loss of the half-space \n$|D|-Q_{H_{x}}^{(\\alpha)}(t)$ for  \n$x$ near $\\partial D$ (see Proposition \\ref{prop:ub12}) \nand show that the approximation error is of order $o(t^{1\/\\alpha})$ in Lemma \\ref{lemma:near boundary}, \nwhich is similar to tools exposed in the trace estimate results in \\cite{BK08, PS14}.\nHowever, these tools do not work when $\\alpha=1$ due to the non-integrability of \n${\\mathbb P}(\\overline{Y}^{(1)}_{1}>u)$ over $(1,\\infty)$, where $ \\overline{Y}^{(1)}_t$ stands for the\nsupremum of the Cauchy process up to time $t$, \nand the proof for $\\alpha=1$ requires new ideas and is considerably more difficult. \n\nIn case of $\\alpha=1$, we prove that the coefficient of the second term \nof the asymptotic expansion of $Q_{D}^{(1)}(t)$ is $-\\frac{|\\partial D|}{\\pi}$, \nwhich is the same as for the regular heat content $H^{(1)}_{D}(t)$ \nwhich represents the total heat in $D$ without the Dirichlet exterior condition \n(see \\eqref{e:rhc} below for the definition of regular heat content and \\cite[Theorem 1.2]{Val2017} \nfor the two-term asymptotic expansion for the regular heat content $H_{D}^{(1)}(t)$). \nIf there is no Dirichlet exterior condition on $D^{c}$, or equivalently if the heat moves freely in and out of $D$,\nthe heat loss of the regular heat content must be smaller than that of the spectral heat content and we obtain the lower bound for free in \\eqref{eqn:Cauchy lb}. The proof for the upper bound is much more demanding. \nThe crucial ingredient for the upper bound is the spectral heat content for subordinate killed Brownian motions in \\cite{PS19}. \nAn isotropic $\\alpha$-stable process $X_{t}^{(\\alpha)}$ can be realized as a subordinate Brownian motion $W_{S_{t}^{(\\alpha\/2)}}$, where $S_{t}^{(\\alpha\/2)}$ is an independent $(\\alpha\/2)$-stable subordinator.\nHence, the spectral heat content $Q_{D}^{(\\alpha)}(t)$ is the spectral heat content for the \\textit{killed subordinate Brownian motion} via the independent $(\\alpha\/2)$-stable subordinator $S_{t}^{(\\alpha\/2)}$. \nWhen one reverses the order of killing and subordination, one obtains the \\textit{subordinate killed Brownian motion}. \nThis is the process obtained by subordinating the killed Brownian motion in $D$ via the independent $(\\alpha\/2)$-stable subordinator $S_{t}^{(\\alpha\/2)}$. \nLet $\\widetilde{Q}_{D}^{(\\alpha)}(t)$ be the spectral heat content of the subordinate killed Brownian motion in $D$\n(see \\eqref{eqn:SKBM} for the precise definition).\nBy construction,\n $\\widetilde{Q}_{D}^{(\\alpha)}(t)$ is always smaller than $Q_{D}^{(\\alpha)}(t)$, \nor equivalently the heat loss $|D|-\\widetilde{Q}_{D}^{(\\alpha)}(t)$ of the subordinate killed Brownian motion provides a natural upper bound for \nthat of the killed subordinate Brownian motion $|D|-Q_{D}^{(\\alpha)}(t)$. \nWe use two independent $\\alpha\/2$ and $\\beta\/2$ stable subordinators \nwith $\\alpha\\beta=2$ \nand consider the $\\alpha$-stable process $X^{(\\alpha)}_{t}=W_{S_{t}^{(\\alpha\/2)}}$ and killed it upon exiting $D$. \nThen we time-change the killed $\\alpha$-stable process by the $(\\beta\/2)$-stable subordinator. \nBy using the heat loss of the resulting process, we obtain in Lemma \n\\ref{lemma:Cauchy ub} that, for $\\alpha\\in (1,2)$, \n$$\n\\limsup_{t\\to 0}\\frac{|D|-Q_{D}^{(1)}(t)}{t\\ln(1\/t)}\\leq\n\\frac{|\\partial D|}{\\pi} +\n\\frac{\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}<u )du }\n{\\Gamma(1-\\frac{1}{\\alpha})}\\cdot |\\partial D|,\n$$\nwhere $Y^{(\\alpha)}_t$ is a 1 dimensional symmetric $\\alpha$-stable process\nand $\\overline{Y}_{t}^{(\\alpha)}=\\sup_{s\\le t}Y^{(\\alpha)}_s$. \nWe would like to show that the second summand on the right hand side of the previous inequality converges to 0 as $\\alpha\\downarrow 1$. \nWe know that\n$\\frac{1}{\\Gamma(1-\\frac{1}{\\alpha})}=O(\\alpha-1)$ as $\\alpha\\downarrow 1$. \nThe integrand in the  numerator can be written as \n$$\n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}<u )du\n={\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u)-{\\mathbb P}(Y^{(\\alpha)}_{1}\\geq u)\n$$\nand it can be shown that \n${\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u)\\sim {\\mathbb P}(Y^{(\\alpha)}_{1}\\geq u) \\sim cu^{-\\alpha}$ \nas $u\\to \\infty$, \nhence\n$\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u)du$ and $\\int_{0}^{\\infty}{\\mathbb P}(Y^{(\\alpha)}_{1}\\geq u)du$ \nshould be of order $\\frac{1}{\\alpha-1}$ as $\\alpha\\downarrow 1$.\nWe show that\n$\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u)du$ and $\\int_{0}^{\\infty}{\\mathbb P}(Y^{(\\alpha)}_{1}\\geq u)du$ \nhave \\textit{the exactly same leading coefficients} and this gives a cancellation of the main terms of order $\\frac{1}{\\alpha-1}$. \nWe still need to show that the sub-leading terms are of order $o(\\alpha-1)$ as $\\alpha\\downarrow 1$ and we show this by establishing uniform heat kernel upper and lower bounds in Lemma \\ref{lemma:uniform ub} \nfor the heat kernel  of the supremum process $\\overline{Y}_{t}^{(\\alpha)}$ \nand Lemma \\ref{lemma:uniform lb} for the heat kernel of $Y_{t}^{(\\alpha)}$.\n\n\nThis paper deals with asymptotic behavior of the spectral heat content for isotropic $\\alpha$-stable processes. \nIt is natural and interesting to try to find the  asymptotic behavior of the spectral heat content for more general  L\\'evy processes. We intend to deal with this topic in a future project.\nIn the recent paper \\cite{P20}, a three-term asymptotic expansion of the spectral heat content of 1 dimensional symmetric $\\alpha$-stable processes, $\\alpha\\in [1,2)$, was established.\nWe believe that a similar result should hold true for $d\\geq 2$.\n\nThe organization of this paper is as follows. \nIn Section \\ref{section:preliminaries}, we introduce the setup. In Section \\ref{section:alpha12},\nwe deal with the case $\\alpha\\in (1,2)$ and the main result of that section is Theorem \\ref{thm:main12}.  \nThe case $\\alpha=1$ is dealt with in Section \\ref{section:alpha1} and  the main result there is Theorem \\ref{thm:main1}. \nIn this paper, we use $c_i$ to denote constants whose values are unimportant and may change from one appearance to another.\nThe notation ${\\mathbb P}_{x}$ stands for the law of the underlying processes started at $x\\in {\\mathbb R}$, and ${\\mathbb E}_{x}$ stands for expectation with respect to ${\\mathbb P}_{x}$. For simplicity, we use ${\\mathbb P}={\\mathbb P}_{0}$ and ${\\mathbb E}={\\mathbb E}_{0}$.\n\n\n\\section{Preliminaries}\\label{section:preliminaries}\n\nIn this paper,  unless explicitly stated otherwise, we assume $d\\geq 2$.\nLet $X_{t}^{(\\alpha)}$, $\\alpha\\in (0, 2]$, be an isotropic $\\alpha$-stable process with \n$$\n{\\mathbb E}[e^{i\\xi X^{(\\alpha)}_{t}}]=e^{-t|\\xi|^{\\alpha}}, \\quad \\xi \\in {\\mathbb R}^{d}, \\alpha\\in (0,2].\n$$\n$X_{t}^{(2)}$ is a Brownian motion $W_t$ with transition density given by $(4\\pi t)^{-d\/2}e^{-\\frac{|x|^2}{4t}}$. \nLet $S_{t}^{(\\alpha\/2)}$, $\\alpha\\in (0,2)$, be an $(\\alpha\/2)$-stable subordinator with\n$$\n{\\mathbb E}[e^{-\\lambda S_{t}^{(\\alpha\/2)}}]=e^{-t \\lambda^{\\alpha\/2}}, \\quad \\lambda>0.\n$$\nAssume that $S_{t}^{(\\alpha\/2)}$ is independent of the Brownian motion $W_t$. \nThen, the subordinate Brownian motion $W_{S_{t}^{(\\alpha\/2)}}$ is a realization of the process $X_{t}^{(\\alpha)}$.\nWe will reserve $Y_{t}^{(\\alpha)}$ for the 1 dimensional symmetric $\\alpha$-stable process.\nWe define the running supremum process $\\overline{Y}^{(\\alpha)}_{t}$ of  $Y_{t}^{(\\alpha)}$ by\n\\begin{equation}\\label{eqn:running}\n\\overline{Y}^{(\\alpha)}_{t}=\\sup\\{Y^{(\\alpha)}_{u} : 0\\leq u\\leq t\\}.\n\\end{equation}\n\nRecall that an open set $D$ in ${\\mathbb R}^d$ is said to be a $C^{1, 1}$ open set if \nthere exist a localization radius $R_0>0$ and a constant $\\Lambda_0>0$ such that, \nfor every $z\\in \\partial D$, there exist a $C^{1, 1}$ function $\\phi=\\phi_z: {\\mathbb R}^{d-1}\\to {\\mathbb R}$ satisfying $\\phi(0)=0$, $\\nabla \\phi(0)=(0, \\cdots, 0)$, $\\|\\nabla\\phi\\|_\\infty\\le \\Lambda_0$,\n$|\\nabla \\phi(x)-\\nabla \\phi(y)|\\le \\Lambda_0 |x-y|$ and an orthonormal coordinate system $CS_z: y=(\\widetilde y, y_d)$\nwith origin at $z$ such that\n$$\nB(z, R_0)\\cap D=B(z, R_0)\\cap \\{ y=(\\widetilde y, y_d) \\mbox{ in } CS_z: y_d>\\phi(\\widetilde y)\\}.\n$$\nThe pair $(R_0, \\Lambda_0)$ is called the $C^{1,1}$ characteristics of the $C^{1, 1}$ open set $D$. \nIt is well known that any $C^{1, 1}$ open set $D$  in ${\\mathbb R}^d$ satisfies the uniform interior and exterior $R$-ball condition: for any $z\\in\\partial D$,  there exist balls $B_1$ and $B_2$ of radii $R$ with $B_1\\subset D$, $B_2\\subset{\\mathbb R}^d\\setminus\\overline{D}$ and $\\partial B_1\\cap\\partial B_2=\\{z\\}$.\n\nWe recall from \\cite{BD89} a useful fact about open sets $D$ satisfying the uniform interior and exterior $R$-ball condition.\nLet $D_{q}=\\{x\\in D : \\text{dist}(x, \\partial D) >q\\}$.\nWe will use $\\partial D_{q}$ denote the portion of the boundary of $D_q$ contained in $D$, that is, $\\partial D_{q}=\\{x\\in D: \\text{dist}(x, \\partial D)=q\\}$.\nIt follows from \\cite[Lemma 6.7]{BD89}\nthat\n\\begin{equation}\\label{eqn:stability}\n|\\partial D|\\left(\\frac{R-q}{R}\\right)^{d-1}\\leq |\\partial D_{q}|\\leq |\\partial D|\\left(\\frac{R}{R-q}\\right)^{d-1}, \\quad 0< q<R.\n\\end{equation}\n\nIn the remainder of this paper, $D$ stands for a bounded $C^{1,1}$ open set in ${\\mathbb R}^{d}$, $d\\geq 2$.\nLet $\\tau_{D}^{(\\alpha)}=\\inf\\{t>0: X^{(\\alpha)}_{t}\\notin D\\}$ be the first exit time of $X^{(\\alpha)}_{t}$ from $D$. \nThe spectral heat content  of $D$ for $X^{(\\alpha)}_{t}$  is defined to be\n$$\nQ_{D}^{(\\alpha)}(t)=\\int_{D}{\\mathbb P}_{x}(\\tau_{D}^{(\\alpha)}>t)dx.\n$$\nThe (regular) heat content $H_{D}^{(\\alpha)}(t)$ of $D$  for $X^{(\\alpha)}_{t}$  is defined to be\n\\begin{equation}\\label{e:rhc}\nH_{D}^{(\\alpha)}(t)=\\int_{D}{\\mathbb P}_{x}(X_{t}^{(\\alpha)}\\in D)dx.\n\\end{equation}\nThe spectral heat content $\\widetilde{Q}_{D}^{(\\alpha)}(t)$  of $D$ for the subordinate killed Brownian motion is defined as \n\\begin{equation}\\label{eqn:SKBM}\n\\widetilde{Q}_{D}^{(\\alpha)}(t)=\\int_{D}{\\mathbb P}_{x}(\\tau_{D}^{(2)}>S_{t}^{(\\alpha\/2)})dx.\n\\end{equation}\nNote that we always have \n\\begin{equation}\\label{eqn:two heat contents}\n\\{\\tau_{D}^{(2)}>S_{t}^{(\\alpha\/2)}\\} \\subset \\{\\tau_{D}^{(\\alpha)}>t\\}\\subset \\{X_{t}^{(\\alpha)}\\in D\\}.\n\\end{equation}\nWe sometimes use the terminology \\textit{heat loss} and this will mean either\n$$\n|D|-Q_{D}^{(\\alpha)}(t)=\\int_{D}{\\mathbb P}_{x}(\\tau_{D}^{(\\alpha)}\\leq t)dx,\n$$\nor \n$$\n|D|-\\widetilde{Q}_{D}^{(\\alpha)}(t)=\\int_{D}{\\mathbb P}_{x}(\\tau_{D}^{(2)}\\leq S_{t}^{(\\alpha\/2)})dx,\n$$\ndepending on which process we are dealing with. \nIntuitively, these quantities represent the total heat loss caused by heat particles jumping out of $D$ up to time $t$.\nFrom \\eqref{eqn:two heat contents}, we have \n$$\n|D|- H_{D}^{(\\alpha)}(t)\\leq |D|-Q_{D}^{(\\alpha)}(t) \\leq |D|-\\widetilde{Q}_{D}^{(\\alpha)}(t), \\quad t>0.\n$$\n\n\n\n\\section{The case $\\alpha\\in (1,2)$}\\label{section:alpha12}\nThroughout this section, we assume $\\alpha\\in(1,2)$.\nFor any $x\\in D$, we use $\\delta_D(x)$ to denote the distance between $x$ and $\\partial D$.\nWe start with a lower bound. \nRecall that $D_{q}=\\{x\\in D : \\text{dist}(x, \\partial D) >q\\}$.\n\n\n\n\\begin{lemma}\\label{lemma:lb}\nLet $D$ be a bounded $C^{1,1}$ open set in ${\\mathbb R}^{d}$. Then, we have\n$$\n\\liminf_{t\\to 0}\\frac{|D|-Q^{(\\alpha)}_{D}(t)}{t^{1\/\\alpha}}\\geq |\\partial D|\n{\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}],\n$$\nwhere $\\overline{Y}_{t}^{(\\alpha)}$ is defined in \\eqref{eqn:running}.\n\\end{lemma}\n\\noindent{\\bf Proof.} \nLet $D$ satisfy the uniform interior and exterior $R$-ball condition.\nFix $a\\leq R\/2$. \nFor $x\\in D\\setminus D_{a}$, let \n$z_x\\in \\partial D$ be such that $|x-z_x|=\\delta_D(x)$ and  let ${\\bf n}_{z_x}$ be the outward unit normal vector to the boundary $\\partial D$ at the point $z_x$.\n\nFor $X$ starting from $x$, we define \n$Y^{(\\alpha)}_{t}:=(X_{t}^{(\\alpha)}-x)\\cdot {\\bf n}_{z_x}$, where $\\cdot$ stands for usual scalar product in ${\\mathbb R}^{d}$. Obviously, $Y^{(\\alpha)}_{0}=0$.\nNote that the characteristic exponent of $Y^{(\\alpha)}_{t}$ is given by \n$$\n{\\mathbb E}[e^{i\\eta Y^{(\\alpha)}_{t}}]=\n{\\mathbb E}_x[e^{i\\eta (X_{t}^{(\\alpha)}-x)\\cdot{\\bf n}_{z_x}}]\n=e^{-t|\\eta|^{\\alpha}}, \n\\quad \\eta \\in {\\mathbb R},\n$$\nand this shows that $Y^{(\\alpha)}_{t}$ is a one dimensional stable process starting from 0.\nLet $r\\le a$ and $x\\in  \\partial D_{r}$, and let $H_{x}$ be the half-space containing the interior $R$-ball at the point  $z_x$ and tangent to $\\partial D$ at $z_x$.\nWhen the process $X$ starts from $x$, we have\n$$\n\\{\\overline{Y}_{t}^{(\\alpha)} >r\\}=\\{\\tau_{H_x}^{(\\alpha)}\\leq t\\}\\subset \\{\\tau_{D}^{(\\alpha)} \\leq t\\} \n$$\nand this implies \n$$\n{\\mathbb P}(\\overline{Y}_{t}^{(\\alpha)} >r)\\leq {\\mathbb P}_{x}(\\tau_{D}^{(\\alpha)} \\leq t).\n$$\nHence, by the coarea formula\nand \\eqref{eqn:stability}, we have\n\\begin{align*}\n&|D|-Q^{(\\alpha)}_{D}(t)=\\int_{D}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{D}\\leq t)dx\n\\geq \\int_{D\\setminus D_{a}}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{D}\\leq t)dx\\\\\n&=\\int_{0}^{a}\\int_{\\partial D_{r}}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{D}\\leq t)dSdr\n\\geq \\int_{0}^{a}\\int_{\\partial D_{r}}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{t}>r)dSdr\\\\\n&=\\int_{0}^{a}|\\partial D_{r}|{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{t}>r)dr\n\\geq |\\partial D|\\left(\\frac{R-a}{R}\\right)^{d-1}\\int_{0}^{a}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{t}>r)dr,\n\\end{align*}\nwhere  $dS$ represents the surface measure on $\\partial D_{r}$.\nNow it follows from the scaling property of $Y^{(\\alpha)}$ and the change of variables $t^{-1\/\\alpha}r=s$ that\n\\begin{align*}\n&\\int_{0}^{a}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{t}>r)dr\n=\\int_{0}^{a}{\\mathbb P}(t^{1\/\\alpha}\\overline{Y}^{(\\alpha)}_{1}>r)dr\n=t^{1\/\\alpha}\\int_{0}^{t^{-1\/\\alpha}a}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>s)ds.\n\\end{align*}\nHence, we have\n$$\n\\frac{|D|-Q^{(\\alpha)}_{D}(t)}{t^{1\/\\alpha}}\\geq |\\partial D|\\left(\\frac{R-a}{R}\\right)^{d-1}\\int_{0}^{t^{-1\/\\alpha}a}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>s)ds\n$$\nand by taking $\\liminf$ we obtain\n$$\n\\liminf_{t\\to 0}\\frac{|D|-Q^{(\\alpha)}_{D}(t)}{t^{1\/\\alpha}}\\geq |\\partial D|\\left(\\frac{R-a}{R}\\right)^{d-1}{\\mathbb E}[\\overline{Y}^{(\\alpha)}_1].\n$$\nSince $a>0$ is arbitrary and \n${\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}]<\\infty$ \nwhen $\\alpha\\in (1,2)$,  the conclusion of the theorem is true.\n{\\hfill $\\Box$ \\bigskip}\n\n\n\n\nNow we proceed to prove the opposite inequality of Lemma \\ref{lemma:lb}. \nFor a bounded $C^{1,1}$ open set $D$ satisfying the uniform \ninterior and exterior $R$-ball condition\nand $x\\in D$ with $\\delta_{D}(x)<R\/2$, we let $z_x\\in \\partial D$ be the point on $\\partial D$ such that $|x-z_x|=\\delta_D(x)$ and \n$H_{x}$ be the half-space containing the interior \n$R$-ball at the point  $z_x$ and tangent to $\\partial D$ at $z_x$.\nFor $x\\in D$ with $\\delta_{D}(x)<R\/2$, we have\n\\begin{eqnarray}\\label{eqn:ub cases2}\n\\{\\tau_{D}^{(\\alpha)}\\leq t\\} \n&=& \\{\\tau_{D}^{(\\alpha)}\\leq t \\text{ and } \\tau_{H_{x}}^{(\\alpha)}\\leq t\\} \\cup \\{\\tau_{D}^{(\\alpha)}\\leq t\\text{ and } \\tau_{H_{x}}^{(\\alpha)}>t\\}\\nonumber\\\\\n&\\subset&\\{\\tau_{H_{x}}^{(\\alpha)}\\leq t\\} \\cup \\{\\tau_{D}^{(\\alpha)}\\leq t <\\tau_{H_{x}}^{(\\alpha)}\\}.\n\\end{eqnarray}\nHence, we have\n\\begin{eqnarray}\\label{eqn:ub cases}\n&&|D|-Q^{(\\alpha)}_{D}(t)=\\int_{D}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{D}\\leq t)dx=\\int_{D_{R\/2}}{\\mathbb P}_{x}(\\tau_{D}^{(\\alpha)}\\leq t)dx +\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}(\\tau_{D}^{(\\alpha)}\\leq t)dx\\nonumber\\\\\n&\\leq &\\int_{D_{R\/2}}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{D}\\leq t)dx+\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{H_{x}}\\leq t)dx +\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{D}\\leq t<\\tau^{(\\alpha)}_{H_{x}})dx.\n\\end{eqnarray}\n\nWe deal with the first expression of \\eqref{eqn:ub cases} first.\nRecall the following facts from \\cite[(2.2)]{Val2017} and \\cite[Corollary 6.4]{BD89}:\n\\begin{equation}\\label{eqn:exp moment}\n{\\mathbb E}\\left[\\exp\\left(-\\frac{\\kappa^{2}}{S_{1}^{(\\alpha\/2)}}\\right)\\right]\\leq c_{\\alpha}\\kappa^{-\\alpha}\n\\end{equation}\nand\n\\begin{equation}\\label{eqn:BM exit}\n{\\mathbb P}\\left(\\tau^{(2)}_{B(0,R)}<t\\right)\\leq 2^{1+d\/2}e^{-\\frac{R^2}{8t}}.\n\\end{equation}\n\n\\begin{lemma}\\label{lemma:inside}\nLet $D\\subset {\\mathbb R}^{d}$, $d\\geq 2$, be a bounded $C^{1,1}$ open set satisfying the uniform \ninterior and exterior $R$-ball condition. \nThen, we have\n$$\n\\int_{D_{R\/2}}{\\mathbb P}_{x}\\left(\\tau_{D}^{(\\alpha)}\\leq t\\right)dx\\leq ct.\n$$\n\\end{lemma}\n\\noindent{\\bf Proof.} \nFor $\\delta_{D}(x)\\geq \\frac{R}{2}$, it follows from \\eqref{eqn:exp moment} and \\eqref{eqn:BM exit} that \n\\begin{align*}\n&{\\mathbb P}_{x}\\left(\\tau_{D}^{(\\alpha)}\\leq t\\right)\n\\leq {\\mathbb P}_{x}\\left(\\tau^{(\\alpha)}_{B(x,\\delta_{D}(x))} \\leq t\\right)\n\\leq {\\mathbb P}_{x}\\left(\\tau^{(2)}_{B(x,\\delta_{D}(x))} \\leq S^{(\\alpha\/2)}_{t}\\right)\\\\\n\\leq& 2^{1+\\frac{d}2}{\\mathbb E}[\\exp(-\\frac{\\delta_{D}(x)^{2}}{8S_{t}^{(\\alpha\/2)}})]\n\\leq 2^{1+\\frac{d}2}{\\mathbb E}\\left[\\exp(-\\frac{\\delta_{D}(x)^{2}}{8t^{2\/\\alpha}S_{1}^{(\\alpha\/2)}})\\right]\n\\leq\\frac{ c(\\alpha,d)t}{\\delta_{D}(x)^{\\alpha}}\n\\leq \\frac{2^{\\alpha}c(\\alpha,d)t}{R^{\\alpha}}.\n\\end{align*}\nHence we have\n$$\n\\int_{D_{R\/2}}{\\mathbb P}_{x}\\left(\\tau_{D}^{(\\alpha)}\\leq t\\right)dx\\leq \\frac{2^{\\alpha}c(\\alpha,d)|D|t}{R^{\\alpha}}.\n$$\n{\\hfill $\\Box$ \\bigskip}\n\nThe second expression of \\eqref{eqn:ub cases} is handled in the following proposition. \n\\begin{prop}\\label{prop:ub12}\nLet $D\\subset {\\mathbb R}^{d}$, $d\\geq 2$, be a bounded $C^{1,1}$ open set satisfying the uniform \ninterior and exterior $R$-ball condition. \nThen, we have\n$$\n\\lim_{t\\to 0}\\frac{\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{H_{x}}\\leq t)dx}{t^{1\/\\alpha}}=|\\partial D|\n\\int_{0}^{\\infty}{\\mathbb P}_{(0,r)}(\\tau^{(\\alpha)}_{H}\\leq 1)dr=|\\partial D|\n{\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}],\n$$\nwhere $\\overline{Y}_{t}^{(\\alpha)}$ is defined in \\eqref{eqn:running} and $H=\\{x=(x_{1},\\cdots ,x_{d}) : x_{d}>0\\}$.\n\\end{prop}\n\\noindent{\\bf Proof.} \nFor $\\delta_{D}(x)<R\/2$, we have by the scaling property and the change of variables $v=t^{-1\/\\alpha}u$,\n\\begin{align*}\n&\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}\\left(\\tau^{(\\alpha)}_{H_{x}}\\leq t\\right)dx\n=\\int_{0}^{R\/2}|\\partial D_{u}|\n{\\mathbb P}_{(\\widetilde 0, u)}\\left(\\tau_{H}^{(\\alpha)}\\leq t\\right)du\\\\\n&=\\int_{0}^{R\/2}|\\partial D_{u}|\n{\\mathbb P}_{(\\widetilde 0, t^{-1\/\\alpha}u)}\n\\left(\\tau_{t^{-1\/\\alpha}H}^{(\\alpha)}\\leq 1\\right)du\n=t^{1\/\\alpha}\\int_{0}^{R\/2t^{1\/\\alpha}}|\\partial D_{vt^{1\/\\alpha}}|\n{\\mathbb P}_{(\\widetilde 0, v)}\\left(\\tau_{H}^{(\\alpha)}\\leq 1\\right)dv.\n\\end{align*}\n\n\nRecall that $Y^{(\\alpha)}_t=(X_{t}^{(\\alpha)}-x)\\cdot {\\bf n}_{z_x}$ is a one dimensional $\\alpha$-stable process starting from 0. \nHence, for any starting point $(\\tilde{0},r)$  \nwe have $\\{\\tau^{(\\alpha)}_{H}\\leq 1\\}=\\{\\overline{Y}^{(\\alpha)}_{1}>r\\}$, \nand this implies\n${\\mathbb P}_{(\\widetilde 0, r)}(\\tau^{(\\alpha)}_{H}\\leq 1)=\n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>r)$. \nHence, we have\n$$\n\\int_{0}^{\\infty}{\\mathbb P}_{(\\widetilde 0, v)}(\\tau^{(\\alpha)}\\leq 1)dv\n=\\int_{0}^{\\infty}{\\mathbb P}(\n\\overline{Y}^{(\\alpha)}_{1}\\geq v)dv={\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}].\n$$\nBy \\cite[Corollary 2.1 (ii)]{Val2016}, ${\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}]<\\infty$ if $\\alpha\\in (1,2)$.\nHence, it follows from the Lebesgue dominated convergence theorem that\n$$\n\\lim_{t\\to 0}\\frac{\\int_{D\\setminus D_{R\/2}} {\\mathbb P}_{x}(\\tau^{(\\alpha)}_{H_{x}} \\leq t)dx}{t^{1\/\\alpha}}=|\\partial D|\\int_{0}^{\\infty}\n{\\mathbb P}_{(\\widetilde 0, r)}(\\tau^{(\\alpha)}_{H}\\leq 1)dr=\n|\\partial D|{\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}].\n$$\n{\\hfill $\\Box$ \\bigskip}\n\n\n\nFinally, we estimate the last expression of \\eqref{eqn:ub cases}.\n\\begin{lemma}\\label{lemma:near boundary}\nLet $D\\subset {\\mathbb R}^{d}$, $d\\geq 2$, be a bounded $C^{1,1}$ \nopen set. \nThen, we have\n$$\n\\lim_{t\\to 0}\\frac{\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}\\left(\\tau_{D}^{(\\alpha)}\\leq t< \\tau^{(\\alpha)}_{H_{x}}\\right)dx}{t^{1\/\\alpha}}=0.\n$$\n\\end{lemma}\n\\noindent{\\bf Proof.} \nBy rotational invariance, we have\n\\begin{align}\\label{eqn:approx1}\n&\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}\\left(\\tau_{D}^{(\\alpha)}\\leq t< \\tau^{(\\alpha)}_{H_{x}}\\right)dx\n\\leq\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}\\left(\\tau^{(\\alpha)}_{B_{x}}\\leq t< \\tau^{(\\alpha)}_{H_{x}}\\right)dx\\nonumber\\\\\n&\\leq \\int_{0}^{R\/2}|\\partial D_{u}|{\\mathbb P}_{(\\widetilde 0, u)}\n\\left(\\tau^{(\\alpha)}_{B((\\widetilde 0, R), R)}\n\\leq t< \\tau^{(\\alpha)}_{H}\\right)du,\n\\end{align}\nwhere $H=\\{x=(x_{1},\\cdots ,x_{d}) : x_{d}>0\\}$.\nIt follows from \\cite[Lemma 6.7]{BD89} that  for $u<R\/2$,\n\\begin{equation}\\label{eqn:approx2}\n|\\partial D_{u}| \\leq 2^{d-1}|\\partial D|.\n\\end{equation}\n\nNote that it follows from the scaling property that the law of $a\\tau^{(\\alpha)}_{D}$ under ${\\mathbb P}_{x}$ is equal to the law of $\\tau^{(\\alpha)}_{a^{1\/\\alpha}D}$ under ${\\mathbb P}_{a^{1\/\\alpha}x}$.\nHence, it follows from the scaling property, together with  \\eqref{eqn:approx2}, that \\eqref{eqn:approx1} is bounded above by\n\\begin{eqnarray}\\label{eqn:approx3}\n&&2^{d-1}|\\partial D|\\int_{0}^{R\/2}\n{\\mathbb P}_{(\\widetilde 0,u)}\\left(\\tau^{(\\alpha)}_{B((\\widetilde 0, R), R)}\\leq t< \\tau^{(\\alpha)}_{H}\\right)du\\nonumber\\\\\n&=&2^{d-1}|\\partial D|\\int_{0}^{R\/2}\n{\\mathbb P}_{(\\widetilde 0,u)}\\left(t^{-1}\\tau^{(\\alpha)}_{B((\\widetilde 0, R), R)}\\leq 1< t^{-1}\\tau^{(\\alpha)}_{H}\\right)du\\nonumber\\\\\n&=&2^{d-1}|\\partial D|\\int_{0}^{R\/2}\n{\\mathbb P}_{(\\widetilde 0, t^{-1\/\\alpha}u)}\\left(\\tau^{(\\alpha)}_{B((\\widetilde 0, t^{-1\/\\alpha}R), t^{-1\/\\alpha}R)}\\leq 1< \\tau^{(\\alpha)}_{H}\\right)du\\nonumber\\\\\n&=&2^{d-1}|\\partial D|t^{1\/\\alpha}\\int_{0}^{2^{-1}t^{-1\/\\alpha}R}\n{\\mathbb P}_{(\\widetilde 0, v)}\\left(\\tau^{(\\alpha)}_{B((\\widetilde 0, t^{-1\/\\alpha}R), t^{-1\/\\alpha}R)}\\leq 1< \\tau^{(\\alpha)}_{H}\\right)dv\\nonumber\\\\\n&=&2^{d-1}|\\partial D|t^{1\/\\alpha}\\int_{0}^{\\infty}1_{(0,2^{-1}t^{-1\/\\alpha}R)}(v){\\mathbb P}_{(\\widetilde 0, v)}\\left(\\tau^{(\\alpha)}_{B((\\widetilde 0, t^{-1\/\\alpha}R), t^{-1\/\\alpha}R)}\\leq 1< \\tau^{(\\alpha)}_{H}\\right)dv\\nonumber,\n\\end{eqnarray}\nwhere we used the change of variables $v=t^{-1\/\\alpha}u$.\n\nNow we will show that there exists a non-negative function $f$ on $(0, \\infty)$ such that \n$$\n1_{(0,R\/2t^{1\/\\alpha})}(v)\n{\\mathbb P}_{(\\widetilde 0, v)}\\left(\\tau^{(\\alpha)}_{B((\\widetilde 0, t^{-1\/\\alpha}R), \nt^{-1\/\\alpha}R)}\\leq 1< \\tau^{(\\alpha)}_{H}\\right)\\leq f(v) \\quad \\text{and }\\int_{0}^{\\infty}f(v)dv<\\infty.\n$$\n\nAssume $1\\leq v$. \nSimilarly as in the proof of Lemma \\ref{lemma:inside}, by using \\eqref{eqn:exp moment} and \\eqref{eqn:BM exit} we find that \n\\begin{eqnarray*}\n&&{\\mathbb P}_{(\\widetilde 0, v)}\\left(\\tau^{(\\alpha)}_{B((\\widetilde 0, t^{-1\/\\alpha}R),\nt^{-1\/\\alpha}R)}\\leq 1< \\tau^{(\\alpha)}_{H}\\right)\n\\leq {\\mathbb P}_{(\\widetilde 0, v)}\\left( \\tau_{B((\\widetilde 0, v),v)}^{(\\alpha)}\\leq 1\\right)\n\\leq c(\\alpha,d)v^{-\\alpha}.\n\\end{eqnarray*}\nHence, we have\n$$\n1_{(0,2^{-1}t^{-1\/\\alpha}R)}(v)\n{\\mathbb P}_{(\\widetilde 0, v)}\\left(\\tau^{(\\alpha)}_{B((\\widetilde 0, t^{-1\/\\alpha}R), \nt^{-1\/\\alpha}R)}\\leq 1< \\tau^{(\\alpha)}_{H}\\right)\\leq\n1_{(0,1)}(v)+c(d,\\alpha)1_{(1,\\infty)}(v)\\cdot v^{-\\alpha}:=f(v).\n$$\nStarting from $(\\widetilde 0, v)$ with $v\\in (0,2^{-1}t^{-1\/\\alpha}R)$, as  $t\\to 0$, \n$B((\\widetilde 0, t^{-1\/\\alpha}R), t^{-1\/\\alpha}R)$ increases to $H$, and this implies $\\tau^{(\\alpha)}_{B((\\widetilde 0, t^{-1\/\\alpha}R), t^{-1\/\\alpha}R)}\\uparrow \\tau^{(\\alpha)}_{H}$.\nHence, we have\n$$\n\\lim_{t\\to 0}{\\mathbb P}_{(\\widetilde 0, v)}\\left(\\tau^{(\\alpha)}_{B((\\widetilde 0, t^{-1\/\\alpha}R), \nt^{-1\/\\alpha}R)}\\leq 1< \\tau^{(\\alpha)}_{H}\\right)=0.\n$$\nHence, it follows from the Lebesgue dominated convergence theorem that\n\\begin{align*}\n&\\lim_{t\\to 0}\\frac{\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}\\left(\\tau_{D}^{(\\alpha)}\\leq t< \\tau^{(\\alpha)}_{H_{x}}\\right)dx}{t^{1\/\\alpha}}\\\\\n\\leq&\\lim_{t\\to 0}|\\partial D|\\int_{0}^{\\infty}1_{(0,2^{-1}t^{-1\/\\alpha}R)}(v)\n{\\mathbb P}_{(v,0)}\\left(\\tau^{(\\alpha)}_{B((\\widetilde 0, t^{-1\/\\alpha}R), \nt^{-1\/\\alpha}R)}\\leq 1< \\tau^{(\\alpha)}_{H}\\right)dv\n=0.\n\\end{align*}\n{\\hfill $\\Box$ \\bigskip}\n\n\n\n\n\\begin{thm}\\label{thm:main12}\nLet $D\\subset {\\mathbb R}^{d}$, $d\\geq 2$, be a bounded $C^{1,1}$ open \nset. \nThen, we have\n\\begin{equation}\\label{eqn:main:12}\n\\lim_{t\\to0}\\frac{|D|-Q_{D}^{(\\alpha)}(t)}{t^{1\/\\alpha}}=|\\partial D|\n{\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}],\n\\end{equation}\nwhere $\\overline{Y}_{t}^{(\\alpha)}$ is defined in \\eqref{eqn:running}.\n\\end{thm}\n\\noindent{\\bf Proof.} \nThe lower bounded is proved in Lemma \\ref{lemma:lb}.\nNow we establish the upper bound. \nAssume $D$ satisfies the uniform interior and exterior $R$-ball condition.\nFor $\\delta_{D}(x)<R\/2$, it follows from \\eqref{eqn:ub cases2} that\n$$\n\\{\\tau_{D}^{(\\alpha)}\\leq t\\} \\subset \\{\\tau_{H_{x}}^{(\\alpha)}\\leq t\\} \\cup \\{\\tau^{(\\alpha)}_{D}\\leq t<\\tau^{(\\alpha)}_{H_{x}}\\},\n$$\nand \n$$\n\\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}(\\tau_{D}^{(\\alpha)}\\leq t)\\leq \\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}(\\tau_{H_{x}}^{(\\alpha)}\\leq t)dx + \\int_{D\\setminus D_{R\/2}}{\\mathbb P}_{x}(\\tau^{(\\alpha)}_{D}\\leq t<\\tau^{(\\alpha)}_{H_{x}})dx.\n$$\nCombining inequality in \\eqref{eqn:ub cases}, Lemmas  \\ref{lemma:inside}, \\ref{lemma:near boundary} and Proposition \\ref{prop:ub12},  \nwe arrive at \n$$\n\\limsup_{t\\to 0}\\frac{|D|-Q_{D}^{(\\alpha)}(t)}{t^{1\/\\alpha}}\\leq |\\partial D|\n{\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}],\n$$\nwhich together with Lemma \\ref{lemma:lb} leads to the desired limit. \n{\\hfill $\\Box$ \\bigskip}\n\n\n\n\\section{The case of Cauchy processes}\\label{section:alpha1}\nIn this section, we establish \nthe two-term asymptotic expansion for the spectral heat content of the Cauchy process.\nSince $|D|-H_{D}^{(1)}(t)\\leq |D|-Q_{D}^{(1)}(t)$, by \\cite[Theorem 1.2 (ii)]{Val2017} we have\n\\begin{equation}\\label{eqn:Cauchy lb}\n\\liminf_{t\\to 0}\\frac{|D|-Q_{D}^{(1)}(t)}{t\\ln(1\/t)}\\geq\\frac{|\\partial D|}{\\pi}.\n\\end{equation}\n\\begin{lemma}\\label{lemma:exp sup}\nSuppose $\\alpha \\in (1,2)$. It holds that\n$$\n{\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}]\n=\\frac{\\Gamma(1-\\frac{1}{\\alpha})}{\\pi} +\\int_{0}^{\\infty}\n{\\mathbb P}(\\overline{Y}_{1}^{(\\alpha)}>u, Y_{1}^{(\\alpha)}<u)du,\n$$\nwhere $\\overline{Y}_{t}^{(\\alpha)}$ is defined in \\eqref{eqn:running}.\n\\end{lemma}\n\\noindent{\\bf Proof.} \nConsider the spectral heat content $Q_{(a,b)}^{(\\alpha)}(t)$ of \n$(a,b)\\subset {\\mathbb R}$\nfor $Y_{t}^{(\\alpha)}$. It holds that\n\\begin{eqnarray}\\label{eqn:referee}\n&&\\frac{(b-a)-Q_{(a,b)}^{(\\alpha)}(t)}{t^{1\/\\alpha}}\n=\\frac{\\int_{a}^{b}{\\mathbb P}_{x}(\\tau_{(a,b)}^{(\\alpha)}\\leq t)dx}{t^{1\/\\alpha}}\\nonumber\\\\\n&=&\\frac{\\int_{a}^{b}{\\mathbb P}_{x}(Y_{t}^{(\\alpha)}\\notin (a,b))dx}{t^{1\/\\alpha}}+\\frac{\\int_{a}^{b}{\\mathbb P}_{x}(\\tau_{(a,b)}^{(\\alpha)}\\leq t, Y_{t}^{(\\alpha)}\\in (a,b))dx}{t^{1\/\\alpha}}.\n\\end{eqnarray}\nWe will show that \n\\begin{equation}\\label{eqn:Cauchy aux1}\n\\lim_{t\\to 0}\\frac{\\int_{a}^{b}{\\mathbb P}_{x}(\\tau_{(a,b)}^{(\\alpha)}\\leq t, \nY_{t}^{(\\alpha)}\\in (a,b))dx}{t^{1\/\\alpha}}=2\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}_{1}^{(\\alpha)}>u, Y_{1}^{(\\alpha)}<u)du.\n\\end{equation}\nTogether with $\\lim_{t\\to 0}t^{-1\/\\alpha}\\int_{a}^{b}{\\mathbb P}_{x}(Y_{t}^{(\\alpha)}\\notin (a,b))dx=2\\pi^{-1}\\Gamma(1-\\frac{1}{\\alpha})$ from \\cite[Theorem 1.1. (i)]{Val2017} and\n$\\lim_{t\\to 0}t^{-1\/\\alpha}\\int_{a}^{b}{\\mathbb P}_{x}(\\tau_{(a,b)}^{(\\alpha)}\\leq t)dx=2{\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}]$ from \\cite[Theorem 1.1]{Val2016}, \nthis will establish the conclusion by taking limits at both sides of \\eqref{eqn:referee}. \n\nNote that we have \n\\begin{eqnarray*}\n&&\\int_{a}^{b}{\\mathbb P}_{x}(\\tau_{(a,b)}^{(\\alpha)}\\leq t, Y_{t}^{(\\alpha)}\\in (a,b))dx\\nonumber\\\\\n&=&\\int_{a}^{b}{\\mathbb P}_{x}(\\overline{Y}^{(\\alpha)}_{t} >b, Y_{t}^{(\\alpha)}\\in (a,b))dx +\\int_{a}^{b}{\\mathbb P}_{x}(\\underline{Y}^{(\\alpha)}_{t}<a, Y_{t}^{(\\alpha)}\\in (a,b))dx\\nonumber\\\\\n&&-\\int_{a}^{b}{\\mathbb P}_{x}(\\overline{Y}^{(\\alpha)}_{t} >b, \\underline{Y}^{(\\alpha)}_{t}<a, Y_{t}^{(\\alpha)}\\in (a,b))dx,\n\\end{eqnarray*}\nwhere $\\underline{Y}^{(\\alpha)}_{t}=\\inf\\{ Y_{u}^{(\\alpha)}: 0\\leq u\\leq t\\}$ is the running infimum process of $Y_{t}^{(\\alpha)}$.\nIt follows from \\cite[Lemma 3.2]{P20} that\n$$\n\\int_{a}^{b}{\\mathbb P}_{x}(\\overline{Y}^{(\\alpha)}_{t} >b, \\underline{Y}^{(\\alpha)}_{t}<a, Y_{t}^{(\\alpha)}\\in (a,b))dx=O(t^{1+\\frac{1}{\\alpha}}) \\text{ as } t\\downarrow 0.\n$$\nBy the change of variables $u=(b-x)t^{-1\/\\alpha}$ and the scaling property we have\n\\begin{eqnarray*}\n&&\\int_{a}^{b}{\\mathbb P}_{x}(\\overline{Y}^{(\\alpha)}_{t} >b, Y_{t}^{(\\alpha)}\\in (a,b))dx\\\\\n&=&t^{1\/\\alpha}\\int_{0}^{(b-a)t^{-1\/\\alpha}}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, -(b-a)t^{-1\/\\alpha}+u <Y_{1}^{(\\alpha)}<u)du.\n\\end{eqnarray*}\nNote that we have\n$$\n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, -(b-a)t^{-1\/\\alpha}+u <Y_{1}^{(\\alpha)}<u)\\leq {\\mathbb P}(\\overline{Y}^{(\\alpha)}>u)\n$$\nand\n$$\n\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u)du={\\mathbb E}[\\overline{Y}^{(\\alpha)}_{1}]<\\infty.\n$$\nHence, it follows from the Lebesgue dominated convergence theorem that the limit is\n\\begin{align*}\n&\\lim_{t\\to 0}\\frac{\\int_{a}^{b}{\\mathbb P}_{x}(\\tau_{(a,b)}^{(\\alpha)}\\leq t, Y_{t}^{(\\alpha)}\\in (a,b))dx}{t^{1\/\\alpha}}\\\\\n=&\\lim_{t\\to 0}\\frac{t^{1\/\\alpha}\\int_{0}^{(b-a)t^{-1\/\\alpha}}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, -(b-a)t^{-1\/\\alpha}+u <Y_{1}^{(\\alpha)}<u)du}{t^{1\/\\alpha}}\\\\\n=&\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}<u)du.\n\\end{align*}\nThe second term can be handled in a similar way using the symmetry of $Y^{(\\alpha)}_{t}$, and this proves \\eqref{eqn:Cauchy aux1}.\n{\\hfill $\\Box$ \\bigskip}\n\n\nWe need a simple lemma which is similar to \\cite[Lemma 3.2]{PS19}. The proof is essentially the same with obvious modifications. We provide the details for the reader's convenience. \n\\begin{lemma}\\label{lemma:SS expectation}\nFor any $\\delta>0$ and $\\alpha\\in (0,2)$, we have\n$$\n\\lim_{t\\to 0}\\frac{{\\mathbb E}[\\left(S_{1}^{(\\alpha\/2)}\\right)^{\\alpha\/2}, 0<S_{1}^{(\\alpha\/2)}<\\delta t^{-2\/\\alpha} ]}{\\ln(1\/t)}=\\frac{1}{\\Gamma(1-\\frac{\\alpha}{2})}.\n$$\n\\end{lemma}\n\\noindent{\\bf Proof.} \nNote that the stable subordinator $S_{t}^{(\\alpha\/2)}$ has a continuous transition density $g^{(\\alpha\/2)}(t,u)$ and it follows from \\cite[(2.5)]{PS19} that\n\\begin{equation}\\label{eqn:SS limit}\n\\lim_{u\\to\\infty}g^{(\\alpha\/2)}(1,u)u^{1+\\frac{\\alpha}{2}}=\\frac{\\alpha}{2\\Gamma(1-\\frac{\\alpha}{2})}.\n\\end{equation}\nAlso it follows from \\cite[Proposition 2.1]{Val2017} that \n$$\n\\lim_{t\\to 0}{\\mathbb E}[\\left(S_{1}^{(\\alpha\/2)}\\right)^{\\alpha\/2}, 0<S_{1}^{(\\alpha\/2)}<\\delta t^{-2\/\\alpha} ]=\\infty.\n$$\nHence, it follows from L'H\\^opital's rule and \\eqref{eqn:SS limit} that\n\\begin{eqnarray*}\n&&\\lim_{t\\to 0}\\frac{{\\mathbb E}[\\left(S_{1}^{(\\alpha\/2)}\\right)^{\\alpha\/2}, 0<S_{1}^{(\\alpha\/2)}<\\delta t^{-2\/\\alpha} ]}{\\ln(1\/t)}=\\lim_{t\\to 0}\\frac{\\int_{0}^{\\delta t^{-2\/\\alpha}} u^{\\alpha\/2}g^{(\\alpha\/2)}(1,u)du}{\\ln(1\/t)}\\\\\n&=&\\lim_{t\\to 0}\\frac{(\\delta t^{-2\/\\alpha})^{\\alpha\/2}g^{(\\alpha\/2)}(1,\\delta t^{-2\/\\alpha})(-\\frac{2}{\\alpha}\\delta t^{-\\frac{2}{\\alpha}-1}) }{-1\/t}=\\lim_{t\\to0}\\frac{2}{\\alpha}g^{(\\alpha\/2)}(1,\\delta t^{-2\/\\alpha})(\\delta t^{-2\/\\alpha})^{1+\\frac{\\alpha}{2}}\\\\\n&=&\\frac{1}{\\Gamma(1-\\frac{\\alpha}{2})}.\n\\end{eqnarray*}\n{\\hfill $\\Box$ \\bigskip}\n\n\\begin{lemma}\\label{lemma:Cauchy ub}\nLet $D$ be a bounded $C^{1,1}$ open set in ${\\mathbb R}^{d}$, $d\\geq 2$. For any $\\alpha\\in (1,2)$, we have\n$$\n\\limsup_{t\\to 0}\\frac{|D|-Q_{D}^{(1)}(t)}{t\\ln(1\/t)}\\leq \\frac{|\\partial D|}{\\pi} +\n\\frac{\\int_{0}^{\\infty}{\\mathbb P}(\n\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}<u )du }\n{\\Gamma(1-\\frac{1}{\\alpha})}\\cdot |\\partial D|,\n$$\nwhere $\\overline{Y}_{t}^{(\\alpha)}$ is defined in \\eqref{eqn:running}.\n\\end{lemma}\n\\noindent{\\bf Proof.} \nThe main tool for this proof is the spectral heat content for \\textit{subordinate killed stable processes}. \nLet $\\alpha\\in(1,2)$ and $\\alpha\\beta=2$. Let $S_{t}$ and $T_{t}$ be independent $\\frac{\\alpha}{2}$ and $\\frac{\\beta}{2}$ stable subordinators which are independent of the Brownian motion $W_{t}$. \nLet $X^{(\\alpha)}_t=W_{S_{t}^{(\\alpha\/2)}}$, $X^{(\\alpha), D}_t$ be the process $X^{(\\alpha)}_t$ killed upon exiting $D$, and\n$Z_{t}=X^{(\\alpha), D}_{T_{t}^{(\\beta\/2)}}$ be \nthe subordinate killed $\\alpha$-stable process.\nLet $\\widetilde{Q}^{(\\alpha,\\beta)}_{D}(t)$ be the spectral heat content for $Z$, that is, \n$$\n\\widetilde{Q}^{(\\alpha,\\beta)}_{D}(t)=\\int_{D}{\\mathbb P}_{x}(\\tau_{D}^{(\\alpha)}>T_{t}^{(\\beta\/2)})dx.\n$$\n\nSince $\\{\\tau_{D}^{(1)} \\leq t\\}\\subset \\{\\tau_{D}^{(\\alpha)} \\leq T_{t}^{(\\beta\/2)}\\}$, we have $|D|-Q_{D}^{(1)}(t)\\leq |D|-\\widetilde{Q}^{(\\alpha,\\beta)}_{D}(t)$. \nWe will show that \n\\begin{equation}\\label{eqn:Cauchy aux7}\n\\limsup_{t\\to 0}\\frac{|D|-\\widetilde{Q}^{(\\alpha,\\beta)}_{D}(t)}{t\\ln(1\/t)}\\leq \n\\frac{|\\partial D|}{\\pi} +\n\\frac{\\int_{0}^{\\infty}\n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}<u )du }\n{\\Gamma(1-\\frac{1}{\\alpha})}\\cdot |\\partial D|.\n\\end{equation}\nNote that \n\\begin{eqnarray}\\label{eqn:Cauchy aux3}\n&&|D|-\\tilde{Q}^{(\\alpha,\\beta)}_{D}(t)=\\int_{0}^{\\infty}(|D|-Q_{D}^{(\\alpha)}(u)){\\mathbb P}(T_{t}^{(\\beta\/2)}\\in du).\n\\end{eqnarray}\nIt follows from Lemma \\ref{lemma:exp sup} and \\eqref{eqn:main:12} that, for any $\\varepsilon>0$,\nthere exists $\\delta=\\delta(\\varepsilon)$ such that for all $u\\leq \\delta$,\n\\begin{eqnarray*}\n\\frac{|D|-Q_{D}^{(\\alpha)}(u)}{u^{1\/\\alpha}} \n\\leq \\left(\\frac{\\Gamma(1-\\frac{1}{\\alpha})}{\\pi} + \\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}<u )du\\right)|\\partial D| +\\varepsilon.\n:=C_{1}+\\varepsilon.\n\\end{eqnarray*}\nHence, \\eqref{eqn:Cauchy aux3} can be written as\n\\begin{eqnarray}\\label{eqn:Cauchy aux4}\n&&\\int_{0}^{\\infty}(|D|-Q_{D}^{(\\alpha)}(u)){\\mathbb P}(T_{t}^{(\\beta\/2)}\\in du)\\nonumber\\\\\n&=&\\int_{0}^{\\delta}(|D|-Q_{D}^{(\\alpha)}(u)){\\mathbb P}(T_{t}^{(\\beta\/2)}\\in du)+\\int_{\\delta}^{\\infty}(|D|-Q_{D}^{(\\alpha)}(u)){\\mathbb P}(T_{t}^{(\\beta\/2)}\\in du)\\nonumber\\\\\n&=&\\int_{0}^{\\delta}(\\frac{|D|-Q_{D}^{(\\alpha)}(u)}{u^{1\/\\alpha}})u^{1\/\\alpha}{\\mathbb P}(T_{t}^{(\\beta\/2)}\\in du)+\\int_{\\delta}^{\\infty}(|D|-Q_{D}^{(\\alpha)}(u)){\\mathbb P}(T_{t}^{(\\beta\/2)}\\in du)\\nonumber\\\\\n&\\leq&\\int_{0}^{\\delta}( C_{1}+\\varepsilon)u^{1\/\\alpha}{\\mathbb P}(T_{t}^{(\\beta\/2)}\\in du)+\\int_{\\delta}^{\\infty}(|D|-Q_{D}^{(\\alpha)}(u)){\\mathbb P}(T_{t}^{(\\beta\/2)}\\in du).\n\\end{eqnarray}\nIt follows from \\cite[(2.8)]{PS19} that the second term in \n\\eqref{eqn:Cauchy aux4} can be estimated by\n\\begin{eqnarray*}\n&&\\int_{\\delta}^{\\infty}(|D|-Q_{D}^{(\\alpha)}(u)){\\mathbb P}(T_{t}^{(\\beta\/2)}\\in du)\\leq|D|\\int_{\\delta}^{\\infty}{\\mathbb P}(t^{2\/\\beta}T_{1}^{(\\beta\/2)}\\in du)\\\\\n&=&|D|\\int_{\\delta t^{-2\/\\beta}}^{\\infty}{\\mathbb P}(T_{1}^{(\\beta\/2)}\\in dv)\\leq c|D|(\\delta t^{-2\/\\beta})^{-\\frac{\\beta}{2}}=c|D|\\delta^{-\\frac{\\beta}{2}}t,\n\\end{eqnarray*}\nand \n\\begin{equation}\\label{eqn:Cauchy aux5}\n\\limsup_{t\\to 0}\\frac{\\int_{\\delta}^{\\infty}(|D|-Q_{D}^{(\\alpha)}(u)){\\mathbb P}(T_{t}^{(\\beta\/2)}\\in du)}{t\\ln(1\/t)}\\leq \\limsup_{t\\to 0}\\frac{c|D|\\delta^{-\\frac{\\beta}{2}}t}{t\\ln(1\/t)}=0.\n\\end{equation}\n\nRecall that $\\alpha\\beta=2$. \nHence, by the change of variables $u=t^{2\/\\beta}v$, the first expression in \\eqref{eqn:Cauchy aux4} can be written as\n\\begin{eqnarray*}\n&&\\int_{0}^{\\delta}u^{1\/\\alpha}{\\mathbb P}(T_{t}^{(\\beta\/2)}\\in du)\\nonumber=\\int_{0}^{\\delta}u^{1\/\\alpha}{\\mathbb P}(t^{2\/\\beta}T_{1}^{(\\beta\/2)}\\in du)\\nonumber\\\\\n&=&\\int_{0}^{\\delta t^{-2\/\\beta}}(t^{2\/\\beta}v)^{1\/\\alpha}{\\mathbb P}(T_{1}^{(\\beta\/2)}\\in dv)\\nonumber=t\\int_{0}^{\\delta t^{-2\/\\beta}}v^{\\beta\/2}{\\mathbb P}(T_{1}^{(\\beta\/2)}\\in dv).\n\\end{eqnarray*}\nIt follows from Lemma \\ref{lemma:SS expectation} that\n\\begin{equation}\\label{eqn:Cauchy aux6}\n\\limsup_{t\\to 0}\\frac{t\\int_{0}^{\\delta t^{-2\/\\beta}}v^{\\beta\/2}{\\mathbb P}(S_{1}^{(\\beta\/2)}\\in dv)}{t\\ln(1\/t)}=\\frac{1}{\\Gamma(1-\\frac{\\beta}{2})}.\n\\end{equation}\n\nCombining \\eqref{eqn:Cauchy aux5} and \\eqref{eqn:Cauchy aux6} we conclude that \n$$\n\\limsup_{t\\to 0}\\frac{|D|-\\widetilde{Q}^{\\alpha,\\beta}(t)}{t\\ln(1\/t)}\\leq ( C_{1} +\\varepsilon) \\times \\frac{1}{\\Gamma(1-\\frac{\\beta}{2})}=\\frac{|\\partial D|}{\\pi} \n+\\frac{\\int_{0}^{\\infty}\n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}<u )du \\cdot |\\partial D|}\n{\\Gamma(1-\\frac{1}{\\alpha})}\n+\\frac{\\varepsilon}{\\Gamma(1-\\frac{1}{\\alpha})}.\n$$\nSince $\\varepsilon>0$ is arbitrary, this establishes \\eqref{eqn:Cauchy aux7}.\n{\\hfill $\\Box$ \\bigskip}\n\n\nWe now recall the definition for the double gamma function from \\cite[(4.4)]{Ku11}. \nFor $z\\in \\mathbb{C}$ and $\\tau \\in \\mathbb{C}$ with $|\\text{arg}(\\tau)|<\\pi$, we define \n$$\nG(z;\\tau)=\\frac{z}{\\tau}e^{\\frac{az}{\\tau} + \\frac{bz^2}{2\\tau}}\n{\\textstyle \\prod}_{m\\geq 0}{\\textstyle \\prod}^{'}_{n\\geq 0}\\left( 1+\\frac{z}{m\\tau +n}\\right)\\exp\\left(-\\frac{z}{m\\tau +n} +\\frac{z^2}{2(m\\tau +n)^2}\\right),\n$$\nwhere the prime in the second product means that the term corresponding to $m=n=0$ is omitted. \n\n\\begin{lemma}\\label{lemma:double gamma bounded}\nLet $K$ be a compact set in $\\mathbb{C}$. \nThen there exists a constant $c=c(K)>0$ such that \n$$\n\\left| G(z;\\tau) \\right| \\leq c \\text{ for all } z\\in K \\text{ and } \\tau\\in [1,2].\n$$\n\\end{lemma}\n\\noindent{\\bf Proof.} \nOn the compact set $K\\times [1, 2]$, $\\frac{z}{\\tau}e^{\\frac{az}{\\tau} + \\frac{bz^2}{2\\tau}}$ is continuous and hence bounded. \nWe only need to prove that the double infinite product in $G(z, \\tau)$ is bounded. \nRecall the canonical factor $E_{2}(z)$ (see \\cite[p. 145]{SS complex}):\n$$\nE_{2}(z)=(1-z)e^{z+\\frac{z^2}{2}}.\n$$\nThe double infinite product in $G(z;\\tau)$ can be written as\n$$\n {\\textstyle \\prod}_{m\\geq 0}{\\textstyle \\prod}^{'}_{n\\geq 0}\\left( 1+\\frac{z}{m\\tau +n}\\right)\\exp\\left(-\\frac{z}{m\\tau +n} +\\frac{z^2}{2(m\\tau +n)^2}\\right)\n = {\\textstyle \\prod}_{m\\geq 0}{\\textstyle \\prod}^{'}_{n\\geq 0}E_{2}(-\\frac{z}{m\\tau +n}).\n$$\nIt follows from \\cite[Lemma 4.2]{SS complex} for $|w|\\leq 1\/2$, $|1-E_{2}(w)|\\leq c_{1}|w|^{3}$ for some constant $c_{1}$.\nSince $|-\\frac{z}{m\\tau +n}| \\leq \\frac{|z|}{m+n}$, \nall but finitely many terms of the form $|-\\frac{z}{m\\tau +n}|$ in the double infinite product are less than $\\frac12$, and since each of these finitely many terms are bounded on $K\\times [1, 2]$, we may assume $|-\\frac{z}{m\\tau +n}|\\leq \\frac12$ on $K\\times [1, 2]$ for all $n,m$.\n\nNow it follows from \\cite[Lemma 4.2]{SS complex} for all \n$(z, \\tau)\\in  K\\times [1, 2]$ that\n$$\n\\left| 1-E_{2}(-\\frac{z}{m\\tau +n})\\right| \\leq c_{1}\\left |\\frac{z}{m\\tau +n}\\right|^{3}\\leq \\frac{c_{2}}{(m+n)^{3}} \\text{ for all } n,m.\n$$\nIt is easy to see that \n$$\n\\sum_{m\\geq 0, n\\geq 0, (m,n)\\neq (0, 0)}\\frac{1}{(m+n)^{3}}<\\infty.\n$$\nNote that for any $|w|\\leq \\frac{1}{2}$ we have $\\ln |1-w|\\leq 2|w|$.\nThere exist positive integers $N_1$ and  $M_1$ such that $|1-E_{2}(-\\frac{z}{m\\tau +n})|\\leq \\frac{1}{2}$ on $K\\times [1, 2]$ for all $n\\geq N_{1}$ and $m\\geq M_{1}$.\nHence, for any $M, N \\in {\\mathbb N}$ large we have \n\\begin{align*}\n&\\left|{\\textstyle \\prod}_{M_{1}\\leq m \\leq M}{\\textstyle \\prod}_{N_{1}\\leq n\\leq N} E_{2}(-\\frac{z}{m\\tau +n}) \\right|\n={\\textstyle \\prod}_{M_{1}\\leq m \\leq M}{\\textstyle \\prod}_{N_{1}\\leq n\\leq N}\\left|1- (1- E_{2}(-\\frac{z}{m\\tau +n}) )\\right|\\\\\n&={\\textstyle \\prod}_{M_{1}\\leq m \\leq M}{\\textstyle \\prod}_{N_{1}\\leq n\\leq N}\\exp \\left( \\ln  \\left| (1- (1- E_{2}(-\\frac{z}{m\\tau +n}) )) \\right| \\right)\\\\\n&= \\exp \\left( \\sum_{M_{1}\\leq m \\leq M,N_{1}\\leq n\\leq N} \\ln \\left|(1- (1- E_{2}(-\\frac{z}{m\\tau +n}) )) \\right| \\right)\\\\\n&\\leq \\exp \\left( 2\\sum_{M_{1}\\leq m \\leq M,N_{1}\\leq n\\leq N} \\left| 1- E_{2}(-\\frac{z}{m\\tau +n})  \\right| \\right)\\leq \\exp \\left( 2c_2\\sum_{M_{1}\\leq m \\leq M,N_{1}\\leq n\\leq N} \\frac{1}{(m+n)^{3}} \\right)\\\\\n&\\leq \\exp \\left( 2c_2\\sum_{M_{1}\\leq m, N_{1}\\leq n} \\frac{1}{(m+n)^{3}} \\right).\n\\end{align*}\nBy letting $M, N\\to \\infty$ we see that the double infinite product is bounded on \n$K\\times [1, 2]$.\n{\\hfill $\\Box$ \\bigskip}\n\nIt follows from \\cite[Proposition VIII.1-4]{Ber} that there exists a constant $C>0$ such that\n\\begin{equation}\\label{eqn:same asymptotic}\n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_1>u)\\sim{\\mathbb P}(Y^{(\\alpha)}_1>u)\\sim Cu^{-\\alpha} \n\\text{ as } u\\to \\infty.\n\\end{equation}\nLet $\\rho:={\\mathbb P}(Y_1^{(\\alpha)}>0)$.  We say $Y^{(\\alpha)}\\in C_{k,l}$ if\n\\begin{align}\\label{eqn:ckl}\n&(\\alpha, \\rho)\\in \\{\\alpha\\in (0,1), \\rho\\in(0,1)\\} \\cup \\{\\alpha=1, \\rho=1\/2\\}\\cup \\{\\alpha\\in (1,2), \\rho\\in [1-\\frac{1}{\\alpha}, \\frac{1}{\\alpha}]\\} \\text{ and}\\nonumber\\\\\n&\\frac{1}{2}+k=\\frac{l}{\\alpha}, \\text{ or equivalently } \\alpha=\\frac{2l}{1+2k} \\text{ for some } k,l\\in {\\mathbb N}.\n\\end{align}\nNote that this condition already appeared in \\cite[\\text{Definition 1}]{Ku11}. \n\n\\begin{lemma}\\label{lemma:uniform ub}\nSuppose that $Y^{(\\alpha)}\\in C_{k,k+1}$\nfor some $k\\in \\mathbb{N}$.\nThen $\\overline{Y}_1^{(\\alpha)}$ has a density $\\overline{p}^{(\\alpha)}(x)$ and there exists \na constant $A>0$, \nindependent of all $\\alpha\\in(1,2)$ satisfying $Y_{t}^{(\\alpha)}\\in C_{k,k+1}$, such that\n$$\n\\overline{p}^{(\\alpha)}(x)\\leq C\\alpha x^{-1-\\alpha} +Ax^{-3}\n$$\nfor all $x>0$, where $C$ is the constant in \\eqref{eqn:same asymptotic}.\n\\end{lemma}\n\\noindent{\\bf Proof.} \nThe proof is similar to that of \\cite[Theorem 9]{Ku11} with a focus on establishing \na uniform constant $A$. \nIt follows from the proof of \\cite[Theorem 9]{Ku11} that $\\overline{Y}_1^{(\\alpha)}$ has a density $\\overline{p}^{(\\alpha)}(x)$ given by the inverse Mellin transform\n$$\n\\overline{p}^{(\\alpha)}(x)=\\frac{1}{2\\pi i}\\int_{1+i{\\mathbb R}}M(s,\\alpha)x^{-s}ds,\n$$\nwhere $M(s,\\alpha)$ is the Mellin transform given by\n$$\nM(s,\\alpha)={\\mathbb E}[(\\overline{Y}^{(\\alpha)}_{1})^{s-1}], \\quad s\\in \\mathbb{C}.\n$$\nWe remark here that we write the Mellin transform as $M(s,\\alpha)$ instead of $M(s)$ to emphasize its dependence on $\\alpha$.\nIt follows from \\cite[Lemma 2]{Ku11} that $M(s,\\alpha)$ can be extended to a meromorphic function on $\\mathbb{C}$ whose simples poles are at $s_{m,n}:=m+\\alpha n$, where $m\\leq 1-(k+1)$ and $n\\in \\{0,1,\\cdots, k\\}$, or $m\\geq 1$ and $n\\in \\{1,2,\\cdots, k\\}$ with residues\n$$\n\\text{Res}(M(s,\\alpha), s_{m,n})=c_{m-1,n}^{+},\n$$\nwhere $c^{+}_{m-1,n}$ is the constant defined in \\cite[(7.5)]{Ku11}.\n\nSince $Y^{(\\alpha)}\\in C_{k,k+1}$, \nit follows from \\eqref{eqn:ckl} that $\\alpha=\\frac{2+2k}{1+2k}\\in (1,2)$.\nIf $m\\geq 1$ and $n\\in \\{1,\\cdots, k\\}$ then $s_{m,n}=m+\\alpha n$ has a nonempty intersection with  $[1,3]$ if and only if $m=n=1$. In particular, $s_{1,1}=1+\\alpha$. \nIf $m\\leq 1-(k+1)=-k$ and $n\\in \\{0,\\cdots, k\\}$, we have \n$$\ns_{m,n}=m+\\alpha n \\leq -k +\\frac{2+2k}{1+2k}k=\\frac{k}{1+2k} <\\frac{1}{2}.\n$$\nHence, the only simple pole of $M(s,\\alpha)$ with $\\text{Re}(s)\\in [1,3]$ is $s_{1,1}=1+\\alpha$.\n\n\nWe note here that although it is written as $y\\to \\infty$ in \\cite[Lemma 3]{Ku11}, it is actually true for all $|y|\\to \\infty$ by checking the proof there as \\cite[(7.15)]{Ku11} is true for all $z\\to\\infty$ with $|\\text{arg}(z)|<\\pi$.\nHence, by taking a rectangle with vertices at $1\\pm Pi$ and $3\\pm Pi$ with the help  of the residue theorem, then letting $P\\to \\infty$ and using \\cite[Lemma 3]{Ku11},  we get\n\\begin{eqnarray*}\n&&\\overline{p}^{(\\alpha)}(x)=\\frac{1}{2\\pi i}\\int_{1+i{\\mathbb R}}M(s,\\alpha)x^{-s}ds \\\\\n&=&\\text{Res}(M(s,\\alpha), s_{1,1}^{+})x^{-1-\\alpha} +\\frac{1}{2\\pi i}\\int_{3+i{\\mathbb R}}M(s,\\alpha)x^{-s}ds\\\\\n&=&c_{0,1}^{+}x^{-1-\\alpha}+\\frac{1}{2\\pi i}\\int_{3+i{\\mathbb R}}M(s,\\alpha)x^{-s}ds,\n\\end{eqnarray*}\nwhere $\\text{Res}(M(s,\\alpha), s_{1,1}^{+})$ is the residue of $M(s,\\alpha)$ at $s_{1,1}^{+}$.\n\nWe claim that the constant $c_{0,1}^{+}$ must be $C\\alpha$, where $C$ is from \\eqref{eqn:same asymptotic} as we will show that the reminder is $O(x^{-3})$, which will in turn imply that there exist constants $c_{1}, c_{2}\\in {\\mathbb R}$ such that \n$$\nc_{0,1}^{+}x^{-1-\\alpha} +c_1x^{-3} \\leq \\overline{p}^{(\\alpha)}(x) \\leq c_{0,1}^{+}x^{-1-\\alpha} +c_2x^{-3}\n$$\nfor all sufficiently large $x$.\nBy integrating on $(u,\\infty)$ we obtain \n$$\n\\frac{c_{0,1}^{+}}{\\alpha}u^{-\\alpha} +\\frac{c_1}{2} u^{-2}\\leq \n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u) \n\\leq \\frac{c_{0,1}^{+}}{\\alpha}u^{-\\alpha} +\\frac{c_2}{2}u^{-2},\n$$\nfor all sufficiently large $u>0$.\nComparing the equation above with \\eqref{eqn:same asymptotic} we conclude that \n$\\alpha^{-1}c_{0,1}^{+}=C$.\nThis shows that the leading term of $\\overline{p}^{(\\alpha)}(x)$ is $C\\alpha x^{-1-\\alpha}$.\n\nHence, the proof will be complete once we show that \n\\begin{equation}\\label{eqn:Mellin error}\n\\left| \\frac{1}{2\\pi i}\\int_{3+i{\\mathbb R}}M(s,\\alpha)x^{-s}ds\\right| \n\\leq Ax^{-3},\n\\end{equation}\nwhere the constant $A$ is independent of all $\\alpha\\in (1,2)$ satisfying \n$Y^{(\\alpha)}\\in C_{k,k+1}$.\n\nFrom \\cite[Lemma 3]{Ku11} we know that \n\\begin{equation}\\label{eqn:Mellin}\n\\ln|M(3+iy,\\alpha)|=-\\frac{\\pi |y|}{2\\alpha} +o(y) \\text{ as } |y|\\to\\infty.\n\\end{equation}\nThe proof of \\eqref{eqn:Mellin} in \\cite[Lemma 3]{Ku11} depends on \\cite[(4.5)]{BK97}, which is a uniform estimate of the double gamma functions. Hence, the error term in \\eqref{eqn:Mellin} is uniform for all $\\alpha\\in (1,2)$ and \nthere exists $N>0$, independent of  $\\alpha\\in (1,2)$, such that \n$$\n\\ln|M(3+iy,\\alpha)|\\leq -\\frac{\\pi |y|}{5} \\text{ for all } |y|\\geq N \\text{ and } \\alpha\\in(1,2).\n$$\nHence, we have \n\\begin{equation}\\label{eqn:Mellin ub}\n|M(3+iy, \\alpha)|\\leq e^{-\\frac{\\pi|y|}{5}} \\text{ for all } |y|\\geq N \\text{ and } \\alpha\\in(1,2).\n\\end{equation}\nIt follows from \\cite[Theorem 8]{Ku11} that the Mellin transform $M(s,\\alpha)$ can be written as\n$$\nM(s,\\alpha)=\\alpha^{s-1}\\frac{G(\\alpha\/2;\\alpha)}{G(\\alpha\/2+1;\\alpha)}\\frac{G(\\alpha\/2+2-s;\\alpha)}{G(\\alpha\/2-1+s;\\alpha)}\\frac{G(\\alpha-1+s;\\alpha)}{G(\\alpha+1-s;\\alpha)},\n$$\nwhere $G(z;\\tau)$ is the double gamma function.\n$G(z;\\tau)$ has simple zeroes on the lattice $m\\tau+n$, $m,n\\leq 0$.\nBy a simple calculation one can check that the double gamma functions in the denominators above\nhave no zero with $s=3+iy$ for $|y|\\leq N$.\nIt follows from Lemma \\ref{lemma:double gamma bounded} that there exists a constant $c>0$ such that\n$$\n|M(3+iy,\\alpha)|\\leq ce^{-\\frac{\\pi|y|}{5}} \\text{ for all } |y|\\leq N \\text{ and } \\alpha \\in (1,2),\n$$\nand combining with \\eqref{eqn:Mellin ub} we have \n$$\n|M(3+iy,\\alpha)|\\leq ce^{-\\frac{\\pi|y|}{5}} \\text{ for all } y\\in {\\mathbb R} \\text{ and } \\alpha \\in (1,2).\n$$\n\nHence, \\eqref{eqn:Mellin error} can be estimated as \n$$\n\\left| \\frac{1}{2\\pi i}\\int_{3+i{\\mathbb R}}M(s,\\alpha)x^{-s}ds\\right| \\leq \\frac{x^{-3}}{2\\pi}\\int_{-\\infty}^{\\infty}|M(3+it, \\alpha)|dt\\leq\\frac{x^{-3}}{2\\pi}\\int_{-\\infty}^{\\infty}ce^{-\\frac{\\pi |t|}{5}}dt:=\nAx^{-3}.\n$$\n{\\hfill $\\Box$ \\bigskip}\n\nLet $p^{(\\alpha)}(x)$ be the transition density of $Y^{(\\alpha)}_1$.\nThe following lemma is a variation of \\cite[Propositions 7.1.1 and 7.1.2]{Ko} with an explicit error estimate. \n\\begin{lemma}\\label{lemma:uniform lb}\nLet $\\alpha\\in(1,2)$ and $n\\in{\\mathbb N}$. Then, we have \n$$\np^{(\\alpha)}(x)=\n\\frac{1}{\\pi}\\sum_{k=1}^{n}\\frac{(-1)^{k+1}\\Gamma(1+k\\alpha)\\sin(\\frac{k\\alpha\\pi}{2}) }{k!}x^{-1-\\alpha k}+E(x,\\alpha),\n$$\nwhere \n$$\n\\left|E(x,\\alpha)\\right|\\leq \\frac{2\\Gamma(2n+2)}{\\pi n!}x^{-2-n} \\text{ for all } x\\geq 1 \\text{ and } \\alpha \\in(1,2).\n$$\nIn particular, for any $x\\geq 1$ and $\\alpha\\in (1,2)$ we have\n$$\n \\frac{\\Gamma(1+\\alpha)\\sin\\frac{\\pi\\alpha}{2}}{\\pi}x^{-1-\\alpha}-\\frac{12}{\\pi}x^{-3} \\leq p^{(\\alpha)}(x)\\leq \\frac{\\Gamma(1+\\alpha)\\sin\\frac{\\pi\\alpha}{2}}{\\pi}x^{-1-\\alpha}+\\frac{12}{\\pi}x^{-3},\n$$\nand in fact \n$$\n \\frac{\\Gamma(1+\\alpha)\\sin\\frac{\\pi\\alpha}{2}}{\\pi}=C\\alpha,\n$$\nwhere $C$ is the constant in \\eqref{eqn:same asymptotic}.\n\\end{lemma}\n\\noindent{\\bf Proof.} \nTo prove this lemma, we consider a 1-dimensional $\\alpha$-stable processes $Y^{(\\alpha), \\gamma}_{t}$ with\n$$\n{\\mathbb E}[e^{iy Y_{t}^{(\\alpha), \\gamma}}]=\\exp(-t|y|^{\\alpha}e^{\\frac{i\\pi\\gamma}{2}\\text{sgn}(y)}),\n$$\nwhere $\\gamma$ represents the skewness of $Y^{(\\alpha), \\gamma}_{t}$  and $\\text{sgn}(y):=1_{\\{y\\geq 0\\}}- 1_{\\{y<0\\}}$. \nWe use \n$p^{(\\alpha)}(x, \\gamma)$\nto denote the density of $Y^{(\\alpha), \\gamma}_{1}$. When $\\gamma=0$, $Y^{(\\alpha), \\gamma}_{t}$ reduces to $Y^{(\\alpha)}_{t}$.\nBy the Fourier inversion we have \n\\begin{eqnarray*}\n&&p^{(\\alpha)}(x,\\gamma)\n=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{-ixy}\\exp(-|y|^{\\alpha}e^{\\frac{i\\pi\\gamma}{2}\\text{sgn}(y)})dy\\\\\n&=&\\frac{1}{2\\pi}\\int_{0}^{\\infty}e^{-ixy}\\exp(-|y|^{\\alpha}e^{\\frac{i\\pi\\gamma}{2}})dy+\\frac{1}{2\\pi}\\int_{-\\infty}^{0}e^{-ixy}\\exp(-|y|^{\\alpha}e^{-\\frac{i\\pi\\gamma}{2}})dy\\\\\n&=&\\frac{1}{2\\pi}\\int_{0}^{\\infty}e^{-ixy}\\exp(-|y|^{\\alpha}e^{\\frac{i\\pi\\gamma}{2}})dy+\\frac{1}{2\\pi}\\int^{\\infty}_{0}e^{ixy}\\exp(-|y|^{\\alpha}e^{-\\frac{i\\pi\\gamma}{2}})dy\\\\\n&=&\\frac{1}{\\pi}\\text{Re}\\int_{0}^{\\infty}e^{-ixy}\\exp(-y^{\\alpha}e^{\\frac{i\\pi\\gamma}{2}})dy,\n\\end{eqnarray*} \nwhere $\\text{Re}(\\cdot)$ represents the real part of the argument. \nBy the Taylor expansion with remainder we can write $e^{-ixy}$ as\n$$\ne^{-ixy}=\\sum_{k=0}^{n-1}\\frac{(-ix)^{k}}{k!}y^{k} +\\frac{x^{n}y^{n}}{n!}R,\n$$\nwhere $|R|\\leq 1$.\nSince, \n$$\n\\int_{0}^{\\infty}y^{\\beta}\\exp(-\\lambda y^{\\alpha})dy=\\alpha^{-1}\\lambda^{-\\frac{\\beta+1}{\\alpha}}\\Gamma(\\frac{\\beta+1}{\\alpha}),\n$$\nwe have \n\\begin{align}\\label{eqn:HK}\n&p^{(\\alpha)}(x,\\gamma)\n=\\frac{1}{\\pi}\\text{Re}\\int_{0}^{\\infty}\\left( \\sum_{k=0}^{n-1}\\frac{(-ix)^{k}}{k!}y^{k} +\\frac{x^{n}y^{n}}{n!}R\\right)\\exp(-y^{\\alpha}e^{\\frac{i\\pi\\gamma}{2}})dy\\nonumber\\\\\n&=\\frac{1}{\\pi}\\text{Re}\\sum_{k=0}^{n-1}\\frac{(-ix)^{k}}{k!}\\frac{1}{\\alpha}(e^{\\frac{i\\gamma\\pi}{2}})^{-\\frac{k+1}{\\alpha}}\\Gamma(\\frac{k+1}{\\alpha})+\n\\text{Re}\\left(\\frac{x^{n}}{\\pi\\alpha n!}\\Gamma(\\frac{n+1}{\\alpha})(e^{\\frac{i\\gamma\\pi}{2}})^{-\\frac{n+1}{\\alpha}}R\\right)\\nonumber\\\\\n&=\\frac{1}{\\alpha\\pi}\\text{Re}\\sum_{k=1}^{n}\\frac{(-ix)^{k-1}}{(k-1)!}e^{-\\frac{i\\gamma\\pi k}{2\\alpha}}\\Gamma(\\frac{k}{\\alpha})\n+\\text{Re}\\left(\\frac{x^n}{\\pi\\alpha n!}\\Gamma(\\frac{n+1}{\\alpha})e^{-\\frac{i\\gamma\\pi(n+1)}{2\\alpha}}R\\right)\\nonumber\\\\\n&=\\frac{-1}{\\alpha\\pi x}\\text{Re}\\sum_{k=1}^{n}\\frac{(-x)^{k}}{(k-1)!}\\exp\\left(-i(\\frac{k\\pi(\\gamma-\\alpha)}{2\\alpha} +\\frac{\\pi}{2})\\right)\\Gamma(\\frac{k}{\\alpha})+\\text{Re}\\left(\\frac{x^n}{\\pi\\alpha n!}\\Gamma(\\frac{n+1}{\\alpha})e^{-\\frac{i\\gamma\\pi(n+1)}{2\\alpha}}R\\right)\\nonumber\\\\\n&:=\\frac{1}{\\pi x}\\sum_{k=1}^{n}\\frac{(-x)^{k}}{k!}\\sin(\\frac{k\\pi(\\gamma-\\alpha)}{2\\alpha})\\Gamma(1+\\frac{k}{\\alpha})+R(x,\\alpha),\n\\end{align}\nwhere \n\\begin{equation}\\label{eqn:remainder}\n\\left| R(x,\\alpha)\\right|\\leq \\frac{x^n}{\\pi\\alpha n!}\\Gamma(\\frac{n+1}{\\alpha}).\n\\end{equation}\n\nIt follows from Zolotarev's identity (see \\cite[(7.10)]{Ko}) for $\\alpha\\in(\\frac12,1)\\cup (1,2)$,\n$$\np^{(\\alpha)}(x,\\gamma)=x^{-1-\\alpha}p^{(1\/\\alpha)}(x^{-\\alpha}, \\frac{\\gamma+1}{\\alpha}-1).\n$$\nHence, from \\eqref{eqn:HK} we have \n\\begin{eqnarray*}\n&&p^{(\\alpha)}(x,\\gamma)=x^{-1-\\alpha}p^{(1\/\\alpha)}(x^{-\\alpha}, \\frac{\\gamma+1}{\\alpha}-1)\\\\\n&=&x^{-1-\\alpha}\\left(\\frac{1}{\\pi x^{-\\alpha}}\\sum_{k=1}^{n}\\frac{(-x^{-\\alpha})^{k}}{k!}\\sin(\\frac{k\\pi(\\gamma-\\alpha)}{2})\\Gamma(1+k\\alpha) +R(x^{-\\alpha}, \\frac{1}{\\alpha})\\right).\n\\end{eqnarray*}\nNow let $\\gamma=0$ and define $E(x,\\alpha):=x^{-1-\\alpha}R(x^{-\\alpha}, \\frac{1}{\\alpha})$. From \\eqref{eqn:remainder} we have \n$$\n\\left|E(x,\\alpha)\\right|\\leq \\frac{\\alpha x^{-(n+1)\\alpha-1}}{\\pi n!}\\Gamma(\\alpha(n+1)).\n$$\nHence, for all $\\alpha\\in (1,2)$ and $x\\geq 1$ we have \n$$\n\\left|E(x,\\alpha)\\right|\\leq \\frac{2 x^{-n-2}}{\\pi n!}\\Gamma(2(n+1)).\n$$\n\nThe fact $\\frac{\\Gamma(1+\\alpha)\\sin\\frac{\\pi\\alpha}{2}}{\\pi}=C\\alpha$ can be proved using  a similar argument as in the proof of Lemma \\ref{lemma:uniform ub}.\n{\\hfill $\\Box$ \\bigskip}\n\n\n\\begin{lemma}\\label{lemma:Cauchy cancel}\nLet $\\alpha\\in (1,2)$ and suppose that \n$Y^{(\\alpha)}\\in C_{k,k+1}$ for some $k\\in \\mathbb{N}$. \nThere exists a function $\\phi$, independent of $\\alpha\\in (1,2)$, such that \n$$\n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_1>u, Y^{(\\alpha)}_1\\leq u)\\leq \\phi(u) \\text{ with } \\int_{0}^{\\infty}\\phi(u)du<\\infty.\n$$\n\\end{lemma}\n\\noindent{\\bf Proof.} \nNote that \n$$\n{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}\\leq u)={\\mathbb P}(\\overline{Y}^{(\\alpha)}_1>u)-{\\mathbb P}(Y_{1}^{(\\alpha)}>u).\n$$\nIt follows from Lemmas \\ref{lemma:uniform ub} and \\ref{lemma:uniform lb} that\n$$\n\\overline{p}^{(\\alpha)}(x)\\leq C\\alpha x^{-1-\\alpha} +\nAx^{-3} \\text{ and } \np^{(\\alpha)}(x)\\geq C\\alpha x^{-1-\\alpha} -\\frac{12}{\\pi}x^{-3},\n$$\nwhere $A$ is independent of $\\alpha\\in (1,2)$.\n\nHence, we have \n\\begin{eqnarray*}\n&&{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}\\leq u)={\\mathbb P}(\\overline{Y}^{(\\alpha)}_1>u)-{\\mathbb P}(Y_{1}^{(\\alpha)}>u)\\\\\n&\\leq&\\int_{u}^{\\infty}\\left(C\\alpha x^{-1-\\alpha} +\nAx^{-3}-C\\alpha x^{-1-\\alpha} +\\frac{12}{\\pi}x^{-3} \\right)dx=\\frac{A}{2}u^{-2}+\\frac{6}{\\pi}u^{-2}.\n\\end{eqnarray*}\nFinally, we set \n$$\n\\phi(u):=1_{\\{0<u\\leq 1\\}} + 1_{\\{u>1\\}}\n(\\frac{A}{2}+\\frac{6}{\\pi})u^{-2}.\n$$\n{\\hfill $\\Box$ \\bigskip}\n\n\\begin{prop}\\label{prop:Cauchy ub}\nLet $D$ be a bounded $C^{1,1}$ open set in ${\\mathbb R}^{d}$, $d\\geq 2$. Then, we have\n$$\n\\limsup_{t\\to 0}\\frac{|D|-Q_{D}^{(1)}(t)}{t\\ln(1\/t)}\\leq \\frac{|\\partial D|}{\\pi}.\n$$\n\\end{prop}\n\\noindent{\\bf Proof.} \nFrom Lemma \\ref{lemma:Cauchy ub} for any $\\alpha\\in(1,2)$ we have \n\\begin{equation}\\label{eqn:Cauchy ub main}\n\\limsup_{t\\to 0}\\frac{|D|-Q_{D}^{(1)}(t)}{t\\ln(1\/t)}\\leq \\frac{|\\partial D|}{\\pi} +\n\\frac{\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}<u )du }{\\Gamma(1-\\frac{1}{\\alpha})}\\cdot |\\partial D|.\n\\end{equation}\n\nNote that \n$$\n\\Gamma(x)=\\int_{0}^{\\infty}e^{-u}u^{x-1}du\\geq \\int_{0}^{1}e^{-u}u^{x-1}du\\geq e^{-1}\\int_{0}^{1}u^{x-1}du=\\frac{1}{ex},\n$$\nand this implies \n\\begin{equation}\\label{eqn:Gamma}\n\\frac{1}{\\Gamma(1-1\/\\alpha)}\\leq e(1-\\frac{1}{\\alpha})=\\frac{e(\\alpha-1)}{\\alpha} \\text{ for all } \\alpha>1.\n\\end{equation}\nNote that from \\eqref{eqn:ckl} for \n$Y^{(\\alpha)}\\in C_{k,k+1}$\nwe have $\\alpha \\in (1,2)$ and as $k\\to \\infty$, $\\alpha\\downarrow 1$.\nHence, it follows from Lemma \\ref{lemma:Cauchy cancel} and the Lebesgue dominated convergence theorem that\n$$\n\\lim_{Y_{t}^{(\\alpha)}\\in C_{k,k+1}, k\\to \\infty}\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}\\leq u)du\n=\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(1)}_{1}>u, Y_{1}^{(1)}\\leq u)du.\n$$\nNote that the density $p^{(1)}(x)$ of $Y^{(1)}_{1}$ is given by \n$$\np^{(1)}(x)=\\frac{1}{\\pi(1+x^2)}\n$$\nand $\\int_{u}^{\\infty}{\\mathbb P}(Y_{1}^{(1)}>u)du=\\int_{u}^{\\infty}\\frac{1}{\\pi (1+x^2)}dx=\\frac{1}{\\pi}(\\frac{\\pi}{2}-\\arctan(u))=\\frac{1}{\\pi}\\arctan(\\frac{1}{u})$.\nBy the Taylor expansion of $\\arctan x =\\sum_{n=1}^{\\infty}(-1)^{n+1}\\frac{x^{2n-1}}{2n-1}$ for $|x|<1$ we have $\\frac{1}{\\pi}\\arctan(1\/u)\\geq \\frac{1}{\\pi u} -\\frac{1}{2\\pi u^3}$ for all sufficiently large $u$.\nHence, it follows from \\cite[(3.5)]{P20} that \n\\begin{eqnarray*}\n&&{\\mathbb P}(\\overline{Y}^{(1)}_{1}>u, Y_{1}^{(1)}\\leq u ) ={\\mathbb P}(\\overline{Y}^{(1)}_{1}>u)-{\\mathbb P}(Y_{1}^{(1)}> u)\\\\\n&=&{\\mathbb P}(\\overline{Y}^{(1)}_{1}>u)-\\frac{\\arctan(1\/u)}{\\pi}\\leq{\\mathbb P}(\\overline{Y}^{(1)}_{1}>u)-\\frac{1}{\\pi u} +\\frac{1}{2\\pi u^3}\\leq \\frac{4}{\\pi^2}\\frac{\\ln u}{u^3} +\\frac{1}{2\\pi u^3}\n\\end{eqnarray*}\nfor all sufficiently large $u>0$, and this implies \n\\begin{equation}\\label{eqn:Cauchy finite}\n\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(1)}_{1}>u, Y_{1}^{(1)}\\leq u)du<\\infty.\n\\end{equation}\n\nHence, if follows from \\eqref{eqn:Gamma} and \\eqref{eqn:Cauchy finite} that\n$$\n\\lim_{Y^{(\\alpha)}\\in C_{k,k+1}, k\\to \\infty}\n\\frac{\\int_{0}^{\\infty}{\\mathbb P}(\\overline{Y}^{(\\alpha)}_{1}>u, Y_{1}^{(\\alpha)}<u )du }{\\Gamma(1-\\frac{1}{\\alpha})}=0.\n$$\nHence, from \\eqref{eqn:Cauchy ub main} we reach the conclusion of the theorem.\n{\\hfill $\\Box$ \\bigskip}\n\n\\begin{thm}\\label{thm:main1}\nLet $D$ be a bounded $C^{1,1}$ open set in ${\\mathbb R}^{d}$, $d\\geq 2$. Then, we have\n$$\n\\lim_{t\\to 0}\\frac{|D|-Q_{D}^{(1)}(t)}{t\\ln(1\/t)}= \\frac{|\\partial D|}{\\pi}.\n$$\n\\end{thm}\n\\noindent{\\bf Proof.} \nIt follows immediately from \\eqref{eqn:Cauchy lb} and Proposition \\ref{prop:Cauchy ub}.\n{\\hfill $\\Box$ \\bigskip}\n\n\\noindent\n\\textbf{Acknowledgments:} \nWe thank the referee for helpful comments and suggestions.\nThe first-named author is grateful to Professor Kyeong Hun Kim (Korea University, South Korea) for his encouragement while this project was in progress. \nPart of the research for this project was done while the second-named author was visiting Jiangsu Normal University, where he was partially supported from the National Natural Science Foundation of China (11931004) and by the Priority Academic Program Development of Jiangsu Higher Education Institutions.\n\\bigskip  \n\n\n\n\n\\begin{singlespace}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper we revisit the findings reported in \\cite{isochronous\ncosmologies} and report some additional considerations relevant for a better\nunderstanding of the validity---in the context of theoretical and\nmathematical physics---of those results.\n\nIt has been recently shown \\cite{CL2007,calogero} that---given a general\nautonomous dynamical system $D$, other autonomous dynamical systems $\\tilde{\n}$ can be manufactured, featuring two additional \\textit{arbitrary} positive\nparameters $T$ and $\\tilde{T}$ with $T>\\tilde{T}$ and possibly also two\nadditional dynamical variables---which are characterized by the following\ntwo properties:\n\n(i) For the same variables of the original dynamical system $D$ the new\ndynamical system $\\tilde{D}$ yields, over the time interval $\\tilde{T}$,\nhence over an \\textit{arbitrarily long} time, a dynamical evolution which\nmimics \\textit{arbitrarily closely} that yielded by the original system $D$;\nup to corrections of order $t\/\\tilde{T}$, or possibly even \\textit\nidentically}.\n\n(ii) The system $\\tilde{D}$ is \\textit{isochronous}: for arbitrary initial\ndata all its solutions are completely periodic with a period $T$, which can\nalso be arbitrarily assigned, except for the\\ (obviously necessary)\ncondition $T>\\tilde{T}$.\n\nMoreover it has been shown \\cite{CL2007,calogero} that, if the dynamical\nsystem $D$ is a many-body problem characterized by a (standard, autonomous)\nHamiltonian $H$ which is translation-invariant (i. e., it features no\nexternal forces), other (also autonomous) Hamiltonians $\\tilde{H}$\ncharacterizing \\textit{modified} many-body problems can be manufactured\nwhich feature the \\textit{same} dynamical variables as $H$ (i. e., in this\ncase there is no need to introduce two additional dynamical variables) and\nwhich yield a time evolution \\textit{quite close}, or even \\textit{identical\n, to that yielded by the original Hamiltonian $H$ over the \\textit\narbitrarily assigned} time $\\tilde{T}$, while being \\textit{isochronous}\nwith the \\textit{arbitrarily assigned} period $T$, of course with $T>\\tilde{\n}$.\n\nLet us emphasize that the class of Hamiltonians $H$ for which this result is\nvalid is quite general. In particular it includes the standard Hamiltonian\nsystem describing an \\textit{arbitrary} number $N$ of point particles with\n\\textit{arbitrary} masses moving in a space of \\textit{arbitrary} dimensions\n$d$ and interacting among themselves via potentials depending \\textit\narbitrarily} from the interparticle distances (including the possibility of\nmultiparticle forces), being therefore generally valid for any realistic\nmany-body problem, hence encompassing most of nonrelativistic physics. This\nresult is moreover true, \\textit{mutatis mutandis}, in a quantal context.\n\nFor instance let us tersely review here the case of the standard Hamiltonian\ndescribing the many-body problem, but focusing---merely for notational\nsimplicity---on the case with equal particles (and setting their mass to\nunity) and a one-dimensional setting, which reads (in self-evident standard\nnotation)\n\\begin{subequations}\n\\begin{equation}\nH\\left( \\underline{p},\\underline{q}\\right) =\\frac{1}{2}\\sum_{n=1}^{N}\\left(\np_{n}^{2}\\right) +V\\left( \\underline{q}\\right)  \\label{H}\n\\end{equation\nwith the potential being translation-invariant, $V\\left( \\underline{q\n+a\\right) =V\\left( \\underline{q}\\right) $ (i. e., no external forces) but\notherwise unrestricted (except for the condition---again, for\nsimplicity---that the time-evolution yielded by this Hamiltonian be\nnonsingular). The standard approach to treat this problem is to introduce\nthe center-of-mass coordinate $Q\\left( \\underline{q}\\right) =\\frac{1}{N\n\\sum_{n=1}^{N}q_{n}$ and the total momentum $P\\left( \\underline{p}\\right)\n=\\sum_{n=1}^{N}p_{n},$ and to then focus on the coordinate and momenta in\nthe center of mass, $x_{n}=q_{n}-Q\\left( \\underline{q}\\right) ,$ \ny_{n}=p_{n}-P\\left( \\underline{p}\\right) \/N$, which characterize the physics\nof the problem and whose evolution is determined by the Hamiltonian $h\\left(\n\\underline{y},\\underline{x}\\right) $ defined as follows\n\\begin{equation}\nH\\left( \\underline{p},\\underline{q}\\right) =\\frac{\\left[ P\\left( \\underline{\n}\\right) \\right] ^{2}}{2N}+h\\left( \\underline{y},\\underline{x}\\right) ~,\n\\label{Hh}\n\\end{equation\n\\begin{equation}\nh\\left( \\underline{y},\\underline{x}\\right) =\\frac{1}{2}\\sum_{n=1}^{N}\\left(\ny_{n}^{2}\\right) +V\\left( \\underline{x}\\right) ~,  \\label{h}\n\\end{equation\nwhere $V\\left( \\underline{x}\\right) =V\\left( \\underline{q}\\right) $ thanks\nto the assumed translation-invariance of $V\\left( \\underline{q}\\right) $. An\n\\textit{isochronous} Hamiltonian $\\tilde{H}\\left( \\underline{p},\\underline{q\n;T\\right) $ reads then as follows,\n\\end{subequations}\n\\begin{subequations}\n\\begin{equation}\n\\tilde{H}\\left( \\underline{p},\\underline{q};T\\right) =\\frac{1}{2\n\\sum_{n=1}^{N}\\left\\{ \\left[ P\\left( \\underline{p}\\right) +h\\left(\n\\underline{y},\\underline{x}\\right) \\right] ^{2}+\\left( \\frac{2\\pi }{T\n\\right) \\left[ Q\\left( \\underline{q}\\right) \\right] ^{2}\\right\\} ~.\n\\label{Htilde}\n\\end{equation\nIt can indeed be shown \\cite{CL2007,calogero} that it entails an \\textit\nisochronous} evolution (with period $T$) of the center of mass coordinate $Q\n, of the total momentum $P$, and---most importantly---of all the particle\ncoordinates, whose evolution then read\n\\begin{equation}\n\\underline{\\tilde{x}}\\left( t\\right) =\\underline{x}\\left( \\tau \\left(\nt\\right) \\right) ~,~~~\\underline{\\tilde{y}}\\left( t\\right) =\\underline{y\n\\left( \\tau \\left( t\\right) \\right) ~,  \\label{xytildet}\n\\end{equation\nwhere we indicate as $\\underline{\\tilde{x}}\\left( t\\right) ,$ $\\underline\n\\tilde{y}}\\left( t\\right) $ the time evolution yielded by the tilded\nHamiltonian $\\tilde{H}\\left( \\underline{p},\\underline{q};T\\right) $ and as \n\\underline{x}\\left( t\\right) ,$ $\\underline{y}\\left( t\\right) $ the time\nevolution yielded by the original Hamiltonian $H\\left( \\underline{p}\n\\underline{q}\\right) ,$ an\n\\begin{equation}\n\\tau \\left( t\\right) =A~\\sin \\left( \\frac{2\\pi t}{T}+\\Phi \\right)\n\\label{taut}\n\\end{equation\nwhere $A$ and $\\Phi $ are \\textit{constant} parameters given by simple\nexpressions in terms of the initial values, $Q\\left( 0\\right) $ and $P\\left(\n0\\right) $, of the position of the center of mass of the system and of its\ntotal momentum, and of the Hamiltonian $\\tilde{H}\\left( \\underline{p}\n\\underline{q};T\\right) $ (which is of course a constant of motion for this\ntime evolution). Note that, for $\\left\\vert t\\right\\vert <<T,\n\\begin{equation}\n\\tau \\left( t\\right) =\\alpha +\\beta t+O\\left[ \\left( \\frac{t}{T}\\right) ^{2\n\\right] ~~~\\text{with \\hspace{0in} }\\alpha =A~\\sin \\left( \\Phi \\right)\n~,~~~\\beta =\\frac{2\\pi A}{T}\\cos \\left( \\Phi \\right) ~;\n\\end{equation\nhence up to small corrections (\\textit{arbitrarily small} for an \\textit\narbitrarily large} assignment of the period $T$ as long as $\\left\\vert\nt\\right\\vert $ is in an assigned interval $\\tilde{T}$, of course such that \n\\tilde{T}<<T$) the time evolution of the tilded coordinates approximates\nthat of the untilded coordinates, up to a constant shift and rescaling of\ntime.\n\nNote that in this case---again, for the sake of simplicity---the modified,\n\\textit{isochronous} Hamiltonian $\\tilde{H}\\left( \\underline{p},\\underline{q\n;T\\right) $ features only one additional constant parameter, $T$; but it has\nbeen shown \\cite{CL2007,calogero} that an analogous treatment---involving a\nsomewhat more complicated definition of a modified, \\textit{isochronous}\nHamiltonian, featuring then two arbitrarily assigned parameters, $T$ and \n\\tilde{T}<T$---allows to manufacture a modified Hamiltonian that reproduces\n\\textit{exactly} (up to a constant rescaling and shift of the time variable)\nthe same time evolution yielded by the original Hamiltonian over the\ninterval $\\tilde{T}$ but is \\textit{isochronous} with period $T$---and let\nus again emphasize that this can be done for a many-body problem involving\nan \\textit{arbitrary} number of (possibly different) particles, moving in a\nspace of \\textit{arbitrary} dimension $d$ and interacting via \\textit\narbitrary} interparticle forces. Which therefore includes most of\n(nonrelativistic) physics.\n\nSince it is impossible to distinguish experimentally dynamical systems that\nbehave \\textit{arbitrarily closely}, or even \\textit{identically}, over an\n\\textit{arbitrarily long} period of time, this finding has various\nremarkable implications in the context of theoretical and mathematical\nphysics. In particular it raises \\cite{CL2007,calogero} interesting\nquestions about the distinction between integrable and nonintegrable\nevolutions, about the definition of chaotic behavior, and---for macroscopic\nsystems featuring, say, a number of particles of the order of Avogadro's\nnumber---about statistical mechanics and the second principle of\nthermodynamics. And for, say, $10^{85}$ particles it seems to have some\ncosmological relevance.\n\nBut the proper setting of cosmological theories is general relativity. For\nthis reason in our previous paper \\cite{isochronous cosmologies} we\ninvestigated whether and how this result could be extended to that context.\nWe concluded that this is possible provided one extends general relativity\nby allowing \\textit{degenerate} (i.e. non invertible) metrics. In particular\nwe have shown \\cite{isochronous cosmologies} that for any homogeneous,\nisotropic and spatially flat metric $g_{\\mu \\nu }$ satisfying Einstein's\nequations and providing a model of the universe, it is possible to find a\ndifferent (also homogeneous, isotropic and spatially flat) metric solution \n\\tilde{g}_{\\mu \\nu }$ which is \\textit{locally} (in time) diffeomorphic to \ng_{\\mu \\nu }$---hence yields the same cosmology as $g_{\\mu \\nu }$ for any\nobservation over an \\textit{arbitrary} time interval $\\tilde{T}$---and is\ncyclic\\footnote\nWe prefer to talk of cyclic instead of periodic solutions due to the fact\nthat periodicity is not an invariant concept, since it is not invariant\nunder a redefinition of time. For a review of cyclic models see \\cite{cyclic\n.} (in fact periodic with an \\textit{arbitrary} period $T>\\tilde{T}$ in the\ntime coordinate $t\\,$); but it is \\textit{degenerate} at an infinite,\ndiscrete sequence of times $t_{n}=t_{0}\\pm nT\/2,$ $n=0,1,2,...$ . We\ninterpreted these metrics as corresponding to \\textit{isochronous\ncosmologies }\\cite{isochronous cosmologies}.\n\nLet us also recall \\cite{isochronous cosmologies} that this result is not\nrestricted to homogeneous, isotropic and spatially flat metrics: it can be\neasily extended to any synchronous metric, therefore it is quite general\nsince most metrics can be written in synchronous form by a diffeomorphic\nchange of coordinates.\n\nDue to the diffeomorphic correspondence between $g_{\\mu \\nu }$ and $\\tilde{g\n_{\\mu \\nu }$ \\textit{locally} in time---for time intervals of order $\\tilde{\n}$---these two metrics give the same physics locally in time, and therefore\nthere is no way to distinguish them using observations local in time; which\nis essentially the same finding valid in the context of the Hamiltonian\nsystems considered in \\cite{CL2007,calogero} and tersely recalled above. In\nparticular, the metric $\\tilde{g}_{\\mu \\nu }$---while being cyclic on time\nscales larger than $\\tilde{T}$---may yield over the time interval $\\tilde{T}$\njust the standard sequence of domination firstly by radiation, then by\nmatter and dark energy, as well as the cosmological perturbations consistent\nwith all observational tests \\cit\n{cmb,leansing,bao,lss,supernovae,ref0,ref1,ref2,ref3,ref4,bicep2}, which\ncharacterize the metric $g_{\\mu \\nu }$ of the standard $\\Lambda $-CDM\ncosmological model (see for instance \\cite{mukhanov} for a review).\nFurthermore, it has been shown that such isochronous metrics $\\tilde{g}_{\\mu\n\\nu }$ can be manufactured to be geodesically complete \\cite{isochronous\ncosmologies} and therefore singularity-free\\footnote\nA spacetime is singularity-free if it is geodesically complete, i.e. if its\ngeodesics can be always past- and future-extended \\cite{poisson,MTW}.}, so\nthat the geodesic motion as well as all physical quantities described by\nscalar invariants are always well defined \\cite{isochronous cosmologies} and\nthe Big Bang singularity may be avoided---even when some of the\nphenomenological observations can nevertheless be interpreted as remnants of\na past Big Bang, which however may never be actually attained by the metric \n\\tilde{g}_{\\mu \\nu }$, neither in the past nor in the future. This implies\nthat these isochronous metrics $\\tilde{g}_{\\mu \\nu }$ may describe a\nsingularity-free universe which---while featuring a time evolution which\nreproduces identically (up to diffeomorphic time reparameterization; \\textit\nlocally} in time, except at a discrete set of instants $t_{n}$) the standard\ncosmological model characterized by the metric $g_{\\mu \\nu }$---features an\nexpansion which stops at some instant, to be followed by a period of\ncontraction, until this phase of evolution stops and is again followed by an\nexpanding phase, this pattern being repeated \\textit{ad infinitum}. Let us\nre-emphasize that the universe characterized by the metric $\\tilde{g}_{\\mu\n\\nu }$ may thereby avoid to experience the Big Bang singularity, even if the\nuniverse which it mimics locally in time, characterized by the metric \ng_{\\mu \\nu },$ does encounter that singularity. Let us also note that this\nclass of metrics realize \\textit{de facto} the reversal of time's arrow, as\ndiscussed for instance in \\cite{sakharov}.\n\nBut in order to do so, these isochronous metrics must be \\textit{degenerate}\nat a discrete set of instants $t_{n}$ when the time reversals occur, say\nfrom expansion to contraction and viceversa; a phenomenology whose inclusion\nin general relativity might be considered problematic, because at these\ninstants $t_{n}$---when the metric is \\textit{degenerate}---the \"equivalence\nprinciple\" corresponding to the requirement that the metric tensor have a\nMinkowskian signature is violated. On the other hand a \\textit{physically\nunobservable} violation of a \"principle\" can be hardly considered \\textit\nphysically relevant}.\n\nPurpose and scope of this paper is to better clarify the meaning and the\ninterpretation of the \\textit{isochronous spacetimes} introduced in \\cit\n{isochronous cosmologies}, and to discuss some additional technical aspects.\nAs stated above, these isochronous metrics have the property to be \\textit\ndegenerate} at an infinite set of times $t_{n}$, i.e. over an infinite\nnumber of $3$-dimensional hypersurfaces, where generally the scale factor\nreaches its maximum and minimum values and the spacetime changes from an\nexpanding to a contracting phase and vice versa \\cite{isochronous\ncosmologies}.\n\nThe inclusion of degenerate metrics in general relativity is not a trivial\nmatter; indeed, it is forbidden if it were required---as an absolute (as it\nwere, \"metaphysical\") rule---that general relativity be consistent with the\n\"equivalence principle\", i. e. that spacetime \\textit{always} be a\nPseudo-Riemannian manifold \\textit{with Minkowskian signature}. This would\nindeed imply that the solutions of Einstein's general relativity, to be\nacceptable, must be \\textit{nondegenerate} metric tensors, since wherever in\nspacetime a metric is \\textit{degenerate} it is not possible to define its\ninverse, hence the Christoffel symbols as well as the Riemann, Ricci and\nEinstein tensors are not defined there. Hence---so the argument of some\ncritics goes---our isochronous metrics, which are not Minkowskian (locally,\non the hypersurfaces $t=t_{n}$) should be discarded in general relativity\nbecause they violate the equivalence principle at those degeneracy surfaces.\nOur rejoinder is that the metrics we introduced \\cite{isochronous\ncosmologies} are solutions of a version of general relativity whose\ndynamics, while still governed by Einstein's equations\\footnote\nEinstein's equations by itself do not determine the signature of spacetime,\nsee the discussion in \\cite{teitelboim,ellis1}.}, does allow the violation\nof the equivalence principle at a discrete set of hypersurfaces. It can be\nargued that such theories are therefore \\textit{different} from standard\ngeneral relativity, if validity of the equivalence principle---enforced as\nan absolute, universal rule---is considered an essential element of that\ntheory; although the difference is in fact \\textit{physically unobservable}.\nThis situation is indeed quite analogous with the findings described above,\nvalid in the nonrelativistic context of Hamiltonian many-body problems\n(classical or quantal), where the solutions of a physical theory described\nby a Hamiltonian $H$ are arbitrarily well approximated or even identically\nreproduced over an arbitrary time interval by periodic solutions of a\n\\textit{different} Hamiltonian $\\tilde{H}$, i.e. by a \\textit{different}\nphysical theory. The main difference is that in the general relativistic\ncontext in\\textbf{\\ }the two \\textit{different} physical theories the\ndynamic is given by the\\textit{\\ same} Einstein's equations, only the class\nof acceptable solutions is \\textit{different}.\n\nThe consideration in the context of general relativity of degenerate metrics\nis in any case not a novelty. For instance they were already introduced \\cit\n{ellis1,ellis2}, and subsequently investigated in a series of papers \\cit\n{elliscarfora,elliscomment,ellis3,ellis4,ellis5,ellis6,ellis8,ellis9,ellis10,ellis11,ellis12,ellis13,ellis14,ellis15,ellis16,ellis17,ellis18,ellis19}\n, in order to investigate signature-changing spacetimes. This class of\nsignature-changing metrics corresponds to a classical realization of the\nchange of signature in quantum cosmology conjectured by Hartle and Hawking\n\\cite{hartlehawking3} (see also \\cit\n{hartlehawking1,hartlehawking2,hartlehawking4}), which has philosophical\nimplications on the origin of the universe \\cite{hawkinghistory}.\n\nBelow we will also consider a different realization of isochronous\ncosmologies via non-degenerate metrics featuring a jump in their first\nderivatives at the inversion times $t_{n}$, which then implies a\ndistributional contribution in the stress-energy tensor at $t_{n}$. These\nmetrics are diffeomorphic to the continuous and degenerate isochronous\nmetrics everywhere except on the inversion hypersurfaces $t=t_{n}$, hence\nthey describe the same physics except at the infinite set of discrete times \nt_{n};$ therefore they may also be manufactured so as to agree with all\ncosmological observations (\\textit{locally} in time). Since these two\nrealizations of isochronous cosmologies are \\textit{locally} but \\textit{not\nglobally} diffeomorphic, they correspond to different spacetimes and may be\nconsidered to emerge from \\textit{different} theories: the nondegenerate\nones are generalized (in the sense of distributions) solutions of\nEinsteinian general relativity (including universal validity of the\nequivalence principle, but allowing jumps---whose physical significance and\njustification is moot---at the discrete times $t_{n}$); the degenerate ones\nfeature no mathematical or physical pathologies but fail to satisfy the\n\"equivalence principle\" at the discrete times $t_{n}$. They are essentially\nindistinguishable---among themselves and from non-isochronous\ncosmologies---by feasible experiments (unless one considers feasible an\nexperiment lasting an arbitrarily long time...).\n\n\\section{Isochronous Cosmologies}\n\nThe procedure used to construct, for an autonomous dynamical system $D$,\nanother autonomous dynamical system $\\tilde{D}$ whose solutions \\textit\napproximate} arbitrarily closely, or even reproduce exactly, those of the\nsystem $D$ over an \\textit{arbitrary} time interval $\\tilde{T}$ but are\n\\textit{isochronous} with an \\textit{arbitrary} period $T>\\tilde{T}$, is\nbased on the introduction of an auxiliary variable $\\tau $ in place of the\nphysical time variable $t$ \\cite{CL2007,calogero}. This procedure formally\nentails a change in the time dependence of any physical quantity $f(t)$, so\nthat $f(t)$ gets replaced by $f(\\tau \\left( t\\right) )$ with $\\tau \\left(\nt\\right) $ a periodic function of $t$ (with period $T$: $\\tau \\left( t\\pm\nT\\right) =\\tau \\left( t\\right) $), hence the new dynamics entails that all\nphysical quantities evolve periodically with period $T$. This change, in the\ncase of a nonrelativistic dynamical system, implies a change in the\ndynamical equations: for instance, in the case of the (quite general)\nmany-body problem described above, the theory is characterized by a new\n(autonomous) Hamiltonian $\\tilde{H}$---different from the original\n(autonomous) Hamiltonian $H$---which causes any physical variable originally\nevolving as $f\\left( t\\right) $ under the dynamics yielded by $H$, to evolve\ninstead as $f(\\tau (t))$ under the dynamics yielded by $\\tilde{H}$. This\nchange of the physical laws determining the time evolution of the system\nproduces the \\textit{isochronous} evolution \\cite{calogero,CL2007}. But let\nus stress---as we already did in \\cite{isochronous cosmologies}---that any\nattempt to attribute to the new variable $\\tau $ the significance of \"time\"\nwould be improper and confusing: the variable playing the role of \"time\" is\nthe same, $t$, for both the original dynamical system $D$ and the modified\ndynamical system $\\tilde{D}$; in particular, for that characterized by the\nstandard many-body Hamiltonian $H,$ see (\\ref{H}), as well as for that\ncharacterized by the modified many-body Hamiltonian $\\tilde{H}$, see (\\re\n{Htilde}).\n\nThe dynamical systems $D$ and $\\tilde{D}$ mentioned above feature the same\ntrajectories in phase space, but while the time evolution of the dynamical\nsystem $D$ corresponds to a \\textit{uniform} forward motion along those\ntrajectories, the time evolutions of the modified dynamical systems $\\tilde{\n}$---although produced by time-independent equations of motion---correspond\nto a \\textit{periodic} (with assigned period $T$), forward and backward,\ntime evolution along those same trajectories, exploring therefore only a\nportion of them; with the possibility to manufacture $\\tilde{D}$ in such a\nway that, for an \\textit{arbitrary} subinterval $\\tilde{T}<T,$ the dynamics\nof $D$ and $\\tilde{D}$ be \\textit{very close} or even \\textit{identical}.\n\nIn the context of general relativity, an analogous phenomenology exists \\cit\n{isochronous cosmologies}. However, in this case the situation is a bit\ndifferent, due to the physical invariance of Einstein's equations under\ndiffeomorphisms, which implies that the procedure used to construct the\nisochronous solutions, while again formally entailing the introduction of a\nnew auxiliary variable $\\tau \\left( t\\right) $, does not imply any change of\nthe fundamental Einstein equations, corresponding instead to the\nidentification of an enlarged class of solutions of these equations. But\nagain any attempt to attribute \\textit{globally} to this auxiliary variable \n\\tau $ the significance of \\textit{time} would be improper and confusing;\nwhile by attributing to it \\textit{locally }(in time) the significance of\n\\textit{reparameterized time}, it is immediately seen that the enlarged\nclass of metrics entails dynamical evolutions which, \\textit{locally in time\n, are \\textit{physically equivalent} to that yielded by the original,\nunmodified metric.\n\nOur point of departure is to assume a homogeneous, isotropic and spatially\nflat metric $\\ \\tilde{g}_{\\mu \\nu }$ which, in a given reference frame $(t\n\\vec{x})$, is defined by the line element\n\\end{subequations}\n\\begin{equation}\nds^{2}=b(t)^{2}~dt^{2}-a(t)^{2}~d\\vec{x}^{2}~,  \\label{newmetric}\n\\end{equation\nwhere $b\\left( t\\right) ^{2}$ is commonly termed the \"lapse\" function\n(although we did not use this term in \\cite{isochronous cosmologies}).\n\nThe lapse function is not determined by Einstein's equations; its\nintroduction in (\\ref{newmetric}) corresponds to the freedom to\nreparametrize time, which in general relativity is permissible, via a\ndiffeomorphic transformation, with no physical effect:\\ indeed, in general\nrelativity it is the \\textit{geometry} of spacetime that characterizes the\nphysics, and this geometry is not affected by the coordinates used to\ndescribe it. An additional requirement which is generally considered\nessential---amounting to the \"equivalence principle\"---is that the metric\nhave a \\textit{Lorentzian signature} (everywhere in spacetime). This implies\nthat, in the case of the metric (\\ref{newmetric}), the lapse function $b(t)$\nmust never vanish, so that one can always put this metric in its synchronous\nform via the reparameterization of the time variable from $t$ to $\\tau $\nsuch that\n\\begin{equation}\nd\\tau =b(t)~dt~,~\\hspace{0in}~\\tau \\left( t\\right) =B\\left( t\\right) \\equiv\n\\int_{0}^{t}b\\left( t^{\\prime }\\right) dt^{\\prime }~,  \\label{repar}\n\\end{equation\nwhich maps the metric (\\ref{newmetric}) into the Friedmann-Robertson-Walker\nmetric $g_{\\mu \\nu }$ with line element\n\\begin{subequations}\n\\begin{equation}\nds^{2}=d\\tau ^{2}-\\alpha (\\tau )^{2}~d\\vec{x}^{2}~,  \\label{FRWmetric1}\n\\end{equation\nwhere\n\\begin{equation}\n\\alpha (\\tau (t))\\equiv a(t)~.  \\label{alphadefinition}\n\\end{equation}\n\nAs stated above, our trick to construct \\textit{isochronous} dynamical\nsystems is \\textit{formally} based on the eventual replacement of the\nphysical time variable $t$ with another variable $\\tau $. In a general\nrelativistic context this corresponds to have a \\textit{periodic} lapse\nfunction (with an \\textit{\\ arbitrarily assigned} period $T$) having\nmoreover a vanishing mean value, so that its integral $B\\left( t\\right) ,$\nsee (\\ref{repar}), is also periodic with period $T$:\n\\end{subequations}\n\\begin{equation}\nb(t+T)=b(t)~;~~~\\tau \\left( t+T\\right) =\\tau \\left( t\\right) ~.\n\\label{bBperiodic}\n\\end{equation}\n\nThe choice (\\ref{bBperiodic})---when inserted in (\\ref{newmetric})---implies\nthe following consequences \\cite{isochronous cosmologies}.\n\n(i) For any FRW metric (\\ref{FRWmetric1}), the corresponding---via the\nrelation (\\ref{bBperiodic})---metric (\\ref{newmetric}) is periodic in the\ntime variable $t$, and therefore it is cyclic, defining thereby an \\textit\nisochronous spacetime}.\n\n(ii) Since the lapse function is periodic with period $T$ and has zero mean\nvalue, it must vanish at an infinite set of times $t=t_{n}\\equiv t_{0}\\pm\nnT\/2$ with $n=0,1,2,3,...$, implying that the metric (\\ref{newmetric}) is\ndegenerate at these times.\n\n(iii) If the metric (\\ref{FRWmetric1}) is a solution of Einstein's equations\n(namely, of the FRW equations) corresponding to the stress energy tensor of\na perfect fluid (or a mixture of different perfect fluids) $T_{\\mu \\nu }\n\\left[ r(\\tau )+p(\\tau )\\right] u_{\\mu }u_{\\nu }-g_{\\mu \\nu }p(\\tau )$,\nwhere $r(\\tau )$ and $p(\\tau )$ are the energy density and pressure of the\nfluid and $u_{\\mu }=\\delta _{\\mu 0}$ is its 4-velocity, then (\\ref{newmetric\n) is also a solution of Einstein's equations corresponding to a stress\nenergy tensor $T_{\\mu \\nu }=\\left[ \\rho (t)+P(t)\\right] U_{\\mu }U_{\\nu\n}-g_{\\mu \\nu }P(\\tau )$ of a perfect fluid with energy density $\\rho\n(t)=r(\\tau (t))$, pressure $P(t)=p(\\tau (t))$ and 4-velocity $U_{\\mu\n}=|b(t)|\\delta _{\\mu 0}$. Therefore the metrics (\\ref{newmetric}) and (\\re\n{FRWmetric1}) describe two universes filled with the same perfect fluid(s).\n\n(iv) Since for $t\\neq t_{n}$ the two metrics (\\ref{newmetric}) and (\\re\n{FRWmetric1}) are connected by a diffeomorphic transformation, they give the\nsame physics in any time interval within which $b(t)$ does \\textit{not}\nvanish, so that they are in fact experimentally indistinguishable there.\n\n(v) One can conveniently choose the lapse function in such a way that the\nscale factor is always positive, i.e. $a(t)=\\alpha (\\tau (t))>0$ for any $t\n. This choice implies that all the scalar invariants as well as the pressure\nand energy density of the fluid remain finite at any $t$ and the spacetime\nis geodesically complete, thus singularity free.\n\n(vi) Geodesics have, in spacetime, a helical structure.\n\nThese solutions of course violate the \"equivalence principle\" at the\ninfinite set of discrete times $t_{n}.$ It is thus seen that, with the\nchoice (\\ref{bBperiodic}), the class of spacetime solutions of the equations\nof general relativity is enlarged by allowing unobservable violations of the\n\"equivalence principle\" at a discrete, numerable set of times $t_{n}$. This\nmore general class of spacetimes do not seem to entail any \\textit\nexperimentally observable} differences; although they might entail a quite\ndifferent structure of spacetime, for instance absence versus presence of\nBing Bang singularities.\n\nThe problem of finding a generalization of general relativity which includes\ndegenerate metrics has been studied in the past, motivated by the\nconsideration of a class of metrics, the so called signature-changing\nmetrics \\cit\n{ellis1,ellis2,elliscarfora,elliscomment,ellis3,ellis4,ellis5,ellis6,ellis8,ellis9,ellis10,ellis11,ellis12,ellis13,ellis14,ellis15,ellis16,ellis17,ellis18,ellis19}\n, which give a classical realization of the change of signature in quantum\ncosmology conjectured by Hartle and Hawking \\cit\n{hartlehawking3,hartlehawking1,hartlehawking2,hartlehawking4,hawkinghistory\n. The classical change of signature for a homogeneous, isotropic and\nspatially flat universe is realized by a metric tensor defined by the\nfollowing line elemen\n\\begin{equation}\nds^{2}=N(t)~dt^{2}-R(t)^{2}~d\\vec{x}^{2}~,\n\\label{changeofsignaturedegenerate}\n\\end{equation\nwhere $N(t)$ is a continuous function changing sign at some time $t_{0}$,\nfor instance $N(t)$ is positive for $t>t_{0}$, negative for $t<t_{0}$ and\nzero for $t=t_{0}$ in correspondence to the hypersurface of change of\nsignature where the metric (\\ref{changeofsignaturedegenerate}) is degenerate\n(note the difference from (\\ref{newmetric}), where the coefficient \nb^{2}\\left( t\\right) $ of $dt^{2}$ might vanish at some points but never\nbecomes \\textit{negative, }$b^{2}\\left( t\\right) \\geq 0$).\n\nThese generalizations of standard general relativity including degenerate\nmetrics as (\\ref{newmetric}) and (\\ref{changeofsignaturedegenerate}) are not\nunique since they depend on their defining prescriptions: \\textit{different}\nprescriptions yield \\textit{different} physical theories \\cite{elliscomment\n. In particular, it has been pointed out that different generalizations\nbased on different prescriptions give different junction conditions on the\ndegeneracy hypersurfaces, see for instance the discussion in \\cit\n{elliscarfora,elliscomment}. In fact \"one can also consider discontinuities\nin the extrinsic curvature\" \\cite{elliscarfora} on the degeneracy\nhypersurface of a degenerate solution of Einstein's equations, \"but attempts\nto relate these to a distributional matter source at the boundary require\nsome form of field equations valid on the surface\", see discussion in \\cit\n{elliscomment}, and this would require some special prescription on what\nEinstein's equations are on the degeneracy hypersurfaces. Thus, if one\naccepts degenerate solutions of Einstein's equations as physically\nacceptable spacetimes, there is no reason to infer that a discontinuity of\nthe extrinsic curvature on a degeneracy surface implies that the stress\nenergy tensor associated to such a metric tensor must have a distributional\nform there. What is more, delta functions with support on degeneracy\nhypersurfaces have no significant effect in covariant integrals \\cit\n{ellis15,ellis19}, due to the fact that the covariant volume element $dx^{4\n\\sqrt{-g}$ associated to a degenerate metric is null on its degeneracy\nhypersurface, because the determinant $g$ vanishes there. For instance, for\nthe metric (\\ref{newmetric}) one has, for any integrable function $f\\left(\nx\\right) $\n\\begin{equation}\n\\int dx^{4}\\sqrt{-g}\\delta (t-t_{n})f\\left( x\\right) =\\int\ndtdx^{3}|b(t)||a(t)|^{3}\\delta (t-t_{n})f\\left( x\\right)\n=|b(t_{n})|~|a(t_{n})|^{3}\\int \\,dx^{3}f\\left( x\\right) =0\n\\end{equation\n(because $b\\left( t_{n}\\right) =0$). Hence delta-like distributions centered\non the degeneracy hypersurfaces $t=t_{n}$ disappear from integrals (see also\nthe discussion in \\cite{ellis15}).\n\nSo, the question of how to include degenerate metrics in general relativity\nis moot.\n\nIn order to formulate a generalized gravitational theory which includes such\nmetrics, it is convenient to focus directly on the Einstein's equations\ncharacterizing the geometry of spacetime, which themselves admit degenerate\nmetrics as their solutions. If, however, one considers desirable to derive\nthis theory from a variational principle, a possibility is to assume the\nstandard Einstein-Hilbert gravitational action of general relativity, such\nthat the total action of gravitation plus the matter (nongravitational)\nfields i\n\\begin{equation}\nS_{Tot}=\\int_{M}\\sqrt{-g}\\left( -\\frac{1}{2\\mathit{k}}R+\\mathit{L\n_{Mat}\\right) d^{4}x~,  \\label{actiontotal}\n\\end{equation\nwhere $\\mathit{k}=8\\pi G\/c^{4}$, $G$ is the gravitational constant, $c$ the\nspeed of light, $M$ is the manifold defining the spacetime and $\\mathit{L\n_{Mat}$ is the Lagrangian density of matter fields; and add the requirement\nthat, for a metric tensor which is degenerate on some hypersurface $\\Sigma \n, the variation of the metric $g_{\\mu \\nu }$ vanish on $\\Sigma $, i.e. \n\\delta g_{\\mu \\nu }|_{\\Sigma }=0$. With such an assumption the variation of\nthe action (\\ref{actiontotal}) i\n\\begin{equation}\n\\delta S_{Tot}=\\frac{1}{2}\\int_{M\/\\Sigma }\\sqrt{-g}\\left( -\\frac{1}{\\mathit{\n}}G^{\\mu \\nu }+T^{\\mu \\nu }\\right) d^{4}x~,  \\label{actiontotalvariation}\n\\end{equation\nwhere $G^{\\mu \\nu }$ is the Einstein tensor and $T^{\\mu \\nu }$ is the\nstress-energy tensor of matter fields, which gives the standard equations of\ngeneral relativity in the region where the metric tensor is nondegenerate.\n\nTo bypass the problems related to degenerate signature-changing metrics, it\nhas been proposed \\cite{ellis1,ellis2} to consider a different,\nnondegenerate but discontinuous, realization of the classical change of\nsignature, as given by the metri\n\\begin{equation}\nds^{2}=f(\\tau )~d\\tau ^{2}-R(\\tau )^{2}~d\\vec{x}^{2},\n\\label{changeofsignaturediscontinuous}\n\\end{equation\nwith a discontinuous lapse function $f(\\tau )=1$ for $\\tau >\\tau _{0}$ and \nf(\\tau )=-1$ for $\\tau <\\tau _{0}$. It has been shown \\cite{ellis1,ellis2}\nthat in this case it is possible to introduce a smooth (generalized)\northonormal reference frame which allows a variational derivation of\nEinstein's equations and an extension of the Darmois formalism \\cite{ellis16}\nto discontinuous metrics, in such a way that, also in the case of a\ndiscontinuous metric tensor, the discontinuity of the extrinsic curvature on\na hypersurface is related to a distributional stress energy tensor \\cit\n{ellis15}. This is achieved by adding a surface term to the Einstein-Hilbert\naction, in such a way that the gravitational action become\n\\begin{equation}\nS_{g}=\\int_{M\/\\Sigma }\\sqrt{-g}Rd^{4}x+\\oint_{\\Sigma }\\sqrt{-g}Kd\\Sigma\n\\label{actionboundary}\n\\end{equation\nwhere $M$ is the manifold defining the spacetime, $\\Sigma $ is the boundary\nof $M$ where the metric tensor is discontinuous, $R$ is the Ricci scalar\ncurvature and $K$ the extrinsic curvature of $\\Sigma $. The addition of such\na boundary term is always necessary when one considers manifolds with\nboundaries \\cite{HawkingHorowitz}. Incidentally we note that this action\ncannot be used in the case of degenerate metrics, because in this case the\ndegeneracy surface $\\Sigma $ has no unitary normal vector hence the\nextrinsic curvature $K$ does not exist.\n\nThe two realizations of the classical change of signature, the continuous\nand degenerate one, see (\\ref{changeofsignaturedegenerate}), and the\ndiscontinuous one, see (\\ref{changeofsignaturediscontinuous}), are \\textit\nlocally but not globally} diffeomorphic, since they are related by a change\nof the time variable $dt=d\\tau \/\\sqrt{|N(t)|}$ which is not defined at the\ntime of signature change $t_{0}$ when $N(t)=0$. Hence they represent two\ndifferent spacetimes. However, except for the instants when $N(t)$ vanishes,\nthey are \\textit{locally } diffeomorphic and thus they describe---excepts at\nthose instants---the same physics .\n\nAs in the case of signature-changing metrics, it is also possible to\nconsider a realization of \\textit{isochronous} cosmologies different from\nthat we introduced in \\cite{isochronous cosmologies} and reviewed above (see\n(\\ref{newmetric}) and the subsequent treatment), via \\textit{nondegenerate}\nmetrics featuring a finite jump of their first derivatives at an infinite,\ndiscrete set of equispaced times. Such a realization is given by the\nfollowing metric\n\\begin{equation}\nds^{2}=d\\eta ^{2}-\\tilde{a}(\\eta )^{2}d\\vec{x}^{2}\n\\label{newmetricnondegenerate}\n\\end{equation\nwith\n\\begin{equation}\n\\alpha (\\tau (\\eta ))\\equiv \\tilde{a}(\\eta )\n\\label{newscalarfactornondegenerate}\n\\end{equation\nand\n\\begin{equation}\nd\\tau =C(\\eta )~d\\eta ,  \\label{newCndegenerate}\n\\end{equation\nwith a periodic function $C(\\eta )$ of period $T$ given for instance by \nC(\\eta )=1$ for $nT<\\eta <(2n+1)T\/2$ and $C(\\eta )=-1$ for $(2n+1)T\/2<\\eta\n<(n+1)T$ and $n$ integer, hence such that\n\\begin{subequations}\n\\begin{equation}\n\\tau \\left( \\eta \\right) \\equiv \\int_{0}^{\\eta }C\\left( \\eta ^{\\prime\n}\\right) d\\eta ^{\\prime }=\\eta -nT~~~\\text{for}~~~nT<t<\\left( 2n+1\\right)\nT\/2~,\n\\end{equation\n\\begin{equation}\n\\tau \\left( \\eta \\right) \\equiv \\int_{0}^{\\eta }C\\left( \\eta ^{\\prime\n}\\right) d\\eta ^{\\prime }=\\left( n+1\\right) T-\\eta ~~~\\text{for}~~~\\left(\n2n+1\\right) T\/2<t<\\left( n+1\\right) T~.\n\\end{equation}\n\nThe metric (\\ref{newmetricnondegenerate}) is periodic with period $T$ and is\nnot degenerate at the inversion times $\\eta _{n}=nT\/2$, where its first\nderivatives feature a finite jump. Since (\\ref{newmetricnondegenerate}) is\nnondegenerate, now the Darmois formalism \\cite{ellis16} applies and one can\nrelate the discontinuity of the first derivatives of the metric, hence the\ndiscontinuity of the extrinsic curvature of the hypersurfaces $\\Sigma _{\\eta\n_{n}}$ defined by $\\eta =\\eta _{n}$ (which is now well defined), with a\ndistributional stress-energy tensor. In fact defining the unitary normal\nvector $n_{\\alpha }=\\delta _{\\alpha 0}$, the tangent vectors $e_{\\alpha\n}^{a}=\\delta _{\\alpha }^{a}$ and the induced metric $h_{ab}=-\\tilde{a}(\\eta\n)^{2}\\delta _{ab}$ of a hypersurface $\\Sigma _{\\eta }$ of equation $\\eta\n=constant$, which also implies $n^{\\alpha }=\\delta ^{\\alpha 0}$, \ne_{a}^{\\alpha }=\\delta _{a}^{\\alpha }$ and $h^{ab}=-\\delta _{ab}\/\\tilde{a\n(\\eta )^{2}$, the extrinsic curvature of $\\Sigma _{\\eta }$ is given by \\cit\n{poisson}\n\\end{subequations}\n\\begin{equation}\nK_{ab}\\equiv n_{\\alpha ;\\beta }e_{a}^{\\alpha }e_{b}^{\\beta }= \\alpha (\\tau\n(\\eta )) \\, \\alpha^{\\prime }(\\tau (\\eta )) \\, C(\\eta ) \\,\\delta _{ab}~,\n\\label{extrinsiccurvaturenondegenerate}\n\\end{equation\nwhere $\\alpha^{\\prime }(z)\\equiv d\\alpha(z)\/dz$ \\footnote\nNote that in \\cite{isochronous cosmologies} we have calculated the extrinsic\ncurvature of the hypersurfaces $t=const$ of the metric (\\ref{newmetric})\ndefining the normal vector of such hypersurfaces as $n^{\\alpha }=\\delta\n^{\\alpha }\/b(t)$, the tangent vectors as $e_{\\alpha }^{a}=a(t)\\delta\n_{\\alpha }^{a}$ and $h_{ab}=-\\delta _{ab}$. Instead, (\\re\n{extrinsiccurvatureJUMPnondegenerate}) corresponds to the choice $n^{\\alpha\n}=\\delta ^{\\alpha 0}\/|b(t)|$, $e_{\\alpha }^{a}=\\delta _{\\alpha }^{a}$ and \nh_{ab}=-a(t)^{2}\\delta _{ab}$.}.\n\nTherefore the discontinuity of the extrinsic curvature on the hypersurface \n\\Sigma _{\\eta _{n}}$ i\n\\begin{equation}\n\\lbrack K_{ab}]|_{\\Sigma _{\\eta _{n}}}=2\\,(-1)^{n}\\,\\alpha (\\tau (\\eta\n))\\,\\alpha ^{\\prime }(\\tau (\\eta ))\\,\\delta _{ab}~,\n\\label{extrinsiccurvatureJUMPnondegenerate}\n\\end{equation\nand this implies that the stress-energy tensor has, on the numerable set of\nhypersurfaces $\\Sigma _{\\eta _{n}}$, a distributional contribution given b\n\\begin{equation}\nT_{\\alpha \\beta }^{distr}=\\sum_{n}\\delta (\\eta -\\eta _{n})\\,S_{ab}|_{\\Sigma\n_{\\eta _{n}}}e_{\\alpha }^{a}e_{\\beta }^{b}~,\n\\label{StressEnergyTensorNondegenerate}\n\\end{equation\nwher\n\\begin{equation}\nS_{ab}|_{\\Sigma _{\\eta _{n}}}\\equiv \\frac{1}{8\\pi }\\left( [K_{ab}]|_{\\Sigma\n_{\\eta _{n}}}-[K]|_{\\Sigma _{\\eta _{n}}}h_{ab}\\right) =\\frac{(-1)^{n+1}}\n2\\pi }\\,\\alpha (\\tau (\\eta ))\\,\\alpha ^{\\prime }(\\tau (\\eta ))\\,\\delta _{ab}\n\\end{equation\nand $[K]|_{\\Sigma _{\\eta _{n}}}\\equiv \\lbrack K_{ab}]|_{\\Sigma _{\\eta\n_{n}}}h^{ab}$. We note that this distributional stress-energy tensor is\nabsent in the case of a vacuum Minkowskian universe, since in this case \n\\alpha (\\tau )$ is constant and its derivative vanishes.\n\nAgain, let us point out that the two metrics (\\ref{newmetric}) and (\\re\n{newmetricnondegenerate}) are two \\textit{different} realizations of an\n\\textit{isochronous} cosmology, since they are \\textit{locally}, but \\textit\nnot globally}, diffeomorphic via the change of variable $d\\eta =b(t)dt$\nwhich is singular at $t_{n}$. In particular the statements (i,iii,iv,v,vi),\nsee above, which are valid for the metric (\\ref{newmetric}), also apply to\nthe metric (\\ref{newmetricnondegenerate}). Moreover, both (\\ref{newmetric})\nand (\\ref{newmetricnondegenerate}) are \\textit{locally} (but \\textit{not\nglobally}) diffeomorphic to the FRW metric (\\ref{FRWmetric1}) and therefore\nthey gives an \\textit{identical} dynamics in any time interval such that \nb(t)\\neq 0$ or $C(\\eta )\\neq 0$. The main difference is that the isochronous\nevolution given by (\\ref{newmetric}) entails the introduction of degenerate\nmetrics and the nondegenerate realization of the isochronous dynamics given\nby (\\ref{newmetricnondegenerate}) implies the existence of a distributional\ncontribution to the stress-energy tensor. In both cases the evolution of the\nuniverse is locally the same as that described by the FRW equations,\nimplying that it is not possible to discriminate by experiments local in\ntime between an isochronous and a noncyclic evolution of the universe.\n\n\\section{Conclusions}\n\nIn this paper we have reviewed and further discussed the idea introduced in\n\\cite{isochronous cosmologies}, itself being an extension to a general\nrelativity context---hence relevant to discuss cosmological models---of a\nfinding valid in the wide context of nonrelativistic Hamiltonian dynamics.\nLet us emphasize that---as in that context---the essence of our observation\nis to point out the disturbing possibility, given a physical theory that\ndescribes reality, to manufacture variations of it---i. e., different\ntheories, characterized by the introduction of an arbitrary parameter $T\n---which describe essentially the \\textit{same} reality over time intervals\nsmaller than $T$ but are \\textit{cyclic} with period $T$.\n\nAnd let us conclude this paper by emphasizing that we are \\textit{not}\nourselves arguing that our Universe evolves \\textit{cyclically} indeed\n\\textit{isochronously}, much less wish we to enter the related philosophical\nissues (not our cup of tea!); we have merely pointed out the (disturbing!)\nfact that, in the context of theoretical and mathematical physics, such a\npossibility does not seem to be excluded; nor does it seem feasible to\nexclude it by realizable experiments.\n\n\\bigskip\n\n\\section{Acknowledgements}\n\nThis paper was completed while FB was a Marie Curie Fellow of the Istituto\nNazionale di Alta Matematica \"Francesco Severi\".\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThough the linguistic functions of speech prosody are nowadays well documented in a large number of languages (phonology, syntax\/prosody interface, etc...), its expressive or para-linguistic functions, such as speaking style or speech emotions, does not benefit from the same amount of attention from the linguistic community.\nMeanwhile, speech engineers have realized spectacular advances in the past decade creating extremely realistic synthetic voices \\cite{wang17_tacotron} which are now integrated into voice interfaces that are increasingly present in our everyday lives, such as the voice assistants and conversational\/virtual agents. However, these intelligible and natural voices still clearly lack expressiveness and adaptability which greatly limits the interaction between humans and machines (see for instance \\cite{Cas12}). \nExpressivity is the next frontier of speech research at the interface of linguistics and technology, as shown by the recent increase of research in this domain \\cite{wang2018style}. Consequently, there is a clear need to better understand the expressivity of the human voice which is mainly conveyed through speech prosody. In particular, there is a growing interest of the engineering community in the statistical modelling of prosody \\cite{Lat08, Obi11c, Yin16, Wan17, Luo2017-qn, Obi18, Gerazov2018b}, expressive speech synthesis \\cite{ObinPhD, Eyben12, Akuzawa19}, and conversion of neutral to expressive speech \\cite{Veaux2011-pq, Rob01, Luo_2019_0}. \\\\\n\nNowadays, speech expressivity is generally equated to speech emotions though the scope of expressivity includes but is not restricted to the primary emotions as denoted by Ekman \\cite{Ekman92}. This limitation is probably due to the difficulty of converging to an agreement on the terminology used to describe the various and subtle forms of expressivity in speech.\nAccordingly, the study of speech expressivity is generally limited to dedicated speech emotion databases as interpreted by actors or to audio books read by professional readers (mostly in English, and sometimes in French or German \\cite{Chen}). In the past decade, speech emotion research has mainly focused on acted emotional speech: from its original form in which an actor is asked to interpret a short text with a given emotion \\cite{Bur05} to more open and spontaneous forms in which two actors freely improvise based on a given scenario and then asked to rate their own speech emotions with categories or on valence\/arousal continuous scales \\cite{Bus08, Keo10}.  Moreover, these databases have been created only for the purpose of speech emotion recognition, but not for the modelling of speech prosody and neither for speech synthesis and conversion. \\\\\n\nHowever, speech expressivity is not restricted to primary emotions. For instance, to deal with more subtle variation, the circumplex model, in which emotions are categorized in a bidimensional space, was proposed by psychologists \\cite{russel80}. Attitude was firstly equated to the first dimension (valence) of this model \\cite{ajzen80}. A distinction between emotion and attitude was done in \\cite{Couper} by defining emotion as a speaker state and attitude as a kind of behaviour. This distinction was later refined in \\cite{Wic00}, by defining attitude as a predictor of social behaviour. The attitudinal aspect of expressiveness of course differs from the primary emotions, and create a distinction between the propositional attitude (towards an utterance: irony, doubt, etc... \\cite{Lee83}) and the social attitudes (towards a person: dominant, friendly, seductive, distant, etc...). This last dimension has been recently investigated in the study of the role of speech prosody in neurosciences \\cite{Pon18}. Moreover, all of the existing speech databases miss an essential aspect of speech prosody: its {\\em variety} \\cite{Obi12}. There is always only one realization of each utterance, while any utterance can be obviously realized with many prosodies, some being functionally equivalent, some having various degrees of expressivity. There is a clear need for speech database that would tackle these issues and allow a diversification of the research in speech prosody and expressivity, and with sufficient data to allow learning generative models. \\\\\n\n\nThis paper presents Att-HACK, the first large database of acted speech with social attitudes. This database comprises the recordings of 25 speakers (M\/F) interpreting 100 utterances in 4 different social attitudes: friendly, seductive, dominant, distant.\nIn a given attitude, each utterance is repeated differently between 3 and 5 times in order to provide access to the inherent prosodic variety of speech.\nThis represents a total of about 30 hours of speech, which comes with audio signals, orthographic text transcription, F0 analysis, and phonetic alignments. The Att-HACK \\footnote{\\url{http:\/\/www.openslr.org\/88\/}} is freely available for research under a Creative Commons (BY\/NC\/ND) Licence. The remaining of the paper is organized as follows: in Section \\ref{sec:conception} we describe the conception of Att-HACK. A full description of the database is proposed in Section \\ref{sec:analysis}, with some illustrations of the F0 obtained in the 4 social attitudes by some of the speakers.\n\n\\section{Conception of Att-HACK}\n\\label{sec:conception}\n\nIn this Section, the full conception of the Att-HACK speech database of social attitudes is described, from the initial choice of social attitudes as a continuation of speech expressivity research and including the details of the speech database design.\n\n\\subsection{Models of Emotions}\n\nThe means by which humans construct representations of emotions is still an open and debated issue in the field of psychology and affective sciences: today, two fundamental models are used for the description of emotions. The first approach is \\emph{categorial} and supports the idea that emotions are discrete and relate to concepts that are essentially distinct. A discrete emotion theory was developed by Ekman et al. in 1992 \\cite{Ekman92} in which six basic emotions were proposed: anger, disgust, fear, happiness, sadness, and surprise. The second approach is dimensional and supports that emotions are defined according to at least one dimension. In the circumplex model \\cite{russel80}, emotion is defined on two dimensions: valence (positive\u2013negative) and arousal (passive-active). A third dimension denoted dominance (submissiveness-dominance) was added to compose the PAD (Pleasure, Arousal, Dominance) three-factors model for speech emotions \\cite{Russ01}.\n\n\\subsection{Definition and Choice of Social Attitudes}\n\nThe choice of social attitudes is methodologically rooted with the idea of extending the study of expressivity from primary emotions to more subtle forms. As initially inspired by \\cite{Wic00}, social attitudes allows to represent the attitude of a speaker towards its interlocutor during an interaction. Such social attitudes differ from emotions which are internal state of a speaker and from propositionnal attitudes which are the attitude of a speaker towards an utterance. \n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[scale=0.49]{figures\/attitudes.png}\n    \\caption{Social attitudes represented in a 2-D space inspired by Jean Julien Aucouturier's categorization for musicians playing attitudes \\cite{auc17}.}\n    \\label{fig:att}\n\\end{figure}\n\n\n\n\\vspace{1cm}\n\nFollowing this initial definition, Jean-Julien Aucouturier et al. \\cite{auc17} recently proposed an original categorization for musicians' playing attitudes inspired by Leary's rose model \\cite{Leary57}. The attitudes are described in a bidimensional space, the first dimension reflects the hostility\/friendliness (alternatively, negative\/positive) towards the other musician while the second dimension reflect the position of the musician in a social hierarchy (subordination\/dominance). \nOur proposed choice of social attitudes is in the continuation of these two precursory works. In this paper, four social attitudes were defined by sampling the valence and dominance dimensions during a speech interaction: friendly, seductive, dominant and distant (figure \\ref{fig:att}). This includes one negative (distant) and three positives (friendly, seductive, dominant) attitudes sampled in the semi-space of neutral to high-hierarchy in the dominance space. \\\\\n\n\\noindent The four attitudes are described as follows :\n\n\\begin{itemize}\n    \\item \\textbf{friendly}: you are pleasant and benevolent, you care about others' preferences, you act towards the others independently from your own situation.\n    \\item \\textbf{seductive}: everything in your behaviour aims at charming the others, to make them love you, you do not care about others' preferences but you are ready for anything to seduce them even if you have to fake benevolence.\n    \\item \\textbf{dominant}: you are self confident, sure of your own superiority, you do not care about others' preferences, everything in your behaviour is dedicated to make the others obey and listen to you without imposing anything explicitly.\n    \\item \\textbf{distant}: you (barely) do not care about the others, you are uncommunicative, you do not care about others' preferences.\n\\end{itemize}\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[scale=0.5]{figures\/PAD.png}\n    \\caption{Social attitudes represented with the PAD model proposed by Russel \\& Mehrabian \\cite{Russ01}}\n    \\label{fig:dom}\n\\end{figure}\n\nAn attempt to describe our four attitudes through using the Russell's PAD model has been initiated by recomposing those attitudes with some of the 151 emotion-denoting terms subjects were asked to rate in \\cite{Russ01} using valence, arousal, and dominance. Let us assume that our \\text{friendly} is a mix of $\\{$friendly, humble, curious, respectful, kind$\\}$, \\textit{seductive} a mix of $\\{$affectionate, sexually excited, controlling$\\}$, \\textit{dominant} a mix of $\\{$influential, domineering, bold$\\}$ and \\textit{distant} a mix of $\\{$timid, bored, inhibited, uninterested, detached, shy, snobbish lonely$\\}$. The figure \\ref{fig:dom} shows our four social attitudes categorized according to the three Russel's dimensions (by averaging of the results obtained by Russel for the considered groups).\n\n\\subsection{Set of sentences}\n\nThe set of sentences used for the creation of the database has been designed in French as inspired by the corpus of propositionnal attitudes proposed by Morlec in \\cite{Mor97}. The proposed sentences have been designed according to the following criteria: 1) to avoid introducing a bias due to the expressive content of the text, the sentences were designed to be as neutral as possible; 2) to reduce the prosodic variability due to the text structure, the sentences were designed with a limited and controlled linguistic complexity, by using simple syntactic structures and short sentences; 3) Finally, the  sentences were designed in a way they remain plausible in each social attitude.  \n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[scale=0.5]{figures\/phrase_generator.pdf}\n    \\caption{Phrase generator functioning for the above quoted phrases, chosen words are in bold, dotted lines represent possible choices for the algorithm}\n    \\label{fig:gen}\n\\end{figure}\n\nAccordingly, we constructed a set of 100 sentences (from 2 to 8 syllables) corresponding to simple everyday life situations  (classic situations in classic socialization places like home, restaurant, workplace, ...).  For this purpose, we designed a phrase generator that builds phrases from semantic nucleus (\\{pronoun\/noun + verb\\} or \\{pronoun\/noun + auxilliary\\}) by randomly picking words in dictionaries in order to guarantee the phrases are always conceived the same way. We randomly kept 100 phrases among the 10000 generated ones to build our set of sentences, a sample of 10 sentences is listed below. \n\n\\begin{center}\n    \\emph{Oui} \\\\\n    \\emph{Bonjour} \\\\\n    \\emph{C'est vrai} \\\\\n    \\emph{A demain Paul} \\\\\n    \\emph{Bonne journ\u00e9e Marie} \\\\\n    \\emph{Il est tard \u00e0 Londres} \\\\\n    \\emph{Vous \u00eates all\u00e9s \u00e0 la plage} \\\\\n    \\emph{Vous \u00eates partis rapidement} \\\\\n    \\emph{Impossible, attendons un peu} \\\\\n    \\emph{C'est vrai, allons prendre un caf\u00e9} \\\\\n\\end{center}\n\nFigure \\ref{fig:gen} depicts the functioning of the automatic sentence generator considering the example of two sentences created from the nucleus ''Vous \u00eates ...\" (''You are ...'').\n\n\n\\section{Description of Att-HACK}\n\\label{sec:analysis}\n\n\\subsection{Audio recordings}\n\nTo feed this database, twenty-five actors were recorded in professional studios at Ircam. It consisted of 4 hours sessions during which one actor had to play 100 sentences in the four different social attitudes, proposing from 3 to 6 different versions of each sentence in each attitude. At the beginning of each session, the four attitudes were shortly described as stipulated above. Those descriptions have been used as acting options, actors were told to act the way they felt regarding each attitude denomination and not necessary in regards to those descriptions. The actors were told to be as natural as possible, no other information was given during the session. \\\\\n\nFor the recording, we used a Neumann U87 static microphone plugged into a RME fireface interface synchronized with the ProTools software. All audio recordings were made with a sampling rate frequency of 44.1 kHz and quantization of 16 bits per sample. The recording were made in two different studios depending on their availability. A patch implemented with the Max for Live software \\footnote{\\url{https:\/\/www.ableton.com\/en\/live\/max-for-live\/}} was used to provide a visual interface to the actor, displaying on a screen in front of the actor the sentence to be read and the expected attitude, this display being monitored by a sound engineer. The patch also allowed to store the time codes corresponding to each sentence, which were used after the session to segment and name automatically the continuous recording made with the ProTools software. At the end of a session, we had at least 2500 audio files for each actor. This amount of recordings has been manually sorted to remove corrupted files and utterances that were judged as being not natural enough or badly acted. At the time of writing, the Att-HACK database is composed of more than 22,000 expressive utterances, which will be completed in the months to come to reach around 50,000 utterances and 30 hours of speech. \n\n\\subsection{Speech to text alignment}\n\nSpeech-to-text alignment was performed by using the ircamAlign software \\cite{Lan08} which is based on the HTK toolbox and the Lia\\_phon French phonetizer and learned on the BREF French database of read speech. Phonetic alignment is first processed by ircamAlign and followed by rule-based syllable segmentation and a simple phrase segmentation. The resulting alignment are stored in dedicated lab files, a text format indicating at each line the starting time, ending time, and label corresponding to the text sentence and audio recording.  \n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=0.39\\textwidth]{figures\/sig_f0.png}\n    \\caption{Syllable segmentation (above) and F0 values as estimated on voiced frames (below)}\n    \\label{fig:sig_f0}\n\\end{figure}\n\n\\newpage\n\n\\subsection{F0 estimation}\n\nThe fundamental frequency of the speakers was estimated by using the SWIPEP algorithm \\cite{Cam07} with a minimum pitch value of  $75$ Hz, a maximum pitch value of $450$ Hz, and a hop size of $5$ ms, without any post-processing for correcting or smoothing the raw pitch values. The voiced\/unvoiced decision was computed from the pitch strength associated with the pitch value estimate with a threshold of $0.25$ (the pitch strength being a value between 0 and 1 corresponding to the periodicity of the speech frame).\n\nFigure \\ref{fig:sig_f0} presents the phonetic segmentation of the speech waveform (above) in syllables and the corresponding F0 values (below) for the sentence \\textit{\"Attendons un peu\"} using the Audiosculpt software \\footnote{\\url{https:\/\/forum.ircam.fr\/projects\/detail\/audiosculpt\/}}.\n\n\\section{Preliminary investigation}\n\\label{sec:investigation}\n\nA preliminary investigation was conducted to illustrate the content of the Att-HACK speech database of social attitudes.\n\n\\subsection{Extraction of F0 contours}\n\nA F0 contour was extracted for each syllable in order to illustrate the F0 patterns realized by the actors for the different social attitudes. The F0 contour of a syllable was identified as the one corresponding the longest sequence of F0 values considered as voiced over the syllable, i.e. the longest F0 segment for which each F0 value corresponds to a pitch strength value which is above a given  threshold (in this paper, $0.3$).\n\n\n\\begin{figure}[!h]\n    \\centering\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/Valerie_Fontaine_Oui.pdf}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.48\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/Benoit_Berthon_Oui.pdf}\n    \\end{subfigure}\n    \\caption{F0 contours mean (solid line) and standard deviation (filled with color) for the phrase \\textit{\"Oui\"} for a female speaker (above) and a male speaker (below)}\n    \\label{fig:contours}\n\\end{figure}\n\n\\subsection{F0 Statistics vs Attitudes}\n \n A preliminary investigation was conducted to compare the F0 statistics of the actors across the social attitudes.  We computed mean and standard deviation statistics on F0 segments extracted as stipulated above, for each speaker and attitude. These statistics are reported in Table \\ref{tab:means_and_stds}, including global statistics (female, male and mixed-gender). It is to be noted that only the first part of the recordings has yet been post processed at the time of writing. \\\\\n \nFigure \\ref{fig:contours} illustrates F0 pattern distributions obtained for a given sentence with the four attitudes, each attitude being represented by a dedicated color. In each color, the solid line represents the F0 pattern obtained by averaging the variations realized by the actor, the area filled with color represents the corresponding standard deviation, and the length of the pattern the corresponding mean duration. This illustration reveals that distinctive F0 patterns are associated with the social attitudes, and also highlights the diversity of strategies employed by actors to communicate a social attitude. \n\n\\begin{table}[tb]\\centering\n\\scalebox {0.75}[0.75]{\n\\begin{tabular}{p{1cm}p{1cm}cccccccc}\n\\hline\n\\textbf{gender} & \\multicolumn{2}{c|}{\\textbf{Friendly}} & \\multicolumn{2}{|c|}{\\textbf{Seductive}} & \\multicolumn{2}{|c|}{\\textbf{Dominant}} & \\multicolumn{2}{|c}{\\textbf{Distant}} \\\\ \n\n \\cline{2-9}\n    & Mean & Std & Mean & Std & Mean & Std & Mean & Std\\\\\n   \n\\hline\n\\textit{female} & 208 & 11 & 186 & 12 & 207 & 10 & 186 & 10\\\\\n\\textit{male} & 112 & 8 & 113 & 10 & 119 & 8 & 104 & 7\\\\\n\\hline\n\\textit{global} & 160 & 10 & 150 & 11 & 163 & 9 & 145 & 9\\\\\n\\hline\n\\end{tabular}}\n\\caption{Syllable F0 contours means and standard deviations (in Hz) for female and male speakers of Att-HACK}\n\\label{tab:means_and_stds}\n\\end{table}\n\n\\begin{table}[tb]\\centering\n\\scalebox {0.75}[0.75]{\n\\begin{tabular}{p{1cm}p{1cm}cccccccc}\n\\hline\n\\textbf{gender} & \\multicolumn{2}{c|}{\\textbf{Friendly}} & \\multicolumn{2}{|c|}{\\textbf{Seductive}} & \\multicolumn{2}{|c|}{\\textbf{Dominant}} & \\multicolumn{2}{|c}{\\textbf{Distant}} \\\\ \n\n \\cline{2-9}\n    & Mean & Std & Mean & Std & Mean & Std & Mean & Std\\\\\n   \n\\hline\n\\textit{female} & 404 & 151 & 426 & 141 & 399 & 143 & 440 & 169\\\\\n\\textit{male} & 410 & 133 & 440 & 145 & 413 & 137 & 471 & 177\\\\\n\\hline\n\\textit{global} & 407 & 144 & 431 & 142 & 405 & 141 & 452 & 172\\\\\n\\hline\n\\end{tabular}}\n\\caption{Syllable durations means and standard deviations (in ms) for female and male speakers of Att-HACK}\n\\label{tab:means_and_stds}\n\\end{table}\n\n\\section{Conclusion}\n\nThis paper presents Att-HACK, a first attempt to widen the scope of expressivity in speech, by providing a database of acted speech with social attitudes: friendly, seductive, dominant, and distant. The proposed database comprises 25 speakers interpreting 100 utterances in 4 social attitudes, with 3-5 repetitions each per attitude proving a great prosodic variety for a total of around 28 hours of expressive speech. The Att-HACK is freely available for academic research under Creative Commons Licence.\n\n\\section{Acknowledgements}\n\nThis research has been supported by the French Ph2D\/IDF MoVE project on MOdelling of speech attitudes and application to an expressive conversationnal agent and funded by the Ile-De-France region and by the French ANR project TheVoice: ANR-17-CE23-0025.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgments}\nWe acknowledge helpful conversations with Pianpian Qin, Sixue Qin, Craig Roberts and Minggang Zhao.\nThis work is supported by: the Chinese Government Thousand Talents Plan for Young Professionals and the National Natural Science Foundation of China under contracts No. 11435001, and No. 11775041, the National Key Basic Research Program of China under contract No. 2015CB856900.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and Background}\n\nSince their first prediction about four decades ago, the interest in axions has been continually growing \\cite{review,amoreview,casper,irastorza}. Light axions or axion-like-particles with a mass in the $10^{-6} $~eV $< m<$ $10^{-2} $~eV range form a compelling candidate for the dark matter in the universe. The existence of axions would also solve one of the longstanding theoretical problems in the standard model of particle physics; the so-called strong CP problem \\cite{wilczek,pq1,pq2}. Not surprisingly, there has been a large number of experimental efforts to detect this elusive particle. In this paper, we will be focusing on experimental detection schemes that rely on the interaction of the axions with electromagnetic waves. The theoretical proposals exploring axion-photon coupling date back to 80's \\cite{sikivie,krauss,maiani}, spurring much experimental work. ADMX (Axion Dark Matter eXperiment) and related experiments  \\cite{admx1,admx2,admx3,admx4,admx5} rely on converting low-energy axions that form the background dark matter to microwave photons using an intense DC magnetic field. The detection sensitivity of the microwave photons can be enhanced by many orders of magnitude using microwave cavities with a high quality factor (high-$Q$). Solar helioscopes such as CAST (Cern Axion Solar Telescope) \\cite{cast1,cast2} and IAXO (International AXion Observatory) \\cite{iaxo} use a similar mechanism, but instead search for the conversion of high energy solar axions into x-ray photons. The strategy in these experiments is to search for a change in the x-ray flux as the detector points to and away from the sun, or as earth rotates around the sun. Detecting axionic dark matter using possible resonance effects in superconducting Josephson junctions has also been suggested \\cite{josephson1,josephson2}. \n\nThe experiments mentioned above aim to detect axions that are naturally present in the environment. Another set of experiments work towards generating and detecting axions in the lab, and have greater control of their experimental parameters since they do not rely on an external source of axions. One idea is to produce axions using the interaction of an intense laser inside a Fabry-Perot cavity with a DC magnetic field, and detect the generated axions at a different location by back-converting them into photons. This set of experiments are cordially referred to as {\\it light shining through a wall (LSW)}, and their sensitivity has been steadily increasing over the last few decades \\cite{lsw1,lsw_new}. Other experiments along similar lines have aimed to observe a change in the phase, intensity, or the polarization of a laser, as the beam propagates through a region of space and generates axions through its interaction with a strong DC magnetic field \\cite{lsw2,lsw_new2}. Currently, these laboratory-based experiments do not have detection sensitivities comparable to microwave cavity experiments. Detecting axion-like particles using interaction of multiple high-peak-power pulsed laser beams in vacuum has also been discussed \\cite{mendonca,dobrich,homma1,homma2,nobuhiro}. \n\nIn this letter, we suggest a laser-based scheme that belongs to this second group of experiments (i.e., generating and detecting axions in the lab), but also has quite high sensitivity over a broad axion mass range. As we discuss below, the scheme relies on using laser-based four-wave mixing, which is mediated by the hypothetical axion field. Intense pump and Stokes laser beams with appropriately chosen frequencies resonantly drive axion generation. The central idea of our technique is that with the pump and Stokes lasers confined to a waveguide, for example an optical fiber, the generated axions can also be guided. The spatial profiles of the driving laser beams confine the axion generation, producing propagating axion modes, which we refer to as axitons.  The physics of confinement for axitons is similar to electromagnetic modes of a fiber; instead of the radial refractive index profile, the confinement relies on the electric and magnetic beam profiles of the driving pump and Stokes lasers. These confined axitons can then couple and mix with another laser (which we term as the mixing beam), affecting the propagation of a probe laser. In the spirit of LSW experiments, the detection of the axion using mixing and probe lasers can be accomplished in a separate fiber. We predict that our scheme will be able to detect axions with a sensitivity at the level of $10^{-12}$~GeV$^{-1}$ for the axion-photon coupling constant over an axion mass range $10^{-6} $~eV $< m<$ $10^{-2} $~eV. \n\nCompared to traditional LSW experiments, there are four unique advantages of our scheme: (1) The mass range of the hypothetical axions can be scanned by appropriately tuning the frequencies of the pump and the Stokes laser beams. (2) Because we utilize guided modes of a fiber, the interaction length can be as much as hundreds of kilometers, which is much longer than what can be achieved using other approaches. (3) We point out the use of near-zero refractive index for the probe laser beam, which may possibly be achieved using an engineered (metamaterial) waveguide, to further enhance the detection sensitivity \\cite{caspani,liberal,lawson}. To our knowledge, this is the first suggestion of using metamaterials for axion detection. (4) The scheme is purely laser based and does not rely on interaction using a Tesla-level DC magnetic field. These four advantages may prove to be critical in generating and detecting this elusive particle in the near future. \n  \n\\section{Formalism}\n\nWe proceed with a detailed description of our scheme. Were it to exist, a scalar axion field $\\phi (\\vec{r},t)$  would interact with electric and magnetic fields of electromagnetism through the Lagrangian \\cite{gasperini,visinelli}:\n\\begin{eqnarray}\n\\Delta \\mathcal{L} = \\frac{g_{a \\gamma \\gamma}} {\\mu_0c} \\phi \\vec{E}  \\cdot \\vec{B} \\quad , \n\\end{eqnarray}\n\n\\noindent where the quantity $g_{a \\gamma \\gamma}$ is the axion-EM coupling constant. The factor $1\/(\\mu_0 c)$ assures that the product $g_{a \\gamma \\gamma} \\phi$ is dimensionless.  In order to make the physics more explicit, we will use mks units in our analysis (except when we present the bounds on the coupling constant  $g_{a \\gamma \\gamma} $  in unit of GeV$^{-1}$, which is the tradition in the discipline). With the Lagrangian of Eq.~(1), the modified Maxwell's equations in a medium with current density $\\vec{J}$ can be found through the Euler-Lagrange equations and they are:\n\\begin{eqnarray}\n\\nabla \\cdot \\vec{E} & = & - g_{a \\gamma \\gamma} c \\nabla \\phi \\cdot \\vec{B} \\quad , \\nonumber \\\\\n\\nabla \\cdot \\vec{B} & = &  0 \\quad , \\nonumber \\\\\n\\nabla \\times \\vec{E} & = & -\\frac{ \\partial \\vec{B}}{\\partial t}  \\quad , \\nonumber \\\\\n\\nabla \\times \\vec{B} & = &  \\frac{1}{c^2} \\frac{ \\partial \\vec{E}}{\\partial t}  + \\mu_0 \\vec{J} + \\frac{g_{a \\gamma \\gamma}}{c} \\left( \\frac{\\partial \\phi}{\\partial t} \\vec{B} + \\nabla \\phi \\times \\vec{E} \\right) \\quad . \n\\end{eqnarray}\n\n\\noindent The interaction with the axion field produces additional charge and current densities, and modifies the Maxwell's equations for $\\nabla \\cdot \\vec{E}$ and $\\nabla \\times \\vec{B}$, while leaving the other two equations unchanged. As a result of the electromagnetic interaction, the Klein-Gordon equation for the axion field  $\\phi (\\vec{r},t)$  is also modified with a driving term that involves the dot product of the electric and magnetic fields:\n\\begin{eqnarray}\n \\nabla^2 \\phi - \\frac{1}{c^2} \\frac{ \\partial^2 \\phi}{\\partial t^2}- \\left( \\frac{m c} {\\hbar} \\right)^2  \\phi =  \\frac{g_{a \\gamma \\gamma}} {\\mu_0 c} \\vec{E} \\cdot \\vec{B} \\quad .\n\\end{eqnarray}\n\n\\noindent Here $m$ is the mass of the hypothetical axion and $\\nabla^2$ is the Laplacian operator. The $ \\vec{E} \\cdot \\vec{B} $ driving term produces an effective second-order nonlinearity for the vacuum and in our scheme we borrow many ideas from nonlinear optics, specifically from four-wave mixing processes \\cite{boyd}. \n\n\n\\begin{figure}[tbh]\n\\vspace{-0cm}\n\\begin{center}\n\\includegraphics[width=15cm]{axion_generation_scheme.pdf}\n\\vspace{-0.5cm} \n\\caption{\\label{eit_scheme} \\small   Energy level diagram and simplified schematic for producing guided axitons. Pump and Stokes laser beams whose frequency difference, $\\omega_P - \\omega_S$, is tuned close to the rest mass energy of the axion, resonantly drive axion generation. The two lasers are confined to an optical fiber and the solid curves are cartoon schematics for the radial profiles of the two lasers. The spatial profiles of the two beams then confine axion generation, producing guided axitons, which are shown in dashed curve.}\n\\end{center}\n\\vspace{-0cm}\n\\end{figure}\n\n\n\\section{Guided axion waves: axitons}\n\nWe first discuss the generation of axions that are confined by the spatial profiles of the pump and Stokes laser beams through the Klein-Gordon equation of Eq.~(3). Figure~1 shows the relevant energy level diagram and cartoon schematic for producing guided axion waves, which we refer to as axitons. We consider pump and Stokes laser beams propagating through a wave-guide, for example, an optical fiber. Through the $ \\vec{E} \\cdot \\vec{B} $ term, the electric field of the pump  and the magnetic field of the Stokes will drive the axion excitation. We take the fiber to be cylindrically symmetric and assume the pump and Stokes lasers to be in specific modes of the fiber. We work in cylindrical coordinates $(r, \\varphi, z)$, and take the two lasers to be of the form:\n\\begin{eqnarray}\n\\vec{E}_P (r, \\varphi, z) & = & E_P u_P(r) \\exp{ \\left(i l_P \\varphi \\right)} \\exp{ \\left( i \\beta_P z - i \\omega_P t \\right) } \\hat{e} + c. c.  \\quad , \\nonumber \\\\\n\\vec{B}_S (r, \\varphi, z) & = & B_S u_S(r) \\exp{ \\left(i l_S \\varphi \\right)} \\exp{ \\left( i \\beta_S z - i \\omega_S t \\right) } \\hat{e} + c. c.  \\quad . \n\\end{eqnarray}\n\n\\noindent Here $c. c.$ refers to the complex conjugate of the whole expression that is written before. $E_P$ and $B_S$ are the electric and magnetic field amplitudes for the pump and Stokes laser beams, and $\\hat{e}$ denotes the common polarization direction for the two vectors (which is any direction orthogonal to the propagation direction $\\hat{z}$). The quantities $u_P(r)$ and $u_S(r)$ are the radial mode functions of the corresponding lasers, and the integers $l_P$ and $l_S$ are typically referred to as orbital angular momentum numbers for the associated photons \\cite{boyd}. $\\beta_P$ and $\\beta_S$ are the propagation constants of the modes along the fiber $z$ axis (i. e., the longitudinal $k$-vector) and the quantities $\\omega_P$ and $\\omega_S$ are the angular frequencies. Noting the energy level diagram of Fig.~1, the frequency difference of the pump and Stokes laser beams, $\\omega_P-\\omega_S$ is tuned close to the rest-mass energy of the hypothetical axion. By tuning this frequency difference, axions at different masses can be searched for. In general, there is another contribution to the generation of the axion wave, which is driven by the magnetic field of the pump beam and the electric field of the Stokes laser. For plane-wave like lasers propagating inside a bulk material, this second contribution (which is proportional to $B_P E_S^*$), would interfere destructively with the main contribution that we consider below (which is proportional to $E_P B_S^*$), reducing the produced axion amplitude. Here, we take that the relevant modes of the fiber are chosen appropriately so that this second contribution can be ignored compared to the first one. For example, one could use a TE (transverse electric) mode for the pump, whereas a TM (transverse magnetic) mode for the Stokes laser. This way, the vectors $E_P$ (electric field of the pump) and $B_S$ (magnetic field of the Stokes) can be aligned, maximizing the dot product. However, an angle would be present between $B_P$ and $E_S$, which can be tuned minimizing their dot product using specific choice of the modes.  \n\nBecause the axion generation is driven by the  $\\vec{E} \\cdot \\vec{B} $ term, we are interested in the axion field solutions of the form:\n\\begin{eqnarray}\n\\phi (r, \\varphi, z)  =  u_{\\phi} (r) \\exp{ \\left[i (l_P -l_S) \\varphi \\right]} \\exp{ \\left[ i (\\beta_P- \\beta_S) z - i (\\omega_P-\\omega_S) t \\right] } + c. c.  \\quad . \n\\end{eqnarray}\n\n\\noindent If the ansatz solution of the form Eq.~(5) exists, it would indicate confined axions propagating without any change in their profile along the fiber. Using Eqs.~(4) and (5), the Klein-Gordon equation for the axion field can be reduced to a single ordinary differential equation for the radial axion mode function $u_{\\phi}(r)$:\n\\begin{eqnarray}\n\\frac{d^2 u_{\\phi}}{d r^2} + \\frac{1}{r} \\frac{d u_{\\phi}}{d r} - \\frac{(l_P - l_S)^2}{r^2} u_{\\phi}  + \\left[ \\frac{(\\omega_P - \\omega_S)^2} {c^2} - (\\beta_P - \\beta_S)^2 - \\left(\\frac{m c }{\\hbar} \\right)^2 \\right] u_{\\phi} = \\frac{g_{a \\gamma \\gamma}} {\\mu_0c} E_P B_S^* u_P(r) u_S^*(r) \\quad . \\nonumber \\\\\n\\end{eqnarray}\n\n\\noindent We next define the quantity $ \\Delta k ^2 \\equiv \\frac{(\\omega_P - \\omega_S)^2} {c^2} - (\\beta_P - \\beta_S)^2 - \\left(\\frac{m c }{\\hbar} \\right)^2$, which can be thought of as an ``energy detuning\": i.e., the detuning (difference) between the pump and Stokes laser energy difference (i.e., $\\omega_P -\\omega_S$) and the total energy (kinetic  + rest mass) of the excited axions. To elucidate the physics, we normalize the above equation through the definitions $\\tilde{r} \\equiv \\kappa_{axion} r $  and $ \\tilde{ \\Delta k } \\equiv \\Delta k \/ \\kappa_{axion}$. Here, the quantity $\\kappa_{axion}$ is the $k$-vector of the axion particles if their kinetic energy equaled their rest-mass energy, $\\kappa_{axion} = m c \/ \\hbar$. With these definitions, Eq.~(6) reads:\n\\begin{eqnarray}\n\\frac{d^2 u_{\\phi}}{d \\tilde{r}^2} + \\frac{1}{\\tilde{r}} \\frac{d u_{\\phi}}{d \\tilde{r}} - \\frac{(l_P - l_S)^2}{\\tilde{r}^2} u_{\\phi}  + \\tilde{ \\Delta k}^2 u_{\\phi} = \\frac{1}{\\kappa_{axion}^2} \\frac{g_{a \\gamma \\gamma}} {\\mu_0c} E_P B_S^* u_P(\\tilde{r}) u_S^*(\\tilde{r}) \\quad . \n\\end{eqnarray}\n\n\\noindent Given the parameters of the system, Eq.~(7) can now be numerically integrated to find the radial profile of the axion excitation, $u_{\\phi} (\\tilde{r})$. We look for physical solutions where the axion field is a maximum at $ \\tilde{r} =0$, which gives the following initial condition for the first derivative:  $ \\frac{d u_{\\phi}}{d \\tilde{r}} (\\tilde{r} = 0 ) =0$. We then numerically integrate Eq.~(7), with a trial initial condition $ u_{\\phi} (\\tilde{r} = 0) $. For a given initial condition, the integration will typically not result in a bounded axion field, i.e., $u_{\\phi}  (\\tilde{r} \\rightarrow \\infty ) \\neq 0 $, which is not physical. Given the parameters for the system, we vary the initial condition $ u_{\\phi} (\\tilde{r} = 0) $ until a bounded solution with $u_{\\phi}  (\\tilde{r} \\rightarrow \\infty ) = 0$ is found. \n\n\\begin{figure}[tbh]\n\\vspace{-0cm}\n\\begin{center}\n\\includegraphics[width=15cm]{axiton_mode_profiles.pdf}\n\\vspace{-0.5cm} \n\\caption{\\label{eit_scheme} \\small   Numerically calculated normalized axiton profiles, $u_{\\phi}  (\\tilde{r})$, for $ \\tilde{ \\Delta k }^ 2 = -1 $ (solid black),   $ \\tilde{ \\Delta k }^ 2 = -0.1 $ (solid red), and $ \\tilde{ \\Delta k }^ 2 = -0.01 $ (solid green), respectively. For comparison, the dashed blue line shows the mode profiles for the driving laser beams, $ u_P (\\tilde{r}) $ and $ u_S (\\tilde{r}) $. For a bounded physical solution such that $u_{\\phi}  (\\tilde{r} \\rightarrow \\infty ) = 0$, we require  $ \\tilde{ \\Delta k }^ 2 <0 $. As the quantity  $ \\tilde{ \\Delta k }^ 2 $ gets closer to 0, the axion mode profile gets broader and extends significantly beyond the confinement of the driving lasers.  }\n\\end{center}\n\\vspace{-0.5cm}\n\\end{figure}\n\n\nFigure 2 shows numerically calculated normalized axiton profiles, $u_{\\phi}  (\\tilde{r} )$, for $ \\tilde{ \\Delta k }^ 2 = -1 $ (solid black),   $ \\tilde{ \\Delta k }^ 2 = -0.1 $ (solid red), and $ \\tilde{ \\Delta k }^ 2 = -0.01 $ (solid green), respectively. For comparison, the dashed blue line shows the mode profiles for the driving laser beams, $ u_P (\\tilde{r}) $ and $ u_S (\\tilde{r})  $. Here, for simplicity, we take the profiles for the pump and Stokes laser beams to be Gaussian with unity width, $ u_P (\\tilde{r}) = u_S^* (\\tilde{r}) = \\exp{(-\\tilde{r}^2})$. We also take the angular momentum numbers for the pump and Stokes fields to be the same, $l_P=l_S$.  Due to well-known Bessel function solutions to differential equations of the from Eq.~(7), for a physical bounded solution such that $u_{\\phi}  (\\tilde{r} \\rightarrow \\infty ) = 0$, we require  $ \\tilde{ \\Delta k }^ 2 <0 $. As the quantity  $ \\tilde{ \\Delta k }^ 2 $ gets closer to 0, the axiton radial mode profile gets broader and extends significantly beyond the confinement of the driving lasers. This is well-illustrated in the solid green curve in Fig.~2. The quantity $ \\tilde{ \\Delta k }^ 2$  will likely be important parameter to tune in future experiments, since the broader the axion profile, the more it will leak to the second detection fiber as we will discuss in the detection scheme below. The numerically found initial amplitudes for the axitons, each multiplied by the scaling factor in the right hand side of Eq.~(7) (i.e., $\\frac{1}{\\kappa_{axion}^2} \\frac{g_{a \\gamma \\gamma}} {\\mu_0c} E_P B_S^*$) are, $ u_{\\phi} (\\tilde{r} = 0)= 0.23$ (for  $ \\tilde{ \\Delta k }^ 2 = -1 $), $ u_{\\phi} (\\tilde{r} = 0)= 0.48$ (for  $ \\tilde{ \\Delta k }^ 2 = -0.1 $), and $ u_{\\phi} (\\tilde{r} = 0)= 0.76$ (for  $ \\tilde{ \\Delta k }^ 2 = -0.01 $), respectively. Once the radial mode profile for the axion field is found numerically as shown in Fig.~2, the solution of Eq.~(5) gives the full description of the axiton mode. \n\n\\section{Detection of axions using the guided-wave geometry}\n\nWe have so far focused on generation of confined axitons that propagate along the fiber. Now, we discuss the second half of the problem, namely how to detect these guided axitons and prove their existence. The vision that we have for a future experiment is shown in Fig. 3. Inspired by the LSW experiments, we use  a separate fiber, which we refer to as the detection fiber. This is necessary, since any material will posses a four-wave mixing nonlinearity, which would completely overwhelm the four-wave mixing interaction mediated by the axion field. We, therefore, need to make sure that the four involved laser beams do not spatially overlap. The central idea in Fig.~3 is that the axiton mode produced in the generation fiber overlaps with the detection fiber, while the pump and Stokes laser beam profiles do not. If necessary, the extinction of the pump and Stokes lasers at the detection fiber can be guaranteed by putting a metal shield between the two fibers. In the detection fiber, the axion field mixes with the magnetic field of another laser that we refer to as the mixing laser. Through the axion interaction, the mixing laser then affects the propagation (both phase and intensity) of a probe laser beam. The search for the axion relies on detecting this change on the probe laser. \n\n\\begin{figure}[tbh]\n\\vspace{-0cm}\n\\begin{center}\n\\includegraphics[width=15cm]{axion_generation_and_detection_scheme.pdf}\n\\vspace{-0.5cm} \n\\caption{\\label{eit_scheme} \\small  The left panel shows the energy level diagram for the four-wave mixing scheme for generating and detecting the axions. The axion field, $\\phi$, produced by the pump and Stokes laser beams mix with the mixing laser, affecting the propagation of the probe laser at frequency $\\omega_0$. The four-wave mixing interaction forms a closed loop: $\\omega_P-\\omega_S+\\omega_M = \\omega_0$. The mixing and the probe lasers propagate along a separate fiber, which we refer to as the detection fiber. Inspired by the LSW experiments, the axion field produced in the generation fiber (by the pump and Stokes lasers), overlaps with the detection fiber and mediates the interaction between the mixing and probe lasers.  }\n\\end{center}\n\\vspace{-0cm}\n\\end{figure}\n\n\nSimilar to above, we will now take mixing and probe lasers to be modes of the detection fiber and assume the following forms for the two waves:\n\\begin{eqnarray}\n\\vec{E}_{probe} (r, \\varphi, z) & = & E_0(z) u_0(r) \\exp{ \\left(i l_0 \\varphi \\right)} \\exp{ \\left( i \\beta_0 z - i \\omega_0 t \\right) } \\hat{e} + c. c.  \\quad , \\nonumber \\\\\n\\vec{B}_M (r, \\varphi, z) & = & B_M u_M(r) \\exp{ \\left(i l_M \\varphi \\right)} \\exp{ \\left( i \\beta_M z - i \\omega_M t \\right) } \\hat{e} + c. c.  \\quad . \n\\end{eqnarray}\n\nHere, the quantity $B_M$ is the magnetic field amplitude for the mixing laser beam. To allow for a change in the phase and the intensity of the probe wave, we have explicitly made its electric field amplitude to be a function of $z$, $E_0(z)$. Without the axion interaction, the amplitude for the probe mode would be independent of distance, i.e., $E_0(z)=E_0(z=0)$. This amplitude gets modified due to axion interaction and the basic idea of the scheme is to infer the existence of axions using this amplitude modification. In above, the quantities $u_0(r)$ and $u_M(r)$ are the radial mode functions of the corresponding lasers (now in the detection fiber) and the integers $l_0$ and $l_M$ are the corresponding angular momentum numbers. $\\beta_0$ and $\\beta_M$ are the propagation constants of the modes along the fiber $z$ axis  and the quantities $\\omega_0$ and $\\omega_M$ are the angular frequencies. \n\nWe next solve for the propagation of the probe laser field in the presence of the axion field $\\phi$ of Eq.~(5). The detailed procedure is outlined in the Appendix~A. Briefly, we first reduce Maxwell's equations for the probe laser beam [Eq.~(2)] into a single wave equation. We then solve this wave equation in the detection fiber, while taking into account the interaction of the probe laser with the axion field and the mixing laser beam. We make several reasonable simplifications and three key assumptions: (i) {\\it Energy conservation:} we take the frequencies of the interacting laser beams in the four-wave mixing process to form a closed loop: $\\omega_P- \\omega_S + \\omega_M = \\omega_0$. (ii) {\\it Angular momentum conservation:} we take the angular momentum numbers for the four laser beams to form a closed loop as well: $l_P - l_S + l_M = l_0$. (iii) We make the Slowly Varying Envelope Approximation (SVEA) for the probe laser field, $ \\vert \\frac{d E_0} { d z } \\vert << \\beta_0 E_0$. Under these assumptions and simplifications, the propagation of the probe laser electric field amplitude, $E_0(z)$, can be reduced to a single differential equation:\n\\begin{eqnarray}\n 2 i \\beta_0 \\frac{ d E_0} {d z} = \\frac{g_{a \\gamma \\gamma}^2} {c} \\frac{1}{\\kappa_{axion}^2}  (\\omega_P - \\omega_S) \\omega_0  \\left(\\frac{1}{2 \\mu_0} B_M B_S^* \\right) E_P \\exp{[i (\\beta_P - \\beta_S + \\beta_M - \\beta_0) z]} \\quad . \n\\end{eqnarray}\n\n\\noindent Here, the quantity $\\beta_P - \\beta_S + \\beta_M - \\beta_0$ is the phase-mismatch of the four-wave mixing interaction. We next define $\\Delta k _{FWM} \\equiv \\beta_P - \\beta_S + \\beta_M - \\beta_0$ to simplify the notation in Eq.~(9) further. We also define $n_{eff} \\equiv \\frac{\\beta_0} {\\omega_0\/c}$, which can be thought as the effective refractive index for the probe laser mode as it is propagating through the detection fiber. With these definitions, the differential equation that describes the propagation of the probe laser reduces to: \n\\begin{eqnarray}\n\\frac{d E_0}{d z} =  i \\xi E_P  \\exp{(i \\Delta k_{FWM} z)} \\quad , \n\\end{eqnarray} \n\n\\noindent where the quantity $\\xi$ essentially summarizes the whole interaction and is given by:\n\\begin{eqnarray}\n\\xi= g_{a \\gamma \\gamma}^2 \\frac{1}{n_{eff}} \\frac{1}{c}  \\frac{1}{\\kappa_{axion}^2} \\left( \\frac{1}{2 \\mu_0} B_M B_S^* \\right) (\\omega_P - \\omega_S)    \\quad . \n\\end{eqnarray} \n\n\nWe next focus on the ideal phase-matched case where we assume that the mode propagation constants can be adjusted such that $\\Delta k_{FWM} \\rightarrow 0$ (the case of finite $\\Delta k_{FWM}$ is discussed in Appendix~B). In this limit, the probe propagation equation has a particularly simple form and can immediately be solved:\n\\begin{eqnarray}\n\\frac{d E_0}{d z} & = &  i \\xi E_P   \\nonumber \\\\\n\\Rightarrow E_0(L) & = & E_0(0) + i  \\xi E_P L  \\quad . \n\\end{eqnarray} \n\n\\noindent Here, $L$ is the total length of each fiber, which can be viewed as the interaction length. Equation~(12) describes the change in the electric field of the probe laser beam due to the axion interaction. Using this change in the electric field, we can also find the corresponding fractional change in the intensity of the probe laser beam:\n\\begin{eqnarray}\n\\frac{I(L)}{I(0)} = 1 + 2 \\xi L \\frac{\\Im{(E_P)}}{E_0(0)} \\quad .  \n\\end{eqnarray}\n\n\\noindent Here, for simplicity, we have taken the quantities $\\xi$ and $E_0(0)$ to be real. In Eq.~(13), the symbol $\\Im$ stands taking the imaginary part of the quantity inside the brackets. Note that by changing the phase of the pump beam, $E_P$, we can change the sign of the quantity $\\Im{(E_P)}$, and thereby control whether the probe beam will experience absorption or amplification due to axion-mediated four-wave mixing interaction. \n\n\\section{Bounds on the coupling constant}\n\nWe next discuss the bounds on the axion coupling constant that such an experiment can place, given specific experimental conditions. The sensitivity of such an experiment will critically depend on to what precision we can measure the change in the intensity of the probe laser beam. For this purpose, to first order, we assume shot-noise limited detection, with fluctuations of order $\\sqrt{N_{photon}}$, for a total detected number of probe photons of $N_{photon}$. Under these assumptions and simplifications, given an experimental set of parameters, the bound on the square of axion-photon coupling constant from Eq.~(13) is:\n\\begin{eqnarray}\ng_{a \\gamma \\gamma}^2 = \\frac{ n_{eff} \\kappa_{axion} }{2 L  \\left( \\frac{1}{2 \\mu_0} B_M B_S^* \\right) \\sqrt{N_{photon}} \\sqrt{\\frac{P_P}{P_0}}} \\quad .\n\\end{eqnarray}\n\n\\noindent Here, the quantity  $\\frac{1}{2 \\mu_0} B_M B_S^*$ can be thought as the magnetic energy density, $P_P$ is the optical power in the pump laser and $P_0$ is the optical power in the probe beam. The details of deriving Eq.~(14) is given in Appendix~B.  The dependence of the coupling constant on the probe refractive index, $n_{eff}$, is in the spirit of enhanced nonlinearities in near-zero index materials \\cite{caspani,liberal}.  \n\n We next evaluate the bounds on the coupling constant for four envisioned phases of the experiment. The parameters that are used in these four phases are given below, in Table-I. The envisioned parameters for the lasers are well within the current the state of the art of high-power fiber lasers \\cite{laser_review}. We envision that the parameters that are used in the first two phases of the experiment ({\\it phase-1} and {\\it phase-2}) can be achieved in a few year time-scale, while the last two phases of the experiment ({\\it phase-3} and {\\it phase-4}) can be performed within the next 5 to 10 years. There will likely be two main systematics in the experiment: (1) Optical four-wave mixing interaction due to possible residual leakage of the pump and Stokes laser beams to the detection fiber. (2) The change in the intensity of the probe laser beam due to off-resonant Raman scattering in the detection fiber.  To put the parameters that are listed in Table~1 into perspective, the ALPS~II project, which is the next generation LSW experiment currently at construction at DESY, will use a cavity with a finesse of $F=4 \\times 10^4$, and a cavity length of $L=88$~m, with a quite stringent DC magnetic-field of 5.3~T to be maintained within the whole length using dipolar magnets \\cite{review}. The first results from the ALPS~II project is expected in 2022. \n\n\n\\vspace{0.5cm}\n\n\\begin{table}\n\n\\begin{tabular}{ || c | c | c | c | c ||}\n\\hline\nParameter & Phase-1 & Phase-2 & Phase-3 & Phase-4 \\\\\n\\hline\nLength of the fiber ($L$) & $1$~km & $10$~km & $100$~km & $1000$~km \\\\ \n\\hline\nOptical power in probe ($P_0$) & $1$~mW & $10$~mW & $100$~mW & $1$~W  \\\\\n\\hline\nIntegration time ($T$) & $100$~s & $10^3$~s & $10^4$~s & $10^6$~s \\\\\n\\hline\nOptical power in pump ($P_P$) & $1$~W & $10$~W & $100$~W & $10$~kW \\\\\n\\hline\nOptical intensity of Stokes  ($I_S$) & $0.1$~GW\/cm$^2$ & $1$~GW\/cm$^2$ &  $10$~GW\/cm$^2$ &  $100$~GW\/cm$^2$ \\\\\n\\hline \nOptical intensity of mixing ($I_M$) & $0.1$~GW\/cm$^2$ & $1$~GW\/cm$^2$ &  $10$~GW\/cm$^2$ &  $100$~GW\/cm$^2$ \\\\\n\\hline\nThe refractive index for probe ($n_{eff}$) & $1$ & $10^{-1}$ & $10^{-2}$ & $10^{-4}$ \\\\\n\\hline\n\\end{tabular}\n\n\\caption{The set of parameters that are used for the four envisoned phases of the experiment.}\n\n\\end{table}\n\n\\vspace{0.5cm}\n\nThe calculated bounds for the axion-photon coupling for these four envisioned phases of the experiment are shown in Fig.~4. For comparison the predicted sensitivities of the next generation LSW experiment (ALPS~II), as well  the next generation solar helioscope (IAXO) are also plotted (dashed brown lines) \\cite{review}. Solar helioscope IAXO is proposed to utilize 25-m-long, 5.2-m-diameter toroid magnet assembly. The envisioned {\\it phase-4} experiment of our scheme (solid black line) is quite competitive with planned ALPS~II and IAXO \\cite{review}. \n\n\\begin{figure}[tbh]\n\\vspace{0cm}\n\\begin{center}\n\\includegraphics[width=19cm]{bounds_for_four_phases.pdf}\n\\vspace{-1cm} \n\\caption{\\label{eit_scheme} \\small  The calculated detection sensitivity for the axion-photon coupling constant for four different phases of the experiment. The {\\it phase-4} experiment is quite competitive with several planned experiments, such as the next generation LSW experiment (ALPS~II) and the next generation solar helioscope (IAXO).   }\n\\end{center}\n\\vspace{-0cm}\n\\end{figure}\n\n\n\\section{Conclusions and future directions}\n\nIn summary, we have suggested a new approach for generating and detecting axions using lasers in a guided wave geometry. As we mentioned above, there are important advantages of our scheme including the ability to scan a wide mass-range for the axion, and the long interaction length. Furthermore, our scheme does not rely on interaction of the axions with a Tesla level DC magnetic field. The calculations of Fig.~4 predict that our scheme can achieve sensitivities quite competitive to several planned experiments in the near future. While the parameters that are used for the {\\it phase-4} calculation in Fig.~4 are stringent, they are within the state of the art. \n\nWe believe there are several promising avenues for future research which may result in significant enhancement of the detection sensitivity in our scheme: (1) The interaction length can further be enhanced by integrating a high-finesse cavity with the detection and generation fibers. Using a distributed Bragg reflector at each end of the fiber (which may be achieved by modulating the refractive index over a section near the beginning and the end each fiber), the lasers can be made to bounce back and forth along the fiber, thereby significantly increasing the interaction length. For this case, the sensitivity bounds for the axion-photon coupling constant $g_{a \\gamma \\gamma}$ would reduce by a factor $\\sqrt{F}$, where $F$ is the cavity finesse. (2) For the calculation of Fig.~4, we assumed shot noise limited detection for the probe laser beam. If squeezed light for the probe laser beam is used, the fluctuations in either the phase or the intensity of the probe laser can be reduced substantially below the shot-noise limit, thereby resulting in improved sensitivity for axion detection. (3) To our knowledge, our work is the first to suggest using a low refractive index for the probe laser beam to increase the detection sensitivity. It may also be possible to utilize metamaterials in other ways to increase the detection sensitivity. For example, waveguides with a high effective magnetic susceptibility may be used to increase the magnetic field values for the Stokes and Mixing lasers [i.e., it may be possible to enhance the values of the quantities $B_S$ and $B_M$ in Eq.~(14)] \\cite{caspani,liberal}. \n\n\nWe thank Ben Lemberger for many helpful discussions. This work was supported by the University of Wisconsin-Madison through the Vilas Associates Award. \n\n\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThere has been significant interest, especially in recent years, to develop\nspatially periodic band gap materials and structures, based upon Floquet-Bloch\ntheory \\citep{floquet1883, bloch1929}. Recent developments in the field of\narchitectured materials aimed at achieving novel mechanical properties often\nrely on enhancements that include effects neglected by classical theories.\nContinuum models with local microstructural interactions have become\nincreasingly popular after the advance and growth in the field of metamaterials,\nas summarized in the monograph by  \\cite{book:banerjee_metamaterials}. A family\nof models that has regained popularity in the last few years is the so-called\nCosserat-based theories, which are mainly founded on the formulation by\n\\cite{Cosserat1909}. In a wide sense, these material models consider\nmicrostructural effects through a generalization of Cauchy's postulate to\ninclude additional mechanical interactions involving couples per unit surface\nor couple-stresses. In the present work, we focus on a pure continuum mechanics\nrepresentation by widening the modeling capabilities of consistent-couple stress\ntheory (C-CST), originally formulated in \\cite{hadjesfandiari2011couple}. In\nparticular, we establish for the first time a principle of stationary correlated\naction for the corresponding reduced wave equation of elastodynamics and extend\nthe theory to spatially periodic materials, thus providing an objective physical\nbasis to characterize material through its dispersive behaviour.\n\nThe entire family of Cosserat elasticity models depart from the classical Cauchy\nmodels in the consideration of microstructural effects, which are unavoidably\nexpected to occur once the specimen dimensions become comparable to the material\nmicrostructural features. These effects cannot be addressed in classical\ntheories. On the other hand, microstructural effects are introduced through the\nextension of dynamic and kinematic descriptors from classical continuum mechanics\non a range of alternative models. \\cite{Voigt1910} was probably the first to\npostulate a model with asymmetric mechanical interactions in terms of\ncouple-stresses: the interaction between two material points in this continuum\nencompassed couples per unit contact surface in addition to the classical Cauchy\nforces per unit surface. In a landmark contribution, the Cosserat brothers\n\\citep{Cosserat1909} formulated a mathematical theory involving couple-stresses\nin which new kinematic variables were introduced in the form independent\nmicro-rotations. Various later extensions from these theories were also\ndeveloped by \\cite{Eringen1966, book:nowacki1986, Mindlin1964,\neringen1964nonlinear} in micropolar, microstretch and micromorphic theories. \nAlongside, a different branch of developments resulted in a set of couple stress\ntheories in the work by \\cite{Toupin1962}, \\cite{MindlinAndTiersten1962} and\n\\cite{Koiter1964}, who used the gradients of the true continuum rotation field\nto provide the required kinematic enrichment.\n\nDevelopments from \\cite{hadjesfandiari2011couple} have resulted in a\nconsistent version of the models by \\cite{Toupin1962},\n\\cite{MindlinAndTiersten1962} and \\cite{Koiter1964} in terms of couple-stresses.\nThe consistency of this model is reflected in the determinacy of all the\nforce-stress and couple-stress components, the identification of the necessary\nand sufficient set of natural and essential boundary conditions and the\nelimination of redundant force components. An approach for evaluating the\nusefulness and robustness of a continuum mechanics model is through the\ndetermination of its band structure in terms of its dispersion relationships.\nThese indicate the kinematic response of the material through an identification\nof the wave propagation modes that can exist within the model and the frequency\ndependency of the group and phase velocities of these potential waves. An\neffective technique, relying on the assumption of spatial periodicity, is based\non Bloch's theorem from solid state physics \\citep{book:brillouin}, where the\nproblem of finding the band structure reduces to solving a series of generalized\neigenvalue problems for a variation of the wave vector in the reciprocal space.\nIn the case of the C-CST model, this problem poses several computational\nchallenges. First, since the enriched kinematic variables are now curvatures,\ncorresponding to particular second order gradients of the displacement field,\nthe displacement-based finite element formulation now would require \\(C^1\\)\ninterelement continuity. As shown by \\cite{darrall2014}, this numerical issue\ncan be resolved by introducing Lagrange multiplier techniques, however it is\nnot obvious how to incorporate these within Bloch analysis. Second, as a result\nof enforcing the kinematic constraint in terms of Lagrange multipliers, the\ncomputational framework lacks inertial components associated with the rotational\ninteractions. Since there is only a mass matrix associated with the\ntranslational degrees of freedom, special attention is needed in solving the\neigenproblem. Both of these issues are resolved in the present\nwork.\n\nThe characterization of the bulk properties of periodic materials is commonly\ndone finding the band structure or dispersion relations \\citep{hussein2014dynamics}.\nCommonly, this band structure is obtained using a numerical method such as \nthe Boundary Element Method \\citep{li2013bandgap, li2013boundary}, the Finite\nDifference Method \\citep{tanaka2000band, su2010postprocessing,\nisakari2016periodic}, the Finite Element Method \\citep{langlet95, guarin2015, \nvalencia_uel_2019, mazzotti2019modeling, guarin2020,chin2021spectral},  or\nthe Plane Wave Expansions \\citep{cao2004convergence, xie2017improved,\ndal2020elastic}. We favor the use of the Finite Element Method because of its\nmaturity and versatily to represent arbitrary geometries and boundary\nconditions. In this work, we find the dispersion relations modeling\na single unit cell of the material and using Bloch's theorem. There have been\nfew works on periodic materials involving generalized continua and these have\nbeen related to micropolar elasticity \\citep{zhang2018, guarin2020}. To the best\nof our knowledge, this is the first work using a higher-order elasticity model\nfor phononic crystals.\n\nHere we establish a new variational principle in the temporal frequency domain\nfor reduced couple stress elastodynamics and then extend the finite element\nalgorithm from \\cite{darrall2014} to the case of spatially periodic material\ncells with Bloch boundary conditions. We examine first the closed form\ndispersion relationships for the homogeneous version of the model. This\nhomogeneous model already involves micromechanical effects through a length\nscale material parameter, however additional effects can be considered in terms\nof explicit representations of geometric features at the fundamental material\ncell level. We then formulate a variational statement together with the\nimposition of an extended version of the usual Bloch periodic boundary\nconditions that satisfies Hermiticity and positive definiteness for C-CST.\nSubsequently, this statement is modified by introducing an artificial\nindependent rotation field tied to the continuum displacement field through the\nenforcement of a Lagrange multiplier field that is shown to equal the\nskew-symmetric part of the force-stresses. The resulting numerical framework is\ntested by comparing its results with those obtained in closed form\nfor the homogeneous case and by applying it to a porous periodic material cell\ndesign, which displays interesting bandgap behavior that has\nnot been resported previously.\n\n\n\\section{Governing equations}\n\n\\subsection{Forces and moments in the C-CST solid}\nThe fundamental signature of the extended continuum model considered in this\nwork is the presence of rotational mechanical interaction, in addition to the\nclassical translational interaction between material points in the continuum.\nFollowing a generalized Cauchy's postulate \\citep{MindlinAndTiersten1962,\nKoiter1964} we define force and couple traction vectors \\(t_i^{(\\hat{n})}\\)\nand \\(m_i^{(\\hat{n})}\\) respectively as\n\\begin{subequations}\n\\label{eq:tracts}\n\\begin{align}\n        &t_i^{(\\hat{n})} = \\lim_{\\Delta S(\\hat{n}) \\rightarrow 0} \\frac{\\Delta \n        R_i}{\\Delta S(\\hat{n})}\\label{eq:for_T} \\\\\n        &m_i^{(\\hat{n})} = \\lim_{\\Delta S(\\hat{n}) \\rightarrow 0} \\frac{\\Delta \n        M_i}{\\Delta S(\\hat{n})} \\, , \\label{eq:for_M}       \n\\end{align}\n\\end{subequations}\nand where \\(\\Delta S(\\hat{n})\\) is a small element of area oriented with unit\nnormal \\(\\hat{n}\\) while \\(\\Delta R_i\\) and \\(\\Delta M_i\\) are the resultant\nforce and couple moment, respectively. However, only the tangential components of\n\\(m_i^{(\\hat{n})}\\) exist as independent bending couple tractions\n(\\cref{fig:cauchy_principle}).\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=5 cm]{img\/cauchy_principle}\n\\caption{In the C-CST model mechanical effects are described through linear and\nrotational interactions in terms of resultant forces and moments at the material\npoint. These resultants act on the surface element \\(\\Delta S\\) over a plane\nwhose unit outward normal is \\(\\hat{n}\\).}\n\\label{fig:cauchy_principle}\n\\end{figure}\n\nNote that while the force-tractions vector \\(t_i^{(\\hat{n})}\\) is a polar vector,\nthe couple-tractions vector \\(m_i^{(\\hat{n})}\\) is an axial vector.\nForce-tractions and couple-tractions are also described by projections of the\nnon-symmetric force-stress tensor \\(\\sigma_{ij}\\) and the couple-stress\n\\(\\mu_{ij}\\) tensors according to:\n\\begin{subequations}\n\\label{eq:tensions}\n\\begin{align}\n        &t_i^{(\\hat{n})} = \\sigma_{ji} n_j\\, ,\\label{eq:for_sigma}\\\\\n        &m_i^{(\\hat{n})} = \\mu_{ji} n_j=\\epsilon_{ijk}\\mu_k n_j\\, , \\label{eq:cou_sigma}       \n\\end{align}\n\\end{subequations}\nwhere \\(\\mu_{ij}\\) is skew-symmetric. Thus, \\(\\mu_{ij} = -\\mu_{ji}\\) and the\ncouple-stress tensor can be written as a polar vector with\n\\[\\mu_k = \\frac{1}{2}\\epsilon_{kji}\\mu_{ji}\\, ,\\]\nwhere \\(\\epsilon_{ijk}\\) is the Levi-Civita permutation symbol leading to the\nlast form in \\eqref{eq:cou_sigma}, which clearly shows that \\(m_i^{(\\hat{n})}\\)\nis tangential to the surface.\n\nConsideration of the linear and angular balance equations for an arbitrary part\nof the material continuum of volume \\(V\\), bounded by external surface \\(S\\)\nleads to the following force-stress and couple-stress equilibrium equations for\nthe C-CST model:\n\\begin{equation}\n\\begin{split}\n&\\sigma_{ji,j} + f_i = \\rho \\ddot{u}_i\\, ,\\\\\n&\\mu_{ji,j} + \\epsilon_{ijk}\\sigma_{jk} = 0\\, ,\n\\end{split}\n\\label{eq:balance_eqs}\n\\end{equation}\nwhere \\(f_i\\) are forces per unit volume, and \\(\\rho\\) is the mass density.\nNotice that, in contrast to micropolar models \\citep{guarin2020} where there is\na rotational inertial density and a body couple term, in this model the balance\nequations already include those contributions. This particular aspect of the\nC-CST model is discussed in the original paper by \\cite{hadjesfandiari2011couple}\nwhere it is also proved that from\n\\begin{equation}\n\\epsilon_{ijk}(\\mu_{k,j} + \\sigma_{jk}) = 0\\, ,\n\\label{eq:skew_symmetry}\n\\end{equation}\nit follows that \\(\\mu_{k,j} + \\sigma_{jk}\\) is symmetric and as a result its\nskew-symmetric part is zero leading to\n\\[\\sigma_{[ji]} = - \\mu_{[i,j]}\\, .\\]\nThis gives the skew-symmetric part of the force-stress tensor in terms of the\ncouple-stress vector, which also can be described by its dual vector\nrepresentation\n\\[s_i = \\frac{1}{2} \\epsilon_{ijk} \\mu_{k, j}\\, .\\]\n\n\n\\subsection{Kinematics and constitutive relations}\nIn the linear C-CST model, kinematics is described by the classical infinitesimal\nstrain (\\(e_{ij}\\)) and rotation (\\(\\theta_{ij}\\)) tensors\n\\begin{subequations}\n\\label{eq:kinematics}\n\\begin{align}\ne_{ij} = \\frac{1}{2}(u_{i,j} + u_{j,i})\\, , \\label{eq:strain}\\\\\n\\theta_{ij} = \\frac{1}{2}(u_{i,j} - u_{j,i})\\, , \\label{eq:rotation}\n\\end{align}\n\\end{subequations}\nand by the mean curvature tensor\n\\begin{equation}\n\\kappa_{ij} = \\frac{1}{2}(\\theta_{i,j} - \\theta_{j,i})\\, ,\n\\label{eq:curv_tensor}\n\\end{equation}\nwhere\n\\[\\theta_i = \\frac{1}{2}\\epsilon_{ijk}\\theta_{kj}\\, .\\]\n\nEquation \\eqref{eq:curv_tensor} can also be written in polar form, as an\nengineering curvature vector \\citep{darrall2014}\n\\begin{equation}\n\\kappa_i = \\epsilon_{ijk} \\theta_{j,k} =\\frac{1}{2}(u_{i,kk} - \nu_{k,ik})\n\\label{eq:curv}\n\\end{equation}\nsince\n\\[\\kappa_i = \\epsilon_{ijk}\\kappa_{jk}\\, .\\]\n\nFor a linear elastic centrosymmetric C-CST continuum, the constitutive equations\ncan be written as\n\\begin{equation}\n\\begin{split}\n&\\sigma_{(ij)} = C_{ijkl} e_{kl}\\, ,\\\\\n&\\mu_i = D_{ij} \\kappa_j\\, ,\n\\end{split}\n\\label{eq:constitutive_aniso}\n\\end{equation}\nwhere \\(C_{ijjkl}\\) is the stiffness tensor as in classical (anisotropic)\nelasticity, and \\(D_{ij}\\) is an additional material tensor that accounts for \ncouple-stress effects. In the expressions above, parentheses as subindices are\nused to indicate the symmetric part of the tensor. In the case of a linear\nisotropic elastic C-CST continuum,\n\\begin{equation}\n\\begin{split}\n&C_{ijkl} = \\lambda \\delta_{ij} \\delta_{kl} + \\mu ( \\delta_{ik} \\delta_{jl}\n   + \\delta_{il} \\delta_{jk} ), \\\\\n&D_{ij} = 4\\eta \\delta_{ij},\n\\end{split}\n\\label{eq:isotropic_tensors}\n\\end{equation}\nwhere \\(\\mu\\) and \\(\\lambda\\) are the Lam\u00e9 parameters as in classical \nelasticity, while \\(\\eta\\) is the additional material coefficient that accounts\nfor couple-stress effects. Then, the constitutive equations for isotropy can be\nsimplified to\n\\begin{equation}\n\\begin{split}\n&\\sigma_{(ij)} = \\lambda e_{kk} \\delta_{ij} + 2\\mu e_{ij}\\, ,\\\\\n&\\mu_i = 4\\eta \\kappa_i\\, .\n\\end{split}\n\\label{eq:constitutive}\n\\end{equation}\n\n\n\\subsection{Displacement equations of motion}\nAt this point it may be convenient to alternate between index and explicit\nvector notation. In the latter, the gradient operator reads\n\\(\\nabla = \\frac\\partial{\\partial x_i}\\) in Cartesian coordinates. In these\nterms, the time domain displacement equations of motion are obtained after\nusing the constitutive relations \\eqref{eq:constitutive} in the equilibrium\nequations \\eqref{eq:balance_eqs} yielding\n\\begin{equation}\n(\\lambda + 2\\mu)\\nabla (\\nabla \\cdot \\mathbf{u}) - \\mu\\nabla\\times \\nabla \n\\times \\mathbf{u} + \\eta \\nabla^2 \\nabla \\times \\nabla \\times \\mathbf{u} = \n\\rho\\ddot{\\mathbf{u}}\\, .\n\\label{eq:wave_eq}\n\\end{equation}\n\n\nDefining the phase\/group speed for the longitudinal (P) wave \\(c_1\\) (which is\nnot dispersive), the low-frequency (\\(k\\rightarrow 0\\)) phase\/group speed for\nthe transverse wave (S) \\(c_2\\) (which is dispersive) and the intrinsic material\nlength scale parameter \\(l\\) (which is not present in classical elasticity),\nsuch that\n\\begin{equation}\nc_1^2 = \\frac{\\lambda + 2\\mu}{\\rho}\\, ,\\qquad\nc_2^2 = \\frac{\\mu}{\\rho}\\, ,\\qquad\nl^2 = \\frac{\\eta}{\\mu}\\, ,\n\\label{eq:c1c2l2}\n\\end{equation}\nallows us to write \\eqref{eq:wave_eq} in the form\n\\begin{equation}\nc_1^2 \\nabla (\\nabla \\cdot \\mathbf{u}) - c_2^2 (1-l^2 \\nabla^2) \\nabla\\times \\nabla \\times \n\\mathbf{u} = \\ddot{\\mathbf{u}}\\, .\n\\label{eq:wave_eq_speeds}\n\\end{equation}\n\n\n\\subsection{Dispersion relations for unbounded domains}\nUsing a Helmholtz decomposition,  the displacement field can be written in terms\nof the scalar and vector potentials \\(\\varphi\\) and \\(\\mathbf{H}\\)\n\\citep{book:arfken} as\n\\[\\mathbf{u} = \\nabla\\varphi + \\nabla\\times\\mathbf{H}\\, ,\\quad \n\\nabla\\cdot\\mathbf{H} = 0\\, ,\\]\nand replacing this in \\eqref{eq:wave_eq_speeds} gives the following set of\nuncoupled wave equations\n\\begin{align}\n&c_1^2\\nabla^2\\varphi = \\ddot{\\varphi}\\, ,\\\\\n&c_2^2(1 - l^2\\nabla^2)\\mathbf{H} = \\ddot{\\mathbf{H}}\\, ,\n\\end{align}\nwhere it is observed that the equation for the rotational potential follows a\nhigher-order wave equation that is inherently dispersive.  This becomes evident\nafter assuming a solution of the form \\(\\mathbf{u} = \\mathbf{\\tilde u}e^{ik x - \ni\\omega t}\\) which gives the dispersion relations\n\\begin{align}\\label{eq:dispersion}\n&\\omega_P^2 = c_1^2 k^2\\, ,\\\\\n&\\omega_S^2 = c_2^2 k^2 (1 + k^2 l^2)\\, . \\label{eq:dispersion_SV}\n\\end{align}\n\nSolving the above for \\(k\\), we have in each case\n\\begin{align*}\nk_P^2 &= \\frac{\\omega^2}{c_1^2}\\, , &k_S^2 &= \\frac{1}{2 l^2}\\left[\\pm\\sqrt{1 \n+ \\frac{4\\omega^2 l^2}{c_2^2}} - \n1\\right]\\, .\n\\end{align*}\n\nNoticing that the quantity inside the square root is always greater than 1\nindicates that we should consider only the positive root, while the negative\nroot corresponds to an evanescent wave that should arise under certain boundary\nconditions. The phase and group speeds are now given by\n\\begin{equation}\n\\begin{aligned}\nv_P &= c_1\\, , & g_P &= c_1\\, ,\\\\\nv_S(k) &= c_2\\sqrt{1 + k^2 l^2}\\, , & g_S &= c_2 \\frac{1 + 2k^2 l^2}{\\sqrt{1 + \nk^2 l^2}}\\, .\n\\end{aligned}\n\\label{eq:speeds}\n\\end{equation}\n\nTaking the low and high frequency limits \\(k \\rightarrow 0\\) and\n\\(k \\rightarrow \\infty\\) gives\n\\begin{align*}\n&\\lim_{k \\rightarrow 0} v_S = \\lim_{k \\rightarrow 0} g_S = \nc_2\\, ,\\\\\n&\\lim_{k \\rightarrow \\infty} v_S = \\lim_{k \\rightarrow \\infty}g_S \n\\rightarrow  \\infty\\, ,\n\\end{align*}\nwhich shows how the speed of energy flow increases with frequency. All of these \nrelations are displayed in \\cref{fig:dispersion_analytic}.\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[height=1.8 in]{img\/disp_cst_freqs.pdf}\n    \\includegraphics[height=1.8 in]{img\/disp_cst_phase_speed.pdf}\n    \\includegraphics[height=1.8 in]{img\/disp_cst_group_speed.pdf}\n    \\caption{Dispersion relations for a homogeneous C-CST material \n        with properties: \\(\\rho = 1\\times 10^5\\), \\(\\lambda=2.8 \\times 10^{10}\\),\n        \\(\\eta=1.62 \\times 10^9\\), \\(\\mu = 4\\times 10^9\\). The plot  on \n        the left shows the frequency-wave number relation for the \n        non-dispersive P-wave (continuous line) and the dispersive SV (dashed\n        lines). The plots in the middle and right part of the  figure show the\n        phase and group speeds for the dispersive modes.}\n    \\label{fig:dispersion_analytic}\n\\end{figure}\n\n\n\\subsection{Frequency domain equations}\nBloch analysis considering spatial periodicity of the material is naturally\nconducted in the Fourier domain, involving both the temporal frequencies and\nspatial wave numbers.  After performing a Fourier transform of the linear and\nangular momentum equations  \\eqref{eq:balance_eqs} to the temporal frequency\ndomain, these become\n\\begin{equation}\n\\begin{split}\n&\\tilde\\sigma_{ji,j} + \\tilde f_i = - \\rho \\omega^2 \\tilde{u}_i\\, ,\\\\\n&\\tilde\\mu_{ji,j} + \\epsilon_{ijk}\\tilde\\sigma_{jk} = 0\\, ,\n\\end{split}\n\\label{eq:fourier_balance_eqs}\n\\end{equation}\nwhere the superposed tilde denotes a complex Fourier amplitude.\n\nAfter introducing the constitutive equations \\eqref{eq:constitutive_aniso} for\ncentrosymmetric materials into \\eqref{eq:fourier_balance_eqs} and then combining\nthe angular momentum and linear momentum balance laws into a single set in terms\nof displacement, one finds:\n\\begin{equation}\n( C_{ijkl} \\tilde u_{k,l} )_{,j} + \\frac{1}{4}\\epsilon_{pij}\\epsilon_{pmn}\\{D_{nk} (\\tilde u_{k,ll}-\\tilde u_{l,kl})\\}_{,mj} + f_i = -\\rho \\omega^2 \\tilde u_i\n\\label{eq:fourier_disp_aniso}\n\\end{equation}\n\nSubstituting \\eqref{eq:isotropic_tensors} and \\eqref{eq:c1c2l2} for isotropic\nmaterials into \\eqref{eq:fourier_disp_aniso} provides the corresponding Fourier\ndomain reduced wave equations in the absence of body forces, which can be written\n\\begin{equation}\nc_1^2 \\nabla (\\nabla \\cdot \\mathbf{\\tilde u}) - c_2^2 (1-l^2 \\nabla^2) \\nabla\\times \\nabla \\times\n\\mathbf{\\tilde u}  = -\\omega^2 \\mathbf{\\tilde u}\\, .\n\\label{eq:wave_eq_freq}\n\\end{equation}\nNotice that \\eqref{eq:wave_eq_freq} is the temporal Fourier transform of\n\\eqref{eq:wave_eq_speeds}.\n\n\\section{Variational principles}\nWe will describe next a variational formulation for the elastodynamic C-CST\nmodel. An inherent complexity is the presence of second order displacement\ngradients arising in the curvatures \\eqref{eq:curv}, which requires \\(C^1\\)\ncontinuity of the displacement field.\n\nLet us consider a volume \\(V\\) with boundary \\(S\\), having specified body forces\n\\(f_i\\), force-tractions \\(t_i\\), and couple-tractions \\(m_i\\) (see\n\\cref{fig:domain}). The boundary \\(S\\) is split into different segments, where\n\\(S_u\\) represents the portion of \\(S\\) with specified displacements, \\(S_t\\)\nrepresents the surface with prescribed tractions, \\(S_\\theta\\) represents the\nsegment with enforced rotations, and \\(S_m\\) the boundary with prescribed\ncouple-tractions. Additionally, \\(S = S_u \\cup S_t = S_\\theta \\cup S_m\\) and\n\\(S_u \\cap S_t = S_\\theta \\cap S_m = \\emptyset\\). In general, \\(S_u\\) and \\(S_t\\)\nmight overlap with \\(S_\\theta\\) and \\(S_m\\). This is an important aspect of the\nC-CST model that is relevant in the solution of boundary value problems, as we\nshall see later.\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=4 in]{img\/domain.pdf}\n    \\caption{Schematic representation of the domain and boundary conditions for the C-CST model.}\n    \\label{fig:domain}\n\\end{figure}\n\nWe begin with the following couple stress elastodynamic action functional in the\nfrequency domain, as an extension of the elastostatic formulations introduced\nby \\cite{hadjesfandiari2011couple} and \\cite{darrall2014}:\n\\begin{equation}\n\\mathcal{A}[u;\\omega] = \\mathcal{U}[u;\\omega]+ \\mathcal{T}[u;\\omega]+\\mathcal{V}[u;\\omega],\n\\label{eq:Lagrangian_functional}\n\\end{equation}\nHere, and in the remainder of this paper, the superposed tilde has been\nsuppressed for notational convenience.  Meanwhile, the elastic, kinetic and\napplied load actions can be written in explicit form, respectively, as\n\\begin{equation}\n\\mathcal{U}[u;\\omega] = \\frac{1}{2}\\int\\limits_{V} e_{ij}^* C_{ijkl} e_{kl} \\dd{V}\n+ \\frac{1}{2}\\int\\limits_{V} \\kappa_{i}^* D_{ij} \\kappa_{j} \\dd{V},\n\\label{eq:delta_functional_U}\n\\end{equation}\n\\begin{equation}\n\\mathcal{T}[u;\\omega] = -\\frac{\\omega^2}{2} \\int\\limits_{V} u_i^* \\rho u_i \\dd{V},\n\\label{eq:delta_functional_T}\n\\end{equation}\n\\begin{equation}\n\\mathcal{V}[u;\\omega] =- \\int\\limits_{V} u_i^* f_i \\dd{V}\n- \\int\\limits_{S_t}  u_i^* t_i^{(\\hat{n})} \\dd{S}  - \\int\\limits_{S_m} \\theta_i^* m_i^{(\\hat{n})} \\dd{S}. \n\\label{eq:delta_functional_V}\n\\end{equation}\nwith the asterisk denoting complex conjugate.\n\nThe stationarity of this action becomes\n\\begin{equation}\n\\delta \\mathcal{A}[u;\\omega] = \\delta\\mathcal{U}[u;\\omega]\n+ \\delta\\mathcal{T}[u;\\omega]+\\delta\\mathcal{V}[u,\\omega]) = 0,\n\\label{eq:action_variation}\n\\end{equation}\nor\n\\begin{equation}\n\\begin{split}\n&\\delta\\mathcal{A}[u;\\omega] = \n \\int\\limits_{V} \\delta e_{ij}^* C_{ijkl} e_{kl} \\dd{V}\n + \\int\\limits_{V} \\delta\\kappa_{i}^* D_{ij} \\kappa_{j} \\dd{V}\n -\\omega^2 \\int\\limits_{V} \\delta u_i^* \\rho u_i \\dd{V} \\\\\n&- \\int\\limits_{V} \\delta u_i^* f_i \\dd{V}\n- \\int\\limits_{S_t}  \\delta u_i^* t_i^{(\\hat{n})} \\dd{S}\n- \\int\\limits_{S_m} \\delta\\theta_i^* m_i^{(\\hat{n})} \\dd{S} = 0, \\,\n\\label{eq:delta_action_functional}\n\\end{split}\n\\end{equation}\nwhich can serve as the weak form for a finite element formulation in reduced\nelastodynamics.  With the appearance of mean curvature in\n\\eqref{eq:delta_action_functional}, this would require \\(C^1\\) spatial continuity\nof displacements \\(u_i\\).\n\nNext, let us derive the Euler-Lagrange equations associated with the functional\n\\(\\mathcal{A}[u; \\omega]\\). Starting from the first variation in\n\\eqref{eq:delta_action_functional}, we repeatedly apply integration-by-parts\noperations and the divergence theorem to shift all of the spatial derivatives\nfrom the variations to the true fields. This leads to the following\nstatement:\n\\begin{equation}\n\\begin{split}\n&\\int\\limits_{V} \\delta u_i^* [ ( C_{ijkl} u_{k,l} )_{,j}\n  + \\frac{1}{4}\\epsilon_{pij}\\epsilon_{pmn}\\{D_{nk} (u_{k,ll}-u_{l,kl})\\}_{,mj}\n  + f_i + \\rho \\omega^2 u_i] \\dd{V} \\\\\n&+ \\int\\limits_{S_t}  \\delta u_i^* [ t_i^{(\\hat{n})} - \\sigma_{ji} n_j ] \\dd{S} \\\\\n&+ \\int\\limits_{S_m} \\delta\\theta_i^* [ m_i^{(\\hat{n})} -  \\epsilon_{ijk}\\mu_k n_j ] \\dd{S} = 0. \\, \n\\label{eq:delta_action_E-L_functional}\n\\end{split}\n\\end{equation}\nFor arbitrary variations, each set of terms inside the square brackets must be\nzero. Thus, the Euler-Lagrange equations can be written:\n\\begin{equation}\n( C_{ijkl} u_{k,l} )_{,j} + \\frac{1}{4}\\epsilon_{pij}\\epsilon_{pmn}\\{D_{nk} (u_{k,ll}-u_{l,kl})\\}_{,mj}\n  + f_i = -\\rho \\omega^2 u_i \\quad {\\rm in} \\ V \\\\\n\\label{eq:action_E-L_aniso_momentum}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\n&t_i^{(\\hat{n})} = \\sigma_{ji} n_j \\quad {\\rm on} \\ S_t \\\\\n&m_i^{(\\hat{n})} = \\epsilon_{ijk}\\mu_k n_j \\quad {\\rm on} \\ S_m \\,\n\\label{eq:action_E-L_aniso_traction}\n\\end{split}\n\\end{equation}\nNotice that \\eqref{eq:action_E-L_aniso_momentum} are the reduced wave equations\nfrom \\eqref{eq:fourier_disp_aniso}, while \\eqref{eq:action_E-L_aniso_traction}\nrepresent the corresponding natural boundary conditions for C-CST.  In the\nisotropic case, substituting \\eqref{eq:isotropic_tensors} into\n\\eqref{eq:action_E-L_aniso_momentum} produces\n\\begin{equation}\nc_1^2 u_{j,ji}-c_2^2 \\epsilon_{ijk} \\epsilon_{kmn} (u_{n,mj}-l^2 u_{n,mjll})\n+ f_i = -\\omega^2 u_i\\quad {\\rm in} \\ V\n\\label{eq:action_E-L_iso_momentum}\n\\end{equation}\nwhich is the equivalent of \\eqref{eq:wave_eq_freq} in index notation.\n\nPerforming an inverse Fourier transform of individual terms in\n\\eqref{eq:Lagrangian_functional}-\\eqref{eq:delta_functional_V} back to the time domain, one finds\n\\begin{equation}\n\\mathcal{F}^{-1}[u^* v] = (u \\star v)(t)\n\\label{eq:time_domain_functional}\n\\end{equation}\nwhere the $\\star$ operator denotes correlation over time, such that\n\\begin{equation}\n(u \\star v)(t) = \\int_{-\\infty}^\\infty u(\\tau) v(t+\\tau) d\\tau.\n\\label{eq:correlation}\n\\end{equation}\n\nConsequently, we have established the following stationary {\\it Principle of\nCorrelated Action} for couple stress elastodynamics: Of all the possible\ndisplacement fields in \\(V\\) that satisfy the frequency domain kinematic boundary\nconditions on \\(S_u\\) and \\(S_\\theta\\), the one that renders the action\n\\(\\mathcal{A}[u; \\omega]\\) in \\eqref{eq:Lagrangian_functional} stationary\ncorresponds to the solution of the reduced wave equations\n\\eqref{eq:action_E-L_aniso_momentum} and traction boundary conditions\n\\eqref{eq:action_E-L_aniso_traction}.\n\nWe should emphasize that this stationary Principle of Correlated Action also\nholds for classical theory, if one neglects contributions from mean curvature\nand moment tractions.  Thus, the classical correlated action for reduced\nelastodynamics can be written:\n\\begin{equation}\n\\begin{split}\n&\\mathcal{A}_{\\rm cl}[u;\\omega] = \\frac{1}{2}\n \\int\\limits_{V} e_{ij}^* C_{ijkl} e_{kl} \\dd{V}\n -\\frac{\\omega^2}{2} \\int\\limits_{V} u_i^* \\rho u_i \\dd{V} \\\\\n&\\qquad\\qquad- \\int\\limits_{V} u_i^* f_i \\dd{V}\n- \\int\\limits_{S_t} u_i^* t_i^{(\\hat{n})} \\dd{S} . \\,\n\\label{eq:classical_action_functional}\n\\end{split}\n\\end{equation}\n\n\n\\section{Response of Periodic Materials}\nThis section summarizes the most relevant theoretical aspects for the numerical\nanalysis of periodic materials. An in-depth treatment of the subject can be\nfound in classical textbooks, such as \\cite{book:brillouin} and \\cite{book:kittel},\nwhile a comprehensive review is provided in \\cite{hussein2014dynamics}. In our\ndiscussion we will use a generalized form of the reduced wave equation, however\nwe will provide the Bloch-Floquet boundary conditions for the particular case of\nthe C-CST model.\n\n\\subsection{Bloch's theorem}\nConsider a reduced elastodynamic wave equation in the frequency domain of the form\n\\begin{equation}\n  \\mathcal{L} \\mathbf{u}(\\mathbf{x}) = -\\rho \\omega^2 \\mathbf{u}(\\mathbf{x})\\ \n  \\label{eq:reduced}\n\\end{equation}\nvalid for a  field \\(\\mathbf{u}\\) at a spatial point \\(\\mathbf{x}\\). Here\n\\(\\mathcal{L}\\) is a positive definite linear differential operator\n\\citep{book:reddy_functional, book:kreyszig_functional, johnson2007waves},\nwhile \\(\\rho\\) is the mass density and \\(\\omega\\) the corresponding angular\nfrequency. Bloch's theorem \\citep{book:brillouin} establishes that solutions to\n\\eqref{eq:reduced} are of the form\n\\begin{equation}\n  \\mathbf{u}(\\mathbf{x}) = \\mathbf{w}(\\mathbf{x}) e^{i\\mathbf{k}\\cdot \\mathbf{x}}\\\n  \\label{eq:bloch}\n\\end{equation}\nwhere $\\mathbf{w}(\\mathbf{x})$ is a Bloch function carrying with it the same\nperiodicity as the material. Since the spatial period in\n\\(\\mathbf{w}(\\mathbf{x})\\) is the lattice parameter \\(\\mathbf{a}\\), it follows that\n\\[\\mathbf{w}(\\mathbf{x} + \\mathbf{a}) = \\mathbf{w}(\\mathbf{x}).\\]\n\nAccordingly, \\eqref{eq:bloch} is the product of a spatially periodic function\n\\(\\mathbf{w}(\\mathbf{x})\\), with the periodicity of the lattice, and a plane\nwave (of wave vector \\(\\textbf{k}\\)), which is also periodic. As a result, field\nvariables \\(\\mathbf{\\Phi}\\) at opposite sides of the unit cell and separated by\nthe lattice vector \\(\\mathbf{a}\\) are related through\n\\begin{equation}\n\\mathbf{\\Phi}(\\mathbf{x} + \\mathbf{a}) = \\mathbf{\\Phi}(\\mathbf{x})e^{i\\mathbf{k}\\cdot\\mathbf{a}} .\n\\label{eq:gen_bloch}\n\\end{equation}\n\nIn this case, $\\mathbf{\\Phi}$ refers to the principal variable involved in the\nphysical problem, or to any of its spatial derivatives. From a physical point\nof view, \\eqref{eq:gen_bloch} means that a field variable \\(\\mathbf{\\Phi}\\) at\npoints \\(\\mathbf{x}\\) and \\(\\mathbf{x} + \\mathbf{a}\\) differ only by the phase\nshift \\(e^{i\\mathbf{k}\\cdot\\mathbf{a}}\\).\n\nIn the classical elastodynamic case in which \\(\\mathcal{L}\\) is the Navier\noperator of order 2, the generalized Boundary Value Problem (BVP) considering\nBloch boundary conditions (BBCs) takes the form:\n\\begin{subequations}\n\\label{eq:bvp_bloch}\n\\begin{align}\n&\\mathcal{L} \\mathbf{u}(\\mathbf{x}) = -\\rho \\omega^2 \\mathbf{u}(\\mathbf{x})\\, ,\\\\\n&\\mathbf{u}(\\mathbf{x} + \\mathbf{a}) = \\mathbf{u}(\\mathbf{x})e^{i\\mathbf{k}\\cdot\\mathbf{a}}\\, ,\\\\\n&\\mathbf{\\sigma}(\\mathbf{x} + \\mathbf{a}) \\cdot \\hat{\\mathbf{n}}\n  = - \\mathbf{\\sigma}(\\mathbf{x}) \\cdot \\hat{\\mathbf{n}}\\,  e^{i\\mathbf{k}\\cdot\\mathbf{a}}\\, ,\n\\end{align}\n\\end{subequations}\nwhere \\(\\mathbf{u}(\\mathbf{x} + \\mathbf{a})\\) and \\(\\mathbf{u}(\\mathbf{x})\\)\ngive the field at \\(\\mathbf{x} + \\mathbf{a}\\) and \\(\\mathbf{x}\\), respectively,\nand \\(\\mathbf{\\sigma}(\\mathbf{x})\\) is the corresponding stress.  Meanwhile, \n\\(\\mathbf{a} = \\mathbf{a}_1 n_1 + \\mathbf{a}_2 n_2 + \\mathbf{a}_3 n_3\\) is the\nlattice translation vector and \\(n_i\\) are the lattice normal parameters.\n\nNote that the BVP encompassed by \\eqref{eq:bvp_bloch} simultaneously describes\nthe space-time periodicity of the solutions in the cellular material. Time\nperiodicity is present in the frequency-domain nature of the reduced wave\nequation, while space periodicity explicitly appears in the wave number\nrepresentation of the boundary conditions. The periodic relationship between\nopposite sides of the fundamental cell, appearing in the boundary terms, allows\ncharacterization of the fundamental properties of the material with the analysis\nof a single cell. At the same time the wave vector \\(\\textbf{k}\\) in\n\\eqref{eq:bvp_bloch} simultaneously describes: (i) the propagation direction of\na plane wave traveling through the unit cell and (ii) the spatial periodicity of\nthe plane wave. In consequence, finding solutions to the Bloch-BVP amounts to\nfinding those tuples \\((\\omega, \\mathbf{k}, \\mathbf{u})\\) satisfying\n\\eqref{eq:bvp_bloch} when \\(\\mathbf{k}\\) is varied in the dual Fourier based\nrepresentation of the fundamental material cell. This dual space corresponds to\nthe \\textit{reciprocal space} and since it carries with it the periodic\ncharacter of the physical space it suffices to consider values (and directions)\nof \\(\\mathbf{k}\\) within this reciprocal space representation of the unit cell.\n\nIn the case of the C-CST medium, Bloch's theorem states that the eigenfunctions\nof \\eqref{eq:wave_eq_freq} can be expressed in the form\n\\[\\mathbf{u}(\\vb{x}) = \\mathbf{u}(\\vb{x}+\\vb{a})e^{i\\vb{k}\\cdot\\vb{a}} \\]\nwhere $\\vb{a}$ is a vector that represents the periodicity of the material.\nThat is, the solution is the same at opposite sides of the unit cell, except \nfor a phase shift factor $e^{i\\vb{k}\\cdot\\vb{a}}$. Due to the linearity of the\ndifferential equations we also have Bloch-periodic boundary \nconditions for the corresponding rotation and traction vectors.\nThus, in the case of the C-CST elastic solid, Bloch's theorem reduces to the\nfollowing set of boundary conditions for displacements, rotations,\nforce-tractions and couple-tractions in index notation:\n\\begin{subequations}\n\\begin{align}\n    &u_i(\\vb{x}) = u_i(\\vb{x}+\\vb{a})e^{i\\vb{k}\\cdot\\vb{a}}\\, , \\\\ \n    &\\theta_i(\\vb{x}) = \\theta_i(\\vb{x}+\\vb{a})e^{i\\vb{k}\\cdot\\vb{a}}\\, ,\\\\\n    &t_i(\\vb{x}) = -t_i(\\vb{x}+\\vb{a})e^{i\\vb{k}\\cdot\\vb{a}}\\, , \\label{eq:bloch_bcs.3}\\\\ \n    &m_i(\\vb{x}) = -m_i(\\vb{x}+\\vb{a})e^{i\\vb{k}\\cdot\\vb{a}}\\, .\n    \\label{eq:bloch_bcs.4}\n\\end{align}\n\\label{eq:bloch_bcs}\n\\end{subequations}\n\n\\noindent The set of conditions summarized in \\eqref{eq:bloch_bcs} will be\nsatisfied in a variational sense using a finite element formulation, where the\nfirst two are essential boundary conditions and the other two natural boundary\nconditions. Subsequently, a numerical model of the unit cell resulting in a\ngeneralized eigenvalue problem will be solved for various specifications of the\nwave vector.\n\n\\subsection{Hermiticity}\nOur finite element algorithm follows from the action functional formulated in\n\\eqref{eq:Lagrangian_functional}. As discussed previously this amounts to the\nsolution of the weak form of the frequency domain reduced wave equations subject\nto Bloch-periodic boundary conditions, as given by \\eqref{eq:bloch_bcs}.\nNeglecting body forces in \\eqref{eq:Lagrangian_functional}, we have:\n\\begin{equation}\n\\begin{split}\n\\mathcal{A}[u; \\omega] = &\\frac{1}{2}\\int\\limits_{V} e_{ij}^* C_{ijkl} e_{ij} \\dd{V}\n  + \\frac{1}{2}\\int\\limits_{V} \\kappa_{i}^* D_{ij} \\kappa_{i } \\dd{V}\n  -\\frac{\\omega^2}{2}\\int\\limits_{V} u_i^* \\rho u_i \\dd{V}\\\\\n&- \\int\\limits_{S_t}  u_i^* t_i \\dd{S}  - \\int\\limits_{S_m} \\theta_i^* m_i \\dd{S}\\, ,\n\\end{split}\n\\label{eq:hamilton_freq_free}\n\\end{equation}\n\nTo obtain real eigenvalues that correspond to propagating waves in the band\nstructure of the material, the matrices resulting from the finite element\ndiscretization must be Hermitic. Equivalently, we must prove  Hermiticity\n(self-adjointness) in the action functional. This amounts to showing that the\nboundary terms in \\eqref{eq:hamilton_freq_free} vanish under Bloch periodic\nboundary conditions.\n\nSubstitution of \\eqref{eq:bloch_bcs} into surface integral terms of\n\\eqref{eq:hamilton_freq_free} yields\n\\begin{equation}\n\\begin{split}\n&\\int\\limits_S u_i^{*}(\\vb{x})t_i(\\vb{x}) \\dd{S} + \\int\\limits_S \n\\theta_i^{*}(\\vb{x}) m_i(\\vb{x}) \\dd{S} = \\\\\n&\\sum\\limits_q \\left\\lbrace \\int\\limits_{S_q}\\left[u_i^{*}(\\vb{x})t_i(\\vb{x}) + \nu_i^{*}(\\vb{x} + \\vb{a}_q)t_i(\\vb{x} + \\vb{a}_q)\\right]\\dd{S}_q + \\right.\\\\\n&\\left. \\int\\limits_{S_q}\\left[\\theta_i^{*}(\\vb{x}) m_i(\\vb{x}) + \n\\theta_i^{*}(\\vb{x} + \\vb{a}_q) m_i(\\vb{x} + \\vb{a}_q)\\right]\\dd{S}_q \n\\right\\rbrace \\, ,\n\\end{split}\n\\end{equation}\nwith the index $q$ referring to each pair of opposite sides of the boundary.\nIntroducing the phase shifts and pulling out the common factors give:\n\\begin{equation}\n\\begin{split}\n&\\int\\limits_S u_i^{*}(\\vb{x})t_i(\\vb{x}) \\dd{S} + \\int\\limits_S \n\\theta_i^{*}(\\vb{x}) m_i(\\vb{x}) \\dd{S} = \\\\\n&\\sum\\limits_q \\left\\lbrace \\int\\limits_{S_q} u_i^*(\\vb{x})\\left[t_i(\\vb{x}) \n+ e^{i\\vb{k}\\cdot\\vb{a}}t_i(\\vb{x} + \\vb{a}_q)\\right]\\dd{S}_q + \\right.\\\\\n&\\left. \\int\\limits_{S_q}\\theta_i^*(\\vb{x})\\left[m_i(\\vb{x}) + \ne^{i\\vb{k}\\cdot\\vb{a}}m_i(\\vb{x} + \\vb{a}_q)\\right]\\dd{S}_q \\right\\rbrace \n\\, ,\n\\end{split}\n\\end{equation}\nwhich after substituting \n\\eqref{eq:bloch_bcs.3} and \\eqref{eq:bloch_bcs.4}\nleads to the vanishing of the boundary terms, thus proving the Hermiticity\ncondition.\n\n\\subsection{Positive definiteness}\nSimilarly, the proof for positive (semi)-definiteness reduces to showing that\nthe action functionals are related in such a way that:\n\\begin{equation}\n\\omega^2 = \n\\frac{\\mathcal{U}[u;\\omega]}{\\mathcal{\\tilde T}[u;\\omega]}  \\geq 0 \\, ,\n\\label{eq:definiteness}\n\\end{equation}\nwhere\n\\[\\mathcal{U}[u;\\omega] = \\frac{1}{2}\\int\\limits_{V} e_{ij}^* \nC_{ijkl}e_{kl}\\dd{V} + \\frac{1}{2}\\int\\limits_{V} \\kappa_{i}^* D_{ij} \\kappa_{j}\\dd{V} \n\\, \\]\nand\n\\[\\mathcal{\\tilde T}[u;\\omega] = \\frac{1}{2}\\int\\limits_V u_i^* \\rho u_i \\dd{V} \\,  , \\]\nwith the latter deriving directly from $\\mathcal{T}[u;\\omega]$.\n\nNote that we have used the general representation \\(C_{ijkl}\\) and \\(D_{ij}\\) for\nthe constitutive tensors. The functional \\(\\mathcal{U}[u;\\omega]\\) is positive as\nlong as these constitutive tensors are positive definite, which holds true if\nthey satisfy\n\\begin{align*}\nC_{ijkl}  e_{ij} e_{kl} \\geq 0\\quad \\forall \\ e_{mn}\\, ,\\\\\nD_{ij}  \\kappa_{i} \\kappa_{j} \\geq 0\\quad \\forall \\ \\kappa_{m}\\, .\n\\end{align*}\n\nFor isotropic materials, this implies the following constraints for the material\nparameters:\n\\[\\mu >0\\, ,\\quad   3\\lambda + 2\\mu > 0\\, ,\\quad \\eta > 0\\, .\\]\nOn the other hand, the condition  \\(u_i \\neq 0\\), requires \\(\\mathcal{\\tilde T}\\)\nto be different from zero and thus the condition required by\n\\eqref{eq:definiteness}. In the case of rigid body motion, \\(\\mathcal{U}\\)\ncould be zero implying that the form  is positive semi-definite, while the form\n\\(\\mathcal{\\tilde T}\\) is positive definite.\n\n\\section{Finite element formulation}\nIn this section, we derive a consistent finite element formulation for periodic\ncouple stress elastodynamics, as an extension of those formulated by\n\\cite{darrall2014} for the corresponding quasistatic problem and by\n\\cite{guarin2020} for periodic micropolar Bloch analysis.  In particular,\nthe \\(C^1\\) displacement continuity requirement is avoided by using a Lagrange\nmultiplier approach. Other finite element solutions in C-CST include a penalty\nmethod for isotropic elastostatics \\citep{chakravarty2017}, Lagrange multipliers\nfor centrosymmetric anisotropic elastostatics \\citep{pedgaonkar2021} and mixed\nvariable methods for isotropic elastodynamics \\citep{deng2016, deng2017}.\n\n\\subsection{Lagrange multiplier reformulation}\nConsider now a modification of the action given in\n\\eqref{eq:hamilton_freq_free} to include Lagrange multipliers \\(\\lambda_i\\) that\nenforce compatibility between the displacement field \\(u_i\\) and an assumed\nindependent rotation field \\(\\theta_i\\).  Thus, the modified action becomes\n\\begin{equation}\n\\begin{split}\n\\mathcal{\\hat A}[u; \\omega] = &\\frac{1}{2}\\int\\limits_{V} e_{ij}^* C_{ijkl} e_{ij} \\dd{V}\n  + \\frac{1}{2}\\int\\limits_{V} \\kappa_{i}^* D_{ij} \\kappa_{i } \\dd{V}\n  -\\frac{\\omega^2}{2}\\int\\limits_{V} u_i^* \\rho u_i \\dd{V}\\\\\n&- \\int\\limits_{S_t}  u_i^* t_i \\dd{S}  - \\int\\limits_{S_m} \\theta_i^* m_i \\dd{S}\\\\\n&+ \\int\\limits_{V}\\lambda_i^* (\\epsilon_{ijk} u_{k,j} - 2\\theta_i )\\dd{V}\\, .\n\\end{split}\n\\label{eq:hamilton_freq_with_multipliers}\n\\end{equation}\n\nFor stationarity, we require\n\\[\\var{\\hat{\\mathcal{A}}} = \\pdv{\\hat{\\mathcal{A}}}{u_i} \\var{u_i}\n+ \\pdv{\\hat{\\mathcal{A}}}{\\theta_i}\\var{\\theta_i}\n+ \\pdv{\\hat{\\mathcal{A}}}{\\lambda_i} \\var{\\lambda_i} = 0\\, , \\]\nwhich is equivalent to\n\\begin{equation}\n\\begin{split}\n&\\int\\limits_{V} \\delta e_{ij}^* C_{ijkl} e_{ij} \\dd{V}\n  + \\int\\limits_{V} \\delta \\kappa_{i}^* D_{ij} \\kappa_{i } \\dd{V}\n  -\\omega^2 \\int\\limits_{V} \\delta u_i^* \\rho u_i \\dd{V}\\\\\n&\\qquad- \\int\\limits_{S_t}  \\delta u_i^* t_i \\dd{S}  - \\int\\limits_{S_m} \\delta \\theta_i^* m_i \\dd{S}\\\\\n&\\qquad+ \\int\\limits_{V}\\delta \\lambda_i^* (\\epsilon_{ijk} u_{k,j} - 2\\theta_i )\\dd{V}\\\\\n&\\qquad+ \\int\\limits_{V} (\\epsilon_{ijk} \\delta u_{k,j}^* - 2\\delta \\theta_i^* ) \\lambda_i \\dd{V}\\, .\n\\end{split}\n\\label{eq:weak_form_with_multipliers}\n\\end{equation}\n\n\\Cref{eq:weak_form_with_multipliers} is the modified weak form that will be used\nhere as the basis for the finite element Bloch analysis of an elastic\ncouple-stress solid. The Lagrange multiplier terms enforce the required\nkinematic constraint between the continuum rotations \\(\\epsilon_{ijk} u_{k,j}\\)\nof the material point and the independent rotational variables \\(\\theta_i\\).\n\nFrom \\eqref{eq:weak_form_with_multipliers}, we obtain the following\nEuler-Lagrange equations\n\\begin{equation}\n\\begin{split}\n&(C_{ijkl} e_{kl} + \\epsilon_{ijk} \\lambda_k)_{,j} = -\\rho \n\\omega^2u_i\\quad \\text{in } V,\\\\\n&\\epsilon_{ijk} (D_{kl} \\kappa_{l})_{,j} - 2\\lambda_i = 0\\quad \\text{in } V,\\\\\n&\\theta_i = \\frac{1}{2}\\epsilon_{ijk} u_{k,j}\\quad \\text{in } V,\\\\\n&t_i = (C_{ijkl} e_{kl} + \\epsilon_{ijk}\\lambda_k)n_j\\quad   \\text{on } S_t,\\\\\n&m_i = \\epsilon_{ijk} D_{kl} \\kappa_{l} n_j\\quad   \\text{on } S_m\\, ,\n\\end{split}\n\\label{eq:bvp}\n\\end{equation}\nComparing this with \\eqref{eq:skew_symmetry}, we can conclude that the Lagrange\nmultipliers equal the skew-symmetric part of the force-stress tensor, i.e.,\n\\[\\lambda_i = s_i\\, .\\]\n\n\\subsection{Discretization}\nTo discretize \\eqref{eq:weak_form_with_multipliers}, we use for the\nelement-based shape functions second-order Lagrange interpolation for the\ndisplacements and rotations and constant skew-symmetric stresses. This\ntranslates into \\(C^0\\) inter-element displacement and rotation continuity, and\nskew-symmetric stresses that are constant within the element but discontinuous\nbetween elements. \\Cref{fig:element} depicts a typical element for the\ndiscretization and the degrees of freedom used in two-dimensional idealizations.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=2 in]{img\/element.pdf}\n\\caption{Finite element used for the finite element discretization of the\nC-CST material model. A second-order Lagrange interpolation is used for\ndisplacements and rotations while a constant is used for the skew-symmetric\nstresses. The nodal degrees of freedom are depicted as black disks while the\nwhite disk describes the element skew-symmetric force-stress degree of freedom.}\n\\label{fig:element}\n\\end{figure}\n\nTo write the discretized equations, we will use a combined index notation. In\nthis context subscripts will still make reference to scalar components of\ntensors while capital superscripts will indicate interpolation operations. For\ninstance in the expression\n\\[u_i = _{u}\\hspace{-4pt}N_{i}^{Q}u^{Q}\\]\nsubscripts $i$ indicate the scalar components of the vector \\(u\\). To facilitate\nfurther operations this subscript is also placed in the shape function resulting\nin terms like \\(_{u}N_{i}^{Q}\\) and where the term \\(u^{Q}\\)  represents the\nnodal point displacement associated to the $Q$th nodal point. This nodal vector\nimplicitly considers horizontal and vertical rectangular components. To clarify,\nthe displacement interpolation scheme written here as\n\\(u_i = _{u}\\hspace{-4pt}N_{i}^{Q}u^{Q}\\) takes the following explicit form for\nthe single nodal point Q:\n\\begin{equation}\n\\begin{bmatrix}u_x\\\\ u_y\\end{bmatrix}=\n\\left[\\cdots\\begin{array}{cc}N^Q &0\\\\ 0 &N^Q\\end{array}\\cdots\\right]\n\\begin{Bmatrix}\\vdots\\\\ u_x^Q\\\\ u_y^Q\\\\\\vdots\\end{Bmatrix} \\, .\n\\label{eq:interp_nota}\n\\end{equation}\n\nWith this notation we write for the primary variables\n\\((u_i, \\theta_i, s_i )\\) the following interpolated versions\n\\begin{equation}\nu_i = _{u}\\hspace{-4pt}N_{i}^{Q}u^{Q},\\quad \\theta_i\n  = _{\\theta}\\hspace{-4pt}N_i^{Q}\\theta^{Q} ,\\quad s_i\n  = _{s}\\hspace{-4pt}N_{i}^{Q} s^{Q} \\, ,\n\\label{eq:interp_fun_prim}\n\\end{equation}\nand similarly for the secondary kinematic descriptors\n\\(e_{ij} , \\epsilon_{ijk} u_{i,j}\\) and \\(\\kappa_{i}\\) \n\\begin{equation}\ne_{ij} = _{e}\\hspace{-4pt}B_{ij}^{Q}u^{Q} ,\\quad \\epsilon_{ijk} u_{i,j}\n  = _{\\nabla}\\hspace{-4pt}B_{k}^{Q}u^{Q} ,\\quad \\kappa_{i}\n  = _{\\kappa}\\hspace{-4pt}B_{i}^Q \\theta^Q \\, ,\n\\label{eq:interp_fun_second}\n\\end{equation}\ntogether with the constitutive equations\n\\begin{equation}\n\\begin{aligned}\n  &\\sigma_{ij} = C_{ijkl}e_{kl}\\, , \\\\\n  &\\mu_{i} = D_{ij}\\kappa_{j}\\, .\n\\end{aligned}\n\\label{eq:const}\n\\end{equation}\n\n\nSubstitution of the above relations in \\eqref{eq:weak_form_with_multipliers}\ngives the discrete version of the first variation of the modified correlated\naction;\n\\begin{equation}\n\\begin{split}\n&\\var{\\hat{\\mathcal{A}}} = \\delta u^{Q*} \\int\\limits_{V} (_{e}B_{ij}^{Q}) (C_{ijkl}) (_{e}B_{kl}^{P}) \\dd{V} u^{P} - \\rho \\omega ^2 \\delta u^{Q*}  \\int\\limits_{V} (_{u}N_{i}^{Q}) (_{u}N_{i}^{P}) \\dd{V} u^{P} \\\\\n&- \\delta u^{Q*}\\int\\limits_{V}\\ _{u}N_{i}^Q f_i \\dd{V} - \\delta u^{Q*}\\int\\limits_{S}\\ _{u}N_{i}^Q t_i \\dd{S} \n+ \\delta \\theta^{Q*} \\int\\limits_{V} (_{\\kappa}B_{i}^{Q}) (D_{ij}) (_{\\kappa}B_{j}^{P}) \\dd{V} \\theta^{P}  \\\\\n&-\\delta \\theta^{Q*}\\int\\limits_{S}\\ _{\\theta}N_i^Q m_i \\dd{S} + \\delta s^{Q*} \\int\\limits_{V} (_{s}N_{k}^{Q}) (_{\\nabla}B_{k}^{P}) \\dd{V}  u^{P}\n+ \\delta u^{Q*} \\int\\limits_{V} (_{\\nabla}B_{k}^{Q}) (_{s}N_{k}^{P})  \\dd{V} s^{P} \\\\\n&-\\delta s^{Q*} \\int\\limits_{V} 2 (_{s}N_{k}^{Q}) (_{\\theta}N_{k}^{P}) \\dd{V} \\theta^{P} -\\delta \\theta^{Q*} \\int\\limits_{V} 2(_{\\theta}N_{k}^{Q}) (_{s}N_{k}^{P}) \\dd{V} s^{P} = 0 \\, .\n\\end{split}\n\\label{eq:discrete_PVW}\n\\end{equation}\n\nThe explicit form of the interpolators defined above is given in the appendix.\n\n\\subsection{Discrete equilibrium equations}\nFrom the arbitrariness in the variations \\(\\delta u^Q\\) , \\(\\delta \\theta^Q\\)\nand \\(\\delta s^Q\\) in \\eqref{eq:discrete_PVW} it follows that:\n\\begin{equation*}\n\\begin{split}\n&\\int\\limits_{V} (_{e}B_{ij}^{Q}) (C_{ijkl}) (_{e}B_{kl}^{P}) \\dd{V} u^P\n  - \\int\\limits_{V} (_{u}N_{i}^{Q}) (_{u}N_{i}^{P}) \\dd{V} u^P\n  - \\int\\limits_{V}\\ _{u}N_{i}^Q f_i \\dd{V}\n  - \\int\\limits_{S}\\ _{u}N_{i}^Q t_i \\dd{S} = 0\\,, \\\\\n&\\int\\limits_{V} (_{\\kappa}B_{i}^{Q}) (D_{ij}) (_{\\kappa}B_{j}^{P}) \\dd{V} \\theta^P \n  - \\int\\limits_{S}\\ _{\\theta}N_i^Q m_i \\dd{S}\n  - \\int\\limits_{V} 2(_{\\theta}N_{k}^{Q}) (_{s}N_{k}^{P}) \\dd{V} s^P=0\\, , \\\\\n&\\int\\limits_{V} (_{s}N_{k}^{Q}) (_{\\nabla}B_{k}^{P}) \\dd{V}  u^P\n  - \\int\\limits_{V} 2 (_{s}N_{k}^{Q}) (_{\\theta}N_{k}^{P}) \\dd{V} \\theta^P = 0\\, ,\n\\end{split}\n\\end{equation*}\nwhich can be written in the standard finite element form for dynamic equilibrium\n\\begin{equation}\n\\begin{bmatrix}\nK_{uu}^{QP} &0 &K_{us}^{QP}\\\\\n0 &K_{\\theta\\theta}^{QP} &-K_{\\theta s}^{QP}\\\\\nK_{s u}^{QP} &-K_{s\\theta}^{QP} &0\n\\end{bmatrix}\n\\begin{Bmatrix}\nu^P\\\\\n\\theta^P\\\\\ns^P\n\\end{Bmatrix}\n=\n\\omega^2\\begin{bmatrix}\n M_{uu}^{QP} &0 &0\\\\\n0 &0 &0\\\\\n0 &0 &0\n\\end{bmatrix}\n\\begin{Bmatrix}\nu^P\\\\\n\\theta^P\\\\\ns^P\n\\end{Bmatrix}\n+ \\begin{Bmatrix}\nF_u^Q\\\\\nm_\\theta^Q\\,\\\\\n0\\end{Bmatrix}\n\\label{eq:mat_fem}\n\\end{equation}\nwhere the individual terms are defined as\n\\begin{align*}\nK_{uu}^{QP} &= \\int\\limits_{V} (_{e}B_{ij}^{Q}) (C_{ijkl}) (_{e}B_{kl}^{P}) \\dd{V}\\, ,\n&M_{uu}^{QP} &= \\rho \\omega ^2 \\int\\limits_{V} (_{u}N_{i}^{Q}) (_{u}N_{i}^{p}) \\dd{V}\\, ,\\\\\nK_{u s}^{QP} &= \\int\\limits_{V} (_{\\nabla}B_{k}^{Q}) (_{s}N_{k}^{P})  \\dd{V}\\, ,\n&F_{u }^{Q} &= \\int\\limits_{V}\\ _{u}N_{i}^Q f_i \\dd{V} + \\int\\limits_{S}\\ _{u}N_{i}^Q t_i \\dd{S}\\, ,\\\\\nK_{\\theta \\theta}^{QP} &= \\int\\limits_{V} (_{\\kappa}B_{i}^{Q}) (D_{ij}) (_{\\kappa}B_{j}^{P}) \\dd{V}\\, ,\n&K_{\\theta s}^{QP} &= \\int\\limits_{V} 2(_{\\theta}N_{k}^{Q}) (_{s}N_{k}^{P}) \\dd{V}\\, ,\\\\\nm_{\\theta}^{Q} &= \\int\\limits_{S}\\ _{\\theta}N_i^Q m_i \\dd{S}\\, ,\n&K_{s u}^{QP} &= \\int\\limits_{V} (_{s}N_{k}^{Q}) (_{\\nabla}B_{k}^{P}) \\dd{V}\\, ,\\\\\nK_{s \\theta}^{QP} &= \\int\\limits_{V} 2 (_{s}N_{k}^{Q}) (_{\\theta}N_{k}^{P}) \\dd{V}\\, .\n\\end{align*}\n\n\\Cref{eq:mat_fem} can be rewritten in the following set of equilibrium equations\nin terms of nodal forces and couples\n\\begin{equation}\n\\begin{aligned}\nf_{(\\sigma)}^Q + f_{s}^Q - f_I^Q - T^Q &= 0\\, ,\\\\\nm_{\\mu}^Q + m_{s}^Q - q^Q &= 0\\, ,\\\\\ns(\\theta - \\hat{\\theta}) = 0\\, ,\n\\end{aligned}\n\\label{eq:discrete_balance}\n\\end{equation}\nwhere the subindex $(\\sigma)$ refers to the symmetric part of the stress tensor,\nand \\(I\\) to inertial forces. Notice that we do not have an\ninertial term for the second equation as is the case for the micropolar model\n\\citep{guarin2020}. We also have a third equation reflecting the kinematic\nrestriction, between the rotation \\(\\theta\\) and the introduced degree of\nfreedom \\(\\hat{\\theta}\\),  imposed via the Lagrange-multiplier term \\(s\\) in\neach element.\n\nWhen using a Lagrange multiplier formulation as in\n\\eqref{eq:weak_form_with_multipliers} the equations are still self-adjoint, as\ncan be seen in the structure of \\eqref{eq:mat_fem}. Nevertheless, the stiffness\nmatrix is indefinite and the solution of the problem represents a saddle-point\ninstead of a minimum \\citep{arnold_mixed_1990, darrall2014}.\n\n\\subsection{Eigenvalue problem}\nIn finding the dispersion relations, we are interested in the free wave motion\nin the media. This leads to the following eigenvalue problem\n\\begin{equation}\n[K]\\{U\\} = \\omega^2 [M]\\{U\\}\n\\label{eq:eigenvalue_fem}\n\\end{equation}\nwith\n\\[[K] = \\begin{bmatrix}\nK_{uu}^{QP} &0 &K_{us}^{QP}\\\\\n0 &K_{\\theta\\theta}^{QP} &-K_{\\theta s}^{QP}\\\\\nK_{s u}^{QP} &-K_{s\\theta}^{QP} &0\n\\end{bmatrix} ,\\,\n[M] = \\begin{bmatrix}\nM_{uu}^{QP} &0 &0\\\\\n0 &0 &0\\\\\n0 &0 &0\n\\end{bmatrix}, \\,\n\\{U\\} = \n\\begin{Bmatrix}\nu^P\\\\\n\\theta^P\\\\\ns^P\n\\end{Bmatrix}\\, .\\]\n\nIn \\eqref{eq:eigenvalue_fem} Bloch-periodic boundary  conditions are yet to be\nimposed. This can be done in two ways \\citep{valencia_uel_2019}: (i) modifying\nthe connectivity of the elements; and (ii) assembling the matrices without\nconsidering boundary conditions and impose the Bloch-periodicity through\nrow\/column operations. In this work, we follow the second approach as it\nrequires the stiffness and mass matrices to be assembled once and the\ntransformation matrices are computed for every wavenumber in the first\nBrillouin zone. This process results in the following eigenvalue problem\n\\begin{equation}\n[K_R(\\vb{k})]\\{U\\} = \\omega^2 [M_R(\\vb{k})]\\{U\\}\n\\label{eq:reduced_eigenvalue_fem}\n\\end{equation}\nwith\n\\[[K_R(\\vb{k})] = [T(\\vb{k})^H K T(\\vb{k})]\\, ,\\quad [M_R(\\vb{k})] = [T(\\vb{k})^H M T(\\vb{k})]\\, ,\\]\nwhere \\([T(\\vb{k})]\\) represents the transformation matrix for a given \\(\\vb{k}\\),\nand the \\([T^H]\\) refers to the Hermitian transpose of \\([T]\\). For an explicit\nform for the matrices \\([T]\\) refer to \\cite{hussein2014dynamics} or\n\\cite{guarin2012_msc}.\n\nWe conducted the implementation on top of the in-house finite element code\nSolidsPy \\citep{solidspy} and used SciPy to solve the eigenvalue problem\n\\citep{scipy}. To take advantage of the sparsity of the matrices the problem\nshould be written as matrix-vector multiplications, such as\n\\begin{align*}\n\\{x\\} = [T]\\{U\\}\\, ,\\\\\n\\{y\\} = [K]\\{x\\}\\, ,\\\\\n\\{z\\} = [T^H] \\{y\\}\\, ,\n\\end{align*}\nwith $\\{z\\}$ representing the image of the linear operator \\([K_R]\\) over\n\\(\\{U\\}\\). The same procedure can be applied for the right-hand side of\n\\eqref{eq:reduced_eigenvalue_fem}.\n\nThe Lagrange-multiplier approach represents a saddle-point instead of a minimization\nproblem \\citep{arnold_mixed_1990}. This can be seen in the structure of the stiffness\nmatrix obtained in equation \\eqref{eq:eigenvalue_fem}. Furthermore the mass matrix\nis not positive definite anymore. This structure for the eigenvalue problem requires\nthe use of a specific solver such as the LOBPCG method \\citep{knyazev2001}\ninstead of the classical Arnoldi method \\citep{book:arpack}.\n\n\\section{Results: Dispersion relations for C-CST cellular materials}\nIn this section we conduct a series of dispersion analyses intended to show the\neffectiveness of our mixed finite element implementation of the C-CST material\nmodel in predicting the correct wave propagation properties of the material. All\nthe dispersion graphs use the dimensionless frequency\n\\begin{equation}\n\\Omega = \\frac{2d\\omega}{c_2}\\, ,\n\\end{equation}\nfor the vertical axis, where \\(2d\\) is the dimension of the unit cell and\n\\(c_2^2 = \\mu\/\\rho\\) is the speed of the shear wave for a classical elastic material.\nThe Poisson ratio for all the simulations is \\(\\nu = 1\/4\\).\n\nAs a first instance we find the response of a homogeneous periodic material\nwhich has also a closed form solution. We will then continue to study a second\nprototypical example corresponding to a homogeneous material with a circular\npore. These two problems exhibit two different levels of dispersive behavior. In\nthe homogeneous material cell, dispersion is due to the kinematic enrichment of\nthe model associated to the length scale parameter, while in the porous material\nmodel additional dispersion arises due to the explicit microstructural feature.\n\n\\subsection{Homogeneous material}\nAs a test of accuracy and effectiveness of our implementation we consider the\ncase of a homogeneous material cell with the same mechanical properties of the\nmaterial reported previously and with closed form dispersion relations from\n\\eqref{eq:dispersion} and \\eqref{eq:dispersion_SV}. In this model microstructural\neffects are introduced through the material length parameter \\(\\ell\\).\nRecall that \\(\\ell^2\\) is defined by the ratio \\(\\frac{\\eta}{\\mu}\\) where\n\\(\\eta\\) is the curvature-couple-stress module while \\(\\mu\\) is the shear\nmodulus from Cauchy elasticity. The results in terms of the resulting band\nstructure are shown in \\cref{fig:homogeneous}, where we used a\n\\(16\\times16\\) mesh and \\(\\ell^2\/d^2 = 3\/8\\). For a conceptual description of\nthe reciprocal space and a guide on how to interpret the results in a Bloch\nanalysis the reader is referred to \\cite{Valencia_periodic}. Note that this set\nof results is directly comparable with the curves from the closed form solutions\nfrom \\cref{fig:dispersion_analytic}. Since the material is isotropic there are\nno directional effects and, as discussed previously, the only difference between\nthis model and the result from classical elasticity is the dispersive behavior\nof the shear wave. In contrast with the micropolar model \\citep{guarin2020},\nthe present C-CST model does not exhibit additional rotational waves.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[height=2.8 in]{img\/comparison_analytical_mod.pdf}\n\\caption{Dispersion relations for a homogeneous material model. Solid lines\nrepresent FEM results while markers correspond to the analytic solution.\nTriangular and filled-dots describe the P and SV wave modes, respectively.}\n\\label{fig:homogeneous}\n\\end{figure}\n\n\\Cref{fig:homog_lengths} shows the results for the same\nmaterial cell but now we have considered 4 different values of the length scale\nparameter corresponding to \\(\\ell\/d \\in [0.01, 0.1, 1, 10]\\). The mesh in this\ncase is \\(16\\times16\\). Notice that, as expected, the increasing value of this\nparameter only affects the dispersive response of the shear waves while the\nP-waves retain their classical non-dispersive behavior. As seen in\n\\eqref{eq:dispersion_SV} the dispersion increases for higher values of \\(\\ell\\)\ndue to the factor \\(\\sqrt{1 + k^2 \\ell^2}\\) in the dispersion relation. This\nbehavior is closely followed by the numerical results presented in\n\\cref{fig:homog_lengths}.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=5.5 in]{img\/homogeneous_lengths.pdf}\n\\caption{Dispersion relations for a homogeneous material model with varying\nlength scales, \\(\\ell\/d \\in [0.01, 0.1, 1, 10]\\). Solid lines represent FEM\nresults while markers correspond to analytic solutions. As expected, for\nincreasing \\(\\ell\/d\\) the S-wave presents more dispersion due to the factor\n\\(\\sqrt{1 + k^2 \\ell^2}\\), as presented in \\eqref{eq:dispersion_SV}.}\n\\label{fig:homog_lengths}\n\\end{figure}\n\nAs an additional verification, we also tested the convergence\nin the  calculation of the dispersion relations after considering the first 8\nmodes  for a sequence of meshes of \\(1\\times1\\),  \\(2\\times2\\), \\(4\\times4\\),\nand \\(8\\times8\\) elements for \\(\\ell^2\/d^2 = 3\/8\\). The error in the eigenvalue\ncomputation was measured according to\n\\[e = \\frac{\\Vert\\vb*{\\omega}_\\text{ref} - \n\\vb*{\\omega}_h\\Vert_2}{\\Vert\\vb*{\\omega}_\\text{ref}\\Vert_2}\\, ,\\]\nwhere \\(\\vb*{\\omega}_h\\) is the set of eigenvalues (dispersion relation) for a \nmesh of characteristic element size \\(h\\) and \\(\\vb*{\\omega}_\\text{ref}\\) is the \nsolution corresponding to the finer \\(16\\times16\\) elements mesh, which has been\ntaken as reference. The results for this sequence, together with the variation\nin the  error parameter, are displayed in \\cref{fig:convergence}. The estimated \nconvergence rate for the eigenvalues is 2.32. We see that when\nwe refine the mesh it can reproduce the dispersion curves better for higher\nfrequencies. There are still some differences between the \\(8\\times8\\) and\n\\(16\\times16\\) meshes around the dimensionless frequency of 15 but these\ndifferences will disappear with further refinement. Nevertheless, opposed to\nwhat happens in classical continua we would need more points per wavelength\nevery time that we want to increase the maximum frequency. This is due to the\ninherent dispersive behavior of S-waves as can be seen in equation\n\\eqref{eq:dispersion_SV}. Thus, we would expect to need more than 10 points per\nwavelength, customary for finite element methods, or 5, customary for spectral\nelement methods \\citep{komatitsch1999introduction, ainsworth2009dispersive,\nguarin2015}. Again, we should emphasize the dispersive nature of the SV-waves,\nwhile the P-waves remain non-dispersive.\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=5.75 in]{img\/convergence.pdf}\n\\caption{Convergence of the first 8 modes in the dispersion relations at\n\\(\\ell^2\/d^2 = 3\/8\\) for a sequence of meshes with: \\(1\\times1\\), \\(2\\times2\\),\n\\(4\\times4\\), and \\(8\\times8\\) elements --- presented as solid blue lines in the \nbackground. The results are compared with a mesh that has \\(16\\times16\\)\nelements --- presented as dots in the foreground. The estimated convergence rate\nfor the eigenvalues in the 2-norm is 2.32.}\n\\label{fig:convergence}\n\\end{figure}\n\n\n\\subsection{Dispersion in cellular material with a circular pore}\nNow, we consider a C-CST composite material cell configured by a circular pore\nembedded in a homogeneous matrix. The presence of the pore provides the model\nwith a second length-scale due to the microstructure, in addition to the one\nintroduced by the material length-scale parameter \\(\\ell\\). For\nillustration, we assume a pore diameter \\(a\\) that is half the cell length\n(i.e., \\(a=d\\)). Thi is equivalent to a porosity of \\(\\pi\/16\\) or approximately\n0.196, which is kept fixed as we modify the size of the unit cell to control\n\\(\\ell\/a\\).\n\nThe resulting dispersion curves for this cellular material\nwith four different length scale ratios are shown in \\cref{fig:pore}\nwith \\(\\ell\/a = [0.01, 0.1, 1, 10]\\). In contrast to the results from the fully\nhomogeneous material cell, the presence of the circular pore introduces\nscattering effects inside each cell and the composite shows much more\ncomplicated elastodynamic behavior.  Most importantly, however, the dispersion\ncurves become more regular with increased \\(\\ell\/a\\) and partial bandgaps\nopen up along the \\(\\Gamma M\\) and \\(X\\Gamma\\) directions, especially for\n\\(\\ell\/a = 1\\) and \\(\\ell\/a = 10\\). This type of band structure is not\nobserved for classical elastodynamic cells with a similar geometric periodicity,\nwhich would exhibit behavior close to that obtained here with \\(\\ell\/a = 0.01\\).\nIn fact, as \\(\\ell\/a \\rightarrow 0\\), C-CST theory recovers the classical\nresult.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=5.5 in]{img\/cpore_lengths_mod.pdf}\n\\caption{Dispersion for a cellular material with circular pores for varying \nlength scales, $\\ell\/a = [0.01, 0.1, 1, 10]$.}\n\\label{fig:pore}\n\\end{figure}\n\nFurthermore, it is important to note that this interesting band structure is\nobtained with a consistent continuum mechanics formulation, which requires only\na single additional material parameter, \\(\\ell\\), beyond those needed in the\nclassical elastic case. When this intrinsic length scale is on the order of the hole\ndiameter, the dispersive SV wave has a fundamental group velocity for the cell\nwith dimension \\(2d\\) approximately equal to the group (and phase) velocity of\nthe non-dispersive P wave, which allows the SV and P branches to follow a\nsimilar path, causing band gaps to open. This behavior, which is not seen in\nclassical elastodynamics, occurs in the regions near to where the two branches\nintersect in \\cref{fig:dispersion_analytic}. Consequently, under C-CST,\nthese band gaps will originate whenever the size of the cellular structure is\ntuned to the material length scale, a potentially significant phenomenon that\nhas not been recognized previously. With further tuning of the porosity level\nand cell size, it may be possible to achieve even a complete bandgap at\nrelatively low non-dimensional frequency \\(\\Omega\\).\n\n\n\\section{Conclusions}\n\nThe present work incorporates several innovative aspects. First of all, we\nhave developed a novel frequency domain correlated action principle for the\nconsistent couple stress theory (C-CST) of \\cite{hadjesfandiari2011couple} and\nused that to extend the Lagrange multiplier finite element algorithm of\n\\cite{darrall2014} to study periodic cellular materials through Floquet-Bloch\ntheory from solid state physics. Particularly, we have addressed the imposition\nof extended Bloch boundary conditions for this material model where in\naddition to force tractions and displacements there are also couple tractions\nand rotations. Secondly, we also discussed numerical aspects related to the\nsolution of the wavenumber dependent generalized eigenvalue problem resulting\nfrom the imposition of the Bloch periodic boundary conditions, overcoming\ncomplications arising from the inclusion of Lagrange multipliers and a\nnon-positive definite mass matrix. The implementation was\nshown to give accurate results for homogeneous and porous unit cells and for\nvarying couple stress material length-scale parameters.\n\nThe analysis of the first cell was used mainly to test the correctness of our\nimplementation as this material has a closed-form dispersion relation. The\nalgorithm was shown to correctly capture the non-dispersive P-wave as well as\nthe dispersive SV-wave. This analysis was complemented by a convergence analysis\nwith four different meshes of increasing refinement for the material cell. The\nobserved convergence rate shows that the Lagrange multiplier algorithm is\neffective in maintaining continuity by imposing the newly introduced kinematic\nconstraint implicit in the mean curvature tensor definition. As the final\ncontribution, we have discovered the interesting bandgap structure of a material\ncell with a circular pore embedded in a homogeneous matrix, which reveals the\nappearance of bandgaps introduced by the kinematic features of C-CST and the\ndispersive behavior of the SV-waves defined in terms of the microstructural\nlength scale parameter.\n\nFrom a general perspective, C-CST is a true size-dependent continuum theory,\nwhich is intended here for periodic elastic material cells at scales for which a\ncontinuum representation is appropriate.  From the results shown in the paper,\nC-CST becomes important when the size of the cell in on the order of the\nintrinsic length scale parameter or smaller.  For larger cells, the classical\ntheory can be used instead. On the other hand, micropolar theory disconnects the\nrotational field from the displacements, which can lead to approximations that\nmay or may not be physical.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\tConvolutional Neural Networks (CNNs) have been successfully used in various tasks with Euclidean structure data in recent years. For non-Euclidean structure datasets, graph convolutional networks (GCNs) use the same idea to extract the topological structure information.\n\t\n\t\\cite{bruna2014spectral} firstly proposed two GCN models with spectral and spatial construction respectively. The spectral model using a Laplacian matrix to aggregate neighborhood information of each node in a graph. The spatial model partition graph into clustering and update them by aggregating function. In order to extract deeper features, model based on CNNs usually deepen the model's layers. While in GCNs, Deepening layers will cause a lot of problems. In spectral construction method, too many layers lead to \\textbf{``over smoothing''}\\cite{li2018deeper,zhou2018graph}: the features of nodes in graph will tend to be the same. In spatial construction method, it will cause exponential growth of the number of sampled neighbor nodes, called \\textbf{``neighbor explosion''}. Node sampling and layer sampling\\cite{hamilton2018inductive,chen2018fastgcn,gao2018large,huang2018adaptive,ying2018hierarchical} were proposed to handle this problem, but due to incomplete sampling, the inaccuracy of nodes' representation accumulates errors between layers.\n\t\n\tTo address these two problems, in this paper, we propose a model called PathSAGE, which can learn high-order topological information by expanding the receptive field.\n\tFirstly, We design a path sampling technique based on random walk to sample paths starting from central nodes with different lengths, then the sequences of paths are fed into Transformer encoder\\cite{vaswani2017attention}, which can extract the semantic and distance information in sequence effectively. As shown in Fig. 1, We view the sequences of paths from the tail nodes to the central node as the central node' neighbors. Secondly, following this idea, we take the average of paths of the same length as the representation of the central node in this level of reception field. Finally, after concatenating the aggregated features of paths with different lengths, the final representation of the central node are used for downstream tasks.\n\t\n\tFor the two problems mentioned earlier, on the one hand, the aggregation of the central node only perform once in training of a sample, which never cased ``over-smoothing''. On the other hand, all the paths were sampled with a fixed number for representing the central node in our model, instead of recursively sampling exponentially growing neighbor nodes. And each path only contributes to the central node, we do not need to calculate and store the temporary representations of nodes from the middle layer. Furthermore, it prevents the error propagation caused by incomplete sampling. \n\t\n\tOur contribution can be summarized in three points:\n\t\n\t\\begin{itemize}\n\t\t\n\t\t\\item We utilize the path sampling to take place of the node sampling to avoid error accumulation caused by incomplete node sampling.\n\t\t\n\t\t\\item We propose and evaluate our model Path-SAGE to solve the existing ``neighour explosion'' and ``over-smoothing'' problems. the model can capture richer and more diverse patterns around the central node with only one layer of structure to the central node.\n\t\t\n\t\t\\item We evaluate our model on three inductive learning tasks, and it reaches the state-of-the-art performance on two of them. We analyze the attention weights of Transformer encoder and detect some patterns in the attention mechanism, which can further illustrate how the model works.\n\t\t\n\t\\end{itemize}\n\t\\section{Related work}\n\tGNNs model was initially proposed by \\cite{bruna2014spectral}, and the convolution operation in the traditional Euclidean structure was introduced into the graph network with the non-Euclidean structure in this article. They \\cite{bruna2014spectral} divided GNN into two construction methods: spectral construction and spatial construction. Subsequently, many studies are carried out around these two aspects. \n\t\n\tIn spectral construction, \\cite{defferrard2017convolutional} used Chebyshev polynomials with learnable parameters to approximate a smooth filter in the spectral domain, which improves computation efficiency significantly. \\cite{kipf2017semisupervised} further reduced the computational cost through local first-order approximation. In spatial construction, MoNet was proposed in \\cite{monti2017geometric}, developing the GCN model by defining a different weighted sum of the convolution operation, using the weighted sum of the nodes as the central node feature instead of the average value. \\cite{hamilton2018inductive} attempted various aggregator to gather the features of neighbor nodes. \\cite{velivckovic2017graph,zhang2018gaan} defined the convolution operation with a self-attention mechanism between the central node and neighbor nodes, \\cite{liu2019geniepath} brought the LSTM(Long Short Term Memory networks)\\cite{hochreiter1997long} from NLP (Natural Language Processing) to GNN, and built an adaptive depth structure by applying the memory gates of LSTM. \\cite{yang2021spagan} sampled the shortest path based on their attention score to the central node and combined them in a mixed way. PPNP and APPNP\\cite{klicpera2018predict} used the relationship between GCN and PageRank to derive an improved communication scheme based on personalized PageRank. \n\t\n\tWith the scale of graph data increasing, the full-batch algorithm is no longer applicable, and the mini-batch algorithm using stochastic gradient descent is applied. In recent years, some studies based on different sampling strategies have been proposed. \\cite{hamilton2018inductive} tried random neighbor node sampling for the first time to limit large amounts of nodes caused by recursive node sampling. Layer-wise sampling techniques was applied in \\cite{chen2018fastgcn,gao2018large,huang2018adaptive,ying2018hierarchical}, which only consider fixed number of neighbors in the graph to avoid ``neighbor explosion''. \\cite{chen2018fastgcn,chen2018stochastic} used a control variate-based algorithm that can achieve good convergence by reducing the approximate variance. Besides, subgraph sampling was first introduced by \\cite{zeng2019accurate}, which used a probability distribution based on degree. \\cite{chiang2019cluster} performed clustering decomposition on the graph before the training phase and randomly sampled a subgraph as a training batch at every step. \\cite{zeng2020graphsaint} sampled subgraph during training based on node and edge importance. All these sampling techniques are based on GCNs, how to extract deeper topology information from large graph datasets is still a problem.\n\t\\section{Proposed method}\n\tIn this section, we present the PathSAGE. Firstly, the sampling algorithm is introduced in 3.1. Secondly, we detail the aggregator in 3.2. Finally, we discuss the difference between our model and the related models.\n\t\\begin{algorithm} \n\t\t\\SetAlgoLined\n\t\t\\DontPrintSemicolon\n\t\t\\caption{Random Path Sampling}  \n\t\t\t\\KwIn{garph $G(V,E)$\\\n\t\t\tcentral node $c$\\\\\n\t\t\tsample depth $s$\\\\\n\t\t\tsample num each length $L =\\{n_1, n_2, \u2026, n_s\\}$\\} }\n\t\t\t\\KwOut{path sequcences with different length\\\\ \n\t\t\t$\\{\\bm{P}_1, \\bm{P}_2, ..., \\bm{P}_s\\}$}\n\t\t\t\\SetKwFunction{Fun}{Ramdom Path Sampling}\n\t\t\t\\Fun{$G$, $c$, $s$, $L$}{\n\t\t\t\t\n\t\t\t\\ForEach{$l = 0 \\to s$}{\n\t\t\t$i\\gets0$;\n\t\t\t\n\t\t\t\\While{$i < n_l$}{\n\t\t\t $u\\gets c$;\n\t\t\t \n\t\t\t $P\\gets \\{u\\}$;\n\t\t\t \n\t\t\t\\For{$j = 0 \\to l$}{\n\t\t\t $u \\gets$ Node randomly selected from $u$'s neighbors;\n\t\t\t \n\t\t\t $P\\gets P \\cup \\{u\\}$}\n\t\t \n\t\t\t $\\bm{P}_l \\gets \\bm{P}_l \\cup \\{P\\}$;\n\t\t\t \n\t\t\t $i \\gets i+1$;}\n\t\t \n\t\t\t}\n\t\t\\Return $\\{\\bm{P}_1, \\bm{P}_2, ..., \\bm{P}_s\\}$\n\t\t}\n\t\\end{algorithm}\n\t\\subsection{Path Sampling}\n\tExcept deepening model, another way to expand receptive field of CNNs is to increase the size of convolution kernel. Following this idea, we sample the node sequences starting from central node and regard these paths as the context of the central node. Therefore, the receptive field can be expanded by extending these paths.\n\t\n\tWe use a straightforward random sampling algorithm based on random walk, shown in \\textbf{Algorithm 1}: for a central node, a sampling starts from it and randomly selects a neighbor node each time until reaching the preset path length, and multiple path sequences for the corresponding central node of various lengths can be obtained in this way, constituting a training sample.\n\n\t\n\t\\begin{figure*}[htb]\n\t\t\\begin{center}\n\t\t\t\\includegraphics[height=4.5cm]{.\/fig1.pdf}\n\t\t\\end{center}\n\t\t\\caption{Mechanism of sampling and aggregation. (a) and (b) are two different possible training samples with the same central node. Paths with same colors are with same lengths same length and share the same aggregators. }\n\t\n\t\t\\label{fig1}\n\t\\end{figure*}\n\t\n\t\\begin{figure}[htbp]\n\t\t\\begin{center}\n\t\t\t\\includegraphics[height=4.5cm]{.\/fig2.pdf}\n\t\t\\end{center}\n\t\t\\caption{Structure of first aggregation in aggregator. A specific sampled path's features adds the position embedding, through Transformer encoder layers to fuse the information.\n\t\t}\n\t\t\\label{fig2}\n\t\\end{figure}\n\t\n\t\\subsection{Aggregator}\n\tThere are two aggregations in the aggregator: the first one aggregate path sequences; the second one aggregate different paths as the final representation of the central node. \t\n\tFor the first aggregation, We formulate each path sequence as:\n\t\\begin{equation}\n\t\t{\\bm{P}}_{ij}^{}\\; = \\;\\{ {\\vec a_{ij}^1,\\;\\vec a_{ij}^2,\\;\\vec a_{ij}^3,\\;...,\\;\\vec a_{ij}^j\\;} \\}\n\t\\end{equation}\n\n\t\n\tTo utilize the position information of path sequence, we define positions of nodes in paths as their distances to the central nodes. As same as Transformer, we add nodes' features and their positional vectors together as the input of the structure:\n\t\\begin{equation}\\label{eq6}\n\t\t\\bm{\\tilde{P}}_{ij}\\; = \\;\\{ {a_{ij}^1\\; + \\;pos\\_emb\\left( 1 \\right),...,\\;a_{ij}^j\\; + \\;pos\\_emb\\left( j \\right)} \\}\n\t\\end{equation}\n\tFor the $pos\\_emb(\\cdot)$, we use $sine$ and $cosine$ functions of different frequencies to generate the positional vectors. In each dimension, the positional embedding is:\n\t\n\t\\begin{equation}\\label{eq7}\n\t\tpos\\_emb(p)_{2i} = sin(\\frac{{p}}{{{{10000}^{\\frac{{2i}}{{{d}}}}}}})\n\t\\end{equation}\n\t\n\t\\begin{equation}\\label{eq8}\n\t\tpos\\_emb(p)_{2i + 1} = cos(\\frac{{p}}{{{{10000}^{\\frac{{2i}}{{d}}}}}})\n\t\\end{equation}\n\tWhere $p$ is the position in the sequences, $2i$ and $2i+1$ mean the odd and even dimensions of position embedding, and $d$ is the number of features. \n\t\n\tAfter that, we apply the Transformer encoder on each path: \n\t\n\t\\begin{equation}\\label{eq9}\n\t\t\\bm{\\tilde{P}}_{ij}^{k} = transformer\\_bloc{k^k}(\\bm{\\tilde{P}}_{ij}^{k-1})\n\t\\end{equation}\n\tWhere $k$ means $k$-th Transformer encoder layer. Noted that, in a $m$-layer Transformer encoder, the output of the last layer $\\bm{\\tilde{P}}_{ij}^{m}$ is a sequence of features. We only take the output at position 0 as the final representation of the path sequence.\n\t\n\t\\begin{equation}\\label{eq10}\n\t\t\\tilde{P}_{ij}' = \\left[\\bm{\\tilde{P}}_{ij}^{m}\\right]_{0}\n\t\\end{equation}\t\n\t\n\tFollowing these equations (\\ref{eq6})-(\\ref{eq10}), we can obtain representations of paths with different length to the central node, as shown in Fig. 1. In the second aggregation, we apply an average pooling layer to aggregate the central node paths. Then we concatenate all the path representation and apply a feed-forward layer with non-linearity to fuse these features. The final output of a central node $C'$ is computed as following:\n\t\n\t\\begin{equation}\n\t\t\\begin{array}{c}\n\t\t\tC = \\;concat\\left( {{C_1},\\;...,\\;{C_s}} \\right)\\\\\n\t\t\t\\\\\n\t\t\twhere\\;{C_i}\\; = \\;Average\\left( \\tilde{P}_{i1}',\\;...,\\;\\tilde{P}_{in}' \\right)\n\t\t\\end{array}\n\t\\end{equation}\n\t\n\t\\begin{equation}\n\t\t{C'}\\; = \\;max\\left( {0,\\;C{\\bm{W}_1}\\; + \\,{b_1}} \\right){\\bm{W}_2}\\; + \\;{b_2}\n\t\\end{equation}\n\twhere $s$ denotes the sample depth, $n$ denote the number of paths sampled in\n\t\n\t\\subsection{Comparisons to related work}\n\n\t\\begin{itemize}\n\t\t\\item We introduce a sequence transduction model into our structure, but it is distinct from related work based on these models. LSTM\\cite{hochreiter1997long} was also used in GraphSAGE\\cite{zeng2020graphsaint} to aggregate node's features, it is very sensitive to the order of the input sequence. In contrast, neighbor nodes are disordered, and the authors have rectified it by consistently feeding randomly-ordered sequences to the LSTM. In our model, paths to the central node already have orders, which are naturally suitable for sequence processing model. \n\t\t\\item The difference between our attention mechanism and GAT is that we collect all the mutual attention information of each node in the path. The output of our model is integrated information of entire sequence, instead of only attend to the central node.\n\t\t\\item Path sampling method is also used in SPAGAN\\cite{yang2021spagan}. But each sampled path in SPAGAN is the shortest one to the central node, this sample technique will leads to a high computational overhead and limits the ability of the model to be applied to large graph datasets. By contrast, we sample paths randomly, which greatly save the computation overhead. Besides, in our sample algorithm, the same node may have a different path to the central node, which may help the model acquire more diverse patterns around the central node while saving the computational overhead. \n\t\\end{itemize}\n\t\\section{Experiments}\n\tIn this section, we introduce the datasets and the experiment setting in 4.1 and 4.2 respectively. we present the results of evaluation in 4.3. \n\t\\begin{table}[htbp]\n\t\t\\caption{Summary of inductive learning tasks' statistics.}\n\t\t\\begin{center}\n\t\t\n\t\t\t\\centering\n\t\t\t\\renewcommand{\\arraystretch}{2.2}\n\t\t\n\t\t\t\\setlength{\\tabcolsep}{4mm}{\n\t\t\t\t\\begin{tabular}{|c|c|c|c|}\n\t\t\t\t\t\\hline\n\t\t\t\t\t&     \\textbf{Reddit} &       \\textbf{Yelp} &     \\textbf{Flickr} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\ttype & single-label & multi-label & multi-label \\\\\n\t\t\t\t\t\n\t\t\t\t\t\\#Node &    232,965 &    716,847 &     89,250 \\\\\n\t\t\t\t\t\n\t\t\t\t\t\\#Edges & 11,606,919 &  6,977,410 &    899,756 \\\\\n\t\t\t\t\t\n\t\t\t\t\t\\#Features &        602 &        300 &        500 \\\\\n\t\t\t\t\t\n\t\t\t\t\t\\#Classes &         41 &        100 &          7 \\\\\n\t\t\t\t\t\n\t\t\t\t\t\\makecell[c]{Train \/ Val \\\\\/ Test} & \\makecell[c]{66\\% \/ 10\\% \\\\\/ 24\\%} & \t\\makecell[c]{75\\% \/ 10\\% \\\\\/ 15\\%} & \\makecell[c]{50\\% \/ 25\\% \\\\\/ 25\\%} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\\end{tabular} }\n\t\t\\end{center}\n\t\t\\label{tab2}\n\t\\end{table}\n\t\\subsection{Dataset}\n\tThree large-scale datasets are used to evaluate our model: 1) \\textbf{Reddit}\\cite{hamilton2018inductive}: a collection of monthly user interaction networks from the year 2014 for 2046 sub-reddit communities from Reddit that connect users when one has replied to the other, 2) \\textbf{Flickr}: a social network built by forming links between images sharing common metadata from Flickr. 3) \\textbf{Yelp}\\cite{zeng2020graphsaint}: a network that links up businesses with the interaction of its customers. Reddit is a multiclass node-wise classification task; Flickr and Yelp are multilabel classification tasks. \n\tThe detail statistics of these datasets are shown in Tab 1.\n\t\n\t\\begin{table}[t]\n\t\t\n\t\t\\caption{Perforamnce on inductive learning tasks(Micro-F1).}\n\t\t\\begin{center}\n\t\t\n\t\t\t\\centering\n\t\t\t\n\t\t\t\\renewcommand{\\arraystretch}{2}\n\t\t\t\\setlength{\\tabcolsep}{4mm}{\n\t\t\t\t\\begin{tabular}{|c|c|c|c|}\n\t\t\t\t\t\\hline\n\t\t\t\t\t\\textbf{Model} &     \\textbf{Reddit} &       \\textbf{Yelp} &     \\textbf{Flickr} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\tGCN & 0.933$\\pm$0.000 & 0.378$\\pm$0.001 & 0.492$\\pm$0.003 \\\\\n\t\t\t\t\t\n\t\t\t\t\tGraphSAGE & 0.953$\\pm$0.001 & 0.634$\\pm$0.006 & 0.501$\\pm$0.013 \\\\\n\t\t\t\t\t\n\t\t\t\t\tFastGCN & 0.924$\\pm$0.001 & 0.265$\\pm$0.053 & 0.504$\\pm$0.001 \\\\\n\t\t\t\t\t\n\t\t\t\t\tS-GCN & 0.964$\\pm$0.001 & 0.640$\\pm$0.002 & 0.482$\\pm$0.003 \\\\\n\t\t\t\t\t\n\t\t\t\t\tAS-GCN & 0.958$\\pm$0.001 &     -      & 0.504$\\pm$0.002 \\\\\n\t\t\t\t\t\n\t\t\t\t\tClusterGCN & 0.954$\\pm$0.001 & 0.609$\\pm$0.005 & 0.481$\\pm$0.005 \\\\\n\t\t\t\t\t\n\t\t\t\t\tGraphSAINT & 0.966$\\pm$0.001 & \\bf{0.653$\\pm$0.003} & 0.511$\\pm$0.001 \\\\\n\t\t\t\t\t\n\t\t\t\t\tours & \\bf{0.969$\\pm$0.002} & 0.642$\\pm$0.005 & \\bf{0.511$\\pm$0.003} \\\\\n\t\t\t\t\t\\hline\n\t\t\t\\end{tabular} }\n\t\t\t\\label{tab4}\n\t\t\\end{center}\n\t\\end{table}\n\t\\subsection{Experiment setup}\n\t\n\tWe build our model on Pytorch framework\\cite{paszke2017automatic} and construct the Transformer encoder based on UER\\cite{zhao2019uer}. For all tasks, We train the model with Adam SGD optimizer\\cite{kingma2014adam} with learning rate 1e-3 and a learning rate scheduler with 0.1 warmup ratio. We use two layers Transformer, each of which has 8 attention heads to gather the features of paths. The batch size is 32. The dropout layers are applied between each sub-layer in Transformer layer. We use different dropout rates in the output layer and Transformer encoder, which are 0.3 and 0.1 respectively. We set the sampling length of the path ranging from 1 to 8 (depth $s = 8$), and the number of paths sampled in each length are [5, 5, 5, 5, 5, 10, 10, 10]. For multi-label classification task Flickr and Yelp, the final output is obtained through a $sigmoid$ activation, and in Reddit, the final output is obtained through a $softmax$ activation. The hidden dimension is 128 for Reddit and Flickr, 512 for Yelp. \n\t\n\t\\subsection{Result}\n\tWe compare our model with seven state-of-the-art model:  GCN\\cite{kipf2017semisupervised}, GraphSAGE\\cite{hamilton2018inductive}, FastGCN\\cite{chen2018fastgcn}, S-GCN\\cite{chen2018stochastic}, AS-GCN\\cite{huang2018adaptive}, ClusterGCN\\cite{chiang2019cluster}, GraphSAINT\\cite{zeng2020graphsaint}. GraphSAGE uses a random node sampling and LSTM aggregator. FastGCN and S-GCN use a control variate-based algorithm that can achieve good convergence by reducing the sampling variance. ClusterGCN performs clustering decomposition on the graph before the training phase and randomly sampling a subgraph as a training-batch at every step. GraphSAINT samples subgraph while training with several sampling methods based on node and edge importance. The evaluation on these tasks uses the $Micro-F1$ metric, and report the mean and confidence interval of the metrics by five runs.\n\t\n\tThe results of inductive learning experiments are shown in Tab 2. As we can see from the table, For Reddit, our model outperforms all the baseline models. For Flickr, we achieve a comparable F1-score with the top-performing model. For Yelp, we surpass most GCN models, second only to GraphSAINT. One hypothesis to explain the difference of the results is: Reddit and Flickr have more training samples and number of nodes features, which makes the attention mechanism have enough data to capture the relationships between nodes.\n\t\n\t\\begin{figure*}[t]\n\t\t\\begin{center}\n\t\t\t\\includegraphics[height=4.7cm]{.\/emp.pdf}\n\t\t\\end{center}\n\t\t\\caption{Detection of attention weights in Transformer that node have the same label in the path sequence may receive higher weights at some attention heads. The heatmap in the upper-left corner is the weight of position 4 of different layers attend to position 2 and attention heads, the right one is the complete heatmap of the selected head. The lower-left picture shows the labels of the nodes in the sequence. }\n\t\t\\label{fig3}\n\t\\end{figure*}\n\t\n\t\\section{Attention Analysis}\n\tWe observed the attention weight of the trained model in the PubMed test set. At some attention heads, we find that nodes with the same labels get very high attention scores to each other. We visualize an example in Fig 3. Central node 0 and node 4 have the same label and receive extremely high attention scores on second attention heads of the first layer. By observing the attention scores for the whole sequence, we can see that this score also occupies a significant share in the sequence. \n\t\n\tThis observation proves that the attention mechanism can successfully capture the information that is helpful to the downstream tasks in the aggregation of paths.\n\t\n\t\\section{Conclusion}\n\tIn this paper, we propose a model---PathSAGE, which can expand the size of receptive field without stacking layers which solved the problem of ``neighbor explosion'' and ``over smoothing''. Regarding all paths as the neighbor nodes of the central node, PathSAGE samples paths starting from the central node based on random walk and aggregates these features by a strong encoder---Transformer. Consequently, the model can obtain more features of neighbor nodes and more patterns around the central node. In our experiment, PathSAGE achieves the state-of-the-art on inductive learning tasks. Our model provides a novel idea to handle large-scale graph data.\n\t\n\t\\section*{Acknowledgments}.\n\tThis work is supported by National Key R\\&D Program of China (No.2017YFB1402405-5), and the Fundamental Research Funds for the Central Universities (No.2020CDCGJSJ0042). The authors thank all anonymous reviewers for their constructive comments.\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Conclusion}\nIn this paper, we propose a Spatio-Temporal feature auto-selective (STAS) approach that can automatically extract optimal Spatio-Temporal (ST) features hidden in EC for Bias Correcting on Precipitation (BCoP). Experiments on EC benchmark datasets in Eastern China indicate that STAS achieves the highest threat score (TS) on BCop than other 8 algorithms, and has a strong correcting ability in dealing with different degree of precipitation, especially for heavy precipitation. In the future, we will study how to employ ST mechanisms on more complex precipitation scenarios such as squall line, severe convection and thunderstorm.   \n\n\n\n\\section{Experiments}\nWe conduct all experiments on time-series EC benchmarks collected from 1) the high-resolution version of the public European Centre (EC) dataset and 2) Meteorological Information Comprehensive Analysis and Process System (MICAPS)~\\cite{luo2006introduction} that can provide the labels of 5 Meteorological Elements (MEs) including precipitation, temperature, pressure, wind, and dew. Our experiments on Bias Correcting on Precipitation (BCoP) mainly contain two parts. The first part compares the Spatio-Temporal feature Auto-Selective (STAS) model with 8 published machine learning (ML) methods. The second part is a set of ablation experiments on STAS. \n\n\\begin{table}[htp]\n\\LARGE\n\\caption{The details of STAS. $N \\bigtimes C \\bigtimes \\ell \\bigtimes(h \\bigtimes w)_{29\\sim3}$ are multi-scale (from 29*29 to 3*3) dimensions of inputs. $(*)_{[.]}$ represent a operator layer and its parameters setting such as filter size from CNN and output scale from Adaptive Pooling (ADP) or Up-Sampling (UpSp). Specifically, the last parameter of Deformable CNN (D-CNN) is spatial dilation rate. Besides, $\\oplus Noisy$ is addition operation of Gaussian noise, and $(\\cdot)$ is shape of outputs in current module along with pipeline $\\rightarrow$.}\n\\smallskip\n\\centering\n\\resizebox{230pt}{25mm}{\n\\smallskip\\begin{tabular}{l|l|l}\n\\hline\n\\multicolumn{3}{c}{{\\bf Inputs:} $N \\bigtimes C \\bigtimes \\ell \\bigtimes(h \\bigtimes w)_{29\\sim3}$} \\\\\n\\hline \n\\multirow{2}*{SFM} & $CNN_{[1\\times1,0]\\rightarrow[3\\times3,0]}$ & $(N\\bigtimes C\\bigtimes (\\ell-1)\\bigtimes (h\\bigtimes w))$ \\\\\n~ & $D-CNN_{[3\\times3,1,0.8]\\rightarrow[3\\times3,1,0.6]} \\rightarrow ADPooling_{[1\\times1]} \\rightarrow FC$ & $\\rightarrow (N\\bigtimes C \\bigtimes(\\ell-1)\\bigtimes(1\\bigtimes1)) \\rightarrow(N)\\bigtimes(\\ell-1) \\Rightarrow \\vct{s}^*$\\\\\n\\hline\n\\hline\n\\multirow{2}*{Encoder} & $CNN_{[1\\times1,0]\\rightarrow[3\\times3,0]} \\rightarrow ADP_{[16\\times16]}$ & $ \\rightarrow (N\\bigtimes C \\bigtimes \\ell \\bigtimes16\\bigtimes16)$ \\\\\n~ & $\\oplus Noisy$ & $\\oplus \\rightarrow (N\\bigtimes C \\bigtimes \\ell \\bigtimes18\\bigtimes18)$ \\\\\n\\hline\n\\hline\n\\multirow{2}*{TFM} & $3DCNN_{[3\\times3,1]} \\rightarrow (ADP_{[1\\times1]}+FC) \\bigtimes 3$ & $\\rightarrow(N\\bigtimes C\\bigtimes18\\bigtimes18) \\rightarrow(N\\bigtimes6) \\bigtimes 3$  \\\\\n~ & $Concat\\rightarrow FC\\bigtimes2$ & $\\rightarrow (N\\bigtimes18) \\rightarrow (N) \\Rightarrow l^{*}$\\\\\n\\hline\n\\hline\nDecoder & $UpSp_{[18\\times18]}$ & $\\rightarrow (N \\bigtimes C\\bigtimes \\ell \\bigtimes 18 \\bigtimes 18)$ \\\\\n\\hline\n\\hline\nRC & $ CNN_{[3\\times3,1]}\\rightarrow ADP_{[1\\times1]} \\rightarrow FC$ & $(N\\bigtimes C\\bigtimes29\\bigtimes29) \\rightarrow (N\\bigtimes C\\bigtimes 1\\bigtimes1) \\rightarrow (N)$ \\\\\n\\hline\n\\multicolumn{3}{c}{{\\bf Output:} $N$} \\\\\n\\hline\n\\end{tabular}\n}\n\\label{details}\n\\end{table}  \n\n\\subsection{Datasets and Training Details}\n\\noindent {\\bf EC benchmarks (ECb)} are sliced from a high-resolution version of the public EC dataset~\\cite{berrisford2009era}, only covering Eastern China between ranging from June 1st to August 31st for three years (2016-2018). Concretely, ECb consists of 57 weather features (channels) worldwide selected from 601 meteorological factors by Pearson correlation analysis~\\cite{benesty2009pearson}. Every feature stems from a grid where each pixel in the grid means a specific location. The spatial and temporal resolutions of ECb are about $111km$ per pixel and 6h per time-level respectively. Then ECb is divided into four datasets for different experiments. The first three datasets are 1) ECb only including Tiny rainfall (ECbT), 2) ECb only including Moderate rainfall (ECbM) and 3) ECb only including Heavy rainfall (ECbH) separated into half-open range of precipitation intensity interval ($[0,1mm)$, $[1-10mm)$, $[10mm,+\\infty)$). The last one is ECb Mixed 3 rainfall above (ECbMi) and its mixture ratio of samples is $T:M:H = 9:3:1$.\n\n\\noindent {\\bf Labels} are observations of 5 Meteorological Elements (MEs) from MICAPS in the specific locations in Eastern China every $6h$.\n\n\\noindent {\\bf Training details} are shown in Tab.~\\ref{details}. The table includes the main structure of layers and initial settings of modules in STAS. Besides, the batch size of training is 256 and the testing is 64. We employ Adam as optimizers for all modules in STAS and the learning rate is $1e-4$. Besides, the weight ratios on 5MEs for spatio-temporal losses are set as $2:1:1:1:1$ in which rainfall is 2 and other elements are 1. The uniform scale after upsampling and downsampling is set as $16*16$. For ordinal regression and C3AE, the ranking intervals for precipitation value are set as 0.5 and 1.5 respectively. Specifically, we set a constant standard $1e-3$ as white Gaussian noise. We test our model in every 6 epochs on training and the max epoch of training is 80. Finally, all experiments are conducted in 8 NVIDIA GPUs.\n\\begin{table}[htp]\n\\caption{The 5 criteria between 8 machine learning methods and STAS on ECbMi as in Tab.~\\ref{contrastiveI}. $TS_{0.1}$ is TS score in $PI>0.1$. $TS_{1} | TS_{10}$ is $PI>1$ and $PI>10$ respectively. The ECb forecasts are results of predictions from EC benchmarks themselves. SVR is support vector regression, LR is linear regression, MLP is multilayer perceptron, FCN is full convolutional network, FPN is feature pyramid network, LSTM long short-term memory, OBA is ordinal boosting auto-encoder.}\n\\smallskip\n\\centering\n\\resizebox{240pt}{20mm}{\n\\smallskip\\begin{tabular}{c|c|c|c|c|c}\n\\hline \n       &\\multicolumn{5}{|c}{Criteria} \\\\\n\\hline \nMethods& MAE & MAPE & $TS_{0.1}$ & $TS_{1}$ & $TS_{10}$ \\\\\n\\hline \nECb forecasts~\\cite{ran2018evaluation}  & 1.76 & 17.09 & 0.41 & 0.31 & 0.19 \\\\\n\\hline\nSVR~\\cite{srivastava2015wrf} & 1.67 & 15.81 & 0.48 & 0.37 & 0.1 \\\\\n\\hline\nLR~\\cite{hamill2012verification} & 1.73 & 16.90 & 0.35 & 0.35 & 0.21 \\\\\n\\hline\nMLP~\\cite{yuan2007calibration} & 1.59 & 15.13 & 0.46 & 0.39 & 0.21 \\\\\n\\hline\nFCN~\\cite{xu2019towards} & 1.26 & 12.30 & 0.49 & 0.48 & 0.24 \\\\\n\\hline\nFPN~\\cite{lin2017feature} & 1.15 & 7.38 & 0.56 & 0.51 & 0.27 \\\\\n\\hline\nLSTM~\\cite{xingjian2015convolutional} & 1.21 & 9.8 & 0.52 & 0.48 & 0.24 \\\\\n\\hline\nOBA~\\cite{xu2019towards} & 1.01 & 8.96 & 0.58 & 0.53 & 0.25 \\\\\n\\hline\nSTAS(ours) & {\\bf 0.98} & {\\bf 5.84} & {\\bf 0.75} & {\\bf 0.69} & {\\bf 0.38} \\\\\n\\hline\n\\end{tabular}\n}\n\\label{contrastiveI}\n\\end{table} \n\\begin{table*}[t]\n\\centering\n\\caption{Ablation experiments conducted on ECbMi using $TS_{1}$ and $MAPE$. $\\surd$ is defined as an existing component in current ablated STAS for every line of Table. Instead, $\\times$ is no this component in the ablation. SFM-MSMs and TFM-MTMs are spatial and temporal meteorological elements modules severally. Besides, D-CNN is deformable CNN and R, T, P, W, and D represents the module of Rainfall, Temperature, Pressure, Wind and Dew respectively.}\n\\fontsize{8}{9}\\selectfont\n    \\begin{tabularx}{15.5cm}{@{\\extracolsep{\\fill}}|c|c|ccccc|ccccc|c|c||c|c|}\n    \\hline\n    \\multirow{2}{*}{SFM} & \\multirow{2}{*}{TFM} & \\multicolumn{5}{c|}{SFM-MSMs} & \\multicolumn{5}{c|}{TFM-MTMs} & \\multirow{2}{*}{D-CNN} & \\multirow{2}{*}{C3AE} & \\multirow{2}{*}{$TS_1$} & \\multirow{2}{*}{$MAPE$} \\\\\n\n\n   \n          &  & R & T & P & W & D & R & T & P & W & D &  &  &  &  \\\\\n    \\hline\n    $\\surd$ & $\\times$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & \\multicolumn{5}{c|}{N\/A} &$\\surd$ & N\/A & 0.60 & 8.58 \\\\\n    \\hline\n    $\\times$ & $\\surd$ & \\multicolumn{5}{c|}{N\/A} & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ &N\/A & $\\surd$ & 0.64 & 8.01 \\\\\n    \\hline\n    $\\surd$ & $\\surd$ & $\\surd$ & $\\times$ & $\\times$ & $\\times$ & $\\times$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & 0.62 & 7.97 \\\\\n    \\hline        \n    $\\surd$ & $\\surd$ & $\\times$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & 0.65 & 8.15 \\\\\n    \\hline        \n    $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\times$ & $\\times$ & $\\times$ & $\\times$ & $\\surd$ & $\\surd$ & 0.65 & 7.03 \\\\    \n    \\hline        \n    $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\times$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & 0.66 & 7.94 \\\\   \n    \\hline        \n    $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\times$ & $\\surd$ & 0.67 & 6.08 \\\\\n    \\hline        \n    $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\times$ & 0.69 & 6.81 \\\\    \n    \\hline        \n    $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & $\\surd$ & 0.70 & 5.96 \\\\ \n     \\hline           \n    \\end{tabularx}\n  \\label{ablation}\n\\end{table*}\n\\subsection{Evaluation Metrics}\nMAE and MAPE are regarded as two evaluation criteria for training model.  Here MAE~\\cite{willmott2005advantages} is defined as the Mean Absolute Error of corrected precipitation, and MAPE is a variant of MAE without rainless sampless ($<1mm$). In numerical weather prediction, threat score (TS) is a standard criterion for evaluating the accuracy of forecast~\\cite{mesinger2008bias} as follows:\n\\begin{equation} \\label{reconsLoss}\nTS_{\\rho} = H_{\\rho}\/(H_{\\rho} + M_{\\rho} +FA_{\\rho})\n\\end{equation}where $H_{.}$ is Hit (correction = 1, truth = 1), $M_{.}$ is Miss (correction = 0, truth = 1), and $FA_{.}$ is False Alarm (correction = 1, truth = 0), in which 1 is rainfall and 0 is rainless. Specifically, $\\rho$ is a threshold for splitting the range of Precipitation Intensity (PI) into two intervals and set $[0.1,1,10]$ for three different rainfall cases.\n\\subsection{Contrastive Experiments on ECbMi}\nWe list the assessment results of 8 methods and our models on the ECbMi show in Tab.~\\ref{contrastiveI}. The reported results are the average of 20 repetitions, each of which is the mean of predicted results on all batches. STAS outperforms all the other methods on five criteria and its $TS_{10}$ is $28.94\\%$ higher than the second highest result from OBA in this case. Meanwhile, the performance of FPN and LSTM can extract the spatial and temporal features severally beyond the traditional methods from third line to seventh line. Furthermore, FPN has preferable performance than LSTM. There are mainly two reasons for these phenomena above. 1) the performance of BCoP can be promoted either by learning temporal features or spatial features. 2) As for BCop, the spatial features are more important than temporal features. Furthermore, one possible reason is that FPN can predict rainfall utilized adaptive feature layer that has maximum likelihood~\\cite{lin2017feature}, but only learning temporal features in a fixed time scale for LSTM. Besides, the machine learning methods from the fourth to sixth line have somewhat better performance than original ECb forecasts because of utilizing more information from EC data.\n\\begin{table}[htp]\n\\caption{The 4 criteria between 3 machine learning methods and STAS on ECbT\/ECbM\/ECbH divided by precipitation intensity. N\/A(Not Applicable) is none of the samples in the current condition.}\n\\smallskip\n\\centering\n\\resizebox{150pt}{20mm}{\n\\smallskip\\begin{tabular}{c|c|c|c|c|c}\n\\hline \n       &\\multicolumn{1}{|c}{Methods} &\\multicolumn{4}{|c}{Criteria} \\\\\n\\hline \nEcb & & MAPE & $TS_{0.1}$ & $TS_{1}$ & $TS_{10}$ \\\\\n\\hline \n\\multirow{4}*{EcbT} & OBA & 3.81 & 0.65 & N\/A & N\/A \\\\\n~ & LSTM  & 3.79 & 0.55 & N\/A & N\/A \\\\\n~ & FPN  & 2.14 & 0.60 & N\/A & N\/A \\\\\n~ & STAS & {\\bf 2.01} & 0.78 & N\/A & N\/A \\\\\n\\hline\n\\multirow{4}*{EcbM} & OBA & 8.39 & 0.53 & 0.49 & N\/A \\\\\n~ & LSTM  & 8.01 & 0.51 & 0.47 & N\/A \\\\\n~ & FPN  & 6.93 & 0.54 & 0.50 & N\/A  \\\\\n~ & STAS & {\\bf 4.43} & 0.61 & {\\bf 0.59} & N\/A  \\\\\n\\hline\n\\multirow{4}*{EcbH} & OBA & 13.44 & 0.21 & 0.21 & 0.09 \\\\\n~ & LSTM  & 12.81 & 0.24 & 0.24 & 0.16 \\\\\n~ & FPN  & 10.79 & 0.28 & 0.28 & 0.20  \\\\\n~ & STAS & {\\bf 7.05} & 0.38 & 0.38 & {\\bf 0.28}  \\\\\n\\hline\n\\end{tabular}\n}\n\\label{contrastiveII}\n\\end{table}\n\\subsection{Contrastive Experiments on ECbT\/ECbM\/ECbH}\nFor investigating the influence of ST representation in different precipitation intensity, we compare the performance of the 3 targeted methods and STAS on ECBT, ECBM and ECbH as shown in Tab.~\\ref{contrastiveII}.\n\nWe prefer to select $MAPE$ instead of $MAE$ because correcting rainfall samples are our main purpose. Overall, The performance of all methods on $TS_{1}$ and $TS_{10}$ decreases, compared with the same $TS$s of these methods in Tab.~\\ref{contrastiveI}. The possible reason is that the large rainfall value is hard to correct because of the distribution of longspan and the few limited numbers of samples. Both FPN and LSTM have better performance in $TS_{10}$ than OBA because FPN can capture richer multi-scale spatial features and LSTM can obtain temporal dependency in EC. However, OBA only encodes deep representation. Furthermore, the $TS_{10}$ of LSTM on EcbH is $0.04$ lower than FPN in the same case. One possible explanation is that FPN can automatically select a scale layer with the largest confidence level as the predictive layer in the testing phase, whereas LSTM is a fixed ST scale before testing. Finally, It is worth noting that the $MAPE$s of all methods sharply rise with an increment of precipitation intensity. The reason is that the more samples heavy rainfall has, the bigger contribution MAPE according to its equation. Nevertheless, STAS nearly obtains all SOTA results on 3 Ecbs owing to the learning ability of optimized ST representation.\n\n\\subsection{Ablation Experiments}\nHere we perform ablation experiments to verify the effectiveness of each new-introduced components in STAS.\n\n\\noindent {\\bf Impact of SFM and TFM} \\quad The first two lines of Tab.~\\ref{ablation} show the effectiveness of SFM and TFM. It is obvious that the $TS_{1}$ sharply decreases after removing SFM or TFM, and the $TS_{1}$ of without ($w\/o$) SFM is $0.4$ lower than TFM in the same case. One possible reason is that the role of spatial scale is more important in BCoP than that of temporal scale. \n\n\\noindent {\\bf SFM vs TFM for ME modules}\\quad A similar conclusion can be obtained when we evaluate the SFM\/TFM impacts in 5ME modules if we see the results shown in the third line to the sixth line of Tab.~\\ref{contrastiveII}. Furthermore, it is obvious that the rainfall (R) module has a greater impact on $MAPE$ than other modules since it utilizes the historical observations of precipitation.     \n  \n\\noindent {\\bf Impact of deformable CNN and C3AE}\\quad The influences of Deformable CNN (D-CNN) and C3AE are shown in the seventh and eighth lines. MAE of $w\/o$ C3AE increases because C3AE is a regression method, quite like ordinal regression for solving the longspan range of precipitation distribution. Besides, the decrement of $TS_{1}$ on $w\/o$ D-CNN indicates that D-CNN does work in selecting optimized pixels in the process of learning spatial features.        \n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\columnwidth]{figure7.png} \n\\caption{ The visualized comparisons between predictions of precipitation and corresponding ground-truth in the same regions of specific stations. From left to right: the predicted precipitation on OBA and STAS respectively, and observed precipitation. The color-patch bar on the right of Figure is used for distinguishing precipitation intensity by changing light color to high color.}\n\\label{EC_vs}\n\\end{figure}   \n\n\\subsection{Qualitative Analysis}\nWe visualize predictive results from 2 methods and corresponding observations in several examples shown in Fig.~\\ref{EC_vs}. Among the visualizations, the 2 red ovals on the left and middle pictures show that STAS can almost correct heavy rain (mazarine) accurately while OBA cannot correct exactly. Besides, the 2 orange ovals on the 2 same pictures above reveal that OBA has some errors on calibrating moderate rain (green), but STAS is successful for correcting it. All in all, STAS has better prediction performance than OBA in forecasting mesoscale or large-scale precipitation.   \n\n\n\n\n\n\n\n\\iffalse\n\\noindent {\\bf Typhoon dataset} is the real-time observed typhoon data from MICAPS and records typhoon features such as maximum wind velocity and typhoon radius every $6h$. Specifically, we manually label these data according to maximum wind velocity~\\cite{cao2009wind}.\n\\fi\n\n\\iffalse\n\\noindent {\\bf Pre-processing} is divided into 3 procedures including feature selecting, slicing time-series ECMWF dataset in the given stations, and slicing the ECMWF data with typhoon flags. First, we utilize the Pearson Correlation to select the top 57 features that have the highest coefficients on BCoP from 601features in ECMWF. Second, we fix precipitation stations from MICAPS, and empirically slice the effective EC features around one station that are composed of the refined grid points in the geographical range of $\\mbox{[lat-2, lon-2]}$ to $\\mbox{[lat+2, lon+2]}$, then perform the time-slider in the time dimension to require time-series ECMWF dataset. Third, we select the ECMWF data that have the same time with typhoon and these ECMWF data are also included the $\\pm4$ range around the Lat-Lon of this typhoon radius. Meanwhile, we save the label and radius of the typhoon to guide the selection of ECMWF data and flag this typhoon data.\n\\fi\n\n\\iffalse\n\\begin{table}[htp]\n\\caption{The 5 criteria between 8 machine learning methods and STAS on ECbMi. $TS_{0.1}$ is TS score in $PI>0.1$. $TS_{1} | TS_{10}$ is $PI>1$ and $PI>10$ respectively. SVR is support vector regression, LR is linear regression, MLP is Multilayer Perceptron, LSTM is long short-term memory, and FCN is full convolutional network.}\n\\smallskip\n\\centering\n\\resizebox{240pt}{12mm}{\n\\smallskip\\begin{tabular}{c|c|c|c|c|c}\n\\hline \n       &\\multicolumn{5}{|c}{Criterion($TS_{20\/30\/40}$regarding)} \\\\\n\\hline \nMethods& MAE & MAPE & $TS_{0.1}$ & $TS_{1}$ & $TS_{10}$ \\\\\n\\hline \nBCoP Benchmark Method~\\cite{ran2018evaluation}  & 1.76$\\pm$0.13 & 3.18$\\pm$0.21 & 0.41$\\pm$0.04 & 0.35$\\pm$0.03 & 0.19$\\pm$0.04 \\\\\n\\hline\nSVR~\\cite{srivastava2015wrf} & 1.67$\\pm$0.12 & 3.04$\\pm$0.19 & 0.48$\\pm$0.04 & 0.41$\\pm$0.03 & 0.0$\\pm$0.0 \\\\\n\\hline\nLR~\\cite{hamill2012verification} & 1.73$\\pm$0.12 & 2.77$\\pm$0.15 & 0.35$\\pm$0.04 & 0.23$\\pm$0.03 & 0.21$\\pm$0.04 \\\\\n\\hline\nMLP~\\cite{yuan2007calibration} & 1.59$\\pm$0.11 & 2.54$\\pm$0.15 & 0.51$\\pm$0.04 & 0.41$\\pm$0.04 & 0.21$\\pm$0.04 \\\\\n\\hline\nLSTM~\\cite{yu2018coordinate} & (1.39$\\pm$0.08) & 2.37$\\pm$0.12 & 0.58$\\pm$0.04 & 0.48$\\pm$0.02 & 0.24$\\pm$0.02 \\\\\n\\hline\nFCN~\\cite{hu2018soil} & 1.26$\\pm$0.11 & 2.29$\\pm$0.14 & 0.61$\\pm$0.07 & 0.54$\\pm$0.04 & 0.27$\\pm$0.03 \\\\\n\\hline\nOBA\\iffalse~\\cite{hu2018soil}\\fi & 1.01$\\pm$0.05 & 2.21$\\pm$0.14 & 0.58$\\pm$0.03 & 0.53$\\pm$0.01 & 0.29$\\pm$0.01 \\\\\n\\hline\nSTAS(ours) & {\\bf 0.98$\\pm$0.03} & {\\bf 1.98$\\pm$0.10} & {\\bf 0.67$\\pm$0.03} & {\\bf 0.58$\\pm$0.02} & {\\bf 0.35$\\pm$0.02} \\\\\n\\hline\n\\end{tabular}\n}\n\\label{contrastive}\n\\end{table}\n\n\\fi\n\n\\iffalse The form of mean$\\pm$standard deviation. \\fi\n\\iffalse and changing curves on key hyperparameters respectively.      \\fi\n\n\n\n\n\n\\section{Introduction}\n\nWeather forecast plays a crucial role in disaster monitoring and emergency disposal. Numerical Weather Prediction (NWP) based on the equations set of kinetic and thermodynamics~\\cite{liu2017kinetic} is often used to cope with sudden climate change and extreme weather beforehand. One of the representatives of progressive NWP in the global is the European Centre for medium-range weather forecasts (EC)~\\cite{ran2018evaluation}. However, the predictions of precipitation from EC suffer from some uncertainty intrinsic mechanisms of rainfall, e.g., the elusive physical process of rainfall. Therefore, Bias Correcting on Precipitation (BCoP) is the need to improve the forecast level of EC around a local area. \n\\begin{figure}[htbp]\n\\centering\n\\subfigure[$t-2$]{\n\\begin{minipage}[t]{0.3\\linewidth}\n\\centering\n\\includegraphics[width=1\\columnwidth,height=1.2\\textwidth]{figure1}\n\n\\end{minipage}%\n}%\n\\subfigure[$t-1$]{\n\\begin{minipage}[t]{0.3\\linewidth}\n\\centering\n\\includegraphics[width=1\\columnwidth,height=1.2\\textwidth]{figure2}\n\\end{minipage}%\n}%\n\\subfigure[$t$]{\n\\begin{minipage}[t]{0.3\\linewidth}\n\\centering\n\\includegraphics[width=1\\columnwidth,height=1.2\\textwidth]{figure3}\n\\end{minipage}\n}%\n\\centering\n\\caption{The visualization of EC precipitation of Eastern China in the 3 continuous timestamps with region segmentation, which can embody the spatiality and temporal granularity of precipitation.}\n\\label{ST}\n\\end{figure}\n\nClassical BCoP can be roughly divided into two categories: regression and parameter estimation. Regression methods can be regarded as a probabilistic model obeying a credible distribution judged by historical prior information such as expert experience~\\cite{hamill2008probabilistic}. And the parameter estimating methods figure out a set of optimal parameters in some functions such as Kalman filter~\\cite{monache2008kalman} via historical observation for better correcting. Note that in the era of restricted prior information or history observations, these two correcting methods less utilize the spatio-temporal weather regularity in EC~\\cite{hamill2012verification}.    \n\nA feasible way to refine the performance of BCoP is to continuously learn the weather dynamic features from EC data and boosting the correcting ability with machine learning algorithms~\\cite{srivastava2015wrf}.\n\nNevertheless, shallow or low-level dynamic representation is not enough to significantly improve the performance of BCoP. Therefore, it is necessary to capture the high-level representation such as spatial and temporal-dependencies from EC data. To clarify this point, we visualize the colored EC precipitation region in the 3 continuous timestamps shown in Fig.~\\ref{ST}, and employ a Simple Linear Iterative Clustering (SLIC) is employed for segmenting different precipitation subregions by clustering the pixels in similar semantic information~\\cite{achanta2012slic}. From Fig.~\\ref{ST}, we can observe that pixels with the same color are naturally segmented into the same region, separated by the purple line, which reflects the spatiality of precipitation. Besides, an obvious movement for the positions of precipitation regions over time reflects varied temporal granularity. \n\nWe thus propose a deep ST Feature Auto-Selective (STAS) Model for learning ST representation. Further, we add two pre-trained modules termed Spatial Feature-selective Mechanism (SFM) and Temporal Feature-selective Mechanism (TFM) to STAS, in which five observations of meteorological elements (precipitation\/temperature\/pressure\/wind\/dew) are used to guide the selection of optimal ST scales shown in Fig.~\\ref{pipeline} for better extracting ST features. Besides, we integrate a binary classifier so that the precipitation prediction can be more precise. The final prediction is thus achieved by multiplying regression and classified results. The contributions of STAS are summarized as follows:\n\n\\begin{itemize}\n\\item {\\bf Spatial Adaptivity} The Spatial Feature-selective Mechanism (SFM)  adaptively selects an optimal spatial scale of specific EC data for capturing the richer spatial representation, which can refine the performance of correcting, especially in the heavy rainfall, indicating its practicability in forecasting mesoscale or large-scale precipitation.  \n\\item {\\bf Temporal Adaptivity} The Temporal Feature-selective Mechanism (TFM) is utilized for automatically choosing the optimal time-lagging sequence of time-series features of EC in line with the minimal loss value for acquiring the better temporal representation.      \n\\item {\\bf Effective} Experiments indicate that our model achieved better prediction performance than the other 8 published methods on a BCOP benchmark dataset, especially for correcting the large-scale precipitation.\n\\end{itemize}\n\n\n\n\\section{STAS: A Spatio-Temporal Feature Auto-Selective Model}\n\nIn this section, we will introduce our proposed STAS for automatically selecting the spatial and temporal scales of meteorological features from European Centre for Medium-Range Weather Forecasts (EC) in detail. For better illustration, an overall pipeline of STAS is shown in Fig.~\\ref{pipeline}.\n\n\\begin{figure*}[!thbp] \n\\centering\n\\includegraphics[width=0.8\\textwidth,height=0.28\\textwidth]{figure4.png}\n\\caption{The framework is STAS. SFM and TFM is the mechanism of spatial and temporal feature-selective respectively. $s^{*}$ is adaptive spatial size of an EC data in one batch and $\\ell^{*}$ is adaptive time-lagging length of the encoded time-series features. $s^{*}$ and $\\ell^{*}$ are adaptively adjusted by SFM and TFM respectively. $\\mathcal{L}$s are main loss functions in STAS and RC is rainfall classifier. $\\mathcal{L}_{min}$s are minimal losses from the process of STAS.}\n\\label{pipeline}\n\\end{figure*}\n\n\\subsection{Notations and Methodology}\nFirst, we define several basic symbols for our method. Assuming that we have a total of $N$ surface observation stations from Eastern China. The EC data around one station can be set as the combination of the refined grid points in the geographical range from \\mbox{[$\\emph{da} - \\omega$, $\\emph{do} - \\omega$]} to \\mbox{[$\\emph{da} + \\omega$, $\\emph{do} + \\omega$]}, where \\emph{da} and \\emph{do} are the latitude and longitude of this station respectively, and $\\omega$ is the degree. Furthermore, the time-series EC data from the $i$th station at time level $t$ are defined as $(\\mat{X}^{i}_{t})_{\\ell} = [\\mat{X}_{t}^{u}, \\mat{X}_{t-1}, \\ldots, \\mat{X}_{t-\\tau}]_{\\ell}$, where $i \\in N$, and $\\ell$ is the length of time sequence and $\\ell = \\tau + 1$. $u$ is uniform spatial scale. In this study, the interval of the sequence $\\ell$ is 6h. Therefore, $(\\mat{X}^{i}_{t})_{\\ell}$ can be regarded as a four-dimensional tensor-form input including the dimensions of the channels of features, the length of the sequence, the height and width of features.\n\nWith these notations, we roughly build 3 subdivisions for performing Bias Correcting on Precipitation (BCoP) in order as follows:\n\\begin{eqnarray}\\label{eq_pipeline}\n \\hat{y}_{tp} &= & OR \\left \\{LSTM\\left ( E\\left ([\\mat{X}^{u}_{t}, \\ldots, \\mat{X}^{s^{*}_{\\tau}}_{t-\\tau}]\\right)_{\\ell^{*}}\\right)  \\right\\}  \\\\\n \\hat{y}_{rc} &= & RC(\\mat{X}^{u}_{t})  \\\\\n \\hat{y}_{t} &= & \\hat{y}_{tp} \\otimes \\hat{y}_{rc}\n\\end{eqnarray}\nwhere $s^{*}_{\\tau}$ denotes the optimized size (height $\\times$ width) of the EC data $\\mat{X}_{t-\\tau}$ through spatial feature-selective mechanism (SFM), which will be introduced in Sec.3.2. Parameter $\\ell^{*}$ denotes the refined time-lagging length for time-series features $[(\\mat{X}^{i}_{t})_{\\ell}]^{u}$ that are encoded (Sec.3.3) to a uniform ($u$) size via temporal feature-selective mechanism (TFM), which will be discussed in Sec.3.4. $E(\\cdot)$ is an encoder backbone and $LSTM(\\cdot)$ is a stacked ConvLSTM~\\cite{xingjian2015convolutional}. The acronym $OR(\\cdot)$ means an ordinal regression model~\\cite{zhu2018facial} is used for regressing corrected precipitation value in the end. Besides, we utilize the precipitation binary classifier $RC(\\cdot)$ for classifying raining or rainless samples. Finally, predicted result $\\hat{y}_{t}$ is obtained by multiplying the predicted precipitation value $\\hat{y}_{tp}$ and the classified result $\\hat{y}_{rc}$.\n\n\\subsection{Spatial Feature-Selective Mechanism}\n\\vspace{-0.1em}\nWhen lacking an instructive spatial scale, the EC features centered on all stations are empirically set to be a fixed spatial scale for prediction. However, these rules-of-thumb may impair predictive accuracy when there is a strong connection between features scale and precipitation intensity~\\cite{mu2003conditional}. Therefore, we propose a Spatial Feature-selective Mechanism (SFM) to adaptively search the optimal spatial scales of specific EC data based on observations of 5 Meteorological Elements (MEs) including precipitation, temperature, pressure, wind, dew. Concretely, we can find the optimal spatial scales by minimizing the total spatial losses in different scales, which are the summation of 5 spatial MSE losses between predictive MEs and their observations shown in Fig.~\\ref{frameSm}. The selection is formulated as:\n\\begin{eqnarray}\\label{spatialMe}\ns^{*}  &= & \\mathop{\\arg\\min}_{s}\\mathcal{L}_{s} \\\\ \n\\mathcal{L}_{s}  & = & \\sum_{i=0}^{n(MEs)}\\mathcal{L}_{MSE}(MSM_i(\\mat{X'}^{s}_{t}), y_{t}^{i})\n\\end{eqnarray}where $\\mathcal{L}_{s}$ is the spatial total loss in the scale $s$. $n(MEs)$ is the number of MEs. $MSM_i(\\cdot)$ is $i$-th ME Spatial Module and $\\mat{X'}^{s}_{t}$ is a given EC data that is scale $s$ in timestamp $t$. $y_{t}^{i}$ are the labels of $i$-th ME in timestamp $t$. Therefore, the goal is to search an optimal scale $s^{*}$ shown in Eq \\eqref{spatialMe}. Specifically, a deformable CNN layer is introduced for boosting the rep- resentational ability of 5 modules via learning the offsets of filters that are appropriate for capturing better features~\\cite{dai2017deformable}. \n\n\\subsection{Backbone with Denoising}\n\\vspace{-0.1em}\nThe EC features contain numerous noises~\\cite{xu2019towards} that can produce negative side effects for correcting. To solve this issue, we introduce a denoised Encoder-Decoder (E-D) as backbone shown in Fig.~\\ref{pipeline}. In the process of encoder, the Gaussian white noise is added to the encoded features that have uniform size $u$ via upsampling and downsampling. We also introduce a reconstruction loss utilized for calculating difference between encoded features and reconstructed features by decoder to optimize denoised ability of encoder as follows:\n\\begin{equation} \\label{reconsLoss}\n\\mat{W}_{E}^*, \\mat{W}_{D}^* = \\mathop{\\arg\\min}_{\\mat{W}_{E},\\mat{W}_{D}}{\\Vert E([\\mat{X}^{u}_{t}, \\ldots, \\mat{X}^{s^{*}_{\\tau}}_{t-\\tau}], \\mat{\\epsilon}) - D(Z_{t}) \\Vert}\n\\end{equation}where $\\mat{W}_{E}$ and $\\mat{W}_{D}$ are parameter matrices from encoder $E(\\cdot)$ and decoder $D(\\cdot)$ respectively. $[\\mat{X}^{u}_{t}, \\ldots, \\mat{X}^{s^{*}_{\\tau}}_{t-\\tau}]$ is time-series features with optimized spatial scale $\\vct{s^{*}}$ through SFM. $Z_{t}$ is hidden features by $E(\\cdot)$. Moreover, $\\mat{\\epsilon}$ is Gaussian noise.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.4\\columnwidth, height=0.28\\textwidth]{figure5.png} \n\\caption{The structure of Spatial Feature-selective Mechanism (SFM), the visualizations of 5 Meteorological Elements (ME) are temperature, pressure, wind, dew and precipitation. $\\{ \\mathcal{L}_{s1}, \\ldots, \\mathcal{L}_{sw} \\}$ are MSE spatial total losses that are the sum ($\\oplus$) of 5 spatial MSE losses from ME spatial modules in different scales. SFM can select the adaptive spatial scale $s^{*}$ of specific EC data in $\\mat{X'}_{t}$. The deformable CNN is utilized for helping filters to operate the given pixels that can capture the better representation.}\n\\label{frameSm}\n\\end{figure}    \n\\subsection{Temporal Feature-Selective Mechanism}\n\\vspace{-0.1em}\nIt is obvious that rainfall patterns in one station are not only related to EC features around this station, but also closely connected with historical features~\\cite{ciach2006analysis}. Therefore, the temporal Feature-selective Mechanism (TFM) is proposed for adaptively acquiring an optimized encoded features $[(\\mat{X}^{i}_{t})_{\\ell^*}]^{u} = [\\mat{X}^{u}_{t}, \\ldots, \\mat{X}^{u}_{t-\\tau}]_{\\ell^*}$, , from which more useful temporal representation can be learned. Formally, it is defined as: \n\\begin{eqnarray}\\label{temporalMe}\nl^{*}  &= & \\mathop{\\arg\\min}_{\\ell}\\mathcal{L}_{T} \\\\\n\\mathcal{L}_{T}  & = & \\sum_{i=0}^{n(MEs)}\\mathcal{L}_{MAE}(MTM_i([(\\mat{X}^{i}_{t})_{\\ell}]^{u}), y_{t}^{i})\n\\end{eqnarray}where $\\ell^{*}$ is the optimal time-lagging length and $y_{t}^{i}$ are labels of $i$-th MEs in timestamp $t$. $MTM_i(\\cdot)$ is $i$-th ME Temporal Module. Same as SFM, TFM can select optimal time-lagging sequence $\\ell^{*}$ by finding out minimum temporal total $MAE$ loss, which are the sum of 5 temporal $MAE$ losses between predictions of $MTM(\\cdot)$ and labels of MEs in different time-lagging sequences. The detailed structure of C3AE~\\cite{zhang2019c3ae} and 3DCNN~\\cite{zhang2017learning} are shown in Fig.~\\ref{frameTm}. In consideration of maldistribution of precipitation values~\\cite{xu2019towards}, an ordinal regression (OR) method~\\cite{zhu2018facial} is utilized for outputing regressing value of precipitation $\\hat{y}$ shown in Fig.~\\ref{pipeline}. OR may solve the problem of longspan range of precipitation values and convert a regression task into a multi-binary classification one to reduce the complexity of regression.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.3\\columnwidth,height=0.28\\textwidth]{figure6.png} \n\\caption{ The structure of Temporal Feature-selective Mechanism (TFM). TFM can select adaptive time-lagging length $\\ell^*$ of one time-series features $(\\mat{X}^{i}_{t})_{\\ell}$. $\\{ \\mathcal{L}_{T1}, \\ldots, \\mathcal{L}_{Th} \\}$ are MAE temporal total losses that are the sum ($\\oplus$) of 5 MAE temporal losses from ME temporal modules in different time-lagging sequence. 3D CNN is utilized for capturing the patial-temporal dependency. Specifically, C3AE is a lightweight rank learning module and suited for regression distribution that has longspan range.}\n\\label{frameTm}\n\\end{figure}   \n\n\\subsection{Training and Testing}\n{\\bf Training}\\quad In the training phase, we firstly pre-train the Spatial Feature-selective Module (SFM) and Temporal Feature-selective Module (TFM) so that these two modules can predict the MEs accurately. Secondly, we integrate SFM into our framework shown in Fig.~\\ref{pipeline} for predicting the MEs from historical EC data $\\mat{X'}_{t}^{s}$ and selecting the optimal scale $s^{*}$ that has minimal MSE loss. Similarly, we begin to train Temporal Feature-selective Module (TFM) and Encoder-Decoder (E-D) together, and use a specific ADAM optimizer for BP when adaptive time-lagging length $\\ell^{*}$ in one batch is selected by TFM. Meanwhile, the rainy classifier (RC) is cross-trained along with TFM\/SFM and E-D.\n\n\\noindent{\\bf Testing}\\quad In the testing phase, SFM plays a role in selecting the optimal scale $s^{*}$ and TFM selects the adaptive time-lagging length $\\ell^{*}$, then SFM is forward to calculate probabilities of all classifiers from ordinal regression. We select specific classifiers with large probabilities according to initialized interval $\\xi$ and transform these probabilities into regression value of precipitation $\\hat{y}_{tp}$ formulated as:\n\\begin{equation} \\label{temporalMe}\n\\hat{y}_{tp} = \\xi * \\sum\\limits_{v=0}^{c-1}(p_{v}\\ge\\xi)\n\\end{equation} where $p_{v}$ is classified probability of the $v$-th binary classifier. Besides, RC is forward to obtain the classified result $\\hat{y}_{rc}$. Final predicted result $\\hat{y}_{t}$ is required by multiplying SFM result $\\hat{y}_{tp}$ and RC result $\\hat{y}_{rc}$ as shown in the third equation of Eq \\eqref{eq_pipeline} in Sec.3.1.\n\n\n\n\n\n\\iffalse\nThe conditional nonlinear optimal perturbation (CNOP)~\\cite{mu2003conditional} is employed for searching a sensitive area in time dimension of history time level that can has an important effect of current prediction of precipitation. Some methods based on CNOP such as Ensemble Transforming Kalman Filter (ETKF) have some effect on finding sensitive area. However, these methods embrace excess intervention from experts and manual features, so increasing the overhead. \n\\fi  \n\n\n\\iffalse\nTheoretically, the classified accuracy of RC is higher than OR because of only having one binary label. \n\\fi\n\n\\iffalse\n~\\cite{ciach2006analysis}\n\\fi\n\n\\iffalse\nSpecifically, the key role of RC is to enhance error-tolerant rate of SFM for improving performance of our model.\n\\fi\n\n\\section{Related Work}\n\n\\subsection{Bias Correcting on Precipitation} \nIn this section, we will give a brief survey on Bias Correcting on Precipitation (BCoP) and spatio-temporal (ST) pattern selection. Classical BCoP can be grouped into regression and parameter estimating methods. Regression methods~\\cite{hamill2008probabilistic} heavily depend on expert experience, which requires manually setting a threshold for the generation of probability, losing their flexibility and adaptivity. Meanwhile, the parameter estimating methods heuristically assess the key parameters set by trial and error in specific models. Nevertheless, traditional BCoP methods suffer from limited available priors and historical observations, and less utilize the clue of possible motion or dynamics existed in European Centre for medium-range weather forecasts (EC) data. To address these issues, some recently proposed models attempt to capture complex climate patterns from the large-scale EC data and optimize the model in line with observations~\\cite{srivastava2015wrf}. However, these methods neglect the potential dependency among variables in EC data, especially the ST dependency.      \n\n\\subsection{Spatio-Temporal Pattern on EC Precipitation}\nExcept for qualitative analysis for reflecting the ST dependencies shown in Fig.~\\ref{ST}, it is proved that rainfall value in one location correlates with weather indicators such as temperature, pressure, wind and dew~\\cite{yapp1982model}. However, we do not know which scale around this location has a strong connection with the precipitation. Moreover, we know that the system on atmospheric dynamics is a  spatio-temporal evolution system, in which physical field changes over time~\\cite{mu2003conditional}. Hence, capturing the ST representation would be a feasible way to improve the performance of BCoP. We consider employing an end-to-end deep-learning model based on multi-scale EC feature~\\cite{lin2017feature}, and adaptively select the features that have an optimal scale and take advantage of these features for correcting precipitation more accurately. To our knowledge, there is no report on how to adaptively extract the ST features based deep-learning model for BCoP in literature. This is the first time to adaptively extract ST representation end-to-end.\n      \n\n\n\\iffalse \n\nIn deep learning domain, as we know, ST Long Short-Term Memory (ST-LSTM) is applied for action recognition by capturing the spatial features from joints of the trunk and temporal features from consecutive frames of action~\\cite{liu2016spatio}. Besides, a ST residual network is employed for traffic flow prediction~\\cite{zhang2017deep}.\n\nMoreover, conditional nonlinear optimal perturbation (CNOP), is utilized for identifying a sensitive time area that has a key influence on specific precipitation along the time dimension of this precipitation~\\cite{mu2003conditional}. \n\\fi \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Sec. Introduction}\n\nIn his pioneering 1968 Ph.D. thesis, Leonard Parker discovered the surprising phenomenon that evolving cosmological spacetimes can produce quantum particles \\cite{Parker69, Parker71, ZeldovichStarobinsky77}. This work laid much of the theoretical framework for our current understanding of quantum fields in curved spacetimes (see, e.g., the textbooks \\cite{ParkerToms09, BirrellDavies82}). The phenomenon has since been investigated in  astrophysical and cosmological contexts, leading to fundamental theoretical results including the emission of Hawking radiation by blackholes \\cite{Hawking75, Unruh76, Davies76, Page82, Brown86, Frolov87, Anderson93, Anderson94, Anderson95} and the generation of primordial fluctuations during inflation \\cite{Starobinsky79, Allen88, Sahni90, Mukhanov92, Souradeep92, Glenz09, Agullo11}.\n\nCosmological quantum particle production results from the shifting quantum vacuum state due to spacetime expansion. A given mode of a quantum field can begin in a state with no particles, but at a later time have a non-zero particle number expectation value. Parker showed that while this does not happen for massless conformally-coupled fields, it is generic for massive scalar fields of arbitrary coupling to the spacetime curvature. Typically, the particle production is significant when the particle mass $m$ is of the order of the expansion rate $H$, or when $m^{2} \\simeq \\dot{H}$. Simple dimensional arguments show that in a Universe with critical density $\\rho_c \\simeq 3H^{2}\/8\\pi G$, quantum particle production can contribute significantly to the energy density of the Universe at early epochs when $H$ is not too far below the Planck scale. \n\nThis effect naturally suggests that quantum particle production may have a significant impact on the expansion rate of the Universe at early times. The back-reaction problem consists in understanding how the energy density generated through this process alters the evolution of the background spacetime. This seemingly straightforward calculation is actually subtle due to two technical challenges of quantum fields in curved spacetimes. First, the definition of the quantum vacuum and particle are not formally well defined in time-varying spacetimes. Second, the energy-momentum tensor of a quantized field propagating on a curved background possesses a formally infinite expectation value, and must be regularized to yield physically sensible results. In homogeneous and isotropic spacetimes, both problems can be addressed via the method of adiabatic regularization \\cite{ParkerFulling74, FullingParker74, Bunch80, Fulling74, Christensen78, AndersonParker87}, which is particularly useful for numerical computations. The adiabatic notion of particle offers a clear way to track particle number in a spacetime which is expanding sufficiently slowly. This representation has the special property of defining a vacuum state which comes closest to the Minkowski vacuum when the background expansion is sufficiently slow. However, adiabatic regularization introduces several technical complications to the back-reaction problem. As a consequence, ambiguities arise in specifying initial conditions, and computationally the problem becomes very complicated with potential numerical instabilities \\cite{Anderson83, Anderson84, Anderson85, Anderson86, SuenAnderson87}. In practice, the ambiguities and complexities together have prevented any general numerical solution, although the test-field limit in which back-reaction effects are neglected has been investigated in several studies \\cite{Birrell78, Anderson00, Anderson05, Bates10, Habib99}.\n\nFormal developments clarified aspects of the adiabatic regularization approach, offering a clear separation between the energy density due to the field particle content and the divergent contributions from the zero-point energy \\cite{Habib99, AndersonMottola14, AndersonMottola14b, Anderson18}. However, the notion of adiabatic particle appearing in the energy density suffers from ambiguities. For a typical mode of a quantum field, its associated particle number during times of significant particle production depends on the perturbative order of adiabatic regularization employed.  This is obviously an unphysical result, since the dominant term in the energy density is often just a simple function of the particle number density. Typically, successive orders of adiabatic regularization gives the particle number in a given mode as a divergent asymptotic series. Recent important papers by Dabrowski and Dunne \\cite{DabrowskiDunne14, DabrowskiDunne16} employed a remarkable result of asymptotic analysis \\cite{Dingle73, Berry90, Berry82, Berry89, Berry88} to sum the divergent series and provide a sensible notion of particle which is valid at all times. Since any remaining physical ambiguity is then removed from the problem, this result points the way to a general numerical solution.\n\nWe combine these recent results into an approach which numerically solves the quantum back-reaction problem in regimes with field energy density dominated by particle production. We then apply this technique to a toy cosmological model, that of a positive-curvature spacetime with a constant energy density (the closed de Sitter model). In the absence of any quantum fields or other particle content, this spacetime exhibits a bounce behavior, contracting to a minimum scale factor and then expanding again. Here we show explicitly that the existence of a massive scalar field in a particular mass range will cause large changes in the spacetime evolution: even if the contracting spacetime initially contains a quantum field in its adiabatic vacuum state, quantum particle production can create enough energy density to push the spacetime into a radiation crunch. Special values of the field mass can also delay but not eliminate the bounce. This appears to be the first general solution for the quantum back-reaction problem in cosmology.\n\nIn Section II we review standard results for quantized scalar fields propagating in cosmological spacetimes, while Section III discusses the adiabatic field representation and the semi-classical notion of adiabatic particle number. Section IV recasts quantum particle production in terms of the Stokes phenomenon of the complex-plane wave equation for specific modes, including interference between different modes. Section V formulates the back-reaction problem for scenarios in which the field particle content or particle production dominates the field energy density. Section VI outlines our numerical implementation of the mathematical results in Sections IV and V. Physical results for a closed de Sitter model are presented in Section VII. Finally, in Section VIII we discuss the prospects for more general situations, including quantum fields with spin and interacting quantum fields, and the possible relevance of quantum particle production to early-Universe models, including inflationary and bounce scenarios. Salient technical details are summarized in the Appendix. Natural units with $\\hbar = c = 1$ are adopted throughout.\n\n\\section{Scalar Fields in FLRW Spacetimes}\\label{Sec. QFT in FLRW}\n\nWe first summarize basic results for scalar fields in spatially isotropic and homogeneous spacetimes (see, e.g., \\cite{ParkerToms09, BirrellDavies82}). \nConsider a Universe described by the Friedmann-Lema\\^{i}tre-Robertson-Walker (FLRW) metric\n\\begin{equation}\\label{Eq. FLRW Metric}\n\t\\mathrm{d}s^{2} = g_{ab}\\mathrm{d}x^{a}\\mathrm{d}x^{b} = -\\mathrm{d}t^{2} + a^{2}(t) g_{ij}\\mathrm{d}x^{i}\\mathrm{d}x^{j} \\, ,\n\\end{equation}\nwith\n\\begin{equation}\\label{Eq. FLRW Spatial Metric}\n\tg_{ij}\\mathrm{d}x^{i}\\mathrm{d}x^{j} =\\frac{\\mathrm{d}r^{2}}{1 - K r^{2}} + r^{2} \\mathrm{d}\\theta^{2} + r^{2} \\sin^{2}\\theta \\, \\mathrm{d}\\varphi^{2} \\, .\n\\end{equation}\nHere $a(t)$ is the scale factor which describes the cosmological expansion history, and $K = -1, \\, 0, \\, +1$ corresponds to the curvature parameter of an open, flat, and closed Universe, respectively.\nThe non-vanishing components of the Ricci tensor $R_{ab}$ are\n\\begin{subequations}\\label{Eq. Ricci Tensor}\n\\begin{align}\n\tR_{00}(t) &= 3\\Big[ \\dot{H}(t) + H^{2}(t) \\Big] g_{00} \\, ,\\\\[8pt]\n\tR_{ij}(t) &= \\Big[ \\dot{H}(t) + 3 H^{2}(t) + 2K\/a^{2}(t) \\Big] g_{ij} \\, ,\n\\end{align}\n\\end{subequations}\nand the Ricci scalar $R = g^{ab}R_{ab}$ is\n\\begin{equation}\\label{Eq. Ricci Scalar}\n\tR(t) = 6 \\Big[ \\dot{H}(t) + 2 H^{2}(t) + K\/a^{2}(t) \\Big] \\, \n\\end{equation}\nwhere\n\\begin{equation}\\label{Eq. Hubble Parameter}\n\tH(t) \\equiv \\frac{\\dot{a}(t)}{a(t)} \n\\end{equation}\nis the Hubble parameter. Overdots indicate differentiation with respect to proper time $t$.\n\nWe are interested in the evolution of a free scalar field $\\Phi(t,\\, \\mathbf{x})$ of arbitrary mass and curvature coupling in this spacetime. The action for such a field can be expressed generically as\n\\begin{align}\\label{Eq. Field Action}\n\tS = -\\frac{1}{2} \\int \\sqrt{-g} \\, \\mathrm{d}^{4}x \\Big[ \\big( \\nabla_{a} \\Phi \\big) g^{ab} \\big( \\nabla_{b} \\Phi \\big) + \\Big. & \\\\\n\t \\Big. + m^{2} \\Phi^{2} + \\xi R \\Phi^{2} \\Big] & \\, , \\nonumber\n\\end{align}\nwhere $\\nabla_{a}$ is the covariant derivative, $g = \\det{(g_{ab})}$, $m$ is the field mass, and $\\xi$ is the field coupling to the spacetime curvature. Applying the variational principle to this action yields the equation of motion \n\\begin{equation}\\label{Eq. Field Equation of Motion}\n\t\\Big[ \\Box - m^{2} - \\xi R(t) \\Big]\\Phi(t,\\, \\mathbf{x}) = 0 \\, ,\n\\end{equation}\nwhere $\\Box = g^{ab}\\nabla_{a}\\nabla_{b}$ is the d'Alembert operator associated with the spacetime.\n\nDue to the homogeneity and isotropy of the background metric, the solutions of Eq.~(\\ref{Eq. Field Equation of Motion}) \ncan be separated into purely temporal and spatial parts. As a consequence, the quantized field operator can be written as\n\\begin{align}\\label{Eq. Field}\n\t\\hat{\\Phi}(t,\\, \\mathbf{x})=a^{-3\/2}(t) \\! \\int \\mathrm{d}\\mu(k) \\Big[ a_{\\mathbf{k}}^{\\phantom{\\dagger}}f_{k}(t)Y_{\\mathbf{k}}(\\mathbf{x}) + \\Big. & \\\\\n\t\\Big. + a_{\\mathbf{k}}^{\\dagger}f_{k}^{\\ast}(t)Y_{\\mathbf{k}}^{\\ast}(\\mathbf{x}) \\Big] & \\, , \\nonumber\n\\end{align}\nwhere the raising and lowering operators $a_{\\mathbf{k}}^{\\dagger}$ and $a_{\\mathbf{k}}^{\\phantom{\\dagger}}$ satisfy the canonical commutation relations\n\\begin{equation}\\label{Eq. Commutation Relations}\n\\left[ a_{\\mathbf{k}}^{\\phantom{\\dagger}} \\, , a_{\\mathbf{k^{\\prime}}}^{\\dagger} \\right] = \\delta_{\\mathbf{k},\\, \\mathbf{k^{\\prime}}} \\, ,\n\\end{equation}\nand $\\mathrm{d}\\mu(k)$ is a geometry-dependent integration measure given by \n\\begin{equation}\\label{Eq. Integration Measure}\n\t\\int \\mathrm{d}\\mu(k) =\n\t\\left\\{\\begin{aligned}\n\t\t\t&\\sum_{k=1}^{\\infty} k^2\\, , && \\text{for } K = +1 \\\\\n\t\t\t&\\int_{0}^{\\infty} k^2 \\,\\mathrm{d}k \\, , && \\text{for } K = 0,\\,-1 \\, .\n\t\t\\end{aligned}\\right.\n\\end{equation}\nThe functions $Y_\\mathbf{k}(\\mathbf{x})$ and $f_k(t)$ contain the spatial and temporal dependence of each $\\mathbf{k}$-mode. The harmonic functions $Y_\\mathbf{k}(\\mathbf{x})$ are eigenfunctions of the Laplace-Beltrami operator associated with the geometry of spatial hypersurfaces, while the mode functions $f_k(t)$ obey the harmnonic oscillator equation\n\\begin{equation}\\label{Eq. Mode Equation}\n\\ddot{f}_{k}(t) + \\Omega_{k}^{2}(t) f_{k}(t) = 0 \\, \n\\end{equation}\nwith the time-dependent frequency function \n\\begin{equation}\\label{Eq. Mode Frequency}\n\\Omega_{k}^{2}(t) = \\omega_{k}^{2}(t) + \\bigg( \\xi - \\frac{1}{6} \\bigg) R(t) - \\Bigg[ \\frac{\\dot{H}(t)}{2} + \\frac{H^{2}(t)}{4} \\Bigg] \\, ,\n\\end{equation}\nwhere\n\\begin{equation}\\label{Eq. Minkowski Frequency}\n\t \\omega_{k}(t) = \\Bigg[\\frac{k^{2}}{a^{2}(t)} + m^{2}\\Bigg]^{\\!1\/2} \\, .\n\\end{equation}\nThe complex mode functions $f_{k}(t)$ and $f^{\\ast}_{k}(t)$ also satisfy the Wronskian condition\n\\begin{equation}\\label{Eq. Wronskian Condition}\nf_{k}(t)\\dot{f}^{\\ast}_{k}(t) - \\dot{f}_{k}(t) f^{\\ast}_{k}(t) = i \\, .\n\\end{equation}\nIf Eq.~(\\ref{Eq. Wronskian Condition}) holds at some particular time $t$, then Eq.~(\\ref{Eq. Mode Equation}) guarantees it will also hold at all future\ntimes.\n\nThe quantization procedure outlined above naturally leads to the construction of the Fock space of field states. The base element of this space is the vacuum state, which is defined as the normalized state that is annihilated by all lowering operators:\n\\begin{equation}\\label{Eq. Vacuum}\na_{\\mathbf{k}}^{\\phantom{\\dagger}} \\left| 0 \\right\\rangle = 0 \\ \\ \\mathrm{and} \\ \\ \\left\\langle 0 | 0 \\right\\rangle = 1 \\, .\n\\end{equation}\nAll remaining states are generated from the vacuum by the successive application of raising operators, such as\n\\begin{equation}\n\\left| \\mathbf{k}_1 ,\\, \\mathbf{k}_2 ,\\, \\dots \\right\\rangle = a_{\\mathbf{k}_1}^{\\dagger} a_{\\mathbf{k}_2}^{\\dagger} \\dots \\left| 0 \\right\\rangle \\, ,\n\\end{equation}\nand normalized by the requirement of mutual orthonormality. In what follows, we will be interested in the family of field states which are spatially isotropic and homogeneous, as these constitute viable sources of the FLRW metric.\n\nThe field operator $\\hat{\\Phi}(t,\\, \\mathbf{x})$ admits numerous representations of the form shown in Eq.~(\\ref{Eq. Field}), each of which is associated with a different mode function pertaining to the set of solutions of Eq.~(\\ref{Eq. Mode Equation}). These representations are related: the complex mode functions $f_{k}(t)$ and $h_{k}(t)$ belonging to any two different representations can be expressed in terms of one another through the Bogolyubov transformations\n\\begin{subequations}\\label{Eq. Bogolyubov Modes}\n\\begin{align}\n\t{f}_{k}(t) &= \\alpha_{k}h_{k}(t) + \\beta_{k}h^{\\ast}_{k}(t) \\, ,\\\\[8pt]\n\t{f}_{k}^{\\ast}(t) &= \\beta_{k}^{\\ast}h_{k}(t) + \\alpha_{k}^{\\ast}h^{\\ast}_{k}(t) \\, ,\n\\end{align}\n\\end{subequations}\nwhere $\\alpha_{k}$ and $\\beta_{k}$ are known as Bogolyubov coefficients. Due to homogeneity and isotropy, these coefficients depend only on ${k = \\left| \\mathbf{k} \\right|}$. Substituting these expressions into Eq.~(\\ref{Eq. Field}) leads to similar transformations relating the raising and lowering operators belonging to these representations:\n\\begin{subequations}\\label{Eq. Bogolyubov Operators}\n\\begin{align}\na_{\\mathbf{k}}^{\\phantom{\\dagger}} &= \\alpha_{k}^{\\ast} b_{\\mathbf{k}}^{\\phantom{\\dagger}} - \\beta_{k}^{\\ast} b_{\\mathbf{k}}^\\dagger\\, , \\\\[8pt]\na_{\\mathbf{k}}^\\dagger &= \\alpha_{k} b_{\\mathbf{k}}^\\dagger - \\beta_{k} b_{\\mathbf{k}}^{\\phantom{\\dagger}} \\, ,\n\\end{align}\n\\end{subequations}\nfrom which it follows that the Bogolyubov coefficients must satisfy\n\\begin{equation}\\label{Eq. Bogolyubov Constraint}\n\t\\big| \\alpha_{k} \\big|^{2} - \\big| \\beta_{k} \\big|^{2} = 1 \\,\n\\end{equation}\nin order to guarantee that the commutation relations of Eq.~(\\ref{Eq. Commutation Relations}) are valid across all representations.\n\nA direct consequence of Eqs.~(\\ref{Eq. Bogolyubov Operators}) is that the notion of vacuum is not unique for a quantized field defined on a FLRW spacetime \\cite{BirrellDavies82, ParkerToms09}. This is evident from the following simple calculation, which shows that the vacuum defined in Eq.~(\\ref{Eq. Vacuum}) is not necessarily devoid of particles according to the number operator belonging to a different field representation:\n\\begin{align}\n\t\\mathcal{N}_{k} &= \\big\\langle 0 \\big| b_{\\mathbf{k}}^{\\dagger} b_{\\mathbf{k}}^{\\phantom{\\dagger}} \\big| 0 \\big\\rangle \\nonumber \\\\\n\t&= \\big| \\alpha_{k} \\big|^{2} \\big\\langle 0 \\big| a_{\\mathbf{k}}^{\\dagger} a_{\\mathbf{k}}^{\\phantom{\\dagger}}  \\big| 0 \\big\\rangle + \\big| \\beta_{k} \\big|^{2} \\big\\langle 0 \\big| a_{\\mathbf{k}}^{\\phantom{\\dagger}} a_{\\mathbf{k}}^\\dagger \\big| 0 \\big\\rangle \\\\\n\t&= \\big| \\beta_{k} \\big|^{2} \\nonumber \\, .\n\\end{align}\nTherefore, different choices of representation inevitably lead to distinct notions of vacuum and, consequently, to distinct notions of particle. This result is a quite general feature of quantum field theory defined on curved spacetimes, and although it initially seems troublesome, it actually becomes useful in numerical back-reaction calculations. To that end, we introduce in the next section a particularly useful representation which defines the most physical notion of particle in a FLRW spacetime.\n\n\\section{Adiabatic Representation}\\label{Sec. Adiabatic Representation}\n\nDespite the multitude of available representations for a quantized scalar field defined on a FLRW spacetime, one particular choice referred to as the adiabatic representation stands out. This representation has the special property of defining a vacuum state which comes closest to the Minkowski vacuum when the background expansion is sufficiently slow. As a consequence, the adiabatic representation provides the most meaningful notion of physical particle in an expanding homogeneous and isotropic Universe. Here we discuss this representation \nclosely following Ref.~\\cite{Habib99}.\n\nThe adiabatic representation is characterized by mode functions which are the phase-integral solutions \\cite{Parker69, Parker71, ZeldovichStarobinsky77} of Eq.~(\\ref{Eq. Mode Equation}):\n\\begin{equation}\\label{Eq. Adiabatic Modes}\n\th_{k}(t) = \\frac{1}{\\sqrt{2 W_{k}(t)}} \\exp{\\!\\bigg( \\! -i \\! \\int^{t} W_{k}(s) \\, \\mathrm{d}s \\bigg)} \\, ,\n\\end{equation}\nwhere the integral in the exponent can be computed from any convenient reference time, and the function $W_{k}(t)$ is given by the formal asymptotic series\n\\begin{equation}\\label{Eq. Adiabatic W}\n\t W_{k}(t) \\equiv \\Omega_{k}(t) \\sum_{n=0}^{\\infty} \\varphi_{k,\\,2n}(t) \\, .\n\\end{equation}\nThe terms $\\varphi_{k,\\,2n}(t)$ are obtained by substituting Eqs.~(\\ref{Eq. Adiabatic W}) and (\\ref{Eq. Adiabatic Modes}) into Eq.~(\\ref{Eq. Mode Equation}). The expressions which ensue from these substitutions are standard results of the phase-integral method \\cite{FromanFroman96, FromanFroman02}; up to fourth order they are\n\\begin{subequations}\n\\begin{align}\n\t&\\varphi_{k,\\,0}(t) = 1 \\, , \\\\[8pt]\n\t&\\varphi_{k,\\,2}(t) = \\frac{1}{2} \\varepsilon_{k,\\,0}(t) \\, , \\\\[8pt]\n\t&\\varphi_{k,\\,4}(t) = -\\frac{1}{8}\\Big[ \\varepsilon_{k,\\,0}^{2}(t) + \\varepsilon_{k,\\,2}(t) \\Big] \\, ,\n\\end{align}\n\\end{subequations}\nfor which the quantities appearing on the right-hand sides are given by\n\\begin{subequations}\n\\begin{align}\n\t\\varepsilon_{k,\\,0}(t) &\\equiv \\Omega_{k}^{-3\/2}(t) \\, \\frac{d^{2}}{dt^{2}} \\! \\bigg[ \\Omega_{k}^{-1\/2}(t) \\bigg]\\, , \\\\[8pt]\n\t\\varepsilon_{k,\\,m}(t) &\\equiv \\bigg[ \\Omega_{k}^{-1}(t) \\, \\frac{d}{dt} \\bigg]^{\\!m} \\!\\!\\varepsilon_{k,\\,0}(t) \\, .\n\\end{align}\n\\end{subequations}\n\nIn a sense, the functions $W_{k}(t)$ capture the overall time dependence of each $\\mathbf{k}$-mode due the evolving FLRW metric, leaving behind only the Minkowski-like mode oscillations which take place on top of this background \\cite{DabrowskiDunne14, DabrowskiDunne16}. It is this property that makes the adiabatic mode functions $h_{k}(t)$ and $h^{\\ast}_{k}(t)$ such good templates for probing the particle content of fields evolving in cosmological spacetimes. This template role is made precise by the following time-dependent generalization of Eqs.~(\\ref{Eq. Bogolyubov Modes}) \\cite{Parker69, Parker71, ZeldovichStarobinsky77}, which expresses the field modes $f_{k}(t)$ as linear combinations of the adiabatic mode functions:\n\\begin{equation}\\label{Eq. Adiabatic Bogolyubov Mode}\n\t{f}_{k}(t) = \\alpha_{k}(t)h_{k}(t) + \\beta_{k}(t)h^{\\ast}_{k}(t) \\, ,\n\\end{equation}\nwhere the Bogolyubov coefficients $\\alpha_{k}(t)$ and $\\beta_{k}(t)$ are analogous to those appearing in Eqs.~(\\ref{Eq. Bogolyubov Modes})~and~(\\ref{Eq. Bogolyubov Operators}), but are here regarded as time-dependent quantities due to the fact that $h_{k}(t)$ and $h_{k}^{\\ast}(t)$ are merely approximate solutions of Eq.~(\\ref{Eq. Mode Equation}). In order to completely specify these coefficient functions, an additional expression must be provided. For that purpose, it is common to introduce a condition on the time derivative of the mode function which preserves the Wronskian relation of Eq.~(\\ref{Eq. Wronskian Condition}). In its most general form, this condition can be stated as \\cite{Habib99}\n\\begin{align}\\label{Eq. Adiabatic Bogolyubov Mode Derivative}\n\t\\dot{f}_{k}(t) = \\bigg[ -iW_{k}(t) + \\frac{V_{k}(t)}{2}  \\bigg] \\alpha_{k}(t)h_{k}(t) \\,\\, & \\\\\n\t + \\, \\bigg[ iW_{k}(t) + \\frac{V_{k}(t)}{2}  \\bigg] \\beta_{k}(t)h^{\\ast}_{k}(t)& \\, . \\nonumber\n\\end{align}\nHere the arbitrary function $V_{k}(t)$  contains the residual freedom in the definition of the adiabatic vacuum. In this work we will choose this function to be\n\\begin{equation}\\label{Eq. Adiabatic V}\n\tV_{k}(t) = -\\frac{\\dot{W}_{k}(t)}{W_{k}(t)} \\, ,\n\\end{equation}\nas this choice leads to important simplifications in the back-reaction problem. \n\nGathering Eqs.~(\\ref{Eq. Field}) and (\\ref{Eq. Adiabatic Bogolyubov Mode}), we find that the ladder operators associated with the adiabatic representation satisfy the transformations\n\\begin{subequations}\\label{Eq. Bogolyubov Adiabatic Operators}\n\\begin{align}\na_{\\mathbf{k}}^{\\phantom{\\dagger}} &= \\alpha_{k}^{\\ast}(t) b_{\\mathbf{k}}^{\\phantom{\\dagger}}(t) - \\beta_{k}^{\\ast}(t) b_{\\mathbf{k}}^{\\dagger}(t)\\, , \\\\[8pt]\na_{\\mathbf{k}}^\\dagger &= \\alpha_{k}(t) b_{\\mathbf{k}}^{\\dagger}(t) - \\beta_{k}(t) b_{\\mathbf{k}}^{\\phantom{\\dagger}}(t) \\, ,\n\\end{align}\n\\end{subequations}\nwhich in turn imply a time-dependent version of Eq.~(\\ref{Eq. Bogolyubov Constraint}), \n\\begin{equation}\\label{Eq. Adiabatic Bogolyubov Constraint}\n\t\\big| \\alpha_{k}(t) \\big|^{2} - \\big| \\beta_{k}(t) \\big|^{2} = 1 \\, .\n\\end{equation}\n\nFinally, it is useful to characterize field states according to the values of the non-trivial adiabatic bilinears $\\big\\langle b_{\\mathbf{k}}^{\\dagger}(t) b_{\\mathbf{k}}(t) \\big\\rangle$ and $\\big\\langle b_{\\mathbf{k}}(t) b_{\\mathbf{k}}(t) \\big\\rangle$. The first of these bilinears tracks the adiabatic particle content per comoving volume in the $\\mathbf{k}$-mode under consideration. Using the transformations established above by Eqs.~(\\ref{Eq. Bogolyubov Adiabatic Operators}), it follows that\n\\begin{align}\\label{Eq. Adiabatic Particles}\n\t\\mathcal{N}_{k}(t) &= \\big\\langle b_{\\mathbf{k}}^{\\dagger}(t) b_{\\mathbf{k}}(t) \\big\\rangle \\nonumber \\\\\n\t&= \\big| \\alpha_{k}(t) \\big|^{2} \\big\\langle a_{\\mathbf{k}}^{\\dagger} a_{\\mathbf{k}}^{\\phantom{\\dagger}} \\big\\rangle + \\big| \\beta_{k}(t) \\big|^{2} \\big\\langle a_{\\mathbf{k}}^{\\phantom{\\dagger}} a_{\\mathbf{k}}^\\dagger \\big\\rangle \\\\\n\t&= N_{k} + \\sigma_{k} \\, \\big| \\beta_{k}(t) \\big|^{2} \\nonumber \\, ,\n\\end{align}\nwhere $N_{k} = \\big\\langle a_{\\mathbf{k}}^{\\dagger} a_{\\mathbf{k}}^{\\phantom{\\dagger}} \\big\\rangle$ is a constant of motion which can be understood as the initial number of adiabatic particles per comoving volume populating the field mode of wavenumber $k$, and $\\sigma_{k} = 1 + 2N_{k}$ is the Bose-Einstein parameter responsible for stimulated particle production. The second bilinear can be expressed as\n\\begin{align}\\label{Eq. Adiabatic Interference}\n\t\\mathcal{M}_{k}(t) &= \\big\\langle b_{\\mathbf{k}}(t) b_{\\mathbf{k}}(t) \\big\\rangle \\nonumber \\\\\n\t&= \\alpha_{k}(t) \\beta^{\\ast}_{k}(t) \\big\\langle a_{\\mathbf{k}}^{\\dagger} a_{\\mathbf{k}}^{\\phantom{\\dagger}} \\big\\rangle + \\alpha_{k}(t) \\beta^{\\ast}_{k}(t) \\big\\langle a_{\\mathbf{k}}^{\\phantom{\\dagger}} a_{\\mathbf{k}}^\\dagger \\big\\rangle \\\\\n\t&= \\sigma_{k} \\, \\alpha_{k}(t) \\beta^{\\ast}_{k}(t) \\nonumber \\, .\n\\end{align}\n\nIn principle, these bilinears contain all the required information to track the field evolution and, consequently, the time dependence of the field energy density and pressure. In practice, however, these quantities suffer from an irreducible ambiguity which is particularly pronounced when $\\mathcal{N}_{k}(t)$ and $\\mathcal{M}_{k}(t)$ incur rapid changes, such as when particle production occurs. \nThe root of this issue can be traced back to the asymptotic representation of $W_{k}(t)$, which is usually handled by simply truncating the series in Eq.~(\\ref{Eq. Adiabatic W}) at a finite order. \nHowever, the values of the bilinears depend strongly on where the series is truncated if they are rapidly changing (see Ref.~\\cite{DabrowskiDunne14} for striking graphical representations).\nIn the next Section, we discuss a technique for finding the exact universal evolution for both adiabatic bilinears which the asymptotic series represents.\n\n\n\\section{Particle Production and the Stokes Phenomenon}\\label{Sec. Stokes}\n\nThe adiabatic representation introduced in the previous section provides an accurate description of the bilinears $\\mathcal{N}_{k}(t)$ and $\\mathcal{M}_{k}(t)$ whenever $\\left| \\varepsilon_{k,\\,0} \\right| \\ll 1$. The more severely this condition is violated, the more unreliable these adiabatic quantities become. Adiabatic particle production, for instance, coincides with the momentary violation of this condition, implying that the notion of particle remains uncertain until particle production ceases. Nonetheless, a universal notion of particle can be restored for all times when particle production events are understood in terms of the Stokes phenomenon.\n\nThe sharp transitions between asymptotic solutions of a given differential equation which are valid in different regions of the complex plane are termed the Stokes phenomenon. These regions are bounded by the so-called Stokes and anti-Stokes lines. In the context of a scalar field evolving in a FLRW spacetime, the differential equation of interest is the equation of motion for a given field mode extended to a complex time variable $z$:\n\\begin{equation}\\label{Eq. Complex Mode Equation}\n{f}^{\\prime\\prime}_{k}(z) + \\Omega_{k}^{2}(z) f_{k}(z) = 0 \\, ,\n\\end{equation}\nin which the primes stand for differentiation with respect to $z$, the proper time is given by $t \\equiv \\mathrm{Re}\\, z$, and $\\Omega_{k}(z)$ represents the analytic continuation of the time-dependent frequency of Eq.~(\\ref{Eq. Mode Frequency}). The Stokes lines associated with Eq.~(\\ref{Eq. Complex Mode Equation}) are those lines which emanate from the zeros (also known as turning points) and poles of $\\Omega_{k}(z)$ and along which $\\mathrm{Re}\\big[\\Omega_{k}\\mathrm{d}z\\big]~\\!\\!=~\\!\\!0$. An illustration of such a line is shown in Fig.~\\ref{Fig. Stokes Line}. The asymptotic solutions susceptible to the Stokes phenomenon are given by\n\\begin{equation}\\label{Eq. Phase-Integral Solution}\n\tf_{k}(z) = \\alpha_{k}(z) h_{k}(z) + \\beta_{k}(z) h^{\\ast}_{k}(z) \\, ,\n\\end{equation}\nwhere $h_{k}(z)$ and $h_{k}^{\\ast}(z)$ are the complex extensions of the adiabatic mode functions defined in the previous section. As this solution evolves across a Stokes line, the values of the Bogolyubov coefficients $\\alpha_{k}(z)$ and $\\beta_{k}(z)$ change abruptly. By Eqs.~(\\ref{Eq. Adiabatic Particles}) and (\\ref{Eq. Adiabatic Interference}), this implies a sudden change in the adiabatic bilinears and, in particular, the production of adiabatic particles. Remarkably, a result from asymptotic analysis guarantees the existence of a smooth universal form for this rapid transition between different asymptotic regimes. Below we outline the derivation of this important result and summarize the quantities which determine the functional form of such smooth Stokes jumps.\n\n\\begin{figure}[t!]\n    \\includegraphics[width=0.49\\textwidth]{StokesLine.pdf}\n    \\caption{A depiction of the Stokes line sourced by a conjugate pair of simple turning points $\\big( z_{0},\\, z_{0}^{\\ast}\\big)$ of a frequency function $\\Omega_{k}(z)$. Here the Stokes line crosses the real axis at the point $s_{0}$, which corresponds to the time at which particle production occurs for the mode of wavenumber $k$. The guiding lines show the directions for which the condition $\\mathrm{Re}\\big[\\Omega_{k}\\mathrm{d}z\\big] = 0$ is locally satisfied.}\n    \\label{Fig. Stokes Line}\n\\end{figure}\n\nWe start by defining Dingle's singulant variable \\cite{Dingle73} anchored at $z_{0}$:\n\\begin{equation}\\label{Eq. Singulant}\nF_{k}^{(0)}(z) = 2i \\! \\int_{z_0}^{z} \\Omega_{k}(w) \\, \\mathrm{d}w \\, ,\n\\end{equation}\nwhere $z_0$ is a solution of $\\Omega_{k}(z) = 0$ which sources the Stokes line of interest, is closest to the real axis, and is located in the upper half-plane. The singulant is a convenient variable for tracking the change incurred by the Bogolyubov coefficients $\\alpha_{k}(z)$ and $\\beta_{k}(z)$ across a Stokes line. Indeed, it was shown by Berry \\cite{Berry90, Berry89, Berry88, Berry82} that these coefficients satisfy the following differential equations in the vicinity of a Stokes line:\n\\begin{subequations}\\label{Eq. Bogolyubov Couplers}\n\\begin{align}\n\t\\frac{d\\beta_{k}}{dF_{k}^{(0)}} &= C_{\\beta,\\,k}^{(0)}\\,\\alpha_{k}\\, , \\\\[8pt]\n\t\\frac{d\\alpha_{k}}{dF_{k}^{(0)}} &= C_{\\alpha,\\,k}^{(0)}\\,\\beta_{k}  \\, ,\n\\end{align}\n\\end{subequations}\nwhere $C_{\\beta,\\,k}^{(0)}$ and $C_{\\alpha,\\,k}^{(0)}$ are coupling functions which depend on the order at which the series representation of $W_{k}(z)$ is truncated. A remarkable discovery by Dingle \\cite{Dingle73} states that the large-order terms in the asymptotic series of Eq.~(\\ref{Eq. Adiabatic W}) have a closed form given by\n\\begin{equation}\n\\varphi_{k,\\, 2n}(z) \\sim -\\frac{(2n - 1)!}{\\pi F_{k}^{(0)\\,2n}} \\quad \\text{for} \\quad n \\gg 1 \\, .\n\\end{equation}\nIt is clear from this result that the smallest term in such a series corresponds to $n \\approx \\big| F_{k}^{(0)} \\big|$. Terminating the series at this order leads to optimal closed form expressions for $C_{\\beta,\\,k}^{(0)}$ and $C_{\\alpha,\\,k}^{(0)}$, which can be substituted in Eqs.~(\\ref{Eq. Bogolyubov Couplers}) to yield the following universal behaviors for $\\beta_{k}(t)$ and $\\alpha_{k}(t)$ along the real axis and across the Stokes line under consideration:\n\\begin{subequations}\\label{Eq. Smooth Bogolyubov}\n\\begin{align}\n\t\\beta_{k}(t) &\\approx \\frac{i}{2} \\, \\mathrm{Erfc}\\Big( \\! -\\vartheta_{k}^{(0)}(t) \\Big) \\delta_{k}^{(0)}\\, , \\\\[8pt]\n\t\\alpha_{k}(t) &\\approx \\sqrt{1 + \\big| \\beta_{k}(t) \\big|^{2}} \\, ,\n\\end{align}\n\\end{subequations}\nwhere $\\vartheta_{k}^{(0)}(t)$ is a natural time evolution parameter which determines the sharpness of the Stokes jump, and $\\delta_{k}^{(0)}$ corresponds to the jump's amplitude. Both of these parameters are expressible in terms of the singulant variable evaluated over the real axis:\n\\begin{subequations}\n\\begin{align}\n\t&\\vartheta_{k}^{(0)}(t) = \\frac{\\phantom{\\sqrt{2 \\,\\,}}\\mathrm{Im} \\Big[ F_{k}^{(0)}(t) \\Big]}{\\sqrt{2 \\, \\mathrm{Re} \\Big[ F_{k}^{(0)}(t) \\Big]}}\\, , \\\\[8pt]\n\t&\\delta_{k}^{(0)} = \\exp\\Big( \\! -F_{k}^{(0)}(s_{0}) \\Big) \\, .\n\\end{align}\n\\end{subequations}\nHere $F_{k}^{(0)}(s_{0})$ is simply the singulant computed at the point $z=s_{0}$, where the Stokes line sourced by $z_{0}$ intersects the real axis, i.e., \n\\begin{equation}\\label{Eq. Amplitude}\nF_{k}^{(0)}(s_{0}) = 2i \\! \\int_{z_0}^{s_{0}} \\Omega_{k}(w) \\, \\mathrm{d}w = i \\! \\int_{z_0}^{z_0^{\\ast}} \\Omega_{k}(w)\\,\\mathrm{d}w \\, ,\n\\end{equation}\nwhere the last equality follows from the reality of $\\Omega_{k}(z)$ over the real axis. Putting together Eqs.~(\\ref{Eq. Adiabatic Particles}), (\\ref{Eq. Adiabatic Interference}), and (\\ref{Eq. Smooth Bogolyubov}) yields a universal functional form which describes the time evolution of the adiabatic bilinears associated with the field mode of wavenumber $k$:\n\\begin{subequations}\\label{Eq. Smooth Bilinears}\n\\begin{align}\n\t\\mathcal{N}_{k} &\\approx N_{k} + \\frac{\\sigma_{k}}{4} \\Big| \\mathrm{Erfc}\\Big( \\! -\\vartheta_{k}^{(0)} \\Big) \\delta_{k}^{(0)} \\Big|^{2}\\, , \\\\[8pt]\n\t\\mathcal{M}_{k} &\\approx -i \\, \\frac{\\sigma_{k}}{2} \\Big[ \\mathrm{Erfc}\\Big( \\! -\\vartheta_{k}^{(0)} \\Big) \\delta_{k}^{(0)} \\Big] \\Big[ 1 + \\mathcal{N}_{k} \\Big]^{\\!1\/2}\n\\, .\n\\end{align}\n\\end{subequations}\n\nThese results can be further generalized to account for multiple Stokes line crossings, as well as the interference effects between them \\cite{DabrowskiDunne14}. Define the accumulated phase between the first and the $p$-th pair of zeros of $\\Omega_{k}(z)$ as\n\\begin{equation}\\label{Eq. Phase}\n\\theta_{k}^{(p)} = \\int_{s_0}^{s_p} \\Omega_{k}(w)\\,\\mathrm{d}w \n\\end{equation}\nwhere $s_p$ corresponds to the point where the Stokes line associated with the $p$-th conjugate pair of zeros crosses the real axis. The functions which describe both adiabatic bilinears are then given by\n\\begin{subequations}\\label{Eq. Bilinears Multiple Crossings}\n\\begin{align}\n\t\\mathcal{N}_{k} &\\approx N_{k} + \\frac{\\sigma_{k}}{4} \\bigg| \\sum_{p} \\mathrm{Erfc}\\Big( \\! -\\vartheta_{k}^{(p)} \\Big) \\delta_{k}^{(p)} \\exp\\Big(2i\\theta_{k}^{(p)} \\Big) \\bigg|^{2}\\, , \\\\[8pt]\n\t\\mathcal{M}_{k} &\\approx -i \\, \\frac{\\sigma_{k}}{2} \\bigg[ \\sum_{p} \\mathrm{Erfc}\\Big( \\! -\\vartheta_{k}^{(p)} \\Big) \\delta_{k}^{(p)} \\exp\\Big( \\! -2i\\theta_{k}^{(p)} \\Big) \\bigg] \\times \\nonumber \\\\\n\t & \\,\\,\\;\\; \\quad \\quad \\times \\! \\bigg[ 1 + \\mathcal{N}_{k} \\bigg]^{\\!1\/2}\n\\end{align}\n\\end{subequations}\nwith $\\delta_{k}^{(p)}$ and $\\vartheta_{k}^{(p)}(t)$ being the amplitude and time evolution parameter associated with the $p$-th Stokes line.\n\nTherefore, by monitoring the turning points and Stokes lines which accompany each mode's frequency function on the complex plane, we can track the evolution of the adiabatic bilinears related to any physically acceptable field state. In the next section we examine how this evolution affects the Universe's scale factor through the semi-classical Einstein equations.\n\n\\onecolumngrid\n\\section{The Semi-Classical Einstein Equations}\\label{Sec. Semi-Classical Einstein}\n\nIf cosmological quantum particle production occurs at a sufficiently high rate, it can in principle back-react on the cosmic evolution through the semi-classical Einstein equations\n\\begin{equation}\\label{Eq. Einstein Equations}\n\tR_{ab} - \\frac{1}{2}R \\, g_{ab} + \\Lambda \\, g_{ab} = M^{-2} \\big\\langle \\hat{T}_{ab}\\big\\rangle \n\\end{equation}\nwhere $\\Lambda$ represents the cosmological constant, ${M = ({8 \\pi G})^{-1\/2}}$ stands for the reduced Planck mass, and $\\big\\langle \\hat{T}_{ab}\\big\\rangle$ corresponds to the expectation value of the energy-momentum tensor operator, including contributions both from the scalar field we are considering plus any other stress-energy sources. The canonical expression for $\\hat{T}_{ab}$ due to the scalar field is constructed by varying the action in Eq.~(\\ref{Eq. Field Action}) with respect to the metric $g_{ab}$, and subsequently substituting the field operator $\\hat{\\Phi}$ from Eq.~(\\ref{Eq. Field}) into the resulting expression:\n\\begin{equation}\\label{Eq. Energy-Momentum Tensor}\n\t\\hat{T}_{ab} = \\big( \\nabla_{a} \\hat{\\Phi} \\big) \\big( \\nabla_{b} \\hat{\\Phi} \\big) - \\frac{1}{2}g_{ab} \\big( \\nabla^{c} \\hat{\\Phi} \\big) \\big( \\nabla_{c} \\hat{\\Phi} \\big) + \\xi \\bigg[  g_{ab} \\, \\Box - \\nabla_{a}\\nabla_{b} + R_{ab} - \\frac{1}{2}R \\, g_{ab} - \\frac{m^{2}}{2}g_{ab} \\bigg] \\hat{\\Phi}^{2} \\, .\n\\end{equation}\nFor a statistically homogeneous and isotropic field state, $\\big\\langle \\hat{T}_{ab}\\big\\rangle$ is equivalent to the energy-momentum tensor of a perfect fluid for which the field energy density and pressure are given, respectively, by $\\rho(t) = \\big\\langle \\hat{T}_{00} \\big\\rangle$ and $P(t) = \\frac{1}{3}g^{ij}\\big\\langle \\hat{T}_{ij} \\big\\rangle$. As a consequence, such a state naturally sources an FLRW metric, reducing Eq.~(\\ref{Eq. Einstein Equations}) to the usual Friedmann equations:\n\\begin{subequations}\\label{Eq. Friedmann Equations}\n\\begin{align}\n\t& H^{2}(t) = \\frac{1}{3} M^{-2} \\rho(t) + \\frac{\\Lambda}{3} - \\frac{K}{a^{2}(t)} \\label{Eq. First Friedmann Equation} \\\\[8pt]\n\t&\\dot{H}(t) + H^{2}(t) = -\\frac{1}{6} M^{-2} \\Big[ \\rho(t) + 3P(t) \\Big] + \\frac{\\Lambda}{3} \\, .\n\\end{align}\n\\end{subequations}\nFurthermore, it can be shown that $\\big\\langle \\hat{T}_{ab}\\big\\rangle$ is covariantly conserved, resulting in the cosmological continuity equation\n\\begin{equation}\\label{Eq. Cosmological Continuity}\n\t\\dot{\\rho}(t) + 3H(t)\\Big[ \\rho(t) + P(t) \\Big] = 0 \\, .\n\\end{equation}\n\nHowever, at this stage these equations are merely formal, because both the pressure and energy density of a quantized field are in general divergent and need to be regularized. \nIn a FLRW spacetime, these divergencies can be partially isolated by expressing $\\rho(t)$ and $P(t)$ in terms of the adiabatic bilinears $\\mathcal{N}_{k}(t)$ and $\\mathcal{M}_{k}(t)$ \\cite{Habib99}. Substituting Eqs.~(\\ref{Eq. Field}), (\\ref{Eq. Adiabatic Bogolyubov Mode}), and (\\ref{Eq. Adiabatic Bogolyubov Mode Derivative}) into the expectation value of Eq.~(\\ref{Eq. Energy-Momentum Tensor}) and collecting terms with the same adiabatic factor gives\n\\begin{subequations}\\label{Eq. Energy Density and Pressure}\n\\begin{align}\n\t&\\rho(t) = \\big\\langle \\hat{T}_{00} \\big\\rangle = \\frac{1}{4 \\pi a^{3}(t)} \\! \\int d\\mu(k) \\Bigg\\{  \\rho^{\\mathcal{N}}_{k}\\!(t) \\bigg[ \\mathcal{N}_{k}(t) + \\frac{1}{2} \\bigg] + \\rho^{\\mathcal{R}}_{k}(t)\\,\\mathcal{R}_{k}(t) + \\rho^{\\mathcal{I}}_{k}(t) \\, \\mathcal{I}_{k}(t) \\Bigg\\} \\label{Eq. Energy Density} \\\\[8pt] \n\t& P(t) = \\frac{1}{3} g^{ij} \\big\\langle \\hat{T}_{ij} \\big\\rangle = \\frac{1}{4 \\pi a^{3}(t)} \\! \\int d\\mu(k) \\Bigg\\{  P^{\\mathcal{N}}_{k}\\!(t) \\bigg[ \\mathcal{N}_{k}(t) + \\frac{1}{2} \\bigg] + P^{\\mathcal{R}}_{k}(t)\\,\\mathcal{R}_{k}(t) + P^{\\mathcal{I}}_{k}(t) \\, \\mathcal{I}_{k}(t) \\Bigg\\} \\label{Eq. Pressure} \\, .\n\\end{align}\n\\end{subequations}\nThe terms proportional to $\\mathcal{N}_{k}(t)$ capture the contribution to the energy density and pressure due to the evolving distribution of adiabatic particles populating the field modes, while the quantum interference terms contain ${\\mathcal{R}_{k}(t) = \\mathrm{Re}\\big[ \\mathcal{M}_{k}(t) \\big]}$ and ${\\mathcal{I}_{k}(t) = \\mathrm{Im}\\big[ \\mathcal{M}_{k}(t) \\big]}$ \\footnote{Here the definitions for $\\mathcal{R}_{k}(t)$ and $\\mathcal{I}_{k}(t)$ might seem to differ from those found in the literature by a phase factor, but this factor is implicit in our definitions for $\\alpha_{k}(t)$ and $\\beta_{k}(t)$ obtained from asymptotic analysis}. The prefactors in each of these terms are defined as\n\\begin{subequations}\\label{Eq. Adiabatic Factors}\n\\begin{align}\n\t&\\rho^{\\mathcal{N}}_{k}\\!(t) \\equiv \\frac{1}{W_{k}(t)} \\Bigg\\{ W^{2}_{k}(t) + \\omega^{2}_{k}(t) + \\frac{1}{4} \\Big[ V_{k}(t) - H(t) \\Big]^{\\!2} + \\big(6\\xi - 1\\big) \\bigg[ H(t) V_{k}(t) - 2H^{2}(t) + \\frac{K}{a^{2}(t)} \\bigg] \\Bigg\\}\\, , \\\\[8pt]\n\t&\\rho^{\\mathcal{R}}_{k}(t) \\equiv \\frac{1}{W_{k}(t)} \\Bigg\\{ - W^{2}_{k}(t) + \\omega^{2}_{k}(t) + \\frac{1}{4} \\Big[ V_{k}(t) - H(t) \\Big]^{\\!2} + \\big(6\\xi - 1\\big) \\bigg[ H(t) V_{k}(t) - 2H^{2}(t) + \\frac{K}{a^{2}(t)} \\bigg] \\Bigg\\}\\, , \\\\[8pt]\n\t&\\rho^{\\mathcal{I}}_{k}(t) \\equiv  V_{k}(t) - H(t) +2H(t)\\big(6\\xi - 1\\big)\\, , \\\\[8pt]\n\t&P^{\\mathcal{N}}_{k}\\!(t) \\equiv \\frac{1}{3W_{k}(t)} \\Bigg\\{ W^{2}_{k}(t) + \\omega^{2}_{k}(t) - 2m^{2} + \\frac{1}{4} \\Big[ V_{k}(t) - H(t) \\Big]^{\\!2} + \\frac{1}{3}\\big(6\\xi - 1\\big)^{\\!2}R(t) + \\Bigg. \\, \\\\\n\t & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\Bigg. + \\big(6\\xi - 1\\big) \\bigg[ -2W^{2}_{k}(t) - \\frac{1}{2}V^{2}_{k}(t) + 4H(t)V_{k}(t) + 2\\omega^{2}_{k}(t) + 2\\dot{H}(t) + \\frac{K}{a^{2}(t)} - \\frac{5}{2}H^{2}(t) \\bigg] \\Bigg\\}\\, , \\nonumber \\\\[8pt]\n\t&P^{\\mathcal{R}}_{k}(t) \\equiv \\frac{1}{3W_{k}(t)} \\Bigg\\{ -W^{2}_{k}(t) + \\omega^{2}_{k}(t) - 2m^{2} + \\frac{1}{4} \\Big[ V_{k}(t) - H(t) \\Big]^{\\!2} + \\frac{1}{3}\\big(6\\xi - 1\\big)^{\\!2}R(t) + \\Bigg.\\,  \\\\\n\t & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\Bigg. + \\big(6\\xi - 1\\big) \\bigg[ 2W^{2}_{k}(t) - \\frac{1}{2}V^{2}_{k}(t) + 4H(t)V_{k}(t) + 2\\omega^{2}_{k}(t) + 2\\dot{H}(t) + \\frac{K}{a^{2}(t)} - \\frac{5}{2}H^{2}(t) \\bigg] \\Bigg\\}\\, , \\nonumber \\\\[8pt]\n\t&P^{\\mathcal{I}}_{k}(t) \\equiv \\frac{1}{3}\\Big[ V_{k}(t) - H(t) \\Big] + \\frac{2}{3} \\big(6\\xi - 1\\big) \\Big[ 4H(t) - V_{k}(t) \\Big] \\, .\n\\end{align}\n\\end{subequations}\nFor adiabatic field states, the contributions to the energy density and pressure due to the real bilinears $\\mathcal{N}_{k}(t)$, $\\mathcal{R}_{k}(t)$, and $\\mathcal{I}_{k}(t)$ are always finite. This implies that the divergencies in Eqs.~(\\ref{Eq. Energy Density}) and (\\ref{Eq. Pressure}) are isolated in the vacuum-like terms characterized by the $\\frac{1}{2}$ factors, henceforth identified as\n\\begin{subequations}\\label{Eq. Energy and Pressure Vacuum}\n\\begin{align}\n\t\\rho_{\\mathrm{vac}}(t) &\\equiv \\frac{1}{8 \\pi a^{3}(t)} \\! \\int d\\mu(k)\\,\\rho^{\\mathcal{N}}_{k}\\!(t) \\label{Eq. Vacuum Energy} \\\\[8pt]\n\tP_{\\mathrm{vac}}(t) &\\equiv \\frac{1}{8 \\pi a^{3}(t)} \\! \\int d\\mu(k)\\,P^{\\mathcal{N}}_{k}\\!(t) \\label{Eq. Vacuum Pressure} \\, .\n\\end{align}\n\\end{subequations}\n\nRegularization consists precisely in controlling the divergent behavior of $\\rho_{\\mathrm{vac}}(t)$ and $P_{\\mathrm{vac}}(t)$ so as to obtain finite expressions for  $\\rho(t)$ and $P(t)$ which still satisfy the cosmological continuity equation. Adiabatic regularization achieves this result by subtracting the fourth-order phase-integral expansions of $\\rho^{\\mathcal{N}}_{k}\\!(t)$ and $P^{\\mathcal{N}}_{k}\\!(t)$ from the integrands of Eqs.~(\\ref{Eq. Vacuum Energy}) and (\\ref{Eq. Vacuum Pressure}), respectively \\cite{ParkerFulling74, AndersonParker87}. From a technical perspective, however, this procedure introduces significant challenges to the numerical implementation of the semi-classical Friedmann equations. Chief among these is the appearance of higher-order time derivatives of $H(t)$ in the integrands of Eqs.~(\\ref{Eq. Energy Density and Pressure}), turning the semi-classical Friedmann equations into a system of integro-differential equations which is not amenable to standard numerical treatments. We circumvent this difficulty by employing an alternative regularization scheme which, albeit cruder, yields a good approximation to $\\rho(t)$ and $P(t)$ in regimes dominated by particle production.\n\nCentral to the regularization approach adopted here is the realization that $\\rho_{\\mathrm{vac}}(t)$ and $P_{\\mathrm{vac}}(t)$ independently satisfy the cosmological continuity equation as long as the function $V_{k}(t)$ has the form established in Eq.~(\\ref{Eq. Adiabatic V}) (see the Appendix). It follows that the vacuum contributions to $\\rho(t)$ and $P(t)$ can be discarded in their entirety while still ensuring that Eq.~(\\ref{Eq. Cosmological Continuity}) remains valid. Despite its simplicity, this procedure yields a good approximation to the field energy density and pressure provided the adiabatically regularized integrands of Eqs.~(\\ref{Eq. Energy Density and Pressure}) are dominated by the real adiabatic bilinears $\\mathcal{N}_{k}(t)$, $\\mathcal{R}_{k}(t)$, and $\\mathcal{I}_{k}(t)$. Therefore, in what follows we take the regularized expressions for the energy density and pressure to be\n\\begin{subequations}\\label{Eq. Reg. Energy Density and Pressure}\n\\begin{align}\n\t\\rho(t) &\\approx \\frac{1}{4 \\pi a^{3}(t)} \\! \\int d\\mu(k) \\Bigg\\{  \\rho^{\\mathcal{N}}_{k}\\!(t) \\, \\mathcal{N}_{k}(t) + \\rho^{\\mathcal{R}}_{k}(t)\\,\\mathcal{R}_{k}(t) + \\rho^{\\mathcal{I}}_{k}(t) \\, \\mathcal{I}_{k}(t) \\Bigg\\} \\label{Eq. Reg. Energy Density} \\\\[8pt] \n\tP(t) &\\approx \\frac{1}{4 \\pi a^{3}(t)} \\! \\int d\\mu(k) \\Bigg\\{  P^{\\mathcal{N}}_{k}\\!(t) \\, \\mathcal{N}_{k}(t) + P^{\\mathcal{R}}_{k}(t)\\,\\mathcal{R}_{k}(t) + P^{\\mathcal{I}}_{k}(t) \\, \\mathcal{I}_{k}(t) \\Bigg\\} \\label{Eq. Reg. Pressure} \\, ,\n\\end{align}\n\\end{subequations}\nwhere the factors $\\rho^{\\mathcal{N}}_{k}\\!(t)$, $\\rho^{\\mathcal{R}}_{k}(t)$, and $\\rho^{\\mathcal{I}}_{k}(t)$, $P^{\\mathcal{N}}_{k}\\!(t)$, $P^{\\mathcal{R}}_{k}(t)$, and $P^{\\mathcal{I}}_{k}(t)$ are computed by truncating the asymptotic series Eq.~(\\ref{Eq. Adiabatic W}) for $W_{k}(t)$ at its optimal order.\n\nFinally, the regularization of $\\big\\langle \\hat{T}_{ab}\\big\\rangle$ also induces the renormalization of the gravitational coupling constants $G$ and $\\Lambda$. Moreover, self-consistency demands the  introduction of a covariantly conserved tensor composed of fourth-order derivatives of the metric into the semi-classical Einstein equations \\cite{FullingParker74, Bunch80}. This tensor is accompanied by a new unknown coupling constant whose renormalization assimilates the ultra-violet divergence in the field energy-momentum tensor. For simplicity, in this work we assume this new coupling constant to be renormalized to zero, thus preserving the form of Eq.~(\\ref{Eq. Einstein Equations}). Non-zero values for this coupling constant will be considered elsewhere.\n\nTaken together, Eqs.~(\\ref{Eq. Bilinears Multiple Crossings}), (\\ref{Eq. Friedmann Equations}), and (\\ref{Eq. Reg. Energy Density and Pressure}) describe the coupled field evolution and cosmic evolution in regimes dominated by particle production. In the next section we present an algorithm which numerically solves this system of equations.\\\\\n\\twocolumngrid\n\n\\section{Numerical Implementation}\\label{Sec. Numerical Implementation}\n\nThe semi-classical Friedmann equations can be formulated as a discretized initial value problem. We take the domain of numerical integration to be a band of the complex plane which is bisected by the real $t$ axis. As illustrated in Figure \\ref{Fig. Numerical Analytic Continuation}, this band is discretized by a uniformly spaced grid where the real-valued entries $t_{j}$ label the physical time. Initial conditions are set by an appropriately chosen functional form for the scale factor $a(t)$ which not only admits an adiabatic field state at the initial time $t_{0}$, but which is also consistent with our choice for the initial distribution of adiabatic particles $\\mathcal{N}_{k}(t_{0}) = N_{k}$ populating the field modes. In addition, we require that\n\\begin{equation}\\label{Eq. Particle Distribution Constraint}\n\tN_{k} < \\mathcal{O}(k^{-3}) \\quad \\text{as} \\quad k \\rightarrow \\infty\n\\end{equation}\nin order to ensure that both the energy density and pressure associated with the initial particle distribution are finite. \n\nWe use a standard finite-difference scheme to step $a(t)$, $H(t)$, and $\\dot{H}(t)$ along the real axis, and employ B-splines to scan the Stokes geometry on the complex plane. The latter is accomplished by generating a numerical sample of $\\Omega_{k}(t)$ through Eq.~(\\ref{Eq. Mode Frequency}), and subsequently performing high-order B-spline interpolations to construct a truncated Taylor polynomial for this function over the real line up to the value of $t$ in the current time step. Due to the analyticity of $\\Omega_{k}(t)$, this series representation is also valid on the complex plane, and thus encodes the analytical continuation of the frequency function. Explicitly, given a grid point $z_{ij}$ on the discretized plane, we compute $\\Omega_{k}(z_{ij})$ through the expression\n\\begin{equation}\\label{Eq. Taylor}\n\\Omega_{k}(z_{ij}) \\approx \\sum_{n = 0}^{T} \\frac{1}{{n!}} \\big(z_{ij} - t_{j} \\big)^{n} \\, \\Omega_{k}^{(n)}(t_{j}) \\, ,\n\\end{equation}\nwhere $t_{j} = \\mathrm{Re}\\,z_{ij}$, as depicted in Figure \\ref{Fig. Numerical Analytic Continuation}. The numerical derivatives $\\Omega_{k}^{(n)}$ are extracted from B-spline interpolations over the real axis, and $T$ corresponds to a truncation order which depends on the density of grid points lying over the real axis. In addition, we feed Eq.~(\\ref{Eq. Taylor}) to a Pad\\'{e} approximant \\cite{Bender13} routine to accelerate its convergence and improve its accuracy. Once this approximate representation of the frequency function has been computed over the discretized plane, it can be interpolated and used in the monitoring of turning points and Stokes lines.\n\n\\begin{figure}[t!]\n    \\includegraphics[width=0.49\\textwidth]{NumericalAnalyticContinuation.pdf}\n    \\caption{A grid of uniformly spaced points covering a band of the complex plane. The grid points lying over the real axis mark the discretization of physical time. Numerically constructing the Taylor polynomial associated with the frequency function around the point $t_{j}$ allows for the optimal evaluation of $\\Omega_{k}(z_{ij})$ at grid points $z_{ij}$ for which $\\mathrm{Re}\\,z_{ij} = t_{j}$.}\n    \\label{Fig. Numerical Analytic Continuation}\n\\end{figure}\n\nWhile the turning points of $\\Omega_{k}(z)$ can be located with the aid of root-finding algorithms designed for multi-valued functions, the problem of determining the Stokes lines sourced by these points requires the numerical integration of an ordinary differential equation. This is evident from the Stokes lines definition ${\\mathrm{Re}\\big[\\Omega_{k}\\mathrm{d}z\\big]=0}$, which implies that, locally, its line element must satisfy ${\\mathrm{d}z \\propto i\/\\Omega_{k}(z)}$. Defining ${t=\\mathrm{Re} \\, z}$ and ${\\tau=\\mathrm{Im} \\, z}$, this condition can be rewritten as\n\\begin{equation}\n\t\\mathrm{d}z = \\mathrm{d}t + i\\,\\mathrm{d}\\tau \\propto \\frac{i}{\\Omega_{k}(z)} \\, .\n\\end{equation}\nTaking the ratio between the matched real and imaginary parts of this proportionality relation leads to the differential equation \n\\begin{equation}\\label{Eq. Stokes ODE}\n\t\\frac{dt}{d\\tau} = \\frac{ \\mathrm{Im} \\, \\Omega_{k}(z) }{ \\mathrm{Re} \\, \\Omega_{k}(z) } \\, \n\\end{equation}\nfor the Stokes line,\nwhich can be numerically integrated from the turning point of interest to yield $t(\\tau)$.\n\nHere is a summary of the minimal set of tasks performed by our algorithm while evolving the physical quantities of interest by one time step:\n\\begin{itemize}\n\t\\item[1.] Take samples of $a(t)$, $H(t)$, and $\\dot{H}(t)$ describing the metric along an interval of the real axis. Over this same interval, sample and interpolate the field energy density $\\rho(t)$ and pressure $P(t)$.\n\t\n\t\\item[2.] Numerically integrate the semi-classical Friedmann equations so as to enlarge the  input metric samples $a(t)$, $H(t)$, and $\\dot{H}(t)$ by a time step $\\Delta t$.\n\n\t\\item[3.] For each field mode, generate a sample of the frequency function $\\Omega_{k}(t)$ over the real axis, and numerically extend this function onto the complex plane to obtain $\\Omega_{k}(z)$.\n\t\n\t\\item[4.] Search for complex turning points of each frequency function $\\Omega_{k}(z)$, and numerically trace their corresponding Stokes lines.\n\n\t\\item[5.] If a Stokes line associated with a mode of wavenumber $k$ is found to intersect the real axis, update the real bilinears $\\mathcal{N}_{k}(t)$, $\\mathcal{R}_{k}(t)$ and $\\mathcal{I}_{k}(t)$ accordingly.\n\t\n\t\\item[6.] For each field mode, compute $W_{k}(t)$ and $V_{k}(t)$ up to the optimal truncation order set by the last Stokes line crossing.\n\t\n\t\\item[7.] Gather the results from all previous steps to evolve the input samples for the field energy density $\\rho(t)$ and pressure $P(t)$ by a time step $\\Delta t$.\n\\end{itemize}\n\nIn general, the Stokes lines associated with field modes of comparable wavenumber will cross the real axis within close proximity of one another, giving rise to overlapping particle production events. In order to correctly capture the influence that such events might have on each other, we apply the stepping algorithm outlined above in an iterative fashion. In other words, once the quantities of interest have been forward-stepped up to $t_{j}$, the following iteration backtracks to $t_{0}$ and then proceeds to step the problem up to $t_{j+1} = t_{j} + \\Delta t$ using as sources for the semi-classical Friedmann equations the field energy density and pressure obtained in the previous iteration.\n\nIn summary, our numerical implementation allows for the scale factor and the Stokes geometry to reconfigure themselves with each iteration and thereby construct a self-consistent solution to the back-reaction problem. \n\n\\section{Numerical Results}\\label{Sec. Results}\n\nTo assess the accuracy of our numerical approach, we first neglect back-reaction effects and compare numerical results to known analytic solutions for a quantized scalar field evolving in a closed de Sitter spacetime \\cite{Mottola85}. This case is characterized by a positive cosmological constant $\\Lambda$ and a curvature parameter of $K = 1$, which together lead to a bouncing scale factor evolution\n\\begin{equation}\\label{Eq. de Sitter}\n\ta(t) = \\bar{H}^{-1} \\cosh{\\left(\\bar{H} t\\right)} \\quad \\text{with} \\quad \\bar{H} = \\sqrt{\\Lambda\/3} \\, .\n\\end{equation}\nHere $\\bar{H}$ is the asymptotic value of the Hubble parameter in the infinite future,\n\\begin{equation}\\label{Eq. Hubble de Sitter}\n\t\\lim_{t \\to \\pm \\infty}{H(t)} = \\pm \\bar{H} \\, .\n\\end{equation}\nThis model Universe contracts for $t < 0$, reaches its minimum size at $t = 0$, and subsequently expands for the $t > 0$.\n\nSubstituting Eq.~(\\ref{Eq. de Sitter}) into Eq.~(\\ref{Eq. Mode Frequency}) yields \n\\begin{equation}\\label{Eq. de Sitter Mode Frequency}\n\\Omega_{k}^{2}(t) = \\bar{H}^{2} \\Bigg[ \\bigg(k^{2} - \\frac{1}{4} \\bigg) \\, \\text{sech}^{2} \\! \\left(\\bar{H} t\\right) + \\frac{m^{2}}{\\bar{H}^{2}} + 12\\,\\xi - \\frac{9}{4} \\Bigg] \\, \n\\end{equation}\nfor the mode frequency function.\nAnalytically extending this function to the complex plane, locating its turning points, and tracing its Stokes lines are straightforward. We verify our numerical calculations against these analytic results. For definiteness, we choose a scalar field of mass $m = 0.1\\,M$ which is conformally coupled to the scalar curvature, $\\xi = \\frac{1}{6}$. We set the cosmological constant to $\\Lambda = 3 \\, m^{2}$, so that $\\bar{H} = 1 \\, m$. All dimensional quantities are thus expressed in terms of the field mass.\n\nA comparison between the analytic extension of Eq.~(\\ref{Eq. de Sitter Mode Frequency}) and the  numerical analytic continuation produced by our algorithm is displayed in Figure \\ref{Fig. Continuation Error} for the field mode of wavenumber $k = 5 \\, m$. The left panel shows the absolute value of the numerically obtained frequency function, while the right panel exhibits how this result deviates from the analytic expression for $\\Omega_{k}(z)$. In addition to correctly reproducing the function's conjugate pair of zeroes $\\big( z_{0},\\, z_{0}^{\\ast}\\big)$ located in this region, the numerical analytic continuation differs from the analytic value by at most 2\\% in the vicinity of these points. As a result, the Stokes lines which occupy this area of the complex plane can be traced with high fidelity. This is demonstrated in the left panel of Figure \\ref{Fig. Frequency Particle}, where the Stokes lines sourced by the pairs of turning points $\\big( z_{0},\\, z_{0}^{\\ast}\\big)$ and $\\big( z_{1},\\, z_{1}^{\\ast}\\big)$ are superimposed over the numerically obtained frequency function. The effects of each Stokes line on the adiabatic bilinear $\\mathcal{N}_{k}(t)$ are displayed in the right panel of Figure \\ref{Fig. Frequency Particle}, wherein this quantity is tracked as a function of time. Each burst of particle production is prompted by a Stokes line crossing, the first of which occurs as the Universe contracts and the field mode under consideration becomes sub-horizon; while the second burst happens after the bounce, when the mode reverts back to being super-horizon due to the Universe's expansion \\cite{Habib99}. Despite the symmetry between these events, constructive interference expressed by Eq.~(\\ref{Eq. Bilinears Multiple Crossings}) causes more particles to be produced in the second burst. The expected values for the particle number plateaus are indicated by the square markers on the vertical axis, both of which agree well with the numerical curve.\n\n\\begin{figure}[t!]\n    \\includegraphics[width=0.49\\textwidth]{ContinuationError.pdf}\n    \\caption{A comparison between the numerical analytic continuation of $\\Omega_{k}(z)$ produced by our algorithm and the expected analytic expression for this function in a closed de Sitter spacetime. The field parameters are $m = 0.1 \\, M$, $\\xi = \\frac{1}{6}$, and $k = 5 \\, m$, while the spacetime is characterized by $\\Lambda = 3 \\, m^{2}$ and $K = 1$. The left panel shows the absolute value of the numerically produced frequency function in the vicinity of the pair of conjugate turning points $\\big( z_{0},\\, z_{0}^{\\ast}\\big)$, while the right panel exhibits the relative difference between the analytic and numerical results.}\n    \\label{Fig. Continuation Error}\n\\end{figure}\n\n\\begin{figure*}[hbtp]\n    \\includegraphics[width=1.0\\textwidth]{FrequencyParticle.pdf}\n    \\caption{The numerically traced Stokes geometry associated with the frequency function $\\Omega_{k}(z)$, and the adiabatic particle number evolution $\\mathcal{N}_{k}(t)$ extracted from it. The field parameters are set to $m = 0.1 \\, M$, $\\xi = \\frac{1}{6}$, $k = 5 \\, m$, and $N_{k} = 0$, while the spacetime is characterized by $\\Lambda = 3 \\, m^{2}$ and $K = 1$. The left panel shows the Stokes lines sourced by the pairs of turning points $\\big( z_{0},\\, z_{0}^{\\ast}\\big)$ and $\\big( z_{1},\\, z_{1}^{\\ast}\\big)$ superimposed over the absolute value of the numerically obtained frequency function. The real axis corresponds to the central dashed line. The effects of each Stokes line on the adiabatic particle number $\\mathcal{N}_{k}(t)$ are illustrated on the right panel, wherein this quantity is tracked as a function of time. Each burst of particle production is prompted by a Stokes line crossing, indicated here by the circular markers on the horizontal axis. The expected values for the particle number plateaus featuring in this image are indicated by the square markers on the vertical axis, both of which show very good agreement with the numerically produced curve for $\\mathcal{N}_{k}(t)$. Constructive interference causes more particles to be produced in the second burst.}\n    \\label{Fig. Frequency Particle}\n\\end{figure*}\n\\begin{figure*}[hbtp]\n    \\includegraphics[width=1.0\\textwidth]{EnergyDensity_NoBackReaction.pdf}\n    \\caption{The evolution of every term appearing on the right-hand side of the semi-classical Friedmann Eq.~(\\ref{Eq. First Friedmann Equation}) in a closed de Sitter spacetime evolution, as well as the quantities describing the metric for this spacetime in the absence of back reaction. The field parameters are set to $m = 0.1 \\, M$ and $\\xi = \\frac{1}{6}$, while the spacetime is characterized by $\\Lambda = 3 \\, m^{2}$ and $K = 1$. The bounce starts at $t_{0} = -5 \\, m^{-1}$ with an initial particle distribution given by $\\mathcal{N}_{k}(t_{0}) = 0$. The left panel follows the evolution of $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ (solid line), $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ (dot-dashed line), $H^{2}_{\\Lambda}$ (dotted line), and $H^{2}_{K}$ (dashed line). While $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ remains negligible throughout, $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ grows exponentially and eventually comes to dominate over all other contributions. The right panels illustrate the scale factor $a(t)$ and Hubble parameter $H(t)$ which describe the de Sitter bounce. Because back-reaction effects are being neglected, the Hubble parameter is just $H^{2} = H^{2}_{\\Lambda} + H^{2}_{K}$.} \n    \\label{Fig. Contributions to H2 - No back-reaction}\n\\end{figure*}\n\nBy tracing the Stokes geometry of every field mode, we can also track the evolution of the field energy density as the spacetime evolves. Even though back-reaction effects are being neglected, this quantity shows whether the effects of particle production will eventually become comparable to the contributions from $\\Lambda$ and $K$ which source the background de Sitter spacetime. To that end, we track every term appearing on the right-hand side of  the semi-classical Friedmann Eq.~(\\ref{Eq. First Friedmann Equation}), identifying each contribution according to the notation\n\\begin{subequations}\\label{Eq. Contributions to H2}\n\\begin{align*}\n\tH^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}} \\equiv \\frac{\\rho}{3M^2}\\,\\,, \\quad H^{2}_{\\Lambda} \\equiv \\frac{\\Lambda}{3} \\,\\,, \\quad \\text{and} \\quad H^{2}_{K} \\equiv - \\frac{K}{a^{2}} \\, .\n\\end{align*}\n\\end{subequations}\nAdditionally, we define $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ as the contribution to the right-hand side of Eq.~(\\ref{Eq. First Friedmann Equation}) which stems solely from terms proportional to the real bilinears $\\mathcal{R}_{k}$ and $\\mathcal{I}_{k}$. The left panel of Figure~\\ref{Fig. Contributions to H2 - No back-reaction} displays the evolution of the above-defined quantities for a bounce that starts at $t_{0} = -5 \\, m^{-1}$ with an initial particle distribution given by $\\mathcal{N}_{k}(t_{0}) = 0$. Being the only true sources in this case, $H^{2}_{\\Lambda}$ and $H^{2}_{K}$ behave in the standard way, acting in concert to produce the de Sitter bounce. Because back-reaction effects are neglected, the Hubble parameter $H$ shown on the right panel of Figure~\\ref{Fig. Contributions to H2 - No back-reaction} is entirely characterized by these two quantities, i.e., $H^{2} = H^{2}_{\\Lambda} + H^{2}_{K}$. On the other hand, the field-related quantities $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ and $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ display an interesting behavior which mirrors the result found in Ref.~\\cite{AndersonMottola14}. While $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ remains negligible throughout, $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ grows exponentially as the Universe progresses toward the bounce. In other words, the field energy density eventually becomes dominated by $\\mathcal{N}_{k}$ -- the field particle content. Physically, the soaring field energy density is due to the blueshift experienced by particles produced in the contracting phase. As a result, the Universe is filled with relativistic particles which effectively behave as radiation, making the field energy density grow as $\\rho \\propto a^{-4}$. This trend is then reversed in the ensuing expanding phase, during which the field energy density drops rapidly as particles are continuously redshifted.\n\\begin{figure*}[hbtp]\n    \\includegraphics[width=1.0\\textwidth]{EnergyDensity_BackReaction.pdf}\n    \\caption{The evolution of every term appearing on the right-hand side of the semi-classical Friedmann Eq.~(\\ref{Eq. First Friedmann Equation}) and the quantities describing the metric evolution in a full back-reacting calculation. The field parameters are $m = 0.1 \\, M$ and $\\xi = \\frac{1}{6}$, while the cosmological constant and curvature parameter are $\\Lambda = 3 \\, m^{2}$ and $K = 1$. The closed de Sitter initial conditions are set at $t_{0} = -5 \\, m^{-1}$, along with an initial particle distribution given by $\\mathcal{N}_{k}(t_{0}) = 0$. The left panel follows the evolution of $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ (solid line), $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ (dot-dashed line), $H^{2}_{\\Lambda}$ (dotted line), and $H^{2}_{K}$ (dashed line). The exponential growth of $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ effectively fills the Universe with relativistic particles, introducing an instability to the initial de Sitter phase. The right panels illustrate the scale factor $a(t)$ and Hubble parameter $H(t)$ transitioning from a Sitter bounce to a radiation dominated phase. Here the solid lines represent the solutions to the back-reaction problem, while the dashed lines trace the pure de Sitter bounce.} \n    \\label{Fig. Contributions to H2 - Back-reaction}\n\\end{figure*}\n\nThe preceding calculations demonstrate that back-reaction effects due to particle production can become dynamically significant in an initially closed de Sitter spacetime. A full account of these effects is shown in Figure~\\ref{Fig. Contributions to H2 - Back-reaction}, using the algorithm for computing back-reaction effects described in the previous Section. In this case, the metric evolution initially matches that of a closed de Sitter spacetime at $t_{0} = -5 \\, m^{-1}$, while the initial particle distribution is given by $\\mathcal{N}_{k}(t_{0}) = 0$. These initial conditions self-consistently satisfy the semi-classical Friedmann equations at the initial time $t_{0}$ within our approximations. As in the case without back reaction, the quantity $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ remains sub-dominant throughout the evolution, while $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ grows exponentially as newly-created particles are continuously blueshifted. Since they quickly become relativistic, these particles behave as an additional radiation-like component, destabilizing the initial de Sitter phase. This is illustrated in the right panels of Figure~\\ref{Fig. Contributions to H2 - Back-reaction}, where the scale factor and Hubble parameter can be seen transitioning from a de Sitter bounce to a radiation-dominated behavior. The contributions from the regularized vacuum terms discarded in our approximations remain negligible at all times. We stop the numerical integration at $t = -1.3 \\, m^{-1}$, since beyond this time the Hubble parameter becomes of order $H \\simeq M^{-1}$, invalidating the semi-classical picture of gravity on which our calculations rely.\n\nThe de Sitter bounce is not always disrupted by particle production. For sufficiently low values of the field mass, the bounce is merely delayed. Figure~\\ref{Fig. Contributions to H2 - Back-reaction 2} illustrates a near-limiting case with ${m = 0.0145 \\, M}$ for which an initial de Sitter evolution is still driven toward a radiation dominated phase. The contributions due to particle production $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ only come to dominate over the combined $H^{2}_{\\Lambda}$ and $H^{2}_{K}$ near the bounce at $t = 0 \\, m^{-1}$. For field masses $m \\lesssim 0.0142 \\, M$, the negative curvature contributions $H^{2}_{K}$ neutralize the growth of $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ for long enough to preserve the bounce. The resulting bounce is pushed to a slightly later time and occurs at a smaller value of the scale factor.\n\n\\begin{figure*}[hbtp]\n    \\includegraphics[width=1.0\\textwidth]{EnergyDensity_BackReaction_2.pdf}\n    \\caption{The evolution of every source term featuring on the right-hand side of the semi-classical Friedmann Eq.~(\\ref{Eq. First Friedmann Equation}) and the quantities describing the metric evolution in a full back-reacting calculation. The field parameters are set to $m = 0.0145 \\, M$ and $\\xi = \\frac{1}{6}$, while the cosmological constant and curvature parameter are characterized by $\\Lambda = 3 \\, m^{2}$ and $K = 1$. The closed de Sitter initial conditions are set at $t_{0} = -5 \\, m^{-1}$, along with an initial particle distribution given by $\\mathcal{N}_{k}(t_{0}) = 0$. The left panel follows the evolution of $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ (solid line), $H^{2}_{\\mathcal{R},\\,\\mathcal{I}}$ (dot-dashed line), $H^{2}_{\\Lambda}$ (dotted line), and $H^{2}_{K}$ (dashed line). The growth of $H^{2}_{\\mathcal{N},\\,\\mathcal{R},\\,\\mathcal{I}}$ fills the Universe with just enough relativistic particles to destabilize the initial de Sitter phase. The right panels illustrate the scale factor $a(t)$ and Hubble parameter $H(t)$ transitioning from a de Sitter bounce to a radiation dominated phase. Here the solid lines represent the solutions to the back-reaction problem, while the dashed lines trace the pure de Sitter bounce. Had the field mass been set to a value $m \\lesssim 0.0142 \\, M$, a bounce would still take place, albeit at a slightly later time and for a smaller value of the scale factor.}\n    \\label{Fig. Contributions to H2 - Back-reaction 2}\n\\end{figure*}\n\n\\section{Discussion}\\label{Sec. Discussion}\n\nThe back-reaction problem addressed in this work imposes several technical hurdles which have resisted a satisfactory solution for decades. These difficulties stem primarily from the necessity to control the divergent nature of the vacuum energy. Adiabatic regularization accomplishes this at the cost of increasing the problem's complexity. As a result, ambiguities arise in the specification of initial conditions and in the value of physical quantities when particle production is rapid, and computationally the problem becomes susceptible to potential numerical instabilities. In this work we have shown that these issues can be circumvented in scenarios dominated by particle production. Our approach relies on a particular choice of adiabatic mode functions which isolate the vacuum contributions into a separate covariantly conserved component of the total stress-energy. In regimes dominated by particle production, this vacuum component is sub-dominant and can be discarded in its entirety. By definition, the remaining covariantly conserved portion of the stress-energy dominates, as it encapsulates the effects of particle production. This component can be expressed in terms of the particle number density as described by Berry's universal form, resolving the ambiguity in physical quantities, and computed from the analytic continuation of each mode's frequency function onto the complex plane (Figures \\ref{Fig. Frequency Particle} and \\ref{Fig. Contributions to H2 - No back-reaction}). The resulting stress-energy is a calculable source term for the semi-classical Friedmann equations, and can be used to obtain a numerical solution to the back-reaction problem. We have performed this calculation for an initially closed de Sitter spacetime, demonstrating that the effects of particle production in this scenario can become strong enough to drive the cosmic evolution into a radiation-dominated phase (Figures \\ref{Fig. Contributions to H2 - Back-reaction} and \\ref{Fig. Contributions to H2 - Back-reaction 2}). Our results illustrate the reliability of our numerical implementation, and open the possibility of the systematic investigation of cosmological scenarios dominated by quantum particle production.\n\nOn a technical level, our method relies on some previous knowledge of the Stokes geometry associated with the spacetime evolution. For the case studied in this work, all Stokes lines are sufficiently separated from each other so that Berry's universal form for particle production applies without corrections. In general, however, the spacetime evolution might result in near-lying Stokes lines for which higher-order Stokes corrections are required for an accurate description of particle production. Although not included in this work, such corrections are well-documented in the literature \\cite{BerryHowls94, HowlsLongmanDaalhuis04, HowlsDaalhuis12} and could in principle be added to our numerical implementation. More fundamentally, our method is based on a well-defined semi-classical notion of particle. Mathematically, this notion is tied to the existence of a phase-integral expansion for the field mode functions. Such a representation can always be constructed as long as $\\left| \\varepsilon_{k,\\,0} \\right| \\ll 1$. Physically, this requirement typically translates to an approximate bound on the Hubble rate $H \\lesssim m$ set by the mass of the field under consideration. Nonetheless, some scenarios exist for which $\\left| \\varepsilon_{k,\\,0} \\right| \\ll 1$ is satisfied even when $H > m$.\n\nQuantum backreaction  is potentially important in models of the very early Universe. Quantum particle production is actually quite familiar in the context of inflation, as it provides the standard mechanism for the generation of perturbations in an inflating spacetime \\cite{Starobinsky79, Allen88, Sahni90, Mukhanov92, Souradeep92, Glenz09, Agullo11}.  Interestingly, it has been suggested that these same ideas could be applied to the problem of driving inflation itself \\cite{Prigogine89, Calvao92, Lima96, Abramo96, Gunzig98, Lima14, HaroPan16}. Indeed, a phase of accelerated expansion can result if particles are produced at a high enough rate. A time derivative of the usual Friedmann Equation $H^{2}(t) = \\frac{1}{3}M^{-2} \\rho(t)$ shows that an accelerating expansion ${\\ddot a} > 0$ occurs when\n\\begin{equation}\\label{Eq. Accelerated Condition}\n\t\\dot{\\rho}(t) > -\\frac{2}{\\sqrt{3}} \\, M^{-1} \\rho^{3\/2}(t) \\, .\n\\end{equation}\nSuch a scenario has the potential to sidestep some of the conceptual problems of the standard inflationary paradigm. For instance, it has been argued that standard inflation cannot generically start in a patch which is smaller than the cosmological horizon without violating either causality or the weak energy condition \\cite{VachaspatiTrodden99, BereraGordon01}. However, if inflation is initially driven by an increasing energy density due to particle production, the weak energy condition {\\it is} effectively violated. Therefore, inflation driven by such a mechanism could generically start in small patches contained within the cosmological horizon without violating causality. Inflation driven by particle production would also clarify the meaning of the inflaton effective potential by making manifest the high mass-scale physics it represents.\n\nThe same conditions which lead to quantum particle production can also result in particle annihilation. If sufficiently pronounced, this effect can drive a contracting spacetime toward a bounce phase. Indeed, it follows from the cosmological continuity equation that $\\rho(t) + P(t) < 0$ provided the particle annihilation rates are high enough to cause the field energy density to decrease as the Universe contracts. In other words, the null energy condition is effectively violated, making $\\dot{H}(t) > 0$ according to the Friedmann equations \\cite{IjjasSteinhardt18, Ijjas16}. Thus, a classical bounce can emerge provided enough energy density is sequestered by quantum particle annihilation during a phase of cosmological contraction. If realized, such a mechanism could provide a natural description for cosmological bounce scenarios which does not require new physics. Also, successful bounces require constraints on high mass-scale quantum fields, so that quantum back-reaction does not push the contracting phase into a radiation crunch, as with the example solved in this paper.\n\nAnother possibly interesting effect is the production of a relativistic condensate in the early universe. Under certain circumstances, quantum particle production can lead to large occupation numbers for some scalar field modes, representing condensate formation. This phenomenon could lead to additional interesting phenomenology \\cite{AragaoRosa80, ParkerZhang91, ParkerZhang93}. \n\nA number of technical questions remain to be answered. Fermion fields require more complex calculations than scalar fields, and may present some different physics \\cite{Landete13, Landete14}. How to handle interacting fields remains an open question, and multiple fields offer additional possibilities \\cite{Ringwald87, CooperMottola87, PazMazzitelli88, Habib96, Cooper97, MolinaParis00}. We have a long road to travel before the range of interesting early-Universe dynamical scenarios driven by quantum particle production has been fully explored. \n\n\n\\begin{acknowledgments}\nF.Z. thanks D. Boyanovsky, S. Habib, P. Anderson, and E. Mottola for helpful discussions. F.Z. acknowledges support from the Andrew Mellon Predoctoral Fellowship and the A\\&S PITT PACC Fellowship. The authors have been partly supported by the National Science Foundation under the grant AST-1312380. This work made use of many community-developed or community-maintained software packages, including (in alphabetical order): Matplotlib \\cite{Matplotlib}, NumPy \\cite{NumPy}, and SciPy \\cite{SciPy}. Bibliographic information was obtained from the NASA Astrophysical Data System.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nConsider a set of combinatorial objects $\\{\\alpha\\}$ with statistics $wt(\\alpha)$ and $t(\\alpha)$, thinking of $wt(\\alpha)$ as the primary descriptor.  Let $G(z,q)$ be its two-variable generating function, that is, if $p(n,k)$ is the number of objects $\\alpha$ with $wt(\\alpha) = n$ and $t(\\alpha) = k$, \n \\[\n G(z,q) = \\sum_{n,k \\in \\mathbb{N} \\bigcup \\{0\\}} p(n,k) q^n z^k\\,  \\text{.}\n \\]\n    \nAn important example occurs when the generating function\n $G(z,q)$ has the Euler product form \n\\begin{equation}\\label{eq:product}\nG(z,q) = \\prod_{i \\geq 1} \\frac{1}{(1-z q^i)^{a_i}}\n\\end{equation}\nwith $a_i \\in \\mathbb{Z}^+ \\bigcup \\{0\\}$.  \nWe let  $F_n(z)$ denote the $q^n$ coefficient of $G(z,q)$ which is a polynomial in $z$, so\n\\[\nG(z,q) = \\sum_n F_n(z) q^n, \\quad  \\, F_n(z) = \\sum_k p(n,k) z^k \\, .\n\\]\n\nFor $a_i = 1$ this is the generating function for partitions of weight $wt(\\alpha) = n$ with number of parts $t(\\alpha) = k$.  It is a well-known fact in partition theory that $p(n,k)$ for $k$ nearly equal to $n$ has a value independent of $n$: $p(n,n-b)$ is the number of partitions of $b$ for $b\\leq\\frac{n}{2}$.  A similar result holds for plane partitions of $n$ indexed by their trace \\cite[p. 199]{Andrews_book} or \\cite[Corollary 5.3]{Stanley}.\n\nThis phenomenon, which we call {\\sl stabilization,}  is widespread in generating functions of combinatorial interest, even those of greater complexity.  The purpose of this paper is to describe this behavior in more general cases, and consider some illustrative examples and variations.  \nWe found the polynomial framework to be well suited to these problems rather than a direct approach.  The arguments should be adaptable to a wide variety of cases.\n\nWe would like to thank the anonymous referee for a careful readthrough: noting typos, improving exposition, and suggesting occasional strengthening of theorems for which we had been hesitant to extend our reach.  This article is substantially improved from their efforts.\n\n\\section{ Basic Infinite Product Generating Functions }\n\nLet  $G(z,q)$ be an infinite product generating function of the form\n\\begin{equation}\\label{eq:new_genfct}\nG(z,q) = \\sum_{n=1}^\\infty F_n(z) q^n =  \\prod_{j=1}^{\\infty} \\frac{1}{(1-z^{b(j)} q^{c(j)})^{a_j}}.\n \\end{equation}\n where we assume that the number of $j$ for which $c(j) = t$ for any $t$ is finite, so that the series converges. We find that if the $c$ grow sufficiently faster than the $b$, the upper ends of the $F_n(z)$ stabilize, to the coefficients of a single-variable generating function which we can give.\n  \n  Let ${\\mathcal F}$ denote the set of all nonnegative integer sequences with finite support.\n  For ${\\bf e} = (e_1,e_2,\\dots) \\in {\\mathcal F}$\nset\n \\[\n \\mu(\\mathbf{e}) = (c(1)^{e_1}c(2)^{e_2}\\cdots),\\quad\n \\nu(\\mathbf{e}) = (b(1)^{e_1}b(2)^{e_2}\\cdots)\n \\]\nto denote  the partitions with parts $c(j)$ (resp. $b(j)$) appearing $e_j$ times. \nA direct expansion of the generating functions yields\nan explicit form  for the polynomial $F_n(z)$:\n\\begin{eqnarray}\\label{eq:take_away}\nF_n (z)\n &= &\n\\sum_{k=0}^n z^k  \\,   \\sum_{{\\mu(\\mathbf{e}) \\vdash n} \\atop {\\nu(\\mathbf{e}) \\vdash k}} \\prod_{i \\geq 1} \n  \\binom{ a_i+e_i-1}{e_i}\n  \\end{eqnarray}\n  We will compare the coefficients of $F_n(z)$ with those of the expansion of\n  \\[\n  \\prod_{i=1}^\\infty\n  \\frac{1}{ (1-z^{b(i)})^{a_i}} =\n  \\sum_{k=0}^\\infty z^k \\, \\sum_{ \\nu( {\\bf e}) \\vdash k} \\, \\prod_{i\\geq 1} \\binom{ a_i + e_i -1}{ e_i} .\n  \\]\n \n\n\\begin{theorem}\n\\label{thm:stabliztion2}\n Suppose that the exponents  satisfy $a_1 = 1 = b(1) = c(1)$.\n If there exists a positive integer $m \\geq 2$ such that \n \\begin{equation}\\label{eq:stable2}\n m \\cdot b(j) \\leq c (j), \\quad j \\geq 2  ,\n \\end{equation}\n then \n \\\\\n (a)\n for $k > n\/m $, $\\left[z^k \\right] F_n(z) = \\left[z^{k+1} \\right] F_{n+1} (z)$.\n \\\\\n (b) \n  If  $c(j) - b(j) > 0$ for all $j>1$ and the set of $j$ for which $c(j) - b(j)$ takes a given value is finite for any fixed difference, \n  then for $\\ell \\leq \\lfloor n\/m \\rfloor$, \n  \\[\n  \\left[ z^{n-\\ell}\\right]F_n(z) = \\left[z^{\\ell}\\right] \\prod_{j \\geq 2} \\frac{1}{(1-z^{c(j)-b(j)})^{a_j}} .\n  \\]\n\\end{theorem}\n\\begin{proof}\n(a)\nFrom the explicit form (\\ref{eq:take_away}) of the polynomial $F_n(z)$, we can expand it as\n\\begin{eqnarray} \nF_n (z)\n &= &\n\\sum_{k=0}^n z^k  \\,   \\sum_{{\\mu(\\mathbf{e}) \\vdash n} \\atop {\\nu(\\mathbf{e}) \\vdash k}} \\prod_{i \\geq 1} \n  \\binom{ a_i+e_i-1}{e_i}\n  \\nonumber\n\\\\ \n&=& \\sum_{k=0}^n \\sum_{e_1=0}^k z^{b(1) e_1}\n \\sum_{{\\mu^-(\\mathbf{e}) \\vdash n - c(1)e_1} \\atop {\\nu^-(\\mathbf{e}) \\vdash k - b(1)e_1}} z^{k-b(1)e_1}\n  \\prod_{i \\geq 2}  \\left( {{a_i+e_i-1} \\atop {e_i}} \\right)  .\n\\nonumber\n\\end{eqnarray}\n\n\nNow if the integer sequence ${\\bf e}$ gives a contribution to $[z^{k+1}]F_{n+1}(z)$ and $e_1>0$, then we define \n${\\bf e}'$ \nas the integer sequence \nall of whose  terms  agree with ${\\bf e}$ except for $j=1$ where we set $e_1'=e_1-1$.\nIn this way, we obtain all the possible terms contributing to $[z^k]F_n(z)$.\nConversely, any term for $[z^k]F_n(z)$ gives a contribution to $[z^{k+1}]F_{n+1}(z)$ by simply adding\n1 to its first component.\n\nThe result reduces to showing that any contribution to $[z^{k+1}]F_{n+1}(z)$ indexed by ${\\bf e}$\nmust have $e_1>0$.\n\nWe introduce the notation for the modified partitions \n\\[\n \\mu^{-}(\\mathbf{e}) = (c(2)^{e_2} c(3)^{e_3} \\cdots),\\quad\n \\nu^{-}(\\mathbf{e}) = (b(2)^{e_2}  b(3)^{e_3} \\cdots)  .\n \\]\nNow assume that  the partition with $\\mu({\\bf e}) \\vdash n$ and $\\nu({\\bf e}) \\vdash k$ gives a contribution to $[z^{k+1}]F_{n+1}(z)$ but\n  $e_1 = 0$. So $\\mu^-(\\mathbf{e}) \\vdash n$ and $\\nu^-(\\mathbf{e}) \\vdash k$, but if $c(j) \\geq m \\cdot b(j)$ for all $j \\geq 2$, \n  then $\\vert \\mu^-({\\bf e}) \\vert \\geq m \\vert \\nu^-({\\bf e}) \\vert$.\nHence  if $ k > \\frac{n}{m}$, we find that $\\vert \\mu^{-}({\\bf e}) \\vert > n$, a contradiction.  Thus all terms in both expansions are the same, and so the coefficients are equal.  This proves part (a).\n\nFor part (b), we begin by forming a new partition as follows.\nFirst subtract $b(j)$ from each $c(j)$ and consider the partition\n$\\lambda(\\mathbf{e}) = ((c(2)-b(2))^{e_2}\\dots)$.  This removes exactly the amount $\\vert \\nu^{-}(\\mathbf{e})\\vert$ from $\\vert \\mu^{-}(\\mathbf{e})\\vert$, so\n\\begin{eqnarray}\nF_n (z)\n&=& \\sum_{k=0}^n \\sum_{e_1=0}^k z^{k}\n \\sum_{{\\lambda(\\mathbf{e}) \\vdash n - k} \\atop {\\nu^-(\\mathbf{e}) \\vdash k - b(1)e_1}} \\prod_{i \\geq 2}  \\left( {{a_i+e_i-1} \\atop {e_i}} \\right)\n\\nonumber\n\\\\\n&=& \\sum_{k=0}^n z^{k}\n \\sum_{{\\lambda(\\mathbf{e}) \\vdash n - k} \\atop {\\vert \\nu^-(\\mathbf{e}) \\vert \\leq k}} \\prod_{i \\geq 2}  \\left( {{a_i+e_i-1} \\atop {e_i}} \\right) . \\nonumber\n\\end{eqnarray}\n\nBy hypothesis, the parts of $\\lambda$ satisfy $c(j) - b(j) \\geq (m-1)b(j)$.  \nIf $k = n-\\ell > \\lfloor n\/m \\rfloor$, then $\\ell < \\lfloor n \\frac{m-1}{m} \\rfloor$. \n If $\\lambda(\\mathbf{e}) \\vdash n-k = \\ell$, then $\\nu^-(\\mathbf{e}) \\leq \\frac{\\ell}{m-1} \\leq k$.  \n Thus the sum runs over all $\\mathbf{e}$ for which $\\lambda(\\mathbf{e}) \\vdash \\ell$ with parts $c(j) - b(j)$.  But this is exactly the coefficient of $z^{\\ell}$ in the expansion claimed: \n \\[\n \\left[ z^{n-\\ell}\\right]F_n(z) = \\left[z^{\\ell}\\right] \\prod_{j \\geq 2} \\frac{1}{(1-z^{c(j)-b(j)})^{a_j}} \\, \\text{ .}\n \\]\n\\end{proof}\n\nBy the technique of proof, we have a slight improvement in a special case of hand enumerators of prefabs\n (\\cite{Bender_Goodman}, \\cite[page 92]{Wilf}).\n\n\\begin{corollary}\nSuppose $a_1=1$.\nIf  $F_n(z) = [q^n] \\prod_{j=1}^\\infty (1-zq^j)^{- a_j}$,\n then\n  for $k  \\geq n\/2$, $\\left[z^k \\right] F_n(z) = \\left[z^{k+1} \\right] F_{n+1} (z)$.\n\\end{corollary}\n\nBy the proof, we find that the generating function for the stabilized coefficients gives an upper bound outside the range of stability.\n\n\\begin{corollary}\nWith the hypotheses of the Theorem,\n\\[\n \\left[ z^{n-\\ell}\\right]F_n(z) \\leq \\left[z^{\\ell}\\right] \\prod_{j \\geq 2} \\frac{1}{(1-z^{c(j)-b(j)})^{a_j}} ,\n \\quad 0 \\leq \\ell \\leq n. \n \\]\n\\end{corollary}\n\nVariants of stabilization exist in several guises.  If the $b$ grow faster than the bound of the previous theorem, we find that the smaller end of the polynomials stabilize instead of the larger (and $b$ and $c$ staying within a given ratio range will permit both phenomena).  We also note that it is possible for the larger coefficients of a sequence of polynomials to stabilize in periods, i.e., the coefficients match those of a polynomial every 2 or more steps further along.\n\n\\begin{theorem}\\label{thm:stabilization2}\n(a)\nLet $\\{ b(j)\\}$ and $\\{ c(j)\\}$ be two strictly increasing sequences of integers, positive except that \n$b(1)=0$. \n Let $F_n(z) = [q^n]\\prod_{j=1}^\\infty (1-z^{ b(j) } q^{ c(j) })^{  - a_j  }$ with $a_1=1$.\nIf  there exists a positive integer $m $ such that for all $j \\geq 2$\n \\[\n (m+1) b(j) > c(j) ,\n \\]\nthen \n\\[\n[z^k] F_n(z)  = \\, [ z^k] F_{n+ c(1)}(z), \\quad 0 \\leq k \\leq n\/(m+1).\n\\]\n(b)\nLet $a_1=0, a_2=1$. \nIf\n$F_n(z) = [q^n]\\prod_{j=2}^\\infty (1-z q^{ j})^{  - a_j  }$, then\n$\\deg(F_n) = \\lfloor n\/2 \\rfloor$\nand\n\\[\n[z^k] F_n(z) = \\, [z^{k+1}] F_{n+2}(z) , \\quad  k \\geq n\/3 .\n\\]\n\\end{theorem}\n\\begin{proof}\nLet ${\\bf e}$ be a sequence of non-negative integers with finite support.\nBy (\\ref{eq:take_away}), for the coefficients $[z^k]F_n(z)$ and $[z^k] F_{n+c(1)}(z)$, we need to consider the following two sets of \nfinitely supported nonnegative integer sequences:\n\\begin{eqnarray*}\nS_n &=&\n \\{ {\\bf e} \\in {\\mathcal F} : | \\mu({\\bf e}) | = n,  \\, | \\nu({\\bf e} ) | =k \\}, \n \\\\\nS_{n+c(1)} &=&\n  \\{ {\\bf f} \\in {\\mathcal F} : | \\mu({\\bf f}) | = n+c(1),  \\, | \\nu({\\bf f} ) | =k \\} .\n\\end{eqnarray*}\nWe construct a bijection between these two sets. Given ${\\bf e} \\in S_n$, we take the\ncorresponding ${\\bf f}$ with $f_i=e_i$ for $2 \\leq i$ since $b(1)=0$.\nNext write out $ | \\mu({\\bf e}) | = n$\n and $ | \\mu({\\bf f}) | = n+c(1)$:\n\\[\nn= \\sum_{j \\geq 1} c(j) e_j, \\quad n+c(1) = c(1) f_1 + \\sum_{j \\geq 2} c(j) e_j .\n\\]\nThe term $f_1$ uniquely determines a preimage $e_1$ \nprovided $f_1>0$. But $f_1$ must be positive for $k \\leq n\/(m+1)$ since\n\\[\n\\sum_{j\\geq 2} c(j) f_j = \\sum_{j \\geq 2} c(j) e_j < (m+1) \\sum_{j \\geq 2} b(j) e_j  = (m+1)k  \\leq n.\n\\]\nFinally, the coefficients themselves agree; that is,  $[z^k] F_n(z) = [z^k] F_{n+c(1)}(z)$:\n\\[\n  \\prod \\left\\{ \\binom{ a_i + e_i -1}{ e_i} : {\\bf e} \\in S_n \\right\\} =\n  \\prod \\left\\{ \\binom{ a_i + f_i -1}{ f_i} : {\\bf f} \\in S_{n+c(1)} \\right\\} \n\\]\nsince the above binomial coefficients are all equal for $i \\geq 2$ and when $i=1$ they both reduce to $1$ since $a_1=1$.\n\n\nFor part (b), let\n$S_n = \\{ {\\bf e } \\in {\\mathcal F} :\n\\mu({\\bf e}) \\vdash n, \\nu({\\bf e} ) \\vdash k\\}$\nwhile\n$S_{n+2}=\n \\{ {\\bf f } \\in {\\mathcal F} :\n\\mu({\\bf f}) \\vdash n+2, \\nu({\\bf f} ) \\vdash k+1\\}$.\nWe can construct a bijection between these two sets as in part (a) provided $f_2 >0$ when $k \\geq n\/3$.\nAssume that $f_2=0$ is possible. Then $\\sum f_j = k+1$ while $\\sum_{j \\geq 3} j f_j = n+2$. On the other hand,\n$3(k+1) \\geq \\sum_{j \\geq 3} j f_j $ which yields a contradiction.\n\\end{proof}\n\n\n\n  \n \\section{Partitions With Prescribed Subsums}\\label{section:subsums}\n \n  \n \nFix a positive integer $m$  and integer $i$ so $1 \\leq i \\leq m$.  Canfield-Savage-Wilf \\cite[Section 3]{CSW} introduced the generating function\n\\[\nG_{m,i}(z,q) = \\prod_{j=1}^\\infty \\, \n\\prod_{b=1}^{i-1} \\frac{1}{ 1- z^{j-1} q^{ (j-1)m+b}} \\, \\prod_{b=i}^m \\frac{1}{ 1- z^{j} q^{ (j-1)m+b}}\n\\]\nto describe partitions with prescribed subsums. They let\n$\\Lambda_{m,i}(n,k)$ be the number of partitions $\\lambda=(\\lambda_1,\\cdots, \\lambda_n)$ of $n$\nsuch that the sum of those parts $\\lambda_j$ whose indices $j$ are congruent to $i$ modulo $m$ is $k$;\nthat is, \n\\[\n\\sum_{ j :  j \\equiv i \\, ({\\rm mod} \\, m)} \\lambda_j = k .\n\\]\nThen they found\n\\[\n\\sum_{n,k \\geq 0} \\Lambda_{m,i}(n,k) z^k q^n = G_{m,i}(z,q).\n\\]\n\nWe begin by recovering a result for \n$\\Lambda_{2,2}(n,k)$ in \\cite[Theorem 1]{CSW} and \\cite{Yuri}\nby reformulating it  in terms of the generating function\n$G_{2,2}(z,q)$  and stabilization of polynomial coefficients.\n\n\\begin{proposition}\\label{prop:n-mk}\nLet $m \\geq 2$ and $1 \\leq b < m$. Let $G(z,q)= \\prod_{j=1}^\\infty (1-z^{j-1} q^{(j-1)m+b} )^{-1}$\nand $A_n(z) = [q^n] G(z,q)$.  Then for $ 0 \\leq k \\leq n\/(m+1)$  we have\n\\begin{enumerate}\n\\item\\quad\n$[z^k] A_n(z) = [z^k] A_{n+b}(z)$,\n\\item\\quad\nif $n-mk$ is not divisible by $b$, then\n$[z^k] A_n(z) = 0$,\n\\item\\quad\nif $n-mk$ is divisible by $b$ and $bk \\leq n\/(m+1)$, then\n$[z^k] A_n(z) = p(k)$.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nThe first part is a direct consequence of the last theorem.\n\nFor part (2), in Theorem \\ref{thm:stabilization2} let $c(j) = m(j-1)+b$ and $b(j)=j-1$.  When all $a_j =1$, $[z^j]F_n(z)$ is the number of all ${\\bf e} \\in {\\mathcal F}$ such that $\\mu({\\bf e}) \\vdash n$ and $\\nu({\\bf e}) \\vdash k$.\nConsider\n\\begin{eqnarray}\\label{eq:n-mk}\nn\n&=&| \\mu({\\bf e})| = \\sum_{j \\geq 1} e_j [ m(j-1) + b] \n\\nonumber\n\\\\\n&=&\nm \\sum_{j \\geq 1} e_j (j-1) + b  \\sum_{j \\geq 1} e_j \n\\nonumber\n\\\\\n&=&\nmk + b  \\sum_{j \\geq 1} e_j .\n\\end{eqnarray}\nHence $n-mk$ must be divisible by $b$ for any nonzero choice of {\\bf e}.\n\n\nFor part (3),\nassume $0 \\leq bk \\leq n\/(m+1)$ and that $n-mk$ is divisible by $b$. \nLet {\\bf e} be any solution to $\\nu( {\\bf e})=k$. Note that this does not give\na constraint for the choice of $e_1$. On the other hand, the choice of $e_1$ by (\\ref{eq:n-mk}) must be\n\\[\ne_1 = (n-mk)\/b - \\sum_{j\\geq 2} e_j, \\quad e_1 \\geq 0\n\\]\nto yield $\\mu ({\\bf e}) = n$. \nHence, $0\\leq [z^k] F_n(z) \\leq p(k)$.\nFinally,\nthe inequality $0 \\leq bk \\leq n\/(m+1)$ shows that $e_1\\geq 0$ always holds. Hence, \n$[z^k] F_n(z)=p(k)$.\n\\end{proof}\n\n\n\n \\begin{theorem}\n  Let  $G_A(z,q)=\\prod_{k=1}^\\infty ( 1-z^{k} q^{2k-1} )^{-1}$\n  and $G_B(z,q)= \\prod_{k=1}^\\infty ( 1-z^{k} q^{2k} )^{-1}$, so $G_{2,2}(z,q) = G_A(z,q) G_B(z,q)$. Then the coefficients of the polynomials\n $F_n(z)= [q^n]  G_{2,2}(z,q)$ satisfy\n  \\[\n   [z^{k}] F_n(z) = [z^{k}] F_{n+1}(z) = \\sum_{\\ell=0}^k p(\\ell) p(k-\\ell), \\quad 0 \\leq k \\leq n\/3,\n  \\]\n  where $p(\\ell)$ is the number of partitions of $\\ell$, as usual.\n  \\end{theorem}\n  \\begin{proof} \n Let $A_n(z)=[q^n]G_A(z,q)$ and $B_n(z) = [q^n]G_B(z,q)$. The generating function $G_B(z,q)$ has the explicit expansion\n  \\[\n  G_B(z,q) = \\sum_{j=0}^\\infty p(j) z^j q^{2j} \n  \\]\n  so $B_{2k+1}(z)=0$ while $B_{2k}(z) = p(k) z^k$.\n  We also know by Proposition \\ref{prop:n-mk} that\n  \\[\n  [z^j] A_n(z) = p(j), \\quad 0 \\leq j \\leq n\/3.\n  \\]\n  Next we have that\n  \\[\n  F_n(z) = \\sum_{\\ell=0}^{n\/2} A_{n-2\\ell}(z) B_{2\\ell}(z) = \\sum_{\\ell=0}^{n\/2}  p(\\ell) z^\\ell A_{n-2\\ell}(z)\n  \\]\n  Examine the coefficient $[z^k]F_n(z)$ for $0 \\leq k \\leq n\/3$:\n  \\[\n  [z^k]F_n(z) = [z^k] \\sum_{\\ell=0}^{n\/2}  p(\\ell) z^\\ell A_{n-2\\ell}(z)\n  =\\sum_{\\ell=0}^{n\/2} p(\\ell) \\, [z^{k-\\ell} ] A_{n-2\\ell}(z)\n  \\]\n  Since $k-\\ell \\leq (n-2\\ell)\/3$ for $0\\leq \\ell \\leq k$ and $k\\leq n\/3$,\n  we find    $[z^{k - \\ell}] A_{n - 2 \\ell}(z) = p(k - \\ell)$.\n  We conclude that\n    \\[\n  [z^k]F_n(z)= \\sum_{\\ell=0}^{k} p(\\ell) \\, p( k - \\ell) .\n  \\]\n   \\end{proof}\n    \n\n\n  \n\nIn order to investigate more general cases for the\n polynomials $F_n(z) = [q^n] G_{m,i}(z,q)$ it is convenient to rewrite the generating function as\n\\begin{equation}\\label{eq:new_eq}\nG_{m,i}(z,q)=\n\\left( \\prod_{b=1}^{i-1} \\frac{1}{1-q^b} \\, \\right) \\,\n\\prod_{a=1}^\\infty \\, \\prod_{d=0}^{m-1} \\frac{1}{ 1-z^a q^{ i+ (a-1) m + d}}.\n\\end{equation}\nWe wish to find a useful form for the coefficient $[z^j] F_n(z)$.  As usual, we have\n\\begin{eqnarray*}\n[q^n] G_{m,i}(z,q)\n&& = \n\\sum_{s=0}^n [q^{n-s}] \\left( \\prod_{b=1}^{i-1} \\frac{1}{1-q^b} \\, \\right) \\, \n[q^s] \\prod_{a=1}^{\\infty} \\prod_{d=0}^{m-1} \\frac{1}{ 1- z^{a} q^{i+ (a-1)m+d}}\n\\\\\n&&= \\quad\n\\sum_{s=0}^n [q^{n-s}] \\left( \\prod_{b=1}^{i-1}\\, \n \\sum_{ f_b=0}^\\infty \\,  \\left(q^b \\right)^{f_b} \\, \\right) \n\\\\\n&&\n\\qquad\n\\times \\quad\n[q^s] \\prod_{a=1}^{\\infty} \\prod_{d=0}^{m-1} \\, \\sum_{ f_{(a,d)}=0}^\\infty \n\\left(z^{a} q^{ i+(a-1)m+d}  \\right)^{f_{(a,d)}} \n\\end{eqnarray*}\nA typical term of the above expansion is indexed by a pair of partitions $(\\rho, \\mu)$\nwhere $\\rho \\vdash n-s$ where each part of $\\rho$ is $<i$ while $\\mu \\vdash s$ where\neach part of $\\mu$ is $\\geq i$. We write the parts of $\\mu$ as $\\mu_{(a,d)}$ with multiplicity\n$e_{\\mu(a,d)}$ where $a \\geq 1$ and $0 \\leq d < m$. By construction, we find that\n\\[\ns = \\sum_{a,d} e_{\\mu(a,d)} \\mu_{(a,d)}.\n\\]\nwhile, of course, no part of $\\rho$ has a contribution to $z$.\n\nEach part $\\mu_{(a,d)}$ of $\\mu$ with multiplicity $e_{\\mu(a,d)}$ contributes the power of $z$\n\\[\n \\left( z^a \\right)^{e_{\\mu(a,d)}} = z^{ a e_{ \\mu(a,d)}},\n\\]\nso the total contribution from $\\mu$ is\n\\[\n\\sum_{a,d} a e_{\\mu(a,d)}.\n\\]\n\nWe prove a few lemmas in preparation for the next theorem.\n\n\\begin{lemma}\n\\begin{eqnarray}\\label{eq:coeff_F}\n\\lefteqn{\n[z^j] F_n(z)\n}\n\\nonumber\n\\\\\n&& = \\sum_{s=ji}^n  p(n-s, < i) \\,  \\#\\{  \\mu \\vdash s :   \\textrm{all parts of } \\mu \\textrm{ are } \\geq  i, \\sum_{a,d} a e_{\\mu(a,d)}=j   \\}\n\\end{eqnarray}\nwhere $p(n-s, < i)$ denotes the number of all partitions of $n-s$ all of whose parts are strictly less than $i$.\n\\end{lemma}\n\\begin{proof}\nThis expression for $[z^j] F_n(z)$ follows directly from the generating function except for the\nrange of summation.\nLet $\\mu$ be a partition of $s$. Then we have\n\\begin{eqnarray*}\ns &=& \\sum_{a,d} [ i + m (a-1) + d] e_{\\mu(a,d)}\n\\\\\n&\\geq& \\sum_{a,d} [ i + i (a-1) ] e_{\\mu(a,d)}\n\\\\\n&=&\ni \\sum_{a,d} a e_{\\mu(a,d)} = ij .\n\\end{eqnarray*}\n\\end{proof}\n\n\n\nLet $r$ denote the number of  parts of $\\mu$ while $t$ denotes  the number of parts with $a\\geq 2$ so $r-t$ gives the number of \nparts with $a=1$. Note that\n\\[\nr = \\sum_{a,d} e_{\\mu(a,d)}.\n\\]\n\n\\begin{lemma}  Given the partition $\\mu$, we have the bound $j-r \\geq t$, and can rewrite $s$ as \n$$s = i r + m (j-r) + \\sum_{a,d} d e_{\\mu(a,d)}.$$\n\\end{lemma}\n\\begin{proof}\nSince $\\mu \\vdash s$, we find that\n\\begin{eqnarray*}\ns &=&\n\\sum_{a,d} e_{\\mu(a,d)} \\, [ i + (a-1) m + d ]\n\\\\\n&=&\ni \\, \\sum_{a,d} e_{\\mu(a,d)}  + m \\sum_{a,d} (a-1) e_{\\mu(a,d)} + \\sum_{a,d} d e_{\\mu(a,d)}\n\\\\\n&=&\ni r + m (j-r) + \\sum_{a,d} d e_{\\mu(a,d)}.\n\\end{eqnarray*}\nFor the bound, consider\n\\[\nj-r = \\sum_{a,d} (a-1) e_{\\mu(a,d)} \\geq \\sum_{a \\geq 2, d} e_{\\mu(a,d)} \\geq t.\n\\]\n\n\\end{proof}\n\n\n\n\\begin{lemma}\nLet $\\mu$ be a partition of $s$ all of whose parts are $\\geq i$. If\n$m \\geq i+2$ and\n$j > s \/(i+1)$, then $\\mu$ must have at least one part of size $i$.\n\\end{lemma}\n\\begin{proof}\nWe now assume that $m \\geq i+2$ and that the part $i$ does not appear in $\\mu$.\nIn particular,  for any part of $\\mu$ of the form $i +d$, with $a=1$, $d$ must be strictly positive.\nThen we have a refinement of the above bounds:\n\\begin{eqnarray*}\ns &\\geq&\ni r + m (j-r) + \\sum_{a,d} d e_{\\mu(a,d)}\n\\geq\ni r + (i+2) (j-r) + \\sum_{ d} d e_{1,d}\n\\\\\n&\\geq&\ni r + (i+2) (j - r) + (r-t)\n\\\\\n&\\geq&\n i r + (i+1) (j - r) + t + (r-t)\n \\\\\n &=&\n (i+1) [ r + (j-r)] = (i+1) j.\n\\end{eqnarray*}\nHence the partition $\\mu$ must have $i$ as a part; otherwise, $(i+1)j >s$  which contradicts our assumption.\n\\end{proof}\n\n\nWith these preparations, we will show that\n\n\\begin{theorem}\\label{thm:subsums}\nLet $m \\geq 1$ and $1 \\leq i \\leq m$. Set $F_n(z) = [ q^n] G_{m,i}(z,q)$ where\n$G_{m,i}(z,q)$ is given by (\\ref{eq:new_eq}).\n If $m>i+1$  and $j> \\frac{n}{i+1}$, then \n\\[\n[z^j] F_n(z) = [z^{j-1}] F_{n-i}(z)  .\n\\]\n\\end{theorem}\n\\begin{proof}\nBy (\\ref{eq:coeff_F}), we need to show that\n\\begin{eqnarray*}\n&&\n \\sum_{s=ji }^{ n}  p(n-s, < i) \\, \\# \\{   \\mu \\vdash s :  \\textrm{all parts of } \\mu \\textrm{ are } \\geq  i, \\sum_{a,d} a e_{\\mu(a,d)}=j   \\}\n \\\\\n &&\n =\n  \\sum_{s'= (j-1) i }^{n- i }  p(n-s'-i, < i) \\,\n  \\\\\n  &&\n  \\quad\n  \\times  \\quad\n  \\# \\{    \\nu \\vdash s'  :  \\textrm{all parts of } \\nu \\textrm{ are } \\geq  i, \\sum_{a,d} a e_{\\nu(a,d)}=j  -1 \\}\n\\end{eqnarray*}\nThese two coefficients are equal provided we construct a bijection $T$ between\n\\begin{eqnarray*}\nU_s &=&  \\{   \\mu \\vdash s :  \\textrm{all parts of } \\mu \\textrm{ are } \\geq  i, \\sum_{a,d} a e_{\\mu(a,d)}=j   \\}\n \\\\\n V_{s'}\n &=&\n  \\{    \\nu \\vdash s'  :  \\textrm{all parts of } \\nu \\textrm{ are } \\geq  i, \\sum_{a,d} a e_{\\nu(a,d)}=j  -1 \\}.\n\\end{eqnarray*}\nwhen $s' = s-i$. Let $\\mu \\in U_s$. By Lemma 7, the partition $\\mu$ must have a part equal to $i$. \nLet $\\nu=T(\\mu)$ be the partition of $s-i$ obtained by deleting one part from $\\mu$ of size $i$. It is easy\nto verify that $\\nu \\in V_{s'}$. The inverse of $T$ is simply adding a part of size $i$ to $\\nu \\in V_{s'}$.\n \\end{proof}\n \nIn other words, Theorem \\ref{thm:subsums} shows that if $m>i+1$ and $j> n\/(i+1)$, then the subsums satisfy\n$\\Lambda_{m,i}(n,j)= \\Lambda_{m,i}(n-i,j-1)$.\n\n\n\n\\section{Laurent Type Polynomials }\\label{section:laurent}\n\nA more general case consists of generating functions that involve $z$ raised to different powers; \nultimately, we might treat the case of the generating function \n\\[\nG(z,q) = \\prod_{(i,j) \\in \\mathbb{Z}^2 \\setminus (0,0)} {(1 - z^i q^j)}^{a_{ij}} \\, .\n\\]\n\n \nAn important example comes from the generating function for the crank statistic for partitions. \nLet\n\\[\nC(z,q) = \\prod_{k\\geq 1} \\frac{1-q^k}{(1-z q^k)(1-z^{-1} q^k)} = \\sum_{n=0}^\\infty \\, M_n(z) q^n \n\\]\nwhere $M_n(z)$ is a symmetric Laurent polynomial.\nFrom the definition of the crank, the coefficient of $z^{n-k}$ in $M_n(z)$,\nfor $ k \\leq \\frac{n}{2}$, equals  the number of partitions of $k$ that include no 1s.\nIn particular, the coefficients of $M_n(z)$ stabilize in the ranges for powers $n-k$ and $-n+k$\nfor $0\\leq k \\leq \\lfloor n\/2 \\rfloor$.  It is suggested in \\cite{BG2} that the zeros  for the crank polynomial converge to\nthe unit circle. \nAnother example is  the generating\nfunction\n\\[\n\\prod_{k=1}^\\infty \\frac{ 1}{ (1-z q^k)^{k}} \\, \\frac{1}{ (1-z^{-1} q^k)^k } \\, \\frac{1}{ (1-q^k)^{2k}}.\n\\]\nwhich comes from the Donaldson-Thomas Theory in algebraic geometry and whose asymptotics were studied in   \\cite{siam10}.\n\n\n\n \n\\begin{lemma}\\label{thm:freezing1}\n Let $\\{A_n(z)\\}_{n=0}^\\infty$ be a sequence of polynomials, such that the degree of $A_n(z)$ is $n$, whose coefficients satisfy\n \\[\n [z^{n-k}] A_n(z) = [ z^{n+1-k}]A_{n+1}(z), \\quad 0 \\leq k \\leq n\/m\n \\]\n for some integer $m \\geq 2$.\n Let $\\{B_n(z)\\}_{n=0}^\\infty$ be another sequence of polynomials. Then the coefficients of the \n polynomial sequence $\\{F_n(z)\\}_{n=0}^\\infty$ \n where\n \\[\n F_n(z) = \\sum_{\\ell=0}^n \\, A_{\\ell}(z) B_{n-\\ell}(z^{-1})\n \\]\n also satisfy\n \\[\n [z^{n-k}] F_n(z) = [ z^{n+1-k}]F_{n+1}(z),  \\quad 0 \\leq k \\leq n\/m.\n \\]\n  \\end{lemma}\n \\begin{proof}\n Let $0 \\leq k \\leq \\lfloor n\/m \\rfloor$. Consider $[z^{n-k}] F_n(z)$. We have\n \\begin{eqnarray}\n [z^{n-k}] F_n(z)\n &=&\n [z^{n-k}]  \\sum_{\\ell=0}^n A_{\\ell}(z) \\, B_{n-\\ell}(z^{-1})\n \\nonumber\n \\\\\n &=&\n\\sum_{\\ell=0}^n \\, \\sum_{a=0}^k \\, [ z^{ n-k+a}] A_\\ell(z) \\, [ z^{-a}] B_{n-\\ell}(z^{-1})\n  \\nonumber\n \\\\\n &=&\n\\sum_{\\ell=n-k}^n \\, \\sum_{a=0}^{k} \\, [ z^{ n-k+a}] A_\\ell(z) \\, [ z^{-a}] B_{n-\\ell}(z^{-1})\n \\label{eq:first_sum}\n \\end{eqnarray}\nwhere in the last step we note that $[z^{ n-k+a}]A_\\ell(z)=0$ if $\\ell < n-k$.\n\nFor $[z^{n+1-k}]F_{n+1}(z)$, we reindex $\\ell$ by 1 and note that $[z^{n+1-k}]A_0(z)=0$ for all $n$, giving us the expression for $[z^{n+1-k}]F_{n+1}(z)$:\n \\begin{equation}\n   \\label{eq:second_sum}\n\\sum_{\\ell=n-k}^{n} \\, \\sum_{a=0}^{k} \\, [ z^{ n+1-k+a}] A_{\\ell+1}(z) \\, [ z^{-a}] B_{n-\\ell}(z^{-1})\n\\end{equation}\nBy assumption, we know that\n\\[\n        [ z^{ n-k+a}] A_\\ell(z)  =[ z^{ n+1-k+a}] A_{\\ell+1}(z), \\quad n-k \\leq \\ell \\leq n\n\\]\nsince $0 \\leq k \\leq \\lfloor n\/m \\rfloor$.  The coefficients of $[z^{-a}] B_{n-\\ell}(z^{-1})$ are the same, and consequently the two sums (\\ref{eq:first_sum})  and (\\ref{eq:second_sum}) are\nequal.\n \\end{proof}\n \n \n\\begin{lemma}\\label{lemma:convolution}\nGiven the generating function\n$G(z,q) = \\prod_{i \\geq 1} (1-zq^i)^{-a_i}$ where $a_1=1$, let $A_n(z) = [q^n] G(z,q)$.  Let $Q(q) = \\sum_{j=0}^\\infty c_j q^j$.  \nLet $F_n(z) = [q^n] G(z,q)Q(q)$. Then the tail coefficients of $F_n(z)$ stabilize; that is,\n\\[\n[z^{k}] F_n(z) = [z^{k+1}] F_{n+1}(z), \\quad k \\geq n\/2.\n\\]\n\\end{lemma}\n\\begin{proof}\nBy construction, the polynomials $F_n(z)$ have the form\n\\[\nF_n(z) = \\sum_{\\ell=0}^n c_{n-\\ell} A_\\ell(z).\n\\]\nLet $k \\geq n\/2$. Then the coefficients for $[z^{k}] F_n(z)$ and $ [z^{k+1}] F_{n+1}(z)$ are given by\n\\begin{eqnarray*}\n[z^{k}]  F_n(z) = \\sum_{\\ell=k}^n c_{n-\\ell} [z^{k}  ]  A_\\ell(z),\n\\quad\n\\,[z^{k+1}]F_{n+1}(z) = \\sum_{j = k+1}^{n+1} c_{n+1-j} [z^{k+1}] A_{j}(z).\n\\end{eqnarray*}\nAs a consequence of Corollary 1, $c_{n-\\ell} [z^{k}  ]  A_\\ell(z) = c_{n+1-j} [z^{k+1}] A_{j}(z)$ for $j= \\ell+1$ and $k \\leq \\ell \\leq n$.\n\\end{proof}\n \n \\begin{theorem}\n If $a_1 = b_1 = 1$, and \n\\[\nG(z,q) = \\sum_{n=0}^\\infty F_n (z) q^n = \\prod_{i \\geq 1} {(1 - z q^i)}^{-a_i} {(1 - z^{-1} q^i)}^{-b_i} {(1 \\pm q^i)}^{c_i},\n\\]\nthen the coefficients of the Laurent polynomials $F_n(z)$ satisfy\n\\[\n[z^{n-k}] F_n(z) = [z^{n+1-k}] F_{n+1}(z), \\quad [z^{-(n-k)}] F_n(z) = [z^{-(n+1-k)}] F_{n+1}(z),\n\\]\nfor $0 \\leq k \\leq n\/2$.\n \\end{theorem}\n \n \n\n\n\\section{ Plane Overpartition  Stabilization }\n\nA plane partition is an array of positive integers, conventionally justified to the upper left corner of the fourth quadrant, which are weakly descending left in rows and down in columns.  A plane overpartition is a plane partition whose entries may be overlined or not according to\ncertain rules \\cite{CSV}: in each row, the last occurrence of an integer may be overlined\n(or not) and in every column, all but the first occurrence of an integer are overlined, while the first occurrence may or may not be overlined.\nIn \\cite[Proposition 4]{CSV}, the generating function for the weighted plane overpartitions is found to be \n\\[\n\\sum_{ \\Pi \\,\\,\\textrm{is a plane overpartition}}\nz^{ o(\\Pi)} q^{ | \\Pi |} =\n\\prod_{n=1}^\\infty  \\frac{ (1+ zq^n)^n}{  (1-q^n)^{ \\lceil n\/2 \\rceil} (1-z^2q^n)^{ \\lfloor n\/2 \\rfloor } }\n\\]\nwhere $o( \\Pi)$ is the number of overlined parts of the plane\noverpartition $\\Pi$.\n\n\n\n\\begin{theorem}\nLet $G(z,q)$ be the generating function for the polynomials $F_n(z)$:\n\\[\nG(z,q)=\n\\prod_{n=1}^\\infty  \\frac{ (1+ zq^n)^n}{  (1-q^n)^{ \\lceil n\/2 \\rceil} (1-z^2q^n)^{ \\lfloor n\/2 \\rfloor } }\n=\n\\sum_{n=0}^\\infty F_n(z) q^n .\n\\]\nThen the coefficients of the polynomials $F_n(z)$  satisfy the stabilization condition\n\\[\n[z^{k+1}] F_{n+1}(z) = [ z^k] F_n(z),\n\\]\nfor $k\\geq  2n\/3$.\n\\end{theorem}\n\\begin{proof}\nLet $\\{A_n(z)\\}$ be the polynomial sequence with generating function $G_A(z,q)$ where\n\\[\nG_A(z,q) = \\prod_{n=2}^\\infty \\frac{ 1}{ (1-z^2q^n)^{ \\lfloor n\/2 \\rfloor } } =  \\sum_{n=0}^\\infty A_n(z) q^n\n\\]\nand $\\{ B_n(z)\\}$ with generating function $G_B(z,q)$:\n\\[\nG_B(z,q) = \\prod_{n=2}^\\infty (1+ zq^n)^n  = \\sum_{n=0}^\\infty B_n(z) q^n  .\n\\]\nEasily, we have that $\\deg(A_n) = 2 \\lfloor n\/2 \\rfloor$.\nBy Theorem \\ref{thm:stabilization2}, replacing $z$ by $z^2$, we also find\n\\[\n[z^k] A_n = [ z^{k+2}] A_{n+2}, \\quad k \\geq 2n\/3 .\n\\]\n\nThe degree of the polynomial $B_N(z)$ is the largest number of parts in a possible partition of $N$ drawn from a multiset of two 2s, three 3s, etc.  For $N \\geq 21 = 2+2+3+3+3+4+4$, the average size of part is at least 3 and so \n\\[\n\\deg(B_N) \\leq \\frac{1}{3}N \\, \\text{.}\n\\]\n  For smaller $N$, direct calculation shows that $\\deg(B_N) \\leq \\frac{2}{3}N$.\n\n\nIt is more convenient to work with the intermediate polynomials\n\\[\nQ_n(z) = [q^n] G_A(z,q) G_B(z,q) =\\sum_{\\ell=0}^n A_{n-\\ell}(z) \\, B_\\ell(z).\n\\]\n\nWe have the summation formula for $[z^k] Q_n(z)$:\n\\begin{eqnarray*}\n[z^k] Q_n(z)\n&=&\n\\sum_{\\ell=0}^n \\, \\sum_{a=0}^k [ z^{k-a}] A_{n - \\ell}(z) \\cdot [z^a] B_\\ell(z)\n\\\\\n\\\\\n&=&\n\\sum_{\\ell \\in  I_{k,n} } \\,   \\sum_{ a =0 }^{ \\deg(B_\\ell)} \\, [ z^{k-a}] A_{n - \\ell}(z)  \\cdot [z^a] B_\\ell(z) \n\\end{eqnarray*}\nwhere $I_{k,n}$ is the set of indices \n\\[\nI_{k,n} =\n\\left\\{\n\\ell :\n\\deg(B_\\ell) + \\deg(A_{n - \\ell }) \\geq k, \\,  0 \\leq \\ell \\leq n\n\\right\\} .\n\\]\nWe have $I_{k+2, n+2} = I_{k,n} $ since for any $B_\\ell$ the matching $A_{n-\\ell}$ for sums of $k$ map to the matching $A_{n+2-\\ell}$ for sums of $k+2$.\n\n\nWe next show that\n\\[\n\\sum_{\\ell \\in  I_{k,n} } \\,   \\sum_{ a = 0}^{ \\deg(B_\\ell)} \\, [ z^{k-a}] A_{n - \\ell}(z)  \\cdot [z^a] B_\\ell(z) \n=\n\\sum_{\\ell \\in  I_{k+2,n+2} } \\,  \\sum_{ a = 0}^{ \\deg(B_\\ell)} \\, [ z^{k+2-a}] A_{n+2 - \\ell}(z)  \\cdot [z^a] B_\\ell(z) ;\n\\]\nTo do this, we need that  all the terms $[ z^{k-a}] A_{n- \\ell}$ fall into the stable range of indices. \nNow\nto get into the stable range, we need\n\\[\nk-a \\geq \\frac{2}{3} ( n  - \\ell) \n\\]\nwhere $0\\leq a \\leq \\deg B_\\ell$ and $\\ell \\in I_{k,n}$.\nWe consider  the stronger condition\n\\begin{eqnarray*}\n\\frac{2}{3} n - \\deg B_\\ell  &\\geq&  \\frac{2}{3} ( n  - \\ell) ,\n\\end{eqnarray*}\nwhich reduces to \n\\[\n\\deg(B_\\ell) \\leq \\frac{2}{3} \\ell, \\quad \\ell \\in I_{k,n} \n\\]\nwhich does indeed hold.\n\n\nOur next step is to define\n\\[\nP_n(z) = (1+zq) Q_n(z).\n\\]\n(Leaving out this factor earlier simplified the degree analysis, since without it we had the single exceptional case of $\\deg (B_1) = 1$.)  We observe that\n\\begin{eqnarray*}\n\\,\n[z^k] P_n (z) &=& [ z^k] Q_n(z) + [ z^{k-1}  ] Q_{n-1}(z),\n\\\\\n\\,\n[z^{k+1}]  P_{n+1} (z) &=&  [ z^{k+1}] Q_{n+1}(z) + [ z^{k}] Q_{n}(z) .\n\\end{eqnarray*}\nHence, we have\n\\[\n[z^k] P_n(z) = [z^{k+1}] P_{n+1}(z), \\quad k \\geq 2n\/3 .\n\\]\n\nTo finish the proof, we define a polynomial family $C_n(q)$ by\n\\[\n\\prod_{n=1}^\\infty  \\frac{ 1 }{  (1-q^n)^{ \\lceil n\/2 \\rceil} } = \\sum_{n=0}^\\infty C_n(q) .\n\\]\nThen the polynomials $F_n(z)$ are given by\n\\[\nF_n(z) = [q^n] C(q) \\, \\sum_{\\ell =0}^\\infty  P_\\ell (z) q^\\ell .\n\\]\nBy Lemma \\ref{lemma:convolution}, \nwe see that this construction maintains the stability of the coefficients of the polynomial family $\\{ P_\\ell(z)\\}$.\n\\end{proof}\n\n\\begin{corollary}\nLet $\\overline{pp}_k(n)$ be the number of plane overpartitions of $n$ with $k$ overlined parts. If $k \\geq 2n\/3$, then\n\\[\n\\overline{pp}_k(n) =\\overline{pp}_{k+1}(n+1).\n\\]\n\\end{corollary}\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:form}\n\nIn multicellular organisms ranging from cellular amoebae to the human\nbody, it is essential for cells to communicate with each other. One\ncommon mechanism to initiate communication between cells that are not\nin close contact is for cells to secrete diffusible signaling\nmolecules into the extracellular space between the spatially\nsegregated units. Examples of this kind of signaling range from\ncolonies of the amoebae Dictyostelium discoideum, which release cAMP\ninto the medium where it diffuses and acts on each separate colony\n(cf.~\\cite{Goldbeter1990}), to some endocrine neurons that secrete a\nhormone to the extracellular medium where it influences the secretion\nof this hormone from a pool of such neurons\n(cf.~\\cite{Krasmanovic_etal2003}, \\cite{Li2008}), to the effect of\ncatalysts in surface science \\cite{riecke}, and to quorum sensing\nbehavior for various applications (cf.~\\cite{CLL}, \\cite{Muller1},\n\\cite{Muller2}). In many of these systems, the individual cells or\nlocalized units can exhibit sustained temporal oscillations. In this\nway, signaling through a diffusive chemical can often trigger\nsynchronous oscillations among all the units.\n\nIn this paper we provide a theoretical investigation of the mechanism\nthrough which this kind of synchronization occurs for a class of\ncoupled cell-bulk ODE-PDE models in bounded two-dimensional\ndomains. Our class of models consists of $m$ small cells with\nmulti-component intracellular dynamics that are coupled together by a\ndiffusion field that undergoes constant bulk decay. We assume that the\ncells can release a specific signaling molecule into the bulk region\nexterior to the cells, and that this secretion is regulated by both\nthe extracellular concentration of the molecule together with its\nnumber density inside the cells. Our aim is to characterize conditions\nfor which the release of the signaling molecule leads to the\ntriggering of some collective synchronous oscillatory behavior among\nthe localized cells. Our modeling framework is closely related to the\nstudy of quorum sensing behavior in bacteria done in \\cite{Muller1}\nand \\cite{Muller2} through the formulation and analysis of similar\ncoupled cell-bulk models in ${\\mathbb{R}}^3$. For this 3-D case, in\n\\cite{Muller1} and \\cite{Muller2} steady-state solutions were\nconstructed and large-scale dynamics studied in the case where the\nsignaling compartments have small radius of order ${\\mathcal\n  O}(\\epsilon)$. However, due to the rapid ${1\/r}$ decay of the\nfree-space Green's function for the Laplacian in 3-D, it was shown in\n\\cite{Muller1} and \\cite{Muller2} that the release of the signaling\nmolecule leads to only a rather weak communication between the cells\nof the same ${\\mathcal O}(\\epsilon)$ order of the cell radius. As a\nresult, small cells in 3-D are primarily influenced by their own\nsignal, and hence no robust mechanism to trigger collective\nsynchronous oscillations in the cells due to Hopf bifurcations was\nobserved in \\cite{Muller1} and \\cite{Muller2}. We emphasize that the\nmodels of \\cite{Muller1} and \\cite{Muller2} are based on postulating a\ndiffusive coupling mechanism between distinct, spatially segregated,\ndynamically active sites. Other approaches for studying quorum sensing\nbehavior, such as in \\cite{NSSGM2014}, are based on reaction-diffusion\n(RD) systems, which adopt a homogenization theory approach to treat\nlarge populations or colonies of individual cells as a continuum\ndensity, rather than as discrete units as in \\cite{Muller1} and\n\\cite{Muller2}.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.65\\textwidth,height=5.0cm]{twoDdiag.eps}\n\\caption{Schematic diagram showing the intracellular reactions and\n  external bulk diffusion of the signal. The small shaded regions are\n  the signaling compartments or cells.}\\label{fig:schem}\n\\end{center}\n\\end{figure}\n\nBefore discussing our main results, we first formulate and\nnon-dimensionalize our coupled cell-bulk model assuming that there is\nonly one signaling compartment $\\Omega_0$ inside the two-dimensional\ndomain $\\Omega$. The corresponding dimensionless model for $m>1$ small\nsignaling cells is given in (\\ref{mainbd}).  Fig.~\\ref{fig:schem}\nshows a schematic plot of the geometry for $m$ small cells. We assume\nthat the cell can release a specific signaling molecule into the bulk\nregion exterior to the cell, and that this secretion is regulated by\nboth the extracellular concentration of the molecule together with its\nnumber density inside the cell. If ${\\mathcal U}({\\vecb X},T)$\nrepresents the concentration of the signaling molecule in the bulk\nregion $\\Omega\\backslash\\Omega_0$, then its spatial-temporal evolution\nin this region is assumed to be governed by the PDE model\n\\begin{subequations}\\label{f:mainbd}\n\\begin{equation}\\label{f:mainU}\n\\begin{aligned}\n {\\mathcal U}_T &= D_B\\Delta_{\\vecb X} {\\mathcal U} - k_B {\\mathcal U}\\,,\n \\qquad \\vecb X \\in \\Omega\\backslash\\Omega_{0}\\,; \\qquad\n\\partial_{n_{\\vecb X}} {\\mathcal U}= 0\\,,\\qquad \\vecb X \\in\\partial \\Omega\\,,\\\\\n D_B \\partial_{n_{\\vecb X}} {\\mathcal U}&= \\beta_1 {\\mathcal U} -\\beta_2 {\\mu}^1\n \\,,  \\;\\;\\quad \\vecb X\\in\\partial \\Omega_{0}\\,, \\\\\n\\end{aligned}\n\\end{equation}\nwhere, for simplicity, we assume that the signaling compartment\n$\\Omega_0\\in\\Omega$ is a disk of radius $\\sigma$ centered at some\n$\\vecb X_0\\in \\Omega$. Inside the cell we assume that there are $n$\ninteracting species $\\vecb \\mu \\equiv (\\mu^1,\\ldots,\\mu^n)^T$ whose\ndynamics are governed by $n$-ODEs, with a source term representing the\nexchange of material across the cell membrane $\\partial\\Omega_0$, of\nthe form\n\\begin{equation}\\label{f:mainuj}\n\\frac{d \\vecb \\mu}{dT} = k_R \\mu_c \\vecb F\\left({{\\vecb \\mu}\/\\mu_c}\\right)\n+ \\vecb e_1 \\int_{\\partial \\Omega_{0}}\\left( \\beta_1 {\\mathcal U} -\n\\beta_2 \\mu^1\\right) \\, dS_{\\vecb X} \\,,\n\\end{equation}\n\\end{subequations}$\\!$ where $\\vecb e_1\\equiv (1,0,\\dots,0)^T$. Here $\\vecb \\mu$ is the\ntotal amount of the $n$ species inside the cell, while $k_R>0$ is the\nreaction rate for the dimensionless intracellular dynamics ${\\vecb\n  F}(\\vecb u)$. The scalar $\\mu_c>0$ is a typical value for $\\vecb\n\\mu$.\n\nIn this coupled cell-bulk model, $D_B>0$ is the diffusion coefficient\nfor the bulk process, $k_B$ is the rate at which the signaling\nmolecule is degraded in the bulk, while $\\beta_1>0$ and $\\beta_2>0$\nare the dimensional influx (eflux) constants modeling the permeability\nof the cell wall. In addition, $\\partial_{n_{\\vecb X}}$ denotes either\nthe outer normal derivative of $\\Omega$, or the outer normal to\n$\\Omega_{0}$ (which points inside the bulk region).  The flux\n$\\beta_1{\\mathcal U}-\\beta_2\\mu^1$ on the cell membrane models the\ninflux of the signaling molecule into the extracellular bulk region,\nwhich depends on both the external bulk concentration ${\\mathcal\n  U}({\\vecb X},T)$ at the cell membrane $\\partial\\Omega_0$ as well as\non the intracellular concentration $\\mu^1$ within the cell. We assume\nthat only one of the intracellular species, $\\mu^1$, is capable of\nbeing transported across the cell membrane $\\partial\\Omega_0$ into the\nbulk. We remark that a related class of models was formulated and\nanalyzed in \\cite{bm1} and \\cite{bm2} in their study of the initiation\nof the biological cell cycle, where the dynamically active compartment\nis the nucleus of the biological cell.\n\nNext, we introduce our scaling assumption that the radius $\\sigma$ of\nthe cell is small compared to the radius of the domain, so that\n$\\epsilon\\equiv {\\sigma\/L}\\ll 1$, where $L$ is the length-scale of\n$\\Omega$. However, in order that the signaling compartment has a\nnon-negligible effect on the bulk process, we need to assume that\n$\\beta_1$ and $\\beta_2$ are both ${\\mathcal O}(\\epsilon^{-1})\\gg 1$ as\n$\\epsilon\\to 0$. In this way, in Appendix \\ref{app:A} we show that\n(\\ref{f:mainbd}) reduces to the dimensionless coupled system\n\\begin{subequations}\\label{f4:mainbd}\n\\begin{equation}\\label{f4:mainU}\n\\begin{aligned}\n  \\tau { U}_t &= D \\Delta_{\\vecb x} {U} - {U}\\,, \\qquad \\vecb x \\in\n  {\\Omega}\\backslash{\\Omega}_{0}\\,; \\qquad\n  \\partial_{n_{\\vecb x}} {U} = 0\\,,\\qquad \\vecb x \\in\\partial\n  {\\Omega}\\,,\\\\ \\epsilon D \\partial_{n_{\\vecb x}} {U}&= d_1 { U}\n  - d_2 u^1 \\,, \\;\\;\\quad \\vecb x\\in\\partial {\\Omega}_{0}\\,, \\\\\n\\end{aligned}\n\\end{equation}\nwhere ${\\Omega}_0$ is a disk of radius $\\epsilon\\ll 1$ centered\nat some $x_0\\in {\\Omega}$. The bulk process is coupled to the\nintracellular dynamics by\n\\begin{equation}\\label{f4:mainuj}\n  \\frac{d \\vecb u}{dt} = \\vecb F\\left(\\vecb u\\right)\n+ \\frac{\\vecb e_1}{\\tau \\epsilon} \\int_{\\partial {\\Omega}_{0}}\n  \\left( d_1 {U} - d_2 u^1\\right) \\,  dS_{\\vecb x} \\,.\n\\end{equation}\n\\end{subequations}$\\!$\nThe four ${\\mathcal O}(1)$ dimensionless parameters in\n(\\ref{f4:mainbd}) are $\\tau$, $D$, $d_1$, and $d_2$, defined by\n\\begin{equation}\\label{form:par}\n  \\tau \\equiv \\frac{k_R}{k_B}  \\,, \\qquad  D \\equiv \\frac{D_B}{k_B L^2} \\,,\n\\qquad \\beta_1 \\equiv \\left(k_B L\\right) \\frac{d_1}{\\epsilon} \\,, \\qquad\n    \\beta_2 \\equiv \\left( \\frac{k_B}{L} \\right) \\frac{d_2}{\\epsilon} \\,.\n\\end{equation}\nWe remark that the limit $\\tau\\ll 1$ ($\\tau\\gg 1$) corresponds to when\nthe intracellular dynamics is very slow (fast) with respect to the\ntime-scale of degradation of the signaling molecule in the bulk. The\nlimit $D\\gg 1$ corresponds to when the bulk diffusion length\n$\\sqrt{D_B\/k_B}$ is large compared to the length-scale $L$ of the confining\ndomain. \n\nA related class of coupled cell-bulk models involving two bulk\ndiffusing species, and where the cells, centered at specific spatial\nsites, are modeled solely by nonlinear flux boundary conditions, have\nbeen used to model cellular signal cascades (cf.~\\cite{levy1},\n\\cite{levy2}), and the effect of catalyst particles on chemically\nactive substrates (cf.~\\cite{Peirce}, \\cite{Glass}). In contrast to\nthese nonlinear flux-based models, in the coupled cell-bulk models of\n\\cite{Muller1} and \\cite{Muller2}, and the one considered herein, the\ncells are not quasi-static but are, instead, dynamically active\nunits. Our main goal for (\\ref{f4:mainbd}) and (\\ref{mainbd}) is to\ndetermine conditions that lead to the triggering of synchronized\noscillations between the dynamically active cells.  Related 1-D\ncell-bulk models, where the cells are dynamically active units at the\nends of a 1-D spatial domain, have been analyzed in\n\\cite{Gomez-Marin2007}--\\cite{glnw}.\n\nOur analysis of the 2-D coupled cell-bulk model (\\ref{f4:mainbd}), and\nits multi-cell counterpart (\\ref{mainbd}), which extends the 3-D\nmodeling paradigm of \\cite{Muller1} and \\cite{Muller2}, has the\npotential of providing a theoretical framework to model quorum sensing\nbehavior in experiments performed in petri dishes, where cells live on\na thin substrate. In contrast to the assumption of only one active\nintracellular component used in \\cite{Muller1} and \\cite{Muller2}, in\nour study we will allow for $m$ small spatially segregated cells with\nmulti-component intracellular dynamics. We will show for our 2-D case\nthat the communication between small cells through the diffusive\nmedium is much stronger than in 3-D, and leads in certain parameter\nregimes to the triggering of synchronous oscillations, which otherwise\nwould not be present in the absence of any cell-bulk coupling. In\naddition, when $D={\\mathcal O}(1)$, we show that the spatial\nconfiguration of small cells in the domain is an important factor in\ntriggering collective synchronous temporal instabilities in the cells.\n\nThe outline of this paper is as follows. In \\S \\ref{sec:bd} we use the\nmethod of matched asymptotic expansions to construct steady-state\nsolutions to our 2-D multi-cell-bulk model (\\ref{mainbd}), and we\nderive a globally coupled eigenvalue problem whose spectrum\ncharacterizes the stability properties of the steady-state. In our 2-D\nanalysis, the interaction between the cells is of order\n$\\nu\\equiv-1\/\\log\\epsilon$, where $\\epsilon\\ll 1$ is the assumed\ncommon radius of the small circular cells. In the distinguished limit\nwhere the bulk diffusion coefficient $D$ is of the asymptotic order\n$D=\\mathcal{O}(\\nu^{-1})$, in \\S \\ref{sec:largeD} we show that the\nleading order approximate steady-state solution and the associated\nlinear stability problem are both independent of the spatial\nconfigurations of cells and the shape of the domain. In this regime,\nwe then show that the steady-state solution can be destabilized by\neither a synchronous perturbation in the cells or by $m-1$ possible\nasynchronous modes of instability. In \\S \\ref{sec:largeD}\nleading-order-in-$\\nu$ limiting spectral problems when $D={D_0\/\\nu}$,\nwith $\\nu\\ll 1$, for both these classes of instabilities are\nderived. In \\S \\ref{sec:examples}, we illustrate our theory for\nvarious intracellular dynamics. When there is only a single\ndynamically active intracellular component, we show that no triggered\noscillations can occur. For two specific intracellular reaction\nkinetics involving two local species, modeled either by Sel'kov or\nFitzhugh-Nagumo (FN) dynamics, in \\S \\ref{sec:examples} we perform\ndetailed analysis to obtain Hopf bifurcation boundaries, corresponding\nto the onset of either synchronous or asynchronous oscillations, in\nvarious parameter planes. In addition to this detailed stability\nanalysis for the $D=\\mathcal{O}(\\nu^{-1})$ regime, in \\S\n\\ref{sec:odes} we show for the case of one cell that when $D\\gg\n\\mathcal{O}(\\nu^{-1})$ the coupled cell-bulk model is effectively\nwell-mixed and its solutions can be well-approximated by a\nfinite-dimensional system of nonlinear ODEs. The analytical and\nnumerical study of these limiting ODEs in \\S \\ref{sec:odes} reveals\nthat their steady-states can be destabilized through a Hopf\nbifurcation. Numerical bifurcation software is then used to show the\nexistence of globally stable time-periodic solution branches that are\nintrinsically due to the cell-bulk coupling. For the $D={\\mathcal\n  O}(1)$ regime, where the spatial configuration of the cells in the\ndomain is an important factor, in \\S \\ref{sec:finite_d} we perform a\ndetailed stability analysis for a ring-shaped pattern of cells that is\nconcentric within the unit disk. For this simple spatial configuration\nof cells, phase diagrams in the $\\tau$ versus $D$ parameter space, for\nvarious ring radii, characterizing the existence of either synchronous\nor asynchronous oscillatory instabilities, are obtained for the case\nof Sel'kov intracellular dynamics. These phase diagrams show that\ntriggered synchronous oscillations can occur when cells become more\nspatially clustered. In \\S \\ref{sec:finite_d} we also provide a clear\nexample of quorom sensing behavior, characterized by the triggering of\ncollective dynamics only when the number of cells exceeds a critical\nthreshold. Finally, in \\S \\ref{sec:disc} we briefly summarize our main\nresults and discuss some open directions.\n\nOur analysis of synchronous and asynchronous instabilities for\n(\\ref{mainbd}) in the $D={\\mathcal O}(\\nu^{-1})$ regime, where the\nstability thresholds are to, to leading-order, independent of the\nspatial configuration of cells, has some similarities with the\nstability analysis of \\cite{wei1}, \\cite{wei2}, \\cite{rozada}, and\n\\cite{chen} (see also the references therein) for localized spot\nsolutions to various activator-inhibitor RD systems with short range\nactivation and long-range inhibition. In this RD context, when the\ninhibitor diffusivity is of the order ${\\mathcal O}(\\nu^{-1})$,\nlocalized spot patterns can be destabilized by either synchronous or\nasynchronous perturbations, with the stability thresholds being, to\nleading-order in $\\nu$, independent of the spatial configuration of\nthe spots in the domain. The qualitative reason for this similarity\nbetween the coupled cell-bulk and localized spot problems is\nintuitively rather clear. In the RD context, the inhibitor diffusion\nfield is the long-range ``bulk'' diffusion field, which mediates the\ninteraction between the ``dynamically active units'', consisting of\n$m$ spatially segregated localized regions of high activator\nconcentration, each of which is is self-activating. In this RD\ncontext, asynchronous instabilities lead to asymmetric spot patterns,\nwhile synchronous oscillatory instabilities lead to collective\ntemporal oscillations in the amplitudes of the localized spots\n(cf.~\\cite{wei1}, \\cite{wei2}, \\cite{rozada}, and \\cite{chen}). A more\ndetailed discussion of this analogy is given in Remark \\ref{remark}.\n\nFinally, we remark that the asymptotic framework for the construction\nof steady-state solutions to the cell-bulk model (\\ref{mainbd}), and\nthe analysis of their linear stability properties, relies heavily on\nthe methodology of strong localized perturbation theory\n(cf.~\\cite{ward}). Related problems where such techniques are used\ninclude \\cite{ktcw}, \\cite{pill}, \\cite{levy1}, and \\cite{levy2}.\n\n\\setcounter{equation}{0}\n\\setcounter{section}{1}\n\\section{Analysis of the Dimensionless 2-D Cell-Bulk System}\\label{sec:bd}\n\nWe first generalize the one-cell model of \\S \\ref{sec:form} by\nformulating a class of dimensionless coupled cell-bulk dynamics that\ncontains $m$ small, disjoint, cells or compartments that are scattered\ninside the bounded two-dimensional domain $\\Omega$. We assume that\neach cell is a small disk of a common radius $\\epsilon\\ll 1$ that\nshrinks to a point $\\vecb x_j\\in \\Omega$ as $\\epsilon\\to 0$, and that\nare well-separated in the sense that dist$(\\vecb x_i, \\vecb x_j)=O(1)$ for\n$i\\neq j$ and dist$(\\vecb x_j, \\partial \\Omega)=O(1)$ for\n$j=1,\\ldots,m$, as $\\epsilon\\to 0$.\n\nAs motivated by the dimensional reasoning provided in \\S\n\\ref{sec:form}, if $U({\\vecb x},t)$ is the dimensionless\nconcentration of the signaling molecule in the bulk region between the\ncells, then in this region it satisfies the dimensionless PDE \n\\begin{subequations}\\label{mainbd}\n\\begin{equation}\\label{mainU}\n\\begin{aligned}\n\\tau U_t &= D\\Delta U -U\\,,\\qquad \\vecb x\\in \\Omega\\backslash\n\\cup_{j=1}^m\\Omega_{\\epsilon_j}\\,; \\qquad\n\\partial_nU = 0\\,, \\qquad \\vecb x\\in\\partial \\Omega\\,,\\\\\n\\epsilon D \\partial_{n_j} U&=d_1U-d_2u^1_j\\,,  \\;\\;\\quad \\vecb x\\in\\partial \n\\Omega_{\\epsilon_j}\\,, \\quad j=1,\\ldots,m\\,.\n\\end{aligned}\n\\end{equation}\nHere $D>0$ is the effective diffusivity of the bulk, $d_1>0$ and\n$d_2>0$ are the dimensionless influx (eflux) constants modeling the\npermeability of the cell membrane, $\\partial_n$ denotes the outer normal\nderivative of $\\Omega$, and $\\partial_{n_j}$ denotes the outer normal\nto each $\\Omega_{\\epsilon_j}$, which points inside the bulk\nregion. The signaling cell, or compartment,\n$\\Omega_{\\epsilon_j}$ is assumed to lie entirely within $\\Omega$.  The\nflux $d_1U-d_2u^1_j$ on each cell membrane models the influx of the\nsignaling molecule into the extracellular bulk region, which depends\non both the external bulk concentration $U({\\vecb x},t)$ at the cell\nmembrane $\\partial\\Omega_{\\epsilon_j}$ as well as on the amount\n$u_j^1$ of one of the intracellular species within the $j$-th cell.  We\nsuppose that inside each of the $m$ cells there are $n$ interacting\nspecies, with intracellular dynamics\n\\begin{equation}\\label{mainuj}\n\\frac{d \\vecb u_j}{dt} = \\vecb F_j(\\vecb u_j)+ \\frac{\\vecb e_1}{\\epsilon\n\\tau}  \\int_{\\partial \\Omega_{\\epsilon_j}}( d_1 U -d_2 u^1_j)\\, ds \\,,\n\\end{equation}\n\\end{subequations}$\\!$ where $\\vecb e_1\\equiv (1,0,\\dots,0)^T$. Here $\\vecb u_j =\n(u^1_j,\\dots,u^n_j)^T$ is the mass of the $n$ species inside the\n$j$-th cell and $\\vecb F_j(\\vecb u_j)$ is the vector nonlinearity\nmodeling the reaction dynamics within the $j$-th cell. The integration\nin (\\ref{mainuj}) is over the boundary $\\partial\\Omega_{\\epsilon_j}$\nof the compartment. Since its perimeter has length $|\\partial\n\\Omega_{\\epsilon_j}|={\\mathcal O}(\\epsilon)$, this source term for the\nODE in (\\ref{mainuj}) is ${\\mathcal O}(1)$ as $\\epsilon\\to 0$.\nThe dimensionless parameters $D$, $\\tau$, $d_1$, and $d_2$, are related\nto their dimensional counterparts by (\\ref{form:par}). The qualitative\ninterpretation of the limits $\\tau\\ll 1$, $\\tau\\gg 1$, and $D\\gg 1$, were\ndiscussed following (\\ref{form:par}).\n\n\\subsection{The Steady-State Solution for the $m$ Cells System}\nWe construct a steady-state solution to (\\ref{mainbd}) under the\nassumption that the cells are well-separated in the sense described\npreceding (\\ref{mainbd}). In \\S \\ref{sec:stability} we will then\nformulate the linear stability problem for this steady-state solution.\n\nSince in an ${\\mathcal O}(\\epsilon)$ neighborhood near each cell the\nsolution $U$ has a sharp spatial gradient, we use the method of\nmatched asymptotic expansions to construct the steady-state solution\nto (\\ref{mainbd}). In the inner region near the $j$-th cell, we\nintroduce the local variables $U_j$ and $\\vecb y$, defined by $\\vecb y\n= \\epsilon^{-1}(\\vecb x-\\vecb x_j)$ and $U_j(\\vecb y) = U({\\vecb x_j}\n+ \\epsilon {\\vecb y}) \\,,$ so that (\\ref{mainU}) transforms to\n\\begin{equation}\\label{ssy}\nD\\Delta_{\\vecb y} U_j -\\epsilon^2 U_j= 0\\,,\\qquad |\\vecb y|>1\\,; \\qquad\nD\\partial_{n_j} U_j = d_1 U_j-d_2 u^1_j\\,, \\qquad |\\vecb y|=1\\,.\n\\end{equation}\nWe look for a radially symmetric solution to (\\ref{ssy}) in the form\n$U_j=U_j(\\rho)$, where $\\rho\\equiv |{\\vecb y}|$ and $\\Delta_{\\vecb\n  y}=\\partial_{\\rho\\rho}+\\rho^{-1}\\partial_\\rho$ denotes the radially\nsymmetric part of the Laplacian. Therefore, to leading order, we have that\n$U_j(\\rho)$ satisfies\n\\begin{equation}\\label{main:ssrho}\n\\partial_{\\rho\\rho}U_j+\\rho^{-1}\\partial_\\rho U_j = 0\\,,\\qquad 1<\\rho<\\infty\\,;\n\\qquad D\\frac{\\partial U_j}{\\partial \\rho} = d_1 U_j-d_2 u^1_j\\,, \\qquad \n\\rho =1\\,.\n\\end{equation}\nThe solution to (\\ref{main:ssrho}) in terms of a constant\n$S_j$, referred to as the {\\em source strength} of the $j$-th cell, is\n\\begin{equation}\\label{chijs}\nU_j = S_j \\log\\rho +\\chi_j\\,, \\qquad \\chi_j = \n\\frac{1}{d_1}(DS_j+d_2u^1_j)\\,,\\qquad j = 1, \\dots, m\\,.\n\\end{equation} \nThe constant $S_j$ will be determined below upon matching the inner\nsolutions to the outer solution.\n\nFrom the steady-state of the intracellular dynamics (\\ref{mainuj})\ninside each cell, we find that the source strength $S_j$ and the\nsteady-state solution $\\vecb u_j$ satisfy the nonlinear algebraic\nsystem \n\\begin{equation}\\label{bduj}\n\\vecb F_j(\\vecb u_j)+ \\frac{2\\pi D}{ \\tau} S_j \\vecb e_1=0\\,.\n\\end{equation}\nIn principle, we can determine $u^1_j$ in terms of the unknown $S_j$\nas $u^1_j = u^1_j(S_j)$. The other values $u_2^j,\\dots, u_n^j$ also\ndepend on $S_j$. Next, in terms of $u_j^1$, we will derive a system of\nalgebraic equations for $S_1,\\ldots,S_m$, which is coupled to\n(\\ref{bduj}).\n\nUpon matching the far-field behavior of the inner solution\n(\\ref{chijs}) to the outer solution, we obtain the outer problem\n\\begin{equation}\\label{bdouterU}\n\\begin{aligned}\n \\Delta U-\\varphi_0^2 U &=0\\,, \\qquad  \\vecb x\\in\\Omega\\backslash\n\\lbrace{\\vecb x_1,\\ldots,\\vecb x_m\\rbrace}\\,; \\qquad \\partial_n U=0\\,, \n \\qquad \\vecb x\\in\\partial \\Omega\\,,\\\\\n U &\\sim S_j\\log|\\vecb x-\\vecb x_j|+\\frac{S_j}{\\nu}+\\chi_j\\,, \\quad \n \\text{as }\\;\\vecb x\\rightarrow \\vecb x_j\\,, \\quad j=1,\\ldots,m\\,,\n\\end{aligned}\n\\end{equation}\nwhere we have defined $\\varphi_0$ and $\\nu\\ll 1$ by\n\\begin{equation}\n \\varphi_0\\equiv1\/\\sqrt{D}\\,,\\qquad \\nu\\equiv {-1\/\\log\\epsilon}\\,.\n\\end{equation}\nWe remark that the singularity condition in (\\ref{bdouterU}) is\nderived by matching the outer solution for $U$ to the far-field\nbehavior of the inner solution (\\ref{chijs}). We then introduce the\nreduced-wave Green's function $G(\\vecb x;\\vecb x_j)$ satisfying\n\\begin{subequations} \\label{bdgreenss_all}\n\\begin{equation}\\label{bdgreenss}\n\\Delta G-\\varphi_0^2 G = -\\delta(\\vecb x-\\vecb x_j)\\,,\\qquad \\vecb x\\in \n\\Omega \\,; \\qquad \\partial_n G = 0\\,, \\qquad \\vecb x\\in \\partial \\Omega\\,.\n\\end{equation}\nAs $\\vecb x\\rightarrow \\vecb x_j$, this Green's function has the local behavior\n\\begin{equation}\\label{bdgreenss_reg}\nG(\\vecb x;\\vecb x_j)\\sim -\\frac{1}{2\\pi}\\log|\\vecb x - \\vecb x_j|+R_j+o(1)\\,, \n\\qquad \\mbox{as} \\quad \\vecb x\\rightarrow \\vecb x_j\\,,\n\\end{equation}\n\\end{subequations}$\\!$\nwhere $R_j=R_j(\\vecb x_j)$ is called the regular part of $G(\\vecb\nx;\\vecb x_j)$ at $\\vecb x=\\vecb x_j$. In terms of $G(\\vecb x;\\vecb\nx_j)$, the solution to (\\ref{bdouterU}) is\n\\begin{equation}\\label{bdUso}\nU(\\vecb x)=-2\\pi\\sum_{i=1}^{m}S_iG(\\vecb x, \\vecb x_i)\\,.\n\\end{equation}\nBy expanding $U$ as $\\vecb x\\rightarrow \\vecb x_j$, and equating the\nresulting expression with the required singularity behavior in\n(\\ref{bdouterU}), we obtain the following algebraic system\nfor $\\vecb S=(S_1,\\ldots,S_m)^T$, which we write in matrix form as\n\\begin{equation}\\label{bdsysS2m}\n\\left(1+\\frac{D\\nu}{d_1}\\right)\\vecb S+2\\pi\\nu {\\mathcal G}\\vecb S=-\n\\frac{d_2}{d_1}\\nu \\vecb u^1\\,.\n\\end{equation}\nHere the Green's matrix ${\\mathcal G}$, with matrix entries\n$({\\mathcal G})_{ij}$, and the vector $\\vecb u^1$, whose $j$-th element\nis the first local species in the $j$-th cell, are given by\n\\begin{equation*}\n({\\mathcal G})_{ii} = R_{i} \\,, \\qquad ({\\mathcal G})_{ij} = \nG(\\vecb x_i;\\vecb x_j) \\equiv G_{ij} \\,, \\quad i\\neq j\\,; \\qquad \\vecb u^1\\equiv\n\\left(u_1^1,\\ldots,u_m^1\\right)^T \\,.\n\\end{equation*}\nSince $G_{ji}=G_{ij}$, by the reciprocity of the Green's function,\n${\\mathcal G}$ is a symmetric matrix.\n\nTogether with (\\ref{bduj}), (\\ref{bdsysS2m}) provides an approximate\nsteady-state solution for $\\vecb u$, which is coupled to the source\nstrengths $\\vecb S$. It is rather intractable analytically to write\ngeneral conditions on the nonlinear kinetics to ensure the existence\nof a solution to the coupled algebraic system (\\ref{bduj}) and\n(\\ref{bdsysS2m}). As such, in \\S \\ref{sec:examples} below we will\nanalyze in detail some specific choices for the nonlinear kinetics. We\nremark that even if we make the assumption that the nonlinear kinetics\nin the cells are identical, so that $\\vecb F_j=\\vecb F$ for\n$j=1,\\ldots,m$, we still have that $S_j$ and $\\vecb u^1$ depend on $j$\nthrough the Green's interaction matrix ${\\mathcal G}$, which depends\non the spatial configuration $\\lbrace{\\vecb x_1,\\ldots,\\vecb\n  x_m\\rbrace}$ of the cells within $\\Omega$.\n\nIn summary, after solving the nonlinear algebraic system (\\ref{bduj})\nand (\\ref{bdsysS2m}), the approximate steady-state solution for $U$ is\ngiven by (\\ref{bdUso}) in the outer region, defined at ${\\mathcal\n  O}(1)$ distances from the cells, and (\\ref{chijs}) in the\nneighborhood of each cell.  This approximate steady-state solution is\naccurate to all orders in $\\nu$, since our analysis has effectively\nsummed an infinite order logarithmic expansion in powers of $\\nu$ for\nthe steady-state solution. Related 2-D problems where infinite\nlogarithmic expansions occur for various specific applications were\nanalyzed in \\cite{ktcw} and \\cite{pill} (see also the references therein).\n\n\\subsection{Formulation of the Linear Stability Problem}\\label{sec:stability}\nNext, we consider the linear stability of the steady-state solution\nconstructed in the previous subsection. We perturb this steady-state\nsolution, denoted here by $U_e(\\vecb x)$ in the bulk region and\n$\\vecb u_{e,j}$ in the $j$-th cell as $U=U_e + e^{\\lambda\n  t}\\eta({\\vecb x})$ and $\\vecb u_j = \\vecb u_{e,j}+ e^{\\lambda\n  t}\\vecb \\phi_j$. Upon substituting this perturbation into\n(\\ref{mainbd}), we obtain in the bulk region that\n\\begin{subequations}\\label{bdeigall}\n\\begin{equation}\\label{bdeigeta}\n\\begin{aligned}\n\\tau \\lambda \\eta &=D\\Delta \\eta-\\eta\\,,\\qquad \n \\vecb x\\in\\Omega\\backslash\\cup_{j=1}^m\\Omega_{\\epsilon_j}\\,; \\qquad\n\\partial_n \\eta = 0\\,,\\qquad \\vecb x\\in\\partial \\Omega\\,,\\\\\n\\epsilon D\\partial_{n_j}\\eta &= d_1\\eta-d_2\\vecb \\phi_j^1\\,,\n\\qquad \\vecb x\\in\\partial \\Omega_{\\epsilon_j}\\,, \\qquad j=1,\\ldots,m\\,.\n\\end{aligned}\n\\end{equation}\nWithin the $j$-th cell the linearized problem is\n\\begin{equation}\\label{bdeigphij}\n\\lambda \\phi_j=J_j\\phi_j+ \\frac{\\vecb e_1}{\\epsilon  \\tau} \n \\int_{\\partial\\Omega_{\\epsilon_j}}\\left(d_1\\eta-d_2\\vecb \\phi_j^1\\right) \\, ds\\,,\n\\end{equation}\n\\end{subequations}$\\!$ \nwhere $J_j$ denotes the Jacobian matrix of the nonlinear\nkinetics $\\vecb F_j$ evaluated at $\\vecb u_{e,j}$.  We now study\n(\\ref{bdeigall}) in the limit $\\epsilon\\to 0$ using the method of\nmatched asymptotic expansions. The analysis will provide a limiting\nglobally coupled eigenvalue problem for $\\lambda$, from which we can\ninvestigate possible instabilities.\n\nIn the inner region near the $j$-th cell, we introduce the local\nvariables $\\vecb y=\\epsilon^{-1}(\\vecb x-\\vecb x_j)$, with\n$\\rho=|{\\vecb y}|$, and let $\\eta_j(\\vecb y)=\\eta(\\vecb x_j + \\epsilon\n\\vecb y)$. We will look for the radially symmetric\neigenfunction $\\eta_j$ in the inner variable $\\rho$. Then, from\n(\\ref{bdeigeta}), upon neglecting higher order algebraic terms in $\\epsilon$,\nthe leading order inner problem becomes\n\\begin{equation}\\label{bdeig}\n\\partial_{\\rho\\rho}\\eta_j+\\rho^{-1}\\partial_\\rho\\eta_j= 0\\,, \\qquad \n1<\\rho<\\infty\\,; \\qquad\nD\\frac{\\partial \\eta_j}{\\partial \\rho} = d_1\\eta_j-d_2\\phi_j^1\\,,\n\\qquad \\rho=1\\,,\n\\end{equation}\nwhich has the solution\n\\begin{equation}\\label{bdBj}\n\\eta_j=c_j\\log \\rho+B_j\\,, \\qquad B_j=\\frac{1}{d_1}(Dc_j+d_2\\phi_j^1)\\,,\n\\end{equation}\nwhere $c_j$ is an unknown constant to be determined.  Then, upon\nsubstituting (\\ref{bdBj}) into (\\ref{bdeigphij}), we obtain that\n\\begin{equation}\\label{bdeig1}\n(J_j-\\lambda I)\\vecb \\phi_j+ \\frac{2\\pi D}{ \\tau} c_j\\vecb\n  e_1=0\\,, \\qquad j=1,\\ldots, m\\,.\n\\end{equation}\n\nIn the outer region, defined at ${\\mathcal O}(1)$ distances from the cells,\nthe outer problem for the eigenfunction $\\eta(\\vecb x)$ is\n\\begin{equation}\\label{bdeigout}\n\\begin{aligned}\n &\\Delta \\eta-\\frac{(1+\\tau\\lambda)}{D}\\eta =0, \\qquad   \\vecb x\\in \n\\Omega\\backslash\\lbrace{\\vecb x_1,\\ldots,\\vecb x_m\\rbrace} \\,; \\qquad\n \\partial_n \\eta =0,\\qquad \\vecb x\\in \\partial \\Omega\\,,\\\\\n &\\eta \\sim c_j\\log|\\vecb x-\\vecb x_j|+\\frac{c_j}{\\nu}+B_j, \\qquad \n \\mbox{as} \\quad \\vecb x\\rightarrow \\vecb x_j\\,, \\quad j=1,\\ldots, m\\,,\n\\end{aligned}\n\\end{equation}\nwhere $\\nu\\equiv -{1\/\\log\\epsilon}$. We remark that the singularity\ncondition in (\\ref{bdeigout}) as $\\vecb x\\rightarrow \\vecb x_j$ is\nderived by matching the outer solution for $\\eta$ to the far-field\nbehavior of the inner solution (\\ref{bdBj}). To solve\n(\\ref{bdeigout}), we introduce the eigenvalue-dependent Green's\nfunction $G_{\\lambda}(\\vecb x; \\vecb x_j)$, which satisfies\n\\begin{equation}\\label{bdgneig}\n\\begin{aligned}\n&\\Delta G_{\\lambda}-\\varphi_{\\lambda}^2G_{\\lambda}=-\\delta(\\vecb x-\\vecb x_j), \n\\qquad \\vecb x\\in\\Omega\\,; \\qquad \\partial_n G_\\lambda = 0\\,,\\qquad\n \\vecb x\\in\\partial \\Omega\\,,\\\\\n&G_\\lambda(\\vecb x;\\vecb x_j) \\sim -\n\\frac{1}{2\\pi}\\log|\\vecb x-\\vecb x_j|+R_{\\lambda,j}+ o(1)\\,, \\qquad \n\\mbox{as} \\quad \\vecb x\\rightarrow\\vecb x_j\\,,\n\\end{aligned}\n\\end{equation}\nwhere $R_{\\lambda, j}\\equiv R_{\\lambda}(\\vecb x_j)$ is the regular part of\n$G_\\lambda$ at $\\vecb=\\vecb x_j$. Here we have defined $\\varphi_\\lambda$ by\n\\begin{equation}\n\\varphi_\\lambda\\equiv\\sqrt{\\frac{1+\\tau\\lambda}{D}}\\,.\n\\end{equation}\nWe must choose the principal branch of $\\varphi_\\lambda$, which\nensures that $\\varphi_\\lambda$ is analytic in $\\mbox{Re}(\\lambda)>0$.\nFor the case of an asymptotically large domain $\\Omega$, this choice\nfor the branch cut, for which $\\mbox{Re}(\\theta_\\lambda)>0$, also\nensures that $G_\\lambda$ decays as $|\\vecb x-\\vecb x_j|\\to \\infty$.\n\nIn terms of $G_\\lambda(\\vecb x; \\vecb x_j)$, we can represent the\nouter solution $\\eta(\\vecb x)$ satisfying (\\ref{bdeigout}), as\n\\begin{equation}\\label{eig:out}\n\\eta(\\vecb x) = -2\\pi \\sum_{i=1}^mc_iG_\\lambda(\\vecb x,\\vecb x_i)\\,.\n\\end{equation}\nBy matching the singularity condition at $\\vecb x\\rightarrow\\vecb\nx_j$, we obtain a system of equations for $c_j$ as\n\\begin{equation}\\label{bdcj1}\n c_j + \\nu B_j=-2\\pi\\nu\\left(c_j R_{\\lambda_j}+\\sum_{i\\neq j}^{m} c_i \nG_{\\lambda, ij} \\right) \\,, \\qquad j=1,\\ldots,m \\,,\n\\end{equation}\nwhere $G_{\\lambda, ij }\\equiv G_\\lambda(\\vecb x_j;\\vecb x_i)$.\nUpon recalling that $B_j = \\frac{1}{d_1}(Dc_j+d_2\\phi_j^1)$ from (\\ref{bdBj}),\nwe can rewrite (\\ref{bdcj1}) in matrix form in\nterms of $\\vecb c =(c_1,\\ldots,c_m)^T$ as \n\\begin{equation}\\label{bdeig2m}\n\\left(1+\\frac{D\\nu}{d_1}\\right)\\vecb c+\\frac{d_2}{d_1}\\nu\\vecb\\phi^1+2\\pi \\nu\n {\\mathcal G}_\\lambda \\vecb c=0\\,.\n\\end{equation}\nHere we have defined the symmetric Green's matrix ${\\mathcal\n  G}_\\lambda$, with matrix entries $({\\mathcal G})_{\\lambda,ij}$, and\nthe vector $\\vecb \\phi^1$ by\n\\begin{equation}\\label{gcep:green_red}\n({\\mathcal G})_{\\lambda,ii} = R_{\\lambda,i} \\,, \\qquad ({\\mathcal G})_{\\lambda, ij} = \nG_{\\lambda}(\\vecb x_i;\\vecb x_j) \\equiv G_{\\lambda, ij} \\,, \\quad i\\neq j\\,; \\qquad \n\\vecb \\phi^1\\equiv \\left(\\phi_1^1,\\ldots,\\phi_m^1\\right)^T \\,.\n\\end{equation}\nThe $j$-th entry of the vector $\\phi^1=(\\phi^1_1,\\cdots,\\phi^1_m)^T$\nis simply the first element in the eigenvector for the $j$-th cell.\nTogether with (\\ref{bdeig1}), the system (\\ref{bdeig2m}) will yield an\neigenvalue problem for $\\lambda$ with eigenvector $\\vecb c$.\n\nNext, we calculate $\\phi_1$ in terms of $\\vecb c$ from (\\ref{bdeig1})\nand then substitute the resulting expression into (\\ref{bdeig2m}). If\n$\\lambda$ is not an eigenvalue of $J_j$, (\\ref{bdeig1}) yields that\n$\\vecb \\phi_j = 2\\pi D\\tau^{-1} (\\lambda I-J_j)^{-1} c_j\\vecb\ne_1$. Upon taking the dot product with the $n$-vector $\\vecb\ne_1=(1,0,\\ldots,0)^T$, we get $\\phi_j^1 = 2\\pi D \\tau^{-1} c_j {\\vecb\n  e_1}^T (\\lambda I-J_j)^{-1} {\\vecb e_1}$, which yields in vector form that\n\\begin{subequations} \\label{kdef}\n\\begin{equation}\\label{kdef:1}\n  \\phi^1 = \\frac{2\\pi D}{ \\tau} {\\mathcal K} \\vecb c \\,,\n\\end{equation}\nwhere ${\\mathcal K}={\\mathcal K}(\\lambda)$ is the $m\\times m$ diagonal\nmatrix with diagonal entries\n\\begin{equation}\\label{kdef:2}\n     {\\mathcal K}_j = {\\vecb e_1}^T (\\lambda I-J_j)^{-1} {\\vecb e_1} =\n     \\frac{1}{\\det(\\lambda I - J_j)} \\vecb e_1 M_j^T \\vecb e_1 = \n   \\frac{M_{j,11}}{\\det(\\lambda I - J_j)} \\,. \n\\end{equation}\n\\end{subequations}$\\!$\nHere $M_j$ is the $n\\times n$ matrix of cofactors of the matrix\n$\\lambda I-J_j$, with $M_{j,11}$ denoting the matrix entry in the first row\nand first column of $M_{j}$, given explicitly by\n\\begin{equation}\\label{kdef:m11}\nM_{j,11}=M_{j,11}(\\lambda) \n \\equiv \\det\\left(\n\\begin{array}{ccc}\n\\lambda - \n\\frac{\\partial {F}_j^2}{\\partial u_2}\\Big|_{\\vecb{u}=\\vecb{u}_{e,j}},\n&\\cdots, &-\\frac{\\partial {F}_j^2}{\\partial u_n}\n\\Big|_{\\vecb{u}=\\vecb{u}_{e,j}}\\\\\n\\cdots,&\\cdots,&\\cdots\\\\\n-\\frac{\\partial {F}_j^n}{\\partial u_2}\\Big|_{\\vecb{u}=\\vecb{u}_{e,j}}\n ,&\\cdots, &\\lambda \n-\\frac{\\partial {F}_j^n}{\\partial u_n}\\Big|_{\\vecb{u}=\\vecb{u}_{e,j}}\n\\end{array}\n\\right) \\,.\n\\end{equation}\nHere ${F}_j^2,\\ldots,{F}_j^n$ denote the components of\nthe vector $\\vecb F_j \\equiv (F_{j}^1,\\ldots,F_j^n)^T$, characterizing\nthe intracellular kinetics.\n\nNext, upon substituting (\\ref{kdef:1}) into (\\ref{bdeig2m}), we obtain the\nhomogeneous $m\\times m$ linear system\n\\begin{subequations} \\label{gcep:full}\n\\begin{equation}\n   {\\mathcal M} \\vecb c = \\vecb 0 \\,, \n\\end{equation}\nwhere the $m\\times m$ matrix ${\\mathcal M}={\\mathcal M}(\\lambda)$ is\ndefined by\n\\begin{equation}\\label{gcep:mdef}\n\\mathcal{M}\\equiv \\left(1+\\frac{D\\nu}{d_1}\\right)I+2\\pi \\nu \\frac{d_2}\n        {d_1 { \\tau}} D {\\mathcal K} + 2\\pi \\nu {\\mathcal\n          G}_\\lambda\\,.\n\\end{equation}\n\\end{subequations}$\\!$ In (\\ref{gcep:mdef}), the diagonal matrix ${\\mathcal K}$ has\ndiagonal entries (\\ref{kdef:2}), and ${\\mathcal G}_\\lambda$ is the\nGreen's interaction matrix defined in (\\ref{gcep:green_red}), which\ndepends on $\\lambda$ as well as on the spatial configuration\n$\\lbrace{\\vecb x_1,\\ldots,\\vecb x_m \\rbrace}$ of the centers of the\nsmall cells within $\\Omega$.\n\nWe refer to (\\ref{gcep:full}) as the globally coupled eigenvalue\nproblem (GCEP). In the limit $\\epsilon\\to 0$, we conclude that $\\lambda$ is\na discrete eigenvalue of the linearized problem (\\ref{bdeigall}) if\nand only if $\\lambda$ is a root of the transcendental equation\n\\begin{equation}\\label{bdMdet}\n\\det\\mathcal{M}=0\\,.\n\\end{equation}\nTo determine the region of stability, we must seek conditions to\nensure that all such eigenvalues satisfy $\\mbox{Re}(\\lambda)<0$. The\ncorresponding eigenvector $\\vecb c$ of (\\ref{gcep:full}) gives the\nspatial information for the eigenfunction in the bulk via\n(\\ref{eig:out}).\n\nWe now make some remarks on the form of the GCEP. We first observe\nfrom (\\ref{gcep:mdef}) that when $D={\\mathcal O}(1)$, then to\nleading-order in $\\nu\\ll 1$, we have that ${\\mathcal M}\\sim I +\n{\\mathcal O}(\\nu)$. As such, when $D={\\mathcal O}(1)$, we conclude\nthat to leading order in $\\nu$ there are no discrete eigenvalues of\nthe linearized problem with $\\lambda={\\mathcal O}(1)$, and hence no\n${\\mathcal O}(1)$ time-scale instabilities. However, since\n$\\nu={-1\/\\log\\epsilon}$ is not very small unless $\\epsilon$ is\nextremely small, this prediction of no instability in the $D={\\mathcal\n  O}(1)$ regime may be somewhat misleading at small finite\n$\\epsilon$. In \\S \\ref{sec:finite_d} we determine the roots of\n(\\ref{bdMdet}) numerically, without first assuming that $\\nu\\ll 1$,\nfor a ring-shaped pattern of cells within the unit disk $\\Omega$, for\nwhich the Green's matrix is cyclic.  In the next section we will\nconsider the distinguished limit $D={\\mathcal O}(\\nu^{-1})\\gg 1$ for\n(\\ref{gcep:mdef}) where the linearized stability problem becomes\nhighly tractable analytically.\n\n\\setcounter{equation}{0}\n\\setcounter{section}{2}\n\\section{The Distinguished Limit of $D={\\mathcal O}(\\nu^{-1})\\gg 1$}\n\\label{sec:largeD}\n\nIn the previous section, we constructed the steady-state solution for\nthe coupled cell-bulk system (\\ref{mainbd}) in the limit $\\epsilon\\to\n0$ and we derived the spectral problem that characterizes the linear\nstability of this solution. In this section, we consider the\ndistinguished limit where the signaling molecule in the bulk diffuses\nrapidly, so that $D\\gg 1$. More specifically, we will consider the\ndistinguished limit where $D={\\mathcal O}(\\nu^{-1})$, and hence for\nsome $D_0={\\mathcal O}(1)$, we set\n\\begin{equation}\\label{bdD0}\nD={D_0\/\\nu} \\,.\n\\end{equation}\n\nFor $D={\\mathcal O}(\\nu^{-1})$, we determine a leading order\napproximation for the steady-state solution and the associated\nspectral problem. To do so, we first approximate the reduced-wave\nGreen's function for large $D$ by writing (\\ref{bdgreenss}) as\n\\begin{equation}\\label{bdgreenssD}\n\\Delta G-\\frac{\\nu}{D_0}G=-\\delta(\\vecb x-\\vecb x_j)\\,,\\qquad \\vecb \nx\\in\\Omega\\,; \\qquad \\partial_n G=0, \\qquad \\vecb x\\in\\partial\\Omega\\,.\n\\end{equation}\nThis problem has no solution when $\\nu=0$. Therefore, we expand \n$G=G(\\vecb x; \\vecb x_j)$ for $D={D_0\/\\nu} \\gg 1$ as\n\\begin{equation}\\label{bdgexp}\nG=\\frac{1}{\\nu}G_{-1}+G_0+\\nu G_1+\\ldots \\,.\n\\end{equation}\nUpon substituting (\\ref{bdgexp}) into (\\ref{bdgreenssD}), we equate\npowers of $\\nu$ to obtain a sequence of problems for $G_i$ for\n$i=-1,0,1$. This leads to the following two-term expansion for \n$G(\\vecb x;\\vecb x_j)$ and its regular part $R_j$ in the limit\n$D={D_0\/\\nu} \\gg 1$:\n\\begin{equation}\\label{bdgexp1}\nG(\\vecb x; \\vecb x_j)=\\frac{D_0}{\\nu |{\\Omega}|}+\nG_{0}(\\vecb x; \\vecb x_j)+\\cdots\\,, \\qquad R_j =\\frac{D_0}{\\nu |{\\Omega}|}+\nR_{0,j} + \\cdots \\,.\n\\end{equation}\nHere $G_{0}(\\vecb x; \\vecb x_j)$, with regular part $R_{0j}$, is the\nNeumann Green's function defined as the unique solution to\n\\begin{equation}\\label{g0:neum}\n\\begin{aligned}\n\\Delta G_{0}&=\\frac{1}{|{\\Omega}|}-\\delta(\\vecb x-\\vecb x_j)\\,, \\qquad \n\\vecb x\\in\\Omega\\,; \\qquad \\partial_n G_0=0\\,,\\quad\\vecb x\\in \n\\partial\\Omega\\,; \\qquad \\int_{\\Omega} G_0\\: d\\vecb x =0\\,,\\\\\nG_0(\\vecb x;\\vecb x_j) &\\sim -\\frac{1}{2\\pi}\\log|{\\vecb x-\\vecb x_j}|+\n R_{0,j}, \\qquad \\vecb x\\rightarrow\\vecb x_j\\,.\n\\end{aligned}\n\\end{equation}\n\nWe then substitute the expansion (\\ref{bdgexp1}) and $D={D_0\/\\nu}$ into\nthe nonlinear algebraic system (\\ref{bduj}) and (\\ref{bdsysS2m}), which\ncharacterizes the steady-state solution, to obtain that\n\\begin{equation}\\label{bdsysS2m1}\n\\left(1+\\frac{D_0}{d_1}\\right)\\vecb S+\\frac{2\\pi m D_0}{|\\Omega|} \n  {\\mathcal E} \\vecb S + 2\\pi \\nu {\\mathcal G}_0 \\vecb S \n  =-\\frac{d_2}{d_1}\\nu \\vecb u^1\\,; \\qquad \\vecb F_j(\\vecb u_j) + \n  \\frac{2\\pi D_0}{\\tau \\nu} S_j \\vecb e_1 = 0\\,, \\quad j=1,\\ldots, m \\,,\n\\end{equation}\nwhere the $m\\times m$ matrices ${\\mathcal E}$ and the Neumann Green's \nmatrix ${\\mathcal G}_0$, with entries $({\\mathcal G}_0)_{ij}$, are\ndefined by\n\\begin{equation}\\label{bdEJ}\n{\\mathcal E} \\equiv \\frac{1}{m} \\vecb e \\vecb e^T \\,; \\qquad\n\\left({\\mathcal G}_0\\right)_{ij} = G_{0}(\\vecb x_i;\\vecb x_j)\\equiv G_{0,ij} \\,,\n \\quad i\\neq j \\,, \\qquad\n\\left({\\mathcal G}_0\\right)_{ii} = R_{0,i} \\,.\n\\end{equation}\nHere $\\vecb e$ is the $m$-vector $\\vecb e\\equiv (1,\\ldots,1)^T$.  The\nleading-order solution to (\\ref{bdsysS2m1}) when $\\nu\\ll 1$ has the\nform\n\\begin{equation}\\label{dlarge:s}\n   \\vecb S = \\nu \\vecb S_0 + {\\mathcal O}(\\nu^2) \\,, \\qquad\n    \\vecb u_j = \\vecb u_{j0} + {\\mathcal O}(\\nu) \\,.\n\\end{equation}\nFrom (\\ref{bdsysS2m1}) we conclude that $\\vecb S_0$ and $\\vecb u_{j0}$\nsatisfy the limiting leading-order nonlinear algebraic system\n\\begin{equation}\\label{Dlarge:1ss}\n\\left(1+\\frac{D_0}{d_1}\\right)\\vecb S_0+\\frac{2\\pi m D_0}{|\\Omega|} \n  {\\mathcal E} \\vecb S_0 \n  =-\\frac{d_2}{d_1} \\vecb u_0^1\\,; \\qquad \\vecb F_j(\\vecb u_{0j}) + \n  \\frac{2\\pi D_0}{ \\tau} S_{0j} \\vecb e_1 = 0\\,, \\quad j=1,\\ldots, m \\,.\n\\end{equation}\nSince this leading order system does not involve the Neumann Green's\nmatrix ${\\mathcal G}_0$, we conclude that $\\vecb S_0$ is independent of the \nspatial configuration of the cells within $\\Omega$.\n\nFor the special case where the kinetics $\\vecb F_j$ is\nidentical for each cell, so that $\\vecb F_j=\\vecb F$ for\n$j=1,\\ldots,m$, we look for a solution to (\\ref{Dlarge:1ss}) with\nidentical source strengths, so that $S_{0j}$ and \n$\\vecb u_{0j}=\\vecb u_{0}$ are independent of $j$. Therefore, we write\n\\begin{equation}\n    \\vecb S_0 = S_{0c} \\vecb e \\,, \\label{Dlarge:scom}\n\\end{equation}\nwhere $S_{0c}$ is the common source strength.  From\n(\\ref{Dlarge:1ss}), where we use ${\\mathcal E}\\vecb e=\\vecb e$, this\nyields that $S_{0c}$ and $\\vecb u_0$ satisfy the $m+1$ dimensional\nnonlinear algebraic system\n\\begin{equation}\\label{Dlarge:ss}\n\\left(1+\\frac{D_0}{d_1} + \\frac{2\\pi m D_0}{|\\Omega|} \\right) S_{0c}\n  = -\\frac{d_2}{d_1} u_0^1\\,, \\qquad \\vecb F(\\vecb u_{0}) + \n  \\frac{2\\pi D_0}{ \\tau} S_{0c} \\vecb e_1 = 0\\,,\n\\end{equation}\nwhere $u_0^1$ is the first component of $\\vecb u_0$.  This simple\nlimiting system will be studied in detail in the next section for\nvarious choices of the nonlinear intracellular kinetics $\\vecb F(\\vecb\nu_0)$.\n\nNext, we will simplify the GCEP, given by (\\ref{gcep:full}), when\n$D={D_0\/\\nu}\\gg 1$, and under the assumption that the reaction\nkinetics are the same in each cell. In the same way as was derived in\n(\\ref{bdgreenssD})--(\\ref{bdgexp1}), we let $D={D_0\/\\nu}\\gg 1$ and\napproximate the $\\lambda$-dependent reduced Green's function\n$G_\\lambda(\\vecb x;\\vecb x_j)$, which satisfies (\\ref{bdgneig}). For\n$\\tau={\\mathcal O}(1)$, we calculate, in place of (\\ref{bdgexp1}),\nthat the two-term expansion in terms of the Neumann Green's function\n$G_0$ is\n\\begin{equation*}\nG_\\lambda(\\vecb x;\\vecb x_j) = \\frac{D_0}{\\nu (1+\\tau\\lambda) |{\\Omega}|}\n +G_{0}(\\vecb x;\\vecb x_j)+ {\\mathcal O}(\\nu)\\,, \\qquad\nR_{\\lambda,j} = \\frac{D_0}{\\nu (1+\\tau\\lambda)|{\\Omega}|}\n +R_{0,j} + {\\mathcal O}(\\nu)\\,.\n\\end{equation*}\nTherefore, for $D={D_0\/\\nu}\\gg 1$ and $\\tau={\\mathcal O}(1)$, we have in\nterms of ${\\mathcal E}$ and the Neumann Green's matrix ${\\mathcal G}_0$ of\n(\\ref{bdEJ}), that\n\\begin{equation}\\label{dlarge:glam}\n {\\mathcal G}_\\lambda=\\frac{mD_0}{\\nu(1+\\tau\\lambda)|{\\Omega}|}\n  {\\mathcal E} + {\\mathcal G}_0 + {\\mathcal O}(\\nu) \\,.\n\\end{equation}\n\nWe substitute (\\ref{dlarge:glam}) into (\\ref{gcep:mdef}), and set\n$D={D_0\/\\nu}$.  In (\\ref{gcep:mdef}), we calculate to leading order in\n$\\nu$ that the matrix ${\\mathcal   K}(\\lambda)$, defined in (\\ref{kdef:2}), \nreduces to \n\\begin{equation}\\label{klambda:2t}\n   {\\mathcal K} \\sim \\frac{M_{11}}{\\det(\\lambda I - J)} + {\\mathcal O}(\\nu)\\,,\n\\end{equation}\nwhere $J$ is the Jacobian of $\\vecb F$ evaluated at the solution\n$\\vecb u_0$ to the limiting problem (\\ref{Dlarge:ss}), and $M_{11}$ is\nthe cofactor of $\\lambda I - J$ associated with its first row and\nfirst column. The ${\\mathcal O}(\\nu)$ correction in ${\\mathcal\n  K}(\\lambda)$ arises from the higher order terms in the Jacobian\nresulting from the solution to the full system (\\ref{bdsysS2m1}). \n\nIn this way, the matrix ${\\mathcal M}$ in (\\ref{gcep:mdef}), which is\nassociated with the GCEP, reduces to leading order to\n\\begin{subequations} \\label{mdef:dlarge}\n\\begin{equation}\\label{gcep:mdef_large}\n    {\\mathcal M} = a(\\lambda) I + b(\\lambda) {\\mathcal E} + {\\mathcal O}(\\nu)\n  \\,,\n\\end{equation}\nwhere $a(\\lambda)$ and $b(\\lambda)$ are defined by\n\\begin{equation} \\label{mdef:dlarge_ab}\n  a(\\lambda) = 1 + \\frac{D_0}{d_1} + \\frac{2\\pi d_2}{d_1 { \\tau}} \n  \\frac{D_0 M_{11}}\n  {\\det(\\lambda I-J)} \\,, \\qquad b(\\lambda) = \n  \\frac{2\\pi m D_0}{(1+\\tau\\lambda)|\\Omega|} \\,. \n\\end{equation}\n\\end{subequations}$\\!$ We remark that the ${\\mathcal O}(\\nu)$ correction terms in\n(\\ref{gcep:mdef_large}) arises from both $2\\pi\\nu {\\mathcal G}_0$,\nwhich depends on the spatial configuration of the cells within\n$\\Omega$, and the ${\\mathcal O}(\\nu)$ term in ${\\mathcal K}$ as\nwritten in (\\ref{klambda:2t}).\n\nTherefore, when $D={D_0\/\\nu}$, it follows from (\\ref{gcep:mdef_large})\nand the criterion (\\ref{bdMdet}) of the GCEP that $\\lambda$ is a\ndiscrete eigenvalue of the linearization if and only if there exists a\nnontrivial solution $\\vecb c$ to\n\\begin{equation}\\label{dlarge:key}\n     \\left(  a(\\lambda) I + b(\\lambda) {\\mathcal E} \\right) \\vecb c = \n  \\vecb 0\\,.\n\\end{equation}\nAny such eigenvalue with $\\mbox{Re}(\\lambda)>0$ leads to a linear instability\nof the steady-state solution when $D={\\mathcal O}(\\nu^{-1})$.\n\nWe now derive explicit stability criteria from (\\ref{dlarge:key}) by\nusing the key properties that ${\\mathcal E}\\vecb e=\\vecb e$ and\n${\\mathcal E} \\vecb q_j=0$ for $j=2,\\ldots,m$, where $\\vecb q_j$ for\n$j=2,\\ldots,m$ are an orthogonal basis of the $m-1$ dimensional\nperpendicular subspace to $\\vecb e$, i.e $\\vecb q_j^T\\vecb e=0$.\nWe obtain that $\\lambda$ is a discrete eigenvalue for \nthe {\\em synchronous mode}, corresponding to $\\vecb c=\\vecb e$, whenever\n$\\lambda$ satisfies\n\\begin{equation}\\label{dlarge:sync}\n   a(\\lambda)+b(\\lambda) \\equiv \n   1 + \\frac{D_0}{d_1} + \\frac{2\\pi d_2}{d_1 { \\tau}} \\frac{D_0 M_{11}}\n  {\\det(\\lambda I-J)} + \\frac{2\\pi m D_0}{(1+\\tau\\lambda)|\\Omega|} =0 \\,.\n\\end{equation}\nThis expression can be conveniently written as\n\\begin{equation}\\label{dlarge:sync_new}\n   \\frac{M_{11}}{\\det(\\lambda I-J)} = -\\frac{{ \\tau}}{2\\pi d_2} \\left( \n  \\frac{\\kappa_1 \\tau \\lambda + \\kappa_2}{\\tau\\lambda + 1} \\right) \\,, \\qquad\n \\mbox{where} \\qquad \n   \\kappa_1 \\equiv \\frac{d_1}{D_0} + 1 \\,, \\quad \\kappa_2 \\equiv \\kappa_1 + \n  \\frac{2m \\pi d_1}{|\\Omega|} \\,. \n\\end{equation}\nIn contrast, $\\lambda$ is a discrete eigenvalue for \nthe {\\em asynchronous or competition modes}, corresponding to \n$\\vecb c=\\vecb q_j$ for $j=2,\\ldots,m$, whenever $\\lambda$ satisfies\n$a(\\lambda)=0$. This yields for any $m\\geq 2$ that\n\\begin{equation}\\label{dlarge:async}\n   \\frac{M_{11}}{\\det(\\lambda I-J)} = - \\frac{{ \\tau}}{2\\pi\n     d_2} \\left( \\frac{d_1}{D_0} + 1 \\right) \\,.\n\\end{equation}\n\nAny discrete eigenvalue for either of the two modes that\nsatisfies $\\mbox{Re}(\\lambda)>0$ leads to an instability.  If all such\neigenvalues satisfy $\\mbox{Re}(\\lambda)<0$, then the steady-state\nsolution for the regime $D={D_0\/\\nu}$ is linearly stable on an\n${\\mathcal O}(1)$ time-scale.\n\n\\begin{remark}\\label{remark} The spectral problems (\\ref{dlarge:sync_new}) and\n(\\ref{dlarge:async}) have a remarkably similar form to the spectral\n  problem characterizing the linear stability of localized spot\n  solutions to the Gierer-Meinhardt (GM) RD system\n\\begin{equation}\\label{gm}\n  a_t=\\epsilon^2 \\Delta a - a + {a^2\/h} \\,, \\qquad \\tau h_t=D \\Delta h-\nh + \\epsilon^{-2} a^2 \\,,\n\\end{equation}\nposed in a bounded two-dimensional domain $\\Omega$ with $\\partial_n\na=\\partial_n h=0$ on $\\partial\\Omega$. For $\\epsilon\\to 0$, and for\n$D={D_0\/\\nu}$ with $\\nu={-1\/\\log\\epsilon}\\ll 1$, the linear stability\nof an $m$-spot solution, with eigenvalue parameter $\\lambda$,\nis characterized by the roots of a transcendental equation of the\nform (see \\cite{wei1})\n\\begin{equation}\\label{gm:stab}\n     \\frac{a+b \\tau\\lambda}{c+d \\tau \\lambda} = {\\mathcal F}(\\lambda) \\,,\n  \\qquad {\\mathcal F}(\\lambda)\\equiv \\frac{\\int_{0}^{\\infty} w\\left(L_0-\\lambda\n  \\right)^{-1} w^2 \\rho \\, d\\rho}{\\int_{0}^{\\infty} w^2\\rho\\, d\\rho} \\,,\n\\end{equation}\nwhere $w(\\rho)>0$ is the radially symmetric ground-state solution of\n$\\Delta w-w+w^2=0$ with $w\\to 0$ as $\\rho\\to\\infty$, and $L_0$ is the\nlocal operator $L_0\\Phi\\equiv \\Delta \\Phi - \\Phi + 2w\\Phi$, where\n$L_0^{-1}$ is restricted to radially symmetric functions. In direct\nanalogy to our cell-bulk problem, it was shown in \\cite{wei1} that\nthere can be either synchronous or asynchronous instabilities of the\namplitudes of the localized spots. For synchronous instabilities, the\ncoefficients in (\\ref{gm:stab}) are $a=b=1$, $c=2$, and\n$d={2\/(1+\\mu)}$, where $\\mu\\equiv {2\\pi m D_0\/|\\Omega|}$, while for\nasynchronous instabilities we have $a={(1+\\mu)\/2}$, and\n$b=c=d=0$. Since $L_0$ has a unique eigenpair in $H^{1}$ with positive\neigenvalue $\\sigma_0>0$, the function ${\\mathcal F}(\\lambda)$ in\n(\\ref{gm:stab}) is analytic in $\\mbox{Re}(\\lambda)>0$, with the\nexception of a simple pole at $\\lambda=\\sigma_0>0$. In this way,\n(\\ref{gm:stab}) is remarkably similar to our cell-bulk spectral\nproblems (\\ref{dlarge:sync_new}) and (\\ref{dlarge:async}) when there\nis only a single intracellular species that is self-activating in the\nsense that $J\\equiv F_{u}(u_e)>0$ so that ${M_{11}\/\\det(\\lambda\n  I-J)}={1\/(\\lambda-F_u(u_e))}$.  Spectral problems similar to\n(\\ref{gm:stab}) also occur for the Schnakenberg RD\nsystem \\cite{wei2}, the Brusselator \\cite{rozada}, and the Gray-Scott\nmodel \\cite{chen}. For the GM model, it can be shown that there is a\nunique Hopf bifurcation value of $\\tau$ for the synchronous mode\nwhenever $\\mu>1$. However, for our cell-bulk model, below in \\S\n\\ref{one:ode} we prove that Hopf bifurcations are impossible for the\nsynchronous mode whenever there is only one intracellular species.\n\\end{remark}\n\n\n\\setcounter{equation}{0}\n\\setcounter{section}{3}\n\\section{Examples of the Theory: Finite Domain With $D={\\mathcal O}(\\nu^{-1})$}\n\\label{sec:examples}\n\nIn this section we will study the leading-order steady-state problem\n(\\ref{Dlarge:ss}), and its associated spectral problem\n(\\ref{dlarge:sync_new}) and (\\ref{dlarge:async}), for various special\ncases and choices of the reaction kinetics $\\vecb F$. We investigate\nthe stability properties of the steady-state (\\ref{Dlarge:ss}) as\nparameters are varying, and in particular, find conditions for Hopf\nbifurcations to occur.\n\n\\subsection{Example 1: $m$ Cells; One Local Component}\\label{one:ode}\nTo illustrate our theory, we first consider a system such that the\nlocal dynamics inside each cell consists of only a single component\nwith arbitrary scalar kinetics $F$.  For this case, the steady-state\nproblem for $\\vecb u_0=u_0$ and $S$, given by (\\ref{Dlarge:ss}),\nreduces to two algebraic equations.  We will study the stability\nproblem for both the synchronous and asynchronous modes.  We show that\nfor any $F(u)$, the steady-state can never be destablilized by a Hopf\nbifurcation.\n\nFor the one-component case, we calculate $M_{11}=1$ and $\\det(\\lambda\nI-J)=\\lambda-F_u^e$, where $F_u^e$ is defined as the derivative of\n$F(u)$ evaluated at the steady-state $u_0$. From (\\ref{dlarge:sync_new}),\nthe spectral problem for the synchronous mode reduces to \n\\begin{subequations} \\label{d0:one}\n\\begin{equation}\\label{mcell1ode:sync}\n\\lambda^2-\\lambda p_1 + p_2 =0 \\,, \\qquad p_1 \\equiv\nF_u^e-\\frac{\\gamma}{ \\tau} -\\frac{\\zeta}{\\tau} \\,, \\qquad p_2 \\equiv\n\\frac{1}{\\tau}\\left(\\frac{\\gamma}{ \\tau}-\\zeta F_u^e\\right) \\,,\n\\end{equation}\nwhere\n\\begin{equation}\\label{alphabeta}\n\\gamma\\equiv\\frac{2\\pi d_2 D_0}{d_1+D_0}>0\\,, \\qquad\n\\zeta\\equiv1+\\frac{2\\pi md_1D_0}{|\\Omega| (d_1+D_0)}>1\\,.\n\\end{equation}\n\\end{subequations}$\\!$ The following result characterizes the stability properties for\nthe synchronous mode:\n\n\\begin{result}\\label{d0:theorem_1}\nThere can be no Hopf bifurcations associated with the synchronous mode.\nMoreover, suppose that\n\\begin{equation}\\label{d0:stab_one}\n  F_{u}^e < \\frac{\\gamma}{\\zeta  \\tau} = \\frac{2\\pi\n    d_2}{ \\tau} \\left[ 1+ \\frac{d_1}{D_0} + \\frac{2\\pi m\n      d_1}{|\\Omega|} \\right]^{-1} \\,.\n\\end{equation}\nThen, we have $\\mbox{Re}(\\lambda)<0$, and so the steady-state is\nlinearly stable to synchronous perturbations. If $F_{u}^{e} >\n{\\gamma\/\\left(\\zeta \\tau\\right)}$, the linearization has\nexactly one positive eigenvalue.\n\\end{result}\n\n\\begin{proof}\nFor a Hopf bifurcation to occur we need $p_1=0$ and $p_2>0$. Upon\nsetting $p_1=0$, we get $F_u^e={\\left(\\gamma+ \\zeta\\right)\/\\tau} >0$.\nUpon substituting this expression into the formula for\n$p_2$ in (\\ref{mcell1ode:sync}) we get\n\\begin{equation}\np_2 = \\frac{1}{ \\tau}\\left( \\frac{\\gamma}{ \\tau}\n -\\zeta F_u^e\\right)=\\frac{1}{{\\tau^2}} \\left( \\gamma(1-\\zeta)-\n\\zeta^2\\right)<0\\,,\n\\end{equation}\nsince $\\gamma>0$ and $\\zeta>1$. Therefore, there can be no Hopf\nbifurcation for the synchronous mode.\n\nNext, to establish the stability threshold, we note that the\nsteady-state solution is stable to synchronous perturbations if and\nonly if $p_1<0$ and $p_2>0$. From (\\ref{mcell1ode:sync}), we have that\n$p_1<0$ and $p_2>0$ when\n\\begin{equation}\\label{equal}\n   \\tau F_u^e < \\zeta + \\gamma \\,, \\qquad \\tau F_u^e < {\\gamma\/\\zeta}\\,,\n\\end{equation}\nrespectively, which implies that we must have $\\tau F_u^e\n<\\min(\\zeta+\\gamma,{\\gamma\/\\zeta})$.  Since $\\zeta>1$, the two\ninequalities in (\\ref{equal}) hold simultaneously only when $\\tau\nF_u^e <{\\gamma\/\\zeta}$. This yields that\n$\\mbox{Re}(\\lambda)<0$ when (\\ref{d0:stab_one}) holds. Finally, when\n$F_{u}^{e} >{\\gamma\/\\left(\\zeta \\tau\\right)}$, then $p_2<0$,\nand so there is a unique positive eigenvalue. \\end{proof}\n\nThis result shows that the effect of cell-bulk coupling is that the\nsteady-state of the coupled system can be linearly stable even when\nthe reaction kinetics is self-activating in the sense that\n$F_{u}^{e}>0$. We observe that the stability threshold\n${\\gamma\/\\zeta}$ is a monotone increasing function of $D_0$, with\n${\\gamma\/\\zeta}\\to 0$ as $D_0\\to 0$ and ${\\gamma\/\\zeta}$ tending to a\nlimiting value as $D_0\\to \\infty$. This shows that as $D_0$ is\ndecreased, corresponding to when the cells are effectively more\nisolated from each other, there is a smaller range of $F_u^e>0$ where\nstability can still be achieved.\n\nNext, we will consider the spectral problem for the asynchronous mode.\nFrom (\\ref{dlarge:async}), we get\n\\begin{equation}\n\\frac{1}{\\lambda-F_u^e}=-\\frac{{\\tau}}{\\gamma}\\,,\n\\end{equation}\nwhere $\\gamma$ is defined in (\\ref{alphabeta}). Therefore,\n$\\lambda=F_u^e-{\\gamma\/{\\tau}}$, and so $\\lambda$ is real and\nno Hopf bifurcation can occur. This asynchronous mode is stable if\n$F_u^e<{\\gamma\/ \\tau}$. Since $\\zeta>1$, we observe, upon\ncomparing this threshold with that for the synchronous mode in\n(\\ref{d0:stab_one}), that the stability criterion for the synchronous\nmode is the more restrictive of the two stability thresholds.\n\nIn summary, we conclude that a Hopf bifurcation is impossible for\n(\\ref{mainbd}) in the parameter regime $D={D_0\/\\nu}$ when there is only\none dynamically active species inside each of $m$ small cells.\nMoreover, if $F_{u}^e < {\\gamma\/(\\zeta \\tau)}$, where\n$\\gamma$ and $\\zeta$ are defined in (\\ref{alphabeta}), then the\nsteady-state solution is linearly stable to both the synchronous and\nasynchronous modes.\n\n\\subsection{Example 2: $m$ Cells; Two Local Components}\n\nNext, we assume that there are two dynamically active local species\ninside each of $m$ distinct cells. For ease of notation, we write the\nintracellular variable as $\\vecb u=(v,w)^T$ and the local kinetics as\n$\\vecb F(v,w)=(F(v,w),G(v,w))^T$.  In this way, the steady-state\nproblem (\\ref{Dlarge:ss}) becomes\n\\begin{equation}\\label{mcell2ode:ss}\n\\left(1+\\frac{D_0}{d_1} + \\frac{2\\pi m D_0}{|\\Omega|} \\right) S_{0c} =\n-\\frac{d_2}{d_1} v_e\\,, \\qquad F(v_e,w_e)+\\frac{2\\pi D_0}{\n  \\tau} S_{0c}= 0\\,, \\qquad G(v_e,w_e)=0\\,.\n\\end{equation}\nGiven specific forms for $F$ and $G$, we can solve the steady-state\nproblem (\\ref{mcell2ode:ss}) either analytically or numerically.\n\nTo analyze the stability problem, we first calculate the cofactor\n$M_{11}$ as $M_{11}=\\lambda-G_w^e$ and $\\det(\\lambda\nI-J)=\\lambda^2-\\mbox{tr}(J)\\lambda+\\det(J)$, where $\\mbox{tr}(J)$ and\n$\\det(J)$ are the trace and determinant of the Jacobian of $\\vecb F$,\ngiven by\n\\begin{equation}\n \\mbox{tr}(J)=F_v^e+G_w^e\\,,\\qquad \\det(J)=F_v^eG_w^e-F_w^eG_v^e\\,.\n\\end{equation}\nHere $F_i^e$, $G_i^e$ are partial derivatives of $F$, $G$ with respect\nto $i$, with $i\\in(v,w)$, evaluated at the solution to\n(\\ref{mcell2ode:ss}).\n\nNext, we analyze the stability of the steady-state solution with respect to\neither synchronous or asynchronous perturbations. For the synchronous mode, \nwe obtain, after some algebra,  that (\\ref{dlarge:sync_new}) can be\nreduced to the study of the cubic\n\\begin{subequations} \\label{d0:cubic}\n\\begin{equation}\\label{mcell2ode:H3}\n\\mathcal{H}(\\lambda)\\equiv\\lambda^3+\\lambda^2 p_1+\\lambda p_2+p_3=0\\,.\n\\end{equation}\nwhere $p_1$, $p_2$, and $p_3$, are defined in terms of\n$\\gamma$ and $\\zeta$, as given in (\\ref{alphabeta}),  by\n\\begin{equation}\\label{mcell2ode:p123}\n\\begin{aligned}\np_1\\equiv \\frac{\\gamma}{ \\tau}\n+\\frac{\\zeta}{\\tau}-\\mbox{tr}(J)\\,,\\qquad p_2\\equiv\n\\det(J)-\\frac{\\gamma}{ \\tau}\nG_w^e+\\frac{1}{\\tau}(\\frac{\\gamma}{ \\tau}-\\zeta\n\\mbox{tr}(J))\\,,\\qquad p_3\\equiv \\frac{1}{\\tau}(\\zeta \\det(J)-\n \\frac{\\gamma}{ \\tau} G_w^e)\\,.\n\\end{aligned}\n\\end{equation}\n\\end{subequations}$\\!$\n\nTo determine whether there is any eigenvalue in\n$\\mbox{Re}(\\lambda)>0$, and to detect any Hopf bifurcation boundary in\nparameter space, we use the well-known \\textit{Routh-Hurwitz\n  criterion} for a cubic function.  It is well known that all three\nroots to the cubic satisfy $\\mbox{Re}(\\lambda)<0$ if and only if the\nfollowing three conditions on the coefficients hold:\n\\begin{equation}\np_1>0\\,, \\qquad p_3>0\\,,\\qquad p_1p_2>p_3\\,.\n\\end{equation}\nTo find the Hopf bifurcation boundary, we need only consider a special\ncubic equation which has roots $\\lambda_1=a<0$ and $\\lambda_{2,3}=\\pm\ni\\omega$, for which\n$(\\lambda-a)(\\lambda-i\\omega)(\\lambda+i\\omega)=\\lambda^3-\na\\lambda^2+\\omega^2\\lambda-a\\omega^2=0\\,.$ Comparing this expression\nwith (\\ref{mcell2ode:H3}), and using the \\textit{Routh-Hurwitz\n  criterion}, we conclude that on any Hopf bifurcation boundary the\nparameters must satisfy\n\\begin{equation}\np_1>0\\,,\\qquad p_3>0\\,, \\qquad p_1p_2=p_3\\,. \\label{mcell2ode:syn}\n\\end{equation} \nWe will return to this criterion below when we study two specific\nmodels for the local kinetics $(F,G)$.\n\nNext, we consider the spectral problem for the asynchronous mode. Upon\nsubstituting the expressions of $M_{11}$ and $\\det(\\lambda I-J)$ into\n(\\ref{dlarge:async}) and reorganizing the resulting expression,\n(\\ref{dlarge:async}) becomes the quadratic equation \n\\begin{equation}\n\\lambda^2-\\lambda q_1 + q_2 = 0 \\,, \\qquad \\mbox{where} \\qquad q_1\n\\equiv \\mbox{tr}(J)- \\frac{\\gamma}{ \\tau} \\,, \\qquad q_2 \\equiv\n\\det(J)- \\frac{\\gamma}{ \\tau} G_w^e \\,.\n\\end{equation}\nFor a Hopf bifurcation to occur, we require that $q_1=0$ and $q_2>0$, which\nyields that\n\\begin{equation}\\label{mcell2ode:asyn_all}\n\\frac{\\gamma}{ \\tau} =\\mbox{tr}(J)=F_v^e+G_w^e\\,, \\qquad \\mbox{provided that}\n \\qquad  \\det(J)-\\frac{\\gamma}{ \\tau} G_w^e=-G_v^eF_w^e-(G_w^e)^2>0\\,.\n\\end{equation}\nFinally, we conclude that $\\mbox{Re}(\\lambda)<0$ for the\nasynchronous modes if and only if\n\\begin{equation}\n \\mbox{tr}(J)<{\\gamma\/ \\tau} \\,, \\qquad \\mbox{and} \\qquad \n \\det(J) -\\frac{\\gamma}{ \\tau} G_w^e>0\n  \\,.\n\\end{equation}\n\nTo write the stability problem for the asynchronous mode in terms of\n$D_0$, we use (\\ref{alphabeta}) for $\\gamma$ in terms of $D_0$ to\nobtain from the conditions (\\ref{mcell2ode:asyn_all}) that the Hopf\nbifurcation threshold satisfies the transcendental equation\n\\begin{equation}\\label{mcell:d0_async}\nD_0=\\frac{\\tau d_1 \\mbox{tr}(J)}{2\\pi d_2-\\tau \\mbox{tr}(J)}\\,,  \\qquad\n  \\mbox{provided that} \\qquad \n D_0\\left( \\frac{2\\pi d_2}{\\tau} G_w^e-\\det (J)\\right)<d_1\\det (J)\\,.\n\\end{equation}\nWe observe that in this formulation, both $\\mbox{tr}(J)$ and $\\det(J)$\ndepend on the local kinetics and the steady-state solution, with the\nlatter depending on $D_0$ and $\\tau$.  In the next two subsections we\nstudy in some detail two specific choices for the local kinetics, and\nwe calculate phase diagrams where oscillatory instabilities can occur.\n\n\\subsubsection{Local Kinetics Described by the Sel'kov model}\\label{sec:ex:selkov}\n\nWe first consider the Sel'kov model, used in simple models of glycolysis, \nwhere $F(v,w)$ and $G(v,w)$ are given in terms of parameters\n$\\alpha>0$, $\\mu>0$, and $\\epsilon_0>0$ by\n\\begin{equation}\\label{2dsel:fg}\nF(v,w)=\\alpha w+wv^2-v\\,, \\qquad G(v,w) =\\epsilon_0\\left\n(\\mu-(\\alpha w+wv^2)\\right) \\,.\n\\end{equation}\nFirst, we determine the approximate steady-state solution $v_e$ and\n$w_e$ by substituting (\\ref{2dsel:fg}) into (\\ref{mcell2ode:ss}). This\nyields that\n\\begin{equation}\\label{2dsel:ss_sol}\n v_e=\\frac{\\mu}{\\left[1+ {2\\pi D_0\\beta\/{ \\tau}}\\right]}\\,, \\qquad\n w_e=\\frac{\\mu}{\\alpha+v_e^2}\\,, \\qquad S_{0c}= -\\beta v_e\\,, \\qquad\n \\mbox{where} \\qquad \\beta \\equiv \\frac{d_2}{d_1+D_0+2\\pi md_1D_0\/|\\Omega|}\\,.\n\\end{equation}\nAs needed below, the partial derivatives of $F$ and $G$ evaluated at\nthe steady-state solution are\n\\begin{subequations} \\label{2dsel:jac}\n\\begin{equation}\\label{2dsel:fgp}\nF_v^e = 2v_ew_e-1\\,, \\qquad F_w^e=\\alpha+v_e^2\\,, \\qquad\nG_v^e=-2\\epsilon_0 v_ew_e\\,, \\qquad G_w^e=-\\epsilon_0(\\alpha+v_e^2)\\,,\n\\end{equation}\nwhich yields\n\\begin{equation}\n  \\mbox{det}(J) = \\epsilon_0\\left(\\alpha+ v_e^2\\right)=-G_w^e >0 \\,, \\qquad\n  \\mbox{tr}(J)= 2 v_e w_e -1  -\\epsilon_0 \\left(\\alpha + v_e^2\\right) \\,.\n\\end{equation}\n\\end{subequations}$\\!$\n\nTo study possible synchronous oscillations of the $m$ cells, we\ncompute the Hopf bifurcation boundaries as given in\n(\\ref{mcell2ode:syn}), where we use (\\ref{2dsel:jac}). For the\nparameter set $\\tau=1$, $D_0=1$, $|\\Omega|=10$, $\\mu=2$, $\\alpha=0.9$, and\n$\\epsilon_0=0.15$, we obtain the Hopf bifurcation boundary in the\n$d_1$ versus $d_2$ parameter plane as shown by the solid curves in\nFig.~\\ref{d1d2sel} for $m=1,2,3$.\n\nNext, to obtain instability thresholds corresponding to the asynchronous\nmode, we substitute (\\ref{2dsel:jac}) into (\\ref{mcell2ode:asyn_all})\nto obtain that the Hopf bifurcation boundary is given by\n\\begin{equation}\n\\gamma=\\tau \\mbox{tr}(J) \\equiv \\tau\n  \\left[2v_ew_e-1-\\epsilon(\\alpha+v_e^2) \\right]\\,,\n\\label{2dsel:asy_1}\n\\end{equation}\nprovided that $\\mbox{det}(J)-\\mbox{tr}(J) G_{w}^e>0$. This latter\ncondition can be written, using (\\ref{2dsel:jac}), as\n$\\epsilon_0(\\alpha+v_e^2)\\left(1+\\mbox{tr}(J)\\right)>0$, and so is\nsatisfied provided that $\\mbox{tr}(J)>-1$. Since $\\gamma>0$ from\n(\\ref{alphabeta}), (\\ref{2dsel:asy_1}) implies that we must have\n$\\mbox{tr}(J)>0$, which guarantees that $\\mbox{det}(J)-\\mbox{tr}(J)\nG_{w}^e>0$ always holds at a Hopf bifurcation. In this way, and by\nsubstituting (\\ref{2dsel:ss_sol}) for $w_e$ into (\\ref{2dsel:asy_1}),\nwe obtain that the asynchronous mode has a pure imaginary pair of\ncomplex conjugate eigenvalues when\n\\begin{equation}\\label{2dsel:hopf_async}\n\\gamma=  \\tau \\left[\\frac{2 v_e \\mu}{\\alpha + v_e^2} -1 -\n  \\epsilon_0 \\left(\\alpha + v_e^2 \\right) \\right] \\,, \\qquad \\mbox{where}\n  \\qquad v_e = \\frac{\\mu}{\\left[1+ {2\\pi D_0\\beta\/ \\tau}\\right]}\\,.\n\\end{equation}\nHere $\\gamma$ and $\\beta$, depending on $d_1$, $d_2$, $m$,\n$|\\Omega|$, and $D_0$, are defined in (\\ref{alphabeta}) and\n(\\ref{2dsel:ss_sol}), respectively. By using these expressions for\n$\\gamma$ and $\\beta$, we can readily determine a parametric form for\nthe Hopf bifurcation boundary in the $d_1$ versus $d_2$ plane as the\nsolution to a linear algebraic system for $d_1$ and $d_2$ in terms of\nthe parameter $v_e$ with $0<v_e<\\mu$, given by\n\\begin{subequations} \\label{2dcell:a_par}\n\\begin{equation}\n   d_1 = \\frac{D_0(a_{12}-a_{22})}{a_{11}a_{22}-a_{21}a_{12}}\\,, \\qquad\n   d_2 = \\frac{D_0(a_{21}-a_{11})}{a_{11}a_{22}-a_{21}a_{12}}\\,,\n \\label{2dcell:a_par_1}\n\\end{equation}\nwhere $a_{11}$, $a_{12}$, $a_{22}$, and $a_{21}$, are defined in terms of the\nparameterization $v_e$ by\n\\begin{equation}\n  a_{11}\\equiv  1 + \\frac{2\\pi m D_0}{|\\Omega|} \\,, \\qquad\n  a_{12}\\equiv -\\frac{1}{\\beta(v_e)} \\,, \\qquad a_{21}\\equiv 1 \\,, \\qquad\n  a_{22}\\equiv -\\frac{2\\pi D_0}{\\gamma(v_e)} \\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\\gamma(v_e)\\equiv \\tau \\left[\\frac{2 v_e \\mu}{\\alpha + v_e^2}\n  -1 - \\epsilon_0 \\left(\\alpha + v_e^2 \\right)\\right]\\,, \\qquad\n  \\beta(v_e)\\equiv  \\tau \\frac{(\\mu-v_e)}{2\\pi D_0 v_e} \\,.\n\\end{equation}\n\\end{subequations}$\\!$\n\nBy varying $v_e$, with $0<v_e<\\mu$, and retaining only the portion of the\ncurve for which $d_1>0$ and $d_2>0$, we obtain a parametric form for\nthe Hopf bifurcation boundary for the asynchronous mode in the \n$d_1$ versus $d_2$ parameter plane. For $m=2$ and $m=3$, these are\nthe dashed curves shown in Fig.~\\ref{d1d2sel}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{sel_hopfbd.eps}\n\\caption{Hopf bifurcation boundaries for the synchronous (solid curve)\n  and asynchronous (dashed curve) modes for the Sel'kov model for\n  different numbers $m$ of cells in the $d_1$ versus $d_2$ parameter\n  plane. The synchronous mode for $m=1$ is unstable between the two\n  black lines. For $m=2$ and $m=3$ the synchronous mode is\n  unstable in the horseshoe-shaped region bounded by the blue and red\n  solid curves, respectively. Inside the dotted regions for $m=2$ and\n  $m=3$ the asynchronous mode is unstable. For the asynchronous mode,\n  the boundary of these regions is given parametrically by\n  (\\ref{2dcell:a_par}). The parameters used are $\\mu=2$, $\\alpha=0.9$,\n  $\\epsilon_0=0.15$, $\\tau=1$, $D_0=1$, and $|\\Omega|=10$.}\n\\label{d1d2sel}\n\\end{center}\n\\end{figure}\n\nWe now discuss qualitative aspects of the Hopf bifurcation boundaries\nfor both synchronous and asynchronous modes for various choices of $m$\nas seen in Fig.~\\ref{d1d2sel}. For $m=1$, we need only consider the\nsynchronous instability. The Hopf bifurcation boundary is given by the\ntwo black lines, and the region with unstable oscillatory dynamics is\nlocated between these two lines. For $m=2$, inside the region bounded\nby the blue solid curve, the synchronous mode is unstable and under\nthe blue dashed curve, the asynchronous mode is unstable. Similar\nnotation applies to the case with $m=3$, where the Hopf bifurcation\nboundaries for synchronous\/asynchronous mode are denoted by red\nsolid\/dashed curves.\n\nOne key qualitative feature we can observe from Fig.~\\ref{d1d2sel}, for\nthe parameter set used, is that the oscillatory region for a larger\nvalue of $m$ lies completely within the unstable region for smaller\n$m$ for both the synchronous and asynchronous modes. This suggests\nthat if a coupled system with $m_1$ cells is unstable to synchronous\nperturbations, then a system with $m_2<m_1$ cells will also be\nunstable to such perturbations. However, if a two-cell system is\nunstable, it is possible that a system with three cells, with the same\nparameter set, can be stable.  Finally, we observe qualitatively that\nthe Hopf bifurcation boundary of the asynchronous mode always lies\nbetween that of the synchronous mode. This suggests that as we vary\n$d_1$ and $d_2$ from a stable parameter region into an unstable\nparameter region, we will always first trigger a synchronous\noscillatory instability rather than an asynchronous instability. It is\nan open problem to show that these qualitative observations still hold\nfor a wide range of other parameter sets.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{sel_tauD0.eps}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{sel_tauD0m1.eps}\n\\caption{Left panel: Hopf bifurcation boundaries for the synchronous\n  (solid curves) and asynchronous (dashed curves) modes for the\n  Sel'kov model with different numbers $m$ of cells in the $\\tau$\n  versus $D_0$ plane. The synchronous mode for $m=1$ is unstable only\n  inside the region bounded by the two black solid curves. Similarly,\n  in the lobe formed by the blue solid and red solid curves the\n  synchronous mode is unstable for $m=2$ (blue) and $m=3$ (red),\n  respectively. In the region enclosed by the blue (red) dashed curve,\n  the asynchronous mode is unstable for $m=2$ ($m=3$). Right panel:\n  Hopf bifurcation boundaries for the synchronous mode with $m=1$\n  shown in a larger region of the $\\tau$ versus $D_0$ plane, showing\n  that the instability lobe is a bounded region. Parameters used are\n  $\\mu=2$, $\\alpha=0.9$, $\\epsilon_0=0.15$, $d_1=0.5$, $d_2=0.2$ and\n  $|\\Omega|=10$.}\\label{tauDsel}\n\\end{center}\n\\end{figure}\n\n  Next, we show the region where oscillatory instabilities can\n  occur in the $\\tau$ versus $D_0$ parameter plane for the synchronous\n  and asynchronous modes. We fix the Sel'kov parameter values as\n  $\\mu=2$, $\\alpha=0.9$, and $\\epsilon_0=0.15$, so that the uncoupled\n  intracelluar kinetics has a stable steady-state. We then take\n  $d_1=0.5$, $d_2=0.2$, and $|\\Omega|=10$. For this parameter set, we\n  solve the Hopf bifurcation conditions (\\ref{mcell2ode:syn}) by a\n  root finder. In this way, in the left panel of Fig.~\\ref{tauDsel} we\n  plot the Hopf bifurcation boundaries for the synchronous mode in the\n  $\\tau$ versus $D_0$ plane for $m=1,2,3$. Similarly, upon using\n  (\\ref{mcell:d0_async}), in the left panel of Fig.~\\ref{tauDsel} we\n  also plot the Hopf bifurcation boundaries for the asynchronous\n  mode. In the right panel of Fig.~\\ref{tauDsel}, where we plot in a\n  larger region of the $\\tau$ versus $D$ plane, we show that the\n  instability lobe for the $m=1$ case is indeed closed. We observe for\n  $m=2$ and $m=3$ that, for this parameter set, the lobes of\n  instability for the asynchronous mode are almost entirely contained\n  within the lobes of instability for the synchronous mode.\n\n  Finally, we consider the effect of changing $d_1$ and $d_2$ to\n  $d_1=0.1$ and $d_2=0.2$, while fixing the Sel'kov parameters as\n  $\\mu=2$, $\\alpha=0.9$, and $\\epsilon_0=0.15$, and keeping\n  $|\\Omega|=10$. In Fig.~\\ref{tauDsel_2} we plot the Hopf bifurcation\n  curve for the synchronous mode when $m=1$, computed using\n  (\\ref{mcell2ode:syn}), in the $\\tau$ versus $D_0$ plane. We observe\n  that there is no longer any closed lobe of instability. In this\n  figure we also show the two Hopf bifurcation values that correspond\n  to taking the limit $D_0\\gg 1$. These latter values are Hopf\n  bifurcation points for the linearization of the ODE system\n  (\\ref{d:odes}) around its steady-state value. This ODE system\n  (\\ref{d:odes}), derived in \\S \\ref{sec:odes}, describes large-scale\n  cell-bulk dynamics in the regime $D\\gg {\\mathcal O}(\\nu^{-1})$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{sel_tauD0_comp.eps}\n\\caption{Hopf bifurcation boundaries for the synchronous mode for the\n  Sel'kov model with $m=1$ in the $\\tau$ versus $D_0$ plane when\n  $d_1=0.1$ and $d_2=0.2$. The other parameters are the same as in\n  Fig.~\\ref{tauDsel}. Inside the region bounded by the two black solid\n  curves, which were computed using (\\ref{mcell2ode:syn}), the\n  synchronous mode is unstable. The instability region is no longer a\n  closed lobe as in Fig.~\\ref{tauDsel}. The dashed lines represent the\n  two Hopf bifurcation points that are obtained by using the\n  bifurcation software \\cite{xpp} on the steady-state problem for the\n  ODE system (\\ref{d:odes}). As $D_0$ increases, the Hopf bifurcation\n  thresholds in $\\tau$ gradually approach those obtained from the large\n  $D$ approximation. }\\label{tauDsel_2}\n\\end{center}\n\\end{figure}\n\n\n\\subsubsection{Local Kinetics Described by a Fitzhugh-Nagumo Type System}\\label{sec:ex:fn}\n\nNext, for the cell kinetics we take the Fitzhugh-Nagumo (FN) nonlinearities,\ntaken from \\cite{Gomez-Marin2007}, given by\n\\begin{equation}\\label{2dsf:fg}\nF(v,w)=\\epsilon_0 (wz-v)\\,, \\qquad G(v,w) =w-q(w-2)^3+4-v\\,,\n\\end{equation}\nwhere the parameters satisfy $\\epsilon_0>0$, $q>0$, and $z>1$. \n\nUpon substituting (\\ref{2dsf:fg}) into (\\ref{mcell2ode:ss}) we\ncalculate that the steady-state solution $w_e>0$ is given by the unique\nreal positive root of the cubic ${\\mathcal C}(w)=0$ given by\n\\begin{equation}\n {\\mathcal C}(w)\\equiv qw^3-6qw^2+w\\left(12q-1+\\Lambda\\right)-(8q+4)\\,,\\qquad\n \\mbox{where} \\qquad\n\\Lambda\\equiv\\frac{\\epsilon_0 z}{\\left[\\epsilon_0 + {2\\pi\n      D_0\\beta\/ \\tau} \\right]}\\,, \\label{2dfitz:cubic}\n\\end{equation}\nwhere $\\beta$ is defined in (\\ref{2dsel:ss_sol}). The uniqueness of the\npositive root of this cubic for any $\\Lambda>0$ was proved previously\nin \\S 2 of \\cite{gou_ward}. In terms of the solution $w_e>0$ to the\ncubic equation, we calculate $v_e=\\Lambda w_e$ and $S_{0c}=-\\beta \\Lambda w_e$.\n\nAs needed below, we first calculate the partial derivatives of $F$ and $G$\nat the steady-state solution as\n\\begin{subequations} \\label{2dsf:jac}\n\\begin{equation}\\label{2dsf:fgp}\nF_v^e = -\\epsilon_0\\,, \\qquad F_w^e=\\epsilon_0 z\\,, \\qquad\nG_v^e=-1\\,, \\qquad G_w^e=1 - 3q(w_e-2)^2 \\,,\n\\end{equation}\nwhich yields\n\\begin{equation}\n  \\mbox{det}(J) = \\epsilon_0\\left[ z-1 + 3q(w_e-2)^2\\right]>0 \\,, \\qquad\n  \\mbox{tr}(J)= -\\epsilon_0 + 1 - 3q(w_e-2)^2 \\,.\n\\end{equation}\n\\end{subequations}$\\!$\n\nTo determine conditions for which the synchronous mode has a Hopf\nbifurcation we first substitute (\\ref{2dsf:fgp}) into\n(\\ref{mcell2ode:p123}). The Hopf bifurcation boundary is then found by\nnumerically computing the roots of (\\ref{mcell2ode:syn}).\nSimilarly, to study instabilities associated with the asynchronous\noscillatory mode we substitute (\\ref{2dsf:fgp}) into\n(\\ref{mcell2ode:asyn_all}) to obtain that \n\\begin{subequations} \\label{2dfitz:async}\n\\begin{equation}\n\\gamma =  \\tau \\left[-\\epsilon_0 +1-3q(w_e-2)^2\\right]\\,,\n\\end{equation}\nwhich yields the Hopf bifurcation boundary, provided that\n\\begin{equation}\\label{2dsf:asy1}\n\\det(J)-\\frac{\\gamma}{ \\tau} G_w^e=-G_v^eF_w^e-(G_w^e)^2 =\n-G_v^eF_w^e-(G_w^e)^2=\\epsilon_0 z- \\left[1-3q(w_e-2)^2\\right]^2>0\\,,\n\\end{equation}\n\\end{subequations}$\\!$ where $w_e$ is the positive root of the cubic\n(\\ref{2dfitz:cubic}). As was done for the Sel'kov model in \\S\n\\ref{sec:ex:selkov}, the Hopf bifurcation boundary for the\nasynchronous mode in the $d_1$ versus $d_2$ parameter plane can be\nparametrized as in (\\ref{2dcell:a_par_1}) where $a_{11}$, $a_{12}$,\n$a_{22}$, and $a_{21}$, are now defined in terms of the parameter\n$w_e>0$ by \\begin{subequations}\n\\begin{equation}\n  a_{11}\\equiv  1 + \\frac{2\\pi m D_0}{|\\Omega|} \\,, \\qquad\n  a_{12}\\equiv -\\frac{1}{\\beta(w_e)} \\,, \\qquad a_{21}\\equiv 1 \\,, \\qquad\n  a_{22}\\equiv -\\frac{2\\pi D_0}{\\gamma(w_e)} \\,,\n\\end{equation}\nwhere\n\\begin{equation}\n  \\beta(w_e) \\equiv  \\frac{\\epsilon_0\\tau}{2\\pi D_0}\n    \\left( \\frac{z}{\\Lambda(w_e)} -1 \\right) \\,, \\quad \\mbox{with}\n    \\quad \\Lambda(w_e)\\equiv -\\frac{q(w_e-2)^3}{w_e} + 1 +\n    \\frac{4}{w_e} \\,; \\quad \\gamma(w_e) \\equiv  \\tau\n    \\left[-\\epsilon_0 + 1 - 3q(w_e-2)^2 \\right]\\,.\n\\end{equation}\n\\end{subequations}$\\!$\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth,height=5.0cm]{d1d2_slowfast.eps}\n\\caption{Hopf bifurcation boundaries for the synchronous (solid curve)\n  and asynchronous (dashed curve) modes for the FN system \n  (\\ref{2dsf:fg}) with various number $m$ of cells in the $d_1$ versus\n  $d_2$ parameter plane. Between the solid lines the synchronous mode\n  is unstable, whereas between the dashed lines the asynchronous mode\n  is unstable. Notice that the region of instability for the asynchronous\n  mode is contained within the instability region for the synchronous mode.\n  Parameters used are $z=3.5$, $q=5$, $\\epsilon_0=0.5$, $\\tau=1$, $D_0=1$, \n  and $|\\Omega|=10$.}\n\\label{d1d2slowfast}\n\\end{center}\n\\end{figure}\n\nBy varying $w_e>0$ and retaining only the portion of the\ncurve for which $d_1>0$ and $d_2>0$, and ensuring that the constraint\n(\\ref{2dsf:asy1}) holds,  we obtain a parametric form for\nthe Hopf bifurcation boundary for the asynchronous mode in the \n$d_1$ versus $d_2$ parameter plane. For $m=2$ and $m=3$, these are\nthe dashed curves shown in Fig.~\\ref{d1d2slowfast}.\n\nIn this way, in Fig.~\\ref{d1d2slowfast} we plot the Hopf bifurcation\nboundaries for the synchronous mode (solid curves) and asynchronous\nmode (dashed curves) for various values of $m$ for the parameter set\n$z=3.5$, $q=5$, $\\epsilon_0=0.5$, $\\tau=1$, $D_0=1$, and\n$|\\Omega|=10$. As compared to Fig.~\\ref{d1d2sel}, we notice that the\nunstable regions for both modes are not only shrinking but also the\nboundaries shift as the number of cells increases. This feature\ndoes not appear in the previous Sel'kov model.\n\nNext, in Fig.~\\ref{tauD0slowfast} we show the region where oscillatory\ninstabilities occur for either synchronous or asynchronous modes for\n$m=1,2,3$ in the $\\tau$ versus $D_0$ plane. These Hopf bifurcation\nboundaries are computed by finding roots of either\n(\\ref{mcell2ode:syn}) or (\\ref{2dfitz:async}) for various values of\n$D_0$. The other parameter values are the same as those used for\nFig.~\\ref{d1d2slowfast} except $d_1=10$ and $d_2=0.2$. Inside the\nregion bounded by the solid curves the synchronous mode is unstable,\nwhile inside the region bounded by the dashed curves, the asynchronous\nmode is unstable. Similar to that shown in Fig.~\\ref{d1d2slowfast},\nthe regions of instability are shrinking, at the same time as the Hopf\nbifurcation boundaries shift, as $m$ increases. For these parameter\nvalues of $d_1$ and $d_2$, the Hopf bifurcation for the synchronous\nmodel still occurs for large values of $D_0$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth,height=5.0cm]{tauD_slowfast.eps}\n\\caption{Hopf bifurcation boundaries for the synchronous (solid curves)\n  and asynchronous (dashed curves) modes for the FN system \n  (\\ref{2dsf:fg}) with various number $m$ of cells in the $\\tau$ versus\n  $D_0$ parameter plane. Between the solid lines the synchronous mode\n  is unstable, whereas between the dashed lines the asynchronous mode\n  is unstable. Parameters used are $z=3.5$, $q=5$, $\\epsilon_0=0.5$, $d_1=10$,\n  $d_2=0.2$, \n  and $|\\Omega|=10$.}\n\\label{tauD0slowfast}\n\\end{center}\n\\end{figure}\n\n\n\n\\setcounter{equation}{0}\n\\setcounter{section}{4}\n\\section{Finite Domain: Reduction to ODEs for $D\\gg {\\mathcal O}(\\nu^{-1})$}\n\\label{sec:odes}\n\nIn this section we assume that there is one small dynamically active\ncircular cell $\\Omega_\\epsilon$ in the bounded domain $\\Omega$ under\nthe assumption that $D\\gg {\\mathcal O}(\\nu^{-1})$, with\n$\\nu={-1\/\\log\\epsilon}$.  In this limit, in which the bulk region is\neffectively well-mixed, we now show that we can reduce the dynamics for the\ncoupled cell-bulk model to a system of nonlinear ODEs.\n\nFor the case of one cell, the basic model is formulated as\n\\begin{subequations}\\label{d:mainbd}\n\\begin{equation}\\label{d:mainU}\n\\tau U_t = D\\Delta U -U\\,, \\qquad \\vecb x\\in \\Omega\\backslash\n\\Omega_{\\epsilon}\\,; \\qquad \\partial_nU = 0\\,,\\qquad \\vecb x\\in\\partial \\Omega\\,;\n \\qquad\n\\epsilon D \\partial_{n} U =d_1U-d_2u^1\\,,  \\;\\;\\quad \\vecb x\\in\\partial \n\\Omega_{\\epsilon}\\,,\n\\end{equation}\nwhich is coupled to the intracellular cell dynamics, with $\\vecb u =\n(u_1,\\ldots,u_n)^T$ and reaction kinetics $\\vecb F(\\vecb u)$, by\n\\begin{equation}\\label{d:mainuj}\n\\frac{d \\vecb u}{dt} = \\vecb F(\\vecb u)+ \\frac{\\vecb e_1}{\\epsilon  \\tau}\n \\int_{\\partial \\Omega_{\\epsilon}}( d_1 U -d_2 u_1)\\, ds \\,, \\qquad\n\\mbox{where} \\qquad \\vecb e_1\\equiv (1,0,\\dots,0)^T \\,.\n\\end{equation}\n\\end{subequations}$\\!$ \n\nWe will assume that $D\\gg 1$, and so in the bulk region we expand\n\\begin{equation}\\label{d:uout}\n    U=U_0+\\frac{1}{D}U_1+ \\cdots \\,.\n\\end{equation}\nUpon substituting this expansion into (\\ref{d:mainU}), and noting that\n$\\Omega_\\epsilon\\to \\vecb x_0$ as $\\epsilon\\to 0$, we obtain to\nleading order in ${1\/D}$ that $\\Delta U_0=0$ with $\\partial_{n} U_0=0$\non $\\partial\\Omega$. As such, we have that $U_0=U_0(t)$. At next\norder, we have that $U_1$ satisfies\n\\begin{equation}\\label{d:u1}\n\\Delta U_1=U_0+\\tau U_{0t}, \\quad \\vecb x \\in \\Omega\\backslash \\{\\vecb x_0\\}\\,;\n \\qquad \\partial_{\\vecb n} U_1 = 0, \\quad \\vecb x \\in \\partial\\Omega \\,.\n\\end{equation}\nThe formulation of this problem is complete once we determine a matching\ncondition for $U_1$ as $\\vecb x\\rightarrow \\vecb x_0$. \n\nTo obtain this matching condition, we must consider the inner region defined\nat ${\\mathcal O}(\\epsilon)$ distances outside the cell. In this inner\nregion we introduce the new variables \n$\\vecb y=\\epsilon^{-1}(\\vecb x-\\vecb x_0)$ and\n$\\hat{U}(\\vecb y,t) = U(\\vecb x_0+\\epsilon \\vecb y,t)$. From (\\ref{d:mainU}),\nwe obtain that\n\\begin{equation*}\n\\tau \\hat{U_t}=\\frac{D}{\\epsilon^2}\\Delta_{\\vecb y}\\hat{U}-\\hat{U}, \\quad\n \\rho=|\\vecb y|\\geq 1 \\,; \\qquad\nD\\frac{\\partial \\hat{U}}{\\partial \\rho} =d_1 \\hat{U}-d_2 u_1 \\,,\n \\quad \\mbox{on} \\quad \\rho=1 \\,.\n\\end{equation*}\nFor $D\\gg 1$, we then seek a radially symmetric solution to this inner\nproblem in the form\n\\begin{equation}\\label{d:uin}\n    \\hat{U}(\\rho,t)=\\hat{U}_0(\\rho,t) +\\frac{1}{D}\\hat{U}_1(\\rho,t)+ \\cdots \\,.\n\\end{equation}\nTo leading order we obtain $\\Delta_{\\rho}\\hat{U}_0=0$ in $\\rho\\geq 1$,\nwith $\\hat{U}_{0\\rho}=0$ on $\\rho=1$, subject to the matching condition\nto the bulk that $\\hat{U}_0\\to U_0$ as $\\rho\\to \\infty$. We conclude\nthat $\\hat{U}_0=U_0$.  At next order, the problem for $\\hat{U}_1$ is\n\\begin{equation}\\label{di:u1}\n\\Delta_{\\rho}\\hat{U}_1 = 0\\,, \\qquad \\rho\\ge 1\\,; \\qquad\n\\frac{\\partial \\hat{U}_1}{\\partial \\rho}=d_1 U_0-d_2 u_1\\,, \\qquad \\rho=1\\,.\n\\end{equation}\nAllowing for logarithmic growth at infinity, the solution to this problem is\n\\begin{equation}\\label{di:u1_sol}\n\\hat{U}_1=(d_1 U_0-d_2u_1)\\log\\rho+C \\,,\n\\end{equation}\nwhere $C$ is a constant to be found. Then, by writing (\\ref{di:u1_sol})\nin outer variables, and recalling (\\ref{d:uin}), we obtain that the \nfar-field behavior of the inner expansion is\n\\begin{equation}\\label{di:ff}\n  \\hat{U}\\sim U_0 + \\frac{1}{D} \\left[ \\left(d_1U_0-d_2u_1\\right)\n \\log|\\vecb x-\\vecb x_0| + \\frac{1}{\\nu} \\left(d_1U_0-d_2u_1\\right) + C\\right]\n  + \\cdots \\,.\n\\end{equation}\n\nFrom (\\ref{di:ff}) we observe that the term proportional to ${1\/D}$ is\nsmaller than the first term provided that $D\\gg O(\\nu^{-1})$. This\nspecifies the asymptotic range of $D$ for which our analysis will\nhold.  From asymptotic matching of the bulk and inner solutions, the\nfar-field behavior of the inner solution (\\ref{di:ff}) provides the\nrequired singular behavior as $\\vecb x\\to \\vecb x_0$ for the outer bulk\nsolution. In this way, we find from (\\ref{di:ff}) and (\\ref{d:uout}) that\n$U_1$ satisfies (\\ref{d:u1}) subject to the singular behavior\n\\begin{equation}\\label{d:u1_sing}\n   U_1 \\sim \\left(d_1U_0-d_2u_1\\right)\\log|\\vecb x - \\vecb x_0| \\,, \\quad\n  \\mbox{as} \\quad \\vecb x\\to \\vecb x_0 \\,.\n\\end{equation}\nThen, (\\ref{d:u1}) together with (\\ref{d:u1_sing}) determines $U_1$ uniquely.\nFinally, in terms of this solution, we identify that the constant\n$C$ in (\\ref{di:ff}) and (\\ref{di:u1_sol}) is obtained from\n\\begin{equation}\\label{d1:ceval}\n\\lim_{\\vecb x\\to\\vecb x_0}\\left[U_1 -\n \\left(d_1U_0-d_2u_1\\right)\\log|\\vecb x - \\vecb x_0| \\right] =\n \\nu^{-1}  \\left(d_1U_0-d_2u_1\\right) + C \\,.\n\\end{equation}\n\nWe now carry out the details of the analysis.  We can write the problem\n(\\ref{d:u1}) and (\\ref{d:u1_sing}) for $U_1$ as\n\\begin{equation}\\label{d:u1_new}\n\\Delta U_1=U_0+\\tau U_{0t} + 2\\pi \\left(d_1U_0-d_2 u_1\\right) \n  \\delta(\\vecb x-\\vecb x_0)\\,, \\qquad \\vecb x \\in \\Omega\\,; \n \\qquad \\partial_{n} U_1 = 0, \\quad \\vecb x \\in \\partial\\Omega \\,.\n\\end{equation}\nBy the divergence theorem, this problem has a solution only if\n$\\left(U_0+\\tau U_{0t}\\right)|\\Omega|=-2\\pi (d_1U_0-d_2 u_1)$. This leads to \nthe following ODE for the leading-order bulk solution $U_0(t)$:\n\\begin{equation}\\label{dlarge:one}\n    U_0^{\\prime}= -\\frac{1}{\\tau} \\left( 1 + \\frac{2\\pi d_1}{|\\Omega|}\\right)\n  U_0 + \\frac{2\\pi d_2}{\\tau |\\Omega|} u_1 \\,.\n\\end{equation}\nWithout loss of generality we impose $\\int_{\\Omega} U_1\\, d\\vecb x=0$\nso that $U_0$ is the spatial average of $U$. Then, the solution to\n(\\ref{d:u1_new}) is\n\\begin{equation}\\label{d:u1_sol}\n   U_1 = -2\\pi \\left(d_1 U_0 - d_2 u_1\\right) G_0(\\vecb x;\\vecb x_0) \\,,\n\\end{equation}\nwhere $G_0(\\vecb x;\\vecb x_0)$ is the Neumann Green's function defined\nby (\\ref{g0:neum}). We then expand (\\ref{d:u1_sol}) as $\\vecb x\\to\n\\vecb x_0$, and use (\\ref{d1:ceval}) to identify $C$ in terms of\nthe regular part $R_0$ of the Neumann Green's function, defined in\n(\\ref{g0:neum}), as\n\\begin{equation}\\label{d:cres}\n   C = - \\left(d_1 U_0 - d_2 u_1\\right) \\left(\\nu^{-1}+2\\pi R_0\\right)\n   \\,.\n\\end{equation}\n\nIn summary, by using (\\ref{d:uin}), (\\ref{di:u1_sol}), and (\\ref{d:cres}),\nthe two-term inner expansion near the cell is given by\n\\begin{equation}\\label{ds:ui}\n  \\hat{U}\\sim U_0 + \\frac{1}{D}\\left(d_1U_0-d_2u_1\\right)\\left(\n  \\log\\rho - \\frac{1}{\\nu} - 2\\pi R_0 \\right) + \\cdots \\,.\n\\end{equation}\nFrom (\\ref{d:uout}) and (\\ref{d:u1_sol}), the two-term expansion\nfor the outer bulk solution, in terms of $U_0(t)$ satisfying the ODE\n(\\ref{dlarge:one}), is\n\\begin{equation}\n   U \\sim U_0 - \\frac{2\\pi}{D} \\left(d_1 U_0 - d_2 u_1\\right) \n  G_0(\\vecb x;\\vecb x_0) \\,. \\label{ds:uout}\n\\end{equation}\nThe final step in the analysis is to use (\\ref{d:mainuj}) to derive\nthe dynamics inside the cell. We readily calculate that\n\\begin{equation*}\n    \\frac{1}{\\epsilon  \\tau}\n  \\int_{\\partial\\Omega_\\epsilon} \\left(d_1U - d_2u_1\\right)\\, ds\n  \\sim \\frac{2\\pi}{ \\tau} \\left(d_1U_0- d_2u_1\\right) \\,,\n\\end{equation*}\nwhich determines the dynamics inside the cell from (\\ref{d:mainuj}).\n\nThis leads to our main result that, for $D\\gg {\\mathcal O}(\\nu^{-1})$,\nthe coupled PDE model (\\ref{d:mainbd}) reduces to the study of the\ncoupled (n+1) dimensional ODE system for the leading-order average\nbulk concentration $U_0(t)$ and cell dynamics $\\vecb u$ given by\n\\begin{equation}\\label{d:odes}\n    U_0^{\\prime}= -\\frac{1}{\\tau} \\left( 1 + \\frac{2\\pi\n      d_1}{|\\Omega|}\\right) U_0 + \\frac{2\\pi d_2}{\\tau |\\Omega|} u_1\n    \\,, \\qquad \\vecb u^{\\prime} = \\vecb F(\\vecb u) + \\frac{2\\pi}{\n  \\tau} \\left[d_1U_0-d_2u_1\\right] \\vecb e_1 \\,.\n\\end{equation}\n\nBefore studying (\\ref{d:odes}) for some specific reaction kinetics, we\nfirst examine conditions for the existence of steady-state solutions for\n(\\ref{d:odes}) and we derive the spectral problem characterizing the\nlinear stability of these steady-states.\n\nA steady-state solution $U_{0e}$ and $\\vecb u_e$ of (\\ref{d:odes}), if\nit exists, is a solution to the nonlinear algebraic system\n\\begin{equation} \\label{dlarge:ss}\n    \\vecb F(\\vecb u_e) + \\frac{2\\pi}{ \\tau} \\left(d_1 U_{0e}-d_2\\vecb\n    u_{1e}\\right) \\vecb e_1 = 0 \\,, \\qquad rU_{0e}= s u_{1e} \\,, \\qquad\n  \\mbox{where} \\qquad r \\equiv 1 + \\frac{2\\pi d_1}{|\\Omega|} \\,, \\qquad \n   s \\equiv \\frac{2\\pi d_2}{|\\Omega|}\\,.\n\\end{equation}\nTo examine the linearized stability of such a steady-state, we\nsubstitute $U_0=U_{0e}+e^{\\lambda t} \\eta$ and $\\vecb u =\\vecb u_e +\ne^{\\lambda t}\\vecb \\phi$ into (\\ref{d:odes}) and linearize. This\nyields that $\\eta$ and $\\vecb \\phi$ satisfy\n\\begin{equation*}\n   \\lambda \\vecb \\phi = J \\vecb \\phi + \\frac{2\\pi}{ \\tau}\n  \\left(d_1\\eta - d_2 \\phi_1\\right)\n  \\vecb e_1 \\,, \\qquad \\tau \\lambda \\eta = - r\\eta + s\\phi_1 \\,,\n\\end{equation*}\nwhere $J$ is the Jacobian of $\\vecb F$ evaluated at the steady-state\n$\\vecb u=\\vecb u_e$. Upon solving the second equation for $\\eta$, and\nsubstituting the resulting expression into the first equation, we\nreadily derive the homogeneous linear system \n\\begin{equation}\\label{dlarge:eigv}\n    \\left[(\\lambda I - J) - \\mu {\\mathcal E}_1\\right] \\vecb \\phi = 0\n    \\,, \\qquad \\mbox{where} \\qquad \\mu \\equiv \\frac{2\\pi}{\n      \\tau} \\left( \\frac{d_1 s}{\\tau\\lambda + r} - d_2\\right) \\,, \\qquad\n {\\mathcal E}_1 \\equiv \\vecb e_1\\vecb e_1^T \\,.\n\\end{equation}\n\nBy using the matrix determinant lemma we conclude that $\\lambda$ is an\neigenvalue of the linearization if and only if $\\lambda$ satisfies\n$\\vecb e_1^T \\left(\\lambda I - J\\right)^{-1}\\vecb e_1 = {1\/\\mu}$, where\n$\\mu$ is defined in (\\ref{dlarge:eigv}). From this expression, and by\nusing $d_1 s- d_2 r=-d_2$ as obtained from (\\ref{dlarge:ss}), we\nconclude that $\\lambda$ must be a root of ${\\mathcal Q}(\\lambda)=0$,\ndefined by\n\\begin{equation}\\label{dlarge:stab_crit}\n     {\\mathcal Q}(\\lambda) \\equiv \n  \\frac{{ \\tau} (r + \\tau\\lambda)}{2\\pi d_2 \\left(\n  1 + \\tau\\lambda\\right)} + \\frac{M_{11}}{\\det(\\lambda I - J)} \\,,\n\\end{equation}\nwhere $r$ is defined in (\\ref{dlarge:ss}).  Here $M_{11}$\nis the cofactor of the element in the first row and first column of\n$\\lambda I-J$.\n\nNext, we show that (\\ref{dlarge:stab_crit}), which characterizes the\nstability of a steady-state solution of the ODE dynamics\n(\\ref{d:odes}), can also be derived by taking the limit $D_0\\gg 1$ in\nthe stability results obtained in (\\ref{dlarge:sync}) of \\S\n\\ref{sec:largeD} for the $D={\\mathcal O}(\\nu^{-1})$ regime where we\nset $D={D_0\/\\nu}$. Recall from the analysis in \\S \\ref{sec:largeD} for\n$D={D_0\/\\nu}$, that when $m=1$ only the synchronous mode can occur,\nand that the linearized eigenvalue satisfies\n(\\ref{dlarge:sync_new}). By formally letting $D_0\\to \\infty$ in\n(\\ref{dlarge:sync_new}) we readily recover (\\ref{dlarge:stab_crit}).\n\n\\subsection{Large D Theory: Analysis of Reduced Dynamics}\\label{dlarge:examples}\n\nWe now give some examples of our stability theory.  We first consider\nthe case where there is exactly one dynamical species in the cell so\nthat $n=1$. From (\\ref{dlarge:ss}) with $n=1$ we obtain that the\nsteady-state $u_e$ is any solution of\n\\begin{equation}\\label{dlarge:1ode_ss}\n F(u_e) -\\frac{2\\pi d_2}{ \\tau} \\left[1+ \\frac{2\\pi\n     d_1}{|\\Omega|}\\right]^{-1} u_e =0 \\,, \\qquad U_{0e}= \\frac{2\\pi\n   d_2}{|\\Omega|} \\left[1+ \\frac{2\\pi d_1}{|\\Omega|}\\right]^{-1} u_e\n    \\,.\n\\end{equation}\nIn the stability criterion (\\ref{dlarge:stab_crit}) we set $M_{11}=1$\nand $\\det(\\lambda I-J)= \\lambda-F_{u}^{e}$, where\n$F_{u}^{e}\\equiv {dF\/du}\\vert_{u=u_{e}}$, to conclude that the\nstability of this steady-state is determined by the roots of the\nquadratic \\begin{subequations} \\label{dlarge:stab_1cell}\n\\begin{equation}\\label{dlarge:quad}\n   \\lambda^2 - \\lambda p_1 + p_2 =0 \\,,\n\\end{equation}\nwhere $p_1$ and $p_2$ are defined by\n\\begin{equation}\\label{dlarge:1cell_p12}\n  p_1 = -\\frac{1}{\\tau} \\left(1+ \\frac{2\\pi d_1}{|\\Omega|}\\right) +\n  F_u^{e} - \\frac{2\\pi d_2}{ \\tau} \\,, \\qquad p_2 =\n  -\\frac{F_{u}^{e}}{\\tau} \\left(1+ \\frac{2\\pi\n    d_1}{|\\Omega|}\\right) + \\frac{2\\pi d_2}{ \\tau^2} \\,.\n\\end{equation}\n\\end{subequations}$\\!$\nWe now establish the following result based on (\\ref{dlarge:stab_1cell}).\n\n\\begin{result}\\label{dlarge:theorem_1}\nLet $n=1$. Then, no steady-state solution of (\\ref{d:odes}) can undergo\na Hopf bifurcation. Moreover, if\n\\begin{equation}\\label{dlarge:stab_one}\n  \\qquad F_{u}^{e} < F_{\\textrm{th}} \\equiv \\frac{2\\pi d_2}{ \\tau} \n \\left[ 1 + \\frac{2\\pi d_1}{|\\Omega|}\\right]^{-1} \\,,\n\\end{equation}\nthen $\\mbox{Re}(\\lambda)<0$, and so the steady-state is linearly stable.\nIf $F_{u}^{e} > F_{\\textrm{th}}$, the linearization has exactly one positive\neigenvalue.\n\\end{result}\n\n{\\ni \\bf \\underline{Proof:} } We first prove that no Hopf bifurcations are possible for the\nsteady-state.  From (\\ref{dlarge:quad}) it is clear that there exists\na Hopf bifurcation if and only if $p_1=0$ and $p_2>0$ in\n(\\ref{dlarge:1cell_p12}). Upon setting $p_1=0$, we get\n$F_{u}^{e}=2\\pi d_2 { \\tau^{-1}} + \\tau^{-1}\\left(1+{2\\pi\n  d_1\/|\\Omega|}\\right)$. Upon substituting this expression into\n(\\ref{dlarge:1cell_p12}) for $p_2$, we get that\n\\begin{equation*}\n p_2 = -\\frac{1}{\\tau} \\left[ \\frac{4\\pi^2 d_2^2}{{ \\tau}\n |\\Omega|} + \\frac{1}{\\tau}\n  \\left( 1 + \\frac{2\\pi d_1}{|\\Omega|}\\right)\n  \\left( 1 + \\frac{2\\pi d_2}{|\\Omega|}\\right)\\right]<0\\,.\n\\end{equation*}\nSince $p_2<0$ whenever $p_1=0$, we conclude that no Hopf bifurcations\nfor the steady-state are possible.\n\nThe second result follows by showing that $p_1<0$ and $p_2>0$\nwhen $F_{u}^{e} < F_{\\textrm{th}}$. From (\\ref{dlarge:1cell_p12}),\n$p_1<0$ and $p_2>0$ when\n\\begin{equation}\n  \\frac{2\\pi d_2}{ \\tau} - F_u^{e} + \\frac{1}{\\tau} \\left(1+\n  \\frac{2\\pi d_1}{|\\Omega|}\\right) >0 \\,, \\qquad \\frac{2\\pi\n    d_2}{ \\tau} - F_u^{e} -\\frac{2\\pi d_1}{|\\Omega|}\n  F_u^{e}>0 \\,.\n\\end{equation}\nThese two inequalities hold simultaneously only when the second relation\nis satisfied. This yields that when (\\ref{dlarge:stab_one}) holds, we have\n$\\mbox{Re}(\\lambda)<0$. Finally, when $F_{u}^{e} > F_{\\textrm{th}}$, we have\n$p_2<0$, and so there is a unique positive eigenvalue.\\hfill$\\blacksquare$\\vspace{6pt}\n\nThis result shows that with cell-bulk coupling the steady-state of the\nODE (\\ref{d:odes}) can be linearly stable even when the reaction\nkinetics is self-activating in the sense that $F_{u}^{e}>0$. Moreover,\nwe observe that as $\\tau$ increases, corresponding to when the\nintracellular kinetics has increasingly faster dynamics than the time\nscale for bulk decay, then the stability threshold $F_{\\textrm{th}}$\ndecreases. Therefore, for fast cell dynamics there is a smaller\nparameter range where self-activation of the intracelluar dynamics can\noccur while still maintaining stability of the steady-state to the\ncoupled system.\n\nNext, we let $n=2$ where $\\vecb F(\\vecb\nu)=(F(u_1,u_2),G(u_1,u_2))^T$. Then, the steady-state of the ODEs\n(\\ref{d:odes}) satisfies\n\\begin{equation}\\label{d:ss_2}\n  F(u_{1e},u_{2e})-\\frac{2\\pi d_2}{r  \\tau} u_{1e}=0 \\,, \\qquad\n  G(u_{1e},u_{2e})=0 \\,, \\qquad U_{0e}=\\frac{s}{r} u_{1e} \\,,\n\\end{equation}\nwhere $r$ and $s$ are defined in (\\ref{dlarge:ss}). We then\nobserve that the roots of ${\\mathcal Q}(\\lambda)=0$ in\n(\\ref{dlarge:stab_crit}) are roots to a cubic polynomial in $\\lambda$.\nSince $M_{11}=\\lambda-G_{u_2}^{e}$, $\\det(\\lambda\nI-J)=\\lambda^2- \\mbox{tr}(J)\\lambda+\\det J$, where\n\\begin{equation}\n\\mbox{tr}(J)=F_{u_1}^e+G_{u_2}^e\\,, \\qquad \\det\nJ=F_{u_1}^eG_{u_2}^e-F_{u_2}^eG_{u_1}^e\\,,\n\\end{equation}\nand $F_{v}^e$, $G_{v}^e$ are partial derivatives of $F$ or $G$ with\nrespect to $v\\in(u_1,u_2)$ evaluated at the steady-state, we conclude\nthat the linear stability of the steady-state is characterized\nby the roots of the cubic\n\\begin{subequations} \\label{largeD:3ode}\n\\begin{equation}\\label{largeD:H3}\n\\lambda^3+p_1\\lambda^2+p_2\\lambda+p_3=0\\,,\n\\end{equation}\nwhere $p_1$, $p_2$ and $p_3$ are defined as\n\\begin{equation}\\label{largeD:p123}\n\\begin{split}\np_1&\\equiv \\frac{2\\pi d_2}{ \\tau}+\\frac{1}{\\tau}\\left(1+\\frac{2\\pi\n  d_1}{|\\Omega|}\\right)-\\mbox{tr}(J)\\,, \\qquad\np_3\\equiv\\frac{1}{\\tau}\\left(\\left(1+\\frac{2\\pi d_1}{|\\Omega|}\\right)\\det J -\n \\frac{2\\pi d_2}{ \\tau} G_{u2}^e\\right)\\,, \\\\\n  p_2 &\\equiv \\det J- \\frac{2\\pi\nd_2}{ \\tau} G_{u_2}^e+\\frac{1}{\\tau}\\left(\n \\frac{2\\pi d_2}{ \\tau}\n - \\left(1+\\frac{2\\pi\n  d_1}{|\\Omega|}\\right) \\mbox{tr}(J) \\right)\\,.\n\\end{split}\n\\end{equation}\n\\end{subequations}$\\!$\nBy taking $m=1$ and letting $D_0\\rightarrow \\infty$ in (\\ref{mcell2ode:p123}),\nit is readily verified that the expressions above for $p_i$, for $i=1,2,3$,\nagree exactly with those in (\\ref{mcell2ode:p123}). Then, by satisfying\nthe Routh-Hurwitz conditions (\\ref{mcell2ode:syn}), we can\nplot the Hopf bifurcation boundaries in different parameter planes.\n\n\\subsubsection{Example: One Cell with Sel'kov Dynamics}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{d1d2_largeDcomp.eps}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{ode_sim.eps}\n\\caption{Left panel: Comparison of the Hopf bifurcation boundaries for\n  the synchronous mode for the Sel'kov model (\\ref{2dsel:fg}) in the\n  $d_1$ versus $d_2$ plane with $D_0=1,\\ 5,\\ 50$ (solid), as\n  obtained from (\\ref{d0:cubic}), and the large $D$ approximation\n  (dashed), as obtained from (\\ref{largeD:3ode}). Between the outer\n  two black curves, the synchronous mode is unstable for $D_0=1$.\n  In the region bounded by the solid or dashed curve this\n  mode is unstable. As $D_0$ increases, the Hopf\n  boundaries from (\\ref{d0:cubic}) gradually approach the one given in\n  from the large $D$ approximation.  Right panel:\n  Numerical results for the ODE system (\\ref{d:odes}) showing\n  sustained oscillations. In the left and right panels we fixed\n  $\\mu=2$, $\\alpha=0.9$, $\\epsilon_0=0.15$, $\\tau=1$, and\n  $|\\Omega|=10$. In the right panel we took $d_1=0.8$ and\n  $d_2=0.05$ for which the steady-state solution of the ODEs\n  (\\ref{d:odes}) is unstable.}\n\\label{dlarge:hopfbd_sel}\n\\end{center}\n\\end{figure}\n\nWhen the intracellular kinetics is described by the Sel'kov model,\nwhere $F$ and $G$ are given in (\\ref{2dsel:fg}), we obtain from\n(\\ref{d:ss_2}) that the steady-state solution of the ODE dynamics is\n\\begin{equation}\nu_1^e=\\frac{r\\mu}{\\left[r+{2\\pi d_2\/ \\tau} \\right]}\\,,\\qquad \nu_2^e=\\frac{\\mu}{\\alpha+(u_1^e)^2}\\,,\\qquad U_{0e}=\n\\frac{s\\mu}{r+{2\\pi d_2\/\\tau}}\\,,\n\\end{equation}\nwhere $r$ and $s$ are defined in (\\ref{dlarge:ss}). The partial\nderivatives of $F$ and $G$ can be calculated as in\n(\\ref{2dsel:fgp}). \n\nIn the left panel of Fig.~\\ref{dlarge:hopfbd_sel} we plot the Hopf\nbifurcation boundary in the $d_1$ versus $d_2$ plane associated with\nlinearizing the ODEs (\\ref{d:odes}) about this steady-state solution.\nThis curve was obtained by setting $p_1 p_2=p_3$ with $p_1>0$ and\n$p_3>0$ in (\\ref{largeD:3ode}).  In this figure we also plot the Hopf\nbifurcation boundary for different values of $D_0$, with\n${D=D_0\/\\nu}$, as obtained from our stability formulation\n(\\ref{d0:cubic}) of \\S \\ref{sec:examples} for the $D={\\mathcal\n  O}(\\nu^{-1})$ regime. We observe from this figure that the stability\nboundary with $D_0=50$ closely approximates that obtained from\n(\\ref{largeD:3ode}), which corresponds to the $D_0\\to \\infty$ limit of\n(\\ref{d0:cubic}).\n\nWe emphasize that since in the distinguished limit $D\\gg{\\mathcal\n  O}(\\nu^{-1})$ we can approximate the original coupled PDE system\n(\\ref{d:mainbd}) by the system (\\ref{d:odes}) of ODEs, the study\nof large-scale time dynamics far from the steady-state becomes\ntractable. In the right panel of Fig.~\\ref{dlarge:hopfbd_sel}, we plot\nthe numerical solution to (\\ref{d:odes}) with Sel'kov dynamics\n(\\ref{2dsel:fg}), where the initial condition is $u_1(0)=0.01$,\n$u_2(0)=0.2$, and $U_0(0)=0.5$. We observe that by choosing $d_1$ and\n$d_2$ inside the region bounded by the dashed curve in the left panel\nof Fig.~\\ref{dlarge:hopfbd_sel}, where the steady-state is unstable,\nthe full ODE system (\\ref{d:odes}) exhibits a stable periodic orbit,\nindicating a limit cycle behavior.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{u12U_t.eps}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{u12_nocoupling.eps}\n\\caption{Left panel: $u_1$, $u_2$ and $U_0$ versus $t$ showing\n  sustained oscillatory dynamics. Parameters are $\\mu=2$,\n  $\\alpha=0.9$, $\\epsilon_0=0.15$, $\\tau=1$, $|\\Omega|=10$, $d_1=0.8$\n  and $d_2=0.05$. Right panel: $u_2$ versus $u_1$ when the local\n  kinetics is decoupled from the bulk, showing decaying oscillations\n  towards the stable steady-state solution $u_1=\\mu$ and\n  $u_2=\\mu\/(\\alpha+u_1^2)$. Initial values are $u_1(0)=0.01$ and\n  $u_2(0)=0.2$. The parameter values of $\\mu$, $\\epsilon_0$ and $\\alpha$\n  are the same as in the left panel.}\n\\label{dlarge:u12U}\n\\end{center}\n\\end{figure}\n\nIn addition, in the left panel of Fig.~\\ref{dlarge:u12U} we plot the\ntime evolution of $u_1$, $u_2$ and $U_0$, showing clearly the\nsustained periodic oscillations. For comparison, fixing the same\nparameter set for the Sel'kov kinetics (\\ref{2dsel:fg}), in the right\npanel of Fig.~\\ref{dlarge:u12U} we plot the phase plane of $u_2$\nversus $u_1$ when there is no coupling between the local kinetics and\nthe bulk. We now observe that without this cell-bulk coupling the\nSel'kov model (\\ref{2dsel:fg}) exhibits transient decaying oscillatory\ndynamics, with a spiral behavior in the phase-plane towards the\nlinearly stable steady-state.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{bif_sel_largeD.eps}\n\\caption{Global bifurcation diagram of $u_1$ with respect to $d_2$ at\n  a fixed $d_1=0.8$ as computed using XPPAUT \\cite{xpp} from the ODE\n  system (\\ref{d:odes}) for the Sel'kov kinetics (\\ref{2dsel:fg}). The\n  thick\/thin solid line represents stable\/unstable steady-state\n  solutions of $u_1$, while the solid dots indicate a stable periodic\n  solution branch. The parameters used are $\\mu=2$, $\\alpha=0.9$,\n  $\\epsilon_0=0.15$, $\\tau=1$, and $|\\Omega|=10$. }\\label{sel:xpp}\n\\end{center}\n\\end{figure}\n\nFinally, we use the numerical bifurcation software XPPAUT \\cite{xpp}\nto confirm the existence of a stable large amplitude periodic solution\nto (\\ref{d:odes}) with Sel'kov kinetics when $d_1$ and $d_2$ are in the\nunstable region of the left panel of Fig.~\\ref{dlarge:hopfbd_sel}. In\nFig.~\\ref{sel:xpp} we plot a global bifurcation diagram of $u_{1}$\nversus $d_2$ for $d_1=0.8$, corresponding to taking a horizontal\nslice at fixed $d_2=0.8$ through the stability boundaries in the $d_2$\nversus $d_1$ plane shown in Fig.~\\ref{dlarge:hopfbd_sel}. The two\ncomputed Hopf bifurcation points at $d_2\\approx 0.0398$ and\n$d_1=0.1098$ agree with the theoretically predicted values in\nFig.~\\ref{dlarge:hopfbd_sel}. \n\n\\subsubsection{Example: One Cell with Fitzhugh-Nagumo Dynamics}\n\nFinally, we apply our large $D$ theory to the case where the\nintracellular dynamics is governed by the FN kinetics\n(\\ref{2dsf:fg}). From (\\ref{d:ss_2}) we obtain that the steady-state\nsolution of the ODEs (\\ref{d:odes}) with the kinetics (\\ref{2dsf:fg})\nis\n\\begin{equation}\\label{dlarge:fn_eq}\nu_1^e=\\Lambda u_{2}^e \\,, \\qquad U_{0e}=\\frac{s u_{1}^e}{r} \\,, \n \\qquad \\mbox{where} \\qquad\n \\Lambda \\equiv \\frac{\\epsilon_0 z r}{\\left[\\epsilon_0 r + {2\\pi d_2\/{\n \\tau}}\\right]} \\,.\n\\end{equation}\nHere $r$ and $s$ are defined in (\\ref{dlarge:ss}), and $u_{2e}>0$\nis the unique root of the cubic (\\ref{2dfitz:cubic}) where $\\Lambda$\nin (\\ref{2dfitz:cubic}) is now defined in (\\ref{dlarge:fn_eq}). The\npartial derivatives of $F$ and $G$ can be calculated as in\n(\\ref{2dsf:jac}).\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{d1d2_slowfast_largeDcomp.eps}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{ode_sim_slowfast.eps}\n\\caption{Left panel: Comparison of the Hopf bifurcation boundaries for\n  the synchronous mode with FN kinetics (\\ref{2dsf:fg})\n  in the $d_1$ versus $d_2$ parameter plane with $D_0=1,\\ 5,\\ 50$\n  (solid), as obtained from (\\ref{d0:cubic}), and the large $D$\n  approximation (dashed), as obtained from (\\ref{largeD:3ode}). In the\n  wedge-shaped regions bounded by the solid curves the synchronous\n  mode is unstable for the finite values of $D_0$. As $D_0$ increases,\n  the Hopf boundaries obtained from (\\ref{d0:cubic}) becomes rather close\n  to the dashed one obtained from (\\ref{largeD:3ode}) from the large\n  $D$ approximation.  Right panel: Numerical simulation for the ODE\n  system (\\ref{d:odes}), showing sustained oscillations, with initial\n  conditions $u_1(0)=6.0$, $u_2(0)=2.3$, and $U_0(0)=0.05$. In the left\n  and right panels we fixed $z=3.5$, $q=5$, $\\epsilon_0=0.5$,\n  $\\tau=1$, and $|\\Omega|=10$, and in the right panel we took\n  $d_1=15.6$ and $d_2=0.2$ corresponding to a point where the\n  steady-state solution of the ODEs (\\ref{d:odes}) is unstable.}\n\\label{dlarge:hopfbd_fn}\n\\end{center}\n\\end{figure}\n\nIn the left panel of Fig.~\\ref{dlarge:hopfbd_fn} the dashed curve is\nthe Hopf bifurcation boundary in the $d_1$ versus $d_2$ plane\nassociated with linearizing the ODEs (\\ref{d:odes}) about this\nsteady-state solution.  In this figure the Hopf bifurcation boundaries\nfor different values of $D_0$, with ${D=D_0\/\\nu}$, are also\nshown. These latter curves were obtained from our stability\nformulation (\\ref{d0:cubic}) of \\S \\ref{sec:examples}. Similar to what\nwe found for the Sel'kov model, the stability boundary for $D_0=50$ is\nvery close to that for the infinite $D$ result obtained from\n(\\ref{largeD:3ode}). In the right panel of Fig.~\\ref{dlarge:hopfbd_fn}\nwe plot the numerical solution to (\\ref{d:odes}) with FN dynamics\n(\\ref{2dsf:fg}) for the parameter set $d_1=15.6$ and $d_2=0.2$, which\nis inside the unstable region bounded by the dashed curves in the left\npanel of Fig.~\\ref{dlarge:hopfbd_fn}.  With the initial condition\n$u_1(0)=6.0$, $u_2(0)=2.3$, and $U_0(0)=0.05$, the numerical\ncomputations of the full ODE system (\\ref{d:odes}) again reveal a\nsustained and stable periodic solution.\n\nFinally, we use XPPAUT \\cite{xpp} on (\\ref{d:odes}) to compute a\nglobal bifurcation of $u_1$ versus $d_1$ for fixed $d_2=0.2$ for FN\nkinetics. This plot corresponds to taking a vertical slice at fixed\n$d_2=0.2$ through the stability boundaries in the $d_1$ versus $d_2$\nplane shown in Fig.~\\ref{dlarge:hopfbd_fn}. The two computed Hopf\nbifurcation points at $d_1\\approx 15.389$ and $d_1=42.842$ agree with\nthe predicted values in Fig.~\\ref{dlarge:hopfbd_fn}. These results\nconfirm the existence of a stable periodic solution branch induced by\nthe cell-bulk coupling.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{bif_slowfast_largeD.eps}\n\\caption{Global bifurcation diagram of $u_{1}$ versus $d_1$ when\n  $d_2=0.2$, as computed using XPPAUT \\cite{xpp} from the ODE system\n  (\\ref{d:odes}) for the FN kinetics (\\ref{2dsf:fg}).  The thick\/thin\n  solid line represents stable\/unstable steady-state solutions,\n  while the solid dots indicate a stable periodic solution.\n  The other parameters are $z=3.5$, $q=5$, $\\epsilon_0=0.5$,\n  $\\tau=1$, and $|\\Omega|=10$. }\n\\label{fn:xpp}\n\\end{center}\n\\end{figure}\n\n\\setcounter{equation}{0}\n\\setcounter{section}{5}\n\\section{The Effect of the Spatial Configuration of the Small Cells: \nThe $D={\\mathcal O}(1)$ Regime}\\label{sec:finite_d}\n\nIn this section we construct steady-state solutions and study their\nlinear stability properties in the $D={\\mathcal O}(1)$ regime, where\nboth the number of cells and their spatial distribution in the domain\nare important factors. For simplicity, we consider a special spatial\nconfiguration of the cells inside the unit disk $\\Omega$ for which the\nGreen's matrix ${\\mathcal G}$ has a cyclic structure.  More\nspecifically, on a ring of radius $r_0$, with $0<r_0<1$, we place $m$\nequally-spaced cells whose centers are at\n\\begin{equation}\\label{cyc:point}\n   \\vecb x_j = r_0 \\left( \\cos\\left(\\frac{2\\pi j}{m}\\right)\\,, \n \\sin\\left(\\frac{2\\pi j}{m}\\right)\\right)^T\\,, \\qquad\n  \\qquad j=1,\\ldots,m \\,.\n\\end{equation}\nThis ring of cells is concentric with respect to the unit disk\n$\\Omega\\equiv \\lbrace{ \\vecb x \\, \\vert \\, |\\vecb x|\\leq 1\\,\n  \\rbrace}$. We also assume that the intracellular kinetics is the\nsame within each cell, so that $\\vecb F_j=\\vecb F$ for\n$j=1,\\ldots,m$. A related type of analysis characterizing the\nstability of localized spot solutions for the Gray-Scott\nRD model, where localized spots are equally-spaced on\na ring concentric with the unit disk, was performed in \\cite{chen}.\n\nFor the unit disk, the Green's function $G(\\vecb x;\\vecb \\xi)$\nsatisfying (\\ref{bdgreenss_all}) can be written as an infinite sum\ninvolving the modified Bessel functions of the first and second kind\n$I_n(z)$ and $K_n(z)$, respectively, in the form (see Appendix A.1 of\n\\cite{chen})\n\\begin{equation} \\label{cyclic:G}\nG(\\vecb x;\\vecb \\xi) = \\frac{\\,1}{\\,2 \\pi}\nK_0\\left(\\theta_0 |\\vecb x-\\vecb \\xi|\\right)\n-\\frac{\\,1}{\\,2 \\pi} \\sum^{\\infty}_{n=0} \\sigma_n \n\\cos\\left(n(\\psi-\\psi_0)\\right) \\frac{K^{\\prime}_n(\\theta_0)}\n{I^{\\prime}_n(\\theta_0)} I_n\\left(\\theta_0 r\\right) I_n\\left(\\theta_0 r_0\\right)\n\\,; \\quad \\sigma_0=1 \\,, \\quad \\sigma_n=2\\,, \\,\\,\\, n\\geq 1\\,.\n\\end{equation}\nHere $\\theta_0\\equiv D^{-1\/2}$, $\\vecb x=re^{i\\psi}$, $\\vecb\n\\xi=r_0e^{i\\psi_0}$, and $|\\vecb x-\\vecb \\xi|=\\sqrt{r^2+r_0^2 - 2r\n  r_0 \\cos(\\psi-\\psi_0)}$. By using the local behavior\n$K_{0}(z)\\sim -\\log{z} + \\log{2}-\\gamma_e+o(1)$ as $z\\to 0^{+}$, where\n$\\gamma_e$ is Euler's constant, we can extract the regular part $R$ of\n$G(\\vecb x;\\vecb \\xi)$ as $\\vecb x\\to \\vecb \\xi$, as identified in\n(\\ref{bdgreenss_reg}), as\n\\begin{equation}\\label{cyclic:R}\n R = \\frac{1}{2\\pi} \\left(\\log{2} - \\gamma_e + \\frac{1}{2}\\log{D}\n   \\right) -\\frac{\\,1}{\\,2 \\pi} \\sum^{\\infty}_{n=0} \\sigma_n\n \\frac{K^{\\prime}_n(\\theta_0)} {I^{\\prime}_n(\\theta_0)} \\left[\n   I_n\\left(\\theta_0 r_0\\right) \\right]^2 \\,.\n\\end{equation}\n\nFor this spatial configuration of cells, the Green's matrix ${\\mathcal\n  G}$ is obtained by a cyclic permutation of its first row vector $\\vecb\na\\equiv (a_1,\\ldots,a_m)^T$, which is defined term-wise by\n\\begin{equation}\\label{cyclic:G_row}\n  a_1 \\equiv R \\,; \\qquad a_j = G_{j1} \\equiv G(\\vecb x_j;\\vecb x_1) \\,, \n\\quad j=2,\\ldots, m \\,.\n\\end{equation}\nWe can numerically evaluate $G_{j1}$ for $j=2,\\ldots,m$ and $R$ by\nusing (\\ref{cyclic:G}) and (\\ref{cyclic:R}), respectively. Since\n${\\mathcal G}$ is a cyclic matrix, it has an eigenpair, corresponding to\na synchronous perturbation, given by\n\\begin{equation}\\label{cyclic:sync}\n    {\\mathcal G}\\vecb e= \\omega_1 \\vecb e \\,; \\qquad \\vecb e\\equiv\n    (1,\\ldots,1)^T \\,, \\qquad \\omega_1 \\equiv \\sum_{j=1}^{m}a_j = R +\n    \\sum_{j=1}^{m} G_{ji} \\,.\n\\end{equation}\n\nWhen $D={\\mathcal O}(1)$, the steady-state solution is determined by\nthe solution to the nonlinear algebraic system (\\ref{bduj}) and\n(\\ref{bdsysS2m}). Since $\\vecb F_j=\\vecb F$ for $j=1,\\ldots,m$, and\n$\\vecb e$ is an eigenvector of ${\\mathcal G}$ with eigenvalue\n$\\omega_1$, we can look for a solution to (\\ref{bduj}) and\n(\\ref{bdsysS2m}) having a common source strength, so that $\\vecb S=S_c\n\\vecb e$, $\\vecb u_j=\\vecb u$ for all $j=1,\\ldots,m$, and $\\vecb u^1 =\nu_1\\vecb e$. In this way, we obtain from (\\ref{bduj}) and\n(\\ref{bdsysS2m}), that the steady-state problem is to solve the $n+1$\ndimensional nonlinear algebraic system for $S_c$ and $\\vecb\nu=(u_1,u_2,\\ldots,u_n)^T$ given by\n\\begin{equation}\\label{cyclic:ss}\n   \\vecb F(\\vecb u) + \\frac{2\\pi D}{\\tau} S_c \\vecb e = \\vecb 0\\,;\n \\qquad S_c = -\\beta u_1 \\,, \\qquad  \\beta \\equiv \\frac{d_2\\nu}{d_1\n  + 2\\pi \\nu d_1 \\omega_1 + D\\nu} \\,,\n\\end{equation}\nwhere $\\nu\\equiv {-1\/\\log\\epsilon}$ and $\\omega_1$ is defined in\n(\\ref{cyclic:sync}). We remark that $\\omega_1$ depends on $D$, \n$r_0$, and $m$.\n\nTo study the linear stability of this steady-state solution,\nwe write the GCEP, given in (\\ref{gcep:full}), in the form\n\\begin{equation}\\label{cyclic:gcep}\n    {\\mathcal G}_{\\lambda} \\vecb c = - \\frac{1}{2\\pi\\nu} \\left[\n 1+ \\frac{D\\nu}{d_1} + \\frac{2\\pi\\nu d_2 D}{d_1\\tau} \\frac{M_{11}}{\n   \\det(\\lambda I-J)} \\right] \\vecb c \\,,\n\\end{equation}\nwhere $J$ is the Jacobian of $\\vecb F$ evaluated at the steady-state.\nIn terms of the matrix spectrum of ${\\mathcal G}_{\\lambda}$, written as\n\\begin{equation}\\label{cyclic:Glam_mat}\n  {\\mathcal G}_{\\lambda} \\vecb v_j = \\omega_{\\lambda,j} \\vecb v_j \\,, \\qquad\n j=1,\\ldots,m \\,,\n\\end{equation}\nwe conclude from (\\ref{cyclic:gcep}) that the set of discrete eigenvalues\n$\\lambda$ of the linearization of the steady-state are the union of the\nroots of the $m$ transcendental equations, written as \n${\\mathcal F}_j(\\lambda)=0$, where\n\\begin{equation}\\label{cyclic:omega_j}\n  {\\mathcal F}_{j}(\\lambda) \\equiv \\omega_{\\lambda,j} +\n  \\frac{1}{2\\pi\\nu} \\left( 1+ \\frac{D\\nu}{d_1}\\right) + \\left(\n  \\frac{d_2 D} {d_1\\tau} \\right) \\frac{M_{11}}{\\det(\\lambda I-J) } \\,,\n  \\qquad j=1,\\ldots,m\\,.\n\\end{equation}\nAny such root of ${\\mathcal F}_j(\\lambda)=0$ with\n$\\mbox{Re}(\\lambda)>0$ leads to an instability of the steady-state\nsolution on an ${\\mathcal O}(1)$ time-scale. If all such roots satisfy\n$\\mbox{Re}(\\lambda)<0$, then the steady-state is linearly stable on an\n${\\mathcal O}(1)$ time-scale.\n\nTo study the stability properties of the steady-state using\n(\\ref{cyclic:omega_j}), and identify any possible Hopf bifurcation\nvalues, we must first calculate the spectrum (\\ref{cyclic:Glam_mat})\nof the cyclic and symmetric matrix ${\\mathcal G}_{\\lambda}$, whose\nentries are determined by the $\\lambda$-dependent reduced-wave Green's\nfunction $G_{\\lambda}(\\vecb x;\\vecb \\xi)$, with regular part\n$R_{\\lambda}(\\vecb \\xi)$, as defined by (\\ref{bdgneig}).  Since\n${\\mathcal G}_{\\lambda}$ is not a Hermitian matrix when $\\lambda$ is\ncomplex, its eigenvalues $\\omega_{\\lambda,j}$ are in general\ncomplex-valued when $\\lambda$ is complex. Then, by replacing\n$\\theta_0$ in (\\ref{cyclic:G}) and (\\ref{cyclic:R}) with\n$\\theta_\\lambda\\equiv \\sqrt{(1+\\tau\\lambda)\/D}$, we readily obtain\nthat\n\\begin{equation} \\label{cyclic:Glam}\nG_\\lambda(\\vecb x;\\vecb \\xi) = \\frac{\\,1}{\\,2 \\pi}\nK_0\\left(\\theta_\\lambda|\\vecb x-\\vecb \\xi|\\right) -\\frac{\\,1}{\\,2 \\pi}\n\\sum^{\\infty}_{n=0} \\sigma_n \\cos\\left(n(\\psi-\\psi_0)\\right)\n\\frac{K^{\\prime}_n(\\theta_\\lambda)} {I^{\\prime}_n(\\theta_\\lambda)}\nI_n\\left(\\theta_\\lambda r\\right) I_n\\left(\\theta_\\lambda r_0\\right)\n\\,; \\quad \\sigma_0=1 \\,, \\quad \\sigma_n=2\\,, \\,\\,\\, n\\geq 1\\,,\n\\end{equation}\nwith regular part\n\\begin{equation}\\label{cyclic:Rlam}\n R_{\\lambda} = \\frac{1}{2\\pi} \\left[\\log{2} - \\gamma_e +\n   \\frac{1}{2}\\log{D} -\\frac{1}{2}\\log(1+\\tau\\lambda) \\right]\n -\\frac{\\,1}{\\,2 \\pi} \\sum^{\\infty}_{n=0} \\sigma_n\n \\frac{K^{\\prime}_n(\\theta_\\lambda)} {I^{\\prime}_n(\\theta_\\lambda)} \\left[\n   I_n\\left(\\theta_\\lambda r_0\\right) \\right]^2 \\,,\n\\end{equation}\nwhere we have specified the principal branch for $\\theta_\\lambda$.\nThe Green's matrix ${\\mathcal G}_\\lambda$ is obtained by a cyclic\npermutation of its first row $\\vecb a_\\lambda \\equiv\n(a_{\\lambda,1},\\ldots,a_{\\lambda,m})^T$, which is defined term-wise by\n\\begin{equation}\\label{cyclic:G_lam_row}\n  a_{\\lambda,1} \\equiv R_{\\lambda} \\,; \\qquad a_{\\lambda,j} =\n  G_{\\lambda,j1} \\equiv G_{\\lambda} (\\vecb x_j;\\vecb x_1) \\,, \\quad\n  j=2,\\ldots, m \\,.\n\\end{equation}\nAgain we can numerically evaluate $G_{\\lambda,j1}$ for $j=2,\\ldots,m$\nand $R_\\lambda$ by using (\\ref{cyclic:Glam}) and (\\ref{cyclic:Rlam}),\nrespectively. \n\nNext, we must determine the full spectrum\n(\\ref{cyclic:Glam_mat}) of the cyclic and symmetric matrix ${\\mathcal\n  G}_\\lambda$. For the $m\\times m$ cyclic matrix ${\\mathcal\n  G}_{\\lambda}$, generated by permutations of the row vector $\\vecb\na_{\\lambda}$, it is well-known that its eigenvectors $\\vecb v_j$ and\neigenvalues $\\omega_{\\lambda,j}$ are\n\\begin{equation}\\label{cyclic:mat_eig}\n\\omega_{\\lambda, j}= \\sum_{n=0}^{m-1} a_{\\lambda, n+1} e^{2 \\pi i (j-1)n\/m} \\,,\n \\qquad \\vecb v_j = (1, e^{2 \\pi i (j-1)\/m} \\,, \\ldots, e^{2\n \\pi i (j-1) (m-1)\/m})^T \\,, \\qquad j=1,\\ldots, m\\,.\n\\end{equation} \nSince ${\\mathcal G}$ is also necessarily a symmetric matrix it\nfollows that\n\\begin{equation}\\label{cyclic:arow}\n    a_{\\lambda,j} = a_{\\lambda,m+2-j} \\,, \\qquad j=2,\\ldots, \n\\lceil {m\/2} \\rceil \\,,\n\\end{equation}\nwhere the ceiling function $\\lceil x \\rceil$ is defined as the\nsmallest integer not less than $x$. This relation can be used to\nsimplify the expression (\\ref{cyclic:mat_eig}) for\n$\\omega_{\\lambda,j}$, into the form as written below in\n(\\ref{cyclic:spectrum_all}).  Moreover, as a result of\n(\\ref{cyclic:arow}), it readily follows that\n\\begin{equation}\\label{cyclic:omega}\n    \\omega_{\\lambda,j}=\\omega_{\\lambda,m+2-j} \\,, \\qquad \n \\mbox{for} \\quad j=2,\\ldots, \\lceil {m\/2} \\rceil \\,,\n\\end{equation}\nso that there are $\\lceil {m\/2} \\rceil -1$ eigenvalues of multiplicity\ntwo. For these multiple eigenvalues the two independent real-valued\neigenfunctions are readily seen to be $\\mbox{Re}(\\vecb v_j)={(\\vecb\n  v_j +\\vecb v_{m+2-j})\/2}$ and $\\mbox{Im}(\\vecb v_j)={(\\vecb v_j\n  -\\vecb v_{m+2-j})\/(2i)}$. In addition to $\\omega_1$, we also observe\nthat there is an additional eigenvalue of multiplicity one when $m$ is\neven.\n\nIn this way, our result for the matrix spectrum of ${\\mathcal G}_{\\lambda}$\nis as follows: The synchronous eigenpair of ${\\mathcal G}_{\\lambda}$ is \n\\begin{subequations} \\label{cyclic:spectrum_all}\n\\begin{equation}\\label{cyclic:spectrum_1}\n\\omega_{\\lambda, 1} = \\sum_{n=1}^m a_{\\lambda,n}\\,, \\qquad v_1 =\n(1, \\ldots, 1)^T\\,,\n\\end{equation}\nwhile the other eigenvalues, corresponding to the asynchronous modes, are\n\\begin{equation}\\label{cyclic:spectrum_j}\n\\omega_{\\lambda, j} = \\sum_{n=0}^{m-1} \\cos\\left( \\frac{2 \\pi\n(j-1)\\,n}{m}\\right) a_{\\lambda, n+1}\\,, \\qquad j=2,\\ldots,m \\,,\n\\end{equation}\nwhere $\\omega_{\\lambda,j}=\\omega_{\\lambda,m+2-j}$ for $j=2,\\ldots,\n\\lceil {m\/2} \\rceil$. When $m$ is even, we notice that there is an\neigenvalue of multiplicity one given by\n$\\omega_{\\lambda,\\frac{m}{2}+1}=\\sum_{n=0}^{m-1} (-1)^{n}\na_{n+1}$. The corresponding eigenvectors for\n$j=2,\\ldots, \\lceil {m\/2} \\rceil$ can be written as\n\\begin{equation}\\label{cyclic:eigvec}\n\\begin{aligned}\n \\vecb v_j &= \\left(1, \\cos \\left(\\frac{2 \\pi (j-1)}{m}\\right), \\ldots,\n \\cos \\left( \\frac{2 \\pi (j-1)(m-1)}{m} \\right)\\,\\right)^T \\,,\\\\\n \\vecb v_{m+2-j} &= \\left(0, \\sin \\left( \\frac{2 \\pi (j-1)}{m}\\right),\n \\ldots, \\sin \\left( \\frac{2 \\pi (j-1)(m-1)}{m}\\right)\\,\\right)^T \\,.\n\\end{aligned}\n\\end{equation}\n\\end{subequations}$\\!$\nFinally, when $m$ is even, there is an additional eigenvector given by\n$\\vecb v_{\\frac{m}{2}+1}=(1,-1,\\ldots,-1)^T$.\n\nWith the eigenvalues $\\omega_{\\lambda,j}$, for $j=1,\\ldots,m$,\ndetermined in this way, any Hopf bifurcation boundary in parameter\nspace is obtained by substituting $\\lambda=i\\lambda_I$ with\n$\\lambda_I>0$ into (\\ref{cyclic:omega_j}), and requiring that the real\nand imaginary parts of the resulting expression vanish. This yields,\nfor each $j=1,\\ldots,m$, that\n\\begin{equation}\\label{cyclic:real_imag}\n  \\mbox{Re}\\left(\\omega_{\\lambda,j}\\right) + \\frac{1}{2\\pi\\nu} \\left(\n  1+ \\frac{D\\nu}{d_1} \\right) + \\frac{d_2 D} {d_1\\tau} \\mbox{Re}\\left(\n  \\frac{M_{11}}{\\det(\\lambda I-J) }\\right)=0 \\,, \\qquad\n  \\mbox{Im}\\left(\\omega_{\\lambda,j}\\right) + \\frac{d_2 D} {d_1\\tau}\n  \\mbox{Im}\\left( \\frac{M_{11}}{\\det(\\lambda I-J) }\\right)=0 \\,.\n\\end{equation}\nFinally, we can use the winding number criterion of complex analysis\non (\\ref{cyclic:omega_j}) to count the number of eigenvalues of the\nlinearization when the parameters are off any Hopf bifurcation\nboundary.  This criterion is formulated below in \\S \\ref{cyc:sel}.\n\nWe remark that in the limit $D\\gg 1$, we can use $K_{0}(z)\\sim\n-\\log{z}$ together with $I_{0}(z)\\sim 1+{z^2\/4}$ as $z\\to 0$, to\nestimate from the $n=0$ term in (\\ref{cyclic:Glam}) and\n(\\ref{cyclic:Rlam}) that\n$-(2\\pi)^{-1}{K_0^{\\prime}(\\theta_\\lambda)\/I_0^{\\prime}(\\theta_\\lambda)}\\sim\n{D\/\\left[\\pi(1+\\tau\\lambda)\\right]}$ as $D\\to \\infty$. Therefore, for\n$D\\to \\infty$, the Green's matrix ${\\cal G}_{\\lambda}$ satisfies\n${\\mathcal G}_\\lambda \\to {Dm {\\mathcal\n    E}\/{\\left[\\pi(1+\\tau\\lambda)\\right]}}$, where ${\\mathcal\n  E}={{\\vecb e}{\\vecb e}^T\/m}$ and $\\vecb e\\equiv\n(1,\\ldots,1)^T$. This yields for $D\\gg 1$ that\n$\\omega_1={Dm\/{\\left[\\pi (1+\\tau\\lambda)\\right]}}$ and\n$\\omega_j={\\mathcal O}(1)$ for $j=2,\\dots,n$. By substituting these\nexpressions into (\\ref{cyclic:real_imag}), we can readily recover the\nspectral problems (\\ref{dlarge:sync_new}) and (\\ref{dlarge:async}),\nconsidered in \\S \\ref{sec:largeD}, associated with the regime\n$D={\\mathcal O}(\\nu^{-1})$. Therefore, (\\ref{cyclic:real_imag})\nprovides a smooth transition to the leading-order spectral problems\nconsidered in \\S \\ref{sec:largeD} for $D={\\mathcal O}(\\nu^{-1})$.\n\n\\subsection{Example: The Sel'kov Model}\\label{cyc:sel}\n\nWe now use (\\ref{cyclic:real_imag}) to compute phase diagrams in the\n$\\tau$ versus $D$ parameter space associated with $m$ equally-spaced\ncells of radius $\\epsilon$ on a ring of radius $r_0$, with $0<r_0<1$,\nconcentric within the unit disk. For the intracellular dynamics we let\n$n=2$, so that $\\vecb u=(u_1,u_2)^T$, and we consider the Sel'kov\ndynamics $\\vecb F=(F(u_1,u_2),G(u_1,u_2))^T$ as given in\n(\\ref{2dsel:fg}).  For this choice, (\\ref{cyclic:ss}) yields the\nsteady-state solution $(u_{1e},u_{2e})^T$ for the coupled cell-bulk\nsystem given by \\begin{subequations} \\label{cyclic:example}\n\\begin{equation}\n   u_{1e} = \\frac{\\mu}{1+{2\\pi D\\beta\/\\tau}} \\,, \\qquad \n u_{2e} = \\frac{\\mu}{\\alpha + u_{1e}^2} \\,,\n\\end{equation}\nwhere $\\beta$ is defined in (\\ref{cyclic:ss}). Upon using (\\ref{2dsel:jac})\nwe calculate that\n\\begin{equation}\\label{cyc:det_tr}\n   \\mbox{det}(J)=\\epsilon_0 \\left(\\alpha + u_{1e}^2\\right)>0 \\,, \\qquad\n   \\mbox{tr}(J)= \\frac{1}{\\alpha+u_{1e}^2}\\left[ 2 u_{1e}\\mu -\n     (\\alpha+u_{1e}^2) - \\epsilon_0 (\\alpha+u_{1e}^2)^2\\right] \\,.\n\\end{equation}\n\\end{subequations}$\\!$ In this subsection we fix the Sel'kov parameters $\\mu$, $\\alpha$, and\n$\\epsilon_0$, the permeabilities $d_1$ and $d_2$, and the cell radius\n$\\epsilon$ as\n\\begin{equation}\n   \\mu=2 \\,, \\qquad \\alpha=0.9\\,, \\qquad \\epsilon_0=0.15, \\qquad d_1=0.8\\,, \n \\qquad d_2=0.2 \\,, \\qquad \\epsilon=0.05 \\,. \\label{cyc:par}\n\\end{equation}\nWith these values for $\\mu$, $\\alpha$, and $\\epsilon_0$, the\nintracellular dynamics has a stable steady-state when uncoupled from\nthe bulk.\n\nThen, to determine the Hopf bifurcation boundary for the coupled\ncell-bulk model we set $M_{11}=\\lambda-G_{u_{2}}^e$ in\n(\\ref{cyclic:real_imag}), and use $G_{u_{2}}^e=-\\mbox{det}(J)$ as\nobtained from (\\ref{2dsel:jac}). By letting $\\lambda=i\\lambda_I$\nin the resulting expression, we conclude that any Hopf bifurcation\nboundary, for each mode $j=1,\\ldots,m$, must satisfy\n\\begin{equation}\\label{cyclic:hb_ex}\n   \\begin{split}\n  \\mbox{Re}\\left(\\omega_{\\lambda,j}\\right) + \\frac{1}{2\\pi\\nu} \\left(\n  1+ \\frac{D\\nu}{d_1} \\right) - \\left(\\frac{d_2 D} {d_1\\tau}\\right) \\frac{\n  \\left[ \\lambda_I^2 \\mbox{tr}(J) + \\mbox{det}(J) \\left(\\lambda_I^2-\n \\mbox{det}(J)\\right)\\right]}{ \\left[ \\left(\\mbox{det}(J)-\\lambda_I^2\\right)^2\n  + \\left(\\lambda_I \\mbox{tr}(J)\\right)^2 \\right]} =0 \\,,\\\\\n  \\mbox{Im}\\left(\\omega_{\\lambda,j}\\right) + \\left(\\frac{d_2 D} {d_1\\tau}\\right)\n\\frac{  \\left[ \\lambda_I \\left(\\mbox{det}(J)-\\lambda_I^2\\right) + \n   \\mbox{det}(J) \\mbox{tr}(J) \\lambda_I \\right]}\n { \\left[ \\left(\\mbox{det}(J)-\\lambda_I^2\\right)^2\n  + \\left(\\lambda_I \\mbox{tr}(J)\\right)^2 \\right]} =0 \\,.\n   \\end{split}\n\\end{equation}\nFor a specified value of $D$, we view (\\ref{cyclic:hb_ex}) as a\ncoupled system for the Hopf bifurcation value of $\\tau$ and the\ncorresponding eigenvalue $\\lambda_I$, which we solve by Newton's\nmethod. \n\nFor parameter values off of any Hopf bifurcation boundary, we\ncan use the winding number criterion on ${\\mathcal F}_j(\\lambda)$ in\n(\\ref{cyclic:omega_j}) to count the number of unstable eigenvalues\n$N_j$ of the linearization for the $j$-th mode. By using the\nargument principle, we obtain that the number $N_j$ of roots of\n${\\mathcal F}_j(\\lambda)=0$ in $\\mbox{Re}(\\lambda)>0$ is\n\\begin{equation}\\label{wind:last_1}\nN_j=\\frac{1}{2\\pi}[\\mbox{arg} {\\mathcal F}_j]_{\\Gamma}+P\\,, \n\\end{equation}\nwhere $P$ is the number of poles of ${\\mathcal F}_{j}(\\lambda)$ in\n$\\mbox{Re}(\\lambda)>0$, and the square bracket denotes the change in\nthe argument of ${\\mathcal F}_{j}$ over the contour $\\Gamma$. Here the\nclosed contour $\\Gamma$ is the limit as ${\\mathcal R}\\to \\infty$ of\nthe union of the imaginary axis, which can be decomposed as\n$\\Gamma_{I+}=i\\lambda_I$ and $\\Gamma_{I-}=-i\\lambda_I$, for\n$0<\\lambda_I<{\\mathcal R}$, and the semi-circle $\\Gamma_{\\mathcal R}$\ndefined by $|\\lambda|={\\mathcal R}$ with\n$|\\mbox{arg}(\\lambda)|\\leq\\pi\/2$. Since $\\omega_{\\lambda j}$ is\nanalytic in $\\mbox{Re}(\\lambda)>0$, it follows that $P$ is determined\nby the number of roots of $\\det(\\lambda I-J)=0$ in\n$\\mbox{Re}(\\lambda)>0$. Since $\\mbox{det}(J)>0$, as shown in\n(\\ref{cyc:det_tr}), we have that $P=2$ if $\\mbox{tr}(J)>0$ and $P=0$\nif $\\mbox{tr}(J)<0$. Next, we let ${\\mathcal R}\\to \\infty$ on\n$\\Gamma_{\\mathcal R}$ and calculate $[\\mbox{arg}\n  \\mathcal{F}_{j}]_{\\Gamma_{\\mathcal R}}$. It is readily seen that the\nGreen's matrix ${\\mathcal G}_{\\lambda}$ tends to a multiple of a\ndiagonal matrix on $\\Gamma_{\\mathcal R}$ as ${\\mathcal R}\\gg 1$, of\nthe form ${\\mathcal G}_{\\lambda}\\to R_{\\lambda,\\infty}I$, where\n$R_{\\lambda,\\infty}$ is the regular part of the free-space Green's\nfunction $G_{f}(\\vecb x;\\vecb x_0) =\n(2\\pi)^{-1} K_{0}\\left(\\theta_\\lambda|\\vecb x-\\vecb x_0| \\right)$ at\n$\\vecb x=\\vecb x_0$, given explicitly by the first term in the\nexpression (\\ref{cyclic:Rlam}) for $R_{\\lambda}$. Since\n$\\omega_{\\lambda,j}\\to R_{\\lambda,\\infty}$ for $j=1,\\ldots,m$, we estimate\n on $\\Gamma_{\\mathcal R}$ as ${\\mathcal R}\\gg 1$ that\n\\begin{equation*}\n   {\\mathcal F}_{j}(\\lambda) \\sim -\\frac{1}{2\\pi}\n   \\log\\sqrt{1+\\tau\\lambda} + c_0 + {\\mathcal\n     O}\\left({1\/\\lambda}\\right) \\,,\n\\end{equation*}\nfor some constant $c_0$. It follows that $\\mathcal{F}_j(\\lambda)\\sim\nO(\\ln {\\mathcal R})- {i\/8}$ as ${\\mathcal R}\\to \\infty$, so that\n$\\lim_{{\\mathcal R}\\to\n  \\infty}[\\mbox{arg}\\mathcal{F}_{j}]_{\\Gamma_{\\mathcal R}}=0$.\nFinally, since $[\\mbox{arg}\\mathcal{F}_{j}]_{\\Gamma_{I+}}= [\\mbox{arg}\n  \\mathcal{F}_{j}]_{\\Gamma_{I-}}$, as a consequence of ${\\mathcal F}_j$\nbeing real-valued when $\\lambda$ is real, we conclude from\n(\\ref{wind:last_1}) that\n\\begin{equation}\\label{wind:last_2}\nN_j=\\frac{1}{2\\pi}[\\mbox{arg} {\\mathcal F}_j]_{\\Gamma_{I+}}+ P\\,, \\qquad\n  P = \\left\\{\n\\begin{array}{c}\n 2  \\quad \\mbox{if} \\quad \\mbox{tr}J>0 \\\\\n 0  \\quad \\mbox{if} \\quad \\mbox{tr}J<0 \n\\end{array}\n\\right. \\,.\n\\end{equation}\nBy using (\\ref{cyclic:hb_ex}) for the real and imaginary parts of \n${\\mathcal F}_j$, $[\\mbox{arg} {\\mathcal F}_j]_{\\Gamma_{I+}}$ is\neasily calculated numerically by a line sweep over\n$0<\\lambda_I<{\\mathcal R}$. Then, by using (\\ref{cyc:det_tr}) to\ncalculate $\\mbox{tr}(J)$, $P$ is readily determined. In this way, \n(\\ref{wind:last_2}) leads to a highly tractable numerical\nprocedure to calculate $N_j$. This criterion was used for all the\nresults below to identify regions in parameter space where\ninstabilities occur away from any Hopf bifurcation boundary.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{fig_2spot_34.eps}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{fig_2spot_34_large.eps}\n\\caption{Hopf bifurcation boundaries in the $\\tau$ versus $D$ plane\n  for $m=2$, $r_0=0.25$, and with parameters as in (\\ref{cyc:par}), \n  computed from (\\ref{cyclic:hb_ex}). Left panel: the heavy solid\n  curve and the solid curve are the Hopf bifurcation boundaries for\n  the synchronous and asynchronous modes, respectively. Inside the\n  respective lobes the corresponding mode is linearly unstable, as\n  verified by the winding number criterion (\\ref{wind:last_2}).  Right\n  panel: same plot except that we include the Hopf bifurcation\n  boundary for the synchronous mode from the leading-order\n  $D={D_0\/\\nu}\\gg 1$ theory, computed from (\\ref{dlarge:sync_new}).}\n  \\label{fig:m2}\n\\end{center}\n\\end{figure}\n\nIn Fig.~\\ref{fig:m2} we plot the Hopf bifurcation boundaries when\n$m=2$ and $r_0=0.25$. From the left panel of this figure, the\nsynchronous mode is unstable in the larger lobe shaped region, whereas\nthe asynchronous mode is unstable only in the small lobe for small\n$D$, which is contained within the instability lobe for the\nsynchronous mode. In the right panel of Fig.~\\ref{fig:m2} we show the\nHopf bifurcation boundary for the synchronous mode, as obtained from\n(\\ref{dlarge:sync_new}), corresponding to the leading-order\n$D={D_0\/\\nu}\\gg 1$ theory. Since the instability lobe occurs for only\nmoderate values of $D$, and $\\epsilon=0.05$ is only moderately small,\nthe leading-order theory from the $D={D_0\/\\nu}$ regime is, as\nexpected, not particularly accurate in determining the Hopf\nbifurcation boundary. The fact that we have stability at a fixed $D$\nfor $\\tau\\gg 1$, which corresponds to very fast intracellular\ndynamics, is expected since in this limit the intracellular dynamics\nbecomes decoupled from the bulk diffusion. Alternatively, if $\\tau\\ll\n1$, then for a fixed $D$, the intracellular reactions proceed too\nslowly to create any instability. Moreover, in contrast to the large\nregion of instability for the synchronous mode as seen in\nFig.~\\ref{fig:m2}, we observe that the lobe of instability for the\nasynchronous mode only occurs for small values of $D$, where the\ndiffusive coupling, and communication, between the two cells is rather\nweak. Somewhat more paradoxically, we also observe that the\nsynchronous lobe of instability is bounded in $D$. This issue is\ndiscussed in more detail below.\n\nIn Fig.~\\ref{fig:m2_all} we show the effect of changing the ring\nradius $r_0$ on the Hopf bifurcation boundaries. By varying $r_0$, we\neffectively are modulating the distance between the two cells. From\nthis figure we observe that as $r_0$ is decreased, the lobe of\ninstability for the asynchronous mode decreases, implying, rather\nintuitively, that at closer distances the two cells are better able to\nsynchronize their oscillations than when they are farther apart.  We\nremark that results from the leading-order theory of \\S\n\\ref{sec:largeD} for the $D={\\mathcal O}(\\nu^{-1})$ regime would be\nindependent of $r_0$. We further observe from this figure that a\nsynchronous instability can be triggered from a more clustered spatial\narrangement of the cells inside the domain. In particular, for $D=5$\nand $\\tau=0.6$, we observe from Fig.~\\ref{fig:m2_all} that we are\noutside the lobe of instability for $r_0=0.5$, but inside the lobe of\ninstability for $r_0=0.25$ and $r_0=0.75$. We remark that due to the\nNeumann boundary conditions the cells on the ring with $r_0=0.75$ are\nclose to two image cells outside the unit disk, which leads to a\nqualitatively similar clustering effect of these near-boundary cells\nas when they are on the small ring of radius $r_0=0.25$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{fig_2spot_123456.eps}\n\\caption{Hopf bifurcation boundaries for the synchronous mode (larger\n  lobes) and the asynchronous mode (smaller lobes) in the $\\tau$\n  versus $D$ plane for $m=2$ and for three values of $r_0$, with\n  $r_0=0.5$ (heavy solid curves), $r_0=0.75$ (solid curves), and\n  $r_0=0.25$ (dashed curves).  The other parameters are given in\n  (\\ref{cyc:par}). We observe that as $r_0$ decreases, where the two\n  cells become more closely spaced, the lobe of instability for the\n  asynchronous mode decreases.}\n  \\label{fig:m2_all}\n\\end{center}\n\\end{figure}\n\nIn Fig.~\\ref{fig:m3} we plot the Hopf bifurcation boundaries when\n$m=3$ and $r_0=0.5$. For $m=3$, we now observe that the region where\nthe synchronous mode is unstable is unbounded in $D$. The lobe of\ninstability for the asynchronous mode still exists only for small $D$,\nas shown in the right panel of Fig.~\\ref{fig:m3}. In this case, we\nobserve that the Hopf bifurcation boundary for the synchronous mode,\ncorresponding to the leading-order $D={D_0\/\\nu}\\gg 1$ theory and computed\nfrom (\\ref{dlarge:sync_new}), now agrees rather well with results\ncomputed from (\\ref{cyclic:hb_ex}).\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{fig_3spot.eps}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{fig_3spot_zoom.eps}\n\\caption{Left panel: Hopf bifurcation boundaries in the $\\tau$ versus\n  $D$ plane for the synchronous mode for $m=3$ equally-spaced cells on\n  a ring of radius $r_0=0.50$ (heavy solid curves), as computed from\n  (\\ref{cyclic:hb_ex}), with parameters as in (\\ref{cyc:par}). The\n  dashed curve is the Hopf bifurcation boundary from the leading-order\n  $D={D_0\/\\nu}$ theory computed from (\\ref{dlarge:sync_new}).  Right\n  panel: The Hopf bifurcation boundaries for the asynchronous mode\n  (solid curve) and the synchronous mode (heavy solid curve) shown in\n  a magnified region of $D$. The asynchronous mode is linearly\n  unstable only inside this small lobe, which lies within the\n  unstable region for the synchronous mode.}\n  \\label{fig:m3}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{fig_hopf.eps}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{fig_large.eps}\n\\caption{Left panel: Hopf bifurcation boundaries in the $\\tau$ versus\n  $D$ plane for the synchronous mode for $m=5$ equally-spaced cells on\n  a ring of radius $r_0=0.25$ (solid curves) and radius $r_0=0.5$\n  (heavy solid curves) concentric with the unit disk, as computed from\n  (\\ref{cyclic:hb_ex}), with parameters (\\ref{cyc:par}). Right panel:\n  Comparison of the Hopf bifurcation boundaries for the synchronous\n  mode with $r_0=0.5$ (heavy solid curves), as computed from\n  (\\ref{cyclic:hb_ex}), with that obtained from\n  (\\ref{dlarge:sync_new}) for the leading-order $D={D_0\/\\nu}$ theory\n  (solid curves). These curves agree well when $D$ is large.}\n  \\label{fig:m5}\n\\end{center}\n\\end{figure}\n\nIn the left panel of Fig.~\\ref{fig:m5} we plot the Hopf bifurcation\nboundaries for the synchronous mode for $m=5$ when $r_0=0.5$ (heavy\nsolid curves) and for $r_0=0.25$ (solid curves). We observe that for\nmoderate values of $D$, the Hopf bifurcation values do depend\nsignificantly on the radius of the ring. The synchronous mode is\nunstable only in the infinite strip-like domain between these Hopf\nbifurcation boundaries. Therefore, only in some intermediate range of\n$\\tau$, representing the ratio of the rates of the intracellular\nreaction and bulk decay, is the synchronous mode unstable. As\nexpected, the two curves for different values of $r_0$ coalesce as $D$\nincreases, owing to the fact that the leading-order stability theory\nfor $D={D_0\/\\nu}\\gg 1$, as obtained from (\\ref{dlarge:sync_new}), is\nindependent of $r_0$. In the right panel of Fig.~\\ref{fig:m5} we\ncompare the Hopf bifurcation boundaries for the synchronous mode with\n$r_0=0.5$ with that obtained from (\\ref{dlarge:sync_new}),\ncorresponding to the leading-order theory in the $D={D_0\/\\nu\\gg 1}$\nregime.  Rather curiously, we observe upon comparing the solid curves\nin the left and right panels in Fig.~\\ref{fig:m5} that the Hopf\nbifurcation boundaries from the $D={\\mathcal O}(1)$ theory when\n$r_0=0.25$, where the five cells are rather clustered near the origin,\nagree very closely with the leading order theory from the\n$D={D_0\/\\nu}\\gg 1$ regime. Since the clustering of cells is\neffectively equivalent to a system with a large diffusion coefficient,\nthis result above indicates, rather intuitively, that stability\nthresholds for a clustered spatial arrangement of cells will be more\nclosely approximated by results obtained from a large $D$\napproximation than for a non-clustered spatial arrangement of cells.\nIn Fig.~\\ref{fig:asy_m5} we plot the Hopf bifurcation boundaries for\nthe distinct asynchronous modes when $m=5$ for\n$r_0=0.5$ (left panel) and $r_0=0.75$ (right panel), as computed from\n(\\ref{cyclic:hb_ex}) with $j=2,5$ (larger lobe) and with $j=3,4$\n(smaller lobe). The asynchronous modes are only linearly unstable\nwithin these small lobes. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{fig_5asyn_r05.eps}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{fig_5asyn_r075.eps}\n\\caption{Hopf bifurcation boundaries for the two distinct asynchronous\n  modes when $m=5$ for $r_0=0.5$ (left panel) and $r_0=0.75$ (right\n  panel), as computed from (\\ref{cyclic:hb_ex}) with $j=2,5$ (larger\n  solid curve lobe) and with $j=3,4$ (smaller dashed curve lobe). The\n  heavy solid curves are the Hopf bifurcation boundaries for the\n  synchronous mode.  The parameters are as in (\\ref{cyc:par}). The\n  asynchronous mode for $j=2,5$ and $j=3,4$ is linearly unstable only\n  inside the larger and smaller lobe, respectively.}\n\\label{fig:asy_m5}\n\\end{center}\n\\end{figure}\n\nTo theoretically explain the observation that the instability region\nin the $\\tau$ versus $D$ plane for the synchronous mode is bounded for\n$m=2$, but unbounded for $m\\geq 3$, we must first extend the large $D$\nanalysis of \\S \\ref{sec:odes} to the case of $m$ small cells.  We\nreadily derive, assuming identical behavior in each of the $m$\ncells, that the reduced cell-bulk dynamics (\\ref{d:odes}) for one cell\nmust be replaced by\n\\begin{equation}\\label{d:modes}\n    U_0^{\\prime}= -\\frac{1}{\\tau} \\left( 1 + \\frac{2\\pi m\n      d_1}{|\\Omega|}\\right) U_0 + \\frac{2\\pi d_2 m}{\\tau |\\Omega|} u_1\n    \\,, \\qquad \\vecb u^{\\prime} = \\vecb F(\\vecb u) + \\frac{2\\pi}{\n  \\tau} \\left[d_1U_0-d_2u_1\\right] \\vecb e_1 \\,,\n\\end{equation}\nwhen there are $m$ cells. This indicates that the effective domain\narea is ${|\\Omega|\/m}={\\pi\/m}$ when there are $m$ cells. Therefore, to\nexamine the stability of the steady-state solution of (\\ref{d:modes})\nfor the Sel'kov model, we need only replace $|\\Omega|$ with ${|\\Omega|\/m}$\nin the Routh-Hurwitz criteria for the cubic (\\ref{largeD:3ode}).\n\nWith this approach, in Fig.~\\ref{bif:m3m5} we show that there are two\nHopf bifurcation values of $\\tau$ for the steady-state solution of\n(\\ref{d:modes}) when $m=3$ and $m=5$. These values correspond to the\nhorizontal asymptotes as $D\\to\\infty$ in Fig.~\\ref{fig:m3} for $m=3$\nand in Fig.~\\ref{fig:m5} for $m=5$. The numerical results from XPPAUT\n\\cite{xpp} in Fig.~\\ref{bif:m3m5} then reveal the existence of a\nstable periodic solution branch connecting these Hopf bifurcation\npoints for $m=3$ and $m=5$. A qualitatively similar picture holds for\nany $m\\geq 3$. In contrast, for $m=2$, we can verify numerically using\n(\\ref{largeD:3ode}), where we replace $|\\Omega|$ with ${|\\Omega|\/2}$,\nthat the Routh-Hurwitz stability criteria $p_1>0$, $p_3>0$, and\n$p_1p_2>p_3$ hold for all $\\tau>0$ when $m=2$ (and also $m=1$).\nTherefore, for $m=2$, there are no Hopf bifurcation points in $\\tau$\nfor the steady-state solution of (\\ref{d:modes}). This analysis\nsuggests why there is a bounded lobe of instability for the synchronous\nmode when $m=2$, as was shown in Fig.~\\ref{fig:m2}.\n\nWe now suggest a qualitative reason for our observation that the lobe\nof instability for the synchronous mode is bounded in $D$ only when\n$m<m_c$, where $m_c$ is some threshold. We first observe that the\ndiffusivity $D$ serves a dual role. Although larger values of $D$\nallows for better communication between spatially segregated cells,\nsuggesting that synchronization of their dynamics should be\nfacilitated, it also has the competing effect of spatially\nhomogenizing any perturbation in the diffusive signal. We suggest that\nonly if the number of cells exceeds some threshold $m_c$, i.e. if some\n{\\em quorum} is achieved, will the enhanced communication between the\ncells, resulting from a decrease in the effective domain area by\n${|\\Omega|\/m}$, be sufficient to overcome the increased homogenizing\neffect of the diffusive signal at large values of $D$, and thereby\nlead to a synchronized time-periodic response.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{bif_sel_m3.eps}\n\\includegraphics[width=0.45\\textwidth,height=5.0cm]{bif_sel_m5.eps}\n\\caption{Global bifurcation diagram of $u_{1e}$ versus $\\tau$ for the\n  Sel'kov model (\\ref{2dsel:fg}) as computed using XPPAUT \\cite{xpp}\n  from the ODE system (\\ref{d:modes}) characterizing the limiting\n  problem as $D\\to\\infty$ with $m$ small cells in the unit disk\n  $\\Omega$. Left panel: $m=3$. Right panel: $m=5$. The Sel'kov\n  parameters are $\\mu=2$, $\\alpha=0.9$, and $\\epsilon_0=0.15$, while\n  $d_1=0.8$, and $d_2=0.2$. The thick\/thin solid line represents\n  stable\/unstable steady-state solutions, while the solid dots\n  indicate a stable synchronous periodic solution in the\n  cells. For $m=5$ (right panel), there are two Hopf bifurcation points\n  at $\\tau=0.2187$ and $\\tau=0.6238$. For $m=3$ (right panel), the two\n  Hopf bifurcation points are at $\\tau=0.3863$ and\n  $\\tau=0.6815$. These points correspond to the horizontal asymptotes\n  as $D\\to\\infty$ in Fig.~\\ref{fig:m5} for $m=5$ and in\n  Fig.~\\ref{fig:m3} for $m=3$.}\n  \\label{bif:m3m5}\n\\end{center}\n\\end{figure}\n\n\\setcounter{equation}{0}\n\\setcounter{section}{6}\n\\section{Discussion and Outlook}\\label{sec:disc}\n\nWe have formulated and studied a general class of coupled cell-bulk\nproblems in 2-D with the primary goal of establishing whether such a\nclass of problems can lead to the initiation of oscillatory\ninstabilities due to the coupling between the cell and bulk. Our\nanalysis relies on the assumption that the signaling compartments\nhave a radius that is asymptotically small as compared to the\nlength-scale of the domain.  In this limit $\\epsilon\\to 0$ of small\ncell radius we have used a singular perturbation approach to determine\nthe steady-state solution and to formulate the eigenvalue problem\nassociated with linearizing around the steady-state.  In the limit for\nwhich the bulk diffusivity $D$ is asymptotically large of order\n$D={\\mathcal O}(\\nu^{-1})$, we have derived eigenvalue problems\ncharacterizing the possibility of either synchronous and asynchronous\ninstabilities triggered by the cell-bulk coupling. Phase diagrams in\nparameter space, showing where oscillatory instabilities can be\ntriggered, were calculated for two specific choices of the\nintracellular kinetics. Our analysis shows that triggered oscillations\nare not possible when the intracellular dynamics has only one\nspecies. For the regime $D\\gg {\\mathcal O}(\\nu^{-1})$, where the bulk\ncan be effectively treated as a well-mixed system, and for the simple\ncase of one cell, we have reduced the cell-bulk PDE system to a\nfinite-dimensional ODE system for the spatially constant bulk\nconcentration field coupled to the intracellular dynamics. This ODE\nsystem was shown to have triggered oscillations due to cell-bulk\ncoupling, and global bifurcation diagrams were calculated for some\nspecific reaction kinetics, showing that the branch of oscillatory\nsolutions is globally stable. Finally, for the regime $D={\\mathcal\n  O}(1)$, where the spatial configuration of cells is an important\nfactor, we have determined phase-diagrams for the initiation of\nsynchronous temporal instabilities associated with a ring pattern of\ncells inside the unit disk, showing that such instabilities can be\ntriggered from a more clustered spatial arrangement of the cells\ninside the domain.\n\nWe now discuss a few possible extensions of this study, and additional\ndirections that warrant further investigation. For our choices of the\nintracellular kinetics used to illustrate the theory, the coupling\nbetween the bulk and cells leads to a unique steady-state solution. In\ncontrast, it would be interesting to study more elaborate biologically\nrelevant models, such as those in \\cite{tyson2008} and \\cite{Muller1},\nwhere bistable and hysteric behavior of the steady-states is\npossible. In this way, the cell-bulk coupling and the spatial\narrangement of the cells can possibly trigger transitions between\ndifferent steady-state solutions. With regards to our basic model\n(\\ref{mainbd}), it would be interesting to combine fast methods of\npotential theory (cf.~\\cite{mck}) to compute the eigenvalue-dependent\nGreen's function in (\\ref{bdgneig}) so as to obtain a hybrid\nanalytical-numerical method to predict triggered cell-bulk\noscillations from the GCEP (\\ref{gcep:full}). This would then allow us\nto readily consider random spatial configurations of $m$ cells in an\narbitrary domain, rather than the simple ring patterns of cells\nstudied in \\S \\ref{sec:finite_d}.  In addition, it would be\ninteresting to derive amplitude equations characterizing the weakly\nnonlinear development of any oscillatory linear instability for the\ncoupled cell-bulk model. A weakly nonlinear analysis with an\neigenvalue-dependent boundary condition was performed in \\cite{glnw}\nfor the simpler, but related, problem of a coupled membrane-bulk model\nin 1-D.  It would also be worthwhile to analyze large-scale\noscillatory dynamics for the $D={\\mathcal O}(1)$ regime in the limit\n$\\epsilon\\to 0$ in terms of the time-dependent Green's function for\nthe bulk diffusion process. From a numerical viewpoint, we further\nremark that for fixed $\\epsilon$ a full numerical study of\n(\\ref{mainbd}) does not appear to be possible with off-the-shelf PDE\nsoftware owing to the rather atypical coupling between the bulk and\nthe cells.\n\nWe now suggest two extensions of the model (\\ref{mainbd}) that warrant\nfurther study.  We first remark that for the well-mixed regime, where\n$D\\gg {\\mathcal O}(\\nu^{-1})$, we can readily allow for a nonlinear\nbulk degradation, with possibly a Michaelis-Menton saturation, modeled\nby $\\tau U_t = D \\Delta U - \\sigma_B(U)$, where $\\sigma_B(U)={U\/(1+ c\n  U)}$. With this modification, we can readily show, in place of\n(\\ref{d:odes}), that the solution to the one cell-bulk model can be\napproximated by the finite-dimensional dynamics\n\\begin{equation}\\label{dnew:odes}\n    U_0^{\\prime}= -\\frac{1}{\\tau} \\left( \\sigma_B(U_0) + \\frac{2\\pi\n      d_1}{|\\Omega|} U_0\\right) + \\frac{2\\pi d_2}{\\tau |\\Omega|} u_1\n    \\,, \\qquad \\vecb u^{\\prime} = \\vecb F(\\vecb u) + \\frac{2\\pi}{\n  \\tau} \\left[d_1U_0-d_2u_1\\right] \\vecb e_1 \\,.\n\\end{equation}\nIt would be interesting to explore the effect of this nonlinear bulk\ndecay on the possibility of Hopf bifurcations. Finally, as a second\nworthwhile direction, it would be interesting to study the stability\nof steady-state solutions to the coupled cell-bulk model when the\nintracellular dynamics $\\vecb F$ has a time-delay. Time-delays in\nreaction kinetics often occur in many biophysical systems, such as\nthose involved with gene production (see \\cite{bm1}, \\cite{bm2},\n\\cite{levy2}, and the references therein). With such a delay, we\nexpect that Hopf bifurcations can now occur with only one\nintracellular species in the regime $D={\\mathcal O}(\\nu^{-1})$.\n\n\\vspace*{-0.2cm}\n\\section*{Acknowledgements}\nM.~J.~Ward was supported by NSERC (Canada). We are\ngrateful to Prof.~B.~Ermentrout (U.~Pittsburgh), Prof.~T.~Erneux\n(U. Brussels), Prof.~L.~Glass (McGill), and Prof.~J.~Mahaffy (San\nDiego State), for helpful discussions on cell-bulk dynamics.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction}\n\\label{introduction}\n\nDiscovered by Kamerlingh Onnes nearly a century ago~\\cite{onnes}, and first explained \nmicroscopically back in 1957 in terms of electron pairing by Bardeen,Cooper and Schrieffer (BCS)~\\cite{BCS,BCS-1}, \nsuperconductivity is still surprisingly {\\it a' la page}. On one \nhand, superconductivity is being constantly discovered in an ever increasing variety of \nsolid state compounds. On the other hand, it appears more and more difficult to use \nbasically the same standard theory, essentially BCS and its extensions~\\citep[e.g.][chapters~10~and~11]{BCSreview}\nto account for all of them. In this standard, conventional theory, superconductivity\narises from the condensation of electron pairs, the two electrons usually bound in \na pair state of $s$-wave symmetry and held together by exchange of lattice\nphonons. The Coulomb repulsion between the two electrons opposes pair formation, but\nit does not suppress superconductivity because screening makes it weak enough.\n\nThe surprisingly favorable effect of repulsive electron correlations on superconductivity \nfound in some systems, particularly in high-$T_c$ superconducting cuprates, where \nelectron-electron repulsion is dominant, remains a standing puzzle. An immense amount \nof experimental and theoretical work has accumulated over the last two decades in the \nattempt to understand these phenomena, see e.g. the review by \\textcite{hitc}.  Actually, cuprates are but the most \nspectacular members of a wider class of strongly correlated superconductors including \nheavy fermion and organic molecular compounds~\\cite{hitc}, systems for which \nthere is no really comprehensive theory either. Among other factors, theoretical efforts \nhave been hampered by the general {\\it inter-site} nature of electron interactions \nand correlations in many of these systems, a fact that poses large technical \ndifficulties. In this light, identifying a superconductor family where correlations are \nat the same time strong, simple, and {\\it on-site} is welcome.  \n\nA more crucial element is one of perspective. It has been a widespread prejudice to \ndistinguish between superconductors where (as in BCS theory) pairing of electrons \ntakes place in the $s$-wave channel and is mediated by phonons, from those where \nthe mechanism may be electronic and not phononic, and where pairing instead takes\nplace in the $d$-wave channel. Whereas it is widely held that strong repulsive \ncorrelations are essential to superconductivity in the latter~\\cite{PWA}, they \nare not considered crucial in the former. The conventional BCS scenario and its \nextensions, namely the Migdal-Eliashberg theory~\\cite{Migdal,Eliashberg,BCSreview} -- a controlled approximation valid \nwhen the typical phonon frequency is much smaller than the Fermi-energy -- are more \nor less automatically accepted, and used to account for the superconducting properties. \nIn this theory, electron-electron repulsion merely renormalizes the electron-phonon \nparameters, lowering the critical temperature $T_c$ rather than enhancing it.\n\nThe trivalent alkali fullerides superconductors, excellently reviewed, e.g., by \n~\\textcite{Gunnarsson-review,Gunnarsson-book} and \\textcite{Ramirez}, are among the systems where \nthis conventional logic seemingly applies. Fullerides  are solid state compounds of generic \ncomposition A$_3$C$_{60}$, where C$_{60}$ is the fullerene molecule~\\cite{Gunnarsson-book} \nand A = K, Rb, and Cs are alkali cations. The three alkalis donate a total of \n$n$=3 electrons to each fullerene, half filling its threefold degenerate $t_{1u}$ \nmolecular level. Electron hopping between first neighboring fullerenes gives\nrise to a metal, where conduction is restricted to the three narrow $t_{1u}$-derived \nbands, with a total energy bandwidth of no more than 0.6 eV~\\cite{satpathy,erwin}.\nMetallic fullerides are generally superconducting, with critical temperatures \n$T_c$ reaching $\\sim$ 40~K, depending on various factors. An empirically \nimportant factor appears to be the cell volume. When the \nfulleride lattice is chemically expanded, by either increasing cation size or \nby insertion of neutral molecules, or else physically expanded by removing pressure,\n$T_c$ undergoes a definite and systematic change. It rises initially with \na good correlation with the C$_{60}$-C$_{60}$ distance~\\cite{fischer,Gunnarsson-review}. \nFurther expansion however causes $T_c$ to drop, ending eventually, through a first order \ntransition, in an insulating state, as we shall discuss later. \n\nA wealth of evidence indicates that superconducting pairing in fullerides\nis phononic and that the relevant phonons are the stiff intra-molecular $H_g$ vibrations \nof the C$_{60}$ molecule, Jahn-Teller coupled to the $t_{1u}$ conduction \nelectrons~\\cite{Gunnarsson-review}. Further support to the apparent BCS \nnature of superconductivity in fullerides comes from specific-heat jumps \nthat scale linearly with $T_c$ in agreement with BCS theory~\\cite{Ramirez,Ramirez-1,Ramirez-2}, \nas well as a regular (i.e., not exceedingly high) normal phase magnetic susceptibility.~\\cite{Petit-Robert,Ramirez-1} \nSuperconducting energy gaps are less clearly defined,~\\cite{Gunnarsson-review} \nthe gap ratio $2\\Delta\/T_c \\simeq 3.4-4.2$ in K$_3$C$_{60}$ and \nRb$_3$C$_{60}$~\\cite{Gunnarsson-review, Ramirez}, but not far from the BCS value \nof 3.53. These elements suggest viewing the A$_3$C$_{60}$ compounds as weakly \ncorrelated Fermi-liquid conductors~\\cite{Ramirez}, though with unusually narrow \nelectron bands, with a large effective mass roughly three free electron masses~\\cite{Petit-Robert}. \nEven the observed decrease of $T_c$ under applied pressure \nin K$_3$C$_{60}$ and Rb$_3$C$_{60}$ is in qualitative agreement with \nan increasing bandwidth and decreasing density of states at the Fermi level, \nwhich further supports a standard BCS picture.\n\nThese reassuring, conventional looking facts are however contrasted by\na number of conflicting elements that are strong enough to cast serious \ndoubts on the general applicability of the BCS scenario to superconductors\nin this family. These elements are especially apparent in \nthe more expanded fullerides including (NH$_3$)$_x$NaK$_2$C$_{60}$~\\cite{Ricco_intercalato} \nand Li$_3$C$_{60}$~\\cite{Durand2003}, and \nin the alloys Cs$_{3-x}$K$_x$C$_{60}$ and Cs$_{3-x}$Rb$_x$C$_{60}$~\\cite{Dahlke}.\nFor these expanded compounds $T_c$ decreases upon expansion, contrary to BCS \ntheory. The electron density of states extracted by NMR Knight shift \nis at the same time an increasing function of lattice parameter, \nsmoothly connecting with that of the unalloyed compounds under pressure~\\cite{Dahlke}.\nWithin BCS theory, that increase should lead to a rise of $T_c$ and not to a \ndrop as observed.  The same unconventional behavior is observed in Cs$_3$C$_{60}$, the \nfulleride compound with the highest $T_c\\sim 40$~K attained under pressure~\\cite{Palstra}. \nA novel A15 superconducting phase of Cs$_3$C$_{60}$ with expanded structure \nhas very recently been synthesized~\\cite{Ganin} corresponding \nto a body-centered cubic arrangement of fullerenes. Superconductivity \nemerges under pressure through a first order non-structural transition at 4~Kbar.\nThe critical temperature $T_c$ first increases with pressure, reaching a dome-shaped \nmaximum of 38~K around 7~Kbar, above which $T_c$ drops. Since no structural \nchanges are observed under pressure, the appearance of superconductivity as well \nas the dome-shaped $T_c$ vs. pressure behavior must be ascribed solely to the volume \ncontraction~\\cite{Ganin}. This nonmonotonic behavior of $T_c$ with pressure\nfinds no apparent explanation within the conventional theory. \n\nThe basic and striking anomaly of expanded fullerides occurs in the \ncompounds with the largest inter-molecule distances. In these materials a relatively modest additional lattice expansion \n(and minor change of symmetry due to intercalated ammonia) is enough to dramatically \nturn them from metallic and superconducting to antiferromagnetic and insulating~\\cite{Iwasa,Durand2003}.\\footnote\n{We expect that below the superconducting pressure of 4 kbar, \nthe new compound A15 Cs$_3$C$_{60}$~\\cite{Ganin}, yet to be characterized in this respect, \nshould also be an insulating antiferromagnet.}\nWith temperature, antiferromagnetic order in the ammoniated compound NH$_3$K$_3$C$_{60}$\nchanges to paramagnetic disorder at a N\\'eel temperature slightly above $\\sim$ 40~K~\\cite{Prassides1999}.\nEven above the N\\'eel temperature, the microwave conductivity in NH$_3$K$_3$C$_{60}$ remains \nseveral orders of magnitude below that of K$_3$C$_{60}$~\\cite{Kitano}, \ntestifying the Mott insulator (correlation driven) nature of the insulating phase. \nElectrons in a lattice give rise to a Mott insulating state when electron electron \nrepulsion stops their free propagation and the lattice appears as a collection of molecular ions. \nCorrelations lead to an energy gap in their \nspectrum, even if their number density per cell is odd, instead of even as in\nregular band insulators~\\cite{Mott}.\nProximity of a Mott insulator phase in fullerides had long been advocated\nin different contexts~\\cite{Kivelson,Kivelson-1,Baskaran,lof} but was not taken seriously \nby the community prior to this data. Superconductivity next to a Mott insulating phase \nas a function of doping or volume change is the hallmark of strong correlations \nin high temperature superconducting cuprates  and organics. \nOne is thus naturally led to inquire whether superconductivity in expanded \nfullerides might, despite the obvious differences, and despite the phononic mechanism,\nbe somehow related to strong correlations. Our contention is that it is indeed \nclosely related, as outlined in the following.\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig1.eps}\n\\caption{\\label{structure} Schematic representation of a planar projection of the crystal \nstructure of NH$_3$K$_3$C$_{60}$. Green dots are potassium atoms, blue dots surrounded by \nthree grey dots are NH$_3$ molecules, and C$_{60}$ molecules are shown according \nto their actual spatial orientation. (Courtesy of Kosmas Prassides).}\n\\end{figure} \n\n\\section{Electron Interactions in Fullerides}\nThe electrons donated by the alkalis to the C$_{60}$ molecule enter the threefold \ndegenerate $t_{1u}$ former lowest unoccupied molecular orbital (LUMO). In a\ndegenerate molecular orbital, electrons interact through a variety of mechanisms. \nThe first is overall Coulomb repulsion, which we will associate later below with the\nHubbard parameter $U$. The second is Coulomb exchange energy, minimized when the \nmolecular state has the highest total spin, and the highest total orbital angular momentum\ncompatible with it (Hund's rules)~\\cite{landau_lifshitz}. The third is the\nJahn-Teller (JT) interaction, caused by coupling of the electron levels to symmetry lowering \nmolecular distortions~\\citep[chap.~13]{landau_lifshitz}. Contrary to Hund's rules, in a\nJT distorted molecule the ground state maximizes double occupancy of levels, thus\nfavoring low total spin instead of high spin in the isolated molecular ion. \nIn molecular C$_{60}^{3-}$, the strength of these interactions has been evaluated \nin the past, and the JT strength has been estimated to prevail narrowly over Hund's \nrule exchange~\\cite{Manini02}.  This narrow balance favors a low-spin ground state, \nwith a relatively small ``spin gap''-- the energy between the low spin ground \nstate and the lowest high-spin excited state, expected to be of the order of 0.1~eV~\\cite{PRL01,Manini02}. \nIn agreement with this expectation, local moments indicate that in antiferromagnetic \nMott insulating NH$_3$K$_3$C$_{60}$ the  $C_{60}(3-)$ sites are in a {\\em low-spin} \nstate, $S$ =1\/2~\\cite{Prassides1999}, their high spin state $S$ = 3\/2 lying about \n100~meV higher in energy. A low-spin qualifies the overexpanded fullerides as ``Mott-Jahn-Teller'' \ninsulators -- that is, Mott insulators whose sites are in a JT stabilized \nlow-spin state (as opposed to a Hund's rule stabilized high-spin state)~\\cite{PRB97}. \nUnder hydrostatic pressure, NH$_3$K$_3$C$_{60}$ undergoes a \ntransition to a  metallic state where superconductivity re-emerges with a rather \nlarge $T_c$~\\cite{Prassides1999}. It is important to note that this superconducting \nphase still belongs to the ``expanded'' family, as signaled by the fact the $T_c$ here \n\\emph{increases} further with increasing pressure (reaching 28~K at 14.8~kbar~\\cite{Zhou}) \nat variance with non-expanded fullerides, where $T_c$ drops under pressure.  \n\nA lateral, but relevant additional element comes from  {\\em tetravalent} \ncompounds A$_4$C$_{60}$, which are insulators or near insulators. By comparison\nwith the trivalent fullerides, the slight reduction of band-energy gain per particle \ncaused by adding one more electron per molecule and by slightly changing the crystal \nstructure is sufficient to turn the trivalent metals into tetravalent insulators \neven in  non-expanded materials. Careful density-functional electronic structure \ncalculations indicated that it is not possible to describe the tetravalent compounds\nsuch as K$_4$C$_{60}$ as statically distorted Jahn-Teller band insulators~\\cite{PRB2000}.  A \nstatic JT distortion and the associated orthorhombic state is actually present only \nin Cs$_4$C$_{60}$~\\cite{Dahlke&Rosseinsky}, and in Rb$_4$C$_{60}$ above a \ncritical pressure~\\cite{stephens}, while it never shows up in K$_4$C$_{60}$~\\cite{stephens} \n(with the exception of monolayers, see \\textcite{Crommie}). \nThe persistence of insulating or near insulating behavior and the recovery of \nmolecular symmetry observed in the high temperature phase of tetravalent fullerides\nsuggests that these compounds too are Mott-Jahn-Teller insulators~\\cite{PRB2000,Knupfer,Kamaras}, \nlike the overexpanded trivalent materials. The dynamic JT effect in  each \nC$_{60}^{4-}$ ion associated with Mott localization of carriers is crucial \nin explaining the low spin ground state and the spin gap of A$_4$C$_{60}$, \nexactly as for the expanded trivalent compounds.\n\nFrom the above discussion one might be tempted to conclude that strong \ncorrelations play a role only in tetravalent and expanded trivalent compounds, \nwhile face-centered cubic (f.c.c.) K$_3$C$_{60}$ and Rb$_3$C$_{60}$, where superconductivity \nwas originally discovered, could still be viewed as weakly correlated systems, and\nas BCS type superconductors. We do not believe in this conclusion.\nA final, independent and strongly unconventional signal is provided by NMR. \nIn fact, NMR data show direct evidence of a spin gap of order 0.1 eV, appearing as\nan anomalous activated increase of inverse relaxation time. \nMost likely this gap between a low spin ground state and a high spin\nexcited state reflects the multiplet behavior of the localized C$_{60}^{n-}$ molecular ion. \nIt shows up ubiquitously in all alkali doped fullerides, \nincluding superconducting f.c.c. compounds~\\cite{Zimmer,Brouet,Brouet-1}. \nThe existence of the spin gap signifies that the magnetic response \nof fullerides is very far from Fermi-liquid behavior, which has no such feature.\nMagnetically, the fullerides behave as if localized molecular multiplet excitations \ncoexisted with delocalized propagating quasiparticles. As discussed, the recovery \nof molecular physics is characteristic of Mott insulators, \nsuggesting that the fingerprint of Mott physics is strongly present already \nin the non-expanded superconducting f.c.c. compounds. This suggests\nthat the f.c.c. compounds are somehow the analog of the overdoped cuprates, \nwhereas the expanded trivalent materials are analogous to the underdoped cuprates. \nThe conclusion is that both are crucially, even if differently, \ninfluenced by electron correlations. \n\nWe believe that the above elements are strong enough to call for a \nnew physical picture for the whole family of A$_3$C$_{60}$ superconductors. \nProximity of the Mott insulator strongly suggests that the anomalies of expanded \nfullerene superconductors most likely originate from strong repulsive electron correlation \nin the narrow $t_{1u}$ bands. The prevalence in the Mott localized state of molecular \nphysics, with its orbital degeneracy, JT effect and intra-molecular exchange \nmust be taken into account along with itinerant electron band physics. Upon \nexpanding the cell volume, the intermolecular hopping of electron weakens, whereas \nall the on-site correlation terms -- Coulomb and exchange electron-electron interactions \nas well as molecular JT effect -- are likely to become increasingly relevant. We are \nled to a picture where the Mott insulator physics of weakly coupled molecular \nions progressively prevails over band physics for increasing lattice expansion. \nIn particular, superconductors that operate in this regime are bound to \ndeviate from the standard Migdal-Eliashberg scenario, the more so as \nthe lattice spacing increases. To investigate that regime, we need to start \nwith a broader theoretical scheme for trivalent fullerides, capable of \ndescribing their behavior under lattice expansion and near the Mott transition. \nWhile that has been the scope of our work for several years, previous work\nwas for practical technical reasons limited to tetravalent systems \n~\\cite{PRB2000,PRL01,Science}. The study of A$_3$C$_{60}$ systems, computationally \nmuch heavier due to the simultaneous relevance of magnetic and orbital \nordering, has only now been completed, and we offer here an outline\nof the main results.\n\n\\section{Model and Interactions}\nOur theoretical model of trivalent fullerides assumes a lattice of \nmolecular sites, each representing a C$_{60}$ molecule. The C$_{60}$ $t_{1u}$ \nthreefold degenerate LUMO can for all purposes be treated\nas an atomic $p$ level. An average of three electrons per molecule are donated by \nalkali atoms and partially fill these orbitals, which can host up to six electrons.\nThe electrons hop from site to site giving rise to half-filled bands of width \n$W \\sim 0.6 \\mathrm{eV}$. On each site the electrons experience a Hubbard \nrepulsion $U \\sim 1\\mathrm{eV}$ (corresponding to the Slater integral $F_0 \\propto U$),\n\\begin{equation}\n\\mathcal{H}_U = \\frac{U}{2}\\,\\left(n-3\\right)^2,\\label{H-U}\n\\end{equation} \ntogether with a weaker, yet crucial, inter-orbital Hund's rule exchange coupling term\n$J_H$ proportional to the Slater integral $F_2$. Under the sole assumption \nof full rotational orbital symmetry, this exchange term takes the form~\\cite{PRL01,Science}\n\\begin{equation}\n\\mathcal{H}_{J} = J\\,\n\\left(2\\mathbf{S}\\cdot\\mathbf{S} + \\frac{1}{2}\\mathbf{L}\\cdot\\mathbf{L}\n\\right) + \\frac{5}{6}\\,J\\,\\left(n-3\\right)^2,\n\\label{H_el-ph}\n\\end{equation}\nwhere $J=-J_H<0$, while $n$, $\\mathbf{S}$ and $\\mathbf{L}$ are the density, \nspin and orbital angular momentum operators, defined as for $p$-orbitals, \non the given site. This exchange term favors high $\\mathbf{S}$ and $\\mathbf{L}$ \nmolecular multiplets, and has been overlooked or neglected in most treatments \nso far. The next interaction is the JT intra-molecular coupling of electrons \nin the $t_{1u}$ orbital to $H_g$ intra-molecular vibrations. This interaction has\non the contrary been very widely discussed (see e.g.~\\textcite{Schluter,Varma,Manini94,\nGunnarsson1995} and references therein), and we will not dwell too much on its details \nhere. It acts to split the  $t_{1u}$ orbital degeneracy and thus favors low \nspin states, effectively playing the opposite role \nto intra-molecular exchange. A proper treatment of the JT coupling involves \nthe dynamics of carbon nuclei in C$_{60}^{n-}$ ions, including retardation, and\nhas been developed by \\textcite{Gunnarsson1995} \nHowever for expanded fullerides close enough to the Mott transition, retardation\nis not essential and can be omitted. In fact, in this regime the inter-site motion of quasiparticles is severely slowed down~\\cite{DMFT} and the coherent bandwidth \nwill eventually drop from $W$ to $ZW$ ($Z \\ll 1$). When $ZW$ falls enough to\napproach the relatively high $H_g$ vibration frequencies of fullerene \n$\\hbar\\omega \\sim 90$~meV~\\cite{PRB2000,PRL01,Science},\nquasiparticles begin to move on a comparable time scale with the vibrating carbon atoms, and \nnon-adiabatic effects become important~\\cite{Pietronero}. Even closer to the Mott transition, the\nphonon dynamics becomes eventually {\\it faster} than inter-molecular quasiparticle hopping. \nIn this anti-adiabatic limit, molecular vibrations can be integrated \naway, and the resulting unretarded effective JT interaction recovers identically the same \nform as Hund's rule exchange (\\ref{H_el-ph}) except for the sign of $J$, namely\n$J_{JT} >0$~\\cite{PRL01}. In this limit the Hund and JT intra-molecular inter-orbital \ninteraction terms can be directly combined in the form (\\ref{H_el-ph}) \nwith $J= -J_H + J_{JT}$. In $C_{60}$, $J_H \\simeq 0.03-0.1~\\mathrm{eV}$~\\cite{Martin,Manini02}, \nwhereas $J_{JT} \\simeq  0.06-0.12~\\mathrm{eV}$~\\cite{Schluter,Varma,Gunnarsson1995,Manini94,Manini02}.\nThe total result -- and the only one compatible with s-wave superconductivity, \nwith a spin 1\/2 Mott insulator, and with a moderate spin gap near 0.1 eV --\nis a relatively weak unretarded attraction which has the form of an  \n{\\em inverted} Hund's rule coupling~\\cite{Science,Granath}. This is the\napproximation we shall adopt, keeping in mind that it is strictly valid only when \n$ZW < \\hbar\\omega$, that is, close enough to the Mott transition. \n\nThe lattice expansion characterizing the expanded fullerides is believed to\nhave little effect on either $ J_H$ or  $J_{JT}$. On the other hand, expansion will surely \ndecrease $W$ and increase $U$, so it can be modeled as a gradual increase of $U\/W$, \nreflecting both the band narrowing due to smaller overlap between molecular \nwave functions, and a decreased screening strength.  \nAs discussed above, this Hamiltonian model, even if not really simple, has many body \ninteractions that are strictly on site, an ideal situation for Dynamical Mean-Field \nTheory (DMFT)~\\cite{DMFT}, one of the most popular and powerful tools in the field of \nstrongly correlated electron systems that we briefly describe in the next subsection.\n\n\\subsection{Dynamical Mean-Field Theory}\n\\label{Sec:dmft}\nDMFT is a quantum version of classical mean-field theory, which provides \nan exact description of the local dynamics, at the price of freezing away all spatial fluctuations.\nThe mean-field scheme is formulated by a mapping of the lattice model onto an Anderson \nimpurity model (AIM) embedded in a free-electron Fermi ``bath'' subject to a self-consistency condition~\\cite{DMFT}.\nIn our model, the effective AIM is three-fold\norbitally degenerate, with p-like levels representing the $t_{1u}$ orbitals,\neach hybridized with a bath. The Hamiltonian is \n\\begin{equation}\n\\label{aim}\n{\\cal H} = {\\cal H}_U +{\\cal H}_J +\\sum_{ka\\sigma} \\varepsilon_{ka} c^{\\dagger}_{ka\\sigma}c^{\\phantom{\\dagger}}_{ka\\sigma}+\\sum_{ka\\sigma} \nV_{ka} (c^{\\dagger}_{ka\\sigma}p^{\\phantom{\\dagger}}_{a\\sigma} + H.c.),\n\\end{equation}\nwhere ${\\cal H}_U$ and ${\\cal H}_J$ are (\\ref{H-U}) and (\\ref{H_el-ph}) for fermions on the impurity orbitals, \n$\\varepsilon_{ka}$ are the bath energy levels labeled by the index $k$ and by an orbital index $a=x,y,z$, \n$V_{ka}$ are the hybridization parameters between the bath fermions, created by $c^{\\dagger}_{ka\\sigma}$, \nand the impurity fermions, created by $p^{\\dagger}_{a\\sigma}$.\nThe mean-field scheme implies a self-consistency condition that depends on the \nimpurity Green's function and on the bare density of states of the original lattice. \nThroughout our calculations, we use an infinite-coordination Bethe lattice, \nwhose density of states is semicircular. This is a reasonable description of a \nrealistic three-dimensional density of states devoid of accidental features \nsuch as van-Hove singularities. It is moreover particularly convenient since it leads \nto a very simple and transparent form of the self-consistency condition.\nThe Bethe lattice is the $z\\to\\infty$ limit of a Cayley tree of coordination \n$z$, scaling the nearest-neighbor hopping in each one of the $z$ directions as \n$t\/\\sqrt{z}$. The resulting semicircular density of states has bandwidth $=4t$.\n  \nThe self-consistency condition requires the so-called Weiss field \n(i.e., the non-interacting Green's function of the AIM)\n\\begin{equation}\n\\label{hybridization}\n{\\cal G}_0^{-1}(i\\omega_n)_a=(i\\omega_n+\\mu) - \\sum_k\\frac{V_{ka}^2}{i\\omega_n-\\varepsilon_{ka}},\n\\end{equation}\nto be related to the local interacting Green's function, $G_a(i\\omega_n)$, not only \nthrough the Dyson equation for the AIM\n\\[\n{\\cal G}_0^{-1}(i\\omega_n)_a = G_a^{-1}(i\\omega_n) + \\Sigma_a(i\\omega_n),\n\\]\nwhich requires the full solution of the impurity model, but also by the additional \nselfconsistency equation\n\\begin{equation}\n\\label{selfconsistency}\n\\mathcal{G}_{0a}^{-1}(i\\omega_n)=(i\\omega_n+\\mu) -t^2G(i\\omega_n)_a.\n\\end{equation}\n\nDMFT for a given model thus amounts to solve iteratively the AIM until the impurity Green's function satisfies \nEq.~(\\ref{selfconsistency}). In this work, we solved the threefold degenerate AIM by exact diagonalization.\nThat requires truncating the sum over $k$ in Eqs.~(\\ref{aim}) and (\\ref{selfconsistency}) to a finite \nand relatively small number of baths $N_b$, so that the Hamiltonian can be \ndiagonalized in the finite resulting Hilbert space. We generally used $N_b=4$ for each orbital.\nMost of the results we shall present are at zero temperature, where we can use the \nLanczos algorithm to calculate the Green's function without fully diagonalizing the Hamiltonian. \nThe finite temperature results for the specific heat and its jump at $T_c$ reported in \nSec.~\\ref{Sec:Comparison} are the exception. They are obtained by means of the \nfinite-temperature extension of Lanczos~\\cite{caponelanczos},\nwhere the thermal Green's function is expressed as a sum over the low-lying eigenvectors \n$\\vert n\\rangle$ and eigenvalues $E_n$ of the impurity model\n\\begin{equation}\\label{G_sum_Gm}\nG_{a\\sigma}(i\\omega_n)=\\frac{1}{Z}\\sum_m e^{-\\beta E_m} G_{a\\sigma}^{(m)}(i\\omega_n)\n\\end{equation}\nwith\n\\begin{equation}\\label{G_m}\nG_{a\\sigma}^{(m)}(i\\omega_n)\\equiv \\sum_n\n\\frac{\\left\\vert \\langle n\\vert p_{a\\sigma}\\vert m\\rangle\\right\\vert ^2}{E_m-E_n-i\\omega_n} +\n\\sum_n \\frac{\\left\\vert \\langle  n\\vert p^{\\dagger}_{a\\sigma}\\vert m\\rangle\\right\\vert ^2}\n{E_n-E_m-i\\omega_n}\\,.\n\\end{equation}\nThe Boltzmann factor in Eq. (\\ref{G_sum_Gm}) guarantees that the lower the temperature, \nthe smaller the number of excited states \nthat actually contribute to the Green's function. Therefore, for sufficiently low temperature, \nwe can still use the Lanczos algorithm to find the lowest energy eigenstates.\nIn practice, only a limited number of states, $20\\div 25$, can be calculated in \na reasonable time. Unfortunately, this number is insufficient to \nmake the truncation error in Eq.~(\\ref{G_sum_Gm}) negligible for the present model \nin the relevant correlated regime. We estimate the systematic error on the \nGreen's function to be of order of a few per cent, a level of accuracy that does \nnot allow to fully determine thermodynamic properties such as the critical temperature. \nLuckily though, this error affects much more the absolute specific heat value than \nits relative changes, including the superconducting jump in units of $T_c$ \nwhich we shall discuss further below (see  Sec.~\\ref{Sec:Comparison}). We underline that \nthis limitation does not affect by any means the $T=0$ calculations.\n\nThe above DMFT equations refer to a paramagnetic phase where no symmetry breaking occurs. \nHowever, in this work we shall be crucially interested in $s$-wave superconducting \nand in antiferromagnetic solutions, where symmetry is broken.\nSuperconductivity is conveniently studied in the Nambu-Gor'kov representation \nby introducing spinors\n\\[\n\\psi_{\\mathbf{k} a} = \n\\left(\\begin{array}{c}\np^{\\phantom{\\dagger}}_{\\mathbf{k} a\\uparrow}\\\\\np^\\dagger_{-\\mathbf{k} a\\downarrow}\n\\end{array}\\right),\n\\]\n\nand defining accordingly the Green's function $G_{\\mathbf{k} a}(\\tau) = -\\langle T_\\tau\n\\left(\\psi^{\\phantom{\\dagger}}_{\\mathbf{k} a}(\\tau)\\,\\psi^\\dagger_{\\mathbf{k} a}(0)\\right)\\rangle$ in imaginary time as a $2\\times 2$ matrix \nthat satisfies the Dyson equation in Matsubara frequencies\n\n\\begin{equation}\n\\label{dyson}\nG_{\\mathbf{k} a}(i\\omega) = G^0_{\\mathbf{k} a}(i\\omega)+ G^0_{\\mathbf{k} a}(i\\omega)\\,\\Sigma(i\\omega)\\,G_{\\mathbf{k} a}(i\\omega),\n\\end{equation}\nwith $G^0_{\\mathbf{k} a}(i\\omega)$ the non-interacting value. The single-particle self-energy $\\Sigma(i\\omega)$ \nis also a $2\\times 2$ matrix whose off-diagonal element $\\Delta(i\\omega)$, when finite,  \nsignals a superconducting phase. The DMFT self-consistency can be written now as\n\\begin{equation}  \n\\label{selfsuper}  \n\\mathcal{G}_0^{-1}(i \\omega_n)= i\\omega_n \\,\\tau_0 + \\mu \\,\\tau_3 \n -t^2 \\, \\tau_3 \\,G(i\\omega_n)\\, \\tau_3,\n\\end{equation}  \nwhere $\\tau_{0}$ and $\\tau_{3}$ are Pauli matrices. \n\nAnalogously, antiferromagnetism in a bipartite lattice is conveniently described using the spinor \n\\[\n\\psi_{\\mathbf{k} a \\sigma} = \n\\left(\\begin{array}{c}\np^{\\phantom{\\dagger}}_{\\mathbf{k} a\\sigma}\\\\\np^{\\phantom{\\dagger}}_{\\mathbf{k} + \\mathbf{Q} a\\sigma}\\end{array}\\right),\n\\]\nwith $\\mathbf{k}$ in the magnetic Brillouin zone and $\\mathbf{Q}$ the modulation vector. \nThis leads once more to a $2\\times 2$ Green's function and self-energy matrices related by the Dyson equation \n(\\ref{dyson}). Here too, a finite off-diagonal element signals an antiferromagnetically ordered phase.\nThe self-consistency equation for antiferromagnetism exploits the bipartite property of the lattice. \nIndicating one sublattice with A and the other with B, the general self-consistency condition is  \n\\begin{equation}\n\\label{selfconsistencyafm}\n{\\cal G}_0^{-1}(i\\omega_n)_{\\sigma A}=(i\\omega_n+\\mu)-t^2\\,G(i\\omega_n)_{\\sigma B},\n\\end{equation}\nwhich becomes Eq.~(\\ref{selfconsistency}) if $G_{\\sigma A} \\equiv G_{\\sigma B}$, i.e. if \nthe system is nonmagnetic. When the system is antiferromagnetic, then $G_{\\sigma A}\\equiv G_{-\\sigma B}$. \nThus we can eliminate the sublattice B from Eq. (\\ref{selfconsistencyafm}), and obtain the following \nresult for the self-consistency equation: \n\\begin{equation}\n\\label{selfconsistencyafm-bis}\n{\\cal G}_0^{-1}(i\\omega_n)_{\\sigma A}=(i\\omega_n+\\mu)-t^2G(i\\omega_n)_{-\\sigma A}.\n\\end{equation}\nThis equation is valid for a Bethe lattice with nearest neighbor hopping, a case with \nperfect nesting that is rather exceptional in realistic antiferromagnets. \nA way to simulate imperfect nesting typical of more realistic situations, while still taking \nadvantage of the Bethe-lattice simplifications, is to add a next-nearest-neighbor \nhopping $t^\\prime\/z$ in the Cayley tree (merely a device to eliminate nesting, \nnot meant to suggest next nearest neighbor hopping, small in fullerides). In the limit \n$z\\to\\infty$ of the Bethe lattice, the self-consistency equation becomes\n\\begin{eqnarray}\n{\\cal G}_0^{-1}(i\\omega_n)_{\\sigma A}&=& (i\\omega_n+\\mu)-t^2\\,G(i\\omega_n)_{\\sigma B}-{t^{\\prime}}^2\\,G(i\\omega_n)_{\\sigma A}\n\\nonumber\\\\\n&=& (i\\omega_n+\\mu)-t^2\\,G(i\\omega_n)_{-\\sigma A}-{t^{\\prime}}^2\\,G(i\\omega_n)_{\\sigma A}.\n\\label{selfconsistency2}\n\\end{eqnarray}\n\nFor both broken-symmetry phases, if the diagonal elements of the self-energy matrix \nat low Matsubara frequencies  follow the conventional Fermi-liquid behavior -- which \nwe always find to be the case,  \n\\begin{equation}\n\\Sigma_{diagonal}(i\\omega_n) \\simeq \\left(1-\\frac{1}{Z}\\right)\\,i\\omega_n,\n\\label{zeta}\n\\end{equation}\nthe actual value of the spectral gap in the single particle spectrum is given by \n$Z\\,\\Delta(0)$, the zero frequency anomalous (superconducting or antiferromagnetic) \nself-energy multiplied by the so-called quasiparticle weight $Z$.\n\nWe end by noting that, within DMFT, one can search for solutions with different \nsymmetries by simply allowing\/preventing symmetry breaking order parameters, even \nin regions where the chosen phase is not the most stable. When two or more \nsolutions coexist, the stable one is determined by an explicit energy \ncalculation. As we shall discuss, for a wide range of $U\/W$ values \nwe do find coexisting superconducting and antiferromagnetic solutions,\nthe former prevailing at smaller $U$, the latter at larger $U$. \nThe physical phase diagram thus exhibits a first-order transition \nbetween these two symmetry broken phases -- a nonmagnetic $s$-wave superconductor,\nand an insulating spin 1\/2 antiferromagnet -- taking place when \nthe respective energy curves intersect.\n\n\n\\subsection{$T =$ 0 Phase Diagram}\nModeling lattice expansion of fullerides as a gradual increase of $U\/W$, \nwe can proceed to analyze the theoretical zero temperature ``phase diagram'' of our model\nobtained by DMFT as a function of $U\/W$.\nStarting with the uncorrelated system with $U = 0$ the model initially exhibits \nstraight BCS superconductivity driven by JT phonons \n(i.e., the attractive $J$ we introduced above) with an $s$-wave ($S = L = 0$) \norder parameter\\footnote{Since gauge symmetry is broken, we are allowed to assume $P_{SC}$ real}  \n\\begin{equation}\nP_{SC} = \\frac{1}{N}\\sum_i\\sum_{a=x,y,z}\\,\n\\langle p^\\dagger_{i\\,a\\uparrow}\\, p^\\dagger_{i\\,a\\downarrow}\\rangle,\n\\label{Cooper-channel}\n\\end{equation}\nwhere $N$ is the number of molecules and $p^\\dagger_{i\\,a\\sigma}$ creates an electron on molecule $i$, with spin $\\sigma$ and in orbital   \n$a=x,y,z$.  The Fermi-liquid scattering amplitude in \nthe Cooper channel, measuring the strength of\nthe effective attraction, is $A=-10 J\/3$. We note that owing to fairly strong JT interactions\n~\\cite{Manini94, Gunnarsson1995}, if Hund's exchange $J_H$ were neglected, the\ndimensionless JT coupling constant of fullerene controlling superconductivity would be \nnumerically very large, $\\lambda = \\rho_0\\vert\\,A\\, \\vert\\simeq 0.6 - 1.0$, where \n$\\rho_0$ ($\\simeq 2.4~\\mbox{eV}^{-1}$) is the bare density of states per spin and orbital.\nTurning on a weak Coulomb repulsion $U$ on top of that will reduce the pairing attraction \nin this regime. Perturbatively one obtains for small $U$ that $A= -10\\, J\/3 + U$. \nSince $J$ is insensitive to expansion while $U\/W$ increases, this implies \nthat in this picture, where Hund's rule exchange is neglected, $T_c$ should \nalways {\\it decrease} upon expansion, a prediction which is at odds with experiments. \n\nIn fact, as anticipated, Hund's rule exchange is not negligible, and its \neffect is to introduce a substantial cancellation in $J$ leading to a largely\nreduced effective coupling $\\lambda_{eff} \\simeq \\frac{10}{3}\\left(J_{JT}-J_H \\right)\\,\\rho_0 $.\nShould we interpret the ubiquitous spin gap of 0.1~eV~\\cite{Zimmer,Brouet,Brouet-1} as \nthe C$_{60}^{n-}$ molecular excitation energy between its low spin ground state \nwith $S= 1\/2$ and $S=0$ for $n=3$ and $n=4$, respectively, and its high spin excited \nstate, with $S= 3\/2$ and $S=1$ for $n=3$ and $n=4$, respectively, we would conclude \nthat, to be consistent with our model where both gaps are equal to $5J$~\\cite{Capone04},   \nthe total effective inverted exchange $J$, comprehensive of both JT and Hund's rule\nexchange, should be approximately $J=0.02$~eV. In reality, the \nqualitative scenario we will describe is relatively independent of the precise \nvalue of $J$ provided it is inverted, i.e., negative and small. \n\nOnce exchange is included, the resulting $\\lambda_{eff} \\simeq 0.16$ is now much \nsmaller -- in fact way too small to explain within conventional BCS or Migdal-Eliashberg theory  \nany of the observed values of $T_c$ in fullerides (let alone the non-monotonic behavior \nof $T_c$ versus the density of states for expanded fullerides~\\cite{Dahlke,Ganin}). \nThings get worse when we increase the on-site Coulomb repulsion $U$ closer to\nrealistic values, $U\\sim W$ and beyond. In the conventional weakly correlated picture \n$U$ would provide in the electron pairing problem a repulsive ``Coulomb pseudopotential'' whose \nbare value is $\\mu_* = U\\,\\rho_0\\simeq 3$~\\cite{BCSreview}. Simply comparing \nthese bare values of $\\lambda$ and $\\mu_*$ we should conclude that $s$-wave BCS \nsuperconductivity  in fullerenes is simply impossible (with the obvious proviso that for small $U$ \nan unretarded treatment of JT phonon interactions is not really justified).\n\nThe full DMFT solution of the model for $J=0.05~W$ and increasing $U\/W$  \nyields the phase diagram in Fig.~\\ref{phase-diagram}. While confirming the above\nexpectations for moderate $U$, it has a surprise in reserve at \nlarger $U$ values, where the Mott transition is approached.\n\n\\begin{figure}\n\t\\includegraphics[width=8.5cm]{fig2.eps}\n\t\\caption{\\label{phase-diagram} Superconducting solution (triangles) and antiferromagnetic solution (squares) \n\tzero-frequency anomalous self-energies $\\Delta(0)$ as function of $U\/W$. Solid symbols are used when the \n        corresponding symmetry broken phase is stable, while open symbols when it is metastable, i.e. has lower energy \n        then the other phase.  The first-order transition between the two phases is indicated by a \n        vertical line separating the superconductor from the antiferromagnetic Mott insulator. \n        Panel (b) corresponds to the case in the presence of a frustrating next-nearest neighbor \n        hopping $t'=0.3 t$, absent in (a).}\n\t\\end{figure} \n\nFig.~\\ref{opdelta} shows the zero frequency anomalous single-particle self-energies calculated within DMFT \nfor the superconducting and for the antiferromagnetic solution. At $U=0$, the model is \nan $s$-wave BCS superconductor, with an exponentially small superconducting $\\Delta(0)$. \nIt is too small to be visible in the figure, since the effective exchange-reduced $\\lambda \\sim 0.2$ \nis as was said very weak. Beginning from zero, the increase of $U$ first rapidly \ndestroys the weak BCS superconductivity. The superconducting $\\Delta(0)$ vanishes at \nroughly the mean-field value $U=10\/3 J$, and above this value of $U$ the ground state \nbecomes a normal metal as expected. Upon further increasing $U\/W$ the model remains \na normal metal -- no superconductivity, no antiferromagnetism. However, the importance \nof electron correlations increases with $U$, as signaled for example in the DMFT \nspectral function (not shown) by the gradual\nformation of incoherent Hubbard bands on both sides of the Fermi level. The metallic \ncharacter persists until a hypothetically continuous Mott transition eventually \nreached near $U\/W \\sim 1.5$, where $Z=0$ and the metallic character is extinguished. \n\nBefore this happens however, $s$-wave superconductivity re-enters from the normal metal\nstate. The anomalous self energy $\\Delta(0)$, proportional to the superconducting \norder parameter has, as a function of $U$, a bell-shaped behavior --\na ``superconducting dome'' as it is called in cuprates -- hitting a large maximum before dropping again. The \nre-entrant superconductive behavior is a clear realization of phonon-induced strongly \ncorrelated superconductivity (SCS)~\\cite{Science}. The sharply rising order parameter\nedge with increasing $U\/W$ can in our view explain the strong rise of $T_c$ upon \nlattice expansion in non-expanded compounds, previously (and we believe incorrectly)\nattributed to a BCS-like increase of density of states upon band narrowing. \nPast the dome maximum, and upon increasing expansion, the SCS superconducting order\nparameter declines, and would eventually drop to zero at the continuous metal \ninsulator transition near  $U\/W \\sim 1.5$. This continuous decline of superconductivity\nis preempted by a first order transition to a lower energy antiferromagnetic \nMott insulating phase, with order parameter\n\\begin{equation}\nP_{AFM}=\\frac{1}{N}\\sum_{i}(-1)^i(n_{i\\uparrow}-n_{i\\downarrow}),\n\\end{equation}\nwhere $n_{i\\sigma}=\\sum_{a=x,y,z} p^\\dagger_{i\\,a\\sigma}p^{\\phantom{\\dagger}}_{i\\,a\\sigma}$ is the full occupation number \nwith spin $\\sigma$ at molecule $i$. \nThe exact location of the superconductor-insulator transition depends on details. \nFor strong nesting ($t^\\prime$=0) it takes place even before the superconducting dome maximum. \nIn Fig.~\\ref{opdelta} we show $\\Delta(0)$ of Fig.~\\ref{phase-diagram} \nin comparison with the spectral gaps $Z\\Delta(0)$, as well as the order \nparameters $P_{SC}$ and $P_{AFM}$.  Notice that $Z$ for the superconductor is smaller and vanishes at the continuous  \nmetal insulator transition, while $Z$ for the antiferromagnet is of order 1. In both \ncases the dimensionless order parameter essentially follows the behavior of the spectral gap.\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig3.eps}\n\\caption{\\label{opdelta} Superconducting solution (SC, left) and antiferromagnetic solution (AFM, right) \nanomalous self-energies (top) and order parameters (bottom) as a function of $U\/W$. \nThe top panels also show (green diamonds) the spectral gaps obtained multiplying $\\Delta$ by the quasiparticle \nweight $Z$ of each solution. The gaps are in units of the bandwidth $W$ (the order parameters are \nby definition dimensionless). Dashed vertical lines mark the first-order \nphase transition between the two solutions.}\n\\end{figure} \n\nWhen we add a next-nearest neighbor hopping $t^\\prime$ to mimic imperfect nesting \n(as we expect to find generically for realistic band structures,  \nin particular in the face-centered cubic A$_3$C$_{60}$ materials), \nwe find that the superconducting phase is only weakly affected but antiferromagnetism is \nstrongly frustrated. As a result the superconducting region expands at expense of the \nmagnetic insulator, and the superconducting dome may emerge in full \n[panel (b) of Fig.~\\ref{phase-diagram}]. We propose that the gradual drop of superconducting\norder parameter past the dome maximum now naturally explains the decline of $T_c$\nof expanded fullerides~\\cite{Dahlke&Rosseinsky, Durand2003,Ganin}.\n\nFinally, past the first-order Mott transition, we find that the antiferromagnetic insulator \nis formed mainly by spin-1\/2 local configurations, which is in agreement with experiments \nin NH$_3$K$_3$C$_{60}$~\\cite{Iwasa}. We also predict that ambient pressure \nA15 Cs$_3$C$_{60}$, yet to be characterized, should similarly be a spin-1\/2\nantiferromagnetic insulator. Besides spin rotational symmetry, this kind of\nstate also breaks orbital rotational symmetry, signaling that spin ordering \nmust be generally accompanied by orbital ordering. In ammoniated fullerides that\nagain is consistent with experiment~\\cite{Iwasa}. We conclude that, in spite of strong \nsimplifying assumptions, our model seems able to reproduce very important features of the \nphase diagram of expanded fullerides. In the following we shall discuss in more\ndetail the strongly correlated superconducting phase near the Mott transition, and \nalso propose experiments that might distinguish it from a standard BCS state.\n\n\n\\section{Understanding Strongly Correlated Superconductivity from DMFT and the Impurity Model}\nThe re-emergence of phonon-driven superconductivity close to the Mott transition -- \nStrongly Correlated Superconductivity (SCS) -- was discussed in \\textcite{Science} \nin terms of Fermi-liquid theory. A key point of that phenomenon is the renormalization of \nthe effective bandwidth and thus of the effective mass, both controlled (in a Bethe \nlattice) by the  quasiparticle weight $Z$, Eq.~(\\ref{zeta}). $Z(U)$ decreases as a function of\n$U\/W$ and vanishes at the continuous metal insulator transition point $U=U_c$, where the effective \nmass $m^*\/m = 1\/Z(U)$ diverges, and the effective quasiparticle bandwidth $W_* = ZW$ vanishes.\nAn estimate of the interaction between charged quasiparticles requires the evolution of fluctuations that \ntake place in charge space. Because charge fluctuations are gradually frozen away near the Mott transition, \nthe effective repulsion between quasiparticles is also renormalized down to some smaller value $U_*<U$. \nIn particular, the Fermi-liquid description provided, e.g., by the Gutzwiller variational \napproach~\\cite{gutzwiller} and supported by the DMFT behavior of the average charge \nfluctuations $\\langle (n-3)^2\\rangle$, suggests that $U_*\\simeq U\\,\\Gamma_U\\,Z(U)^2 \\sim U\\,Z(U)$, where \n$\\Gamma_U$ includes all the so-called vertex corrections~\\cite{abrikosov}. This implies that  \nthe vertex function $\\Gamma_U$ diverges close to the Mott transition, but does not compensate the vanishing of $Z$~\\cite{Science}. \nThe pairwise Jahn-Teller and exchange based attraction $J$ between quasiparticles, \neven if small, here is restricted to spin and orbital space, and has nothing to do with charge\nfluctuations. In other words, Hund's rule exchange and the JT coupling only influence \nthe internal splitting of each molecular multiplet without affecting its center of gravity.  \nAs a result, this attraction should remain unrenormalized $J_* = J\\,\\Gamma_J\\,Z(U)^2 \\sim J$ close to the \nmetal insulator transition, thus implying a strongly divergent vertex correction $\\Gamma_J$ that \ncancel the vanishing $Z$. Therefore the electron pair scattering amplitude $A_*$ in the\nCooper channel should renormalize as $A = U - \\frac{10}{3}J \\rightarrow A_* = Z(U)\\,U - \\frac{10}{3}J$.\nWhen $U\/W$ is small, $Z \\lesssim 1$, the main effect of $U$ is to suppress \nsuperconductivity, as was noted earlier. However, if $U$ is close to the critical \nmetal-insulator value $U_c$,  then $Z \\sim (U_c-U)\/U_c\\ll 1$ and the scattering amplitude \nturns negative in spite of a large $U$. This is qualitatively the reason for the SCS re-entrance \nof superconductivity (though in this region of course the actual pair scattering amplitude \nmight deviate from this simple formula~\\cite{Science}). \n\nIn addition, the Fermi-liquid \nargument suggests an explanation for the large value of superconducting order parameter,\nimplying a large $T_c$, in the SCS regime, see Fig.~\\ref{phase-diagram}, compared to the $U = 0$ BCS \nvalues. In fact, when $A_* \\simeq Z\\,W$, and $Z$ is dropping sharply, the quasiparticle \nattraction $A_*$ will at some point $U=U*$ equal the coherent quasiparticle bandwidth $ZW$. That very \nuncommon situation, of metallic quasiparticles with an pair attraction equal to their\nenergy bandwidth, is known to yield maximum superconductivity for a given attraction.\nAs shown by studies of purely attractive models~\\cite{Robaszkiewicz},  the \nmaximum superconducting temperature $k_B T_c$ attainable in that case is about 7\\% of \nthe pair attraction energy itself. In our model of trivalent fullerides, this estimate yields \n$k_BT_c \\sim 0.07 A_* \\sim 0.2 |J|$, which has the correct magnitude of roughly \n40 K for $J \\sim$ 20 meV -- a value in turn fully consistent with the observed spin gap\n0.1 eV $\\sim 5 J$. While this coincidence of numbers is probably fortuitous, it does indicate \nthat orders of magnitude implied by our model with realistic parameters are quite \nconsistent with experimental facts. At face value, it also suggests that \n0.07 $|J|$ = 0.07 $(J_{JT} -J_H)$ could be the maximum attainable $k_BT_c$ in fullerides.\nWe conclude that strong correlations play a crucial \nrole in bringing the superconducting gap magnitude to the right range of values \nas compared with the experimental ones, see Fig.~\\ref{gap}. Such values and\nlarge critical temperatures would never be attained within conventional BCS theory \nusing a value of $\\lambda \\simeq 0.16-0.2$, including as it should the large cancellation \nof JT by exchange. They could in point of fact be attained if the cancellation due to \nexchange were (incorrectly) neglected; but then a lattice expansion should always \nlead to a decrease of $T_c$, contrary to experiment. \n\n\\begin{figure}[t]\n\\includegraphics[width=8cm]{fig4.eps}\n\\caption{\\label{gap} Quasiparticle superconducting energy gap in units of the bandwidth $W\\simeq 0.6$~eV and on a larger scale than \nFig.~\\ref{opdelta}, computed through \nthe anomalous self-energy, panel (a) in Fig.~\\ref{phase-diagram}, multiplied by \nthe quasiparticle residue $Z(U)$. Notice that, above $U\/W\\simeq 1.1$, \nthe superconducting solution is metastable since the antiferromagnetic one has \nlower energy, see Fig.~\\ref{phase-diagram}. We note that the maximum gap $Z\\,\\Delta_c\\simeq 0.045~W\\simeq 27$~meV.}\n\\end{figure}\n\nTo appreciate further the effect of exchange-JT cancellation, it is instructive \nto consider, as was done for a simplified model by \\textcite{Capone04}, \nthe behavior with $U\/W$ of the superconducting self-energy $\\Delta(0)$ (proportional to the \n$T=0$ gap, and roughly speaking to $T_c$) starting with pure JT and without exchange, \nand then proceeding to turn on exchange and gradual cancellation, see Fig.~\\ref{VariousJ}.  \nFor the bare JT,  $\\lambda\\simeq 1$ (a strong coupling value) the superconducting \nself-energy decreases monotonically with increasing $U$, in agreement with the \nMigdal-Eliashberg prediction of an increasing Coulomb pseudo-potential. Above a \ncritical value, the system turns directly, via a second-order or weakly first-order \nphase transition, to a Mott insulating phase. This result is fully consistent with \nprevious calculations by Han {sl et al.}~\\cite{Han2003}, where the same type of \nHubbard model was studied within DMFT at finite temperature.\nTreating explicitly the electron-phonon coupling (including the full phonon dynamics) \nwith $\\lambda=0.6$ and neglecting exchange they obtained a superconductor with \nmonotonically decreasing gap. \n\nThrough a progressive reduction of $\\lambda$ (mimicking JT cancellation by \nexchange) we find that a non-monotonic superconducting behavior makes its appearance\nas a function of  $U$.  Initially there is still a single superconducting phase for all \n$U\/W$ values; but two different regions near zero and near $U_c$ begin to materialize.\n(Note that $U_c$ simultaneously shifts to higher $U$ as $\\lambda$ decreases).\nWhen the cancellation is so strong that $\\lambda$ is still positive but small, \nthe two superconducting regions break apart to form two separate pockets, \nleaving a normal metal phase in between. In the leftmost pocket near $U\/W =0$\nthe anomalous self-energy has a BCS-like exponential dependence on $\\lambda$ \nand indeed superconductivity in this corner is BCS. Superconductivity in \nthe rightmost pocket near the metal insulator transition behaves quite \ndifferently. Here the $\\lambda$ dependence of superconductivity is much weaker, \nand the superconductive gap much stronger, than in the BCS pocket.  Superconductivity \nin this pocket can in fact be characterized as SCS~\\cite{Science}, due to \nnarrow quasiparticle pairing as described above. A similar behavior to Fig.~\\ref{VariousJ}, \nwith two separate BCS and SCS regimes emerging from a single initial one when  \nthe effective pairing attraction is progressively weakened by exchange, was derived and illustrated \nin a simpler twofold degenerate model in \\textcite{Capone04}.\n \n\n\\begin{figure}[t]\n\\includegraphics[width=8cm]{fig5.eps}\n\\caption{\\label{VariousJ} Anomalous self-energy $\\Delta(0)$ (related to the superconducting \ngap by a factor $Z^{-1}$) for the superconducting solution and different values of the coupling \nparameter $J=0.02$, $0.05$, $0.1$ and $0.2$, which correspond to \n$\\lambda=0.09,~0.21,~0.42,~0.85$, respectively. Note that for large $J$, corresponding\nto a case where the Jahn Teller coupling is not canceled by Hund's rule exchange, superconductivity\nis strongest at $U$ = 0. For increasing cancellation (decreasing $J$), two separate superconducting\npockets emerge, a BCS pocket near $U$ = 0, and a SCS pocket near the Mott insulator. When the \ncancellation is nearly complete, SCS is many orders of magnitudes stronger than BCS. This is the situation we propose in our model of fullerides. Similar physics was described for a simpler model in \\textcite{Capone04}.}\n\\end{figure}\n\nThe SCS superconducting pocket near the Mott transition is expected to differ from the BCS pocket\neven in its normal state properties.  The normal state underlying a BCS superconductor \nis Fermi-liquid-like. On the other hand, previous analysis suggest that the Fermi-liquid picture \nis likely to break down in our model when the Mott transition is approached. \nThe key reason for the breakdown of the Fermi-liquid is precisely that, when \n$Z \\rightarrow$ 0, the attraction between quasiparticles must eventually reach and exceed \nin magnitude the quasiparticle bandwidth $ZW$, a situation difficult to sustain.\\footnote{Note that the high-energy Hubbard bands are \nunaffected by the small attraction $J$; superconductivity is just a matter of \nquasiparticles~\\cite{Science}. Therefore, a quasiparticle attraction exceeding their bandwidth cannot correspond to an instability \ntowards a Mott insulator, which must involve also the Hubbard bands, but at most towards a breakdown of a quasiparticle-based Fermi liquid.} \nPossible deviations from the Fermi-liquid paradigm were\nin fact overlooked in Ref.~\\cite{Science} as they are related to the very-low energy behavior of \nthe normal phase close to the Mott transition, not explored in that work. Later, \nthe non-Fermi-liquid behavior was discovered in the two-orbital model \nwhere the physics is very similar~\\cite{Capone04}.\n\n\\subsection{Anderson Impurity With a Rigid Bath}\n\nThe DMFT calculations described above involved two steps, one solving the\nAnderson impurity model (AIM), the other making that selfconsistent with the\nbath. Following a reasoning proposed by \\textcite{DeLeo2003}, one \nmay start off with the first step alone, namely analyzing the bare AIM \nwithout imposing any self-consistency constraint. The conduction bath\ncan be assumed to have a flat density of states, and the bath-impurity hybridization\nto be structureless, a situation which avoids numerical uncertainties \nand yields accurate low-energy properties. This kind of analysis applied to the  \nAIM (\\ref{aim}) shows~\\cite{DeleoPRL} that two different impurity \nphases are stabilized according to the ratio between the attraction $J$, \nand the Kondo temperature $T_K$~\\cite{Kondo}. Below this temperature and when $J=0$, the spin of an impurity\ncoupled to a Fermi sea is screened out and absorbed in the conduction sea~\\cite{Kondo}. \nIn the lattice context within DMFT, the Kondo scale measures metallic coherence and  \ncorresponds to the renormalized quasiparticle bandwidth $ZW$. For finite $J\\not =0$ but smaller \nthan $T_K$, Kondo screening remains, thus still implying a  Fermi-liquid behavior in DMFT.\nIn fullerides, the impurity represents the C$_{60}^{3-}$ ion, carrying three\norbitals and three spins. In the Kondo phase each of the three spins is separately \nscreened by the bath and thus incorporated in the Fermi sea. \nConversely, when $J > T_K$ the Kondo screening is lost, and that \nwas shown to imply a non Fermi-liquid phase characterized by a pseudogap in the \nsingle-particle spectrum and by several other singular properties~\\cite{DeleoPRL}.  \nA very qualitative description of this phase is that, unlike the Kondo phase, \ntwo spins out of three pair off antiferromagnetically at any given time, leaving \nout a single spin 1\/2 available for Kondo screening. However, since orbital degeneracy is unbroken, \nthis residual spin $S=1\/2$ also carries orbital momentum $L=1$, which corresponds to an overscreened \nnon-Fermi liquid situation~\\cite{DeleoPRL}. \nIn this regime, it was predicted that the impurity contributions\nto the specific-heat coefficient and to the pair susceptibility in the $s$-wave \nchannel (\\ref{Cooper-channel}) diverge as $T^{-1\/5}$ at low temperature $T$. \nIn addition, the conduction electron scattering rate has a non-analytic \ntemperature-behavior $T^{2\/5}$. The local response functions to either\na quadrupolar field, which splits the orbital degeneracy, or to a magnetic \nfield which polarizes both spins and orbitals also diverge as $T^{-1\/5}$.     \nThe two phases, the Kondo screened phase and the pseudogap phase, are separated \nby a critical point at $J = J_{cp} \\simeq T_K$. It is endowed with a finite entropy \n$1\/2\\,\\ln 3$, and with a divergent superconducting susceptibility with an exponent $1\/3$. \nAs shown in Fig.~\\ref{DOS}, the single-particle spectral function displays strong \ndeviations from a normal metal in the pseudogap phase and at the critical point.\n\n\\begin{figure}[t]\n\\includegraphics[width=7cm]{fig6.eps}\n\\caption{\\label{DOS} Impurity spectral function with \na rigid bath characterized by a flat density of states (no DMFT self-consistency) across \nthe critical point: the curves from top to bottom \ncorrespond to the evolution from the Kondo screened regime towards the pseudogap regime. \nThe dotted line is the spectral function \ndeep inside the non-Fermi-liquid phase $J\\gg T_K$. The top inset shows the same curves plotted on a \nlarger energy range, where the pseudogap of the dotted line is visible. The Fermi liquid\nbehavior corresponds to the normal state of the SCS superconducting phase for $J < J_{cp}$ \n(analogous to the overdoped regime of cuprates) whereas the pseudogap behavior \ncorresponds to the SCS normal state for $J > J_{cp}$ (analogous to the underdoped cuprates). \nFigure from \\textcite{DeleoPRL}.}\n\\end{figure}\n\nNear this critical point it has been found that the low-energy dynamics around the impurity is controlled \nby two separate energy scales~\\cite{DeleoPRL}, $T_+$ and $T_-$, \nwhose behavior is very different as a function of increasing $U\/W$. \nA higher energy scale $T_+$ is set by the critical $J_{cp}\\sim T_K$ and represents \nthe width of a broad incoherent resonance; it evolves smoothly and uneventfully\nas the critical point is crossed (See Fig.~\\ref{DOS}). A lower energy scale \n$T_-\\propto |J-T_K|^3$ measures instead the distance from the critical point, \nand leads simultaneously to a narrow resonance in the Fermi-liquid region, and to \nan equally narrow spectral density dip (the ``pseudogap'') in the pseudogap \nphase~\\cite{DeleoPRL}. Since in this phase the impurity still carries a residual \nspin-1\/2, there remains a finite value of the spectral function at the chemical \npotential, and the gap is not complete. The pseudogap widens if $J$ is increased,  \nand the cusp-like dip in the impurity spectral function smoothly turns into a \ncusp-like peak, the value at the chemical potential staying fixed and constant. \nThis behavior is shown by the dotted curve in Fig.~\\ref{DOS} corresponding to a \nvery large pseudogap (see top inset), possessing a very tiny peak at the \nchemical potential. This indicates the existence of yet another energy scale \nbesides $T_+$ and $T_-$ that sets the width of the cusp peak. \n\n\n\\subsection{Anderson Impurity with a Self-Consistent Bath in DMFT}\n\nThe rigid bath AIM behavior and its critical point briefly reviewed above \nprovide a guide to the DMFT results once the impurity-bath coupling is self-consistently \ndetermined. First of all, since $T_K$ coincides within DMFT with the renormalized $ZW$ \nwhich in turn vanishes when the continuous Mott transition is approached, \nthe impurity critical point is inevitably met before the metal insulator transition\nas $U\/W$ is increased, at some $U_{cp} \\alt U_c$. This entails several important consequences:\n\n\\begin{itemize}\n\\item The normal state may be a Fermi liquid only far below the continuous metal insulator\ntransition, such as perhaps may be the case in the non-expanded fullerides. Expanded \ncompounds on the other hand are expected to have a non Fermi-liquid normal state, \nand eventually a pseudogap, possibly developing before the first order transition \nto the antiferromagnetic insulator.\n\n\\item The SCS superconducting pocket near the Mott transition reflects the leading \ninstability of the impurity critical point. In other words SCS superconductivity \nis the way in which the lattice model responds to impurity criticality and avoids it.\n\n\\item The lower energy scale vanishes right at the critical point, $T_-=0$. Here \n$T_+\\simeq J$ thus remains as the only energy scale controlling the magnitude of the \nsuperconducting energy gap. Away from $U=U_{cp}$, $T_-\\not =0$, and the amplitude of the \nsuperconducting gap should decrease monotonically with $(T_+-T_-)\/T_+$~\\cite{Capone04}, since \n$T_-$ cut-offs the local pairing instability. Therefore the gap should be maximum right at the impurity critical point (the top of the dome).\n\n\\item Even though the normal phase is non-Fermi liquid, well defined Bogoliubov \nquasiparticles should exist inside the SCS superconducting pocket. \n\n\\end{itemize}\nThe last statement comes from the fact that, at the impurity critical point, superconductivity \nprovides a new screening channel which helps the system get rid of the finite \nresidual entropy at the critical point, thus eliminating non-Fermi liquid singularities~\\cite{DeleoPRL,SchiroPRB}. \n\n\\begin{figure}[t]\n\\includegraphics[width=7cm]{fig7.eps}\n\\caption{\\label{Drude} Calculated weights of the zero-frequency delta-function contribution \nto the optical conductivity for the superconducting phase (superfluid stiffness, squares) \nand the normal phase (Drude weight, circles). The zero-frequency anomalous self-energy is also plotted.\nNote the higher values in the SCS superconductor, relative to the nonsuperconducting metal phase.}\n\\end{figure}\n\nThe role of superconductivity as a novel screening channel close to the \nimpurity model critical point should reflect, in the lattice model, into  \na gain of band energy (the tight binding ``kinetic energy'') at the onset of superconductivity. \nOwing to a sum rule connecting kinetic energy and zero-frequency optical\nconductivity (sometimes referred to as ``Drude weight'')~\\cite{Drude}\nthe onset of SCS superconductivity close to the Mott transition\nshould lead to a Drude weight increase. This prediction is \nwell borne out by the full DMFT solution of our Hamiltonian.\nIn Fig.~\\ref{Drude} we plot the $\\omega=0$ (d.c.) optical conductivity of \nour model superconductor (where it coincides with the superfluid stiffness), defined by~\\cite{toschi}\n\\begin{equation}\nD_s= -E_{kin} + \\chi_{jj}({\\bf q} \\rightarrow 0, \\Omega=0),\n\\label{ds}\n\\end{equation}   \nwhere $E_{kin}$ is the kinetic energy and $\\chi_{jj}$ the static limit of the paramagnetic\npart of the electromagnetic kernel\n\\begin{eqnarray}\n\\chi_{jj} & = & \\frac{2}{\\beta} \\sum_{n} \\int \\, d\\epsilon \\, N(\\epsilon) \\, \nV(\\epsilon) \\nonumber\\\\\n&\\times & \\left[ G(\\epsilon,\\omega_n) G^{*}(\\epsilon, \\omega_n) +\nF(\\epsilon,\\omega_n)F(\\epsilon,\\omega_n) \\right], \n\\label{chijj} \n\\end{eqnarray}         \nwhere $V(\\epsilon)=(4t^2-\\epsilon^2)\/3$ is the current vertex in the Bethe lattice, while $G(\\epsilon,\\omega_n)$ and  \n$F(\\epsilon, \\omega_n)$ are  the normal and anomalous lattice Green functions, respectively. \nIn the same figure we also plot the zero-frequency d.c. conductivity (the Drude weight) of the  \nunderlying metastable solution where superconductivity is \ninhibited, this state meant to provide a cartoon of the real normal phase above $T_c$. \nThe Drude weight is given by Eq. (\\ref{ds}) and \n(\\ref{chijj}) with $F(\\epsilon,\\omega_n)\\equiv0$. Upon entering the SCS dome \nfrom the low $U\/W$ side, the superconducting phase initially loses kinetic energy \nover the normal state as in ordinary BCS theory. However, upon increasing $U\/W$ at \nand beyond the dome maximum, the loss reverts over to a gain, and indeed \nmost of the SCS superconducting pocket is predicted to have a larger \nweight of the zero-frequency optical absorption than the non superconducting state.\nThe same behavior is displayed (Fig.~\\ref{kinetic.vs.potential}) by the energy balance between \nthe two solutions. Only far below the Mott transition the superconductor is stabilized \nby a potential energy gain, as is the case in BCS theory. In the pseudogap regime,\ncorresponding to the expanded fullerides near the Mott transition, the stabilization \nis associated with a kinetic energy gain.\n\\begin{figure}[t]\n\\includegraphics[width=7cm]{fig8.eps}\n\\caption{\\label{kinetic.vs.potential} Energetic balance underlying superconductivity.  $\\Delta E_{pot} = E^{S}_{pot} - E^{N}_{pot}$ \nis the difference between the potential energies of the superconducting and the normal solution, \nwhile $\\Delta E_{kin} = E^{S}_{kin} - E^{N}_{kin}$ is the same difference between the kinetic energies of the two solutions.}\n\\end{figure}\n\nA similar phenomenon is well known in the optical conductivity of \nhigh-T$_c$ copper oxides~\\cite{VanDerMarel}. Our calculations show that an \nincrease in the zero-frequency optical conductivity or a kinetic \nenergy gain do not actually exclude an electron-phonon pairing mechanism, \nbut rather demonstrates the key importance of strong electronic correlations. \nThanks to pairing, the motion of carriers in the superconducting phase is \nfacilitated with respect to the pseudogap nonsuperconducting metal. In that\nanomalous metal -- a non Fermi liquid-- the interaction constraints jam the \nfree propagation of quasiparticles, causing a kinetic energy cost, partly released\nwith the onset of superconductivity. It would be of extreme interest \nif the optical conductivity increase demonstrated in cuprates could be investigated\nin fullerides, both regular and expanded, since that would help \ndiscriminate between conventional BCS and SCS.\n\n\\begin{figure}[b]\n\\includegraphics[width=8cm]{fig9.eps}\n\\caption{\\label{specific-heat-jump} Specific heat jump, in units of \n$T_c$, $\\gamma_0$, as function of $U\/W$. For comparison we also show the behavior \nof $1\/Z(U)$, which should correspond to the same quantity if Fermi liquid theory were \nvalid.}\n\\end{figure}\n\n\n\\section{Discussion of Experimental Probes}\n\\label{Sec:Comparison}\nIn the following we discuss whether and how the dominance of strong correlation \nwe propose for the superconducting fullerides can be reconciled with the list of experimental facts \ngiven earlier, apparently supporting a conventional BCS behavior. The first quantity \nwe discuss is the specific heat jump at $T_c$. Within BCS theory, the specific \nheat jump $\\Delta C_V$ at $T_c$ is approximately \n\\begin{equation}\n\\frac{\\displaystyle \\Delta C_V }{\\displaystyle T_c} \\simeq 1.52\\,\\gamma_*,\n\\label{heat-jump}\n\\end{equation}\nwhere $\\gamma_*$ is the specific heat coefficient of the metallic phase ($C_V = \\gamma_* T$)), \nproportional to the mass enhancement $m^*\/m$, in our case $1\/Z$.  \nThe measured specific heat jump leads, through (\\ref{heat-jump}), to an estimate \nof $\\gamma_*\\simeq 3\\gamma_0$. This is indeed a rather low and regular value which has been \ncommonly advocated as evidence of weak correlations in fullerenes. However, the standard \nargument only holds provided the normal phase is Fermi-liquid, which is not applicable \nclose to a Mott transition, independently of any models. \n\nIn Fig.~\\ref{specific-heat-jump} we plot the jump in $C_V$ at $T_c$ for \nincreasing $U\/W$, compared with the Fermi-liquid estimate $1\/Z$.\\footnote{As mentioned in Sec. \\ref{Sec:dmft}, \nfor the present three-orbital model it has not been practical \nto include in Eq. (\\ref{G_sum_Gm}) a number of excited states sufficient to make the truncation error \nin the Green's function negligible. \nFor the data reported in Fig.~\\ref{specific-heat-jump} we have an average error of 3-4\\% \nwhich does not allow us to determine $T_c$ with sufficient accuracy. Yet, the jump of the specific heat relative to $T_c$ turns out to be almost \nindependent on the truncation error and on the details of the calculations.}\nAfter a region  \nwhere the calculated quantity closely follows $1\/Z$, $\\Delta C_V\/T_c$ flattens out \nand stays roughly constant around  $4\\,\\gamma_0$ up to the Mott \ntransition, despite a diverging $1\/Z$ (See Fig.~\\ref{specific-heat-jump}). \nThis shows that the energy scale that controls the superconducting instability \nis constant near the Mott transition, consistent with the single-impurity prediction \nthat this scale should be identified by  $T_+$, a quantity of order $J$. \nThe bottom line conclusion here is that a normally sized specific-heat jump does not imply\nBCS superconductivity in fullerides.\n\nAnother physical quantity which seemingly pointed towards weak \ncorrelations in fullerides is the magnetic susceptibility measured in the normal \nphase, apparently consistent with a weakly correlated a Fermi liquid and \na Stoner enhancement of about a factor 2 to 3. \nThis argument again becomes inconclusive once the Fermi-liquid \nscenario is abandoned. In Fig.~\\ref{chi} we plot as function of $U\/W$ the \nuniform magnetic susceptibility in the normal phase calculated by DMFT. \nAfter an initial Stoner enhancement at small $U\/W$, the susceptibility flattens out \nand remains almost constant before a rapid growth which takes places extremely \nclose to the Mott transition. In the plateau region the susceptibility \nenhancement of a factor between 2 and 3 with respect to $U=0$, surprisingly \nclose to the experimental enhancement, covering the whole superconducting domain. \nPhysically, the origin of this susceptibility plateau for increasing $U$\nis quite instructive. It corresponds to the gradual crossover of the maximum \nspin available at each site from S=3\/2 in the Fermi liquid at small $U < U_{cp}$\n(Kondo screened AIM) to S=1\/2 near Mott and large $U > U_{cp}$, where Fermi liquid \nbehavior is lost. In essence, at large $U$ each molecule is effectively in a (dynamically) \nJT distorted state, where two out of three electrons are spin paired~\\cite{Manini94}. \nWe are thus led to conclude that the relatively weak observed enhancement of susceptibility \ndoes not correspond at all to a Stoner enhanced, weakly correlated Fermi liquid -- \nin fact quasiparticles do not even exist in most of the plateau region.\n \n\\begin{figure}\n\\includegraphics[width=7cm]{fig10.eps}\n\\caption{\\label{chi} Normal-phase uniform magnetic susceptibility $\\chi$ normalized to the \nnon-interacting value $\\chi_0$. For $U=0$, $\\chi\\simeq \\chi_0$, the difference being \nextremely small since $J\\ll W$. The plateau between $U\/W$ = 0.5 and 1 signals the \neffective crossover from a Fermi liquid with $S$= 3\/2 per site, to a non Fermi liquid\nwith $S$=1\/2 per site. }\n\\end{figure} \n\n\nFinally, we wish to address signatures of the strongly correlated scenario which \nwe expect should show up in important spectroscopies including \nthe tunneling I-V characteristics and angle resolved photoemission spectroscopy (ARPES). \nThis is initially embarrassing on two accounts. First, ARPES spectroscopies are $\\mathbf{k}$-vector resolved, whereas\nin DMFT we do not have access to any spatial structure. Second, tunneling spectroscopies \nare extremely well resolved near zero voltage, whereas our Lanczos method yields\na much poorer spectral function resolution in this region. \n\nLet us addressing tunneling first. Although Fig.~\\ref{DOS} refers to the impurity \nC$_{60}^{3-}$ molecule, we believe that similar features would remain after full DMFT \nself-consistency in the normal phase -- if we only had a better low-frequency numerical resolution \nthan we presently have. Hence, we suggest that tunneling I-V spectra of expanded \nfullerides be measured and examined, in order to bring out the expected rich structure \nof the kind sketched in Fig.~\\ref{DOS}. \n\nNext, let us consider photoelectron spectroscopy. Again according to the single-impurity \nanalysis~\\cite{DeleoPRL}, the imaginary part of the single-particle self-energy should \nbe finite and of order $T_+$ almost everywhere in the non-Fermi liquid normal phase above $T_c$. \nThis has the following implications for ARPES:\n\n\\begin{itemize}\n\n\\item the fulleride photoemission spectrum should show $t_{1u}$ bands dispersing \nin the Brillouin zone with nonzero bandwidth, governed by the energy scale $T_+$. \nThe value of $T_+$ decreases with increasing $U\/W$ (increasing expansion), from \n$W$ at $U$ = 0 to (larger than) $J$ at the Mott transition; \n\n\\item there should be a spectral peak broadening of the same order \nof magnitude $T_+$ as the apparent $\\mathbf{k}$-resolved band dispersion. In particular the broadening \nshould remain constant approaching the Fermi surface -- unlike a Fermi liquid phase \nwhere quasi-particle peaks become narrower and narrower.\n\n\\end{itemize}\n\\begin{figure}\n\\includegraphics[width=7cm]{fig11.eps}\n\\caption{\\label{Arpes} Simulated photoemission spectra elaborated form the \nDMFT result in the normal phase. The left panel refers to $U\/W = 1.1$ (corresponding \nto unexpanded fullerides), while the right panel is for $U\/W = 1.3$ (corresponding to \nexpanded fullerides).\n}\n\\end{figure} \n\nThe momentum-independence of the DMFT self-energy implies that, in this approximation, the \n$\\mathbf{k}$-modulation of the electronic dispersion is assumed to remain unaffected by interactions. \nWithin this assumption and without requiring too high frequency-accuracy, we can compute a toy $\\mathbf{k}$-resolved spectral function according to \n\n\\begin{equation}\n\\label{eq:arpes}\nA(k,\\omega)= -\\frac{1}{\\pi}  \n\\mbox{Im} \\frac{1}{\\omega-\\varepsilon_{\\bf k} -\\Sigma_{DMFT}(\\omega)},\n\\end{equation}\n\nwhere $\\varepsilon_{\\bf k}$ is the non interacting dispersion and $\\Sigma_{DMFT}(\\omega)$ is the DMFT self-energy calculated with a finite \nnumber of baths. The effect \nof the local self-energy will be to change the effective bandwidth and to give \nrise to finite lifetime effects, even if the $\\mathbf{k}$-modulation of the dispersion \nis unrenormalized. For our Bethe lattice, there is no straightforward definition \nof momentum, and we remedy that by computing $A(\\varepsilon,\\omega)$, \nwhich corresponds to Eq. (\\ref{eq:arpes}) with $\\varepsilon_{\\bf k} \\to \\varepsilon$.\n\nIn Fig.~\\ref{Arpes} we show theoretical ARPES results for some choices of \n$\\varepsilon$ obtained by using the DMFT self-energy for temperatures above \n$T_c$, both for a value of $U\/W =1.1$ which lies close to the maximum of the \nsuperconducting dome, but still on the less correlated side, corresponding \nto unexpanded (or moderately expanded) fullerides, and for a value which \nlies in the downward branch of the dome ($U\/W = 1.3$), corresponding to a\nvery expanded fulleride. We note in both cases the existence of an incoherent \nlow-energy feature dispersing with a reduced but nonzero electron bandwidth \nof $0.1~W$. In the expanded case the pseudogap feature is clearly present.\n\nRecent photoemission spectra of K$_3$C$_{60}$~\\cite{Goldoni} indicate an\noverall dispersion bandwidth of about 160~meV, about a quarter of the bare \ncalculated bandwidth in the local-density approximation. The experimental spectral \npeak does not show the usual Fermi-liquid-like narrowing on approaching the Fermi \nlevel, a fact which is in agreement with our expectation for a non Fermi liquid \n(although a non expanded fulleride like K$_3$C$_{60}$ probably lies at the \nbeginning of the SCS dome, where deviations from Fermi liquid are not massive). \nExperimentally the peak does not appear to cross the Fermi level, and the \nintensity instead drops, suggestive of a pseudogap. Unfortunately the spectrum\nshows very strong vibronic effects, reflecting the retarded strong electron Jahn Teller \ncoupling. This aspect is not covered by our unretarded approximation, but it\nheavily affects the line shape and hampers the extraction of purely electronic features. \nTreatment of the vibronic effects, and a quantitative description\nof dispersion, will require abandoning in the future our approximation of infinitely \nfast phonon dynamics, as well as a possible extension to cluster extensions of DMFT which \nallow for different renormalizations of different momenta~\\cite{cdmft}.\n\n\n\\section{Conclusions}\nSummarizing, we addressed the apparently contradictory properties\nof expanded trivalent fullerides superconductors and insulators -- and to some extent \nof the whole family of fullerides -- and presented \na theoretical scenario emphasizing the role of strong electron correlations. \nThat is especially designed and appropriate for the more expanded\nmembers of the family, such as (NH$_3$)$_x$NaK$_2$C$_{60}$, Li$_3$C$_{60}$, \nCs$_{3-x}$K$_x$C$_{60}$ and Cs$_{3-x}$Rb$_x$C$_{60}$, and the recently discovered A15 $Cs_{3}C_{60}$ \nthat are near or past the Mott transition.\n\nOur model explains the dome-shaped increase and subsequent decrease of $T_c$ \nupon expansion of the lattice spacing in fullerides; the coexistence \nof metallic behavior and of Mott insulator features such as the large NMR spin \ngap in all fullerides, and the $S=1\/2$ spin in the insulating state (identified \nas a Mott-Jahn-Teller insulator). It explains why the $s$-wave $T_c$ can be \nas high as 40~K even though the Coulomb interaction strength is prohibitive, and\nwhy $T_c$ does not automatically decrease upon increase of $U\/W$. It also accounts \nfor more standard observations, such as regular specific heat jumps and moderately \nhigh spin susceptibilities, facts that were so far construed as evidence for conventional \nBCS superconductivity. \n\nBesides those listed in the previous Section, one can anticipate a number of additional experiments \nthat could provide ``smoking gun'' evidence for strongly correlated superconductivity\nin fullerides. The tunneling $I$-$V$ characteristics observable, e.g., by a scanning tunneling spectroscopy tip \nshould, in an expanded fulleride, develop the low energy\nfeatures typical of the Kondo impurity spectral function. The isotope effect\nupon carbon substitution should also behave very unconventionally, and eventually get \nsmaller as the superconducting dome is passed and the Mott transition is approached \nupon expansion. In this regime, as the quasiparticle bandwidth $ZW$ gradually falls below\nthe typical energy $\\hbar\\omega$ of an increasing fraction of the eight $H_g$ \nJahn Teller modes, the associated retardation effect should in fact disappear.\nThe expanded fullerides and related materials, clearly not enough\ninvestigated so far, deserve in our view the strongest experimental attention.\nThey combine elements that make them members at large of the high temperature \nsuperconductor family. They combine neighborhood of the Mott transition and predominance \nof strong electron correlations, with conventional elements such as electron-phonon $s$-wave \npairing, that are typical of BCS systems. Our study identifies a pseudogap and other \nfeatures in the IV tunneling spectrum, an increase of zero-frequency optical weight in the optical \nresponse of the superconducting phase, and the emergence of two separate energy scales \nin ARPES as the most urgent experimental undertakings that could confirm \nof falsify our claims.\n\n\n\\section*{Acknowledgments}\nE.T. thanks Kosmas Prassides, Andrea Goldoni, and Mauro Ricco' for information \nand discussion about fullerides and again K.P. for providing us with Fig. 1. \nM.C. acknowledges discussions with A. Toschi. Work in SISSA was sponsored by PRIN \nCofin 2006022847, as well as by INFM\/CNR ``Iniziativa trasversale calcolo parallelo''. \nWork in Rome was sponsored by PRIN Cofin 200522492. \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe study of the kinetics of diffusion-controlled or \nballisticaly-controlled\nannihilation provides a nice playground on which the tools of \nnonequilibrium statistical mechanics can be tested ~\\cite{redner,born}. \nParticularly in low dimensions, the kinetics of these systems is \ngoverned by the fluctuations and any mean-field like (or Boltzmann-like) \napproximation will not give a satisfactory description of the properties \nof the system.\n\nA lot of efforts have be expended in studying the diffusion-controlled \ncase, both for the situation in which the reactants are uniformely \ndistributed in space~\\cite{wil} and\nfor the one in which they are initially spatially separated, leading to \nreaction-diffusion fronts~\\cite{racz,cordro}. In both cases, the long \ntime behavior of physical observables is characterized by power law. The \nassociated exponents have some universal properties~\\cite{born}.\n\nMore recently, the case of ballisticaly-controlled annihilation has been \ninvestigated for spatially homogeneous conditions~\\cite{reddue}. In \nparticular, exact results have been obtained for one-dimensional, single \nspecies models by Piasecki~\\cite{pia_uno} and Droz, Rey,  Frachebourg \nand Piasecki~\\cite{pia_due,pia_tre}.\n\nIn these models, one considers point particles which move freely, with a\ngiven velocity.  When two particles collide they instantaneously \nannihilate \neach other and disappear from the system. The system with only two \npossible\nvelocities $+c$ or $-c$ has been studied in a pioneering work by Elskens \nand\nFrisch~\\cite{ef}. The case of an arbitrary discrete velocity \ndistributions has been investigated by  Droz, Rey,  Frachebourg and \nPiasecki~\\cite{pia_tre}. An exact equation for  the survival probability \nuntil the time $t$ of a particle moving with velocity $v$ was derived, \nallowing to compute the density and the time dependent velocity \ndistribution in the asymptotic regime $t\\to\\infty$.\nFor a symmetric three velocities distribution,\ndifferent kinetic regimes were found as a function of the initial \nfraction of particles at rest. \nSuch processes can model several physical situations as  a recombination \nreaction in the gas \nphase or the fluorescence of laser excited gas atoms\nwith quenching on contact (the one-dimensional aspect can be obtained \nby working in a suitable porous media~\\cite{kopel}) or, \nthe annihilation of kink-antikink pairs in solid state \nphysics\\cite{buti}.\n\nA related but different class of problems is  the case in which the two \nspecies (called $A$ and $B$), are initially separated in space and \nannihilate on contact. Such process can model the situation in which \nchemical species incorporated in a gel move ballistically under the \naction of a drift~\\cite{zrini}. As the two species cannot penetrate one \ninto the other (as they annihilate on contact), a well defined reaction \nfront is formed.\nWe aim at computing exactly the time dependent properties of this front.\n\nThe paper is organized as follows. In section two, the model is defined.\nIn section 3, we compute the average number of collisions that have \noccured before time $t$. An exact analytic expression for the \nprobability density to find the front at a given point and a given time \nis derived.\nIn section 4, it is shown that the front makes a random walk around its \n(time dependent) average value.\nPossible extensions of this work are discussed in the conclusion.\n\n\n\\section{The model}\n\nWe consider a one-dimensional system formed of two species of particles.\nInitially, particles $A$ are spatially randomly  distributed in the  \nregion $(-\\infty, 0)$\nand the $B$ ones are spatially randomly distributed in the  region $(0, \n\\infty)$. The positions of the particles obey a Poissonian distribution. \nThe\ninitial linear average densities for the $A$ and $B$ particles are \nrespectively $\\mathrel{\\rho^{\\mathrm{A}}}$ and $\\mathrel{\\rho^{\\mathrm{B}}}$.\nThe velocities of each  particle is an independent random variable \ntaking the value $\\pm c$ with even probability. Particles of the same \nkind suffer elastic collisions. When two particles $A$ and $B$ meet, \nthey annihilate.\nThus, practically the $A$ particles with velocity $-c$ and the $B$ \nparticles with velocity $+c$ move freely. Accordingly, the relevant part \nof the dynamics concerns the $A$ particles with velocity $+c$ and the \n$B$ ones with velocity $-c$.\n\nLet $(y_1,y_2,\\ldots,y_k,\\ldots)$ be the initial positions of the \n$A$ particles with velocity $+c$ and \n$(x_1,x_2,\\ldots,x_k,\\ldots)$ the initial positions of the $B$'s with \nvelocity $-c$ (see figure~\\ref{fig:one}). The relative velocity between \nthe $A$ and \n$B$ particles being $2c$, the pair of particles initially at $(x_k, \ny_k)$ will collide at time $t_k=(x_k-y_k)\/2$.\n\\begin{figure}[tbh]\n\\setlength{\\unitlength}{1cm}\n\\centerline{\n\\begin{picture}(12,3.5)(-6,-1)\n\\put(-5,0){\\line(1,0){10}}\n\\put(0,1){\\vector(-1,0){3}}\n\\put(-2.5,1.1){$A$ particles}\n\\put(0,1){\\vector(1,0){3}}\n\\put(2.5,1.1){\\makebox(0,0)[br]{$B$ particles}}\n\\put(0,0.9){\\line(0,1){0.2}}\n\\put(0,-0.2){\\line(0,1){0.4}}\n\\put(0,-0.3){\\makebox(0,0)[t]{$0$}}\n\\put(-0.5,0){\\circle*{0.1}}\n\\put(-0.5,-0.3){\\makebox(0,0)[t]{$y_1$}}\n\\put(-1.3,0){\\circle*{0.1}}\n\\put(-1.3,0.3){\\makebox(0,0)[b]{$y_2$}}\n\\put(-1.7,0){\\circle*{0.1}}\n\\put(-1.7,-0.3){\\makebox(0,0)[t]{$y_3$}}\n\\put(-2.7,-0.3){\\makebox(0,0)[t]{$\\cdots$}}\n\\put(-3.7,0){\\circle*{0.1}}\n\\put(-3.7,-0.3){\\makebox(0,0)[t]{$y_{k-1}$}}\n\\put(-4.4,0){\\circle*{0.1}}\n\\put(-4.4,0.3){\\makebox(0,0)[b]{$y_k$}}\n\\put(-5.1,0){\\makebox(0,0)[r]{$\\ldots$}}\n\\put(0.8,0){\\circle*{0.1}}\n\\put(0.8,-0.3){\\makebox(0,0)[t]{$x_1$}}\n\\put(1.1,0){\\circle*{0.1}}\n\\put(1.1,0.3){\\makebox(0,0)[b]{$x_2$}}\n\\put(1.8,0){\\circle*{0.1}}\n\\put(1.8,-0.3){\\makebox(0,0)[t]{$x_3$}}\n\\put(2.6,-0.3){\\makebox(0,0)[t]{$\\cdots$}}\n\\put(3.4,0){\\circle*{0.1}}\n\\put(3.4,-0.3){\\makebox(0,0)[t]{$x_{k-1}$}}\n\\put(4.2,0){\\circle*{0.1}}\n\\put(4.2,0.3){\\makebox(0,0)[b]{$x_k$}}\n\\put(5.1,0){\\makebox(0,0)[l]{$\\ldots$}}\n\\end{picture}}\n\\caption{Typical initial condition. The particles are labeled such that \n$y_k<\\cdots<y_1<0<x_1<\\cdots<x_k$. \\label{fig:one}}\n\\end{figure}\n\nThe position at which this collision will take place  defines the \nposition of the annihilation front.\nThus, the position of the \nfront at time $t$ is $\\half(x_1-y_1)$, if $2ct\\leq \nx_1+y_1$ (i.e. if no particle have yet collided), $\\half(x_2-y_2)$, if \n$x_1+y_1\\leq 2ct\\leq x_2+y_2$ (i.e.  only \nthe first particles on the right and on the left have collided), and so \non. \nThe dynamics in itself is purely deterministic, the only stochastic \naspect comes from the initial conditions.\nThe properties of the front are completely defined by the  \nprobability density $\\mu(X;t)$ to find the front at the point $X$, \nat time $t$. Moreover, another quantity of interest is  $N(t)$, the \nnumber of  particles that have been annihilated until a given time $t$.\n\n\\section{Analytical solution}\\label{sec:method}\n\nWe first investigate the behavior of $N(t)$.\nFor this purpose, we introduce   the probability $P(k;t)$\nthat exactly $k$ collisions have occured within the time interval \n$[0,t]$. For a given initial configuration, at least $k$ collisions have \noccured before the time $t$, if $2ct\\geq x_k-y_k$, and  at most $k$ \ncollisions have occured before $t$, if $2ct\\leq x_{k+1}-y_{k+1}$, \nwith $x_k<x_{k+1}$ and $y_k>y_{k+1}$. Hence averaging over the initial \nconditions, we find:\n\\begin{eqnarray}\nP(k;t) &=& \\langle\\theta\\left(2ct-[x_k-y_k]\\right)\n                  \\theta\\left([x_{k+1}-y_{k+1}]-2ct\\right)\\rangle        \n           \\nonumber\\\\\n       &=& \\langle\\theta\\left(2ct-[x_k-y_k]\\right)\\rangle -\n           \\langle\\theta\\left(2ct-[x_{k+1}-y_{k+1}]\\right)\\rangle\\, .\n           \\label{eq:kcoll}\n\\end{eqnarray}\nwhere $\\theta$ is the usual Heaviside function and the brackets denote \nthe average over the initial conditions (see Appendix~\\ref{app:B}).\nThus:\n\\begin{equation}\nN(t)=\\sum_{k=1}^\\infty k P(k;t)\\, ,\n\\end{equation}\nwhich, using (\\ref{eq:kcoll}), reduces to \\begin{equation}\nN(t)=\\sum_{k=1}^\\infty \\langle\\theta\\left(2ct-[x_k-y_k]\\right)\\rangle\n\\, .\n\\end{equation}\nThus, using the results of Appendix~\\ref{app:B}, $N(t)$ reads:\n\\begin{eqnarray*}\nN(t)=\\sum_{k=1}^\\infty \\int_{-\\infty}^0 \\d y_k\\mathrel{\\rho^{\\mathrm{A}}} \\int_0^\\infty \\d x_k\n&\\mathrel{\\rho^{\\mathrm{B}}}& \n     \\e^{-\\mathrel{\\rho^{\\mathrm{A}}}|y_k|}\\frac{\\left(|y_k|\\mathrel{\\rho^{\\mathrm{A}}}\\right)^{k-1}}{(k-1)!} \\\\\n&\\times&\n     \\e^{-\\mathrel{\\rho^{\\mathrm{B}}} x_k}\\frac{\\left(x_k\\mathrel{\\rho^{\\mathrm{B}}}\\right)^{k-1}}{(k-1)!}\n     \\theta\\left(2ct-[x_k-y_k]\\right)\\, .\n\\end{eqnarray*}\nNoticing that the sum over $k$ can be exchanged with the two integrals, \nwe \nfind (where $a=\\half(x_k-y_k)$ and $b=\\half(x_k+y_k)$)\n\\[\nN(t)=2\\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}} \\int_0^{ct} \\d a \\int_{-a}^a \\d b\n     \\e^{-a(\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}})} \\e^{b(\\mathrel{\\rho^{\\mathrm{A}}}-\\mathrel{\\rho^{\\mathrm{B}}})}\n     I_0\\left(2\\sqrt{\\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}}(a^2-b^2)}\\right)\\, ,\n\\]\n$I_0$ is the modified Bessel function. The integral over $b$ is  \ncalculated \nin appendix~\\ref{app:A} and the integration on $a$ is straightforward. \nWe obtain\n\\begin{equation}\nN(t)=\\frac{2\\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}}}{\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}}}\\left[ct - \\frac{1}{2(\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}})}\n     \\left(1-\\e^{-2ct(\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}})}\\right)\\right]\\, .\\label{eq:nbcol}\n\\end{equation}\nIn terms of the dimensionless time\n\\[\n\\tau\\equiv ct(\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}})\n\\]\nand the parameter\n\\[\n\\lambda\\equiv\\frac{4\\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}}}{\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}}}\\, ,\n\\]\nwe have:\n\\begin{equation}\nN(\\tau)=\\frac{\\lambda}{2}\\left[\\tau -\n        \\frac{1}{2}\\left(1-\\e^{-2\\tau}\\right)\\right]\\, .\n\\end{equation}\nAfter a transient regime, $N(\\tau)$ grows linearly with time.\n\nThe front position is defined as the point where the next collision \nwill \ntake place. Thus the probability density $\\mu(X;t)$ to find the front at \nthe point $X$, \nat time $t$ is  given by\n\\begin{eqnarray}\n\\lefteqn{\\mu(X;t) =\n    \\left\\langle\\delta\\left(X-\\frac{x_1+y_1}{2}\\right)\n    \\theta(x_1-y_1-2ct)\\right\\rangle}\\nonumber \\\\\n&+& \\left\\langle\\sum_{k=1}^\\infty \\delta\\left(X \n-\n             \\frac{x_{k+1}+y_{k+1}}{2}\\right)\n    \\theta\\left(2ct-[x_k-y_k]\\right)\n    \\theta\\left([x_{k+1}+y_{k+1}]-2ct\\right)\\right\\rangle \\nonumber \\\\\n&=& \n\\left\\langle\\delta\\left(X-\\frac{x_1+y_1}{2}\\right)\\right\\rangle\n    \\label{eq:defmu} \\\\\n&+& \\left\\langle\\sum_{k=1}^\\infty \\left[\n             \\delta\\left(X-\\frac{x_{k+1}+y_{k+1}}{2}\\right) -\n             \\delta\\left(X-\\frac{x_k+y_k}{2}\\right) \\right]\n    \\theta\\left(2ct-[x_k-y_k]\\right)\\right\\rangle\\, . \\nonumber\n\\end{eqnarray}\nThe first term in the right hand side reads simply\n\\begin{eqnarray}\n\\left\\langle\\delta\\left(X - \\frac{x_1+y_1}{2}\\right)\\right\\rangle\n&=& \\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}} \\int_0^\\infty \\d x_1 \\int_{-\\infty}^0 \\d y_1 \n       \\e^{-\\mathrel{\\rho^{\\mathrm{B}}} x_1}\\e^{-\\mathrel{\\rho^{\\mathrm{A}}} y_1}\n       \\delta\\left(X-\\frac{x_1+y_1}{2}\\right) \\nonumber\\\\\n&=& \\frac{2\\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}}}{\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}}}\\left[\\theta(X)\\e^{-2\\mathrel{\\rho^{\\mathrm{B}}} X} +\n       \\theta(-X)\\e^{2\\mathrel{\\rho^{\\mathrm{A}}} X}\\right]\\, .\n\\end{eqnarray}\n\nTo calculate the second term in the right hand side, we remark that one \nhas to\n average over the position $x_k$, $x_{k+1}$, $y_k$ and $y_{k+1}$. We \nshall then first average over particles $k+1$, to obtain an expression \ncontaining only the position $x_k$ and $y_k$. To do this, we need to \ncalculate the following integral\n\\[\nS \\equiv \\mathrel{\\rho^{\\mathrm{B}}}\\int_{x_k}^\\infty \\d x_{k+1} \\mathrel{\\rho^{\\mathrm{A}}}\\int_{-\\infty}^{y_k} \\d \ny_{k+1}\n\\delta\\left(X-\\frac{x_{k+1}+y_{k+1}}{2}\\right)\n\\e^{-\\mathrel{\\rho^{\\mathrm{B}}} x_{k+1}+\\mathrel{\\rho^{\\mathrm{A}}} y_{k+1}}\\, ,\n\\]\nputting $a=\\half(x_{k+1}-y_{k+1})$ and \n$b=\\half(x_{k+1}+y_{k+1})$, we have:\n\\begin{eqnarray*}\nS = 2\\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}}\\int_{-\\infty}^\\infty \\d a \\int_{-\\infty}^\\infty \\d b \n\\theta(a+b-x_k)&\\theta&(a-b+y_k)\\delta(x-b) \\\\\n&\\times&\\e^{-(\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}})a+(\\mathrel{\\rho^{\\mathrm{A}}}-\\mathrel{\\rho^{\\mathrm{B}}})b}\n\\end{eqnarray*}\nThe integral over $b$ is straightforward, and the integral over $a$ \ngives after some algebra\n\\begin{eqnarray*}\nS = \\frac{2\\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}}}{\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}}}\n&& \\left[\\rule{0mm}{5mm}\\theta\\left(\\frac{x_k+y_k}{2}-X\\right)\n   \\e^{-(\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}})x_k+2\\mathrel{\\rho^{\\mathrm{A}}} X}\\right .\\\\\n&& +\\left.\\rule{0mm}{5mm}\\theta\\left(X-\\frac{x_k+y_k}{2}\\right)\n   \\e^{(\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}})y_k-2\\mathrel{\\rho^{\\mathrm{B}}} X}\\right]\\, .\n\\end{eqnarray*}\nHence, the second term in the right hand side of eq.~(\\ref{eq:defmu}) \nbecomes\n\\begin{eqnarray*}\n\\sum_{k=1}^\\infty\\left\\langle\\left\\{\\rule{0mm}{6mm}\n   \\frac{2\\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}}}{\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}}}\\right.\\right.\n&& \\left[\\rule{0mm}{5mm}\\theta\\left(\\frac{x_k+y_k}{2}-X\\right)\n   \\e^{-(\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}})x_k+2\\mathrel{\\rho^{\\mathrm{A}}} X}\\right .\\\\\n&& \n+\\left.\\rule{0mm}{5mm}\\theta\\left(X-\\frac{x_k+y_k}{2}\\right)\n   \\e^{(\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}})y_k-2\\mathrel{\\rho^{\\mathrm{B}}} \nX}\\right]\\\\\n&& -\\left.\\left.\\rule{0mm}{6mm}\n   \\delta\\left(X-\\frac{x_k+y_k}{2}\\right) \\right\\}\n    \\theta\\left(2ct-[x_k-y_k]\\right)\\right\\rangle\\, .\n\\end{eqnarray*}\nNow using the relation (see appendix~\\ref{app:B})\n\\begin{eqnarray}\n\\lefteqn{\\sum_{k=1}^\\infty\\langle f(x_k+y_k,x_k-y_k) \\rangle}\n    \\label{eq:sumav} \\\\\n&=& 2\\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}}\\int_0^\\infty \\d a \\int_{-a}^a \\d b\n    e^{-(\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}})a+(\\mathrel{\\rho^{\\mathrm{A}}}-\\mathrel{\\rho^{\\mathrm{B}}})b} f(2b,2a)\n    I_0\\left(2\\sqrt{\\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}}(a^2-b^2)}\\right)\\, ,\\nonumber\n\\end{eqnarray}\none finally finds:\n\\begin{eqnarray}\n\\lefteqn{\\mu(X;t) = \\frac{2\\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}}}{\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}}}\\left[\\theta(X)\n         \\e^{-2\\mathrel{\\rho^{\\mathrm{B}}} X}+\\theta(-X)\\e^{2\\mathrel{\\rho^{\\mathrm{A}}} X}\\right]} \\\\\n     &+& 2\\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}} \\int_0^{ct} \\d a \\int_{-a}^a \\d b\n         \\e^{-(\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}})a+(\\mathrel{\\rho^{\\mathrm{A}}}-\\mathrel{\\rho^{\\mathrm{B}}})b}\n         I_0\\left(2\\sqrt{\\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}}(a^2-b^2)}\\right)\n         \\nonumber \\\\\n&\\times& \\left\\{-\\delta(X-b)+\\frac{2\\mathrel{\\rho^{\\mathrm{A}}}\\mathrel{\\rho^{\\mathrm{B}}}}{\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}}}\n         \\left[\\theta(b-X)e^{-2\\mathrel{\\rho^{\\mathrm{A}}}(b-X)}+\n         \\theta(X-b)e^{-2\\mathrel{\\rho^{\\mathrm{B}}}(X-b)}\\right]\\right\\}\\, . \\nonumber\n\\end{eqnarray}\nIn terms of the dimensionless  variables $\\tau$ and $\\lambda$, and \n the dimensionless coordinate $\\xi\\equiv(\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}})X$, we can \nwrite\n\\begin{eqnarray}\n\\mu(\\xi;\\tau)\n&=& \\frac{\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}}}{2}\\e^{\\kappa\\xi} \\left\\{\\rule{0mm}{6mm}\n    \\lambda\\e^{-|\\xi|}\\right. \\label{eq:mugen} \\\\\n&+& \\left.\\rule{0mm}{6mm}\\lambda\\int_0^\\tau \\d\\alpha \\e^{-\\alpha}\n     \\int_{-\\alpha}^\\alpha \\d\\beta\n     I_0\\left(\\sqrt{\\lambda(\\alpha^2-\\beta^2)}\\right)\n     \\left[-\\delta(\\beta-\\xi)+\n     \\frac{\\lambda}{2}\\e^{-|\\beta+\\xi|}\\right]\\right\\}\\, ,\\nonumber\n\\end{eqnarray}\nwith\n\\[\n\\kappa\\equiv\\frac{\\mathrel{\\rho^{\\mathrm{A}}}-\\mathrel{\\rho^{\\mathrm{B}}}}{\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}}}\\, .\n\\]\nNote that the function between \nthe curly brackets is an even function of $\\xi$; we shall call it \n$\\hat{\\mu}(\\xi;\\tau)$. The rest of the paper will be devoted to the \nstudy of this function.\n\n\n\\section{Results and interpretation}\\label{sec:results}\n\nStarting from~(\\ref{eq:mugen}) and using the integrals quoted in  \nappendix~\\ref{app:A}, it is possible to verify that $\\mu(x;t)$ is \ncorrectly normalized to $1$. Moreover, we can also calculate its \ndifferent \nmoments. However for this purpose, it is simpler to consider the Fourier \ntransform of $\\mu$.\nLet us define the function ${\\tilde{\\mu}}(p;\\tau)$ as:\n\\begin{eqnarray}\n{\\tilde{\\mu}}(p;\\tau) &=& \\int_{-\\infty}^\\infty \\d X \n\\e^{-(\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}})pX}\n                \\e^{-(\\mathrel{\\rho^{\\mathrm{A}}}-\\mathrel{\\rho^{\\mathrm{B}}})X} \\mu(X;\\tau) \\nonumber \\\\\n            &=& \\frac{1}{2}\\int_{-\\infty}^\\infty \\d\\xi \\e^{-p\\xi}\n                \\hat{\\mu}(\\xi;\\tau)\\, .\\label{eq:defmufou}\n\\end{eqnarray}\nHere again, this integral can be calculated using the formulas quoted in\nappendix~\\ref{app:A}. One finds:\n\\begin{equation}\n{\\tilde{\\mu}}(p;\\tau)=\\lambda\\e^{-\\tau}\\frac{1}{1-p^2}\\left[\n            \\cosh\\left(\\sqrt{p^2+\\lambda}\\tau\\right)+\n            \\frac{\\sinh\\left(\\sqrt{p^2+\\lambda}\\tau\\right)}\n                 {\\sqrt{p^2+\\lambda}}\\right]\\label{eq:mufou}\n\\end{equation}\nIt is now easy  to calculate the different moment of $\\mu$, using the \nfact that\n\\begin{equation}\n\\langle\\xi^n\\rangle\n  \\equiv \\int_{-\\infty}^\\infty \\d X\\left[(\\mathrel{\\rho^{\\mathrm{A}}}+\\mathrel{\\rho^{\\mathrm{B}}})X\\right]^n\\mu(X;t)\n  =(-1)^n\\left[\\frac{\\partial^n}{\\partial p^n}{\\tilde{\\mu}}(p;\\tau)\n  \\right]_{p=-\\kappa}\\, .\\label{eq:defmom}\n\\end{equation}\nNoticing that  $\\kappa^2+\\lambda=1$, it is straightforward to check the \nnormalization. The first moment ${\\bar{\\xi}}(\\tau)=\\langle\\xi\\rangle$ is \ngiven by\n\\begin{equation}\n{\\bar{\\xi}}(\\tau) = \\kappa\\left[\\frac{2}{\\lambda}+\\tau-\\frac{1}{2}\n            \\left(1-\\e^{-2\\tau}\\right)\\right]\n          = \\frac{2\\kappa}{\\lambda}\\left[1+N(\\tau)\\right]\\, ,\n\\end{equation}\nand the variance is\n\\begin{eqnarray}\n\\lefteqn{\\xi_{\\mathrm{rms}}^2(\\tau) = \\langle\\xi^2\\rangle - \n\\langle\\xi\\rangle^2} \\\\\n&=& \\frac{4}{\\lambda^2} - \\frac{2}{\\lambda} + \\frac{3}{4} -\n    \\frac{5\\lambda}{4} \n+\\lambda\\tau \n+\n    \\left[\\lambda-\\frac{1}{2}-2\\tau(1-\\lambda)\\right]\\e^{-2\\tau} -\n    \\frac{1-\\lambda}{4}\\e^{-4\\tau}\\, . \\nonumber\n\\end{eqnarray}\nIf $\\mathrel{\\rho^{\\mathrm{B}}} \\not= \\mathrel{\\rho^{\\mathrm{A}}}$, and   in the long time limit, the front moves \nwith a velocity proportionnal to the density difference.\nThe case $\\lambda=1$ is of particular interest, because it corresponds \nto the symetric case  $\\mathrel{\\rho^{\\mathrm{A}}}=\\mathrel{\\rho^{\\mathrm{B}}}$. As expected: \n\\begin{equation}\n{\\bar{\\xi}}(\\tau)\\equiv 0\n\\end{equation}\nbut\n\\begin{equation}\n\\xi^2_{\\mathrm{rms}}(\\tau) = 2\\left[1+N(\\tau)\\right]\\, .\n\\end{equation}\nIn the long time limit, we see that \n$\\xi_{\\mathrm{rms}}^2(\\tau)\\propto\\tau$:\nthe front move essentially as an unbiased random walker. Thus, we can \nexpect\n $\\mu$ to be essentially a gaussian. In fact, we shall prove that\nfor any $\\lambda$, the long time limit of $\\mu$ is a gaussian.  \nStarting from the definition of an arbitrary moment of $\\mu$ \n(eq.~(\\ref{eq:defmom})), we may write for long time:\n\\begin{equation}\n\\langle\\xi^n\\rangle=(-1)^n\\lambda\\e^{-\\tau}\\left[\\frac{\\partial^n}\n{\\partial p^n}\\e^{\\tau\\sqrt{p^2+\\lambda}}\\right]_{p=-\\kappa}\n+ O(\\tau^{n-1})\\, ,\n\\end{equation}\nusing\n\\begin{equation}\n\\frac{\\partial^n}{\\partial p^n}\\e^{\\tau\\sqrt{p^2+\\lambda}}=\n\\left[\\frac{\\partial}{\\partial p}\\sqrt{p^2+\\lambda}\\right]^n\n\\left[\\frac{\\partial^n}{\\partial \\eta^n}\\e^{\\tau\\eta}\n\\right]_{\\eta=\\sqrt{p^2+\\lambda}} + O(\\tau^{n-1})\\, ,\n\\end{equation}\nwe obtain $\\langle\\xi^n\\rangle=(\\kappa\\tau)^n+O(\\tau^{n-1})$.\n\nIf $\\lambda<1$, $\\kappa\\neq 0$ and we know the leading term for every \nmoment of $\\mu$. If $\\lambda=1$, the situation is different and will be \nconsidered later.\n\nIt is known that for a gaussian distribution with a non vanishing mean \n${\\bar{\\xi}}(\\tau)$ \nand a variance $\\xi_{\\mathrm{rms}}^2(\\tau)$, one has:\n\\begin{equation}\n\\langle\\xi^n\\rangle_{\\mathrm{gauss}}=n!\\xi^n(\\tau)\n\\sum_{k=0}^{[\\frac{n}{2}]}\\frac{1}{(n-2k)!k!}\n\\left(\\frac{\\xi_{\\mathrm{rms}}^2(\\tau)}{2\\xi^2(\\tau)}\\right)^k\\, ,\n\\end{equation}\nIn the long time limit:\n\\begin{equation}\n\\frac{\\xi_{\\mathrm{rms}}^2(\\tau)}{2{\\bar{\\xi}}^2(\\tau)}\n=O\\left(\\frac{1}{\\tau}\\right)\\, ,\n\\end{equation}\nand $\\langle\\xi^n\\rangle_{\\mathrm{gauss}}=(\\kappa\\tau)^n+O(\\tau^{n-1})$.\n\nThus, when $\\lambda <1$  and for sufficiently long times (i.e. when the \ncorrections to the leading \nterm become negligible), the probabilty density $\\mu$ takes the gaussian \nform:\n\\begin{equation}\n\\mu(\\xi;\\tau)\\propto\n\\exp\\left[-\\frac{(\\xi-\\kappa\\tau)^2}{2\\lambda\\tau}\\right]\\, .\n\\end{equation}\nThe front wanders around its mean value with an amplitude of order of\nmagnitude $\\sqrt{\\lambda\\tau}$, like a biased random walker  with a \ndiffusion coefficient of $\\lambda\/2$.\n\nThe $\\lambda=1$ case is different because the amplitude of the $\\tau^n$ \nterm vanishes. However, we can still prove that for large time, the \nprobability density $\\mu$ is a gaussian. More precisely, we shall give \nan upper and a lower bound, both of gaussian form. For this purpose, we \nshall come back to the Fourier transform ${\\tilde{\\mu}}(p;\\tau)$ defined \nby \neq.~(\\ref{eq:defmufou}). It can be shown (see appendix~\\ref{app:C}), \nthat the inverse Fourier transform of ${\\tilde{\\mu}}(p,\\tau)$ is given \nby\n\\begin{eqnarray}\n\\hat\\mu(\\xi;\\tau) &=&\\lambda\\e^{-|\\xi|-\\tau}\n                     \\left[\\cosh\\left(\\sqrt{1+\\lambda}\\tau\\right) +\n                     \\frac{\\sinh\\left(\\sqrt{1+\\lambda}\\tau\\right)}\n                     {\\sqrt{1+\\lambda}}\\right] \\label{eq:muborne}\\\\\n                  &-&\\lambda\\theta(\\tau-|\\xi|)\\e^{-\\tau}\n                     \\left\\{\\rule{0mm}{6mm}\n                     \\sinh\\left(\\sqrt{1+\\lambda}\\tau-|\\xi|\\right) +\n                     \\frac{\\cosh\\left(\\sqrt{1+\\lambda}\\tau-|\\xi|\\right)}\n                     {\\sqrt{1+\\lambda}}\\right.\\nonumber \\\\\n                  &-&\\left.\\rule{0mm}{6mm}\n                     2\\int_0^{\\sqrt{\\lambda}}\\frac{\\d r}{\\pi}\n                     \\frac{\\cos(r\\xi)}{1+r^2}\\left[\n                     \\sinh\\left(\\sqrt{\\lambda-r^2}\\tau\\right)+\n                     \\frac{\\cosh\\left(\\sqrt{\\lambda-r^2}\\tau\\right)}\n                     {\\sqrt{\\lambda-r^2}}\\right]\\right\\}\\, .\\nonumber\n\\end{eqnarray}\nIt is now easy to find an upper and a lower bound for $\\hat\\mu$. Indeed, \n for $|\\xi|\\leq\\frac{\\pi}{2\\sqrt{\\lambda}}$, one has:\n\\[\n\\frac{\\cos(r\\xi)}{1+\\lambda}\\leq\\frac{\\cos(r\\xi)}{1+r^2}\\leq\\cos(r\\xi)\n\\]\nas $r\\in[0,\\sqrt{\\lambda}]$.\nThe remaining integral can be calculated exactly:\n\\begin{eqnarray}\n\\lefteqn{\\frac{2}{\\pi}\\int_0^{\\sqrt{\\lambda}} \\d r \n\\cos(r\\xi)\\left[\n         \\sinh\\left(\\sqrt{\\lambda-r^2}\\tau\\right)+\n         \\frac{\\cosh\\left(\\sqrt{\\lambda-r^2}\\tau\\right)}\n              {\\sqrt{\\lambda-r^2}}\\right]}\\nonumber \\\\\n\\quad\n&=& \\lambda\\tau\\frac{I_1\\left(\\sqrt{\\lambda(\\tau^2-\\xi^2)}\\right)}\n                    {\\sqrt{\\lambda(\\tau^2-\\xi^2)}}+\n    I_0\\left(\\sqrt{\\lambda(\\tau^2-\\xi^2)}\\right)\\, , \\label{eq:borne}\n\\end{eqnarray}\nyielding, then two bounds for $\\mu$. Two conclusions can then be\ndrawn. First for $\\lambda=1$, the asymptotic time behavior \nof~(\\ref{eq:borne}) is\n\\[\n\\e^\\tau\\frac{2}{\\sqrt{2\\pi\\tau}}\\e^{-\\xi^2\/2\\tau}\\, ,\n\\]\nand then, for $\\xi\\leq\\frac{\\pi}{2}$ and $\\tau$ large, we have:\n\\begin{equation}\n\\frac{1}{\\sqrt{2\\pi\\tau}}\\e^{-\\xi^2\/2\\tau}\\leq\\hat\\mu(\\xi;\\tau)\n\\leq\\frac{2}{\\sqrt{2\\pi\\tau}}\\e^{-\\xi^2\/2\\tau}\\, .\n\\end{equation}\nIn the long time limit, the density probability $\\mu$ becomes the \ndensity \nprobability of a random walk, with a diffusion coefficient of $\\half$.\n\nThe second consequence concerns the case $\\xi=0$, for any value of \n$\\lambda$: we have, in the long time limit:\n\\begin{eqnarray}\n\\hat\\mu(0;\\tau) &\\propto& \n\\frac{\\e^{(1-\\sqrt{\\lambda})\\tau}}{\\sqrt{\\tau}}\n                          \\quad\\mbox{if }\\lambda<1\\, , \\\\\n\\hat\\mu(0;\\tau) &\\propto& \\frac{1}{\\sqrt{\\tau}}\n                          \\quad\\mbox{if }\\lambda=1\\, .\n\\end{eqnarray}\nThese results are not really surprising. Indeed, it can be verified that \n$\\xi=0$ is \na maximum of $\\mu$, for $\\lambda=1$ but not for $\\lambda<1$.\n\nIt is useful to get an image of what is going on by plotting the \nfunction\n$\\mu$ for various value of $\\lambda$. However for this purpose, both \nequations~(\\ref{eq:mugen}) and~(\\ref{eq:muborne}) are inadequate for \ntechnical reasons. Equation~(\\ref{eq:mugen}) is not well suited\n because it contains a double integral which requires a lot of \nCPU time and equation ~(\\ref{eq:muborne})  because the integral diverge \nfor $r=\\sqrt\\lambda$, \nleading to numerical difficulties.\nTo avoid these problems, we  have  obtained a different \nintegral representation, starting from~(\\ref{eq:mugen}), and using the \nfollowing relation:\n\\begin{equation}\n\\left[\\frac{\\partial^2}{\\partial\\alpha^2} -\n      \\frac{\\partial^2}{\\partial\\beta^2}\\right]\nI_0\\left(\\sqrt{\\lambda(\\tau^2-\\xi^2)}\\right) =\n\\lambda I_0\\left(\\sqrt{\\lambda(\\tau^2-\\xi^2)}\\right)\\label{eq:Ider} \\, .\n\\end{equation}\nThis relation follows directly from the integral representation for \n$I_0$\n\\[\nI_0\\left(\\sqrt{\\lambda(\\tau^2-\\xi^2)}\\right) =\n\\int_{\\mathrm{C}} \\frac{\\d z}{2\\mathrm{i}\\pi}\\frac{1}{z}\n\\exp\\left\\{\\frac{\\sqrt{\\lambda}}{2}\\left[(\\alpha+\\beta)z+\n           \\frac{\\alpha-\\beta}{z}\\right]\\right\\}\\, ,\n\\]\nwhere $\\mathrm{C}$ is, for example, the circle of radius $1$ centered\nin $z=0$. Inserting~(\\ref{eq:Ider}) in~(\\ref{eq:mugen}), integrating \ntwice by part and using that\n\\[\n\\left[\\frac{\\partial^2}{\\partial\\alpha^2} - \n      \\frac{\\partial^2}{\\partial\\beta^2}\\right]\n\\e^{-\\alpha-|\\beta-\\xi|} = 2\\e^{-\\alpha}\\delta(\\beta-\\xi)\\, ,\n\\]\nwe get\n\\begin{eqnarray*}\n\\hat\\mu(\\xi;\\tau) = \\lambda\\left\\{\\e^{-|\\xi|}\\rule{0mm}{10mm}\\right.\n    + \\frac{1}{2}\\int_0^\\tau\n&\\d&\\alpha\n    \\left[\\e^{-\\tau}\\left(\\e^{-|\\alpha-\\xi|}+\\e^{-|\\alpha+\\xi|}\\right)\n    \\rule{0mm}{9mm}\\right. \\\\\n&\\times&\n    \\left(\\lambda\\tau\n    \\frac{I_1\\left(\\sqrt{\\lambda(\\tau^2-\\alpha^2)}\\right)}\n         {\\sqrt{\\lambda(\\tau^2-\\alpha^2)}}+\n    I_0\\left(\\sqrt{\\lambda(\\tau^2-\\alpha^2)}\\right)\\right)\\\\\n&-& \\left.\\rule{0mm}{10mm}\\left.\\rule{0mm}{9mm}\n    2\\theta(\\alpha-\\xi)e^{\\xi-2\\alpha} \n-\n    2\\theta(\\alpha+\\xi)e^{2\\alpha-\\xi}\\right]\\right\\}\\, .\n\\end{eqnarray*}\nPerforming the integral over the two theta functions, rearranging \nthe terms in the remaining integral and using the integrals of \nappendix~\\ref{app:A}, leads to:\n\\begin{eqnarray}\n\\hat\\mu(\\xi;\\tau)\n&=& \\lambda\\e^{-|\\xi|-\\tau}\n    \\left[\\cosh\\left(\\sqrt{1+\\lambda}\\tau\\right) +\n    \\frac{\\sinh\\left(\\sqrt{1+\\lambda}\\tau\\right)}\n          {\\sqrt{1+\\lambda}}\\right] \\label{eq:muplot}\\\\\n&-& \\lambda\\theta(\\tau-|\\xi|)\\e^{-\\tau}\\left\\{\\rule{0mm}{10mm}\n    \\sinh(\\tau-\\xi)\\right.\\nonumber \\\\\n&-& \\left.\\rule{0mm}{10mm}\n    \\int_{|\\xi|}^{\\tau} \\d\\alpha\\sinh(\\alpha-|\\xi|)\\left[\n    \\frac{I_1\\left(\\sqrt{\\lambda(\\tau^2-\\alpha^2)}\\right)}\n         {\\sqrt{\\lambda(\\tau^2-\\alpha^2)}}+\n    I_0\\left(\\sqrt{\\lambda(\\tau^2-\\alpha^2)}\\right)\\right]\\right\\}\\, .\n    \\nonumber\n\\end{eqnarray}\nWe have not been able to simplify further this expression and in \nparticular to perform analytically the integration. However, this form \ncan be easily integrated numerically. For examples, the function \n$\\mu(\\xi;\\tau)$ is plotted on figure~\\ref{fig:two} and~\\ref{fig:three} \nfor respectively\n$\\lambda=1$ and for $\\lambda=0.5$, $\\mathrel{\\rho^{\\mathrm{A}}}>\\mathrel{\\rho^{\\mathrm{B}}}$.\n\\begin{figure}\n\\epsfysize=8cm\n\\centerline{\\epsfbox{mu_10.eps}}\n\\caption{Plot of $\\mu(\\xi;\\tau)$, for $\\mathrel{\\rho^{\\mathrm{A}}}=1$ and $\\mathrel{\\rho^{\\mathrm{B}}}=1$\n(i.e. $\\lambda=1$). \n\\label{fig:two}}\n\\end{figure}\n\\begin{figure}\n\\epsfysize=8cm\n\\centerline{\\epsfbox{mu_05.eps}}\n\\caption{Plot of $\\mu(\\xi;\\tau)$, for $\\mathrel{\\rho^{\\mathrm{A}}}=1+\\half\\protect\\sqrt{3}$ and \n$\\mathrel{\\rho^{\\mathrm{B}}}=1-\\half\\protect\\sqrt{3}$ (i.e. $\\lambda=0.5$). \n\\label{fig:three}}\n\\end{figure}\n\n\nOne sees that for $\\lambda=1$, the probability density $\\mu$ spreads out \nand approches a gaussian for long times. For the assymetric case \n$\\lambda=0.5$, one sees clearly the drift of the front which beecomes \nlinear in the long time regime.\n\n\\section{Conclusions}\\label{sec:concl}\n\nUsing a combinatorial analysis approach, we have been able to derive exact\nresults for the time dependent number of collisions, the probability\ndensity $\\mu(X;t)$ to find the front at point $X$ at time $t$ and\nits first two moments.\nAlthough simple, this nonequilibrium model leads to nontrivial behavior \nas far as the dynamics of the annihilation front is concerned.  \n\nSeveral extensions of the present work are possible. One is to consider\na similar model in \nwhich the velocities are not restricted to be bimodal, but obey  a more \ngeneral (discrete or continuous) distribution. Another interesting \naspect is given by the generalization of  this model to higher \ndimensions. In this case, some $A$ particles can invade the right part \nof the system whithout being annihilated (and vice-versa for the $B$'s). \nA new definition of the front in term of the annihilation rate should be \nintroduced and different time dependent behavior may occur. These more \ncomplicated problems are under investigation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nElectrokinetic phenomena, which couple fluid flow, ion transport and\nelectric fields in electrolytes, are exploited in a variety of\nmicrofluidic technologies~\\cite{squires2005}.  A solid surface in\ncontact with an electrolyte typically acquires a surface charge and\nforms an electric double layer composed of wall charge and a screening\nlayer of excess counterions.  Electro-osmosis results from the action\nof an applied electric field on this double-layer charge, which sets\nthe bulk fluid into motion. Although the double layer is typically\nvery thin (nm) compared to the channel dimensions ($\\mu$m), it is the\nsource of long-range flow patterns. \n\n\nThe classical continuum theory of\nelectro-osmosis~\\cite{lyklema_book_vol2,hunter_book} suffices to\nunderstand most direct-current (DC) electro-osmotic flows, which are\nlinear in the applied voltage. DC electroosmosis results from\napplication of an external electric field parallel to the surface,\nwhich acts on equilibrium double-layer charge and induces motion in\nthe bulk, neutral liquid. The linear response is due to the assumption\nthat the applied field does not perturb the pre-existing surface\ncharge. When DC electroosmosis is used in microfluidic environments,\nelectrodes are placed at the ends of relatively long (cm) channels,\nand therefore large voltages (kV) are typically needed to generate\nsufficient fields.\n\nElectrokinetic phenomena can be exploited at much lower voltages and\nwith much greater precision by placing electrodes close together {\\em\n  inside} the fluid channels and using alternating current (AC) to\ninhibit Faradaic reactions.  Large electric fields (100 V\/cm) needed to\ninduce adequate flow can thus be generated with only a few volts by\nmicro-electrodes.  The most studied example is alternating-current\nelectro-osmosis (ACEO) \\cite{encyclopedia_ACEO}, discovered by Ramos\net al. ~\\cite{ramos1999,green2000b,gonzalez2000,green2002}.  When a\nperiodic array of interdigitated electrodes is placed inside the\nchannel, an AC signal applied to the electrodes can generate a steady\nset of micro-vortices.  Ajdari showed that if geometrical asymmetry is\nintroduced, then these vortices can be rectified to drive net pumping\nin one direction over the array~\\cite{ajdari2000}. The original\nimplementation of this principle was a planar array of flat electrode\npairs of unequal widths and\ngaps~\\cite{brown2001,ramos2003,studer2004}, as shown in Figure\n~\\ref{fig:schematic}. Recent designs for ``3D ACEO'' pumps have\nachieved much faster flows (mm\/sec) using non-planar \nelectrodes, which more efficiently rectify opposing slip velocities in\na ``fluid conveyor\nbelt''~\\cite{bazant2006,urbanski2006,urbanski2007}. Waves of\nvoltage can also pump fluids over electrode arrays by traveling-wave\nelectro-osmosis (TWEO)~\\cite{cahill2004,ramos2004}.\n\n\\begin{figure}\n\\centering\n  \\epsfig{file=figure1.eps,width=2.8in,clip=} \n  \\caption{\\label{fig:schematic} One period of an asymmetric planar\n    array of microelectrodes in an AC electro-osmotic (ACEO)\n    pump. This geometry has been studied experimentally by several\n    groups~\\cite{brown2001,studer2004,urbanski2006,microTAS2007} and\n    is the subject of the present theoretical study.  The dimensions\n    are; $W1=4.2~ \\mathrm{\\mu m}$ , $W2=25.7~\\mathrm{\\mu m}$, $G1=4.5~\n    \\mathrm{\\mu m}$ , and $G2=15.6 ~\\mathrm{\\mu m}$, using the\n    notation of Ref.~ \\cite{olesen2006}.  }\n\\end{figure}\n\nThese effects exemplify the fundamental nonlinear electrokinetic\nphenomenon~\\cite{encyclopedia_nonlinear} of induced-charge\nelectro-osmosis (ICEO)~\\cite{iceo2004a,iceo2004b,levitan2005}, which\nalso has other applications in microfluidics. The key source of\nnonlinearity is that the diffuse double-layer charge is {\\em induced}\nnear a polarizable surface by the applied electric field, which then\nacts on it to drive nonlinear electro-osmotic flow. Although ICEO\nflows were first described in the 1980s in the context of colloid\nscience~\\cite{murtsovkin1996,gamayunov1986}, they are finding new\napplications in microfluidic devices. As in the case of ACEO pumps,\nbroken spatial symmetries in ICEO flows can also be used to pump\nfluids around polarizable microstructures and to manipulate polarizable\nparticles~\\cite{iceo2004a,squires2006,yariv2006,velev}.\n\nThe current theory of ICEO flow~\\cite{iceo2004b}, including\nACEO~\\cite{gonzalez2000,olesen2006} and\nTWEO~\\cite{cahill2004,ramos2004} microfluidic pumps, is based on the\nclassical electrokinetic\nequations~\\cite{lyklema_book_vol2,hunter_book}. This century-old model\ncomprises the Poisson-Nernst-Planck (PNP) equations of ion transport\ncoupled to the Navier-Stokes equations of viscous fluid flow via\nelectrostatic Maxwell stresses. The crucial modification is in the\nboundary conditions, relaxing the assumption of fixed surface charge\n(or constant zeta potential) to allow for significant electrostatic\npolarizability of the surface (e.g. fixed surface potential). The\nelectrokinetic equations themselves, however, have not been\nquestioned, until very recently~\\cite{large}, which provides the\nmotivation for this work.\n\nICEO flows typically involve {\\it large voltages} induced across the\ndouble layer, greatly exceeding the thermal equilibrium voltage,\n$kT\/e$ ($=25$ mV at room temperature). In particular, experimental\nACEO electro-array\npumps~\\cite{brown2001,studer2004,urbanski2006,urbanski2007,microTAS2007},\nthe subject of our study, usually apply several volts to the double\nlayer. Such large voltages of order $100 kT\/e$ inevitably lead to the\nbreakdown of the PNP equations by violating the fundamental assumption\nof an ideal dilute solution~\\cite{kilic2007a,kilic2007b}.  The\nlimitations of dilute solution theory for double-layer structure are\nwell known, and many attempts have been made to incorporate effects\nsuch as steric exclusion and electrostatic correlations.  (See\nRef.~\\cite{kilic2007a} for a recent review.) ICEO flow, however,\nraises new issues due to its more extreme, dynamical\ncontext~\\cite{large}. Even if the bulk solution is dilute, the surface\nbecomes so highly charged by the induced voltage that counterions\nbecome crowded in the double layer, while simultaneously driving\ntangential fluid flow.\n\n\\begin{figure}\n\\centering\n\\epsfig{file=figure2.eps,width=2.8in}\n\\caption{Pumping velocity vs. frequency for the planar ACEO pump in\n  Figure~\\ref{fig:schematic} from experiments and simulations.\n  {\\it Points:} Experimental data\n  at 3 Vpp in 0.03 mM KCl from Ref.~\\cite{microTAS2007} (taken by\n  J. P. Urbanski); {\\it Solid curve: } Linear Debye-H\\\"uckel-Stern\n  model of the diffuse layer with a constant compact layer capacitance\n  $C_s = \\varepsilon\/(\\delta \\lambda_D)$, where $\\delta=0.5$ is chosen\n  to fit the experimental peak frequency. {\\it Dashed curve: }\n  Nonlinear Gouy-Chapman-Stern (Poisson-Boltzmann) model of the\n  diffuse layer also with $\\delta=0.5$.  }\n     \\label{fig:jpdata}\n\\end{figure}\n\nThe need for improved models is also indicated by some unexplained\nfeatures of ICEO experiments. In this paper, we focus on the tendency\nfor planar ACEO pumps to reverse at high\nvoltage~\\cite{studer2004,chang2004,wu2006,urbanski2006,microTAS2007},\nwhich naturally poses problems for microfluidic applications, if it\ncannot be reliably predicted. The standard model of ACEO based on the\nclassical electrokinetic equations predicts a single peak in the flow\nrate versus frequency at the characteristic ``RC'' charging frequency\nof the electrodes, in both the linear~\\cite{ajdari2000} and\nnonlinear~\\cite{olesen2006,olesen_thesis} regimes.  Flow reversal has\nbeen observed at high voltage ($> 2$ V) and high frequency (10-100\nkHz) in ACEO pumping of dilute\nKCl~\\cite{studer2004,chang2004,microTAS2007} and deionized\nwater~\\cite{urbanski2006} with $10 \\mu$m scale electrode\narrays. \n\nRepresentative examples of the observed and predicted frequency\nresponse are shown in Figure \\ref{fig:jpdata} for the pump geometry of\nFigure~\\ref{fig:schematic}. Here we compare experimental data taken at\n3.0 V $= 120 k T\/e$ in an 0.03 mM KCl solution over gold electrodes\n~\\cite{microTAS2007} to simulations with two commonly used models: one\nassumes the linear Debye-Huckel (DH) model of the double\nlayer~\\cite{ramos1999,gonzalez2000,ajdari2000}, while the other adopts\na non-linear Poisson-Boltzmann (PB) model based on the Gouy-Chapman\nsolution~\\cite{olesen2006}.  Following prior\nwork~\\cite{ajdari2000,green2002,levitan2005}, a constant Stern-layer\ncapacitance is added in order to better {\\em fit} the experimental data by\nmatching the peak frequency in the DH model.  \nBoth models can reasonably fit the data in terms of the \npeak in the forward pumping, though  \nthis becomes more difficult for PB at higher voltages \\cite{olesen2006}. \nEach model predicts a single peak of forward flow,\nwhile the experiments show the pump reversing above 10 kHz.  The\ninability of these classical models to capture, even qualitatively,\nthis high frequency flow reversal motivates us to consider modified\ndouble-layer models.\n\nThe reversal of ACEO pumps has been considered in several previous\nstudies. Flow reversal was first attributed to Faradaic\nreactions~\\cite{chang2004}, and this view of ``Faradaic charging''\npersists in recent work on DC-biased AC\nelectro-osmosis~\\cite{wu2007}. Simulations of ACEO pumps with\nButler-Volmer reaction kinetics, however, have failed to predict the\nobserved flow, especially at high\nfrequency~\\cite{olesen2006,olesen_thesis}. In existing models, a weak\nflow reversal due to reactions can only be observed at low frequency\n(far below the maximum) and for certain sets of\nparameters~\\cite{ajdari2000,olesen2006,olesen_thesis}. This is\nconsistent with the observation of gas bubbles from electrolysis at\nlow frequency and high voltage in dilute KCl with gold\nelectrodes~\\cite{studer2004}.  Recently, higher resolution\nmeasurements of the same pump design with platinum electrodes have\nrevealed very weak ($<10 \\mu$m\/sec) reverse ACEO flow at low frequency\n($< 20$ kHz) and low voltage ($<1.5$ V) and have demonstrated the\nimportance of Faradaic currents through {\\it in situ} impedance\nspectroscopy~\\cite{gregersen2007}. Although the theory predicts\nsimilar flow reversal, it is not in quantitative agreement and does\nnot predict the concentration dependence.\n \nIn addition to Faradaic reactions, various other nonlinear effects in\ndilute solution theory also dominate at low frequencies: The\ndifferential capacitance of the diffuse layer~\\cite{lyklema_book_vol2}\ndiverges, which causes the RC charging time to grow exponentially with\nvoltage~\\cite{olesen2006}, and salt adsorption and tangential\nconduction by the diffuse layer are coupled to (much slower) bulk\ndiffusion~\\cite{bazant2004,chu2006,chu2007}. All of these effects have\nrecently been incorporated in simulations of ACEO at large voltages\nusing the classical electrokinetic equations, but high-frequency flow\nreversal was not observed~\\cite{olesen2006,olesen_thesis}. Another\npossible source of flow reversal is AC electrothermal\nflow~\\cite{gonzalez2006}, which can lead to reverse pumping in\nexperiments with planar electrode arrays~\\cite{wu2007}, but under\ndifferent conditions of much higher salt concentration ($>$ 1 M),\nvoltage ($> 10$ V) and higher frequency ($> 100$ kHz) than ACEO ($<\n0.01$ M, $< 5$ V, $< 100$ kHz).  To date, none of the above effects\nhave been able to explain the experimental data for ACEO pumps.\n\nIn this paper, we propose steric effects of ion crowding in the double\nlayer as a possible cause of the observed flow reversal at high\nfrequency.  Using simple modifications of the electrokinetic equations\nto account for finite ion sizes, we are able to predict a frequency\nresponse of ACEO pumps that is very similar to that observed in\nexperiments. Although our results do not provide a complete\nquantitative theory, we will demonstrate that accounting for steric\neffects can have an important impact on the theory of ACEO.\n\n\n\\section{Steric effects in a thin double layer}\n\\label{sec:steric} \n\nIn the dilute-solution theory of\nelectrolytes~\\cite{lyklema_book_vol2,hunter_book}, the concentration of\neach ionic species in the diffuse part of the electric double layer at\na charged surface is in thermal equilibrium with a Boltzmann\ndistribution,\n\\begin{equation}\n  c_i(x)=c_0 \\mathrm{exp}{\\left( \\frac{-z_ie\\phi(x)}{kT}\\right)},  \n\\label{eq:cminus}\n\\end{equation}\nwhere $k$ is Boltzmann's constant, $T$ the temperature, $e$ the\nelectron charge, $z_i$ the valence, $c_0$ the concentration in the\nneutral bulk electrolyte just outside the double layer, and $\\phi$ is\nthe (mean) electrostatic potential relative to the bulk ($x\\gg\n\\lambda_D$, where $\\lambda_D$ is the Debye-H\\\"uckel screening\nlength). In classical linear electrokinetic phenomena, the\ndiffuse-layer voltage drop $\\phi(0)$ is set by chemical equilibrium at\nthe surface and is thus typically comparable to the thermal voltage\n$kT\/e$. At an electrode driving ACEO flow, however, much larger\ndiffuse-layer voltages $\\phi(0) \\gg kT\/e$ are {\\it induced} by a\napplied voltage of order $100 kT\/e$. Under these conditions, it is\neasy to see that Boltzmann equilibrium (\\ref{eq:cminus}) breaks down\nby predicting diverging concentrations of counterions at the surface,\nof order $e^{100}c_0$. \n\nThe assumption of a dilute solution is thus incompatible with a large\napplied voltage. If we assume ions have a characteristic length scale,\n$a$, then the corresponding cutoff concentration $a^{-3}$ for the\nbreakdown of dilute-solution theory is reached at relatively low\nvoltage, even if the bulk salt solution is very dilute.  For example,\nfor $c_0=10^{-5}$ M, $z=1$, and $a=3$ \\AA (including a solvation\nshell), the cutoff concentration is reached at $0.33$ V.  To account\nfor the excess ions (at typical equilibrium voltages), Stern\npostulated a compact layer of solvated ions of finite size on the\nsurface~\\cite{bockris_book}, which carries most of the double-layer\nvoltage as the diffuse-layer capacitance diverges. Such an intrinsic\n``surface capacitance'' is also invoked in models of ICEO flows, where\nit can also include the effect of a thin dielectric coating on a metal\nsurface~\\cite{ajdari2000,bazant2004}.  It seems unlikely, however,\nthat an atomically thin Stern or coating layer could withstand several\nVolts, e.g. since dielectric breakdown occurs in most materials\n(including water) in fields of order 10 MV\/m $= 0.01$\nV\/nm~\\cite{jones1995}. Instead, it seems the diffuse layer must carry\na substantial voltage $\\phi \\gg kT\/e$ in ACEO experiments, which\ncauses the ions to become highly crowded near the surface.\n\n\\begin{figure}\n(a)\\includegraphics[width=2.8in]{figure3a.eps}\n(b)\\includegraphics[width=2.8in]{figure3b.eps}\n\\caption{\\label{fig:C} (a) Schematic of the \n  equilibrium distribution of counterions\n  near a positively charged surface \n  taking into account a minimum ion spacing $a$ for large applied\n  voltages. As the voltage increases, the width of the condensed\n  layer will increase causing a decrease the capacitance. \n  (b) The voltage dependence\n  of the capacitance  of the diffuse \n  layer for PB and MPB at different values of $\\nu$; \n  where $\\nu = 2a^3 c_0$ is the\n  bulk volume fraction of ions.  }\n\\end{figure}\n\n\n\nA variety of ``modified Poisson-Boltzmann equations'' (MPB) have been\nproposed to describe equilibrium ion profiles near a charged wall, as\nreviewed in Ref.~\\cite{kilic2007a}.  To capture ion crowding effects\nacross a wide range of voltages, we employ the simplest possible MPB\nmodel~\\cite{kilic2007a,kralj-iglic1996,borukhov1997,biesheuvel2007},\nfirst proposed by Bikerman~\\cite{bikerman1942},\nwhich is a continuum approximation of the entropy of ions on a lattice\nof size $a$. As shown in Fig.~\\ref{fig:C}(a), when a large voltage is\napplied, the counterion concentration exhibits a smooth transition\nfrom an outer PB profile to a condensed layer at $c=c_{max}=a^{-3}$\nnear the surface.\n\n\nIn ACEO, charging dynamics are very important, so the double-layer\ncapacitance is an important property.  Dilute-solution theory predicts\nthat the differential capacitance, $C_D$, diverges with the voltage\nas,\n\\begin{equation}\n  C_{D}(\\Psi_D)=\\frac{\\varepsilon_b}{\\lambda_{D}}\\cosh\\left(\\frac{ze\\Psi_{D}}{2kT},\n  \\right) \\label{eq:cdpb}\n\\end{equation}\nwhere $\\Psi_D$ is the voltage applied across the double layer.  For a\nconcentrated solution described by Bikerman's model, we predict the\nopposite trend~\\cite{kilic2007a},\n\\begin{equation}\nC_{D}^{\\nu }=\n\\frac{\\frac{\\varepsilon}{\\lambda_{D}}\\sinh(\\frac{ze\\Psi_{D}}{kT})}\n{[1+2\\nu\\sinh^2\\left(\\frac{ze\\Psi_{D}}{2kT}\\right)]\n\\sqrt{\\frac{2}{\\nu} \\mathrm{ln}[1+2\\nu\\sinh^2\\left(\\frac{ze\\Psi_{D}}{2kT}\\right)]}},\n\\label{eq:cdnu}%\n\\end{equation}\nwhere $\\nu=2 c_0 a^3$ is the bulk volume fraction of\nions~\\cite{kilic2007a}. (Note that the same formula was derived by\nKornyshev~\\cite{kornyshev2007} in the context of ionic liquids around\nthe same time that Kilic et al.~\\cite{kilic2007a} derived it in the\npresent context of electrolytes.) \n\nAs shown in Fig.~\\ref{fig:C}(b), with this model the capacitance\nreaches a maximum near the critical voltage then {\\it decreases} at\nlarger voltages because the capacitor width grows due to steric\neffects.  The same result also follows from a simpler model using PB\ntheory outside a condensed layer of variable width at the maximum\ndensity ~\\cite{kilic2007a}.  We will show that this general decrease\nin capacitance with voltage that can drastically impact the charging\ndynamics in ACEO flows.\n\nAs Biesheuvel {\\it et al.} point\nout~\\cite{biesheuvel2005,biesheuvel2007}, Bikerman's latticed-based\napproach can underestimate the true steric effect to a large degree,\nand more accurate models of the entropy of a hard-sphere liquid, such\nas the Carnahan-Starling (CS) equation of state or various extensions\nto multicomponent mixtures~\\cite{carnahan1969,lue1999}, can be used\ninstead.  The CS equation is not tractable for a closed-form\nanalytical expression for double-layer capacitance, but we can easily\nobtain it numerically as a function of the voltage. For the details,\nsee the Appendix.\n\nAlthough various mean-field models, such as CS, are very accurate for\nequilibrium properties of the hard-sphere gas and liquid at low to\nmoderate volume fractions, $\\Phi < 0.55$, none can claim to accurately\ndescribe the packing limit at large volume fractions, which would\ngenerally lead to the decrease in double-layer capacitance at large\nvoltage~\\cite{kilic2007a}. For example, these models predict a\ndiverging chemical potential only at $\\Phi=1$, and not and not at the\nmathematical packing bound ($\\Phi=0.74$, corresponding to the\nface-centered cubic lattice) or the more relevant bound of the jamming\ntransition (around $\\Phi=0.63$ for hard spheres\n~\\cite{torquato00}). Even in simple model systems in equilibrium, the\njamming transition is poorly understood\n~\\cite{liu98,liu02,ohern02,ohern03,donev04}, so none of these models\ncan possibly be expected to accurately capture the dynamics of jamming\nof hydrated ions very close to a surface driven by a large AC voltage,\nwhile also producing ACEO flow. Of course, under such extreme\nconditions, the mean-field approximation for both steric and\nelectrostatic interactions should also be questioned.\n\nDue to all of these complexities, following Bazant {\\it et\n  al.}~\\cite{large}, we will focus on the simplest approximation of\nBikerman's model in order to explore the qualitative impact of steric\neffects on ACEO flow. It will turn out that this very simple model\nsuffices to predict high frequency flow reversal of ACEO pumps. We\nwill then repeat some calculations with the CS model, which improves\nthe agreement with experiments somewhat, but does not alter the\nqualitative predictions of Bikerman's model.\n\n\n\\section{Model for ACEO pumps}\n\n\nIn our theoretical study of ACEO pumping, we consider the standard\nexperimental geometry of planar, asymmetric electrode-array pumps\nshown in Fig. \\ref{fig:schematic} due to Brown {\\it et\n  al.}~\\cite{brown2001}.  The specific dimensions are selected to\nmatch the experiments Studer \\textit{et. al} \\cite{studer2004} and\nUrbanski {\\it et al.}  ~\\cite{urbanski2006,microTAS2007}.  The basic\nquestion of ACEO flow in such pumps has been thoroughly addressed by\nOlesen {\\em et. al} ~\\cite{olesen2006,olesen_thesis} using the\nclassical electrokinetic equations of dilute solution\ntheory~\\cite{lyklema_book_vol2,hunter_book}.  The significant\ndifference between this work and Ref. \\cite{olesen2006} is our\napplication of the new physical model for the double-layer\ncapacitance, taking into account steric effects at large\nvoltage. Unlike Ref.~\\cite{olesen_thesis}, we also ignore ``strongly\nnonlinear'' effects associated with salt uptake by a highly charged\ndiffuse layer and resulting bulk diffusion~\\cite{bazant2004,chu2006},\nsince these phenomena are greatly reduced by steric\neffects~\\cite{kilic2007a,kilic2007b}.\n\nIn the bulk fluid outside the\nelectric double layers, we assume electroneutrality with a constant\nelectrical conductivity, $\\sigma$.  Under these assumptions, the\nelectric potential in the fluid bulk, $\\phi$, satisfies Laplace's\nequation, which is equivalent to Ohm's law for a constant bulk\nresistance. This corresponds to the ``weakly nonlinear'' regime where\nnonlinear circuit models still\nhold~\\cite{bazant2004,chu2006,olesen_thesis}.  For boundary conditions\non the potential, we assume that there is no normal Ohmic current on\nthe sections of channel substrate at $y=0$ and the upper boundary at\n$y=H$.\n\nFor the boundary condition over the electrodes, we must incorporate a\nmodel of the thin electric double layers that will form.  These double\nlayers are so thin (10-100 nm), that when we apply the boundary\ncondition on $\\phi$ at $y=0$, we assume this is outside the double\nlayer and not at the electrode surface.  The local charge stored in\nthe double layer, $q(x)$, changes in time due to the Ohmic current.\nOur double layer model provides the voltage difference between the\nfluid bulk and the externally applied voltage at the electrode,\n$V_{ext}$, based on the amount of charge stored.  Thus the boundary\ncondition for the electric potential in the bulk is dynamically\ndetermined.  While Faradaic current certainly play a role in ACEO, we\nneglect it in our model to focus on steric effects alone, which have\nnot previously been considered. The role of reactions in classical\ndouble-layer models in ACEO was studied by Olesen {\\em\n  et. al}~\\cite{olesen2006} and shown not to play a role in\nhigh-frequency flow reversal.\n\n\nAs in prior work~\\cite{ajdari2000,olesen2006}, we introduce the following\nscales for the nondimensionalization of the governing equations,\n\\[\n    \\quad[x,y] = L,\\quad[t] = \\frac{L C_o}{\\sigma}, \\quad[\\phi ]=\\frac{k_B T}{z e},\\quad [q]=\\frac{k T }{z e }C_o.\n\\]\nNote that $[t]$ is the ``RC time'' for the equivalent\ncircuit~\\cite{ramos1999,bazant2004}, where the characteristic Debye\ncapacitance is $C_o = \\frac{\\varepsilon}{ \\lambda_D}$ where\n$\\varepsilon$ is the permittivity of the electrolyte and $\\lambda_D$\nis the Debye length.  The characteristic length scale for computing\nthe bulk resistance, $L\/\\sigma$, is the electrode spacing $L=G_1$.  We\nassume no Stern layer with our new double layer model, though its\ninclusion would be trivial and would not significantly alter our\nresults.\n\nThe dimensionless problem reduces to Laplace's equation for the\npotential in the fluid bulk,\n\\begin{equation}\n\\nabla^2 \\phi = 0. \n\\label{eq:lap}\n\\end{equation}\nThe boundary condition on the \nsubstrate (the entire upper wall at $y=H\/L$ and the insulating \nregions along $y=0$ between the electrodes) is\n\\begin{equation}\n\\frac{\\partial \\phi}{\\partial y}=0, \n\\label{eq:bc1}\n\\end{equation}\nThe boundary condition over the electrode double layer\nis given by, \n\\begin{equation}\n  \\phi =    V_{ext} - \\Psi_D(q), \n\\label{eq:bc2}\n\\end{equation}\nwhere  $\\Psi_D(q)$ is the \ntotal potential drop across the  double layer. The functional \nrelationship between   the double layer voltage drop and charge\nis given by the \nmodel from Kilic \\textit{et al.} \\cite{kilic2007a},\n\\begin{equation}\n\\Psi_D =  - 2~ \\mbox{sign}(q)~    \\mbox{sinh}^{-1}  \\left( \\sqrt{ \\frac{1}{2 \\nu} \\left(  \\mbox{e}^{\\frac{q^2 \\nu}{2}}-1 \\right)}  \\right),\n\\label{eq:q_full}\n\\end{equation}\nwhile the local charge is dynamically determined by the ``RC''\ncondition,\n\\begin{equation}\n\\frac{dq}{dt} = \\left. \\frac{\\partial \\phi}{\\partial y} \\right|_{y=0}.\n\\label{eq:charge}\n\\end{equation}\n\n\n\nThe closed set of equations (\\ref{eq:lap})-(\\ref{eq:charge}) form our\nelectrical model.  Note that in the limit of very small $\\nu$ in\nEq. \\ref{eq:q_full} we obtain the classic result from\nPoisson-Boltzmann, $\\Psi_D = -2~ \\mbox{sinh}\\left(\\frac{q}{2}\n\\right).$ In the limit of small potential and small $\\nu$ we obtain\n$\\Psi_D = -q,$ the linear Debye-Huckel model.  The later two limits\nhave been extensively studied \\cite{olesen2006}; it is the impact of\nEq \\ref{eq:q_full} on the flow that is the focus of this paper.\n\nOnce the electrical problem is solved, we compute the electroosmotic\nslip by the Helmholtz-Smoluchowski relationship,\n\\begin{equation}\nU =   \\Psi_D \\frac{\\partial \\phi(y=0)}{\\partial x}.\n\\label{eq:velocity}\n\\end{equation}\nand ignore any modifications at large voltage, e.g. due to crowding\neffects~\\cite{large}, in order to focus on steric effects in the\ndouble-layer charging dynamics. Velocity is made dimensionless by the\nnatural scale~\\cite{ramos1999,iceo2004a},\n\\[\n[U] = \\frac{\\varepsilon V^2}{\\eta L}\n\\]\nwhere $V$ is the magnitude of the applied voltage and $\\eta$ is the\nviscosity.  An interesting fact, exploited by previous\nauthors~\\cite{ajdari2000,gonzalez2000,olesen2006}, is that we can\ncompute the net pumping without actually solving the flow field in the\nchannel, but by taking the time and spatial average of equation\n\\ref{eq:velocity} along the boundary $y=0$.  Further details of this\nderivation can be found elsewhere, e.g. in Olesen {\\em\n  et. al}~\\cite{olesen2006}.\n\nDue to the irregular electrode geometry and the non-linear charging\nprocesses, we must resort to solving Equations\n(\\ref{eq:lap})-(\\ref{eq:charge}) numerically.  We couple the solution\nof Laplace's equation to the dynamic boundary condition for the\ncharge.  The solution is integrated forward in time until an adequate\nsteady state is reached and the net flow is computed by averaging\nEq. \\ref{eq:velocity} in time and space. Details on the numerical\nmethod can be found in the supplementary material \\footnote{ See EPAPS\n  Document No. XXXX for details of the numerical method. For more\n  information on EPAPS, see http:\/\/www.aip.org\/pubservs\/epaps.html }.\nThe simulations were confirmed by an independent method using the \ncommercial software package COMSOL Multiphysics. \n\n\n\n\n\n\\section{Results}\n\n\\begin{figure}\n\\centering\n\\epsfig{file=figure4.eps,width=2.8in}\n\\caption{Average slip velocity as a function of frequency for three different\n  models at 2.5 Volts.  The solid blue curve is for $\\nu=0.01$ with\n  the complete model, the black dash-dot line assumes the linear\n  Debye-Huckel relationship $q=-\\Psi_D$, the red dashed curve is for\n  $\\nu=0$ with a Stern layer capacitance of $\\delta=0.1$, Only the\n  model with steric effects predicts flow in the negative\n  direction. Points A, B, and C will be referred to in\n  Fig. \\ref{fig:flow}.  }\n     \\label{fig:models}\n\\end{figure}\n\nWe start by comparing our results with the steric model for the double\nlayer, Eq. (\\ref{eq:q_full}), to simulations of the dilute-solution\nmodels used in all prior work. In Fig. \\ref{fig:models}, we compare\nthe flow velocity as a function of frequency for the classic linear\nDebye-Huckel model valid only when the applied voltage is much less\nthat $k T\/e$, the Poisson-Boltzmann non-linear capacitance which is\nrecovered from our model when $\\nu=0$, and the model which accounts\nfor steric effects, Eq. \\ref{eq:q_full}.  At a voltage of $100\nkT\/e=2.5$ V, we see that the model which accounts for steric effect\nshows a high frequency flow reversal.  As we note in\nRef.~\\cite{large}, to our knowledge, these results represent the first\ndemonstration of a mathematical model that can predict such reversal\nof flow in ACEO pumps.\n\n\\begin{figure*}\n\\centering\n  \\epsfig{file=figure5.eps,width=5.8in,clip=} \n  \\caption{\\label{fig:mechanism} Basic mechanism for flow in ACEO\n    devices. Here we consider a symmetric array of electrodes under a\n    suddenly applied voltage.  In (a) we show electric field lines at\n    the instance the field is turned on. The electric field points\n    from the positive to the negative electrode.  The location on the\n    electrodes at $y=0$ is shown.  In (b) we show the charge density\n    over the electrodes at $1\/10$ the charging time $t_c$, at the\n    charging time, and at $10$ times the charging time. In (c) we show\n    streamlines of the electric field at the charging time. In (d) we\n    show electric field lines at the charging time. The arrows denote\n    the flow direction.  }\n\\end{figure*}\n\nIn order to understand the physical mechanism of flow reversal, we\nmust first understand why there is flow at all.  The basic mechanism\nfor flow in ACEO can be best understood when considering a\ngeometrically symmetric electrode array, as in Fig\n\\ref{fig:mechanism}, subjected to a suddenly applied DC voltage\n(without any Faradaic current) ~\\cite{encyclopedia_ACEO}.  When the\nvoltage is initially switched on, the electric field lines point from\nthe positive to the negative electrodes as shown in Fig\n\\ref{fig:mechanism}(a).  Since there is initially no double layer\ncharge, the field lines are perpendicular at the electrode surface.  A\ncurrent will begin to flow through the electrolyte with ions migrating\nalong the electric field lines and the double layers over the\nelectrode will begin to charge as shown in Fig \\ref{fig:mechanism}(b).\nThe electric field is strongest at the electrode edges and the\nresulting high current quickly charges the double layer at that\nlocation.  On the order of the characteristic charging time, $t_c$,\n(as computed using the gap length scale) the electrode edges will be\nsignificantly screened.  In Fig \\ref{fig:mechanism}(b) we show the\ncharge at $y=0$ at three instances in time.\n\nAs the double layers charge up, the electric field lines in the fluid\nbulk are diverted around the screened electrode edges, resulting in a\ntangential component of the electric field.  This is shown in Fig.\n\\ref{fig:mechanism}(c) where we can see the field lines are no longer\nperpendicular over the entire electrode, especially near the electrode\nedges.  The tangential electric field exerts a force on the fluid in\nthe charged double layer, causing flow.  Given the direction of the\nfield and sign of the charge, the flow always goes from the edge of\nthe electrodes, inward.  The flow streamlines and flow direction are\nshown in Fig. \\ref{fig:mechanism} (d).  If the applied voltage is\nmaintained, eventually the entire electrode becomes fully screened\nthere is no electric field in the bulk and the flow will stop.  The\nabove argument about the direction of flow did not depend upon the\nsign of the applied voltage, reversing the sign causes the same flow\npattern.  Therefore, if an AC voltage is applied at a frequency\ncorresponding to the charging time, time averaged vortex flow can be\nmaintained.\n\n\n\n\n\\begin{figure*}\n\\centering\n  \\epsfig{file=figure6.eps,width=5.8in,clip=} \n  \\caption{\\label{fig:flow} Streamlines (a,c,e) and slip velocity\n    (b,d,f) for three different cases. The first row is for $V=100$,\n    $\\nu=0.01$, and $\\omega = 0.7$. This frequency corresponds to the\n    peak in the forward flow for the steric model. In (a) we show the\n    streamlines. In (b) we show the time-averaged slip velocity (solid\n    line) and the cumulative spatial integral of this velocity (the\n    dashed curve). The second row, figures (c) and (d) is for $V=100$,\n    $\\nu=0.01$, and $\\omega = 3$. This frequency corresponds to the\n    peak in the reversed flow. The third row, figures (e) and (f) is\n    for $V=100$ and $\\omega = 3$ with the linear Debye-Huckel model\n    $\\Psi_D=-q$.  }\n\\end{figure*}\n\nAjdari introduced the idea of rectifying this symmetric flow pattern\nto create an ACEO pump by introducing various forms of geometrical\nasymmetry within each spatial period of the electrode array\n\\cite{ajdari2000}. This principle was first implemented by Brown {\\it\n  et al.} using flat, co-planar electrodes of different widths and\ngaps~\\cite{brown2001}, as shown in Fig. \\ref{fig:schematic}, and this\ngeneral design has been studied extensively by several\ngroups~\\cite{ramos2003,mpholo2003,studer2004,olesen2006,\nolesen_thesis,gregersen2007,urbanski2006,microTAS2007}.\nIn this geometry, the symmetry breaking causes the flow to go from\nleft to right by upsetting the delicate balance between leftward and\nrightward slip patterns. The fact that pumping comes from a relatively\nsmall bias of opposing, co-planar surface flows makes this design\nsusceptible to flow reversal if nonlinearities in charging dynamics\ntip the balance the other way, as we demonstrate below. Current\ndesigns using stepped, three-dimensional electrodes are much faster\nand more robust against flow reversal, since the opposing slip\npatterns work together to drive pumping in one direction, as a ``fluid\nconveyor\nbelt''~\\cite{bazant2006,urbanski2006,urbanski2007,burch_preprint}. The\nsensitivity to flow reversal in the planar pump, however, allows us a\nbetter opportunity to explore subtle nonlinearites in double-layer\ncharging dynamics, which is why we have chosen it for our theoretical\nstudy.\n\n\n\nIn Fig. \\ref{fig:flow} we demonstrate how the geometrical asymmetry\nresults in forward flow when driven at the charging time and how\nsteric effects can produce high frequency reversal. In\nFig. \\ref{fig:flow}(a) and (b) we show the time averaged streamlines\n(a) and time average slip velocity (b) for the ACEO pump driven at the\ncharging time, corresponding to point A in Fig. \\ref{fig:models}.  We\nnotice a few features of the slip profiles.  First, the magnitude of\nvelocity over the small electrode is greater than the velocity over\nthe large electrode. This difference is due to the fact that the {\\em\n  total} charge contained on both electrodes must be the same, the\nlocal charge density (and thus electroosmotic slip) is always higher\non the small electrode.  Second, we notice slip profiles over either\nelectrode is asymmetric with the greatest slip found at the edges at\nthe small gap. This effect is simply due to high electric field in the\nsmall gap which can exert a larger force.  Finally we see the\nasymmetry in the slip is greater for the large electrode than the\nsmall electrode, an effect due to the geometry.  The result for this\ngeometry and frequency is the net flow goes in the positive $x$\ndirection.  The relative contributions to the flow is easily seen by\nthe cumulative spatial integral of the time averaged slip; the dotted\nline in Fig. \\ref{fig:flow} (b).\n\n\nIn Fig. \\ref{fig:flow} (c) and (d) we show the behavior at the\napproximate peak of the reverse flow in Fig. \\ref{fig:models},\ncorresponding to point B in Fig. \\ref{fig:models}.  The same generic\nfeatures of the slip profiles are the same as in (b). However, the\nrelative contributions have changed at this frequency.  The reason for\nthe reversal can be explained by the change in charging time for the\ndifferent sized electrodes.  When steric effects are included, we must\nremember that the capacitance decreases as more charge is stored\ninside the double layer.  Since the charge density over the small\nelectrode is much greater, the capacitance of the small electrode's\ndouble layer is {\\em decreased} at high voltage due to the steric\neffects. The lower capacitance means that the charging time of the\nsmall electrode is decreased relative to the large electrode. This\nchange in charging time means that at high frequency and high voltage\nthe small electrode has more time to sufficiently charge to generate\nflow.\n\nThis idea is illustrated in Fig. \\ref{fig:flow} (c) and (d).  The\ncumulative spatial integral of the time-averaged slip shows that the\nnegative flow contribution from the small electrode is nearly twice\nthe positive contribution. At this high frequency the large electrode\nis not very highly charged and its forward flow component adds only a\nsmall amount to the net.  That this reversal is caused by the change\nin the charging dynamics can be seen in Fig. \\ref{fig:flow} (e) and\n(f), which shows the same condition as (c) and (d) only with the\nlinear Debye-Huckel model; point C in Fig. \\ref{fig:models}.  Here,\nsince the large electrode has the same charging time as the small\nelectrode, its forward component is significant to provide a net\npositive flow.  It is clear from these pictures that with the\ninclusion of steric effects, the direction and magnitude of flow in\nACEO pumps is determined by a delicate balance influenced {\\em both}\nby the geometry and the charging dynamics.\n\n\\begin{figure*}\n\\centering\n\\epsfig{file=figure7.eps,width=5.8in,clip=}\n\\caption{Flow response as a function of voltage for various values of\n  $\\nu$, the steric parameter. We show values of $\\nu=0.01$ (a),\n  $\\nu=0.001$ (b), $\\nu=0.0001$ (c), $\\nu=0$ (d), The contours show\n  constant levels of total flow rate. The heavy line shows the $U=0$\n  contour and the region above this line denotes the region where flow\n  reversal occurs. }\n     \\label{fig:steric_nu}\n\\end{figure*}\n\nUp to now, we have taken the volume fraction, $\\nu$, to be a constant.\nTo better see the effect of $\\nu$ on the predicted flow, in\nFig. \\ref{fig:steric_nu} we show contours of the flow rate as a\nfunction of voltage and frequency at different values of $\\nu$.  For a\nfixed physical value of $a$, different values of $\\nu$ represent how\nthe flow would change as a function of concentration (in dimensionless\nterms).  The heavy lines denote the region where the flow is negative,\nindicating a reversal of flow.  We see that the frequency response of\nthe flow does not change with voltage at low voltage, as would be\npredicted by the linear Debye-Huckel model.  At low voltage, we see\nonly a single peak in the forward flow at the characteristic\nfrequency.  As the voltage is increased the nonlinearity begins to set\nin which causes the peak of the forward flow to move toward lower\nfrequency as predicted by dilute solution theory.  As the voltage is\nfurther increased the steric effects begin to become important with\nthe reversal eventually setting in.\n\n\n\n\nThe change in frequency of the maximum forward flow at higher voltage\nas $\\nu$ changes is easily understood. As $\\nu$ increases, the value\nof the maximum value of the capacitance decreases, shifting the peak\nresponse to higher frequencies.  For a given $\\nu$, as voltage\ncontinues to increase, the continual decrease in the double layer\ncapacitance can be seen in the shift of the maximum of the positive\nflow to higher frequency at high voltage. When there are no steric\neffects (or Stern layer) the capacitance diverges, which can be seen\nin Figure \\ref{fig:steric_nu}(d).  The inclusion of the constant Stern\nlayer capacitance causes this decrease to limit at the constant Stern\nlayer capacitance \\cite{olesen2006}.\n\nQualitatively, our model can predict a flow \nresponse that looks very similar to experimental\nresults, however the quantitative ability of our model is still problematic. \nIn Fig.  \\ref{fig:studer}(a) we show the model predicting\nthe experimental data of\nStuder {\\it et al.} \\cite{studer2004}. In  Fig. \\ref{fig:studer} we treat\nthe parameter  $\\nu$ as a fitting parameter to approximate the\ndata of Figure 6 in Studer's experimental work \\cite{studer2004}. We\nfind a value of $\\nu=0.01$\nprovides a reasonable fit to their experimental data.\nHowever, given a concentration of 0.1 mM KCl,  this value \ncorresponds to   $a=4.4$ nm, about\nan order of magnitude larger than we would expect based on the hydrated\nradius of the ion. \nThe fact their experiments \\cite{studer2004}\nonly predicts reverse flow at high voltage (their Fig. 7) \nis consistent with the low\nfrequency regime of their work being unaccessible due to\nelectrochemical reactions and degradation of the electrode (see their Fig. 4).\n\n\\begin{figure}\n\\centering\n(a)\\includegraphics[width=2.8in]{figure8a.eps}\n(b)\\includegraphics[width=2.8in]{figure8b.eps}\n\n     \\caption{Flow as a function of frequency at different voltages. The voltages are\n       $V = 0.5,1,1.5,2.9,3.6,4.5,$ and $6$ Volts RMS. The voltages are\n       selected to match Figure 6 in Studer {\\em et al.} \\cite{studer2004}.\n       The data is presented in dimensional form in order to \n       compare the frequency response.  \n       The flow velocity is corrected for the hydraulic resistance of the\n       experimental flow loop. In (a), using the Bikerman model, we \n       used $a=4.4$ nm. \n       In (b), using the Carnahan-Starling model, we \n       used $a=2.2$ nm. The general shape of both frequency response\n       curves have many of the experimentally observed features.  \n     }\n     \\label{fig:studer}\n\\end{figure}\n\n\n\nWe considered whether the Carnahan-Starling (CS) form of the excess\nchemical potential would provide a more reasonable value of $a$.  In\norder to include the CS equation in the model we numerically solve for\nthe relationship of $\\Psi_D(q)$ to replace Eq. \\ref{eq:q_full} in our\nmodel. We follow the work of Bieshuvel and Soestbergen\n\\cite{biesheuvel2007} to obtain this relationship numerically, as\ndescribed in the Appendix.  The result of using the CS equation in our\nACEO model is shown in Fig. \\ref{fig:studer} (b), and it is\nqualitatively similar to our prior results with the Bikerman\nmodel. The CS steric model has somewhat different quantitative\nbehavior but still requires an ion size of approximately $a=2.2$ nm to\nfit the experimental data.  While the magnitude of the flow reversal\nchanges with the CS equation, the use of a more accurate model for the\nexcess chemical potential still require a relatively large ion size.\n\n\n\\section{Discussion}\nIn order to fit the experimental data, the ACEO models modified for\nsteric effects imply an ion length scale, $a$, almost an order of\nmagnitude larger than we would expect.  A factor of 10 in ion size\nresults in a factor of 1000 in $\\nu$. This difference means that while\nthe response in Fig. \\ref{fig:steric_nu}(a) is closer to what is\nobserved, it is Fig. \\ref{fig:steric_nu} (c) that is predicted based\non a reasonable hydrated radius for the ions.  The current model\ncannot yet predict the experimental data from first principles.  It is\nimportant to note that the large value of $\\nu$ is needed not to\npredict flow reversal at all, but to predict flow reversal at the\nfrequencies observed in experiments.\n\nAs discussed in section ~\\ref{sec:steric}, the mean-field models of\nsteric effects which we consider are unlikely to accurately capture\nthe dynamical formation of a condensed layer on the surface. One\nreason is the difficulty in describing the jamming transition, as ions\nand solvent molecules are squashed onto the surface by large,\ntime-dependent normal electric fields. Under such extreme conditions,\nwe should not expect more than qualitative trends from such\nmodels. Another possibly crucial feature missing in these models is\nthe interaction with the surface, which is assumed to be a\nmathematically flat, homogeneously charged, perfectly rigid wall, as\nsketched in Fig. 2. Since the condensed layer of (presumably solvated)\nions is at the scale of only a few molecular lengths, there must be a\nstrong coupling with nanoscale roughness on the surface. This may have\nthe effect of increasing the effective ion size, as we have inferred,\nsince what matters is not the ion density in the solution, but the\ndensity very close to the surface, which is reduced by atomic\nasperities. The viscosity may also diverge near the surface as ions\nbecome jammed among surface inhomogeneities~\\cite{large}. It would be\ninteresting to do experiments with atomically flat electrodes\n(e.g. carbon graphene) or ones with controlled nanoroughness to test\nthis hypothesis.\n\n\nIt is also possible that changes in permittivity, $\\varepsilon$,\nwithin the double layer may also effect the frequency response.\nElectrochemists infer for the Stern layer that $\\varepsilon$ may be\n$1\/10$ of $\\varepsilon$ in the bulk ~\\cite{bockris_book}.  If it were\ntrue that $\\varepsilon$ is reduced by $1\/10$ in the condensed layer,\nthe overall double layer capacitance would be decreased.  Such change\nwould cause the peaks in the frequency response to shift to higher\nfrequency for the same value of $\\nu$.  If we make the assumption that\n$\\varepsilon$ is reduced by a factor of ten in the {\\em entire} double\nlayer, then an ion size of approximately 1 nm is needed to predict\nStuder {\\em et al.}'s \\cite{studer2004} data with the CS equation.  If\n$\\varepsilon$ is reduced only in the condensed layer, the true effect\non the frequency response would be less than predicted with the simple\nassumption. Therefore, it seems likely that accounting for changes in\n$\\varepsilon$ would reduce the needed ion size to correlate the data,\nhowever at most this would reduce the ion size to 1 nm, still much\nlarger than expected.\n\n\nA further complication is that the mean-field approximation breaks\ndown when ion spacings approach the Bjerrum length, $l_B =\n(ze)^2\/4\\pi\\varepsilon kT$, which is 7~ \\AA ~ for bulk water and\nmonovalent ions ($z=1$).  Again, if it were true that $\\varepsilon$ is\nreduced by 1\/10 in the condensed double layer, then $l_B=7$ nm.\nCorrelation effects on electro-osmotic flow (which to our knowledge\nhave never been studied) could be very significant at large voltages,\neven in dilute bulk solutions~\\cite{large}. \n\n\n\nWhile much work remains, we have highlighted an interesting and\npractically important application where dilute solution theory clearly\nbreaks down.  We have demonstrated that accounting for steric effects\nin double-layer models can have a dramatic impact on the predicted\nflow in ACEO pumps.  Because of the large value of $a$ needed to fit\nthe experimental data, we cannot claim that our model definitively\nexplains high-frequency flow reversal in ACEO pumps, but it is the\nonly plausible explanation to date.  The prediction is also quite\nrobust: we have found that flow reversal can be predicted by any model\nwhich accounts steric effects by the generic feature that the double\nlayer capacitance decreases with applied voltage.  \n\nPhysically, it seems that this generic effect of decreasing\ncapacitance must be true for any model of the double layer considering\nfinite sized ions~\\cite{kilic2007a}.  As the voltage increases, more\nions are packed in the double layer. Once the concentration of ions\nbecomes so high that the packing density is reached, the ions would\nhave to pile up forming a condensed layer whose thickness would\nincrease with voltage. As the double layer thickness grows, the\ncapacitance must decrease.  We believe this general effect is true\nregardless of the details of the model, and we have shown that it has\nmajor implications for ACEO pumping and, by extension, all nonlinear\nICEO flows. Steric effects are clearly among the new effects which\nmust be considered in developing more accurate theories of ICEO flow\nat large applied voltages~\\cite{large}.\n\n\\section*{Acknowledgments}\nThis work was supported in part by the National Science Foundation\nunder contract DMS-0707641 and by the U.S. Army through the Institute for\nSoldier Nanotechnologies, under Contract DAAD-19-02-0002 with the\nU.S. Army Research Office. MZB also acknowledges support from ESPCI\nthrough the Joliot Chair.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Introduction}\n\t\t\n\t\tHurwitz numbers are a classical topic in combinatorics and algebraic\n\t\tgeometry; they date back to the work by Adolf Hurwtiz \\cite{AdHurwitz}\n\t\t(1890). A classical definition of the Hurwitz number $h_{m,\\lambda}$\n\t\twhere $m$ is a non-negative integer and $\\lambda=(\\lambda_1 \\DT\\ge\n\t\t\\lambda_s)$ is a partition, is: $h_{m,\\lambda}$ is the number of\n\t\tsequences of $m$ transpositions $(\\sigma_1 \\DT, \\sigma_m)$ such that\n\t\ttheir product $\\sigma_1 \\DT\\circ \\sigma_m$ belongs to a conjugacy\n\t\tclass $C_\\lambda \\in S_n$ in the symmetric group $S_n$. By a classical\n\t\tresult \\cite{GouldJacks}, generating function of the Hurwitz numbers\n\t\tsatisfies a second order parabolic PDE called the cut-and-join\n\t\tequation. Its right-hand side has Schur polynomials as eigenvectors,\n\t\tso the Hurwitz numbers can be expressed via them\n\t\t\\cite{LandoKazarian}. The result by A.\\,Okounkov about Toda lattice\n\t\t\\cite{Okounkov} implies that the same generating function is a\n\t\tsolution of the KP hierarchy (the proof uses tools coming from\n\t\tmathematical physics, namely the boson-fermion\n\t\tcorrespondence). Finally, in a recent work \\cite{TwistedHurwitz} by\n\t\tYu.Burman and the author Hurwitz numbers are shown to solve a problem\n\t\tin low-dimensional topology: they are the numbers of ways to glue $m$\n\t\tribbons to a collection of $n$ disks so as to obtain a surface with a\n\t\tprescribed structure of the boundary.\n\t\t\n\t\tIn this paper we show that there is a parallel theory of Hurwitz\n\t\tnumbers for the reflection groups of series B and D. Section\n\t\t\\ref{Sec:BD} is the starting point where we recall the classical\n\t\tembedding of $B_n$ and $D_n$ into the symmetric group $S_{2n}$; the\n\t\treflections are mapped to transpositions and products of pairs of\n\t\ttranspositions commuting with the involution $\\t = (1,n+1) (2,n+2)\n\t\t\\dots (n,2n)$ (for the group $D_n$, only pairs of transpositions are\n\t\tused). Also we recall a description of the conjugacy classes in these\n\t\tgroups, taken from \\cite{Carter} (see Proposition \\ref{Pp:ConjClass};\n\t\tthe classes are generally indexed by pairs of partitions $\\lambda$ and\n\t\t$\\mu$ such that $\\lmod \\lambda\\rmod + \\lmod \\mu\\rmod = n$).\n\t\t\n\t\tSection \\ref{Sec:HurwNum} gives the actual definition of Hurwitz\n\t\tnumbers for the groups $B_n$ and $D_n$: it is similar to the classical\n\t\tone with reflections instead of transpositions. For the B series, we\n\t\tcount reflections of two classes (transpositions and pairs of\n\t\ttranspositions) separately.\n\t\t\n\t\tIn Sections \\ref{Sec:Expl} and \\ref{Sec:CJD} we study how the\n\t\tmultiplication by a reflection affects conjugacy classes; this gives\n\t\tus an expression for the cut-and-join operator for the groups of\n\t\tseries B and D.  After a suitable change of variables we see that the\n\t\toperator obtained for the B group is actually a tensor square of the\n\t\tclassical cut-and-join. The cut-and-join for the D group is a direct\n\t\tsum of the cut-and-join for B and the classical cut-and-join\n\t\trescaled. This leads to an expression of the Hurwitz numbers for B and\n\t\tD via Schur polynomials.\n\t\t\n\t\tIn Section \\ref{SSec:Ribbon} we provides a model for B-Hurwitz\n\t\tnumbers involving ribbon decomposition of surfaces equipped with an\n\t\tinvolution. Finally, in Section \\ref{SSec:Fermion} we apply the\n\t\tboson-fermion correspondence to the tensor square of the cut-and-join\n\t\tdescribed above; it allows one to show that the generating function of\n\t\tthe B-Hurwitz numbers is a $2$-parameter family of $\\tau$-functions of\n\t\tthe KP hierarchy, independently in $2$ variables, and to prove a\n\t\tsimilar result for D.\n\t\t\n\t\t\\subsection*{Acknowledgements}\n\t\t\n\t\tThe author wants to thank his advisor Yurii Burman for many many\n\t\tfruitful discussions and constant attention to this work.\n\t\t\n\t\tThe research was partially funded by the HSE University Basic Research\n\t\tProgram and by the International Laboratory of Cluster Geometry NRU\n\t\tHSE (RF Government grant, ag.~No.~075-15-2021-608 dated 08.06.2021)\n\t\t\n\t\t\\section{Reflection groups $B_n$ and $D_n$}\\label{Sec:BD}\n\t\t\\subsection{Definition and embedding into $S_{2n}$}\n\t\t\n\t\tLet $B_n$ be a rank $n$ finite reflection group with the mirrors\n\t\t$x_i-x_j = 0$, $x_i+x_j = 0$ ($1 \\le i < j \\le n$) and $x_i = 0$ ($1\n\t\t\\le i \\le n$); the corresponding reflections are denoted $s_{ij}^+$,\n\t\t$s_{ij}^-$, and $\\ell_i$, respectively (see \\cite{Humphreys} for details\n\t\tand standard notation). The subgroup $D_n \\subset B_n$ is generated by\n\t\t$s_{ij}^{\\pm}$ only.\n\t\t\n\t\tConsider an involution $\\t \\in S_{2n}$ without fixed points. All such\n\t\tinvolutions are conjugate, and the results that follow do not depend\n\t\ton a particulate choice; still, fix $\\t \\stackrel{\\mbox{\\tiny def}}{=} (1,n+1)(2,n+2) \\dots\n\t\t(n,2n)$ (that is, $\\t(i) = i+n \\bmod 2n$ for all $i$).\n\t\t\n\t\tGroups $B_n$ and $D_n$ admit embedding into the permutation group\n\t\t$S_{2n}$. Namely,\n\t\t\n\t\t\\begin{proposition}\\label{Pp:Embed}\n\t\t\tThere exists an embedding $\\Phi: B_n \\to S_{2n}$ such that\n\t\t\t$\\Phi(s_{ij}^+) = (ij)(i+n,j+n)$, $\\Phi(s_{ij}^-) = (i,j+n)(i+n,j)$\n\t\t\tand $\\Phi(\\ell_i) = (i,i+n)$. The image $\\Phi(B_n)$ is the normalizer\n\t\t\tof $\\tau$:\n\t\t\t\\begin{equation*}\n\t\t\t\t\\Phi(B_n) = \\{\\sigma \\in S_{2n} \\mid \\sigma\\tau = \\tau\\sigma\\}\n\t\t\t\t\\stackrel{\\mbox{\\tiny def}}{=} \\name{Norm}(\\tau),\n\t\t\t\\end{equation*}\n\t\t\tand the image of the subgroup $D_n \\subset B_n$ under $\\Phi$ is the\n\t\t\tintersection of $\\name{Norm}(\\tau)$ with group $S_{2n}^+  \n\t\t\t\\subset S_{2n}$ of even permutations.  \n\t\t\\end{proposition}\n\t\t\n\t\tIntroduce a notation more convenient for our purposes: let $r_{ij}\n\t\t\\stackrel{\\mbox{\\tiny def}}{=} (i,j)(i+n,j+n)$ for all $1 \\le i \\ne j \\le 2n$, $i \\ne j\\pm n$;\n\t\taddition is modulo $2n$. So if $1 \\le i < j \\le n$ then $r_{ij} =\n\t\tr_{i+n,j+n} = \\Phi(s_{ij}^+)$ and $r_{i,j+n} = r_{i+n,j} =\n\t\t\\Phi(s_{ij}^-)$. Also denote $l_i \\stackrel{\\mbox{\\tiny def}}{=} (i,i+n)$ for all $1 \\le i \\le\n\t\t2n$; so, $l_i = l_{i+n} = \\Phi(\\ell_i)$ if $i \\le n$.\n\t\t\n\t\tA proof (rather elementary) of Proposition \\ref{Pp:Embed} is preceded\n\t\tby an explicit description of $\\name{Norm}(\\tau)$, which will be used\n\t\textensively throughout this article:\n\t\t\n\t\t\\begin{lemma}\\label{Lm:AlphaBeta}\n\t\t\tIf $x \\in \\name{Norm}(\\tau)$ then for every cycle $(\\gamma_1 \\dots \\gamma_m)$\n\t\t\tof its cycle decomposition one of the following is true:\n\t\t\t\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\\label{It:Alpha} The cycle decomposition contains another\n\t\t\t\tcycle $(\\gamma_1' \\dots \\gamma_m')$ of the same length such that\n\t\t\t\t$\\gamma_i' = \\tau(\\gamma_i)$ for $i = 1 \\DT, s$.\n\t\t\t\t\\item\\label{It:Beta} $s = 2t$ is even and $\\gamma_{i+t} =\n\t\t\t\t\\tau(\\gamma_i)$ for all $i = 1 \\DT, t$.\n\t\t\t\\end{enumerate}\n\t\t\\end{lemma}\n\t\t\n\t\tIn the two cases we speak about {\\em $\\alpha$-pairs} of cycles and\n\t\tabout {\\em $\\beta$-cycles}, respectively.\n\t\t\n\t\t\\begin{proof}\n\t\t\tLet\n\t\t\t\\begin{equation*}\n\t\t\t\tx = (\\gamma_{11} \\dots \\gamma_{1m_1}) \\dots (\\gamma_{q1} \\dots\n\t\t\t\t\\gamma_{q m_q})\n\t\t\t\\end{equation*}\n\t\t\tbe the cycle decomposition. If $\\tau x=x\\tau$ then $\\tau x \\tau^{-1}\n\t\t\t= x$. So\n\t\t\t\\begin{equation*}\n\t\t\t\tx = (\\tau(\\gamma_{11}) \\dots \\tau(\\gamma_{1m_1})) \\dots\n\t\t\t\t(\\tau(\\gamma_{q1}) \\dots \\tau(\\gamma_{qm_q}))\n\t\t\t\\end{equation*}\n\t\t\tThe cycle decomposition is unique, so every cycle\n\t\t\t$(\\tau(\\gamma_{i1}) \\dots \\tau(\\gamma_{im_i}))$ must be equal to some\n\t\t\t$(\\gamma_{j_1} \\dots \\gamma_{j m_j})$. If these two cycles are\n\t\t\tdifferent, then they form an $\\alpha$-pair. If these\n\t\t\ttwo cycles are the same then there exists some $t$ such that\n\t\t\t$\\tau(\\gamma_k) = \\gamma_{k+t\\bmod m_i}$ for all $k = 1 \\DT, m_i$. Since\n\t\t\t$\\tau$ is an involution, one has $\\gamma_k = \\tau^2(\\gamma_i)\n\t\t\t=\\tau(\\gamma_{k+t \\bmod m_i}) = \\gamma_{k+2t \\bmod m_i}$ for all $k\n\t\t\t= 1 \\DT, m_i$, which implies $m_i = 2t$, and this is a $\\beta$-cycle.\n\t\t\\end{proof}\n\t\t\n\t\t\\begin{proof}[Proof of Proposition \\ref{Pp:Embed}]\n\t\t\tAny element $x \\in B_n$ is representable as $x = a_1\\dots a_N$ where\n\t\t\tevery $a_k$ is a reflection. Denote by $e_1 \\DT, e_n$ the standard\n\t\t\tbasis in $\\mathbb{R}^n$.\n\t\t\t\n\t\t\t\\begin{lemma}\\label{Lm:ActGen}\n\t\t\t\tFor any $k = 1 \\DT, n$ one has $x(e_k) = \\pm e_\\ell$ for some\n\t\t\t\t$\\ell = 1 \\DT, n$. If $x(e_k) = e_\\ell$ then $(\\tilde a_1 \\DT\n\t\t\t\t\\tilde a_N)(k) = \\ell$, and if $x(e_k) = -e_\\ell$ then\n\t\t\t\t$(\\tilde a_1 \\DT \\tilde a_N)(k) = \\t(\\ell) (= \\ell+n)$.\n\t\t\t\\end{lemma}\n\t\t\t\n\t\t\tThe lemma is proved by an immediate induction by $N$.\n\t\t\t\n\t\t\tLet us define now the map $\\Phi: B_n \\to S_{2n}$ as $\\Phi(x) =\n\t\t\t\\tilde a_1 \\dots \\tilde a_N$. If $x = b_1 \\dots b_M$ is another\n\t\t\trepresentation of $x$ as a product of reflections, then Lemma\n\t\t\t\\ref{Lm:ActGen} implies that $(\\tilde b_1 \\dots \\tilde b_M)(k) =\n\t\t\t(\\tilde a_1 \\dots \\tilde a_N)(k)$ for every $k$. Thus, the map\n\t\t\t$\\Phi$ is well-defined and is a group homomorphism.\n\t\t\t\n\t\t\tSuppose $\\Phi(x) = \\name{id}$ (that is, $x$ belongs to the kernel of\n\t\t\t$\\Phi$). Then $(\\tilde a_1 \\dots \\tilde a_N)(k) = k$ for every $k =\n\t\t\t1 \\DT, n$, and Lemma \\ref{Lm:ActGen} asserts then that $x(e_k) =\n\t\t\te_k$ for every $k$. Thus, $x = \\name{id}$, so $\\Phi$ is an\n\t\t\tembedding.\n\t\t\t\n\t\t\tShow now that $\\Phi(B_n) = \\name{Norm}(\\tau)$. Obviously, $\\Phi(B_n) \\subset\n\t\t\t\\name{Norm}(\\tau)$ because $\\Phi(s^\\pm_{ij})$ and $\\Phi(\\ell_i)$ commute\n\t\t\twith $\\tau$. Also, $\\Phi(D_n) \\subset S_{2n}^+$ because $r_{ij} \\in\n\t\t\tS_{2n}^+$.\n\t\t\t\n\t\t\tProve now that $\\name{Norm}(\\tau) \\subset \\Phi(B_n)$. Let $x \\in\n\t\t\t\\name{Norm}(\\tau)$. By Lemma \\ref{Lm:AlphaBeta}, $x$ is a product of\n\t\t\t$\\alpha$-pairs and of $\\beta$-cycles, so to prove that $x \\in\n\t\t\t\\Phi(B_n)$ it suffices to show that any $\\alpha$-pair and any\n\t\t\t$\\beta$-cycle are products of the elements $r_{ij}$ and\n\t\t\t$l_i$ ($1 \\le i < j\\le 2n$). This is the case:\n\t\t\t\\begin{equation*}\n\t\t\t\t(i_1 \\DT, i_m)(\\tau(i_1) \\DT, \\tau(i_m)) = r_{i_1 i_2}\n\t\t\t\t\\dots r_{i_{m-1} i_m}\n\t\t\t\\end{equation*}\n\t\t\tand\n\t\t\t\\begin{align*}\n\t\t\t\t(i_1 \\DT, i_m \\tau(i_1) \\DT, \\tau(i_m)) &= (i_1 \\DT,\n\t\t\t\ti_m)(\\tau(i_1) \\DT, \\tau(i_m)) (i_1\\tau(i_1))\\\\\n\t\t\t\t&= r_{i_1 i_2} \\dots r_{i_{m-1} i_m}\n\t\t\t\tl_{i_1}.\n\t\t\t\\end{align*}\n\t\t\tTo prove that $\\name{Norm}(\\tau) \\cap S_{2n}^+ \\subset \\Phi(D_n)$ observe\n\t\t\tthat the element $x \\in \\name{Norm}(\\tau)$ is even if and only it contains\n\t\t\tan even number of $\\beta$-cycles. (Indeed, an $\\alpha$-pair is always\n\t\t\tan even permutation, while a $\\beta$-cycle is odd because its length\n\t\t\tis even.) So it suffices to prove that the product of any two\n\t\t\t$\\beta$-cycles is a product of $r_{ij}$ (for\n\t\t\t$\\alpha$-cycles if was proved above). Indeed,\n\t\t\t\\begin{align*}\n\t\t\t\t(i_1 \\dots i_m \\tau(i_1) &\\dots \\tau(i_m)) (j_1 \\dots i_q\n\t\t\t\t\\tau(ij_1) \\dots \\tau(j_q))\\\\\n\t\t\t\t&= (i_1 \\dots i_m)(\\tau(i_1) \\dots \\tau(i_m)) (j_1 \\dots i_q)\n\t\t\t\t(\\tau(j_1) \\dots \\tau(j_q)) (i_1 \\tau(i_1))(j_1 \\tau(j_1).\n\t\t\t\\end{align*}\n\t\t\tThe product of the first four cycles is already proved to be a\n\t\t\tproduct of $r_{ij}$, while for the two remaining cycles\n\t\t\tone has\n\t\t\t\\begin{equation*}\n\t\t\t\t(i_1 \\tau(i_1))(j_1 \\tau(j_1)) = r_{ij} r_{i,\\tau(j)}.\n\t\t\t\\end{equation*}\n\t\t\\end{proof}\n\t\t\n\t\tTo save space, below we will not distinguish $B_n$ and $D_n$ from\n\t\ttheir images $\\Phi(B_n), \\Phi(D_n) \\subset S_{2n}$.\n\t\t\n\t\t\\subsection{Conjugacy classes}\\label{SSec:ConjClass}\n\t\t\n\t\tConjugacy classes in the permutation group $S_m$ are in one-to-one\n\t\tcorrespondence with partitions of $m$: two elements of $S_m$ are\n\t\tconjugate if and only if they have cycles of the same lengths\n\t\t(totalling $m$) in their cycle decomposition. A similar result for the\n\t\treflection groups $B_n$ and $D_n$ can be found in \\cite{Carter}. Quote\n\t\tit here for coompleteness; be warned that the original notation of\n\t\t\\cite{Carter} differ from the one used here.\n\t\t\n\t\tFix two partitions, $\\lambda = (\\lambda_1 \\DT, \\lambda_s)$ and\n\t\t$\\mu = (\\mu_1 \\DT, \\mu_t)$ such that $\\lmod \\lambda\\rmod + \\lmod\n\t\t\\mu\\rmod = \\lambda_1 \\DT+ \\lambda_s + \\mu_1 \\DT+ \\mu_t = n$, and\n\t\tconsider a set $C_{\\lambda\\mid\\mu}$ of elements $x \\in B_n \\subset\n\t\tS_{2n}$ such that their cycle decomposition contains\n\t\t\\begin{enumerate}\n\t\t\t\\item $\\alpha$-pairs of lengths (each cycle) $\\lambda_1, \\lambda_2\n\t\t\t\\DT, \\lambda_s$;\n\t\t\t\n\t\t\t\\item $\\beta$-cycles of lengths $2\\mu_1 \\DT, 2\\mu_t$ (recall that the\n\t\t\tlength should be even).\n\t\t\\end{enumerate}\n\t\t\n\t\t\\begin{proposition}[\\protect{\\cite[Proposition 25]{Carter}}]\\label{Pp:ConjClass}\n\t\t\tThe set $C_{\\lambda\\mid\\mu} \\subset B_n$ is a conjugacy class. Every\n\t\t\tconjugacy class in $B_n$ is $C_{\\lambda\\mid\\mu}$ for some $\\lambda$ and\n\t\t\t$\\mu$ such that $\\lmod\\lambda\\rmod + \\lmod\\mu\\rmod = n$.\n\t\t\\end{proposition}\n\t\t\n\t\tFor $D_n$ the answer is slightly more complicated. Take a partition\n\t\t$\\lambda = (\\lambda_1 \\DT, \\lambda_s)$ such that $\\lmod\\lambda\\rmod =\n\t\tn$ and all $\\lambda_i$ are even. For an element $\\sigma \\in\n\t\tC_{\\lambda\\mid\\emptyset}$ write its cycle decomposition\n\t\t\\begin{equation}\\label{Eq:2LambdaEmpty}\n\t\t\t\\sigma = (\\gamma_{11} \\dots \\gamma_{1\\lambda_1})(\\tau(\\gamma_{11})\n\t\t\t\\dots \\tau(\\gamma_{1\\lambda_1})) \\dots (\\gamma_{s1} \\dots \\gamma_{s\\lambda_s})(\\tau(\\gamma_{s1})\n\t\t\t\\dots \\tau(\\gamma_{s\\lambda_s}))\n\t\t\\end{equation}\n\t\tsatisfying two conditions: cycles forming an $\\alpha$-pair always\n\t\tstand side by side, and their matching elements $\\gamma_{ij} $ and\n\t\t$\\tau(\\gamma_{ij} ) $ occupy the same positions in them. Now consider\n\t\ta permutation $\\gamma = (\\gamma_{11} \\dots \\gamma_{s\\lambda_s} ) \\in\n\t\tS_{2n}$ where the numbers $\\gamma_{ij}$ are written exactly in the\n\t\tsame order as in \\eqref{Eq:2LambdaEmpty}. We write $\\sigma \\in\n\t\tC_{\\lambda\\mid\\emptyset}^+$ if $\\gamma$ is even, and $\\sigma \\in\n\t\tC_{\\lambda\\mid\\emptyset}^-$ if $\\gamma$ is odd. Representation of\n\t\t$\\sigma$ in the form \\eqref{Eq:2LambdaEmpty} is not unique but, as it\n\t\tis easy to see, the parity of $\\gamma$ does not depend on a particular\n\t\tchoice. (Recall, all the cycles in \\eqref{Eq:2LambdaEmpty} have even\n\t\tlength.)\n\t\t\n\t\t\\begin{proposition}[\\protect{\\cite[Proposition 25]{Carter}}]\\label{Pp:ConjD}\\strut\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item If the partition $\\mu$ contains an even number of parts then\n\t\t\t\tthe conjugacy class $C_{\\lambda\\mid\\mu} \\subset B_n$ lies in $D_n$;\n\t\t\t\tif the number of parts is odd then $C_{\\lambda\\mid\\mu}$ does not\n\t\t\t\tintersect $D_n$.\n\t\t\t\t\n\t\t\t\t\\item If $\\mu \\ne \\emptyset$ and the number of parts of $\\mu$ is\n\t\t\t\teven then $C_{\\lambda\\mid\\mu}$ is a conjugacy class in $D_n$.\n\t\t\t\t\n\t\t\t\t\\item If $\\lambda$ is a partition of $n$ containing at least one odd\n\t\t\t\tpart $\\lambda_i$ then $C_{\\lambda\\mid\\emptyset}$ is a conjugacy class\n\t\t\t\tin $D_n$.\n\t\t\t\t\n\t\t\t\t\\item If $\\lambda$ is a partition of $n$ such that all its parts are\n\t\t\t\teven then $C_{\\lambda \\mid \\emptyset}$ splits into two conjugacy\n\t\t\t\tclasses in $D_n$, $C_{\\lambda\\mid\\emptyset}^+$ and\n\t\t\t\t$C_{\\lambda\\mid\\emptyset}^-$.    \n\t\t\t\\end{enumerate}\n\t\t\tAny conjugacy class in $D_n$ is one of the classes listed above.\n\t\t\\end{proposition}\n\t\t\n\t\t\\begin{corollary}\\label{Cr:ConjOdd}\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item Let $\\lambda, \\mu$ be partitions such that $\\lmod\\lambda\\rmod\n\t\t\t\t+ \\lmod\\mu\\rmod = n$, $\\#\\mu$ is even and either $\\mu \\ne\n\t\t\t\t\\emptyset$ or at least one of he parts of $\\lambda$ is odd. Then\n\t\t\t\tfor any $x \\in B_n$ and any $\\sigma \\in C_{\\lambda\\mid\\mu} \\subset\n\t\t\t\tD_n$ one has $x\\sigma x^{-1} \\in C_{\\lambda\\mid\\mu}$. \n\t\t\t\t\\item Let $\\lambda$ be a partition of $n$ such that all its parts\n\t\t\t\tare even, and let $x \\in B_n$. Then for any $\\sigma \\in\n\t\t\t\tC_{\\lambda\\mid\\emptyset}^+$ one has $x \\sigma x^{-1} \\in\n\t\t\t\tC_{\\lambda\\mid\\emptyset}^+$ if $x \\in D_n \\subset B_n$ (that is,\n\t\t\t\t$x \\in S_{2n}$ is an even permutation) and $x \\sigma x^{-1} \\in\n\t\t\t\tC_{\\lambda\\mid\\emptyset}^-$ otherwise; the picture for $\\sigma \\in\n\t\t\t\tC_{\\lambda\\mid\\emptyset}^-$ is symmetric.\n\t\t\t\\end{enumerate}\n\t\t\\end{corollary}\n\t\t\n\t\tIn particular, $C_{{1^{n-2}2^1}\\mid\\varnothing} \\subset D_n \\subset\n\t\tB_n$ consists of all the reflections $r_{ij}$, and $C_{{1^{n-1}}\\mid\n\t\t\t1}\\subset B_n$, of all the reflections $l_i$.\n\t\t\n\t\t\\section{Hurwitz numbers}\\label{Sec:HurwNum}\n\t\t\n\t\t\\subsection{Definitions}\n\t\t\n\t\tFix a pair of partitions $\\lambda, \\mu$ with $\\lmod\\lambda\\rmod +\n\t\t\\lmod\\mu\\rmod = n$. Let $C_{\\lambda\\mid\\mu} \\subset B_n$ be the conjugacy\n\t\tclass defined above.\n\t\t\n\t\t\\begin{definition}\\label{Df:HurwNumb}\n\t\t\tA sequence of reflections $(\\sigma_1 \\DT, \\sigma_{m+\\ell})$ of the\n\t\t\tgroup $B_n$ is said to have {\\em profile} $(\\lambda,\\mu,m,\\ell)$ if\n\t\t\t$\\#\\{p \\mid \\sigma_p = r_{ij}, 1 \\le i < j \\le 2n\\} = m$,\n\t\t\t$\\#\\{p \\mid \\sigma_p = l_i, 1 \\le i \\le 2n\\} = \\ell$ and\n\t\t\t$\\sigma_1 \\dots \\sigma_{m+\\ell} \\in C_{\\lambda \\mid \\mu}$. The {\\em\n\t\t\t\tHurwitz numbers for the group $B_n$} are $h_{m,\\ell,\\lambda,\\mu}^B\n\t\t\t= \\frac{1}{n!} \\#\\{(\\sigma_1 \\DT, \\sigma_{m+\\ell}) \\text{ is a\n\t\t\t\tsequence of profile } (\\lambda,\\mu,m,\\ell)\\}$.\n\t\t\\end{definition}\n\t\t\n\t\tFor the group $D_n$ we use the same numbers, in case they make sense:\n\t\t\n\t\t\\begin{definition}\\label{Df:D-Hurwitz}\n\t\t\tLet $m$ be a positive integer, and $\\lambda$ and $\\mu$, partitions\n\t\t\twhere the number of parts $\\#\\mu$ is even. The Hurwitz number for\n\t\t\tthe group $D_n$ is defined as $h_{m,\\lambda,\\mu}^D =\n\t\t\th_{m,0,\\lambda,\\mu}^B$.\n\t\t\\end{definition}\n\t\t\n\t\tIn other words, $h_{m,\\lambda,\\mu}^D = \\frac{1}{n!} \\#\\{(\\sigma_1 \\DT,\n\t\t\\sigma_m) \\text{ has profile } (\\lambda,\\mu,m)\\}$ where $\\sigma_p$ are\n\t\treflections in the group $D_n$ (that is, $\\sigma_p = r_{ij}$ for some\n\t\t$i$ and $j$), and the profile means that $\\sigma_1 \\dots \\sigma_m \\in\n\t\tC_{\\lambda\\mid\\mu} \\subset D_n$.\n\t\t\n\t\t\\begin{remark}\\label{Rm:DHurwRefine}\n\t\t\tRecall (Proposition \\ref{Pp:ConjD}) that if $\\mu = \\emptyset$ and\n\t\t\tall the parts of the partition $\\lambda$ are even then\n\t\t\t$C_{\\lambda\\mid\\emptyset}$ splits into two conjugacy classes,\n\t\t\t$C_{\\lambda\\mid\\emptyset}^+$ and $C_{\\lambda\\mid\\emptyset}^-$.\n\t\t\t\n\t\t\tDenote by $\\Sigma_{m,\\lambda}^+$ and $\\Sigma_{m,\\lambda}^-$ the sets\n\t\t\tof $m$-tuples of reflections $\\sigma_1 \\DT, \\sigma_m \\in D_n$ such\n\t\t\tthat $\\sigma_1 \\dots \\sigma_m \\in C_{\\lambda\\mid\\emptyset}^+$\n\t\t\t(resp., $C_{\\lambda\\mid\\emptyset}^-$). One can denote\n\t\t\t$h_{m,\\lambda,\\emptyset\\pm}^D \\stackrel{\\mbox{\\tiny def}}{=} \\frac{1}{n!}\n\t\t\t\\#\\Sigma_{m,\\lambda}^\\pm$, so that $h_{m,\\lambda,\\emptyset+}^D +\n\t\t\th_{m,\\lambda,\\emptyset-}^D = h_{m,\\lambda,\\emptyset}^D$. It follows\n\t\t\tfrom Corollary \\ref{Cr:ConjOdd}, though, that if $x \\in B_n$ but $x\n\t\t\t\\notin D_n$ then the map sending an $m$-tuple $(\\sigma_1 \\DT,\n\t\t\t\\sigma_m)$ to $(x\\sigma_1x^{-1} \\DT, x\\sigma_mx^{-1})$ is a\n\t\t\tbijection between $\\Sigma_{m,\\lambda}^+$ and\n\t\t\t$\\Sigma_{m,\\lambda}^-$. Thus, $h_{m,\\lambda,\\emptyset+}^D =\n\t\t\th_{m,\\lambda,\\emptyset-}^D = \\frac12 h_{m,\\lambda,\\emptyset}^D$, so\n\t\t\tconsidering $h_{m,\\lambda,\\emptyset\\pm}^D$ makes little sense.\n\t\t\\end{remark}\n\t\t\n\t\tUp to the end of this section we consider the Hurwitz numbers for the\n\t\tgroup $B_n$ only.\n\t\t\n\t\tDenote by $\\CSum{\\lambda}{\\mu} \\stackrel{\\mbox{\\tiny def}}{=} \\frac{1}{\\#C_{\\lambda\\mid\\mu}}\n\t\t\\sum_{x \\in C_{\\lambda\\mid\\mu}} x \\in \\mathbb{C}[B_n]$ the normalized\n\t\tclass sum. One has $\\CSum{\\lambda}{\\mu} \\in Z[B_n]$ (the\n\t\tcenter of the group algebra of $B_n$); by Proposition\n\t\t\\ref{Pp:ConjClass}, $\\CSum{\\lambda}{\\mu}$ form a basis in\n\t\t$Z[B_n]$. Consider now a ring of polynomials $\\mathbb{C}[p,q]$\n\t\twhere $p = (p_1, p_2, \\dots)$ and $q = (q_1, q_2, \\dots)$ are two\n\t\tinfinite sets of variables. The ring is graded by the total degree\n\t\twhere one assumes $\\deg p_k=\\deg q_k =k$ for all $k=1,2,\\dots$. The\n\t\tmap $\\Psi$ defined by\n\t\t\\begin{equation}\\label{Eq:DefIsoB}\n\t\t\t\\Psi(\\CSum{\\lambda}{\\mu}) = p_\\lambda q_\\mu \\stackrel{\\mbox{\\tiny def}}{=} p_{\\lambda_1}\n\t\t\t\\dots p_{\\lambda_s} q_{\\mu_1} \\dots q_{\\mu_t}\n\t\t\\end{equation}\n\t\testablishes an isomorphism between $Z[B_n]$ and the\n\t\thomogeneous component $\\mathbb{C}[p,q]_n$ of total degree $n$.\n\t\t\n\t\tNow denote by\n\t\t\\begin{equation}\\label{Eq:DefT1}\n\t\t\t\\mathcal T_1 := \\frac12 \\sum_{1 \\le i < j \\le 2n}\n\t\t\tr_{ij} = \\#C_{2 1^{n-2}\\mid\\emptyset} \\cdot \\CSum{1^{n-2}2}{\\emptyset}\n\t\t\\end{equation}\n\t\tand\n\t\t\\begin{equation*}\n\t\t\t\\mathcal T_2 := \\frac12 \\sum_{1 \\le i \\le 2n} l_i = \\#C_{1^{n-1}\\mid1}\n\t\t\t\\cdot \\CSum{1^{n-1}}{1}\n\t\t\\end{equation*}\n\t\tsums of all elements of the conjugacy classes containing reflections\n\t\t(recall that $r_{ij} = r_{i+n,j+n}$, and $l_i = l_{i+n}$, so every\n\t\telement is repeated twice in these sums; hence the factor\n\t\t$\\frac12$). The elements $\\mathcal T_1$ and $\\mathcal T_2$ belong to\n\t\t$Z[B_n]$, so one can consider linear operators $T_1, T_2: Z[B_n] \\to\n\t\tZ[B_n]$ of multiplication by $\\mathcal T_1$ and $\\mathcal T_2$,\n\t\trespectively. Obviously, $T_1$ and $T_2$ commute.\n\t\t\n\t\tConsider now linear operators $\\mathcal{CJ}_1, \\mathcal{CJ}_2: \\mathbb{C}[p,q]_n \\to\n\t\t\\mathbb{C}[p,q]_n$ making the following diagrams commutative:\n\t\t\n\t\t\\begin{equation}\\label{Eq:CJViaGAlg}\n\t\t\t\\xymatrix{\n\t\t\t\tZ\\mathbb{C}[B_n] \\ar[r]^{\\times T_i} \\ar[d]^\\simeq &\n\t\t\t\tZ\\mathbb{C}[B_n] \\ar[d]^\\simeq  \\\\\n\t\t\t\t\\mathbb{C}[p,q]_n \\ar[r]_{\\mathcal{CJ}_i} &\n\t\t\t\t\\mathbb{C}[p,q]_n   \n\t\t\t}, \\qquad i = 1,2.\n\t\t\\end{equation}\n\t\t\n\t\tLet now $\\lambda, \\lambda', \\mu, \\mu'$ be partitions such that\n\t\t$\\lmod\\lambda\\rmod + \\lmod\\mu\\rmod = \\lmod\\lambda'\\rmod +\n\t\t\\lmod\\mu'\\rmod = n$. Take an element $\\sigma_* \\in C_{\\lambda\\mid\\mu}$\n\t\tand define the {\\em multiplicity} $\\langle \\lambda, \\mu \\mid \\lambda',\n\t\t\\mu'\\rangle_1$ as the number of reflections $u \\in C_{2^11^{n-2} \\mid\n\t\t\t\\varnothing}$ (that is, $u = r_{ij}$) such that $u\n\t\t\\sigma_* \\in C_{\\lambda'\\mid\\mu'}$; the multiplicity $\\langle \\lambda,\n\t\t\\mu \\mid \\lambda', \\mu'\\rangle_2$ is defined in the same way with\n\t\t$u\\in C_{1^{n-1}\\mid1}$ (that is, $u = l_i$) instead.\n\t\t\n\t\t\\begin{lemma}\\label{Lm:Conj}\n\t\t\tMultiplicities do not depend on the choice of $\\sigma_*$.\n\t\t\\end{lemma}\n\t\t\n\t\t\\begin{proof}\n\t\t\tDenote by $S_1(\\sigma_*;\\lambda',\\mu') \\subset\n\t\t\tC_{1^{n-2}2^1\\mid\\emptyset}$ the set of reflections $u$ such that\n\t\t\t$u\\sigma_* \\in C_{\\lambda'\\mid\\mu'}$. Take $\\sigma' = x\\sigma_*\n\t\t\tx^{-1} \\in C_{\\lambda\\mid\\mu}$ where $x \\in B_n$.  If $u \\in\n\t\t\tS(\\sigma';\\lambda',\\mu')$ then $u \\sigma' = u x\\sigma_* x^{-1} = x\n\t\t\t((x^{-1}ux) \\sigma_*) x^{-1} \\in C_{\\lambda'\\mid\\mu'}$, which is\n\t\t\tequivalent to $(x^{-1}ux) \\sigma_* \\in C_{\\lambda'\\mid\\mu'}$, that\n\t\t\tis, to $x^{-1}ux \\in S(\\sigma_*;\\lambda',\\mu')$. Thus, conjugation\n\t\t\t$u \\mapsto x^{-1}ux$ is a one-to-one map sending\n\t\t\t$S_1(\\sigma';\\lambda',\\mu')$ to $S_1(\\sigma_*;\\lambda',\\mu')$, and\n\t\t\ttherefore these two sets contain the same number of elements\n\t\t\t$\\langle\\lambda,\\mu \\mid \\lambda',\\mu'\\rangle_1$. The reasoning for\n\t\t\tthe multiplicity $\\langle\\lambda,\\mu \\mid \\lambda',\\mu'\\rangle_2$ is\n\t\t\tthe same.\n\t\t\\end{proof}\n\t\t\n\t\t\\begin{theorem}\\label{Th:TViaMult}\n\t\t\t$T_i\\CSum{\\lambda}{\\mu} = \\sum_{\\lambda',\\mu'} \\langle \\lambda,\\mu\n\t\t\t\\mid \\lambda',\\mu'\\rangle_i \\CSum{\\lambda'}{\\mu'}$ for $i = 1,2$.\n\t\t\\end{theorem}\n\t\t\n\t\t\\begin{proof}\n\t\t\t\\begin{equation}\\label{Eq:ActionT}\n\t\t\t\tT_1 \\CSum{\\lambda}{\\mu} = \\frac{1}{\\#C_{\\lambda\\mid\\mu}}\n\t\t\t\t\\sum_{\\sigma \\in C_{\\lambda\\mid\\mu}} T_1\\sigma =\n\t\t\t\t\\frac{1}{\\#C_{\\lambda\\mid\\mu}} \\sum_{\\sigma \\in\n\t\t\t\t\tC_{\\lambda\\mid\\mu}}\\sum_{u \\in C_{1^{n-2}2^1\\mid\\varnothing}} u\n\t\t\t\t\\sigma .\n\t\t\t\\end{equation}\n\t\t\tIt follows from Lemma \\ref{Lm:Conj} that \\eqref{Eq:ActionT} is an\n\t\t\tarithmetical mean of identical summands, so it is equal to each of\n\t\t\tthem:\n\t\t\t\\begin{equation*}\n\t\t\t\tT_1 \\CSum{\\lambda}{\\mu} = \\sum_{\\lambda',\\mu'} \\sum_{\\substack\n\t\t\t\t\t{u \\in C_{1^{n-2}2^1 \\mid \\varnothing}\\\\\n\t\t\t\t\t\tu\\sigma_* \\in  C_{\\lambda'\\mid\\mu'}}}\n\t\t\t\tu \\sigma_* \n\t\t\t\\end{equation*}\n\t\t\tfor any fixed $\\sigma_* \\in C_{\\lambda\\mid\\mu}$. Using Lemma\n\t\t\t\\ref{Lm:Conj} again, one obtains\n\t\t\t\\begin{align*}\n\t\t\t\tT_1 \\CSum{\\lambda}{\\mu} &= \\sum_{\\lambda',\\mu'} \\sum_{\\substack{\n\t\t\t\t\t\tu \\in C_{1^{n-2}2^1 \\mid \\varnothing}\\\\\n\t\t\t\t\t\tu\\sigma_* \\in C_{\\lambda'\\mid\\mu'}}}\n\t\t\t\t\\frac{1}{\\#C_{\\lambda'\\mid\\mu'}}\n\t\t\t\t\\sum_{\\tau \\in C_{\\lambda'\\mid\\mu'}} \\tau\\\\\n\t\t\t\t&= \\sum_{\\lambda',\\mu'} \\#(\\{u \\in C_{1^{n-2}2^1 \\mid\\varnothing}:\n\t\t\t\tu\\sigma_* \\in C_{\\lambda'\\mid\\mu'}\\} \\CSum{\\lambda'}{\\mu'}\\\\\n\t\t\t\t&= \\sum_{\\lambda',\\mu'} \\langle \\lambda,\\mu \\mid\n\t\t\t\t\\lambda',\\mu'\\rangle \\CSum{\\lambda'}{\\mu'}.\n\t\t\t\\end{align*}\n\t\t\tThe proof for $T_2$ is the same with $C_{1^{n-1}\\mid 1}$ instead of\n\t\t\t$C_{1^{n-2}2^1\\mid \\varnothing}$.\n\t\t\\end{proof}\n\t\t\n\t\t\\subsection{Generating function}\n\t\t\n\t\tConsider the following generating function for Hurwitz numbers of the\n\t\tgroup $B_n$:\n\t\t\\begin{equation*}\n\t\t\t\\mathcal{H}^B(\\beta,\\gamma,p,q) = \\sum_{m,\\ell}\\sum_{\\lambda,\\mu}\n\t\t\t\\frac{ h_{m,\\ell,\\lambda,\\mu}^B}{m!\\ell!}\n\t\t\tp_{\\lambda}q_{\\mu}\\beta^m\\gamma^\\ell.\n\t\t\\end{equation*}\n\t\t\n\t\t\\begin{theorem}\\label{Th:CJequ}\n\t\t\t$\\mathcal{H}^B$ satisfies the {\\em cut-and-join equations}\n\t\t\t\\begin{equation}\\label{Eq:CJ}\n\t\t\t\t\\pder{\\mathcal{H}^B}{\\beta} = \\mathcal{CJ}_1(\\mathcal{H}^B) \\quad \\text{and}\n\t\t\t\t\\quad \\pder{\\mathcal{H}^B}{\\gamma} = \\mathcal{CJ}_2(\\mathcal{H}^B)\n\t\t\t\\end{equation}\n\t\t\\end{theorem}\n\t\t\n\t\t\\begin{proof}\n\t\t\tFix a positive integer $n$ and denote by $\\mathcal{H}_n$ a degree\n\t\t\t$n$ homogeneous component of $\\mathcal{H}^B$. The cut-and-join\n\t\t\toperators preserve the degree, so $\\mathcal{H}^B$ satisfies the\n\t\t\tcut-and-join equations if and only if $\\mathcal{H}_n$ does (for each\n\t\t\t$n$).\n\t\t\t\n\t\t\tLet $i$ be $1$ or $2$, and let\n\t\t\t\\begin{equation*}\n\t\t\t\t\\mathcal{G}_n \\stackrel{\\mbox{\\tiny def}}{=} \\sum_{m,\\ell \\ge 0}\n\t\t\t\t\\sum_{\\lambda,\\mu:\\mid\\lambda\\mid +\\mid\\mu\\mid = n}\n\t\t\t\t\\frac{n!h_{m,\\ell,\\lambda,\\mu}^{B}}{m!\\ell!} \\CSum{\\lambda}{\\mu}\n\t\t\t\t\\beta^m \\gamma^\\ell \\in\\mathbb{C}[B_n]\n\t\t\t\\end{equation*}\n\t\t\tAn elementary combinatorial reasoning gives\n\t\t\t\\begin{equation*}\n\t\t\t\t\\mathcal{G}_n = \\sum_{m\\geq 0}\n\t\t\t\t\\frac{\\beta^m\\gamma^\\ell}{m!\\ell!}T_1^m T_2^\\ell(e_n)\n\t\t\t\\end{equation*}\n\t\t\twhere $e_n \\in B_n$ is the unit element. Clearly\n\t\t\t\\begin{equation*}\n\t\t\t\tT_1(\\mathcal{G}_n) = \\sum_{m\\geq 0}\n\t\t\t\t\\frac{\\beta^m\\gamma^\\ell}{m!\\ell!}(T_1)^{m+1}(e_{n}) = \\sum_{m \\ge\n\t\t\t\t\t1} \\frac{\\beta^{m-1}\\gamma^\\ell}{(m-1)!\\ell!} (T_1)^{m}(e_{n}) =\n\t\t\t\t\\pder{\\mathcal{G}_n}{\\beta}.\n\t\t\t\\end{equation*}\n\t\t\tand, similarly, $T_1(\\mathcal{G}_n) =\n\t\t\t\\pder{\\mathcal{G}_n}{\\gamma}$. Applying the isomorphism $\\Psi$ one\n\t\t\tobtains $\\Psi T_1(\\mathcal{G}_n) = \\Psi(\\pder{\\mathcal{G}_n}{\\beta})\n\t\t\t= \\pder{}{\\beta} \\Psi(\\mathcal{G}_n)$.  One has $\\Psi(\\mathcal{G}_n)\n\t\t\t= \\mathcal{H}_n$, hence $\\pder{}{\\beta} \\Psi(\\mathcal{G}_n) =\n\t\t\t\\pder{\\mathcal{H}_n}{\\beta}$; similarly, $\\pder{}{\\gamma}\n\t\t\t\\Psi(\\mathcal{G}_n) = \\pder{\\mathcal{H}_n}{\\gamma}$. By the\n\t\t\tdefinition of the cut-and-join operators, $\\Psi T_1(\\mathcal{G}_n) =\n\t\t\t\\mathcal{CJ}_1(\\Psi(\\mathcal{G}_n)) = \\mathcal{CJ}_1({\\mathcal{H}_n})$ and the same\n\t\t\tfor $T_2$ and $\\mathcal{CJ}_2$. Equalities \\eqref{Eq:CJ} follow.\n\t\t\\end{proof}\n\t\t\n\t\t\\begin{corollary}\\label{Cr:GenFun}\n\t\t\t\\begin{equation}\\label{Eq:GenFunc}\n\t\t\t\t\\mathcal{H}^B(\\beta,\\gamma,p,q) =\n\t\t\t\te^{\\beta\\mathcal{CJ}_1+\\gamma\\mathcal{CJ}_2}e^{p_1}\n\t\t\t\\end{equation}\n\t\t\\end{corollary}\n\t\t\n\t\t\\begin{proof}\n\t\t\tIt follows from Definition \\ref{Df:HurwNumb} that\n\t\t\t$h^B_{0,0,\\lambda,\\mu} = \\frac{1}{n!}$ if $\\lambda = 1^n$ and\n\t\t\t$\\mu=\\emptyset$, and $h^B_{0,0,\\lambda,\\mu} = 0$ otherwise.  Thus\n\t\t\t$\\mathcal{H}^B(0,0,p,q) = e^{p_1}$ and \\eqref{Eq:GenFunc} follows\n\t\t\tfrom Theorem \\ref{Th:CJequ}.\n\t\t\\end{proof}\n\t\t\n\t\t\\section{Explicit formulas}\\label{Sec:Expl}\n\t\t\n\t\t\\subsection{Multiplication of an element $x \\in B_n \\subset\n\t\t\tS_{2n}$ by a reflection}\\label{SSec:CycleStruct}\n\t\t\n\t\tLet $\\omega \\in S_n$ and $1 \\le a < b \\le n$. The cycle structure of\n\t\tthe product $\\omega' = (ab)\\omega$ depend on the cyclic structure of\n\t\t$\\omega$ and positions of $a$ and $b$ as follows. If $a$ and $b$\n\t\tbelong to the same cycle $(a \\DT, u, b \\DT, v)$ of $\\omega$, then the\n\t\tcycle splits in $\\omega'$ into two: $(a \\DT, u)$ and $(b \\DT, v)$ (``a\n\t\tcut''). If $a$ and $b$ belong to different cycles then the opposite\n\t\tthing happens: they glue together in $\\omega'$ (``a join''). The\n\t\tcycles containing neither $a$ nor $b$ are the same for $\\omega$ and\n\t\t$\\omega'$.\n\t\t\n\t\tLet now $\\omega \\in B_n \\subset S_{2n}$. The cyclic structure of $\\omega'\n\t\t= \\sigma\\omega$ where $\\sigma$ is a reflection (i.e.\\ $\\sigma =\n\t\tr_{ab} = (ab)(a+n,b+n)$ (addition modulo $n$) for $1 \\le a\n\t\t< b \\le 2n$, or $\\sigma = l_a = (a,a+n)$ for $1\\leq\n\t\ta\\leq 2n$) depends on the cyclic structure of $\\omega$ ($\\alpha$-pairs\n\t\tand $\\beta$-cycles) and the position of the points $a, b$ as shown in\n\t\tTable \\ref{Fig:Mult} below.\n\t\t\n\t\tIn this table, hexagons represent $\\beta$-cycles and the pairs of\n\t\ttriangles are $\\alpha$-pairs; numbers inside denote the lengths of the\n\t\tcycles. The two-headed dashed arrows show the position of $a$ and $b$\n\t\tof the reflection $\\sigma$. The solid arrows join cyclic structures of\n\t\t$\\omega$ and $\\sigma\\omega$ for $\\sigma = r_{ab}$, and the empty\n\t\tarrows, for $\\sigma = l_a$ (recall that $\\sigma$ is an involution,\n\t\tso the arrows are two-headed). For example, the first diagram shows\n\t\tthe multiplication by $(a,b)(\\t(a),\\t(b))$ of a pair of $\\beta$-cycles\n\t\tcontaining $a$ and $b$, respectively.\n\t\t\n\t\tThe boxed number on the bottom left corner is the multiplicity\n\t\t$\\langle\\lambda,\\mu \\mid \\lambda',\\mu'\\rangle$ between conjugacy\n\t\tclasses containing $\\omega$ and $\\sigma\\omega$.\n\t\t\n\t\t\n\t\t\\begin{figure} \n\t\t\t\\includegraphics[scale=.6]{ActionFinale.pdf}  \n\t\t\t\\includegraphics[scale=.6]{ActionFinale2.pdf}\n\t\t\t\\caption{Multiplication of cycles by a reflection}\\label{Fig:Mult}\n\t\t\\end{figure}\n\t\t\n\t\t\\subsection{Multiplicities}\\label{SSec:Mult}\n\t\t\n\t\tLet $\\omega \\in C_{\\lambda\\mid\\mu}$ where $\\lambda = 1^{c_1} \\dots n^{c_n}$\n\t\tand $\\mu = 1^{d_1} \\dots n^{d_n}$ --- that is, for every $k = 1 \\DT,\n\t\tn$ the element $u$ contains $c_k$ $\\alpha$-pairs of length $k$ and\n\t\t$d_k$ separate $\\beta$-cycles of length $2k$. Calculate, for all\n\t\t$\\lambda' = 1^{c_1'} \\dots n^{c_n'}$ and $\\mu = 1^{d_1'} \\dots\n\t\tn^{d_n'}$, the multiplicities $\\langle \\lambda',\\mu' \\mid \\lambda,\n\t\t\\mu\\rangle_1$ and $\\langle \\lambda',\\mu' \\mid \\lambda,\n\t\t\\mu\\rangle_2$. In Section \\ref{SSec:CycleStruct} we listed cases when\n\t\tthis multiplicities may be nonzero: these are Cases\n\t\t\\ref{It:OneBeta}--\\ref{It:2join} below for $\\langle\\;\\mid\\;\\rangle_1$ and\n\t\tCases \\ref{It:LCut} and \\ref{It:LJoin} for $\\langle\\;\\mid\\;\\rangle_2$.\n\t\t\n\t\t{\\def Case \\theenumi. {Case \\theenumi.}\n\t\t\t\n\t\t\t\\begin{enumerate}  \n\t\t\t\t\n\t\t\t\t\\item\\label{It:OneBeta} The number of possible positions for $a$ is\n\t\t\t\tthe total number of elements in all the $\\beta$-cycles of length\n\t\t\t\t$2(i+j)$, that is, $2(i+j) d_{i+j}$. Since one knows the\n\t\t\t\tlengths of $\\alpha$-pairs in $\\sigma \\omega$, the position of $b$ is\n\t\t\t\tunique, once the position of $a$ is chosen; the same is for $\\t(a)$\n\t\t\t\tand $\\t(b)$. Doing like this, one counts every reflection $\\sigma =\n\t\t\t\tr_{ab}$ twice, so that the multiplicity is $(i+j)d_{i+j}$.\n\t\t\t\t\n\t\t\t\t\\item Again there are $2i c_i$ possible positions for $a$ and $2j d_j$\n\t\t\t\tpossible positions for $b$; we are to divide by $2$ again by the\n\t\t\t\tsame reason and the multiplicity is $2ijc_i d_j$.\n\t\t\t\t\n\t\t\t\t\\item Like in Case \\ref{It:OneBeta}, the multiplicity is $2ij d_i\n\t\t\t\td_j$.  .\n\t\t\t\t\n\t\t\t\t\\item\\label{It:MatchAlphas} The number of possible positions for $a$\n\t\t\t\tis $2(i+j) c_{i+j}$; the position of $b$ is uniquely determined by\n\t\t\t\tthe position of $a$ like in Case \\ref{It:OneBeta}. Again, divide by\n\t\t\t\t$2$, to obtain the multiplicity $(i+j) c_{i+j}$.\n\t\t\t\t\n\t\t\t\t\\item Same as case \\ref{It:MatchAlphas}, the multiplicity is\n\t\t\t\t$(i+j) c_{i+j}$.\n\t\t\t\t\n\t\t\t\t\\item\\label{It:2join} The number of possible positions for $a$ is the\n\t\t\t\ttotal number of the elements of all the $\\alpha$-pairs of length\n\t\t\t\t$i$, that is, $2i c_i$. Similarly, the number of possible positions\n\t\t\t\tfor $b$ is $2j c_j$, so the multiplicity is $2ij c_i c_j$.\n\t\t\t\t\n\t\t\t\t\\item\\label{It:LCut} The number of possible positions for $a$ is\n\t\t\t\t$2i d_i$, for the same reason as above, we have to divide by $2$,\n\t\t\t\tso that the multiplicity is $i d_i$\n\t\t\t\t\n\t\t\t\t\\item\\label{It:LJoin} The number of possible positions for $a$ is\n\t\t\t\t$2i c_i$, as in the previous cases, we have to divide by $2$, so that\n\t\t\t\tthe multiplicity is $i c_i$\n\t\t\t\\end{enumerate}\n\t\t}\n\t\t\n\t\tWrite now the terms of $\\mathcal{CJ}_1$ and $\\mathcal{CJ}_2$ explicitly. It follows from\n\t\tTheorem \\ref{Th:TViaMult} that $\\mathcal{CJ}_i p_\\lambda q_\\mu =\n\t\t\\sum_{\\lambda',\\mu'} \\langle\\lambda,\\mu\\mid\\lambda',\\mu'\\rangle_i\n\t\tp_{\\lambda'} q_{\\mu'}$ for $i\\in \\{1,2\\}$.\n\t\t\n\t\tLet $\\lambda',\\mu'$ be as at diagram \\ref{It:OneBeta} in Figure\n\t\t\\ref{Fig:Mult}. The monomial $p_\\lambda q_\\mu$ contains\n\t\t$q_{i+j}^{d_{i+j}}$; The exponent at $q_{i+j}$ in the monomial\n\t\t$p_{\\lambda'} q_{\\mu'}$ is less by $1$, and the exponents at $p_i$ and\n\t\t$q_j$ are greater by $1$. We will have $q_{i+j} \\pder{}{p_i}\n\t\t\\pder{}{q_j}$ for this; to get a correct multiplicity, put the\n\t\tcoefficient $(i+j)$ before.\n\t\t\n\t\tSimilar reasoning for the remaining cases gives:\n\t\t\n\t\t\\begin{multline}\\label{Eq:CJ_1}\n\t\t\t\\mathcal{CJ}_1 = \\sum_{i,j=1}^\\infty \\biggl( (i+j) p_iq_j \\pder{}{q_{i+j}} +\n\t\t\t2ij q_{i+j} \\pdertwo{}{p_i}{q_j} + ijp_{i+j} \\pdertwo{}{q_i}{q_j}\\\\\n\t\t\t+ \\frac{1}{2} (i+j) q_iq_j \\pder{}{p_{i+j}} + \\frac{1}{2} (i+j)\n\t\t\tp_ip_j \\pder{}{p_{i+j}} + ij p_{i+j} \\pdertwo{}{p_i}{p_j}\\biggr)\n\t\t\\end{multline}\n\t\tand\n\t\t\\begin{equation}\\label{Eq:CJ_2}\n\t\t\t\\mathcal{CJ}_2 = \\sum_{i=1}^\\infty \\left(i p_i\\frac{\\partial}{\\partial\n\t\t\t\tq_i} + i q_i\\frac{\\partial}{\\partial p_i}\\right)\n\t\t\\end{equation}\n\t\t\n\t\t\\subsection{Change of variables}\\label{SSec:Change}\n\t\t\n\t\tIn this section we reduce, by a suitable change of variables, the\n\t\toperators $\\mathcal{CJ}_1$ and $\\mathcal{CJ}_2$ to classical cut-and-join operators\n\t\t\\begin{equation}\\label{Eq:CJA}\n\t\t\t\\mathcal{CJ} \\stackrel{\\mbox{\\tiny def}}{=}\\sum_{i,j=1}^\\infty \\left(ijp_{i+j} \\pdertwo{}{p_i}{p_j}\n\t\t\t+(i+j) p_ip_j \\pder{}{p_{i+j}}\\right)  \n\t\t\\end{equation}\n\t\tand Euler fields\n\t\t\\begin{equation}\\label{Eq:Euler}\n\t\t\tE\\stackrel{\\mbox{\\tiny def}}{=}\\sum_{i=1}^\\infty ip_i \\pder{}{p_i},\n\t\t\\end{equation}\n\t\trespectively.\n\t\t\n\t\t\\begin{proposition}\\label{Pp:ChVar}\n\t\t\tLet $u_\\ell = \\frac{p_\\ell+q_\\ell}{2}$ and $v_\\ell =\n\t\t\t\\frac{p_\\ell-q_\\ell}{2}$. Then\n\t\t\t\n\t\t\t\\begin{align*}\n\t\t\t\t\\mathcal{CJ}_1 &= \\sum_{i,j=1}^\\infty \\biggl( iju_{i+j}\\pdertwo{}{u_i}{u_j} +\n\t\t\t\t(i+j) u_iu_j \\pder{}{u_{i+j}} + ij v_{i+j} \\pdertwo{}{v_i}{v_j}\\\\\n\t\t\t\t&\\hphantom{\\sum_{i,j=1}^\\infty \\biggl(} + (i+j) v_iv_j \\pder{}{v_{i+j}}\\biggr) = \\mathcal{CJ}_u + \\mathcal{CJ}_v,\\\\\n\t\t\t\t\\mathcal{CJ}_2 &= \\sum_{\\ell=1}^\\infty \\ell \\left(u_\\ell \\pder{}{u_\\ell} -\n\t\t\t\tv_\\ell \\pder{}{v_{\\ell}}\\right) = E_u - E_v.\n\t\t\t\\end{align*}\n\t\t\\end{proposition}\n\t\t\\noindent where by $\\mathcal{CJ}_u$ and $\\mathcal{CJ}_v$ we denote the operator \\eqref{Eq:CJA}\n\t\twith $u_i$ (resp. $v_i$) substituted for $p_i$, and similarly, $E_u$\n\t\tand $E_v$.\n\t\t\n\t\tThe proof is a direct computation.\n\t\t\n\t\t\\begin{theorem}\\label{Th:B-Schur}\n\t\t\tFor all partitions $\\lambda$ and $\\mu$ the polynomials\n\t\t\t\\begin{equation*}\n\t\t\t\ts_{\\lambda\\mid\\mu}(p,q) \\stackrel{\\mbox{\\tiny def}}{=} s_\\lambda((p+q)\/2) s_\\mu((p-q)\/2)\n\t\t\t\\end{equation*}  \n\t\t\twhere $s_\\lambda$ and $s_\\mu$ are Schur polynomials and $(p \\pm\n\t\t\tq)\/2$ means $((p_1 \\pm q_1)\/2, (p_2 \\pm q_2)\/2, \\dots)$, are\n\t\t\teigenvectors of both $\\mathcal{CJ}_1$ and $\\mathcal{CJ}_2$; the respective eigenvalues\n\t\t\tare $\\sum\\limits_{i=1}^\\infty \\left(\\lambda_i(\\lambda_i-2i+1) +\n\t\t\t\\mu_i(\\mu_i-2i+1)\\right)$ for $\\mathcal{CJ}_1$ and $\\sum\\limits_{i=1}^\\infty\n\t\t\t(\\lambda_i - \\mu_i)$ for $\\mathcal{CJ}_2$.\n\t\t\\end{theorem}\n\t\t\n\t\tThe theorem follows immediately from Proposition \\ref{Pp:ChVar} and\n\t\tthe classical fact that Schur polynomials are eigenvectors of the\n\t\tcut-and-join operator \\eqref{Eq:CJA} (see \\cite{LandoKazarian}) and of\n\t\tthe Euler field \\eqref{Eq:Euler} ($s_{\\lambda\\mid\\mu}$ are weighted\n\t\thomogeneous).\n\t\t\n\t\tTheorem \\ref{Th:B-Schur} and the Cauchy identity \\cite{Macdonald}\n\t\timply that\n\t\t\\begin{equation*}\n\t\t\te^{p_1} = e^{u_1} e^{v_1} = \\sum_{\\lambda,\\mu}\n\t\t\ts_{\\lambda\\mid\\mu}(p,q) s_{\\lambda\\mid\\mu}(1, 0, \\dots; 1, 0, \\dots)\n\t\t\\end{equation*}\n\t\t\n\t\tThis allows us to prove the following\n\t\t\n\t\t\\begin{corollary}[of Theorem \\ref{Th:B-Schur}]\\label{Cr:LKGenFunc}\n\t\t\t\\begin{align*}\n\t\t\t\t\\mathcal{H}^B(\\beta,\\gamma,p,q) &= \\sum_{\\lambda,\\mu}\n\t\t\t\t\\exp\\bigl(\\beta \\sum_{i=1}^\\infty (\\lambda_i(\\lambda_i-2i+1) +\n\t\t\t\t\\mu_i(\\mu_i-2i+1) + \\gamma \\sum_{i=1}^{\\infty}\n\t\t\t\t(\\lambda_i-\\mu_i)\\bigr)\\\\\n\t\t\t\t&\\times s_{\\lambda\\mid\\mu}(1,0,\\dots; 1,0,\\dots)\n\t\t\t\ts_{\\lambda\\mid\\mu}(p,q)\n\t\t\t\\end{align*}\n\t\t\\end{corollary}\n\t\tThis is the B-analog of the formula expressing the classical Hurwitz\n\t\tnumbers via the Schur polynomials \\cite{LandoKazarian}\n\t\t\n\t\tTheorem \\ref{Th:B-Schur} allows also to establish a direct relation\n\t\tbetween Hurwitz numbers of the groups $B_n$\n\t\t($h_{m,\\ell,\\lambda,\\mu}^B$, studied here) and $S_n$ (the classical\n\t\tones, denoted $h_{m,\\lambda}^A$ till the end of this section).\n\t\t\n\t\tNamely, it follows from equation \\ref{Eq:CJ_2} that $\\mathcal{CJ}_2(p_\\lambda\n\t\tq_\\mu)= (|\\lambda|+|\\mu|)p_\\lambda q_\\mu$, and therefore, by the\n\t\tcut-and-join equation,\n\t\t\\begin{multline*}\n\t\t\t\\sum_{m,\\ell}\\sum_{\\lambda,\\mu}\n\t\t\t\\frac{h_{m,\\ell,\\lambda,\\mu}^B}{m!(\\ell-1)!}\n\t\t\tp_{\\lambda}q_{\\mu}\\beta^m\\gamma^{\\ell-1} = \\mathcal{CJ}_2(\\mathcal{H}^B) =\n\t\t\t\\pder{\\mathcal{H}^B}{\\gamma}\\\\\n\t\t\t= \\sum_{m,\\ell}\\sum_{\\lambda,\\mu} (\\lmod\\lambda\\rmod +\n\t\t\t\\lmod\\mu\\rmod) \\frac{h_{m,\\ell,\\lambda,\\mu}^B}{m!\\ell!}\n\t\t\tp_{\\lambda}q_{\\mu}\\beta^m\\gamma^\\ell.\n\t\t\\end{multline*}\n\t\tThis implies the equality \n\t\t\\begin{equation*}\n\t\t\th_{m,\\ell,\\lambda,\\mu}^B = (\\lmod\\lambda\\rmod + \\lmod\\mu\\rmod)^\\ell\n\t\t\th_{m,0,\\lambda,\\mu}^B\n\t\t\\end{equation*}\n\t\tIt is thus enough to focus on $h_{m,0,\\lambda,\\mu}^B$.\n\t\t\n\t\tBy Corollary \\ref{Cr:LKGenFunc}, $h_{m,0,\\lambda,\\mu}^B$ is the\n\t\tcoefficient at the monomial $p_\\lambda q_\\mu\\beta^m$ in the expression\n\t\t\\begin{equation*}\n\t\t\t\\sum_{\\lambda',\\mu',m_1,m_2}\n\t\t\th_{m_1,\\lambda'}^A(p+q)_{\\lambda'}  \n\t\t\th_{m_1,\\mu'}^A(p+q)_{\\mu'}\\beta^{m_1+m_2}\n\t\t\t\\frac{m!(\\lmod\\lambda\\rmod + \\lmod\\mu\\rmod)!}{2^{\\#\\lambda\n\t\t\t\t\t+\\#\\mu}\\lmod\\lambda'\\rmod! m_1! \\lmod\\mu'\\rmod! m_2!}\n\t\t\\end{equation*}\n\t\twhere $(p\\pm q)_\\lambda$ is understood as $(p_{\\lambda_1}\\pm\n\t\tq_{\\lambda_1}) \\dots (p_{\\lambda_s}\\pm q_{\\lambda_s})$. The\n\t\tcoefficient may be nonzero only if two conditions are satisfied:\n\t\t$m_1+m_2=m$ and $\\lambda'+\\mu'=\\lambda+\\mu$. Thus the relation can be\n\t\trewritten as\n\t\t\\begin{equation*}\n\t\t\th_{m,0,\\lambda,\\mu}^B =\n\t\t\t\\sum_{\\substack{\\lambda'+\\mu'=\\lambda+\\mu\\\\m_1+m_2=m}}\n\t\t\t\\frac{h_{m_1,\\lambda'}^Ah_{m_2,\\mu'}^A}{2^{\\#\\lambda +\\#\\mu}}\n\t\t\t\\binom{m}{m_1}\\binom{|\\lambda|+|\\mu|}{|\\lambda'|} [p_\\lambda q_\\mu\n\t\t\t:(p+q)_{\\lambda'}(p-q)_{\\mu'}]\n\t\t\\end{equation*}\n\t\t\n\t\tNow let $\\lambda'=(1^{\\alpha_1},2^{\\alpha_2},\\dots)$, $\\mu'=(1^{\\beta_1},2^{\\beta_2},\\dots)$ and\n\t\t$\\mu=(1^{\\gamma_1},2^{\\gamma_2},\\dots)$; it follows from the equality\n\t\t$\\lambda+\\mu = \\lambda' + \\mu'$ that $\\lambda =\n\t\t(1^{\\alpha_1+\\beta_1-\\gamma_1} 2^{\\alpha_2+\\beta_2-\\gamma_2} \\dots)$\n\t\tis fixed. If $f^{\\gamma}_{\\alpha\\beta}$ is a coefficient at\n\t\t$x^\\gamma$ at the polynomial $(1+x)^\\alpha(1-x)^\\beta$ then one has \n\t\t\n\t\t\\begin{equation*}\n\t\t\th_{m,0,\\lambda,\\mu}^B =\n\t\t\t\\sum_{\\substack{\\lambda'+\\mu'=\\lambda+\\mu\\\\m_1+m_2=m}}\n\t\t\t\\frac{h_{m_1,\\lambda'}^A h_{m_2,\\mu'}^A}{2^{\\#\\lambda + \\#\\mu}}\n\t\t\t\\binom{m}{m_1}\\binom{\\lmod\\lambda\\rmod +\n\t\t\t\t\\lmod\\mu\\rmod}{\\lmod\\lambda'\\rmod} f^{\\gamma_1}_{\\alpha_1\\beta_1}\n\t\t\tf^{\\gamma_2}_{\\alpha_2\\beta_2}\\dots\n\t\t\\end{equation*}\n\t\t\n\t\t\\section{Cut-and-join operator for the group $D_n$}\\label{Sec:CJD}\n\t\t\n\t\tLet $Z[D_n]$ be a center of the group algebra of $D_n$; it is a vector\n\t\tspace spanned by the class sums $\\CSum{\\lambda}{\\emptyset}^+$ and\n\t\t$\\CSum{\\lambda}{\\emptyset}^-$ where $\\lmod\\lambda\\rmod = n$ and all\n\t\tthe parts of $\\lambda$ are even, and the class sums\n\t\t$\\CSum{\\lambda}{\\mu}$ for all other $\\lambda$ and $\\mu$ such that \n\t\t$\\lmod\\lambda\\rmod + \\lmod\\mu\\rmod = n$ and $\\#\\mu$ is\n\t\teven. The operator of multiplication by the element $\\mathcal T_1 \\in\n\t\tZ[D_1]$ defined by \\eqref{Eq:DefT1} (a sum of all the reflections in\n\t\t$D_n$) acts on $Z[D_n]$; denote it $\\Theta_D$ for brevity.\n\t\t\n\t\tTake an element $x \\in B_n \\setminus D_n$ (i.e.\\ an odd permutation in\n\t\t$B_n$) and consider the map $\\mathcal{I}:\\mathbb{C}[D_n]\\mapsto\n\t\t\\mathbb{C}[D_n]$ defined by $\\mathcal{I}(a) = xax^{-1}$. Corollary\n\t\t\\ref{Cr:ConjOdd} implies that $\\mathcal I$ maps the center $Z[D_n]$ to\n\t\titself and its restriction to $Z[D_n]$ is an involution. This\n\t\tinvolution defines a splitting $Z[D_n] = V_n^+ \\oplus V_n^-$ where\n\t\t$V_n^{\\pm}$ are eigenspaces of $\\mathcal I$ corresponding to\n\t\teigenvalues $1$ and $-1$.\n\t\t\n\t\tClearly, the vector space $V_n^+$ is spanned by the elements\n\t\t$\\CSum{\\lambda}{\\mu}$ and $\\CSum{\\lambda}{\\emptyset}^++\n\t\t\\CSum{\\lambda}{\\emptyset}^-$, whereas the vector space $V_n^-$ is\n\t\tspanned by the the elements $\\CSum{\\lambda}{\\emptyset}^+-\n\t\t\\CSum{\\lambda}{\\emptyset}^-$. In particular, $V_n^-$ is nonempty if\n\t\tand only if $n = \\lmod\\lambda\\rmod$ is even. Therefore, there exist a\n\t\tlinear injection $\\Phi_+: V_n^+ \\to Z[B_n]$ and a linear\n\t\tisomorphism $\\Phi_-: V_n^- \\to Z[S_{n\/2}]$ (for $n$ even) defined as\n\t\t\\begin{align*}\n\t\t\t\\Phi_+(\\CSum{\\lambda}{\\mu}) &= \\CSum{\\lambda}{\\mu}\\\\\n\t\t\t\\Phi_+(\\CSum{\\lambda}{\\emptyset}^+ + \\CSum{\\lambda}{\\emptyset}^-) &= \\CSum{\\lambda}{\\emptyset},\n\t\t\\end{align*}\n\t\tand\n\t\t\\begin{equation*}\n\t\t\t\\Phi_-(\\CSum{\\lambda}{\\emptyset}^+ - \\CSum{\\lambda}{\\emptyset}^-) = C_{\\lambda\/2}\n\t\t\\end{equation*}\n\t\twhere $\\lambda\/2 = (\\lambda_1\/2 \\DT, \\lambda_s\/2)$ (recall that\n\t\t$\\CSum{\\lambda}{\\emptyset}^\\pm$ are defined only if all the parts\n\t\t$\\lambda_i$ of $\\lambda$ are even). The image of $\\Phi_+$ is a\n\t\tsubspace $Z_e[B_n] \\subset Z[B_n]$ spanned by all the class sums\n\t\t$\\CSum{\\lambda}{\\mu}$ with $\\#\\mu$ even.\n\t\t\n\t\t\\begin{theorem}\\label{Th:CJD}\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\\label{It:Commutes} The operator $\\Theta_D$ commutes with the\n\t\t\t\tinvolution $\\mathcal I$, and therefore, $V_n^+$ and $V_n^-$ are\n\t\t\t\t$\\Theta_D$-invariant.\n\t\t\t\t\n\t\t\t\t\\item\\label{It:CJDonVPlus} For the restriction of $\\Theta_D$ on\n\t\t\t\t$V_n^+$ the following diagram is commutative:\n\t\t\t\t\\begin{equation}\\label{Eq:CJDViaGAlg}\n\t\t\t\t\t\\xymatrix{\n\t\t\t\t\t\tV_n^{+} \\ar[r]^{\\Theta_D} \\ar[d]^{\\Phi_+} & V_n^{+} \\ar[d]^{\\Phi_+} \\\\\n\t\t\t\t\t\tZ[B_n] \\ar[r]_{T_1} & Z[B_n]\n\t\t\t\t\t}\n\t\t\t\t\\end{equation}\n\t\t\t\t\n\t\t\t\t\\item\\label{It:CHDonVMinus} If $n$ is even then for the restriction\n\t\t\t\tof $\\Theta_D$ on $V_n^-$ the following diagram is commutative:\n\t\t\t\t\\begin{equation}\\label{Eq:CJDViaGAlgMinus}\n\t\t\t\t\t\\xymatrix{\n\t\t\t\t\t\tV_n^- \\ar[r]^{\\Theta_D} \\ar[d]^{\\Phi_-} & V_n^- \\ar[d]^{\\Phi_-} \\\\\n\t\t\t\t\t\tZ[S_{n\/2}] \\ar[r]_{4T} & Z[S_{n\/2}]\n\t\t\t\t\t}\n\t\t\t\t\\end{equation}\n\t\t\t\\end{enumerate}\n\t\t\\end{theorem}\n\t\t\n\t\t\\begin{corollary}\n\t\t\t$V_n^+$ is isomorphic to the space of homogeneous polynomials\n\t\t\t$\\mathbb{C}[p,q]_n^e$ of total degree $n$ and even degree in\n\t\t\t$q$; the isomorphism is given by \\eqref{Eq:DefIsoB}. $V_n^-$ is isomorphic to the space $\\mathbb{C}[p]_{n\/2}$ of\n\t\t\thomogeneous polynomials of total degree $n\/2$, the isomorphism is\n\t\t\tgiven by\n\t\t\t\\begin{equation*}\n\t\t\t\t\\Phi_A(\\CSum{\\lambda}{\\emptyset}^+-\\CSum{\\lambda}{\\emptyset}^-) =\n\t\t\t\tp_{\\lambda\/2} \\stackrel{\\mbox{\\tiny def}}{=} p_{\\lambda_1\/2} \\dots p_{\\lambda_s\/2}.\n\t\t\t\\end{equation*}\n\t\t\tThe following diagrams are commutative\n\t\t\t\\begin{equation*}\n\t\t\t\t\\xymatrix{\n\t\t\t\t\tV_n^+ \\ar[r]^{\\Theta_D} \\ar[d]^{\\Psi \\circ \\Phi_+} & V_n^+\n\t\t\t\t\t\\ar[d]^{\\Psi \\circ \\Phi_+} \\\\      \n\t\t\t\t\t\\mathbb{C}[p,q]_n^e \\ar[r]_{\\mathcal{CJ}_1} & \\mathbb{C}[p,q]_n^e\n\t\t\t\t}, \\qquad\n\t\t\t\t\\xymatrix{\n\t\t\t\t\tV_n^- \\ar[r]^{\\Theta_D} \\ar[d]^{\\Psi_A \\circ \\Phi_-} & V_n^-\n\t\t\t\t\t\\ar[d]^{\\Psi_A \\circ \\Phi_-} \\\\    \n\t\t\t\t\t\\mathbb{C}[p]_{n\/2} \\ar[r]_{\\mathcal{CJ}_A} & \\mathbb{C}[p]_{n\/2}\n\t\t\t\t}\n\t\t\t\\end{equation*}\n\t\t\twhere $\\mathcal{CJ}_A$ is the classical cut-and-join operator expressed in\n\t\t\trescaled variables $r_i = p_i\/2$ and multiplied by $4$. \n\t\t\\end{corollary}\n\t\t\n\t\tIn other words, the cut-and-join operator for the group $D_n$ is a\n\t\tdirect sum of the cut-and-join operator for the group $B_n$,\n\t\trestricted to the space of polynomials of even degree in $q$, and, for\n\t\t$n$ even, the cut-and-join operator for the group $S_{n\/2}$, expressed\n\t\tvia rescaled variables $r_i = p_i\/2$ and multiplied by $4$.\n\t\t\n\t\t\\begin{proof}[Proof of Theorem \\ref{Th:CJD} and the corollary]\n\t\t\tThe image of $V_n^+ \\subset Z[D_n] \\subset \\mathbb{C}[D_n]$ under the\n\t\t\tnatural embedding $\\iota: \\mathbb{C}[D_n] \\to \\mathbb{C}[B_n]$ is\n\t\t\t$Z[B_n]$, and the restriction of $\\iota$ to $V_n^+$ coincides with\n\t\t\t$\\Phi_+$. So, assertion \\ref{It:Commutes} follows from the fact that\n\t\t\t$\\Theta_D$ is an operator of multiplication by the element $T_1 \\in\n\t\t\tZ[D_n]$ such that $\\iota(T_1) \\in Z[B_n]$. Assertion\n\t\t\t\\ref{It:CJDonVPlus} is then evident.\n\t\t\t\n\t\t\tInstead of proving assertion \\ref{It:CHDonVMinus} we prove the\n\t\t\tcommutativity of the second diagram from the corollary; it is\n\t\t\tobviously an equivalent fact. The description above implies that\n\t\t\t$\\mathcal{CJ}_A = U \\mathcal{CJ} V$ where the operator $V: \\mathbb{C}[r] \\to\n\t\t\t\\mathbb{C}[p,q]$ is given by $(Vf)(p,q) = f(p_2, p_4, \\dots)$ and $U:\n\t\t\t\\mathbb{C}[p,q] \\to \\mathbb{C}[p]$, by $(Uh)(r) =\n\t\t\t\\left.h\\right|_{\\substack{p_1=p_3 \\DT= 0\\\\ p_2=r_1, p_4=r_2, \\dots\n\t\t\t\t\t\\\\ q_1=q_2 \\DT= 0}}$. Now the required formula follows directly\n\t\t\tfrom \\eqref{Eq:CJ_1} and \\eqref{Eq:CJA}.\n\t\t\\end{proof}\n\t\t\n\t\tHurwitz numbers for the group $D_n$ coincide with (some of) the\n\t\tHurwitz numbers for the group $B_n$ by Definition \\ref{Df:D-Hurwitz};\n\t\tsee also remark \\ref{Rm:DHurwRefine}.\n\t\t\n\t\t\\section{Other properties of B-Hurwitz numbers and B-cut-and-join operator}\n\t\t\n\t\tClassical Hurwitz numbers $h_{m,\\lambda}$ exhibit many interesting\n\t\tproperties. They play an active part in the algebro-geometric theory\n\t\tof moduli spaces of complex curves and holomorphic functions on them\n\t\t(see \\cite{GouldJacks,ELSV}), appear in some problems of\n\t\tlow-dimensional topology \\cite{BurmanZvonkine,TwistedHurwitz} and the\n\t\ttheory of integrable systems \\cite{Kramer,KazarianHodge,FSpace}. The\n\t\tclassical cut-and-join operator also has various interpretations\n\t\tbesides the one described here, including the Fock space\n\t\trepresentation (see \\cite{Johnson} and the references therein).\n\t\t\n\t\tIn this section we describe analogs of some of these results for the\n\t\tB-Hurwitz numbers and B-version of the cut-and-join operator. Proper\n\t\treferences and short explanation of the classical picture is given in\n\t\tthe beginning of each subsection.\n\t\t\n\t\t\\subsection{Ribbon decomposition}\\label{SSec:Ribbon}\n\t\t\n\t\tIn this section we will be using notations and definitions from\n\t\t\\cite{TwistedHurwitz}; to make the article self-contained, we quote\n\t\there the most important definitions from there. A {\\em\n\t\t\tdecorated-boundary surface} (DBS) with $n$ marked points is a\n\t\tsurface with the boundary $\\partial M$ and the marked points $a_1 \\DT, a_n\n\t\t\\in \\partial M$ satisfying some nontriviality conditions (see\n\t\t\\cite{TwistedHurwitz} for details). All DBS in this section are\n\t\tassumed oriented.\n\t\t\n\t\tIf $M$ is an oriented DBS with $n$ marked points then one defines its\n\t\t{\\em boundary permutation} $\\Sigma(M) \\in S_n$ as follows:\n\t\t$\\Sigma(M)(i) = j$ if the marked points $a_i$ and $a_j$ belong to the\n\t\tsame connected component of the boundary $\\partial M$ and $a_j$ immediately\n\t\tfollows $a_i$ as one moves around the boundary in the positive\n\t\tdirection (according to the orientation).\n\t\t\n\t\tA {\\em ribbon} is a long narrow rectangle $\\rho$ with alternating\n\t\tblack and white vertices; black vertices are joined by a diagonal. For\n\t\ta DBS $M$ and $1 \\le i < j \\le n$, denote by $G[i,j] M$ the result of\n\t\tgluing of a ribbon to $M$ such that the black vertices are identified\n\t\twith the marked points $a_i$ and $a_j$, and the short sides are glued\n\t\tto short segments of $\\partial M$ containing $a_i$ and $a_j$ and directed\n\t\taccording to the orientation of $\\partial M$. Apparently, $G[i,j] M$ is an\n\t\toriented DBS with the same marked points; we call $G[i,j]$ an\n\t\toperation of {\\em ribbon gluing}.\n\t\t\n\t\tA {\\em ribbon decomposition} of the DBS $M$ is an\n\t\torientation-preserving diffeomorphism of $M$ with the DBS $G[i_m,j_m]\n\t\t\\dots G[i_1,j_1] E_n$ obtained by consecutive ribbon gluing to a union\n\t\t$E_n$ of $n$ disks each one having one marked point $a_i$ on its\n\t\tboundary. The image in $M$ of the $k$-th ribbon glued is also called a\n\t\tribbon (number $k$) and is denoted $\\rho_k \\subset M$.\n\t\t\n\t\tNow fix positive integers $m$ and $\\ell$. Let $M$ be an oriented\n\t\tdecorated-boundary surface with $2n$ marked points $a_1 \\DT, a_{2n}$,\n\t\tand let $T: M \\to M$ be an orientation-preserving smooth involution\n\t\tmapping marked points to marked points and having no fixed points on\n\t\tthe boundary $\\partial M$. If $T(D) = D$ where $D \\subset \\partial M$ is a connected\n\t\tcomponent then $D$ contains an even number $2\\mu_i$ of marked\n\t\tpoints. If $D$ is not $T$-invariant then $T$ exchanges it with another\n\t\tcomponent $D'$; both contain the same number $\\lambda_i$ of marked\n\t\tpoints. Numbers $\\lambda_1 \\DT, \\lambda_s$ and $\\mu_1 \\DT, \\mu_t$ form\n\t\ttwo partitions denoted $\\Lambda_+(M,T)$ and $\\Lambda_-(M,T)$,\n\t\trespectively; one has $\\lmod\\Lambda_+(M,T)\\rmod +\n\t\t\\lmod\\Lambda_-(M,T)\\rmod = n$.\n\t\t\n\t\tLet now $R$ a ribbon decomposition of $M$ and suppose that it is\n\t\t$T$-invariant: for any ribbon $\\rho$ its image $T(\\rho)$ is also a\n\t\tribbon. Some ribbons are $T$-invariant: $T(\\rho) = \\rho$, while others\n\t\tcome in pairs $\\rho_1 = T(\\rho_2), \\rho_2 = T(\\rho_1)$ (for brevity,\n\t\tcall these $T$-pairs). If the diagonal of a ribbon $\\rho$ joins marked\n\t\tpoints $a_i$ and $a_j$ then the diagonal to the ribbon $T(\\rho)$ joins\n\t\t$T(a_i)$ and $T(a_j)$. In particular, diagonals of $T$-invariant\n\t\tribbons join marked points $a_i$ and $T(a_i)$, for some $i$.\n\t\t\n\t\tA ribbon is homeomorphic to a disk, so, by the Brouwer theorem, $T$\n\t\thas a fixed point on each $T$-invariant ribbon. Ribbons entering a\n\t\t$T$-pair do not contain fixed points: such ribbons, if not disjoint,\n\t\tintersect only at the boundary of $M$ where $T$ has no fixed points.\n\t\t\n\t\tFor any $i$, $1 \\le i \\le 2n$, denote by $\\bar i \\stackrel{\\mbox{\\tiny def}}{=} i + n \\bmod\n\t\t2n$. \n\t\t\n\t\t\\begin{definition}\\label{Def:B-RibDecom}\n\t\t\tA {\\em B-ribbon decomposition} with the profile\n\t\t\t$(\\lambda,\\mu,m,\\ell)$ is a DBS $M$ with $2n$ marked points equipped\n\t\t\twith a smooth orientation-preserving involution $T: M \\to M$ and a\n\t\t\t$T$-invariant ribbon decomposition $R$ such that\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item $T$ sends the marked point $a_i$ to the marked point $a_{\\bar\n\t\t\t\t\ti}$, for all $i = 1 \\DT, 2n$.\n\t\t\t\t\n\t\t\t\t\\item $\\Lambda_+(M,T) = \\lambda$, $\\Lambda_-(M,T) = \\mu$.\n\t\t\t\t\n\t\t\t\t\\item The number of $T$-invariant ribbons in the decomposition $R$\n\t\t\t\tis $\\ell$ and the number of $T$-pairs of ribbons is $m$.\n\t\t\t\t\n\t\t\t\t\\item $T$ sends the ribbon $\\rho_k$ either to itself or to the\n\t\t\t\tribbon $\\rho_{k\\pm1}$, for all $k = 1 \\DT, 2m+\\ell$.\n\t\t\t\t\n\t\t\t\t\\item $T$ has exactly one fixed point inside each $T$-invariant\n\t\t\t\tribbon and no fixed points elsewhere (in particular, it has no\n\t\t\t\tfixed points on the boundary of $M$).\n\t\t\t\\end{enumerate}\n\t\t\\end{definition}\n\t\t\n\t\tThe total space $M$ of a B-ribbon decomposition is thus a DBS with a\n\t\tribbon decomposition $H_1 H_2 \\dots H_{m+\\ell} E_{2n}$ where each\n\t\t$H_k$ is either an operation $G[i_k,\\bar i_k]$ of gluing a\n\t\t$T$-invariant ribbon or an operation $G[i,j] G[\\bar i,\\bar j]$ of\n\t\tgluing a $T$-pair.\n\t\t\n\t\tB-ribbon decompositions are split into equivalence classes in a\n\t\tnatural way: $(M_1,T_1,R_1)$ is said to be equivalent to\n\t\t$(M_2,T_2,R_2)$ if there exists an orientation-preserving\n\t\tdiffeomorphism $f: M_1 \\to M_2$ mapping marked points of $M_1$ to the\n\t\tmarked points of $M_2$ with the same numbers, each ribbon of $R_1$, to\n\t\tthe ribbon of $R_2$ with the same number, and transforming the\n\t\tinvolutions one into the other: $f \\circ T_1 = T_2 \\circ f$.\n\t\t\n\t\tLet $\\xi$ be a quotient of a ribbon $\\rho$ by the symmetry with\n\t\trespect to its center. We call $\\xi$ a {\\em petal}; it is a bigon with\n\t\ta black and a white vertex; one of its sides is long, and the other,\n\t\tshort. The image of the diagonal is a line joining the black vertex\n\t\twith an internal point of the petal.\n\t\t\n\t\tLet $M$ be a DBS with $n$ marked points, and $1 \\le i \\le n$. Denote\n\t\tby $G[i] M$ the result of gluing of a petal to $M$ such that the black\n\t\tvertex is identified with the marked point $a_i$, and the short side,\n\t\twith a short segment of $\\partial M$ containing $a_i$ and directed according\n\t\tto the orientation of $\\partial M$. Apparently, $G[i] M$ is a DBS, and its\n\t\tboundary inherits an orientation from $\\partial M$. We call $G[i]$ an\n\t\toperation of petal gluing; it is similar to the operation $G[i,j]$ of\n\t\tribbon gluing described above. \n\t\t\n\t\tLet $(M,T,R)$ be a B-ribbon decomposition with the profile\n\t\t$(\\lambda,\\mu,m,\\ell)$. Denote by $\\overline M$ the quotient of $M$ by\n\t\tthe involution $T$. Since $T$ has no fixed points on the boundary and\n\t\t$T(a_i) = a_{\\bar i}$, the quotient $\\overline M$ has a\n\t\tnatural structure of a DBS with $n$ marked points $b_i = p(a_i) =\n\t\tp(a_{\\bar i})$, $i = 1 \\DT, n$, where $p: M \\to \\overline M$ is\n\t\tthe natural projection. If the ribbons $\\rho_k$ and $\\rho_{k+1}$ form a\n\t\t$T$-pair then $\\rho = p(\\rho_1) = p(\\rho_2) \\subset \\overline M$ is a\n\t\tribbon; if the ribbon $\\rho_k$ is $T$-invariant then $p(\\rho_k)\n\t\t\\subset \\overline M$ is a petal. In other words, if $M = H_1 \\dots\n\t\tH_{m+\\ell} E_{2n}$ is a ribbon decomposition described above then\n\t\t$\\overline M = \\overline H_1 \\dots \\overline H_{m+\\ell} E_n$ where\n\t\t$\\overline H_k = G[i_k,j_k]$ if $H_k = G[i_k, j_k] G[\\bar i_k, \\bar\n\t\tj_k]$ and $\\overline H_k = G[i_k]$ if $H_k = G[i_k, \\bar i_k]$. \n\t\t\n\t\t\\begin{theorem}\\label{Th:1-1Corresp}\n\t\t\tLet $\\lambda$ and $\\mu$ be two partitions such that\n\t\t\t$\\lmod\\lambda\\rmod + \\lmod\\mu\\rmod = n$, and $m$, $\\ell$, two\n\t\t\tpositive integers. There is a one-to-one correspondences between\n\t\t\tsequences of reflections in the group $B_n$ having profile\n\t\t\t$(\\lambda,\\mu,m,\\ell)$ and equivalence classes of B-ribbon\n\t\t\tdecompositions having the same profile.\n\t\t\\end{theorem}\n\t\t\n\t\t\\begin{proof}\n\t\t\tThe correspondence in question relates a sequence of reflections $h_1\n\t\t\t\\DT, h_{m+\\ell}$ to a ribbon decomposition $H_1 \\dots H_{m+\\ell}\n\t\t\tE_{2n}$ of the DBS $M$ where for every $k$ one has $H_k = G[i,j]G[\\bar\n\t\t\ti, \\bar j]$ if $h_k = (ij)(\\bar i \\bar j)$ and $H_k = G[i,\\bar i]$\n\t\t\tif $h_k = (i \\bar i)$. The involution $T$ exchanges the two ribbons\n\t\t\tattached in the first case and leaves the ribbon invariant in the\n\t\t\tsecond case. By \\cite[Theorem 1.11]{TwistedHurwitz} the boundary\n\t\t\tpermutation of $M$ is equal to $h_1 \\dots h_{m+\\ell} \\in B_n$. It\n\t\t\tfollows from the description of the conjugacy classes in Section\n\t\t\t\\ref{SSec:ConjClass} that $h_1 \\dots h_{m+\\ell} \\in\n\t\t\tC_{\\lambda\\mid\\mu}$ if and only if $\\Lambda_+(M,T) = \\lambda$ and\n\t\t\t$\\Lambda_-(M,T) = \\mu$.\n\t\t\\end{proof}\n\t\t\n\t\t\\subsection{Fermionic interpretation of B-cut-and-join}\\label{SSec:Fermion}\n\t\t\n\t\tFirst, recall some classical results about Fock space,\n\t\ta.k.a.\\ (semi-)infinite wedge product. For a partition $\\lambda =\n\t\t(\\lambda_1 \\DT\\ge \\lambda_s)$ define a {\\em Fock set}\n\t\t\\begin{equation*}\n\t\t\t\\Phi_\\lambda \\stackrel{\\mbox{\\tiny def}}{=} \\{i - \\frac12 - \\lambda_i \\mid i = 1, 2, \\dots\\}\n\t\t\t\\subset \\Integer + \\frac12;\n\t\t\\end{equation*}\n\t\there one assumes $\\lambda_i = 0$ if $i > s$.\n\t\t\n\t\t\\begin{lemma}\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item $\\#\\bigl(\\Phi_\\lambda \\cap \\{x \\mid x < 0\\}\\bigr) =\n\t\t\t\t\\#\\bigl(\\{x \\mid x > 0\\} \\setminus \\Phi_\\lambda =\n\t\t\t\t\\lmod\\lambda\\rmod \\stackrel{\\mbox{\\tiny def}}{=} \\lambda_1 \\DT+ \\lambda_s$.\n\t\t\t\t\n\t\t\t\t\\item If a subset $\\Phi \\subset \\Integer + \\frac12$ is such that $\\Phi \\cap \\{x\n\t\t\t\t\\mid x < 0\\}$ and $\\{x \\mid x > 0\\} \\setminus \\Phi$ are finite sets\n\t\t\t\tof equal cardinality then there exists a unique partition $\\lambda$\n\t\t\t\tsuch that $\\Phi = \\Phi_\\lambda$.\n\t\t\t\\end{enumerate}\n\t\t\\end{lemma}\n\t\t\n\t\tThe lemma is proved by direct induction; see \\cite{FSpace} for\n\t\tdetails.\n\t\t\n\t\tA classical (charge zero) Fock space \\cite{FSpace} (or fermion state\n\t\tspace) is a vector space $\\Lambda^{\\infty\/2}$ with the basis $w_\\lambda$ where\n\t\t$\\lambda$ runs through all partitions, including the empty partition;\n\t\t$w_\\emptyset$ is called the vacuum vector. One usually represents\n\t\t$w_\\lambda$ as an ``infinite wedge product'' $\\bigwedge_{u \\in\n\t\t\t\\Phi_\\lambda} z_u = z_{1\/2-\\lambda_1} \\wedge z_{3\/2-\\lambda_2}\n\t\t\\wedge \\dots$; note that the indices at the product form an increasing\n\t\tsequence.\n\t\t\n\t\tLet $k \\in \\mathbb{Z}$, $k \\ne 0$. Consider a linear operator $a_k:\n\t\t\\Lambda^{\\infty\/2} \\to \\Lambda^{\\infty\/2}$ acting on the basic vectors as follows:\n\t\t\\begin{equation*}\n\t\t\ta_k (\\bigwedge_{u \\in \\Phi_\\lambda} z_u) = \\sum_{v \\in \\Phi_\\lambda}\n\t\t\tz_{v-k} \\wedge (-1)^{d_{\\lambda,k}(v)} \\bigwedge_{u \\in \\Phi_\\lambda\n\t\t\t\t\\setminus \\{v\\}} z_u\n\t\t\\end{equation*}\n\t\twhere $d_{\\lambda,k}(v)$ is defined as $\\#\\{u \\in \\Phi_\\lambda \\mid\n\t\tv-k \\le u < v\\}$ if $k > 0$ and as $\\#\\{u \\in \\Phi_\\lambda \\mid v < u\n\t\t\\le v-k\\}$ if $k < 0$.\n\t\t\n\t\t\\begin{theorem}[\\cite{FSpace}]\\label{Th:Fock}\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item Operators $a_i$ and $a_j$ commute unless $i+j = 0$.\n\t\t\t\t\n\t\t\t\t\\item\\label{It:BFC} For any element $x \\in \\Lambda^{\\infty\/2}$ there exists a\n\t\t\t\tunique polynomial $R_x \\in \\mathbb{C}[p] \\stackrel{\\mbox{\\tiny def}}{=} \\mathbb{C}[p_1, p_2,\n\t\t\t\t\\dots]$ such that $x = R_x(a_1,a_2,\\dots)w_{\\emptyset}$. If $x =\n\t\t\t\tw_\\lambda$ then $R_x = s_\\lambda$ (a classical Schur polynomial).\n\t\t\t\\end{enumerate}\n\t\t\\end{theorem}\n\t\t\n\t\tThe map $R: \\Lambda^{\\infty\/2} \\to \\mathbb{C}[p]$ defined in \\ref{It:BFC} is called\n\t\tboson-fermion correspondence. It is a linear bijection, so for any\n\t\tlinear operator $A: \\Lambda^{\\infty\/2} \\to \\Lambda^{\\infty\/2}$ one can define its polynomial\n\t\tcounterpart $\\hat A \\stackrel{\\mbox{\\tiny def}}{=} R A R^{-1}: \\mathbb{C}[p] \\to \\mathbb{C}[p]$;\n\t\tthe map $A \\leftrightarrow \\hat A$ is an isomorphism of the algebras\n\t\tof linear operators on $\\Lambda^{\\infty\/2}$ and on $\\mathbb{C}[p]$.\n\t\t\n\t\t\\begin{proposition}[\\cite{FSpace}]\\label{Pp:OperHat}\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\\label{It:PdP} If $A = a_k$ then $\\hat A = p_k$ (the operator of\n\t\t\t\tmultiplication) if $k > 0$ and $\\hat A = -k \\pder{}{p_{-k}}$ if $k\n\t\t\t\t< 0$.\n\t\t\t\t\n\t\t\t\t\\item\\label{It:Energy} If $A = \\sum_{i=1}^\\infty a_i a_{-i}$ (called\n\t\t\t\tthe energy operator) then $\\hat A$ is the Euler vector field\n\t\t\t\t$\\sum_{i=1}^\\infty i p_i \\pder{}{p_i}$.\n\t\t\t\t\n\t\t\t\t\\item\\label{It:CJ} If $A = \\sum_{\\substack{i,j,k \\ne 0\\\\ i+j+k = 0}}\n\t\t\t\ta_i a_j a_k$ then $\\hat A$ is the classical cut-and-join operator\n\t\t\t\t$\\mathcal{CJ}$.\n\t\t\t\\end{enumerate}\n\t\t\\end{proposition}\n\t\t\n\t\tIt can be proved (see \\cite{FSpace}) that for any partition $\\lambda$\n\t\tthe basic element $w_\\lambda$ is an eigenvector of both the energy\n\t\toperator and the operator from assertion \\ref{It:CJ} of Proposition\n\t\t\\ref{Pp:OperHat}. The corresponding eigenvalues are $\\lmod\n\t\t\\lambda\\rmod \\stackrel{\\mbox{\\tiny def}}{=} \\sum_i \\lambda_i$ and $\\phi(\\lambda) \\stackrel{\\mbox{\\tiny def}}{=} \\sum_i\n\t\t\\lambda_i(\\lambda_i-2i+1)$. Equivalently, the Schur polynomial\n\t\t$s_\\lambda$ is the eigenvector of the Euler field with the eigenvalue\n\t\t$\\lmod \\lambda\\rmod = \\deg s_\\lambda$ and of the cut-and-join operator\n\t\twith the eigenvalue $\\phi(\\lambda)$.\n\t\t\n\t\tConsider now a space $L \\stackrel{\\mbox{\\tiny def}}{=} \\Lambda^{\\infty\/2} \\otimes \\Lambda^{\\infty\/2}$ (two-particle\n\t\tfermion state space) with the basis $w_\\lambda \\otimes w_\\mu$, where\n\t\t$\\lambda, \\mu$ are partitions. The tensor square of the boson-fermion\n\t\tcorrespondence $R^{\\otimes 2}: L \\to \\mathbb{C}[p]^{\\otimes 2}\n\t\t=\\mathbb{C}[u,v]$ where $u = (u_1, u_2, \\dots)$ and $v = (v_1, v_2,\n\t\t\\dots)$ is a linear isomorphism. Like for the classical case, any\n\t\toperator $A: L \\to L$ has its polynomial counterpart $R^{\\otimes 2} A\n\t\t(R^{\\otimes 2})^{-1}: \\mathbb{C}[u,v] \\to \\mathbb{C}[u,v]$, which we also\n\t\tdenote by $\\hat A$.\n\t\t\n\t\tFor every $k \\ne 0$ consider two linear operators: $b_k \\stackrel{\\mbox{\\tiny def}}{=} a_k\n\t\t\\otimes I + I \\otimes a_k$ and $c_k \\stackrel{\\mbox{\\tiny def}}{=} a_k \\otimes I - I \\otimes\n\t\ta_k$. It follows from Proposition \\ref{Pp:OperHat} (assertion\n\t\t\\ref{It:PdP}) that\n\t\t\\begin{align}\n\t\t\t&\\begin{aligned}\n\t\t\t\t&\\hat b_k = u_k + v_k = p_k,\\\\\n\t\t\t\t&\\hat c_k = u_k - v_k = q_k\n\t\t\t\\end{aligned}\n\t\t\t\\qquad \\text{if } k > 0,\\label{Eq:KPos}\\\\\n\t\t\t&\\begin{aligned}\n\t\t\t\t&\\hat b_k = -k(\\pder{}{u_{-k}} + \\pder{}{v_{-k}}) =\n\t\t\t\t-2k\\pder{}{p_{-k}},\\\\\n\t\t\t\t&\\hat c_k = -k(\\pder{}{u_{-k}} - \\pder{}{v_{-k}}) =\n\t\t\t\t-2k\\pder{}{q_{-k}}\n\t\t\t\\end{aligned}\n\t\t\t\\qquad \\text{if } k < 0.\\label{Eq:KNeg}\n\t\t\\end{align}\n\t\tHere we are using notation from Section \\ref{SSec:Change}.\n\t\t\n\t\tApparently, $b_i$ and $b_j$, as well as $c_i$ and $c_j$, commute unless\n\t\t$i+j = 0$; $b_i$ and $c_j$ commute for all $i, j$.\n\t\t\n\t\t\\begin{proposition}\n\t\t\tIf \n\t\t\t\\begin{align*}\n\t\t\t\tJ_1 &\\stackrel{\\mbox{\\tiny def}}{=} \\frac{1}{12}(\\sum_{\\substack{i+j+k=0\\\\i,j,k \\ne 0}}\n\t\t\t\tb_ib_jb_k + \\sum_{\\substack{i+j+k=0\\\\i,j,k \\ne 0}}b_ic_jc_k +\n\t\t\t\t\\sum_{\\substack{i+j+k=0\\\\i,j,k\\neq0}}\n\t\t\t\tc_ib_jc_k+\\sum_{\\substack{i+j+k=0\\\\i,j,k \\ne 0}}c_ic_jb_k)\\\\\n\t\t\t\t&= \\frac{1}{12}\\sum_{\\substack{i+j+k=0\\\\i,j,k\\neq0}} b_ib_jb_k +\n\t\t\t\t\\frac{1}{4} \\sum_{\\substack{i+j+k=0\\\\i,j,k \\ne 0}} b_ic_jc_k,\\\\\n\t\t\t\tJ_2 &\\stackrel{\\mbox{\\tiny def}}{=} \\sum_{\\ell=1}^\\infty b_\\ell c_{-\\ell} +\n\t\t\t\t\\sum_{\\ell=1}^\\infty c_\\ell b_{-\\ell}\n\t\t\t\\end{align*}\n\t\t\tthen $\\hat{J_1} = \\mathcal{CJ}_1$ and $\\hat{J_2} = \\mathcal{CJ}_2$ (the B-cut-and-join\n\t\t\toperators), and also\n\t\t\t\\begin{align*}\n\t\t\t\tJ_1 &= C \\otimes \\name{Id} + \\name{Id} \\otimes C,\\\\\n\t\t\t\tJ_2 &= E\\otimes \\name{Id} - \\name{Id}\\otimes E\n\t\t\t\\end{align*}\n\t\t\twhere $C$ and $E$ are the operators defined in Proposition\n\t\t\t\\ref{Pp:OperHat}.\n\t\t\\end{proposition}\n\t\t\n\t\tThe proof is straightforward: the first pair of equations is proved by\n\t\ta calculation using \\eqref{Eq:KPos}, \\eqref{Eq:KNeg}, \\eqref{Eq:CJ_1}\n\t\tand \\eqref{Eq:CJ_2}, as well as the commutativity relations. The\n\t\tsecond pair of equations follows from Proposition \\ref{Pp:ChVar} or\n\t\tcan be obtained by a direct calculation, too.\n\t\t\n\t\t\\begin{corollary}\\label{Th:Fock2}\n\t\t\t$w_\\lambda\\otimes w_\\mu$ are eigenvectors of $J_1$ and $J_2$ with the\n\t\t\teigenvalues $\\sum_{i=1}^\\infty (\\lambda_i(\\lambda_i-2i+1) +\n\t\t\t\\mu_i(\\mu_i-2i+1)) $ and $\\sum_{i=1}^\\infty (\\lambda_i-\\mu_i)$,\n\t\t\trespectively.\n\t\t\\end{corollary}\n\t\t\n\t\t\\subsection{KP hierarchy for B-Hutwitz numbers}\n\t\t\n\t\tThe KP hierarchy is one of the best known and studied integrable\n\t\tsystems; see e.g.\\ \\cite{Kramer,KazarianHodge,FSpace} for details. It\n\t\tis an infinite system of PDE applied to a formal series $F \\in\n\t\t\\mathbb{C}[[t]]$ where $t = (t_1, t_2, \\dots)$ is a countable collection\n\t\tof variables (``times''). Each equation looks like\n\t\t$\\pdertwo{F}{t_i}{t_j} = P_{ij}(F)$ where $P_{ij}$ is a homogeneous\n\t\tmultivariate polynomial with partial derivatives of $F$ used as its\n\t\targuments. For example, the first two equations of the hierarchy are\n\t\t\n\t\t\\begin{align*}\n\t\t\t&F_{22}=-\\frac{1}{2}F_{11}^2 + F_{31} -\\frac{1}{12}F_{1111}\\\\\n\t\t\t&F_{32}=-F_{11}F_{21}+F_{41}-\\frac{1}{6}F_{2111};\n\t\t\\end{align*}\n\t\there $F_{i_1i_2\\dots i_n}$ means $\\frac{\\partial^nF}{\\partial\n\t\t\tu_{i_1}\\partial u_{i_2\\dots\\partial u_{i_n}}}$. If $F$ is a solution\n\t\tof the hierarchy, its exponential $\\tau= e^F$ is called a\n\t\t$\\tau$-function.\n\t\t\n\t\t\\begin{example}\\label{Ex:T1Tau}\n\t\t\tEquations of the hierarchy do not contain first-order derivatives,\n\t\t\tso the function $F(t) = t_1$ is a solution of the KP hierarchy;\n\t\t\t$\\tau = e^{t_1}$ is a $\\tau$-function.\n\t\t\\end{example}\n\t\t\n\t\t\\begin{proposition}\\label{Pp:Sato}\n\t\t\tThere exists a Lie algebra $\\mathcal G$ of differential operators on\n\t\t\tthe space $\\mathbb{C}[[p]]$ having the following properties:\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\\label{It:GLInf} $\\mathcal G$ is isomorphic to a\n\t\t\t\tone-dimensional central extension $\\widehat{\\name{gl}(\\infty)}$ of\n\t\t\t\ta Lie algebra $\\name{gl}(\\infty)$ of infinite matrices\n\t\t\t\t$(x_{ij})_{i,j \\in \\mathbb{Z}}$ having nonzero elements at a finite\n\t\t\t\tset of diagonals.\n\t\t\t\t\n\t\t\t\t\\item\\label{It:F2F3} The Euler field $E$ and the cut-and-join\n\t\t\t\toperator $\\mathcal{CJ}$ belong to $\\mathcal G$.\n\t\t\t\t\n\t\t\t\t\\item\\label{It:Orbit} For any $\\tau$-function $f$ and any $X \\in\n\t\t\t\t\\mathcal G$, the power series $e^X f$ is a $\\tau$-function,\n\t\t\t\ttoo. \n\t\t\t\\end{enumerate}\n\t\t\\end{proposition}\n\t\t\n\t\tSee \\cite{Kramer} for the definition of $\\mathcal G$, \\cite{2Sato} and\n\t\t\\protect{\\cite[section 6]{KazarianHodge}} for the proof.\n\t\t\n\t\t\\begin{theorem}\\label{Th:HBsolKP}\n\t\t\tThe generating function $\\mathcal{H}^B(\\beta,\\gamma,u+v,u-v)$ is a\n\t\t\t$2$-parameter family of $\\tau$-functions, independently in the $u$\n\t\t\tand the $v$ variables.\n\t\t\\end{theorem} \n\t\t\n\t\t\\begin{proof}\n\t\t\tEquation \\eqref{Eq:GenFunc} and Proposition \\ref{Pp:ChVar} imply that\n\t\t\t\\begin{align*}\n\t\t\t\t\\mathcal{H}^B(\\beta,\\gamma,u+v,u-v) = e^{\\beta(\\mathcal{CJ}_u+\n\t\t\t\t\t\\mathcal{CJ}_v)}e^{\\gamma(E_u-E_v)} e^{u_1}e^{v_1} =\n\t\t\t\te^{(\\beta\\mathcal{CJ}_{u}+\\gamma E_u)}e^{u_1} e^{(\\beta\\mathcal{CJ}_{v}-\\gamma\n\t\t\t\t\tE_v)}e^{v_1}\n\t\t\t\\end{align*}\n\t\t\t(operators $\\mathcal{CJ}_i$ and $\\mathcal{CJ}_v$ commute because they act on different\n\t\t\tsets of variables).\n\t\t\t\n\t\t\tBy assertion \\ref{It:F2F3} of Proposition \\ref{Pp:Sato},\n\t\t\t$\\beta\\mathcal{CJ}_{u}+ \\gamma E_u \\in \\mathcal G_u$ (the Lie algebra $\\mathcal\n\t\t\tG$ acting by differential operators on the $u$ variables). By Example\n\t\t\t\\ref{Ex:T1Tau}, $e^{u_1}$ is a $\\tau$-function (in the $u$ variables),\n\t\t\tand so is $\\mathcal{H}^B(\\beta,\\gamma,u+v,u-v)$ by assertion\n\t\t\t\\ref{It:Orbit} of the same proposition. The reasoning for the $v$\n\t\t\tvariables is the same.\n\t\t\\end{proof}\n\t\t\n\t\t\\begin{corollary}\n\t\t\tThe generating function $\\mathcal{H}^D(\\beta,u+v,u-v)$ is a 1-parameter\n\t\t\tfamily of $\\tau$-functions independently in the $u$ and the $v$\n\t\t\tvariables.\n\t\t\\end{corollary}\n\t\t\\begin{proof}\n\t\t\tWe apply the same strategy as in the proof of theorem\n\t\t\t\\ref{Th:HBsolKP} to the function $\\mathcal{H}^D(\\beta,u+v,u-v) =\n\t\t\te^{\\beta(\\mathcal{CJ}_u+ \\mathcal{CJ}_v)}e^{u_1}e^{v_1}$.\n\t\t\\end{proof}\n\t\t\n\t\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nHydrogen and helium are the two most abundant elements in giant\nplanets. While Jupiter and Saturn are well characterized on the\nsurface, many basic questions about its interior have not been\nanswered\\cite{Gu02}. Jupiter's surface composition has been measured\n{\\em in situ} by the Galileo probe:\\cite{FH03} H 74.2\\% by weight, He\n23.1\\%, and 0.027\\% heavier elements, which is enhanced compared to\nthe protosolar composition: 0.015\\% heavier elements along with H\n73.6\\% and He 24.9\\%. The abundance of heavy element in the interior\nand their distribution are not well characterized.\\cite{SG04} In\nparticular, it is conversial whether Jupiter has a rocky core. The\ndetection of a core in Jupiter may validate the standard model of\ngiant-planet formation, nucleated capture of nebular\nhydrogen.\\cite{Li93} An alternative scenario was proposed by Alan Boss\nwho suggested that giant planets form directly from spiral\ninstabilities in protostellar disks.\\cite{Boss97,Boss04} Under this\ngravitational instability hypothesis, giant planets would not have a\ncore, or at least a much smaller one.\n\nSince there is no direct way to detect a core in Jupiter, one must\ninstead refer to indirect measurements and to models for the planet\ninterior. Such models are constrained by the available observation\ndata, in particular, the properties at the planet surface and the\ngravitational moments measured through fly-by trajectories. All models\nrely on an equation of state (EOS) of hydrogen-helium\nmixtures. However, the uncertainties in the available EOS are large\nand have not allowed one, among many other questions, to determine\nwhether Jupiter has a core.\n\nIn this article, we present results from path integral Monte Carlo\n(PIMC) simulations\\cite{Ce95} that enable us to study quantum many-body\nsystems at finite temperature from first principles. In this\nsimulation, hydrogen-helium mixtures are represented by an ensemble of\nelectrons, protons and helium nuclei, each described by a path in\nimaginary time to incorporate quantum effects. The electrons are\ntreated as fermions while exchange effects for the nuclei can be\nneglected for the considered thermodynamics conditions.\n\n\\section{PATH INTEGRAL MONTE CARLO}\n\nThe thermodynamic properties of a many-body quantum system at finite\ntemperature can be computed by averaging over the density matrix,\n$\\hat{\\rho} = e^{-\\beta \\hat{H}}, \\beta=1\/k_{\\rm B} T$.  \nPath integral formalism\\cite{Fe53} is based on the identity,\n\\begin{equation}\ne^{-\\beta \\hat{H}} = \\left[ e^{-\\frac{\\beta}{M} \\hat{H}} \\right]^M\n\\end{equation}\nwhere $M$ is a positive integer. Insertion of complete sets of states\nbetween the $M$ factors leads to the usual imaginary time path\nintegral formulation, written here in real space,\n\\begin{equation}\n\\rho({\\bf R},{\\bf R}';\\beta)=\n\\int\\ldots\\int d{\\bf R}_{1}\\ldots d{\\bf R}_{M-1} \\; \\rho({\\bf R},{\\bf R}_{1};\\tau)\\ldots\\rho({\\bf R}_{M-1},{\\bf R}';\\tau)\n     \\label{eq2.5}\n\\end{equation}\nwhere $\\tau=\\beta\/M$ is the time step. Each of the $M$ steps in the\npath now has a high temperature density matrix\n$\\rho({\\bf R}_k,{\\bf R}_{k+1};\\tau)$ associated with it. The integrals are\nevaluated by Monte Carlo methods. \nThe density matrix for bosonic and fermionic systems can be obtained\nby projecting out states of corresponding symmetry. In PIMC, one sums\nup different permutations $P$,\n\\begin{equation}\n\\nonumber\n\\rho_{\\rm b\/f}({\\bf R},{\\bf R}';\\beta) \n= \n\\frac{1}{N!} \\sum_{\\mathcal{P}} \\; (\\pm 1)^{\\mathcal{P}} \\rho({\\bf R},{\\mathcal{P}} {\\bf R}' ; \\beta)\n= \\frac{1}{N!} \\sum_{\\mathcal{P}} \\; (\\pm 1)^{\\mathcal{P}} \\; \\! \\! \\! \\! \\! \\! \\! \\int \\limits_{{\\bf R} \\rightarrow {\\mathcal{P}} {\\bf R}' } \n\\! \\! \\! \\! \\! \\! d{\\bf R}_t \\, e^{-U[{\\bf R}_t] }.\n\\end{equation}\nFor bosons, this is essentially an exact numerical algorithm that has\nfound many applications.\\cite{Ce95} For fermions, the cancellation of\npositive and negative contributions leads to numerically unstable\nmethods, which is known as the {\\em fermion sign problem}. Ceperley\nshowed that this problem can be solved by introducing the {\\em\nrestricted path} approximation,\\cite{Ce91,Ce96}\n\\begin{equation} \n\\rho_f({\\bf R}_0, {\\bf R}' ;\\beta) \\approx\n\\frac{1}{N!}\\; \\sum_{\\mathcal{P}} \\; (-1)^{\\mathcal{P}} \n\\int\\limits_{\\scriptsize {\n\\! \\! \\! \\! \\! \\! \\! \\! \\! \\! \\! \\! \\! \\!\\! \\! \\! \\! \n\\begin{array}{c} \n  {\\bf R}_0 \\rightarrow {\\mathcal{P}} {\\bf R}'\\\\\n  {\\rho_T({\\bf R}(t),{\\bf R}_0;t)>0}\n\\end{array}}\n \\! \\! \\! \\! \\! \\! \\! \\! \\! \\! \\! \\! \\! \\! \\! \\!\n\\! \\! \\! \\! \\! \\! \\! \\! \\! \\! } d{\\bf R}_t \\;\\; e^{-U[{\\bf R}(t)] } \\quad,\n\\label{restricted_PI} \n\\end{equation}\nwhere one only samples path ${\\bf R}(t)$ that stay within the positive\nregion of a {\\em trial} density matrix,\n$\\rho_T({\\bf R}(t),{\\bf R}_0;t)>0$. This procedure leads to an efficient\nalgorithm for fermionic systems. All negative contributions to\ndiagonal matrix elements are eliminated.\\cite{MPC03} Contrary to the\nbosonic case, the algorithm is no longer exact since it now depends on\nthe approximations for the trial density matrix. However, the method\nhas worked very well in many applications.\\cite{PC94,MP02} For\n$\\rho_T$, one can use the free particle or a variational density\nmatrix.\\cite{MP00}\nIn the results presented here, the\nhigh temperature density matrix was taken as a product of exact pair\ndensity matrices,\n\\begin{eqnarray}\n\\frac{\\rho({\\bf R},{\\bf R}';\\tau)}{\\rho_0({\\bf R},{\\bf R}';\\tau)} \n&=& \n\\left< e^{-\\int_0^\\tau dt \\sum_{i<j} V({\\bf r}_{ij}) } \\right>_{\\rm {\\bf R} \\to {\\bf R}'}\n=\n\\left< \\prod_{i<j} e^{-\\int_0^\\tau dt V({\\bf r}_{ij}) } \\right>_{\\rm {\\bf R} \\to {\\bf R}'}\\\\\n&\\approx&\n\\prod_{i<j} \\left< e^{-\\int_0^\\tau dt V({\\bf r}_{ij}) } \\right>_{\\rm {\\bf r}_{ij} \\to {\\bf r}_{ij}'} \n\\equiv\ne^{ - \\sum_{i<j} u({\\bf r}_{ij},{\\bf r}_{ij}'; \\tau) },\n\\end{eqnarray}\nwhere $\\rho_0({\\bf R},{\\bf R}';\\tau)$ is the free particle density matrix and\n$u({\\bf r}_{ij},{\\bf r}_{ij}'; \\tau)$ is the pair action for paths initially\nseparated by ${\\bf r}_{ij}$ and finally at time $\\tau$ by ${\\bf r}'_{ij}$. An\napproximation is introduced by assuming that the different pair\ninteractions can be averaged by independent Brownian random walks that\nare denoted by brackets $\\left<\\ldots\\right>$. This approach is\nefficient but not exact, and therefore puts a limit on the imaginary\ntime step $\\tau$ in many-body simulations. The pair action, $u$, can\nbe computed by different methods.\\cite{St68,Po88,KS95}\nThe following simulation results were derived with $N_e=32$ electrons\nand the corresponding number of protons, $N_p$, and helium nuclei\n$N_{\\rm He}$ to obtain a neutral system ($N_e=N_p+2 N_{\\rm He}$) with\nhelium fraction $x\\equiv 2 N_{\\rm He}\/N_e$. Periodic boundary conditions\nare applied. As imaginary time discretization, we employ\n$\\tau=0.079$. (Atomic units of Bohr radii and Hartrees will be used\nthroughout this article.)\n\nThe electrons are treated as fermions with fixed spin. We use\nvariational nodes to restrict the paths and have therefore extended\nthe approach in Ref.~[\\onlinecite{MP00}] to mixtures. The nuclei\nare also treated as paths but their exchange effects are not relevant\nhere.\n\n\\section{PHASE DIAGRAM OF HOT DENSE HYDROGEN}\n\n\\begin{figure}[htb]\n\\centerline{\\includegraphics[angle=0,width=4in]{log_astro_phase_diagram12.eps}}\n\\caption{Density-temperature phase diagram of hot dense hydrogen. \n       The blue dash-dotted lines separate the molecular, the atomic,\n       the metallic, and the plasma regime. The green solid lines are\n       isentropes for Jupiter and stars with 0.3, 1, and 15 solar\n       masses. Single shock Hugoniot states as well as the inertial\n       confinement fusion path\\cite{lindl} are indicated by dashed\n       lines. The thin solid line show $\\rho$-$T$ conditions of PIMC\n       simulations.}\n\\label{phase}\n\\end{figure}\n\nFigure~\\ref{phase} shows the high temperature phase diagram of dense\nhydrogen beginning with the fluid and reaching up to a highly ionized\nplasma state. The figure includes the isentropes for Jupiter and low\nmass stars\\cite{SC95} and indicates the thermodynamic conditions, at\nwhich PIMC simulations have been applied.\\cite{MC00,MC01,MP02} We are\nnow going to use these simulation results to characterize hot dense\nhydrogen from a path integral perspective.\n\n\\begin{figure}[tb]\n\\centerline{\\includegraphics[angle=0,width=4in]{H024_4.smaller.ps}}\n\\vspace*{3mm}\n\\caption{Snapshot from PIMC simulations of pure hydrogen in the molecular regime at \n         $T=5\\,000\\,\\rm K$ and $r_s=4.0$. The pink spheres denote the\n         protons. The bonds (white lines) were added as a guide to the\n         eye. The electron paths are shown in red and blue [light and\n         dark gray] depending on their spin state. }\n\\label{pict_dilute_H2}\n\\end{figure}\n\nAt low density and temperature, one finds a fluid of interacting\nhydrogen molecules. A PIMC snapshot for $T=5\\,000\\,\\rm K$ and\n$r_s=4.0$ is shown in Fig.~\\ref{pict_dilute_H2}. The proton paths are\nvery localized due to their high mass. Their spread can be estimated\nfrom the de Broglie thermal wavelength, $\\lambda_{d}^{2}=\\hbar^{2}\n\\beta \/2\\pi m$. The electron paths are more spread out but they \nare localized to some extent since two electrons of opposite spin\nestablish the chemical bond in the hydrogen molecule.\n\nIf the temperature is raised from 5000 to $10^6$ K, hydrogen\nundergoes a smooth transition from a molecular fluid though an atomic\nregime and finally to a two-component plasma of interacting electrons\nand protons. Many-body simulations at even higher temperatures are not\nneeded since analytical methods like the Debye-H{\\\"u}ckel theory work\nwell. PIMC with explicit treatment of the electrons can also be\napplied to temperatures below 5$\\,$000$\\,$K but groundstate methods\nare then more practical since excitation become less relevant and the\ncomputational cost scale with the length of paths like $T^{-1}$.\n\nA detailed analysis of the chemical species present in the low density\nregime ($r_s \\ge 2.6$) is given in Ref.~\\onlinecite{MC00}. With\ndecreasing density, one finds that the degree of molecular\ndissociation increases since the atomic state has higher entropy. For\nthe same reason, one observes that the degree of atomic ionization\nincreases with decreasing density. All these low-density effects can\nbe well characterized by analytical models based on approximate free\nenergy expressions for atoms, molecules and ionized\nparticles.\\cite{SC92,Ro90,rotesbuch,Ju01} In this regime, one also\nfinds reasonably good agreement between for the EOS derived from PIMC\nand chemical models.\\cite{MC00}\n\n\\begin{figure}[tb]\n\\centerline{\\includegraphics[angle=0,width=4in]{D023_1.smaller.ps}}\n\\caption{Deuterium in the dense molecular regime ($T=5\\,000\\,\\rm K$ and $r_S=1.86$) \n       Due to the density increase compared to\n       Fig.~\\ref{pict_dilute_H2} (see details there), the electron\n       paths permute with a rising probability (shown as yellow [light\n       gray] lines) but are still localized enough to form a bond\n       between the two protons in the molecule. The electron density\n       average over many electron configurations is indicated in gray\n       color on the blue rectangles.}\n\\label{pict_dense_H2}\n\\end{figure}\n\nIf the density is increased at $T=5\\,000\\,$K, one finds an\nintermediate regime of strongly interacting molecules\n(Fig.~\\ref{pict_dense_H2}). Some electron paths exchange with\nneighboring molecules indicating the importance of fermionic\neffects. However, the electrons are still localized enough to provide\na sufficient binding force for the protons.\n\n\\begin{figure}[tb]\n\\centerline{\\includegraphics[angle=0,width=4in]{D159_1.smaller.ps}}\n\\caption{Deuterium in the metallic regime characterized by unpaired \n\tprotons and a gas of degenerate electrons. The snapshot was\n\ttaken from a PIMC simulation at $T=6\\,250\\,\\rm K$ and\n\t$r_s=1.60$. The electron paths are delocalized and exchange\n\tfrequently (see description of Fig.~\\ref{pict_dense_H2}).}\n\\label{pict_metallic_H}\n\\end{figure}\n\nIf the density is increased further from $r_s=1.86$ to 1.60, this\nbinding force is lost due to further delocalization of the electrons\n(Fig.~\\ref{pict_metallic_H}). Almost all electrons are now involved in\nlong exchange cycles indicating a highly conducting, metallic\nstate. No binding forces of the protons can be observed. One can\ntherefore conclude that hydrogen metallizes in dissociated, or atomic\nform, a conclusion that is consistent with DFT\nsimulations\\cite{Banier2000,Scandolo2003,Desjarlais2003,Bonev2004}. If\nthe density is increased further, the electrons form a rigid\nneutralizing background and one recovers the limit of a one-component\nplasma of protons. If the temperature is increased, the electron paths\nget shorter and shorter, the degree of electron degeneracy decreases\ngradually and one recovers the limit of a two-component plasma at high\ntemperature.\n\nThe nature of the molecular-metallic transition has not yet been\ndetermined with certainty. A large number of analytical\nmodels\\cite{Be99} predict a first order transition, others do\nnot.\\cite{Ro90} PIMC simulation with free particle nodes by\nW. Magro~{\\em et al.}\\cite{Ma96} showed an abrupt transition characterized by\na region $\\left.\\frac{dP}{dT}\\right|_V<0$ for $r_s$=1.86. Later work\nby B. Militzer\\cite{BM00} using more accurate variational nodes did\nnot show such a region for $r_s$=1.86. Whether PIMC simulations with\nvariational nodes predict a gradual molecular-metallic transition for\nall temperatures, or if the region of an abrupt transition has shifted\nto lower, not yet accessible, temperatures has not been\ndetermined. However, recent density functional molecular dynamics\nsimulations by various authors predict a first order transition below\n$5\\,000\\,$K.\\cite{Banier2000,Scandolo2003,Desjarlais2003,Bonev2004}\n\nThere was a lot of recent interest focused on the hydrogen EOS because\nof the {\\em unexpectedly high compressibility} inferred from the\nlaser-driven shock wave experiment by Da Silva~{\\em et al.}\\cite{Si97} and\nCollins~{\\em et al.}\\cite{Co98} using the Nova laser at Lawrence Livermore\nNational Laboratory. The measurements indicated that hydrogen could be\ncompressed to about 6-fold the initial density rather than 4-fold as\nindicated by the Sesame model.\\cite{Ke83,Kerley2003} Even though the\ntemperatures and pressures reached in shock experiments are higher\nthan those in the giant planets these measurements are crucial here\nsince they represent the only way to distinguish between different EOS\nmodels at high temperature.\n\nThe Nova results challenged the existing understanding of high P-T\nhydrogen and triggered many new experimental and theoretical\nefforts. Different chemical models gave rise to very different\npredictions\\cite{Be99,Ro90,Ju00} ranging from 4-fold compression as\nsuggested in\\cite{Ke83,Kerley2003} to 6-fold as predicted by the Ross\nmodel.\\cite{Ro98} While the accuracy of chemical models did not allow\nany conclusive predictions, first principle simulations from\nPIMC\\cite{MC00,Mi01} as well as from density functional molecular\ndynamics\\cite{Le00,Desjarlais2003,Bonev2004} consistently predict a\nlower compressibility of about 4.3.\n\nSince then there have been many attempts to resolve this discrepancy\nbut the most important contributions came from new experiments by\nKnudson~{\\em et al.}\\cite{Kn01,Kn03,Kn04} at Sandia National\nLaboratory. Instead of a laser drive, they used magnetically driven\nshock waves in combination with bigger samples. They found a\nsignificantly lower compressibility quite similar to predictions from\nfirst principles methods. The new results are also supported by a\nthird set of experiments by Russian investigators using spherically\nconverging shock waves.\\cite{Belov2002,Boriskov2003}\n\n\\section{HYDROGEN-HELIUM MIXTURES}\n\nDue to importance for astrophysical applications, EOS models for\nhy\\-dro\\-gen-he\\-lium mixtures have been studied for quite some time. The\nmost widely used EOS was derived by Saumon, Chabrier and van\nHorn (SCH).\\cite{SC95} Like previous chemical models, it is based on a\nhydrogen and a helium EOS that combined using an ideal mixing\nrule. \n\nFollowing the discussion of the hydrogen EOS above, we are now\nanalyzing the second ingredient: the EOS of helium. Given its low\nabundance in giant planets, helium is not expected to be present in\nvery high concentration. However,\nStevenson\\cite{Stevenson77b} has proposed\nthat the hydrogen-helium mixture could phase separate under certain\nhigh pressure conditions. As a result, helium droplets would form and\nfall as rain towards planet core. The associated release of latent\nheat would delay the cooling process of the planet. This was suggested\nas one possible explanation of why standard models predict Saturn to cool\nat a much faster rate than is observed.\\cite{FH04} In order to\ndetect hydrogen-helium phase separation one needs an EOS of pure\nhelium that we will now discuss.\n\n\\begin{figure}[htb]\n\\centerline{\\includegraphics[width=2.48in]{He_P_vs_T06.eps}~~~~\\includegraphics[width=2.4in]{He_E_vs_T08.eps}}\n\\caption{Equation of state comparison for pure helium showing the pressure (left) \n       and the internal energy (right) for different density as\n       derived from path integral Monte Carlo simulations and the\n       chemical model by Saumon, Chabrier, and van Horn.\\cite{SC95}}\n\\label{HeEOS}\n\\end{figure}\n\nFigure~\\ref{HeEOS} shows an EOS comparison of our PIMC calculations\nwith the SCH model. One finds that the pressure is underestimated by\nSCH and that the deviations increase with density. The internal energy\nat high temperature is also underestimated. The analysis suggests a\nmore careful treatment of the helium ionization states is needed to\nimprove the chemical model.\n\nTo conclude the comparison with chemical models, we test one more\nassumption that is generally made: the ideal mixing hypothesis. We\nperformed a large number of PIMC calculations of fully interacting\nhydrogen-helium mixtures at a fixed density of $r_s=1.86$ for various\ntemperatures and mixing ratios. The resulting correction to ideal\nmixing is given by,\n\\begin{equation}\n\\Delta f_{\\rm mix} = f(x) - (1-x)\\,f(x=0) - x\\,f(x=1)\\,.\n\\end{equation} \n\n\\begin{figure}[htb]\n\\centerline{\\includegraphics[width=2.48in]{mixing_pressure02.eps}~~~~\\includegraphics[width=2.4in]{mixing_internal_energy02.eps}}\n\\caption{Excess mixing pressure $\\Delta P_{\\rm mix}$ and internal energy per electron $\\Delta E_{\\rm mix}$\n       are shown for a hydrogen-helium mixture at $r_s=1.86$. Path\n       integral Monte Carlo results are shown for several\n       temperatures. The mixing is performed at constant\n       density. $\\Delta E_{\\rm mix}$ is largest for T=31$\\,$250$\\,$K\n       because at this temperature the two end members are\n       characterized by very different degrees of ionization.}\n\\label{mix}\n\\end{figure}\n\nFigure~\\ref{mix} shows that the corrections to pressure, $\\Delta\nP_{\\rm mix}$, increase with decreasing temperature reaching 10\\% for\n$15\\,625\\,$K. In the considered temperature interval, the corrections\nto the internal energy are largest for $31\\,250\\,$K which can be\nexplained by the different ionization states of the two fluids that\nlead to larger error if ideal mixing is assumed.\n\n\\begin{figure}[htb]\n\\centerline{\\includegraphics[width=4in]{gr05.eps}}\n\\caption{Comparison of different pair correlation functions, $g(r)$, from three PIMC \n       simulations at $r_s=1.86$ and $15\\,625\\,$K: pure hydrogen (red\n       dotted lines), a $x=50\\%$ mixture (black dashed lines) and pure\n       helium (blue solid lines). The six graphs show $g(r)$ for\n       different pairs of protons (p), electrons (e) and helium nuclei\n       ($\\alpha$). For electrons, pairs with antiparallel (upper\n       right) and with parallel (lower right) spins are\n       distinguished.}\n\\label{fig:gr}\n\\end{figure}\n\nFinally, we discuss pair correlation functions, $g(r)$, for hydrogen-helium mixtures,  \n\\begin{equation}\n    g(r)\\equiv \\frac{\\Omega}{N^2}\n          \\left<\\sum_{i\\neq j}\\delta ({\\bf r}-{\\bf r}_{ij})\n         \\right>\\;.\n                       \\label{eq3.1}\n\\end{equation}\n$g(r)$ functions are a standard tool to characterize the short-range\ncorrelation of particles in liquids. In Fig.~\\ref{fig:gr}, we compare\nthe correlation functions for pure hydrogen, a $x$=50\\% mixture, and\npure helium at fixed temperature (15$\\,$625$\\,$K) and density\n($r_s=1.86$). Changes in the pair correlations are now discussed as a\nfunction of helium concentration. The electron-proton $g(r)$ shows a\nstrong peak at the origin due to the Coulomb attraction. The peak is\nnot affected when 50\\% helium is added to a pure hydrogen\nsample. Similarly, the correlation between electrons and helium nuclei\nis not altered by the presence of protons.\n\nThe electron-electron correlation depends strongly on spin. For pairs\nwith parallel spin, exchange effects lead to a strong repulsion. The\nresulting $g(r)$ function does not depend much on whether helium is\npresent or not. The correlation function between pairs of antiparallel\nspins, on the other hand, is strongly affected by the presence of\nhelium nuclei. In the helium atom, two electrons with opposite spin\nare attracted to the core, which indirectly leads to the observed\nincrease in the electron-electron $g(r)$.\n\nIt is interesting to note that proton-proton $g(r)$ changes with the\nhelium concentration but the correlation of helium nuclei does\nnot. With increasing presence of helium, a peak in the proton-proton\n$g(r)$ appears at $r=1.4\\,a_0$ which indicates the formation of H$_2$\nmolecules. Adding helium nuclei leads to the localization of a\nfraction of the electrons. The available space in combination with the\nreduced electronic exchange effects then leads to the formation of hydrogen\nmolecules which are not present in pure hydrogen at the same\ntemperature and density.\n\n\\section{CONCLUSIONS}\n\nThe discussion in this article centered on the phase diagram of hot\ndense hydrogen. The first PIMC results for hydrogen-helium mixtures\nwere presented here, and the accuracy of existing helium EOS models as\nwell as the commonly used ideal mixing rule were analyzed. This\nanalysis revealed substantial inaccuracies in existing EOS\nmodels. These uncertainties prevent us from making reliable models for\nthe interior of giant planets and from drawing conclusions about their\nformation mechanism, either nucleated capture of nebular\nhydrogen\\cite{Li93} or gravitational instability driven formation in\nprotostellar disks.\\cite{Boss97,Boss04}\n\nAn improved EOS will also yield a better characterization of the 120\nextrasolar giant planets that have been discovered so far using radio\nvelocity measurements.\\cite{Ma02} In particular the mass-radius\nrelationship obtained from transit extrasolar giant\nplanets\\cite{brown01} will allow one to draw conclusions about the\nheavy element abundance in those planets. The metallicity of a planet\nrelative to its central star constrains the role of accretion of\nplanetesimals in the planet's formation.\n\n\\section*{ACKNOWLEDGMENTS}\n\nThe author thanks W. Hubbard and D. Stevenson for useful discussions\nand D. Saumon for providing us with the SCH EOS.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe mass distribution of neutron stars contains information about the\nsupernova explosion mechanisms, the equation of state of neutron star\nmatter, and the accretion history of each neutron star since its\nformation. Certain populations of neutron stars such as those in\ndouble neutron stars and in binaries with high mass companions are\nthought to have experienced little-to-no accretion over their\nlifetimes. In contrast, neutron stars in low-mass X-ray binaries and\nfast pulsars, which are typically in close orbits around white dwarfs,\nundergo extended accretion periods that are likely to move the neutron\nstar mass away from its birth value.\n\nThe neutron star mass measurements that were available a decade ago\nallowed a statistical inference of the mass distribution of double\nneutron stars (Finn 1994) or of pulsars in binaries, without\ndistinguishing between subgroups (Thorsett \\& Chakrabarty 1999).  Finn\n(1994) found that neutron star masses fall predominantly in the\n$1.3-1.6~M_\\odot$ range. Thorsett \\& Chakrabarty (1999) found that the\nmass distribution for the combined population is consistent with a\nnarrow Gaussian at $1.35 \\pm 0.04~M_\\odot$. More recently, Schwab,\nPodsiadlowski, \\& Rappaport (2010) argued that the distribution of\nneutron star masses in double neutron stars is actually bimodal, with\none peak centered at $\\sim 1.25~M_\\odot$ and the other at $\\sim\n1.35~M_\\odot$, which they attributed to different supernova explosion\nmechanisms. Kiziltan, Kottas, \\& Thorsett (2010), Valentim, Rangel, \\&\nHorvath (2011), and Zhang et al.\\ (2011), on the other hand, inferred\nthe mass distribution of different neutron star subgroups based either\non the pulsar spin period or the binary companion, both of which were\ntaken to be indicative of the accretion history of the system. All\ngroups found that the neutron stars that are thought to have undergone\nsignificant accretion are on average $0.2-0.3~M_\\odot$ heavier than\nthose that have not.\n\nOne result that is common to all of these studies is the narrowness of\nthe mass distribution of double neutron stars, $\\sigma \\simeq\n0.05~M_\\odot$, which has been taken as indicative of the birth mass\ndistribution of all neutron stars. The mean of the distribution is at\n$1.35~M_\\odot$, which is significantly larger than the mass of the\npre-supernova iron core for neutron stars that form through the\ncore-collapse mechanism.  The Chandrasekhar mass for cores with\nelectron fractions in the range $Y_e = 0.42-0.48$ is\n$1.15-1.34~M_\\odot$. Electrostatic interactions and entropy of the\ncore introduce additional corrections to the pre-collapse mass (see\nTimmes et al.\\ 1996 for a discussion).  Taking into account the\nbinding energy of the neutron star results in gravitational masses for\nthe collapsed cores in the range $1.06-1.22~M_\\odot$. Even the largest\nof these masses is well below the mean of the observed distribution of\ndouble neutron stars. Fallback of stellar matter onto the collapsing\ncore during the supernova explosion allows for the remnant to\nincrease. However, this is also expected to increase the dispersion of\nmasses by a comparable amount (see Zhang et al. 2008), which is\ninconsistent with the narrowness of the inferred mass distribution of\ndouble neutron star masses.\n\nConsidering a bimodal underlying distribution in the population of\ndouble neutron stars, as in Schwab et al.\\ (2010), makes the width of\neach distribution even narrower: $0.008~M_\\odot$ and $0.025~M_\\odot$\nfor the two components. For the lower mass component centered around\n$\\sim 1.25~M_\\odot$, such a narrow distribution may be reasonably\nobtained through an electron capture supernova, the onset of which\noccurs at a particular mass threshold of an ONeMg white dwarf\n(Podsiadlowski et al.\\ 2005). However, the second component, which is\ncentered at $1.35~M_\\odot$ cannot be explained as a result of the\nelectron capture supernovae and poses the same challenge in its\nnarrowness when explained via the core-collapse mechanism.\n\nIn order to model the distribution of neutron star masses both at\ntheir births and throughout their lives, one important question to\naddress is whether double neutron stars are a representative sample\nfor neutron stars at their birth masses. In this paper, we address\nthis question by identifying a different population of neutron stars\nat or near their birth masses and compare the inferred mass\ndistribution with that of double neutron stars. Furthermore, to\npinpoint the effects of subsequent accretion, we compare the inferred\nmass distribution of these neutron stars to that of neutron stars\nwhich have undergone or are currently undergoing extended mass\naccretion.\n\nMaking use of all of the currently available neutron star mass\nmeasurements, we divide the sample into various subgroups based on the\nnature of the companion as well as the neutron star spin. We employ a\nuniform Bayesian statistical approach that utilizes the entire\nposterior likelihood of each mass measurement to infer the parameters\nof the underlying mass distribution without assuming Gaussian errors.\n\nIn Section 2, we present the neutron star mass measurements, grouping\nthe data according to the measurement technique and the type of the\nbinary. In Section 3, we estimate the parameters of the underlying\nmass distribution for each subgroup and assess the sensitivity of our\nresults on the particular measurements and priors. In section 4, we\nsummarize our findings and discuss their implications for the\ndifferent physical mechanisms that determine the neutron star mass\ndistribution. \n\n\\section{Mass Measurements}\n\nMass measurements of neutron stars are carried out in several\ndifferent ways. When neutron stars appear as pulsars, high precision\npulse timing observations lead to a measurement of several orbital\nparameters.  The pulsar's orbit can be described in classical gravity\nby the five Keplerian parameters: the binary period $P_b$, the\neccentricity $e$, the projection of the pulsar's semi-major axis on\nthe observer's line of sight $a \\sin i$, the time of periastron $T_0$,\nand the longitude of periastron $\\omega_0$, where $i$ is the angle\nbetween the orbital angular momentum vector and the line of sight. The\nmass function, which is related to the mass of the pulsar $M_{\\rm psr}$,\nits companion $M_{\\rm c}$, and the inclination angle $i$, \n\\begin{equation}\nf = \\frac{(M_{\\rm c} \\sin i)^3}{M_{\\rm T}^2} = \\left(\\frac{2 \\pi}\n{P_b}\\right)^2 \\frac{(a \\sin i)^3}{G},  \n\\end{equation}\nis, therefore, directly obtained from these orbital parameters, where\n$M_{\\rm T} = M_{\\rm psr} + M_{\\rm c}$ is the total mass of the system.\n\nProceeding from a mass function to a measurement of the mass of the\npulsar and of its companion requires additional information. This\ninformation can come from measurement of relativistic effects in the\nbinary orbits or from independent observations of the companion\nstars. We discuss below the various techniques, the measurements they\nresulted in to date, and the associated uncertainties.\n\nThe majority of the precise neutron star mass measurements come from\nradio pulsar timing techniques and rely on the measurement of\nrelativistic effects in the binary orbits.\n\nThe general relativistic effects can be described by five additional\n``post-Keplerian'' (or PK) parameters.  These are: the advance of\nperiastron $\\dot{\\omega}$, the orbital period decay $\\dot{P}_{b}$, the\ntime dilation-gravitational redshift factor $\\gamma$, as well as the\nrange $r$ and the shape $s$ of Shapiro delay, which are related to the\ncomponent masses, the orbital period, and eccentricity by\n\\begin{equation}\n\\dot{\\omega} =  3\\left(\\frac{P_{b}}{2\\pi}\\right)^{-5\/3}\n\\left( \\frac{G M_{\\rm T}}{c^3} \\right)^{2\/3}\\left(1-e^2\\right)^{-1},\n\\end{equation}\n\\begin{equation}\n\\dot{P}_b = -\\frac{192\\pi}{5}\\left(\\frac{P_{b}}{2\\pi}\\right)^{-5\/3}  \n\\left(\\frac{G}{c^3}\\right)^{5\/3}  \n\\left(1+\\frac{73}{24}e^2+\\frac{37}{96}e^4\\right)\\times \\nonumber\\\\\n(1-e^2)^{-7\/2}\\, M_{\\rm psr} M_{\\rm c}\\,M_{\\rm T}^{-1\/3} ,\n\\end{equation}\n\\begin{equation}\n\\gamma = e\\left(\\frac{P_{b}}{2\\pi}\\right)^{1\/3}\n\\left(\\frac{G}{c^3}\\right)^{2\/3} M_{\\rm T}^{-4\/3}\nM_{\\rm c}\\left(M_{\\rm psr} + 2 M_{\\rm c} \\right) ,\n\\end{equation}\n\\begin{equation}\nr = \\frac{G}{c^3} M_{\\rm c},\n\\end{equation}\n\\begin{equation}\ns = G^{-1\/3} a\\sin i \\left(\\frac{P_{b}}{2\\pi}\\right)^{-2\/3}\nM_{\\rm T}^{2\/3}\\, M_{\\rm c}^{-1}.\n\\label{Eq:PK}\n\\end{equation}\n\nIn highly eccentric systems that have been observed repeatedly over a\nlong period of time, the measurement of $\\dot{\\omega}$ is usually\npossible and leads to a strong constraint on the total mass of the\nbinary $M_{\\rm T}$. The measurement of the parameter $\\gamma$ requires\nsimilarly eccentric systems and long-term monitoring. In high\ninclination systems, on the other hand, it is sometimes possible to\ndetect the Shapiro time delay and obtain the parameters $r$ and\n$s$. The measurement of the rate of orbital period decay typically\nrequires the longest monitoring and timing of the pulsar, sometimes\nover decades. This has been achieved for a handful of pulsars in\nbinaries.\n\nThe precision with which the pulsar mass can be determined ultimately\ndepends on the number of PK parameters that are measured for that\nbinary. In systems where two or more PK parameters are known, the\npulsar mass is precisely determined. In the category where only one PK\nparameter is known in addition to the mass function, the mass of each\nneutron star is not as well constrained.\n\nIn the following, we divide neutron star mass measurements into\ncategories based on the information available for each binary system\nsuch as the number of PK parameters. For each category, we derive the\nlikelihood $P_i({\\rm data} \\vert M_{\\rm NS})$, which measures the\nchance of obtaining the particular set of data for the $i$-th source\nif that source had mass $M_{\\rm NS}$.\n\nWe are ultimately interested in delineating the effects of mass\naccretion from the neutron star birth masses. We, therefore, further\ndivide each category into groups based on the nature of the companion\nstar or the spin of the pulsar.  In particular, neutron star-neutron\nstar (NS-NS) binaries, as well as eclipsing X-ray pulsars in high-mass\nX-ray binaries and slow radio pulsars are expected to have experienced\nlittle-to-no accretion. On the other hand, neutron star-white dwarf\n(NS-WD) binaries are the remnants of a long-lasting low-mass X-ray\nbinary phase, where significant mass accretion may have occurred. We\nalso group millisecond pulsars with main sequence companions (NS-MS)\nalong with the latter group, because of the probable recycling these\nneutron stars underwent to reach millisecond periods. We will refer to\nthe latter group as ``fast pulsars''. Finally, we will also consider\naccreting X-ray bursters, for which the masses have been measured\nprimarily through X-ray spectroscopy.\n\nIn detail, the various categories are:\n\n{\\it (Ia) Double Neutron Stars with at least 2 PK Parameters.---\\\/}\nSix neutron star systems shown in Table~1 have at least 2 measured PK\nparameters leading to well determined masses. In this case, the\nlikelihood of neutron star masses is highly symmetric and narrowly\npeaked, and thus can be described as a Gaussian\n\\begin{equation}\nP_i({\\rm data} \\vert M_{\\rm NS}) = C_i \\exp\\left[-\\frac{(M_{\\rm NS}-M_{0,i})^2}\n{2\\sigma_{M,i}^2} \\right]\n\\end{equation}\nwith a mean $M_{0,i}$ and a standard deviation $\\sigma_{M,i}$. In this\nand the following expressions, $C_i$ is a proper normalization\nconstant such that\n\\begin{equation}\n\\int_0^\\infty P_i({\\rm data} \\vert M_{\\rm NS}) dM_{\\rm NS} = 1.\n\\end{equation} \n\n\\input{table1.tex}\n\nWe plot in Figure~\\ref{fig:psr_gauss} the likelihood of the masses\nof NS-NS binaries that belong to this category.\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f1.ps}\n\\caption{The likelihoods $P_i({\\rm data}\\vert M_{\\rm NS})$ for the 12\ndouble neutron stars with precisely determined masses arising from the\nmeasurement of $\\ge 2$ PK parameters. These systems belong to category\nIa discussed in the text.}\n\\mbox{}\n\\label{fig:psr_gauss} \n\\end{figure}\n\n\n{\\it (Ib) Neutron Stars in Binaries with at least 2 PK\nParameters.---\\\/} To date, observations of nine neutron stars that are\nin close orbits around predominantly white dwarf companions have\nyielded a measurement of at least 2 PK parameters. The known exception\nis PSR~J1903+0327, with a probable main sequence companion, but is\nnevertheless thought to be recycled owing to its millisecond\nperiod. As in the previous category, we assign a Gaussian likelihood\nto each measurement with a mean $M_{0,i}$ and a standard deviation\n$\\sigma_{M,i}$. We present in Table~2 and\nFigure~\\ref{fig:psr_recycl_gauss} the mass measurements and their\nuncertainties for these NS-WD binaries.  Even though it has a white\ndwarf companion, PSR J1141$-$6545 is different from the rest of the\nsources in this category in that it is a slowly spinning neutron\nstar. For this reason, we group it with the accreting and slow pulsars\ndiscussed below and show its likelihood in Figure~\\ref{fig:psr_slow}.\n\n\\input{table2.tex}\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f2.ps}\n\\caption{The likelihoods $P_i({\\rm data}\\vert M_{\\rm psr})$ for nine\nrecycled pulsars with white dwarf companions. The measurement of $\\ge\n2$ PK parameters in these binaries lead to precisely determined\nneutron star masses. These systems belong to category Ib discussed in\nthe text.}\n\\mbox{}\n\\label{fig:psr_recycl_gauss} \n\\end{figure}\n\n{\\it (IIa) Double Neutron Stars with 1 PK Parameter.---\\\/} \nIn this category, there are three double neutron stars, for which \nthe measurement of the advance of periastron $\\dot{\\omega}$\nallows for a precise determination of the total mass of the binary.\nHowever, in the absence of a second PK parameter or a knowledge of the\nsystem inclination $i$, the mass of each neutron star is not well\nconstrained. In these cases, using the total mass of the system as \na constraint allows us to write the likelihood of the mass of the \npulsar as\n\\begin{equation} \nP_i({\\rm data} \\vert M_{\\rm psr}) = C_i \\int dM_{\\rm tot} \n\\exp\\left[-\\frac{(M_{\\rm tot}-M_{{\\rm tot},0})^2}\n{2\\sigma_{M_{\\rm tot},i}^2} \\right] \\int d(\\cos i)\n\\exp\\left[-\\frac{\\left(f_0-\\frac{M^3_{\\rm psr}\\sin^3i}{M^2_{\\rm tot}}\\right)^2}\n{2\\sigma_f^2}\\right]\n\\label{eq:pm_omegadot}\n\\end{equation}\nand of the companion as \n\\begin{equation} \nP_i({\\rm data} \\vert M_{\\rm c}) = C_i \\int dM_{\\rm tot} \n\\exp\\left[-\\frac{(M_{\\rm tot}-M_{{\\rm tot},0})^2}\n{2\\sigma_{M_{\\rm tot},i}^2} \\right] \\int d(\\cos i)\n\\exp\\left[-\\frac{\\left(f_0-\\frac{(M_{\\rm tot}-M_{\\rm c})^3\n\\sin^3i}{M^2_{\\rm tot}}\\right)^2}\n{2\\sigma_f^2}\\right]. \n\\label{eq:pm__c_omegadot}\n\\end{equation}\n\nWe present in Table~3 the relevant pulsar data for the three double\nneutron star binaries that are in this category.\nFigure~\\ref{fig:psr_omega} shows the likelihood of each neutron star\nmass $P_i({\\rm data}\\vert M_{\\rm NS})$ for these sources, with the top\npanel including the pulsars and the bottom panel the companions. Note,\nhowever, that even though they are shown independently for the\npurposes of this figure, the likelihoods of the masses of the pulsar\nand its companion are not independent probabilities. Therefore, when\ninferring the mass distribution of double neutron stars, the\nconstraint over the total mass is incorporated as we will discuss in\n\\S3.\n\n\\input{table3.tex}\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f3a.ps}\n   \\includegraphics[scale=0.75]{f3b.ps}\n\\caption{The likelihoods $P_i({\\rm data}\\vert M_{\\rm NS})$ for the\ndouble neutron stars with 1 PK parameter for (top) the pulsars and\n(bottom) companion neutron stars. These belong to category IIa\ndiscussed in the text.}\n\\mbox{}\n\\label{fig:psr_omega} \n\\end{figure}\n\n{\\it (IIb) Neutron Stars in Binaries with 1 PK Parameter.---\\\/} This\ncategory comprises of eleven neutron star binaries with mostly white\ndwarf companions. In several systems, the companions have not been\nidentified. A constraint on the total mass comes from the measurement\nof the rate of advance of the periastron. We assign to each neutron\nstar mass a likelihood according to equation~(\\ref{eq:pm_omegadot}).\nTable~4 and Figure~\\ref{fig:psr_recycl_omega} shows the relevant\nparameters for these systems. Note that, of this category, PSR B2303+46\nis a slowly spinning neutron star. For this reason, we group it with\nthe accreting and slow pulsars discussed below and show its likelihood\nin Figure~\\ref{fig:psr_slow}.\n\n\\input{table4.tex}\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f4.ps}\n\\caption{The likelihoods $P_i({\\rm data}\\vert M)$ for \nthe recycled neutron stars with 1 PK parameter (blue curves; category\nIIb) or with optical observations of the white dwarf companions (red\ncurves; category III).  }\n\\mbox{}\n\\label{fig:psr_recycl_omega} \n\\end{figure}\n\n{\\it (III) Neutron Stars in Binaries with Optical Observations of\nWhite Dwarf Companions.---\\\/} For two neutron stars in orbit around\nwhite dwarfs, optical observations of the companions have resulted in\nthe measurements of the mass $M_{\\rm WD}$ as well as of the radial\nvelocity amplitude $K_{\\rm opt}$ of the white dwarf. The latter, in\ncombination with the orbital parameters obtained from the radio timing\nsolution gives the mass ratio of the binary according to\n\\begin{equation}\nq = K_{\\rm opt} P_{\\rm b} \\frac{(1-e^2)^{1\/2}}{2 \\pi a_{\\rm psr} \\sin i}.\n\\end{equation}\nUsing this information, we can then calculate the likelihood of the \nneutron star mass as \n\\begin{equation}\nP_i({\\rm data} \\vert M_{\\rm psr}) = C_i \\int dM_{\\rm WD} \\int di \\sin i\n\\exp\\left[-\\frac{(M_{\\rm WD}-M_{{\\rm WD},0})^2}\n{2\\sigma_{M_{\\rm WD},i}^2} \\right] \n\\exp\\left[-\\frac{(\\frac{M_{\\rm psr}}{M_{\\rm WD}}-q_{i,0})^2}\n{2\\sigma_{q,i}^2} \\right] \\delta[f(i)-f_{0,i}],\n\\end{equation}\nwhere we assume that the error in the measurement of each mass function \nis negligible. We perform the integration over inclination making use of \nthe identity\n\\begin{equation}\n\\delta[f(i)-f_{0,i}] = \\frac{\\delta(i-i_0)}{\\vert df\/di\\vert_{i_0}}\n\\end{equation}\nwhere $i_0$ is the solution of the equation $f(i_0) - f_{0,i}=0$, i.e.,\n\\begin{equation}\n\\sin i_0 = \\left[f_{0,i} \\left(1+\\frac{M_{\\rm psr}}{M_{\\rm WD}}\\right)^2 \nM_{\\rm WD}^{-1}\\right]^{1\/3}.\n\\end{equation}\nGiven that \n\\begin{equation}\n\\left\\vert \\frac{df}{di}\\right\\vert_{i_0} = 3 f \\left\\vert \n\\frac{\\cos i_0}{\\sin i_0}\\right\\vert, \n\\end{equation}\nthe likelihood becomes\n\\begin{equation}\nP_i({\\rm data} \\vert M_{\\rm psr}) = \\frac{C_i}{3 f_{0,i}} \n\\int dM_{\\rm WD} \\frac{\\sin^2 i_0}{\\cos i_0}\n\\exp\\left[-\\frac{(M_{\\rm WD}-M_{{\\rm WD},0})^2}\n{2\\sigma_{M_{\\rm WD},i}^2} \\right]\n\\exp\\left[-\\frac{(\\frac{M_{\\rm psr}}{M_{\\rm WD}}-q_{i,0})^2}\n{2\\sigma_{q,i}^2} \\right]\\;.\n\\end{equation}\n\nTable~5 summarizes the relevant data for these two binaries. The\nlikelihoods of the neutron star masses are shown as red curves in\nFigure~\\ref{fig:psr_recycl_omega}.\n\n\\input{table5.tex}\n\n{\\it (IV) Eclipsing X-ray Pulsars.---\\\/} Eclipsing X-ray pulsars\nprovide a wealth of observational information using which we can\nestimate the masses of these neutron stars.  X-ray observations of\neach pulsar give the orbital period of the binary $P_b$, the\neccentricity of the orbit $e$, longitude of periastron $\\omega_0$, the\nsemi-major axis of the neutron star's orbit $a_X\\sin i$, and the\nsemi-duration of the eclipse $\\theta_e$.  In addition, optical\nobservations of the companion star give its velocity amplitude $K_{\\rm\nopt}$, its projected rotational velocity $v_{\\rm rot} \\sin i$, and the\namplitude of ellipsoidal variations $A$. From these observables, it is\npossible to solve for the fundamental parameters of the binary, viz.,\nthe mass of the neutron star $M_{\\rm NS}$, the mass, radius and\nrotational angular velocity of the companion, $M_{\\rm opt}$, $R_{\\rm\nopt}$, $\\Omega_{\\rm opt}$, and the inclination angle of the binary\n$i$.\n\nThe necessary data are available for six eclipsing pulsars: Vela X--1,\n4U1538--52, SMC X--1, LMC X--4, Cen X--3, and Her X--1 (see, e.g., van\nKerkwijk, van Paradijs, \\& Zuiderwijk 1995; van der Meer et\nal. 2007). Recently, Rawls et al. (2011) collected all the available\ndata and presented a detailed analysis of the likelihood of mass for\nthe individual neutron stars; see Table~6 for a compilation of the\nrelevant results. \n\nThere are several sources of potential systematic uncertainties in the\nmasses inferred in these eclipsing binaries. For example, there are\nsignificant residuals in the radial velocity curves of Vela X-1 after\nthe best-fit orbital solution is subtracted (e.g., Barziv et\nal. 2001), in which Quaintrell et al. (2003) noted the presence of a\nperiodicity and suggested modes on the star as a possible origin. More\nrecently, Koenigsberger et al.\\ (2012) developed a model for these\nresiduals based on the interaction between the neutron star and its\ncompanion but did not directly fit the model to the data. Even though\nthey concluded that a 1.55~$M_\\odot$ neutron star is marginally\nconsistent with observations, a higher mass for the neutron star,\ncentered around 1.7~$M_\\odot$, appears to be favored.\n\nA second important source of systematic uncertainty arises from\nmodeling of the ellipsoidal modulations, which includes a contribution\nfrom the accretion disk and may significantly influence the inferred\nneutron star masses.  This disk contribution is often non-negligible\n(see Figs.\\ 8, 9 of Rawls et al.\\ 2011) and involves multiple\nparameters. In order to assess the influence of this additional\ninformation on our results, we reanalyze here the data of eclipsing\npulsars ignoring the ellipsoidal modulations.\n\nFor each of the six systems, we write the likelihood of the data as a\nfunction of the neutron star mass $M_{\\rm NS}$ as follows,\n\\begin{eqnarray}\nP(\\rm{data}|M_{\\rm NS}) &=& C \\int_{M_{\\rm opt,min}}^{M_{\\rm opt,max}} dM_{\\rm opt}\n\\int_{\\beta_{\\rm min}}^{\\beta_{\\rm max}} d\\beta\n\\int_{\\Omega_{\\rm opt,min}}^{\\Omega_{\\rm opt,max}} d\\Omega_{\\rm opt}\n\\int_{0}^{1} d(\\cos i) \\nonumber \\\\\n&~& ~~~\\exp\\left[-\\frac{(f_M-f_0)^2}{2\\sigma_f^2}\n-\\frac{(K_{\\rm opt}-K_0)^2}{2\\sigma_K^2}\n-\\frac{(v_{\\rm rot}\\sin i-v_0)^2}{2\\sigma_v^2}\n-\\frac{(\\theta_e-\\theta_0)^2}{2\\sigma_\\theta^2}\n\\right].\n\\label{eclipse}\n\\end{eqnarray}\nHere $f_M = M_{\\rm opt}^3 \\sin^3i\/(M_{\\rm NS}+M_{\\rm opt})^2$ is the\nmass function of the binary, and $f_0$ and $\\sigma_f$ are the measured\nvalue of $f_M$ and its uncertainty (obtained from X-ray timing\nobservations).  Similarly, $K_0$, $\\sigma_K$ are the measured value of\n$K_{\\rm opt}$ and its uncertainty, $v_0$, $\\sigma_v$ are the measured\nvalue of $v_{\\rm rot}\\sin i$ and its uncertainty, and $\\theta_0$,\n$\\sigma_\\theta$ are the measured semi-duration of the eclipse\n$\\theta_e$ and its uncertainty. All these measurements are listed in\nRawls et al.\\ (2011) for the six systems of interest. Note that there\nis no measurement of $v_{\\rm rot} \\sin i$ for Her X-1, which leads to\nlarger uncertainties in the mass determination of this source.\n\nThe quantity $\\beta$ is equal to $R_{\\rm opt}\/R_L$, where $R_L$ is the\neffective radius of the Roche lobe of the secondary.  For the\nintegration limits in equation (\\ref{eclipse}), we choose $\\beta_{\\rm\nmin}=0$, $\\beta_{\\rm max}=1$, $\\Omega_{\\rm min}=0$, $\\Omega_{\\rm\nmax}=2\\Omega_b$, where $\\Omega_b=2\\pi\/P_b$ is the mean angular\nvelocity of the binary orbit, and a sufficiently generous range of\n$M_{\\rm opt}$.  As indicated in equation (\\ref{eclipse}), we assume a\nflat prior for each of the variables, though we have confirmed that\nthe results are not sensitive to this assumption. We calculate $P({\\rm\ndata}|M_{\\rm NS})$ for each of the six X-ray pulsars by computing the\nintegrals via a Monte Carlo method. We show in\nFigure~\\ref{fig:psr_slow} the resulting likelihoods of mass for each\nof the six systems and summarize the results in Table~6.\n\nWe consider an eccentric orbit in the case of Vela X-1 using the value\nof $e$ given in Rawls et al. (2011), and we assume a circular orbit\nfor the other five systems. Rawls et al. (2011) considered both\neccentric and circular orbits for 4U1538--52. However, the evidence\nfor eccentricity is not very strong. Moreover, the estimate of $K_{\\rm\nopt} =14.1\\pm1.1 ~{\\rm km\\,s^{-1}}$ that they obtain for their\neccentric orbit solution appears to be anomalously low (it does not\nfit the measured velocities very well --- see the lower left panel of\ntheir Fig. 9). For this reason, we consider only a circular orbit for\n4U1538--52, and we take $K_{\\rm opt} = 21.8\\pm3.8 ~{\\rm km\\,s^{-1}}$\nas estimated by Rawls et al. (2011). (The fit to the velocity data\nappears to be better in this case --- see the bottom right panel of\ntheir Fig. 9.)\n\n\\input{table6.tex}\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f5.ps}\n\\caption{The likelihoods $P_i({\\rm data}\\vert M_{\\rm NS})$ for \nthe eclipsing X-ray pulsars in high-mass X-ray binaries, which belong\nto category IV discussed in the text. This figure also includes the\nlikelihoods for the slow pulsars PSR~J1141$-$6545 and PSR~B2303+46,\nwhich belong to categories Ib and IIb, respectively, but have not been\nrecycled.}\n\\mbox{}\n\\label{fig:psr_slow} \n\\end{figure}\n\n{\\it (V) Accreting Bursting Neutron Stars.---\\\/} Spectroscopic\nobservations of a number of neutron stars during thermonuclear X-ray\nbursts have led to measurements of their masses and radii. This\ncategory includes 4U~1745$-$248 (\\\"Ozel et al.\\ 2009), 4U~1608$-$52\n(G\\\"uver et al.\\ 2010a), 4U 1820$-$30 (G\\\"uver et al.\\ 2010b), and\nKS~1731$-$260 (\\\"Ozel et al.\\ 2012). We convert the posterior\nlikelihood of mass and radius for each neutron star reported in\nthese analyses into a likelihood of mass by integrating over radius\nas\n\\begin{equation}\nP_i({\\rm data} \\vert M_{\\rm NS}) = C_i \\int dR_{\\rm NS} P_i({\\rm data} \n\\vert M_{\\rm NS}, R_{\\rm NS}).\n\\end{equation}\nThe resulting likelihoods are shown in Figure~\\ref{fig:bursters}. \n\nThe mass of a fifth burster, Cyg X-2, has been measured in a different\nmanner using the optical observations of its companion star (Orosz \\&\nKuulkers 1999). These observations yield the mass function of the\nbinary $f = 0.69 \\pm 0.03~M_\\odot$, a range of allowed mass ratios $0.3 < q <\n0.38$, as well as a measurement of the binary inclination $i= 61^\\circ\n\\pm 12^\\circ$. Using this information, we calculate the posterior\nlikelihood of the mass of the neutron star according to (\\\"Ozel et\nal.\\ 2010a)\n\\begin{equation}\nP_i({\\rm data} \\vert M_{\\rm NS}) = C_i \\int_{q_{\\rm min}}^{q_{\\rm max}} dq \n\\int_{(\\cos i)_{\\rm min}}^1 \\frac{d(\\cos i)}{1-(\\cos i)_{\\rm min}}\n\\exp \\left\\{- \\frac{[f_{0,i}-M_{\\rm NS} \\sin^3 i\/(1+q)^2]^2} {2 \\sigma_{f,i}^2}\n- \\frac{(i-i_0)^2}{2 \\sigma_i^2}\n\\right\\}\n\\label{eq:cygx2}\n\\end{equation}\nand plot it in Figure~\\ref{fig:bursters}.\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f6.ps}\n\\caption{The likelihoods $P_i({\\rm data}\\vert M_{\\rm NS})$ for the accreting \nbursting neutron stars discussed as category V in the text. The likelihood \nfor Cyg~X-2 was inferred using optical observations of its companion star. }\n\\label{fig:bursters} \n\\mbox{}\n\\end{figure}\n\n\\section{The Intrinsic Distribution of Neutron Star Masses}\n\nThe range of neutron star masses that can be produced in\nastrophysically plausible scenarios is rather narrow, from $\\sim\n1.0~M_\\odot$ to $\\sim 2.5~M_\\odot$. The posterior likelihood functions\nfor the neutron star mass measurements we report in the previous\nsection are indeed within this range. Moreover, earlier studies of the\nneutron star masses indicated a narrowly peaked distribution (Finn\n1994; Thorsett \\& Chakrabarty 1999). For these reasons, we will model\nthe distribution of neutron star masses with a mean $M_0$ and a\ndispersion $\\sigma$, i.e.,\n\\begin{equation}\nP(M_{\\rm NS}; M_0,\\sigma) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} \n\\exp\\left[-\\frac{(M_{\\rm NS}-M_0)^2}{2 \\sigma^2}\\right].\n\\label{eq:gauss}\n\\end{equation} \nOur goal in this section is to obtain the most likely values for the\nparameters $M_0$ and $\\sigma$ of this distribution that are consistent\nwith the measurements. \n\nUsing Bayes' theorem, we can write the probability $P(M_0,\\sigma \\vert\n{\\rm data})$ that measures the likelihood of the parameters of the\nneutron star mass distribution as\n\\begin{equation}\nP(M_0, \\sigma \\vert {\\rm data}) = C P({\\rm data} \\vert M_0,\\sigma) P(M_0) P(\\sigma),\n\\label{eq:bayes}\n\\end{equation}\nwhere $C$ is an appropriate normalization constant and $P(M_0)$ and\n$P(\\sigma)$ are the priors over the parameters of the mass\ndistribution. Hereafter, we will assume a flat prior over $M_0$\nbetween 1~$M_\\odot$ and $3~M_\\odot$ as well as a flat distribution\nover $\\sigma$ between zero and $1~M_\\odot$. \nWe repeated the analysis using a logarithmic prior distribution, which had\nindistiguishable effects on the final results. This is expected, given that\nthe inferred mass distributions are all very narrow and, therefore, are not\nsensitive to any weak priors.\n\nIn equation~(\\ref{eq:bayes}), the quantity $P({\\rm data} \\vert\nM_0,\\sigma)$ measures the posterior probability of having made a\nparticular set of observations for the ensemble of neutron stars given\nthe values of the parameters of the mass distribution. Under the \nassumption that each measurement is independent of all the others, \nwe calculate this quantity using \n\\begin{equation}\nP({\\rm data} \\vert M_0,\\sigma) = \\prod_i \\int dM_{\\rm NS} \nP_i({\\rm data} \\vert M_{\\rm NS}) P(M_{\\rm NS}; M_0, \\sigma).\n\\end{equation}\nThe only case where this assumption is not satisfied is for double\nneutron stars in category IIa, for which the mass measurements of the\npulsar and the companion neutron star are not independent. For these \nthree binary systems, we write instead \n\\begin{eqnarray}\nP({\\rm data} \\vert M_0,\\sigma) &=& \\prod_i^3 \\int dM_{\\rm tot} \n\\exp\\left[-\\frac{(M_{\\rm tot}-M_{\\rm tot,0})^2}{2 \n\\sigma_{M_{\\rm tot}}^2} \\right] \\nonumber \\\\\n&\\times&\n\\int dM_{\\rm NS} \nP_i({\\rm data} \\vert M_{\\rm NS}) P(M_{\\rm NS}; M_0, \\sigma) \nP(M_{\\rm tot}-M_{\\rm NS}; M_0, \\sigma).\n\\end{eqnarray}\n\nIn Section~2, we divided the neutron star mass measurements not only\naccording to the technique by which these measurements were obtained\nbut also by the type of companion and the spin period of the neutron\nstar. We carried out the latter division in anticipation of the fact\nthat fast and slow pulsars are drawn from different parent\npopulations, i.e., from those which have and have not experienced\nsignificant mass accretion phases. In the following, we derive the\nparameters of the intrinsic mass distributions for each of those\npopulations separately. We then address the extent to which the\nparticular evolutionary channels followed by each type of neutron star\nleaves a measurable imprint on the mass distribution.\n\n\\subsection{Neutron Stars at or Near their Birth Masses}\n\nThe low spin periods of a number of pulsars in our sample are\nindicative of mild or even no recycling due to mass accretion. We,\ntherefore, consider the masses of neutron stars in this population\nlikely to be very near their birth values. This sample includes\ncategories Ia and IIa for double neutron stars, category IV for\naccreting pulsars with primarily high-mass companions, as well as one\npulsar each in categories Ib (PSR~J1141$-$6545) and IIb\n(PSR~B2303+46).\n\nWe first study the underlying mass distribution of the double neutron\nstars. The mass measurements in these systems have by far the smallest\nerrors, which can dominate the parameter estimation of the mass\ndistribution of the total ensemble. Furthermore, these binaries have\nfollowed a very particular and highly selective evolutionary path,\nwhich may be evident in their mass distribution. We group the\nremaining sources together as a second sample that consists of neutron\nstars likely to be near their birth masses.\n\n\\subsubsection{Double Neutron Stars}\n\nFigure~\\ref{fig:prob_dns} shows the 68\\% and 95\\% confidence contours\nover the parameters of the intrinsic Gaussian distribution that is\nconsistent with the observed masses of nine double neutron stars. The\nmost likely value of the mean of the Gaussian distribution is\n1.33~$M_\\odot$ and that of the dispersion is 0.05~$M_\\odot$. It is\nevident from the figure that the uncertainties in the parameters of\nthe underlying distribution are very small: the 68\\% errors are\n0.03~$M_\\odot$ in both parameters.\n\nOur results are in agreement with the distribution reported by\nThorsett \\& Chakrabarty (1999), who found a mean of 1.35~$M_\\odot$ and\na dispersion of 0.04~$M_\\odot$, and with the more recent results of\nKiziltan et al.\\ (2010) for this category.\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f7.ps}\n\\caption{The confidence contours over the parameters of a Gaussian \ndistribution for the double neutron stars.}\n\\mbox{}\n\\label{fig:prob_dns} \n\\end{figure}\n\nAn interesting question we can address with the sample of double\nneutron stars is whether the two members of each binary system are\ndrawn from the same underlying population. To study this, we divided\nthe sample into one of pulsars and one of the companions. Note that\nfor the double pulsar, we assigned the faster pulsar to the ``pulsar''\nand the slower to the ``companion'' categories. Repeating the above\ninference for these two subgroups individually, we found that the most\nlikely parameters of the mass distribution for the pulsars are $M_0 =\n1.35~M_\\odot$ and $\\sigma = 0.05~M_\\odot$, whereas for the companions\nthey are $M_0 = 1.32~M_\\odot$ and $\\sigma = 0.05~M_\\odot$.  These\nparameters are the same within the 68\\% confidence ranges of each.\n\nOne further question about the double neutron star population concerns\nthe mass ratio $q$ in each binary. The mass ratios of the neutron\nstars in two of the six binaries with well determined masses are\nwithin one part in $\\sim 5 \\times 10^{-3}$ of unity. This is an order\nof magnitude smaller than the most likely dispersion of the underlying\nmass distribution. We explored whether the distribution of observed\nmass ratios is consistent with the pulsar and the companion being\ndrawn independently from a Gaussian distribution with the parameters\nwe determined above.\n\nThe posterior likelihood of observing a binary with pulsar and\ncompanion masses of $M_{\\rm psr}$ and $M_{\\rm c}$, respectively, is\ngiven by\n\\begin{equation}\nP(M_{\\rm psr}, M_{\\rm c}) dM_{\\rm psr} dM_{\\rm c} = C \\exp \n\\left[-\\frac{(M_{\\rm psr}-M_0)^2}{2\\sigma^2}-\\frac{(M_{\\rm c}-M_0)^2}{2\\sigma^2}\\right]\ndM_{\\rm psr} dM_{\\rm c}\n\\end{equation}\nTo convert this into a distribution over the mass ratio, we set \n$q \\equiv \\min(M_{\\rm psr}\/M_{\\rm c}, M_{\\rm c}\/M_{\\rm psr})$ and write \n\\begin{eqnarray}\nP(q)dq &=& \\int_{M_{\\rm psr}} P(M_{\\rm psr},q) dM_{\\rm psr} dq = \n\\int_{M_{\\rm psr}} P(M_{\\rm psr},M_{\\rm c}) \n\\frac{dM_{\\rm c}}{dq} dM_{\\rm psr} dq \\nonumber \\\\\n       &=& \\int_{M_{\\rm psr}} \\frac{2\\;C}{M_{\\rm psr}} \n\\exp \\left[-\\frac{(M_{\\rm psr}-M_0)^2}{2\\sigma^2}-\n\\frac{(q\\;M_{\\rm psr}-M_0)^2}{2\\sigma^2}\\right]\ndM_{\\rm psr} dq\\;.\n\\end{eqnarray}\nIn Figure~\\ref{fig:dns_massratio}, we compare the cumulative\nlikelihood of the mass ratio\n\\begin{equation}\nC(q>q_0)=\\int_{q_0}^1 P(q)dq\\;,\n\\end{equation}\ncalculated for the most likely values of the parameters of the\nGaussian, to the cumulative distribution of the observed mass ratios\nof the 6 double neutron stars with well determined masses. The\nsimilarity between the two distributions is striking and demonstrates\nthat the pulsar and the companion in each of the double neutron stars\nare consistent with having been drawn independently from the same\nnarrow distribution of masses.\n\nWe also explored whether the observed distribution of mass ratios is\nconsistent with the predicted cumulative distribution for neutron star\npairs drawn independently from the double Gaussian distribution\nsuggested by Schwab et al.\\ (2010; their eq.~(1)). The result is shown\nas a green line in Figure~\\ref{fig:dns_massratio}. The width of the\nindividual components in the bimodal distribution is significantly\nnarrower than the width of the single Gaussian that we infer here.\nThis leads to a larger fraction of double neutron stars with mass\nratios closer to unity for the bimodal distribution, which is not in\nagreement with the observed sample.\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f8.ps}\n\\caption{The histogram shows the cumulative mass ratio distribution for \nthe six double neutron stars with precise mass measurements. The red\nline shows the predicted cumulative distribution for neutron star\npairs drawn independently from a single Gaussian distribution with a\ncentral value and a dispersion equal to the most likely parameters\nshown in Figure~\\ref{fig:prob_dns}. The green line shows the predicted\ncumulative distribution for neutron star pairs drawn independently\nfrom the double Gaussian distribution suggested by Schwab et al.\\\n(2010). The observed distribution of mass ratios is in agreement with\na mass distribution represented by a single Gaussian. Note that, for\nconsistency, we show in this figure the mass ratio histogram generated\nfrom the data used by Schwab et al.\\ (2010).}\n\\mbox{}\n\\label{fig:dns_massratio} \n\\end{figure}\n\n\\subsubsection{Accreting and Slow Pulsars}\n\nThe second subgroup consists of neutron stars accreting from high-mass\ncompanions and slow pulsars, which are likely to be near their birth\nmasses. To infer the neutron star mass distribution, we will use both\nthe numerical results of Rawls et al.\\ (2011) as well as our analytic\nresults discussed in \\S 2, in which the information regarding\nellipsoidal variations in the lightcurves was not taken into account.\n\nIn Figure~\\ref{fig:prob_slow_num}, we show the 68\\% and 95\\%\nconfidence limits on the Gaussian parameters of the underlying mass\ndistribution for the accreting and slow pulsars, using the numerical\nresults of Rawls et al.\\ (2011). For comparison, we overplot the\nequivalent confidence contours for the double neutron stars. There is\na small but statistically insignificant shift in the central mass of\nthe Gaussian between the two populations. On the other hand, the\nGaussian dispersions between the two populations are different to a\nhigh statistical significance. In other words, even though both of\nthese populations are believed to represent neutron stars near their\nbirth masses, the double neutron stars are drawn from a significantly\nnarrower distribution of masses. The most likely values of the central\nmass and dispersion for the accreting and slow pulsars are\n1.28~$M_\\odot$ and 0.24~$M_\\odot$, respectively.\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f9.ps}\n\\caption{The confidence contours over the parameters of a Gaussian \ndistribution for the accreting and slow pulsars using the numerical\ndata from Rawls et al.\\ (2011). The confidence contours for the double\nneutron stars are also shown for comparison. Even though both\npopulations are thought to have masses near their birth masses, the\ndispersion of double neutron star masses is significantly smaller than\nthat of the accreting and slow pulsars.}\n\\mbox{}\n\\label{fig:prob_slow_num} \n\\end{figure}\n\nThe Rawls et al.\\ (2011) analysis depends on a synthesis of a large\nnumber of spectroscopic and photometric measurements of the binaries\nthat are used to infer the binary parameters. Fitting these\nobservations, and especially taking into account the ellipsoidal\nvariations requires complex models of the shape of the companion star\nand of the relative contribution of light from the accretion\ndisk. Moreover, the photometric observations of the ellipsoidal\nvariations typically are the lowest signal-to-noise components of the\nmass measurements. In order to assess the possible influence of the\nmodeling of the ellipsoidal variations on our results, we also infer\nthe underlying mass distribution using our analytical posterior\nprobabilities discussed in \\S 3. Figure~\\ref{fig:prob_slow_anlytc}\nshows the resulting 68\\% and the 95\\% confidence contours for the\nparameters of the mass distribution. In this case, the most likely\nvalues of the central mass and dispersion for the accreting and slow\npulsars are 1.24~$M_\\odot$ and 0.20~$M_\\odot$, respectively. Comparing\nthese to the distribution inferred from the numerical results of Rawls\net al.\\ (2011), we see that the central mass remains unchanged, but\nthe dispersion becomes less constrained and is even statistically\nconsistent with that of double neutron stars.\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f10.ps}\n\\caption{The confidence contours over the parameters of a Gaussian \ndistribution for the accreting and slow pulsars using the analytic\nmass measurements discussed in the text. The confidence contours for\nthe double neutron stars are also shown for comparison. }\n\\mbox{}\n\\label{fig:prob_slow_anlytc} \n\\end{figure}\n\n\\subsection{Recycled Neutron Stars}\n\nWe now focus on the subgroup of neutron stars which have been recycled\nthrough extended mass accretion. Neutron stars with white dwarf\ncompanions, millisecond pulsars, as well as neutron stars in low-mass\nX-ray binaries currently undergoing accretion (categories Ib, IIb,\nIII, and V) belong to this group.  Figure~\\ref{fig:prob_recycl} shows\nthe confidence contours over the parameters of the Gaussian\ndistribution for the recycled neutron stars. The most likely value of\nthe central mass is 1.48~$M_\\odot$ and of the dispersion is\n0.20~$M_\\odot$. Both the dispersion and the mean are similar to those\nfound by Kiziltan et al.\\ (2010) within statistical uncertainties. The\nuncertainties in the mean value quoted by Kiziltan et al.\\ (2010),\nhowever, are significantly smaller than those shown in\nFigure~\\ref{fig:prob_recycl} (c.f. Figure~3 of Kiziltan et al.\\ 2010).\n\nThere are two main differences between our study and that of Kiziltan\net al. (2010) with regard to the recycled neutron star sample. First,\nwe make use of the detailed posterior likelihood for each mass\nmeasurement, whereas Kiziltan et al.\\ (2010) appear to have\napproximated them with asymmetric Gaussians. Second, we include in our\nsample neutron stars in low-mass X-ray binaries for which mass\nmeasurements were performed mostly spectroscopically and typically\nhave larger uncertainties.\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f11.ps}\n\\caption{The confidence contours over the parameters of a Gaussian \ndistribution for the recycled neutron stars. The confidence contours\nfor the double neutron stars are also shown for comparison. As\nexpected, the recycled neutron stars have on average larger masses\nthan those in double neutron stars.}\n\\mbox{}\n\\label{fig:prob_recycl} \n\\end{figure}\n\nIn order to assess the sensitivity of our results to the inclusion of\nthe accreting neutron stars, we repeat the inference of the mass\ndistribution parameters using only the radio pulsars in categories Ib\nand IIb. In Figure~\\ref{fig:prob_fastpsr} we show the resulting\nconfidence contours. The difference with the entire sample is minimal:\nthe most likely mean value and the dispersion are 1.46~$M_\\odot$ and\n0.21~$M_\\odot$, respectively. We, therefore, attribute the small\ndifference with the Kiziltan et al.\\ (2010) results to our handling of\nthe posterior likelihood distributions for each measurement.\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f12.ps}\n\\caption{The confidence contours over the parameters of a Gaussian \ndistribution for a subgroup of the recycled neutron stars that\nincludes only pulsars in orbit around white dwarfs. Considering only\nthese sources with dynamical mass measurements does not alter the\nresults shown in Figure~\\ref{fig:prob_recycl}.  }\n\\mbox{}\n\\label{fig:prob_fastpsr} \n\\end{figure}\n\n\n\\section{Discussion}\n\nIn this paper, we investigated the distribution of neutron star masses\nin different types of binary systems and at different stages of\nevolution based on currently available measurements. We summarize the\nneutron star mass measurements and their uncertainties in each\nsubgroup in Figure~\\ref{fig:nsmasses} and compare them to those of\nblack holes in Figure~\\ref{fig:bhmasses} (compiled and analyzed in\n\\\"Ozel et al.\\ 2010a). In these figures, the error bars correspond \nto a 68\\% confidence level calculated from the detailed likelihood\ndistribution presented for each subgroup of sources in \\S 2.\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f13.ps}\n\\caption{The masses of neutron stars measured in double neutron stars \n(magenta; categories Ia and IIa), in eclipsing binaries with primarily\nhigh mass companions (cyan; category IV; these are the numerical\nvalues from Rawls et al.\\ 2011 given in column~2 of Table~6), with\nwhite dwarf companions (gold; categories Ib and IIb), with optical\nobservations of the white dwarf companions (green; category III), and\nin accreting bursters (purple; category V). }\n\\mbox{}\n\\label{fig:nsmasses} \n\\end{figure}\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f14.ps}\n\\caption{The measured masses of Galactic black holes \n(after \\\"Ozel et al. 2010a).}\n\\label{fig:bhmasses} \n\\mbox{}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.6]{f15a.ps}\n   \\includegraphics[scale=0.6]{f15b.ps}\n\\caption{The inferred mass distributions for the different populations \nof neutron stars (top) and black holes (bottom) discussed in the text.\nThe dashed lines correspond to the most likely values of the\nparameters. For the different neutron star populations these are:\n$M_0=1.33 M_\\odot $ and $\\sigma=0.05 M_\\odot$ for the double neutron\nstars, $M_0=1.28 M_\\odot $ and $\\sigma=0.24 M_\\odot$ for the other\nneutron stars near their birth masses, and $M_0=1.48 M_\\odot $ and\n$\\sigma=0.20 M_\\odot$ for the recycled neutron stars. For the case of\nblack holes, we used the exponential distribution with a low mass\ncut-off at $M_{\\rm c}=6.32 M_\\odot$ and a scale of $M_{\\rm scale}=1.61\nM_\\odot$ obtained in \\\"Ozel et al.\\ (2010a). The solid lines represent\nthe weighted mass distributions for each population, for which\nappropriate fitting formulae are given in the Appendix. The\ndistributions for the case of black holes have been scaled up by a\nfactor of three for clarity. }\n\\mbox{}\n\\label{fig:weighted} \n\\end{figure}\n\nIn the top panel of Figure~\\ref{fig:weighted}, we show the inferred\nmass distributions of the various neutron star populations discussed\nin Section~2. For each population, we present two different\ndistributions. The dashed lines correspond to the most likely\nparameters of the underlying distributions inferred in Section~3. Each\nsolid line represents the weighted distribution over the central mass\nand dispersion for each population. We compute this weighted\ndistribution as\n\\begin{equation}\nP_w(M_{\\rm NS})=\\int dM_0 \\int d\\sigma P(M_{\\rm NS}; M_0,\\sigma)\nP(M_0,\\sigma \\vert {\\rm data}), \n\\end{equation}\nwhere $P(M_{\\rm NS}; M_0,\\sigma)$ and $P(M_0,\\sigma \\vert {\\rm data})$\nis given by equations~\\ref{eq:gauss} and \\ref{eq:bayes},\nrespectively. In the Appendix, we provide approximate analytic fitting\nformulae for these weighted distributions for each population.\n\nIn the bottom panel of Figure~\\ref{fig:weighted}, we compare the\ninferred mass distribution for recycled neutron stars to that of black\nholes reported in \\\"Ozel et al.\\ (2010a). For the latter, we use the\nexponential model with a lower mass cutoff given by\n\\begin{equation}\nP(M_{\\rm BH}; M_{\\rm scale}, M_{\\rm c}) = \n\\frac{\\exp(M_{\\rm c}\/M_{\\rm scale})}{M_{\\rm scale}}\n\\times\\left\\{\n\\begin{array}{ll}\n\\exp(-M_{\\rm BH}\/M_{\\rm scale}), & M_{\\rm BH}>M_{\\rm c} \\\\\n0, & M_{\\rm BH}\\leq M_{\\rm c}\n\\end{array}\n\\right..\n\\end{equation}\nThe most likely values for the parameters of this distribution are\n$M_{\\rm scale}=1.61 M_\\odot$ and $M_{\\rm c} = 6.32 M_\\odot$. In the\nsame panel, we also include the appropriate weighted distribution for\nthe black holes, where we carried out the integration over the\nposterior likelihood of the parameters $M_{\\rm scale}$ and $M_{\\rm\nc}$; we provide an analytic fitting formula for the weighted\ndistribution in the Appendix. This panel highlights the substantial\nmass gap that exists between the black hole population and even the\nheaviest neutron star population (see the discussion in\n\\\"Ozel et al.\\ 2010a and Farr et al.\\ 2011).\n\nWithin the neutron star population, it is evident from these figures\nthat the mass distribution of double neutron star systems is different\nthan those observed in other binary systems, which include both\nneutron stars near their birth masses as well as neutron stars that\nexperienced significant accretion episodes. Indeed, the most likely\nvalues of the mean mass and the dispersion we derived for these\npopulations using the Bayesian inference technique discussed in \\S 3\nare $1.33 \\pm 0.05~M_\\odot$ for double neutron stars, in contrast to\n$1.28 \\pm 0.24~M_\\odot$ for other neutron stars near their birth\nmasses and $1.48 \\pm 0.20~M_\\odot$ for recycled neutron stars. Note\nthat the uncertainties in both the mean mass and the dispersion for\nall of these subgroups are shown in Figures~9-12.\n\nThe narrowness of the mass distribution of double neutron stars is\ndifficult to account for within the current understanding of neutron\nstar formation mechanisms. One possible way to generate a narrow\ndistribution is via electron capture supernovae in ONeMg white dwarfs.\nThe onset of such a supernova occurs at a particular density\nthreshold, which corresponds to a pre-collapse mass of the white dwarf\nin the narrow range $1.36-1.38~M_\\odot$ for different temperatures and\ncompositions (Podsiadlowski et al.\\ 2005). Taking into account a\nbinding mass given by the approximate formula (Lattimer \\& Yahil 1989)\n\\begin{equation}\nE_B = 0.084 \\left(\\frac{M}{M_\\odot}\\right)^2 M_\\odot\n\\end{equation}\nthe gravitational masses of the outcomes of electron capture\nsupernovae become $1.2-1.22~M_\\odot$. This range of masses is compared\nto the parameters of the underlying distributions of double neutron\nstars as well as of the other neutron stars near their birth masses in\nFigure~\\ref{fig:nsmass_theory}. Even though the electron capture\nsupernovae are capable of producing a narrow range of neutron star\nmasses, the mean of the expected distribution is inconsistent with\nthat of double neutron stars to a high confidence level.\n\n\\begin{figure}\n\\centering\n   \\includegraphics[scale=0.75]{f16.ps}\n\\caption{The distribution of neutron star masses at birth expected from \ntheoretical calculations, compared to the observed mass distribution\nof neutron stars that have not accreted significant amounts of mass\n(labeled accreting and slow pulsars) as well as to that of double\nneutron stars. The parameters of the distribution of the former\nsubgroup are consistent with expectations from core-collapse supernova\nand fallback, while the observed double neutron star distribution is \nsignificantly narrower than what is expected.} \n\\mbox{}\n\\label{fig:nsmass_theory} \n\\end{figure}\n\nIn the case of core collapse supernovae, the mean mass of the neutron\nstars produced in the absence of fallback is also expected to be\nsignificantly smaller than that inferred from the double neutron\nstars. We can estimate this mass assuming that the core of the\npre-supernova star collapses when it reaches its Chandrasekhar limit.\nConsidering an electron fraction of $Y_e = 0.42-0.48$, which is\nappropriate for the cores of presupernova stars (Timmes et al.\\ 1996), \nthe Chandrasekhar mass \n\\begin{equation}\nM_{\\rm Ch} = 5.83 Y^2_e\n\\end{equation}\nfalls in the range $1.15-1.34~M_\\odot$. Taking into account the\ngravitational binding energy, the expected range of birth masses for\nneutron stars from core-collapse supernovae is $1.06-1.22~M_\\odot$.\n\nFallback of matter during and immediately following the supernova\nexplosion can naturally lead to neutron stars more massive than the\ncores of the progenitor stars. At the same time, the stochastic nature\nof fallback necessarily leads to an increased dispersion of neutron\nstar masses. In Figure~\\ref{fig:nsmass_theory}, we show the evolution\nof the expected dispersion with central mass assuming that a fallback\nof baryonic mass $\\Delta M_f$ introduces a dispersion of the baryonic\nmass of the neutron star of the same magnitude. (Note that in\nFig.~\\ref{fig:nsmass_theory}, we plot the corresponding gravitational\nmass for the neutron stars.) This simple analytical estimate is in\nagreement with the detailed numerical calculations of Zhang et al.\\\n(2008), which are also shown in the figure. The green hatched region\noutlines the results for different compositions, explosion energies,\nand locations of the pistons for an assumed maximum neutron star mass\nof $2~M_\\odot$. Allowing sufficient fallback to account for the mean\nvalue of the double neutron star masses introduces a dispersion that\nis significantly larger than the observed one. In contrast, the\ninferred mean and dispersion of the mass distribution of other neutron\nstars believed to be near their birth masses (labeled accreting and\nslow pulsars) are in agreement with theoretical expectations of core\ncollapse supernovae.\n\nConsidering a bimodal mass distribution for the double neutron stars,\nas in Schwab et al.\\ (2010), aggravates these challenges. First, as we\nshowed in \\S 3, the cumulative likelihood of the mass ratio for such\na distribution does not agree with the cumulative distribution of the\nobserved mass ratios for the double neutron stars. Second, the\ndispersion in the two components becomes even smaller, $\\simeq\n0.025~M_\\odot$, making the higher mass component even less consistent\nwith the expectations of the core collapse supernovae. All of these\narguments lead to the conclusion that the mass distribution of double\nneutron stars is peculiar and perhaps related to the particular\nevolutionary history that leads to their formation.\n\nThe masses of the population of recycled neutron stars, which include\nfast pulsars with white dwarf companions as well as accreting\nbursters, are consistent with them having undergone extended periods\nof accretion. On average, recycled neutron stars are more massive by\n$\\approx 0.2 M_\\odot$ compared to other accreting and slow\npulsars. Such a mass increase is more than adequate to recycle these\npulsars to millisecond periods. Indeed, assuming that the mass is\ntransferred onto the neutron star via an accretion disk that is\nmagnetically truncated at the corotation radius\n\\begin{equation}\nR_c = \\left(\\frac{GM}{4 \\pi^2 \\nu^2_{\\rm s}}\\right)^{1\/3}, \n\\end{equation}\nwhere $\\nu_{\\rm s}$ is the spin frequency of the neutron star, \nthe angular momentum transferred per unit mass is \n\\begin{equation}\nl= (G M R_c)^{1\/2} = \\left(\\frac{G^2 M^2}{2 \\pi \\nu_{\\rm s}}\\right)^{1\/3}.\n\\end{equation}\nAfter accreting mass $\\Delta M$, the neutron star acquires an angular \nmomentum $\\Delta M \\cdot l$. Equating this to the spin angular momentum of the \nrecycled pulsar $L=2 \\pi I \\nu_{\\rm s}$, where $I$ is its moment of inertia, \nallows us to calculate the mass required to spin up the pulsar as \n\\begin{equation}\n\\Delta M = I (GM)^{-2\/3} (2 \\pi \\nu_{\\rm s})^{4\/3} \n= 0.034 \\left(\\frac{\\nu_{\\rm s}}{300~{\\rm Hz}}\\right)^{4\/3}\n\\left(\\frac{M}{1.48~M_\\odot}\\right)^{-2\/3}\n\\left(\\frac{I}{10^{45}~{\\rm g}~{\\rm cm}^2}\\right)\\;M_\\odot. \n\\end{equation}\n\nIt is interesting that the most likely value of the mean mass of the\nrecycled pulsars is significantly smaller than the $2~M_\\odot$ lower\nbound on the maximum mass of a neutron star (Demorest et al.\\ 2010;\n\\\"Ozel et al.\\ 2010b) as well as the average mass of recycled\nneutron stars predicted by population synthesis studies (e.g., Pfahl,\nRappaport, \\& Podsiadlowski 2003; Lin et al.\\ 2011). This conclusion\ncan be used to refine models of low-mass X-ray binary evolution.\nFurthermore, our analysis shows that a very small fraction of neutron\nstars reach masses comparable to the maximum possible neutron star\nmass and collapse into black holes. Therefore, this channel does not\ncontribute significantly to a putative but still undetected population\nof low-mass black holes in the Galaxy (see the discussion in \\\"Ozel et\nal.\\ 2010a).\n\n\\acknowledgments\n\nF.\\\"O., D.P., and R.N. thank the Institute of Astronomy, where a large\nfraction of this work was carried out, for their hospitality. We thank\nScott Ransom for useful discussions as well as Ingrid Stairs, Michael\nKramer, and Matthew Bailes for detailed comments on the manuscript. We\nthank Laura Kasian for sharing unpublished data. We gratefully\nacknowledge support from NSF grant AST-1108753, NASA ADAP grant\nNNX10AE89G, NSF CAREER award NSF 0746549, and Chandra Theory grant\nTMO-11003X for this work.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nHydrogenic impurities are a central aspect of semiconductor physics and technology.\\cite{pan,yu-car}\nIn recent years, impurity states have been proposed as potential qubits in quantum information \ndevices.\\cite{kan,koi-hu-das}\nIn binary III-V and II-VI semiconductors, the spin-orbit interaction can play an important role\nin the electronic structure of confined electronic states.\\cite{win,int-tam-wei}\nAt the level of the effective-mass approximation, a hydrogenic donor impurity and the hydrogen \natom are almost completely analogous quantum systems.\\cite{koh}\nThus, it is in principle a simple matter to describe at that level the effect of the so-called \nextrinsic or Rashba spin-orbit interaction in the impurity states.\nThis contribution to the spin-orbit interaction in local external potentials in solids is, \nin this context, analogous to the spin-orbit coupling in the hydrogen atom.\nHowever, care must be taken due to the necessary renormalization of the spin-orbit coupling \nconstant and to possible modifications to the spin-orbit formula due to anisotropies of the \ncrystal structure.\nOn the other hand, an additional contribution to the spin-orbit interaction, which is specific \nto the solid-state context must be taken into consideration.\nThis so-called intrinsic or Dresselhaus contribution\\cite{dre} is present in the bulk of\nthe material and reappears in the envelope-function approximation theory of external, \nmesoscopic, potentials, like the one caused by a ionized donor. \n\n\nIn this article we calculate the energy levels of the $n=2$ shell of hydrogenic impurities\nof semiconductors with wurtzite crystal structure in the presence of these two contributions \nto the spin-orbit interaction.\nWhile, as mentioned above, the calculation of the extrinsic contribution is in a sense \na straightforward application of the well-known formulas for the hydrogen atom, we need to\nfill a theoretical gap, caused by the anisotropy of the wurtzite crystal structure.\nTo that effect, we obtain here an expression for the extrinsic spin-orbit interaction\nin wurtzite semiconductors and the effective spin-orbit coupling constant for conduction\nband electrons.\nThe calculation of the effects of the intrinsic spin-orbit interaction in this context\nis entirely new and it is considered acting alone and in combination with the extrinsic\ncontribution.\n\n\nThe article is organized as follows.\nIn Section II we introduce the system and study the effect of the intrinsic spin-orbit interaction.\nIn Section III we derive the effective Hamiltonian of the extrinsic spin-orbit interaction using \nthe Foldy-Wouthuysen transformation adapted to this context \nand in Section IV we calculate the energy corrections that it produces.\nIn Section V we study the combined effect of both spin-orbit interactions, and in Section VI we\nprovide the concluding remarks.\n\n\n\n\\section{Intrinsic spin-orbit interaction in the hydrogenic impurity}\n\\label{sec:efa-impurity}\n\nWe consider an electron bound to a hydrogenic donor impurity in a bulk semiconductor with\nwurtzite crystal structure.\nWorking at the level of the envelope-function approximation (EFA), both the intrinsic \nand the extrinsic spin-orbit couplings appear in the Hamiltonian:\n\\begin{equation} \n  H = H_{\\text{0}} + H_{\\text{int}} + H_{\\text{ext}},\n\\label{eq:Htot}\n\\end{equation}\nwhere\n\\begin{equation} \n  H_0 = \\frac{p^2}{2 m^*} + V(\\mathbf{r}).\n\\label{eq:Hzero} \n\\end{equation}\nHere $V(\\mathbf{r}) = - e^2\/\\epsilon r$ is the effective Coulomb potential of the electron\nbound to the ionized donor impurity.\nWe assume the effective mass $m^*$ and the dielectric constant $\\epsilon$ to be isotropic, \nthereby preserving the spherical symmetry of the hydrogenic Hamiltonian $H_0$.\n\n\nThe extrinsic spin-orbit coupling, $H_{\\text{ext}}$, will be discussed below, and for the\nmoment we focus on the intrinsic contribution, $H_{\\text{int}}$, which for semiconductors\nwith wurtzite crystal structure is given by \\cite{voo-wil-car, fu-wu,wan-wu-tsa}\n\\begin{equation}\n  H_{\\text{int}} = \\alpha \\,( \\sigma_x k_y  - \\, \\sigma_y k_x) \n                   + \\gamma \\,(b \\,k_z^{2}-\\,  k_{\\parallel}^2)\n                              ( \\sigma_x k_y  - \\, \\sigma_y k_x),\n\\label{eq:int_completa}\n\\end{equation}\nwhere $k_{\\parallel}^2=k_x^2+k_y^2$ and $\\alpha$, $b$, and $ \\gamma$ are material-dependent \nparameters which are obtained experimentally or via ab-initio calculations.\n$\\boldsymbol{\\hat{\\sigma}}=(\\sigma_x,\\sigma_y,\\sigma_z)$ are the Pauli matrices.\nWhile $\\alpha$ and $\\gamma$ can vary considerably between different materials,\n$b$ is roughly universal and close to 4 for all materials.\nNote that $H_{\\text{int}}$ has two parts, one of them linear and the other one cubic \nin the wavevector $k$.\nThe cubic-in-k term displays an anisotropy between the $z$-direction and the directions \nin the $xy$-plane.\nThis anisotropy and the presence of the linear term distinguish the intrinsic spin-orbit\nHamiltonian of wurtzite semiconductors from the Dresselhaus coupling of zincblende \nsemiconductors.\\cite{dre}\n\n\nThe hydrogenic Hamiltonian $H_0$ has the renormalized eigenvalues $E_n^*=-E_R^*\/n^2$,\nwhere $E_R^*=m^{\\ast} e^4\/2\\epsilon^2\\hbar^2$ is the effective Rydberg energy.\nThe aim of this study is to obtain the corrections to the $E_2$ energy level due to\nthe intrinsic and extrinsic spin-orbit Hamiltonians.\nWe will work at the level of first-order perturbation theory, which is adequate due\nto the smallness of the spin-orbit couplings compared to the separation of the \nbare $E_n^*$ levels.\nIn order to diagonalize the intrinsic Hamiltonian $H_{\\text{int}}$ in the $E_2$ subspace\nwe use the basis of hydrogenic eigenstates of $\\{L^2,L_z,S^2,S_z\\}$, given by\n\\begin{eqnarray}\n  &&  \\psi_{200 \\eta} = \\left( \\frac{1}{32\\pi a^{*3}} \\right)^{\\frac{1}{2}}\n                        \\left(2 - \\frac{r}{a^*} \\right) \n                        e^{-r\/2a^*} |\\eta\\rangle     \\nonumber \\\\\n  &&  \\psi_{211 \\eta} = \\left( \\frac{1}{64\\pi a^{*3}} \\right)^{\\frac{1}{2}} \n                        \\frac{(x + i y)}{a^*} \\,\n                        e^{-r\/2a^*} |\\eta\\rangle  \\nonumber \\\\\n  &&  \\psi_{210 \\eta} = \\left( \\frac{1}{32\\pi a^{*3}} \\right)^{\\frac{1}{2}} \\frac{z}{a^*} \\,\n                        e^{-r\/2a^*} |\\eta\\rangle        \\nonumber \\\\\n  &&  \\psi_{21-1 \\eta} = -\\left( \\frac{1}{64\\pi a^{*3}} \\right)^{\\frac{1}{2}} \n                         \\frac{(x - i y)}{a^*} \\,\n                         e^{-r\/2a^*} |\\eta\\rangle ,\n\\end{eqnarray}\nwhere $\\eta= \\{\\uparrow, \\downarrow\\}$.\nThe matrix elements of the linear-in-$k$ terms of $H_{\\text{int}}$ in this basis are zero;\nonly the cubic-in-$k$ terms contribute.\nOrdering the basis states as: \n$|200\\uparrow\\rangle$, $|200\\downarrow\\rangle$, \n$|211\\uparrow\\rangle$, $|211\\downarrow\\rangle$, $|210\\uparrow\\rangle$, \n$|210\\downarrow\\rangle$, $|21,-1\\uparrow\\rangle$, and $|21,-1\\downarrow\\rangle$, \nthe matrix of $H_{\\text{int}}$ in the $n=2$ subspace is\n\\begin{equation}\n  \\bar{\\bar H}_{\\text{int}}   \\,=  \\left(\\begin{matrix}  \n                                            \n                         0   &   0   &   0   &   A   &   0   &   0   &    0   &   0 \\\\\n                       \n                         0   &   0   &   0   &   0   &   0   &   0   &    B   &   0  \\\\\n                       \n                         0   &   0   &   0   &   0   &   0   &   0   &    0    &   0  \\\\\n                        \n                         A   &   0   &   0   &   0   &   0   &   0   &    0    &   0\\\\\n                         \n                         0   &   0   &   0   &   0   &   0   &   0   &     0   &   0\\\\\n                        \n                         0   &   0   &   0   &   0   &   0   &   0   &     0   &   0  \\\\\n                        \n                         0   &   B   &   0   &   0   &   0   &   0   &     0   &   0  \\\\\n                        \n                         0   &   0   &   0   &   0   &   0   &   0   &     0   &   0  \\\\\n                                                                       \\end{matrix}\\right) ,\n\\end{equation}\nwhere\n\\begin{eqnarray}\n  && A  = \\frac{\\gamma}{32\\sqrt2 \\, a^{*3}} \\left(\\frac{14}{15} b + \\frac{133}{60} \\right) \\nonumber \\\\\n  && B = \\frac{\\gamma}{32\\sqrt2 \\, a^{*3}} \\left(\\frac{62}{15} b + \\frac{433}{60} \\right) .\n  \\label{eq:AB}\n\\end{eqnarray} \nThe secular equation for $\\bar{\\bar H}_{\\text{int}}$, \n$\\text{det} \\left(\\bar{\\bar H}_{\\text{int}} - \\varepsilon \\bar{\\bar I} \\right) = 0 $, \nyields\n\\begin{equation}\n   \\varepsilon^{4} - \\varepsilon^{2}(A^2 + B^2) + (A\\,\\, B)^2 = 0 .\n   \\label{eq:intAB}\n\\end{equation}\nThe eigenvalues are then $\\varepsilon_{1,2} = \\pm A $ and $\\varepsilon_{3,4} = \\pm B $.\nThe other four eigenvalues are degenerate and equal to zero.\nIn Table I we present the non-zero energy corrections for the materials GaN, ZnO, InN and AlN, \nalong with their $\\gamma$ and $b$ parameters.\nIn the last two columns we present the energy splittings $2A$ and $2B$ as percentages\nof the unperturbed energy $E_2^*$.\n\n\n\\begin{table}[ht]\n\\label{tab:int}\n\\begin{center}\n\\begin{tabular} {| l | c | c | c | c | c| c | c |}\n\\hline\n      &  $\\gamma $   &  $ b $ &  $ \\varepsilon_{1,2} $  & $ \\varepsilon_{3,4} $  & $ E_2^{*} $ & $2A\/E_2^{\\ast} $ & $2B\/E_2^{\\ast} $ \\\\\n      &[$\\text{meV} \\text{\\AA}^3$] &   &    [ $\\mu$eV ]      &      [ $\\mu$eV ]       &   [meV]     &     $ [\\%] $     &     $ [\\%] $   \\\\\n   \\hline \n    GaN  &    400    &  3.954   &   $\\pm$ 13.24           &     $\\pm$  52.79       &    11.97    &     0.22    &    0.88   \\\\\n   \\hline\n    ZnO  &    320    &  3.855   &   $\\pm$ 14.67           &     $\\pm$  58.39       &    14.65    &     0.20    &    0.80   \\\\\n   \\hline\n    InN  &    345    &  4.885   &   $\\pm$ 14.71           &     $\\pm$  59.49       &    16.20    &     0.18    &    0.73   \\\\\n   \\hline \n    AlN  &    6.45   &  3.767   &   $\\pm$ 3.98            &     $\\pm$  15.81       &    70.86    &     0.011   &    0.15  \\\\ \n    \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Intrinsic spin-orbit interaction corrections to the $E_2^*$ energy level of hydrogenic \ndonor impurities in various semiconductors with wurtzite crystal structure.\nThe parameters $\\gamma$ and $b$ are also indicated.\nThe effective masses and dielectric constants used to calculate $E_2^{*}$ are given in Table II.}\n\\end{table}\n\n\\section{Derivation of the Extrinsic spin-orbit interaction}\n\\label{cap:wurtzita-ext}\n\nThe spin-orbit Hamiltonian of an electron in vacuum in the presence of an electrostatic\npotential $V_0(\\mathbf{r})$ is given by\n\\begin{equation}\n  H_{\\text{so}} = \\lambda \\,\\boldsymbol{\\hat{\\sigma}} \\cdot \n                                         \\mathbf{k}\\times \\nabla V_{0}(\\mathbf{r}).\n\\end{equation}\nWhen the electron is immersed in a semiconductor in the presence of a mesoscopic potential \n$V(\\mathbf{r})$, the effective extrinsic spin-orbit Hamiltonian takes the form \n\\begin{equation}\n  H_{\\text{ext}} = \\lambda^{\\ast} \\, \\boldsymbol{\\hat{\\sigma}} \\cdot \n                             \\mathbf{k} \\times \\boldsymbol{\\nabla}V(\\mathbf{r}),\n\\label{eq:HSOeff}\n\\end{equation}\nwhere $\\lambda^{\\ast}$ is an effective coupling constant.\nThis expression is valid for semiconductors with zincblende crystal structure,\nwhich presents a basic cubic symmetry.\nThe wurtzite crystal structure has less symmetry than the zincblende, due to the special role of\nits c-axis.\nThis lack of isotropy is also present in the intrinsic spin-orbit coupling given above, \nEq.\\ \\eqref{eq:int_completa}.\nIn what follows we shall derive an expression analogous to Eq.\\ \\eqref{eq:HSOeff} for semiconductors \nwith wurtzite crystal structure.\n\n\nWe start with the $\\mathbf{k} \\cdot \\mathbf{p}$ crystal Hamiltonian:\n\\begin{equation}\n\\label{eq:kdotp}\n  \\mathcal{H} = \\mathcal{H}_0 + \\mathcal{H}_{\\mathbf{k} \\cdot \\mathbf{p}} + \\mathcal{H}_{\\text{so}}\n\\end{equation}\nwhere \n\\begin{eqnarray}\n  \\mathcal{H}_0 &=& \\frac{P^2}{2 m} + U(r) , \\nonumber \\\\\n  \\mathcal{H}_{\\mathbf{k} \\cdot \\mathbf{p}} &=& \\frac{\\hbar}{m}\\mathbf{k}\\cdot \\mathbf{p} + \n      \\frac{\\hbar^{2} k^{2}}{2\\,m} , \\nonumber \\\\\n  \\mathcal{H}_{\\text{so}} &=& \\frac{\\lambda}{\\hbar} \\, \\boldsymbol{\\hat{\\sigma}} \\cdot \n              \\mathbf{p} \\times \\boldsymbol{\\nabla}U ,\n\\end{eqnarray}\nand $U$ is the periodic crystal potential.\nWe will write the matrix of $\\mathcal{H}$ in the common basis of $H_0$ and $J_z$ given by:\n\\begin{eqnarray}\n  &&  v_1 = \\arrowvert i S \\uparrow \\rangle                                     \\nonumber \\\\\n  &&  v_2 = \\arrowvert i S \\downarrow \\rangle                                   \\nonumber \\\\\n  &&  v_3 = -\\frac{1}{\\sqrt2} \\left \\arrowvert (X+iY) \\uparrow \\right\\rangle    \\nonumber \\\\\n  &&  v_4 = -\\frac{1}{\\sqrt6} \\Big[ \\arrowvert (X+iY)\n                     \\downarrow\\rangle -2\\arrowvert Z \\uparrow \\rangle \\Big]    \\nonumber \\\\\n  &&  v_5 = \\frac{1}{\\sqrt6} \\Big[ \\arrowvert (X-iY) \\uparrow\\rangle \n                    +2\\arrowvert Z \\downarrow \\rangle \\Big]                     \\nonumber \\\\\n  &&  v_6 = \\frac{1}{\\sqrt2}\\arrowvert (X-iY) \\downarrow\\rangle                 \\nonumber \\\\\n  &&  v_7 = -\\frac{1}{\\sqrt3} \\Big[ \\arrowvert (X+iY) \\downarrow\\rangle\n                    +\\arrowvert Z \\uparrow \\rangle \\Big]                        \\nonumber \\\\\n  &&  v_8 = -\\frac{1}{\\sqrt3}\\Big[ \\arrowvert (X-iY) \\uparrow\\rangle \n                    -\\arrowvert Z \\downarrow \\rangle \\Big] . \n\\end{eqnarray}\nHere $\\arrowvert S \\eta \\rangle$ are conduction-band $s$-states, with energy $E_c$, and \n$\\arrowvert X \\eta \\rangle$,  $\\arrowvert Y \\eta \\rangle$, and $\\arrowvert Z \\eta \\rangle$ \nare valence-band $p$-type states, with energy $E_v$.\nThe energy gap is given by $E_g = E_c-E_v$.\nWe calculate the matrix elements \n$\\mathcal{H}_{ij} = \\langle v_i\\,\\arrowvert \\mathcal{H} \\arrowvert v_j\\,\\rangle$,\nwhere $ \\{i,j = 1, \\ldots, 8 \\} $, and obtain\n\n\n\\begin{widetext}\n\\begin{equation}\n  \\mathcal{H} = \\left(\\begin{matrix}  \n     E_c   &    0   &  \\frac{-1}{\\sqrt2}P_2k_+  & \\sqrt{\\frac{2}{3}} P_1k_z   &   \\frac{1}{\\sqrt6}P_2k_-   &   0   &  \\frac{-1}{\\sqrt3}P_1k_z   &   \\frac{-1}{\\sqrt3}P_2k_-  \\\\\n                       \n     0     &   E_c   &   0   &   \\frac{-1}{\\sqrt6}P_2k_+   &   \\sqrt{\\frac{2}{3}} P_1k_z   &   \\frac{1}{\\sqrt2}P_2k_-   &   \\frac{-1}{\\sqrt3}P_2k_+   &   \\frac{1}{\\sqrt3}P_1k_z  \\\\\n                       \n     \\frac{-1}{\\sqrt2}P_2k_-   &   0   &   E_v    &   0   &   0   &   0   &   0   &   0  \\\\\n                        \n     \\sqrt{\\frac{2}{3}} P_1k_z   & \\frac{-1}{\\sqrt6}P_2k_-  &   0   &   E_v   &   0   &   0   &  0  & 0  \\\\\n                        \n     \\frac{1}{\\sqrt6}P_2k_+   &   \\sqrt{\\frac{2}{3}} P_1k_z   &   0   &   0  &   E_v   &   0   & 0  \\\\\n                   \n     0   &   \\frac{1}{\\sqrt2}P_2k_+   &   0   &   0   &   0   &   E_v   &   0   &   0  \\\\\n                        \n     \\frac{-1}{\\sqrt3}P_1k_z    &    \\frac{-1}{\\sqrt3}P_2k_-   &   0   &   0  & 0  & 0 & E_v- \\Delta_0    &   0  \\\\\n                        \n     \\frac{-1}{\\sqrt3}P_2k_+   &   \\frac{1}{\\sqrt3}P_1k_z   &   0   &   0   &   0   &   0   &   0   &   E_v -\\Delta_0  \\\\\n          \\end{matrix}\\right)\n\\end{equation}\n\\end{widetext}\nwhere $k_{\\pm} = k_x \\pm ik_y$ and \n$\\Delta_0 =  \\frac{\\hbar}{4 m^2 c^4} \\langle X | \\frac{\\partial U}{\\partial x} P_y - \n\\frac{\\partial U}{\\partial y} P_x | Y \\rangle$ is the spin-orbit splitting of the valence bands.\nWe have defined the constants $P_1$ and $P_2$ coming from the matrix elements:\n\\begin{equation}\n \\frac{\\hbar}{m}\\langle -iS\\,\\downarrow\\arrowvert\\mathbf{k} \\cdot \\mathbf{p} \n   \\arrowvert Z\\,\\downarrow\\rangle\\,=  \\,-i \\frac{\\hbar}{m} k_z \n   \\langle S\\,\\arrowvert p_z \\arrowvert Z\\rangle \\equiv k_z P_1 , \\nonumber\n\\end{equation}\n\\begin{equation}\n \\frac{\\hbar}{m}\\langle -iS\\,\\downarrow\\arrowvert\\mathbf{k} \\cdot \\mathbf{p} \n    \\arrowvert X\\,\\downarrow\\rangle\\,=  \\, -i \\frac{\\hbar}{m} k_x \n    \\langle S\\,\\arrowvert p_x \\arrowvert X\\rangle \\equiv k_x P_2 .\n\\end{equation}\n\n\nWe now introduce the impurity potential $V(\\mathbf{r})$, which varies slowly in the length \nscale of the lattice constant.\nIts matrix elements in the basis \\{$v_i$\\} are essentially diagonal thanks to the orthogonality \nof the basis set and its slow variation in atomic scale.\nIn short, we are applying here the envelope function approximation.\nThe matrix of $\\mathcal{H}+V$ can be expressed in a compact form using the matrices $\\mathbf{T}$ \nfamiliar from group theory:\\cite{win}\n\\begin{equation}\n T_x  = \\frac{1}{3\\sqrt{2}}  \n \\left(\\begin{matrix} \n    -\\sqrt{3} & 0 & 1 & 0\\\\\n    0 & -1 & 0 & \\sqrt{3}  \n \\end{matrix}\\right),\n\\end{equation}\n\\begin{equation}\n T_y  = \\frac{-i}{3\\sqrt{2}}  \n \\left(\\begin{matrix} \n    \\sqrt{3} & 0 & 1 & 0\\\\\n    0 & 1 & 0 & \\sqrt{3}  \n \\end{matrix}\\right),\n\\end{equation}\n\\begin{equation}\n T_z  = \\frac{\\sqrt{2}}{3}  \n \\left(\\begin{matrix} \n    0 & 1 & 0 & 0 \\\\\n    0 & 0 & 1 & 0  \n \\end{matrix}\\right).\n\\end{equation}\nUsing these matrices, the Hamiltonian matrix becomes:\n\\begin{equation}\n \\left(\\begin{matrix} \n   (E_c+V) \\mathbb{I}_{2\\times 2} & \\sqrt3 P_1\\mathbf{T} \\cdot \\mathbf{k}_{\\alpha} & \n   \\frac{-1}{\\sqrt3}P_1 \\boldsymbol{\\hat{\\sigma}} \\cdot \\mathbf{k_{\\alpha}}      \\\\\n                       \n   \\sqrt3 P_2\\mathbf{T}^{\\dagger} \\cdot \\mathbf{k}_{\\alpha} & (E_v +V) \\mathbb{I}_{4\\times 4} & 0   \\\\\n                       \n   \\frac{-1}{\\sqrt3}P_1 \\boldsymbol{\\hat{\\sigma}} \\cdot \\mathbf{k_{\\alpha}} & 0 & \n   (E_v -\\Delta_0+V) \\mathbb{I}_{2\\times 2} \n \\end{matrix}\\right),\n\\end{equation}\nwhere $\\mathbf{k}_{\\alpha}=(\\alpha k_x, \\alpha k_y, k_z)$ and $\\alpha = P_2\/P_1$.\n\n\nFollowing the application of the Foldy-Wouthuysen transformation described by Winkler for \nzincblende semiconductors,\\cite{win} we obtain an effective equation, restricted to the\nconduction band, for the electronic states in the donor impurity\n\\begin{widetext}\n\\begin{equation}\n  \\left[ \\mathbf{T} \\cdot \\mathbf{k}_{\\alpha} \\frac{3P_1^{2}}{E-V+E_g}\\mathbf{T}^{\\dagger} \n  \\cdot \\mathbf{k}_{\\alpha} + \\boldsymbol{\\hat{\\sigma}} \\cdot \\mathbf{k_{\\alpha}}\n  \\frac{P^{2}_1}{3(E-V+E_g)}\\boldsymbol{\\hat{\\sigma}} \\cdot \\mathbf{k_{\\alpha}} \\right]\\psi_c=(E-V)\\psi_c .\n\\end{equation}\n\\end{widetext}\nUsing the relation \n$ (\\boldsymbol{\\hat{\\sigma}} \\cdot \\mathbf{A}) \n(\\boldsymbol{\\hat{\\sigma}} \\cdot \\mathbf{B}) = \\mathbf{A}\\cdot \\mathbf{B} + i\\boldsymbol{\\hat{\\sigma}}\n\\cdot \\left(\\mathbf{A} \\times \\mathbf{B} \\right)$, \nwe obtain two terms from the second term in the above equation, one of which corresponds to the effective\nspin-orbit interaction in the conduction band:\n\\begin{equation}\n   H_{\\text{ext}} = \\lambda^{*}_w \\, \n   \\boldsymbol{\\hat{\\sigma}} \\cdot \\left(\\mathbf{k_{\\alpha}} \\times \\boldsymbol{\\nabla}_{\\alpha} V \\right).\n\\label{eq:Hext}\n\\end{equation}\nwhere we defined $\\boldsymbol{\\nabla}_{\\alpha} \\equiv (\\alpha \\frac{\\partial}{\\partial x}, \n\\alpha \\frac{\\partial}{\\partial y}, \\frac{\\partial}{\\partial z} )$.\nWe have identified the coupling constant for the extrinsic spin-orbit interaction in wurtzite semiconductors:\n\\begin{equation}\n   \\lambda^{*}_w = \\frac{\\epsilon P_{1}^2}{3}\\left[\\frac{2}{E^{2}_g}  - \n   \\frac{1}{(E_g +\\Delta_0)^{2}} \\right],\n\\label{eq:lambda}\n\\end{equation}\nanalogous to the known coupling constant $\\lambda^{*}$ in Eq.\\ \\eqref{eq:HSOeff} for zincblende materials.\n\n\nNote the factor $\\alpha \\equiv P_2\/P_1$ in Eq.\\ \\eqref{eq:Hext}, which reflects the anisotropy\nof the wurtzite crystal structure.\nThe Coulomb potential of the hydrogenic impurity, $V(\\mathbf{r})$, that appears in Eq.\\ \\eqref{eq:Hext},\nwas introduced in Eq.\\ \\eqref{eq:Hzero}.\nActually, the spherically symmetric form given after Eq.\\ \\eqref{eq:Hzero} is a simplified expression \nwhich does not include the effect of the anisotropic effective mass and dielectric constant of wurtzite\ncrystal structures.\\cite{han}\nAs a first approximation, here we will work with this spherically symmetric Coulomb potential\nand will also disregard the $\\alpha$-dependence of $\\boldsymbol{\\nabla}_{\\alpha}$ and $\\mathbf{k_{\\alpha}}$.\nA complete treatment of the anisotropy effects would require considering the modified\neigenvalue problem of the anisotropic hydrogenic impurity, and then the effect of the\nfactor $\\alpha$ in the spin-orbit interaction.\nWe leave this refined treatment for future work.\nIn Table II we present the values of $\\lambda^*_w$ for GaN, ZnO, InN, and AlN,\nalong with the material parameters needed to evaluate Eq.\\ \\eqref{eq:lambda}.\n\n\n\\begin{table}[ht]\n\\label{tab:lambda}\n\\begin{center}\n\\begin{tabular} {| l | c | c | c | c | c |}\n\\hline\n      & $ m^*\/m_0$      & $\\epsilon$      &  $ E_g $  & $\\Delta_0 $  &  $\\lambda^{*}_w  $\\\\\n      &                 &                 &   [eV]    &  [meV]       & [$10^{-2}$ \\AA$^2$] \\\\\n   \\hline \n    GaN  & 0.32         &    9.53         &   3.51   &  72.9    &    5.95   \\\\\n   \\hline\n    ZnO  & 0.32         &    8.62         &   3.44   &   43     &    3.08   \\\\\n   \\hline\n    InN  & 0.26         &    7.39         &   0.78   &   40     &    1.33   \\\\\n   \\hline \n    AlN  & 0.38         &    4.27         &   5.4    &  -58.5   &   -1.04   \\\\ \n    \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Coupling constant of the effective extrinsic spin-orbit interaction \n and auxiliary material parameters\\cite{han} for wurtzite semiconductors.}\n\\end{table}\n\n\n\n\\section{Extrinsic spin-orbit corrections to the $2p$ level of hydrogenic impurities}\n\nUsing Eqs.\\ \\eqref{eq:Hext} and \\eqref{eq:lambda} we obtain the Hamiltonian of the extrinsic spin-orbit \ninteraction due to the Coulomb potential of the hydrogenic the donor impurity:\n\\begin{equation}\n  H_{\\text{ext}} =  \\frac{24a^{*3}}{r^3} \\xi_{2p}^{\\ast}  \\, \\mathbf{L} \\cdot \\mathbf{S} ,\n\\label{eq:ext_final}\n\\end{equation}\nwhere\n\\begin{equation}\n   \\xi^{\\ast}_{2p} \\equiv \\frac{e^2 \\lambda^{\\ast}_w}{24 \\epsilon \\hbar^2 a^{*3}} .\n\\end{equation}\nAs anticipated above, we simplified the Hamiltonian $H_{\\text{ext}}$ by setting the \nratio $\\alpha=1$ in Eq.\\ \\eqref{eq:Hext}.\nWe thus revert to the standard spin-orbit coupling of the hydrogen atom, but take into account\nthe appropriate, renormalized, coupling constant $\\lambda^*_w$.\nThe calculation of the first-order corrections to the $2p$ energy level of the impurity due\nto $H_{\\text{ext}}$ now follows the standard treatment of spin-orbit interaction in the\nhydrogen atom.\nThe common eigenvalues of $H_{\\text{ext}}$ and $J^2$ are given by:\n\\begin{equation}\n  \\varepsilon_1 = \\frac{1}{2}\\xi_{2p}^{\\ast} \\Big[ \\frac{3}{4}-2-\\frac{3}{4} \\Big]\\hbar^2 = \n        - \\xi_{2p}^{\\ast} \\hbar^2\n\\end{equation}\nfor $j=1\/2$, and\n\\begin{equation}\n  \\varepsilon_2 = \\frac{1}{2} \\xi_{2p}^{\\ast}  \\Big[ \\frac{15}{4}-2-\\frac{3}{4} \\Big]\\hbar^2 =  \n        \\frac{1}{2} \\xi_{2p}^{\\ast} \\hbar^2\n\\end{equation}\nfor $j=3\/2$. \nWe thus obtain for the energy corrections:\n\\begin{eqnarray}\n  &&  \\varepsilon_1 = - \\frac{e^2 \\lambda^{\\ast}}{12 \\epsilon a^{\\ast 3}} \\equiv -2 \\,\\beta     \\nonumber \\\\\n  &&  \\varepsilon_2 =  \\frac{e^2 \\lambda^{\\ast}}{24 \\epsilon a^{\\ast 3}}  \\equiv \\beta . \n  \\label{eq:bett}\n\\end{eqnarray}\nThe numerical values of $\\varepsilon_1$ and $\\varepsilon_2$ are shown in Table III,\ntogether with the parameters needed for their evaluation.\nWe also give the energy variation as a percentage of the unperturbed energy,\n$(\\varepsilon_2 - \\varepsilon_1) \/ E_2^{\\ast}$.\nOne can see that the splitting due to extrinsic spin-orbit interaction is four orders of magnitude\nsmaller than the energy of the original level.\nThis ratio is small but it is not negligible as it is, in fact, one order of magnitude larger than \nthe one obtained for the hydrogen atom, which is equal to 0.00133 \\%.\nIt should be emphasized that this comparison between the hydrogenic impurity and the hydrogen atom\nwas not obvious a priori, since the renormalization of the coupling constant $\\lambda$ is very \npronounced (6 orders of magnitude) and could have produced results radically different.\n\nWe remark that for aluminum nitride (AlN) the relation between $\\varepsilon_1$ and $\\varepsilon_2$ \nis inverted.  \nThis peculiarity originates in the particular characteristics of its electronic structure, \nwhich cause $\\lambda^{\\ast}$ to become negative. \n\n\\begin{table}[ht]\n\\label{tab:energias_SO_ext}\n\\begin{center}\n\\begin{tabular} {| l | c | c | c | c | c| c |}\n\\hline\n      & $ a^{\\ast} $  & $ \\lambda^{\\ast}_w$ &  $ \\text{Ry}^{\\ast} $  & $ \\varepsilon_1 $  & $ \\varepsilon_2 $ & $ (\\varepsilon_2 - \\varepsilon_1)\/E_2^{\\ast} $   \\\\\n      & [\\AA]         &[$10^{-2}$ \\AA$^2$]&    [meV]         & [$\\mu$eV]    & [$\\mu$eV]  &    $ [\\%] $      \\\\\n   \\hline \n    GaN  &    15.8    &      5.95         &    11.97         &     -1.94         &  0.968      &    0.024    \\\\\n   \\hline\n    ZnO  &    14.1    &      3.08         &    14.65         &     -1.56         &  0.779      &    0.016    \\\\\n   \\hline\n    InN  &    15.2    &      1.33         &    16.20         &     -0.625        &  0.313      &    0.0058   \\\\\n   \\hline \n    AlN  &    5.9     &     -1.04         &    70.85         &      14.50        & -7.25       &   -0.031    \\\\ \n    \\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\footnotesize{Extrinsic spin-orbit corrections to the $2p$ energy level of hydrogenic \ndonor impurities for four important wurtzite semiconductores, along with relevant material \nparameters.}}\n\\end{table}\n\n\n\\section{Combined intrinsic and extrinsic spin-orbit interactions}\n\nWe now study the effects of the intrinsic and extrinsic spin-orbit interactions combined on \nthe $E_2^*$ energy level of hydrogenic impurities in wurtzite semiconductors.\nThus, we now consider the complete Hamiltonian, Eq.\\ \\eqref{eq:Htot}, with $H_{\\text{int}}$\nand $H_{\\text{ext}}$ given in Eqs.\\ \\eqref{eq:int_completa} and Eq.\\ \\eqref{eq:ext_final},\nrespectively.\nWe will perform again a first-order perturbative treatment, now considering the full spin-orbit\nHamiltonian $H_{\\text{int}}+H_{\\text{ext}}$ as the perturbation.\nTo that end, we will express the spin-orbit coupling in the so-called uncoupled basis states \nof the $E_2^*$ shell, that is,  $\\left\\lbrace|l, s; m_l,m_s\\rangle\\right\\rbrace$, used in \nSection \\ref{sec:efa-impurity} to treat the intrinsic-alone case.\nThe matrix of the combined spin-orbit Hamiltonian in this basis is:\n\\begin{widetext}\n\\begin{equation}\n  H_{\\text{ext}} + H_{\\text{int}} = \\left(\\begin{matrix}  \n     -\\varepsilon &     0    &       0      &     A         &      0         &       0      &      0         &      0   \\\\\n                       \n     0        & -\\varepsilon &       0      &     0         &      0         &       0      &      B         &      0   \\\\\n                        \n     0        &    0     & \\beta-\\varepsilon&     0         &      0         &       0      &      0         &      0   \\\\\n                        \n     A        &    0     &       0      & -\\beta-\\varepsilon & \\sqrt2\\,\\beta &       0      &       0        &      0   \\\\\n                         \n     0        &    0     &       0      &   \\sqrt2\\,\\beta&  -\\varepsilon     &       0      &       0        &      0   \\\\\n                   \n     0        &    0     &       0      &     0          &      0        &    -\\varepsilon   & \\sqrt2\\,\\beta &      0   \\\\\n                         \n     0        &    B     &       0      &     0          &      0        &     \\sqrt2\\,\\beta & -\\beta-\\varepsilon &     0  \\\\\n                        \n     0        &    0     &       0      &     0          &      0        &       0        &      0       & \\beta-\\varepsilon   \\\\\n \\end{matrix}\\right) ,\n\\end{equation}\n\\end{widetext}\nwhere $\\beta$ was defined in Eq.\\ \\eqref{eq:bett} and $A$ and $B$ have been defined in \nEqs.\\ \\eqref{eq:AB}. \n\nThe characteristic polynomial that solves the eigenvalue problem is:\n\\begin{equation}\n \\varepsilon^2 (\\varepsilon-\\beta)^2(\\varepsilon\\,\\beta+\\varepsilon^2-A^2-2\\,\\beta^2)\n (\\varepsilon\\,\\beta+\\varepsilon^2-B^2-2\\,\\beta^2)=0 ,\n\\end{equation}\nand the corresponding eigenvalues are:\n\\begin{equation}\n \\varepsilon^2 = 0  \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\Rightarrow  \\,\\,\\,\\,\\,  \\varepsilon_{1,2} = 0, \n \\label{eq:rot}\n\\end{equation}\n\\begin{equation}\n \\,\\,\\,(\\varepsilon-\\beta)^2 = 0  \\,\\, \\Rightarrow \\,\\,\\,\\,\\,\\,\\, \\varepsilon_{3,4} = \\beta ,\n \\label{eq:rot1}\n\\end{equation}\n\\begin{equation}\n \\varepsilon^2+\\varepsilon\\beta-(A^2+2\\beta^2) = 0  \\,\\,\\,\\,\\,\\,\\,\\,  \\Rightarrow  \\,\\,\\,\\,\\, \\varepsilon_{5,6} = \n -\\frac{\\beta}{2} \\pm \\sqrt{\\frac{9}{4}\\beta^2 + A^2}, \n \\label{eq:rot2}\n\\end{equation}\n\\begin{equation}\n \\varepsilon^2+\\varepsilon\\beta-(B^2+2\\beta^2) = 0  \\,\\,\\,\\,\\,\\, \\Rightarrow  \\,\\,\\,\\,\\, \\varepsilon_{7,8} = \n -\\frac{\\beta}{2} \\pm \\sqrt{ \\frac{9}{4} \\beta^2 + B^2}.\n  \\label{eq:rot3}\n\\end{equation}\nWe remark that the energy corrections given in Eqs.\\ \\eqref{eq:rot}--\\eqref{eq:rot3} contain the \nprevious cases (intrinsic and extrinsic spin-orbit interactions acting alone) in the appropriate \nlimits, and they are represented schematically in Fig. \\ref{fig:combinado}.\nThis plot offers a qualitative view of the energy splittings and shows the greater breaking of\ndegeneracy caused by the combined action of both spin-orbit couplings.\nFinally, we calculate the eigenvalues $\\varepsilon_i$ ($i=3, \\ldots ,8$) using the values \nof $A$, $B$ y $\\beta$ corresponding to GaN, ZnO, InN y AlN; the results are given in Table IV.\n\n\\begin{figure}\n  \\centerline{\\includegraphics[scale=0.6]{figure1.pdf}}\n  \\caption{\\footnotesize{Schematic representation of the corrections to the energy level $E_2^*$ \n  of hydrogenic donor impurities to first order in the intrinsic and extrinsic spin-orbit \n  interactions combined.}\n  \\label{fig:combinado}}\n\\end{figure}\n\n\n\\begin{table}[ht]\n\\label{tab:comb}\n\\begin{center}\n\\begin{tabular} {| l | c | c | c | c | c |}\n\\hline\n      & $\\varepsilon_{3,4}$ & $\\varepsilon_5$ & $\\varepsilon_6$  & $\\varepsilon_7$ & $\\varepsilon_8$ \\\\\n      &     [$\\mu$eV]    &      [$\\mu$eV]     &    [$\\mu$eV]    &   [$\\mu$eV]      &    [$\\mu$eV]    \\\\\n   \\hline \n    GaN &    0.23       &      -13.36        &    13.13         &    -54.71        &    54.48     \\\\\n   \\hline\n    ZnO  &    4.90      &      -49.39        &     44.94        &    -60.78        &    56.33    \\\\\n   \\hline\n    InN  &    0.76      &      -15.42        &    14.66         &     -61.31       &    60.63     \\\\\n   \\hline \n    AlN  &   -20.15     &      -20.44        &    40.58         &    -17.56        &   37.70      \\\\ \n    \\hline\n\\end{tabular}\n\n\\caption{\\footnotesize{Energy corrections due to the combined intrinsic and extrinsic spin-orbit\ninteractions to the energy level $E_2^*$ of hydrogenic donor impurities for four important  \nbinary semiconductors with wurtzite crystal structure.}}\n\\end{center}\n\\end{table}\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe have studied theoretically the effects of the spin-orbit interaction on the\n$E_2^*$ energy level of hydrogenic donor impurities embedded in semiconductors\nwith wurtzite crystal structure.\nBoth the intrinsic (Dresselhaus) and extrinsic (Rashba) spin-orbit interations\nhave been considered, first acting separately and then together.\nThe study was carried out at the level of first-order perturbation theory,\nwhich turns out to be appropriate given the relative magnitude of the corrections\nto the unperturbed energy spacings.\nFurthermore, in order to evaluate the extrinsic spin-orbit interaction \nit was necessary to calculate the renormalized coupling constant $\\lambda^*$ \nfor wurtzite semiconductors from 8-band Kane theory.\n\n\nWe applied our calculations to four currently important semiconductors, i.e.\\ GaN, ZnO \nInN, and AlN.\nA general conclusion of these calculations is that both spin-orbit couplings produce\nrelative energy corrections that are bigger than the standard spin-orbit corrections\nto the $E_2$ energy level of the hydrogen atom.\nWhile for  GaN, ZnO, InN we conclude that the intrinsic spin-orbit interaction\nproduces larger energy corrections than the extrinsic one, that is not the case for AlN,\nwhere both interactions have comparable effects.\nAnother anomaly shown by AlN is the fact that its effective coupling constant \n$\\lambda^*$ is negative.\nThis causes the eigenvalues $\\varepsilon_{3,4}$, which are positive for GaN, ZnO \nand InN, to become negative for AlN.\nThese anomalies of AlN are due to the specific features of its electronic structure\nwhich determine the relevant parameters $\\Delta_0$, $\\gamma_w$ and $a^{\\ast}$.\nFinally, we have found that the combined action of both types of spin-orbit coupling \nleads to an almost complete breaking of the degeneracy of the unperturbed energy level.\n\n\\bigskip\n\n \n\\acknowledgments\nWe thank Rodolfo Jalabert and Dietmar Weinmann for discussions that motivated this work.\nWe gratefully acknowledge financial support from Projects UBACYT and CONICET PIP.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\subsection{Motivation}\n\\input{sections\/intro\/motivation.tex}\n\\subsection{Related Work and Main Contributions}\n\\input{sections\/intro\/contributions.tex}\n\\subsection{Notation}\n\\input{sections\/intro\/notation.tex}\n\n\\section{Preliminaries} \\label{sec:prelimiaries}\n\\subsection{System Model}\\label{sec:sys_model}\n\\input{sections\/preliminaries\/sys_model.tex}\n\\subsection{Metrics}\n\\input{sections\/preliminaries\/metrics.tex}\n\\section{Jammer Model}\n\\input{sections\/preliminaries\/jammer_model.tex}\n\n\\section{Resilient Receiver Design} \\label{sec:transceiver}\n\\subsection{General Considerations}\\label{sec:look_sinr}\n\\input{sections\/resilient_design\/look_sinr.tex}\n\\input{sections\/resilient_design\/separate_design.tex}\n\n\n\\section{Numerical Results} \\label{sec:results}\n\\subsection{Methodology}\n\\input{sections\/results\/methodology.tex}\n\\subsection{Simulation Studies}\n\\input{sections\/results\/simulation.tex}\n\n\\section{Conclusions} \\label{sec:conclusions}\n\\input{sections\/conclusions.tex}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Closed-Form Design}\\label{subsec:closed_form}\nThe receivers' goal is to ensure reliable communication for all uplink participants,\ni.e. we are interested in maximizing the individual per-stream SINRs $\\gamma_{k}^{B}$ or \nat least ensure a certain quality of service (QoS), e.g. the $\\gamma_{k}^{B}$'s lie above a given threshold $\\gamma_{0}$.\n\nThe maximization of the lower bound in Eq. \\eqref{eq:lb} is a Rayleigh quotient maximization problem with the\nstandard solution being given by:\n\\begin{align}\n    \\label{eq:maxsinr}\n    \\vb_{k} = \\br{\\bm{B}_{k} + \\eta\\bm{A}_B(\\bm{\\theta}_G)\\abg^{\\mathsf{H}}}^{-1}\\ch_{k}\\wa_{k}\n\\end{align} \nfor all $k = 1, \\dots, K$. \nThis is the most simple way to leverage AoA information at the receiver, since the solution is given in closed form\nand computationally tractable, since $\\bm{A}_B(\\bm{\\theta}_G)\\abg^{\\mathsf{H}}$ can be easily computed without matrix multiplications \nby only considering the phase shifts between the impinging jamming signals.\n\nNote, that the resulting filter still depends on $\\eta$, which in general is not available at the receiver.\nThe most easy way to address this problem is to consider $\\eta$ as a hyperparameter which controls how much the\njamming signals are suppressed. \nIndeed, this is justified by the fact that in the jammer-dominant regime, \nthe contribution of the noise variance $\\sigma^{2}$ is small compared to the one of the jammer, i.e\n$\\vb_{k}^{\\mathsf{H}} \\br{\\eta\\bm{A}_B(\\bm{\\theta}_G)\\abg^{\\mathsf{H}} + \\sigma^{2}\\bm{\\mathrm{I}}} \\vb_{k} \\approx \\eta\\vb_{k}^{\\mathsf{H}} \\bm{A}_B(\\bm{\\theta}_G)\\abg^{\\mathsf{H}} \\vb_{k}$.\nIn the following, we will propose other methods to overcome this dependency.\n\n\\subsection{Zero Forcing Design}\nAs it can be seen in Section \\ref{subsec:closed_form}, the main contribution to the jamming signals is given\nby the term \n\\begin{align}\n    \\beta_{k} = \\normsq{\\bm{A}_B(\\bm{\\theta}_G)^{\\mathsf{H}} \\vb_{k}} \n\\end{align}\nwhich can be naturally interpreted as the receive beampattern of the receiver antenna array \\cite{van2004optimum}. \nThis indicates how much the signals coming from the jammer direction are emphasized, thus \nthe anti-jamming criterion can be formalized as a beampattern minimization problem in the same spirit as\n\\cite{mu_mimo_radar_com,su_pls_bf,jcas_sp_survey}.\nFurthermore, a natural way to ensure good per-stream SINRs for all users is to remove the interference,\nwhich can be realized with a zero-forcing (ZF) design. \nThus, the beampattern minimization problem under ZF constraint can be formulated analogously to \\cite{mu_mimo_radar_com} as\n\\begin{align}\n    \\label{eq:bpmin_zf}\n    \\min_{\\substack{\n        \\vb_{k}\\\\\n        k = 1, \\dots, K}\n    }\\sum_{k=1}^{K}\\beta_{k} \\quad \\text{s.t.} \\quad \\bm{P}^{\\mathsf{H}} \\vb_{k} = \\bm{e}_{k} \\, \\forall k\n\\end{align}\nwhere we defined \n\\begin{equation}\n    \\bm{P} = \\begin{bmatrix}\n        \\bm{H}_{1}\\bm{w}_{1} & \\dots & \\bm{H}_{K}\\bm{w}_{K}\n    \\end{bmatrix} \\in \\mathbb{C}^{N_{B} \\times K}\n\\end{equation}\nand $\\bm{e}_{k}$ is $k$-th standard basis vector in $\\mathbb{R}^{K}$.\nNote that this is a convex optimization problem easily solvable in polynomial time. \n\nFurthermore, if the matrix $\\bm{A}_B(\\bm{\\theta}_G)\\abg^{\\mathsf{H}}$ has full-rank, we can also give an analytic solution.\nLetting $\\bm{X} = \\bm{A}_B(\\bm{\\theta}_G)\\abg^{\\mathsf{H}}$, we construct the Lagrangian\n\\begin{align}\n    \\mathcal{L}\\left(\\vb_{k}, \\bm{\\mu}_{k}\\right) = \n    \\sum_{k=1}^{K} \\vb_{k}^{\\mathsf{H}}\\bm{X}\\vb_{k} + 2\\Re\\brcur{\\bm{\\mu}_{k}^{\\mathsf{H}} \\br{\\bm{P}^{\\mathsf{H}} \\vb_{k} - \\bm{e}_{k}}}\n\\end{align}\nwhere $\\bm{\\mu}_{k}\\in\\mathbb{C}^{K}$ are the Lagrange multipliers.\nSince the ZF-constraint is an equality constraint, the Karush-Kuhn-Tucker conditions are both necessary and sufficient.\nComputing the gradients wrt. $\\vb_{k}$ and $\\bm{\\mu}_{k}$, setting them to $\\bm{0}$, and solving for $\\vb_{k}$,\nwe obtain\n\\begin{align}\n    \\label{eq:bpmin_zf_analytic}\n    \\vb_{k} = \\bm{X}^{-1} \\bm{P} \\br{\\bm{P}^{\\mathsf{H}} \\bm{X}^{-1} \\bm{P}}^{-1} \\bm{e}_{k}\n\\end{align}\nBy standard linear algebra $\\bm{X}$ has full rank iff $L_{G}\\geq N_{B}$, which might pose a problem if the propagation channel\nof the jammer shows $L_{G} < N_{B}$ dominant paths. \nIn that case, one can easily ``pad'' the resolved AoAs $\\bm{\\theta_{G}}$ with \na set of different angles $\\{\\phi_{G, l}\\}_{l=1}^{N_{B}-L_{G}}$ thus enlarging the grid over which the beampattern should be minimized.\nThis idea can be naturally extended to the designs considered in Sections \\ref{subsec:closed_form} and \\ref{subsec:min_sinr}.\n\\subsection{Minimum SINR Design}\\label{subsec:min_sinr}\nAnother way to approach this problem is to minimize the influence of the jamming signals, \nwhile ensuring a minimum SINR $\\gamma_{0}$ in the jammer-free case.\nUsing this method one can address the disadvantages ZF-filters generally have in the low SNR regime, i.e. $\\sigma^{2} \\gg 0$.\nRewriting the beampattern $\\beta_{k}$ as\n\\begin{align}\n    \\beta_{k} = \\normsq{\\bm{A}_B(\\bm{\\theta}_G)^{\\mathsf{H}} \\vb_{k}} = \\tr{\\bm{A}_B(\\bm{\\theta}_G)^{\\mathsf{H}} \\bm{V}_{k} \\bm{A}_B(\\bm{\\theta}_G)}\n\\end{align}\nwe cast the resulting problem as\n\\begin{align}\n    \\label{eq:bpmin}\n    \\min_{\n        \\substack{\\bm{V}_{k} \\succcurlyeq \\bm{0}\\, \\bm{V}_{k} = \\bm{V}_{k}^{\\mathsf{H}} \\\\ \\text{rank}(\\bm{V}_{k}) = 1 \\\\\n        k = 1, \\dots, K}\n    } \\,\n    \\sum_{k=1}^{K}\\beta_{k} \\quad \\text{s.t.} \\quad \\dfrac{\\tr{\\bm{A}_{k}\\bm{V}_{k}}}{\\tr{\\bm{B}_{k}\\bm{V}_{k}}} \\geq \\gamma_{0}\n\\end{align}\nwith $\\bm{V}_{k} = \\vb_{k} \\vb_{k} ^{\\mathsf{H}}$ as before and $\\bm{A_{k}}$ and $\\bm{B_{k}}$ as in equations \\eqref{eq:sh1} and \\eqref{eq:sh2}.\nNote that this problem is non-convex due to the rank-$1$ constraint on $\\bm{V}_{k}$. \nIn order to efficiently solve this problem, we apply semidefinite relaxation \\cite{sdr}, i.e. \nwe relax the rank-$1$ constraint, \nsolve the resulting convex problem and recover the solution via eigenvalue decomposition (EVD).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{INTRODUCTION}\n\nThe first step of planet formation in protoplanetary disks is the collisional\ngrowth of (sub-)micron-sized dust particles (or aggregates), and consequent\nsedimentation. Dust settling and the formation of a dust layer at the midplane of\nthe disk play an important role in the subsequent planetesimal formation\nprocess. Enhancement of the particle density in the dust layer accelerates the\ncollisional growth rate. If the density enhancement is high enough,\nplanetesimals may form through the gravitational instability of the dust layer\n(Goldreich \\& Ward 1973; Sekiya 1983). However, the turbulent motion of the gas\nhinders the dust layer from thinning. Dust particles are stirred by\nturbulence and diffuse to high altitudes from the midplane. Thus, turbulence \nis an impediment to planetesimal formation. Turbulence excited by\nmagneto-rotational instability (MRI) is so strong that the dust layer cannot\nbecome thin enough to induce planetesimal formation through gravitational\ninstability (Johansen \\& Klahr 2005; Fromang \\&\\ Papaloizou 2006; Turner et\nal. 2006; Carballido et al. 2006, 2011; Fromang \\& Nelson 2009).  In addition, the\nturbulence-induced collisional velocity can be high enough to destroy dust\naggregates (Carballido et al. 2008, 2010), though several mechanisms have been\nproposed to overcome this difficulty (Johansen et al. 2007, 2011; Lyra et al.\n2008, 2009). Even if the gas disk exhibits an initially laminar flow (such as is expected\nin the dead zone where the ionization fraction of the gas is too low to couple\nto the magnetic field, as mentioned in Gammie 1996 and Sano et al. 2000), dust \nsettling itself induces turbulence. The dust particles tend to rotate around \nthe central star faster than the gas because the gas experiences a pressure gradient \nforce acting in the opposite direction of the gravity of the star. When the dust layer \nthins and the dust-to-gas ratio in the layer approaches unity, the velocity\ndifference from that of the upper gas layer induces Kelvin-Helmholtz\n(KH) instability and excites turbulence in the dust layer (e.g.,\nGoldreich \\& Ward 1973; Sekiya \\& Ishitsu 2000, 2001; Garaud \\&\\ Lin\n2004). The velocity difference between the dust particles and the gas\ninside the dust layer also induces streaming instability (Youdin \\&\nGoodman 2005; Youdin \\& Johansen 2007; Johansen \\& Youdin 2007; see\nChiang \\& Youdin 2010 for review on the various instability in the dust\nlayer). Laminar gas disks are considered excellent sites for planetesimal\nformation.  Thus, it is important to clarify how turbulence induced in\nthe dust layer diffuses the dust particles, versus dust\nsedimentation. Several analytical and numerical studies have focused on\nthis problem (Cuzzi et al. 1993; Champney et al. 1995; Sekiya 1998;\nDobrovolskis et al. 1999; Johansen et al. 2006; Michikoshi \\&\\ Inutsuka\n2006; Weidenschilling 2006, 2010; Bai \\&\\ Stone 2010a, 2010b).\n\nSekiya (1998, hereafter S98) analytically solved for the structure of the dust\nlayer under turbulence induced by KH instability. To derive the density\nprofile of the dust layer analytically, the author adopted several assumptions.  First, the dust particles were assumed to be small enough and were coupled so tightly to the gas that\nthe dust and the gas could be treated as a single fluid.  Second, the density\nstructure was adjusted to keep the dust layer marginally unstable\nto KH instability. Third, the effects of the Coriolis force and of the Keplerian\nshear were neglected. These assumptions allowed the author to determine\nanalytically the density structure of the dust layer and to discuss how far\nfrom the midplane the dust particles diffuse due to turbulence. The\nanalysis in S98 provided a useful guide for subsequent numerical studies\nthat investigated turbulence of the dust layer with more realistic assumptions.\nIn reality, however, dust particles that are not tightly coupled to gas\nplay an important role in inducing turbulence. Also, other kinds of instability,\nsuch as streaming instability, may occur before KH instability sets in (Bai \\&\nStone 2010a). Thus, in addition to performing numerical simulations, it is\ndesirable to include an analytic discussion on the structure of the dust layer using\na more general set of assumptions than adopted in S98.\n\nIn this paper, we revisit the analysis presented in S98 from a different point of view.\nIn S98, the density structure of the dust layer was determined by the condition\nthat the dust layer was marginally unstable to KH instability; i.e., the\nRichardson number, $\\mathrm{Ri}$, had a constant critical value. In the\nstability condition on $\\mathrm{Ri}$, the free energy due to the shear\nvelocity is compared with the energy needed to lift the material against the\nvertical gravity (Chandrasekhar 1961). In the case of a dust layer, the free\nenergy of the azimuthal velocity is compared with the gravitational potential in\nthe vertical direction. We also discuss the energetics of the dust layer to\ndetermine the density structure, but in this paper we focus on the\ngravitational potential in the radial direction. Sustaining steady turbulence\nrequires an energy supply. The source of the free energy for KH instability or\nstreaming instability is the velocity difference between the dust and the gas.\nBecause this free energy is consumed in the process of exciting turbulence, a velocity difference must be continually induced to provide steady turbulence. This velocity difference originates from the force balance in the radial direction\nbetween the stellar gravity, the centrifugal force, and the pressure gradient.\nIn other words, the velocity difference is induced because the gas resides in a slightly shallower effective potential (including both the\nstellar gravity and the gas pressure), than the effective potential for the dust. The dust\nloses its angular momentum due to gas drag and drifts towards the star, while the gas gains \nangular momentum and drifts outward. The total angular momentum remains\nconstant. Because the effective potential for the dust is deeper, this process releases \ngravitational energy that can be a source of the free energy for turbulence. (The relationship between the angular momentum variation, $\\Delta L$, and the energy\nvariation, $\\Delta E$, for circular orbiting material is $\\Delta E=\\Omega\\Delta\nL$, where the angular velocity, $\\Omega$, is different for the gas and for the\ndust.) If the gas disk is initially in a state of laminar flow, i.e., if gas\ndrag is the only process that exchanges angular momentum, then dust accretion is the source of the free energy used to induce turbulence. The accretion\nrate of the dust due to gas drag has been calculated in the literature\n(Nakagawa et al. 1986, hereafter NSH86; Weidenschilling 2003; Youdin \\& Chiang\n2004). Using this dust accretion rate, and assuming that a certain\nfraction of the accretion energy is transferred to turbulence, it is\nstraightforward to calculate the strength of the turbulence and to determine the\ndensity structure of the dust layer. In this paper, we show that an analysis of the\nenergetics based on the gravitational potential in the radial direction results in a dust layer structure that is qualitatively equivalent to the dust layer structure developed in\nthe analysis presented in S98. Most of the important properties of the S98 model\nare reproduced. Our analysis avoids some of the assumptions that were adopted in S98.  \nSpecifically, we do not assume tight coupling between the gas and the dust, nor a\nmarginally unstable structure for KH instability. (The effects of the Coriolis\nforce and of Keplerian shear, which are still neglected in this paper, are\ndiscussed by Ishitsu \\& Sekiya 2002, 2003; G\\'omez \\& Ostriker 2005; Chiang\n2008; Barranco 2009; Lee et al. 2010.) Thus, the results of this paper may be applied to more general situations, including situations in which the\ndust is weakly coupled to the gas, or in which streaming instability acts as a source\nof turbulence.\n\nIn \\S \\ref{sec:assump}, we describe our model assumptions. In\n\\S \\ref{sec:eneinp}, the energy release rate of the accreting dust is\ncalculated. In \\S \\ref{sec:turbstr}, the strength of the turbulence is derived and\nits asymptotic forms in the limits of a small dust-to-gas ratio and a\ntight dust-gas coupling are discussed. In \\S \\ref{sec:comppre}, the results of\nour model are compared with the S98 model and with previous numerical\nsimulations of KH instability and streaming instability. In\n\\S \\ref{sec:dustvelocity}, we discuss how dust layer formation reduces\nthe radial drift velocity of the dust and the relative velocity of the dust\nparticles. In \\S \\ref{sec:ene-into-tur}, we check the applicability of our model to\nturbulence in the dust layer for the KH instability and streaming instability cases.\n\n\\section{DISK MODEL AND ASSUMPTIONS}\n\\label{sec:assump}\n\n\\subsection{Brief Model Description}\n\nWe consider a dust layer that forms after dust particles have settled to the\nmidplane of a gas disk around a star. Before the dust settles, the gas disk is\nassumed to be subject to laminar flow. This means that we consider the dead zone where MRI is\ninactive because of the low degree of ionization. Turbulent gas motion may be present even in the dead zone because sound waves propagate from the\nactive layers at high altitudes and disturb the gas in the dead zone (Fleming \\&\nStone 2003; Suzuki et al. 2010; Okuzumi \\& Hirose 2011). We assume that the\ndead zone is wide enough that the active layer cannot induce strong turbulence\nat the midplane.\n\nEven in an initially laminar disk, dust sedimentation and the formation of the\nmidplane dust layer cannot proceed in a  perfectly undisturbed fashion. \nWhen the midplane dust-to-gas\nratio reaches a critical value, the velocity differences between the dust\nlayer and the upper gas layer, or between the individual dust particles and the\nsurrounding gas, will start to induce hydrodynamical instabilities, such as KH\ninstability or streaming instability (see Chiang \\& Youdin 2010 for review).\nConsequently, turbulent diffusion of the dust particles terminates further\ndust settling. It is expected that a steady dust layer forms, in which \nparticle settling and turbulent diffusion balance each other. In such a\nstate, vertical settling is no longer an energy source for turbulence,\nbut the radial accretion of the dust still proceeds. The turbulence is\nmaintained by the energy liberated from dust accretion to the star.\n\nWe calculate the dust accretion rate and consequent energy release, \nassuming that a steady dust layer has formed. The strength of the\nturbulence is then estimated. The balance between the turbulent diffusion\nand the dust settling determines the dust layer thickness. Thus, the structure of\nthe dust layer, the dust accretion rate, and the turbulence strength must be\ndetermined self-consistently.\n\nFor simplicity, we focus on a narrow axisymmetric ring region at a certain\nradius $r$ from the central star. We consider the dust layer structure only in\nthe vertical direction, neglecting any radial variation in the properties of\nthe gas disk and the dust layer. All the dust particles are assumed to have\na uniform size, i.e., the size distribution of the dust particles is neglected\nin this paper.\n\n\\subsection{Vertical Structure of the Dust Layer}\n\nIf the effect of gas drag on the dust were negligible, the dust particles\n would orbit at the Keplerian velocity \n$\\Omega_{\\mathrm{K}}=\\sqrt{(GM)\/r^{3}}$, where $G$ is the \ngravitational constant and $M$ is the mass of the\ncentral star. However, the orbital velocity of the gas is slightly less\nthan the Keplerian velocity because the direction of the gas pressure\ngradient is usually outward from the star.  This cancels part of the \ncontribution to the orbital velocity from the gravity of the star. Given a negligible influence of \nthe drag force from the dust, the angular velocity of the gas would be\n\\begin{equation}\n\\Omega_{g}=\\sqrt{\\frac{GM}{r^{3}}(1-2\\eta)}\\approx\\Omega_{\\mathrm{K}}(1-\\eta).\n\\label{eq:omega_g\n\\end{equation}\nHere, $\\eta$ represents the deviation of the gas orbital velocity from the\nKeplerian value. It is half of the ratio between the gas pressure gradient\nforce and the gravity of the central star,\n\\begin{equation}\n\\eta=-\\frac{1}{{2\\rho_{g}r\\Omega_{\\mathrm{K}}^{2}}}\\frac{{\\partial P}}{{\\partial r}}\\ ,\n\\end{equation}\nwhere $\\rho_{g}$ and $P$ are the gas density and pressure, respectively.\n\nWe consider a thin dust layer in which most of the dust particles have settled\non the midplane of the gas disk. The gas disk is isothermal in the vertical\ndirection with a scale height of $h_{g}=c_{s}\/\\Omega_{\\mathrm{K}}$, where\n$c_{s}$ is the gas sound speed. Because the thickness of the dust layer is\nmuch smaller than the scale height of the gas disk, the gas density $\\rho_{g}$\nin the dust layer is treated as a constant. Its value is given by the column\ndensity of the gas disk $\\Sigma_{g}$ as\n\\begin{equation}\n\\rho_{g}=\\frac{\\Sigma_{g}}{\\sqrt{2 \\pi} h_{g}}\\ . \n\\label{eq:rhog\n\\end{equation}\nThe dust density profile in the vertical direction, $\\rho_{d}(z)$, is\ndetermined by the balance between the dust settling and turbulent diffusion\n(Youdin \\& Lithwick 2007, hereafter YL07)\n\\footnote{Note that the diffusion coefficient\ngiven by YL07 is derived assuming that the dust particles can be treated as\npassive particles. Problems may arise if this formula is used in a case \nwhere the dust layer is so dense that its dust-to-gas ratio is larger than unity. Although a\ncomparison of our model with the simulation by Johansen et al. (2006) shows \ngood agreement even for a dust-to-gas ratio as large as $40$ (see Fig.\n\\ref{fig:fdmid-joh}), modeling the effect of the dust inertia will be a \nsubject for future investigation using a refinement of the present\nmodel.}.\nThe dust particles have a\nstopping time, $\\tau_{\\mathrm{stop}}=T_{s}\\Omega_{\\mathrm{K}}^{-1}$, in which\nthe velocity difference from the background gas flow becomes $1\/e$ times due\nto gas drag, where $T_{s}$ is the non-dimensional stopping time. In a\nturbulent gas disk with a turbulent diffusion coefficient $D_g$,\nthe density profile of the dust layer composed of single-sized particles is\nGaussian, $\\rho_{d}=\\rho_{d,0}\\exp(-\\tilde{z}^{2})$, where $\\tilde{z\n=z\/(\\sqrt{2}h_{d})$ is the non-dimensional vertical coordinate normalized by\nthe dust scale height $h_d \\approx [D_g \/(\\Omega_{\\rm K} T_s)]^{1\/2}$.\nNote that the turbulent diffusion coefficient in the ``z-direction'',\n$D_g$, may have a significantly different value from the usual turbulent\nviscosity coefficient $\\nu_{\\rm acc}$ in accretion disks, where\n$\\nu_{\\rm acc}$ comes from the ``$r \\theta$-component'' of the Reynolds\nstress (e.g., Lesur \\& Ogilvie 2010). Nevertheless, we follow the conventional\n``$\\alpha$-prescription'', and express the diffusion coefficient as\n$D_g=\\alpha c_g h_g$ for simplicity. This prescription and equation (24)\nof YL07 give\n\\begin{equation}\nh_{d}=\\sqrt{\\frac{\\alpha}{{T_{s}}}}\\sqrt{\\frac{{1+T_{s}}}{{1+2T_{s}}}}h_{g}\\ .\n\\label{eq:hd\n\\end{equation}\nThe dust density profile is written as\n\\begin{equation}\n\\rho_{d}=f_{\\mathrm{mid}}\\rho_{g}\\exp(-\\tilde{z}^{2})\\ ,\n\\end{equation}\nwhere the midplane dust-to-gas ratio $f_{\\mathrm{mid}}$ is increased by a\nfactor $h_{g}\/h_{d}$ from the total dust-to-gas ratio or the\n``metallicity'' of the disk $Z_{\\mathrm{disk}}=\\Sigma_d \/ \\Sigma_g$,\n\\begin{equation}\nf_{\\mathrm{mid}}=Z_{\\mathrm{disk}}\\frac{{h_{g}}}{{h_{d}}}=Z_{\\mathrm{disk}}\\sqrt{\\frac{{T_{s}}}{\\alpha}}\\sqrt{\\frac{{1+2T_{s}}}{{1+T_{s}}}}\\ .\n\\label{eq:fdmid\n\\end{equation}\nWe define $\\beta(\\tilde{z})$ as the ratio of the total (dust and gas)\ndensity to the gas density,\n\\begin{equation}\n\\beta=\\frac{\\rho_{g}+\\rho_{d}}{\\rho_{g}}={1+f_{\\mathrm{mid}}\\exp(-\\tilde\n{z}^{2})}\\ . \n\\label{eq:beta\n\\end{equation}\n\n\\subsection{Typical Values of the Model Parameters}\n\nIn our model, the turbulent parameter $\\alpha$ and the structure of the dust\nlayer (i.e., the thickness, $h_{d}$, and the midplane dust-to-gas ratio,\n$f_{\\mathrm{mid}}$) are determined for a given parameter set (the stopping\ntime of the dust particles $T_{s}$, the disk metallicity\n$Z_{\\mathrm{disk}}$, and the parameters for the gas disk). The\nnon-dimensional variables of the result ($\\alpha$, $h_{d}\/r$, and\n$f_{\\mathrm{mid}}$) depend on the gas disk parameters only through\n\\begin{equation}\n\\tilde{\\eta}=\\eta^{2}\\left(  \\frac{r}{h_{g}}\\right)  ^{2}~, \n\\label{eq:etawave\n\\end{equation}\nwhere $\\tilde{\\eta}\\sim\\eta\\sim(h_{g}\/r)^{2}\\sim10^{-3}-10^{-2}$. The\nnumerical calculations in the subsequent sections mainly use a value\n$\\tilde{\\eta}=(0.05\/\\gamma)^{2}=9 \\times 10^{-4}$, where $\\gamma=5\/3$,\nin order to compare our model with the simulation presented by \nJohansen et al. (2006, hereafter JHK06). \nIn \\S\\ref{sec:BS10}, $\\tilde{\\eta}=0.05^{2}$ is used for comparison\nwith the results by Bai \\& Stone (2010), and in \\S\\ref{sec:dustvelocity},\n$\\tilde{\\eta}=2.92 \\times 10^{-3}$ is used to calculate the radial\ndrift velocity  of the dust in the gas disk model by Hayashi (1981). The\n$\\tilde{\\eta}$ dependence of the results will be discussed in\n\\S\\ref{sec:disketa}.  \n\n\n\\section{TURBULENCE ENERGY SUPPLY}\n\\label{sec:eneinp}\n\nThe energy supply for turbulence comes from liberation of the gravitational\nenergy of the dust. As the dust falls towards the central star, the dust\nparticles penetrate more deeply into the potential well of the star. Although the gas drifts\noutward to conserve the total angular momentum, the difference in the\neffective gravitational potential between the dust ($-GM\/r$) and the gas\n($-GM(1-2\\eta)\/r$), including the work done by the pressure gradient,\ncauses a net energy liberation. Part of the liberated energy is converted\ndirectly into thermal energy, and part is used for supplying energy to\nthe turbulence (see discussion in \\S \\ref{sec:ene-into-tur}).\n\nDust particle accretion occurs either individually or\ncollectively. Individual dust particles suffer gas drag given a\nvelocity difference between the particle and the gas. Usually the orbital\nvelocity of the gas is slower than that of the dust particle, and the gas drag\nforce decelerates the particle orbital motion. Thus, the particle loses its\nangular momentum and drifts inward. In addition to this individual drift,\na collective drag works on the whole dust layer. The dust layer\nat the midplane orbits faster than the gas layer at higher altitudes because in the\ndust layer the increased inertia of the enriched dust particles weakens the\neffect of the gas pressure. The dust layer rotates with a velocity close to\nthe Keplerian value. The upper gas layer, devoid of the inertia of the dust,\norbits more slowly than the midplane dust layer. If the gas exhibits turbulent\nviscosity, then the slower orbiting gas layer exerts a drag force on the dust\nlayer, and consequently the dust layer loses its angular momentum and accretes\ninward. We consider the energy release rate of the dust caused by \nindividual and collective drag.\n\n\\subsection{Dust Accretion Caused by Individual Drag}\n\\label{sec:inddrag}\n\nWe calculate the gravitational energy released when dust\nparticles accrete towards the star due to drag on individual particles. In this\ncalculation, we assume that the gas disk is in a laminar flow state, and that\ndust particles drift inward steadily. Any random motion of particles caused by\ngas turbulence is neglected. We expect that in turbulent gas\ndisks, the average motion of the dust particles can be estimated from the motion in\nthe laminar disk (Bai \\& Stone 2010). Because the individual drag is more\neffective than the collective drag when the midplane dust-to-gas ratio is less\nthan unity (see Fig. \\ref{fig:Sigma} below), we take the $f_{\\mathrm{mid}\n\\ll1$ limit in the following calculation. The calculation for a general $f_{\\mathrm{mid}}$ is described in Appendix \\ref{sec:app-ind}.\n\nIn a laminar disk, the particle drift velocity $v_{d,r}$ is given by equation\n(2.11) of NSH86. For $\\rho_{d}\\ll\\rho_{g}$, the particle radial velocity\nis (see also eq. [23] of Takeuchi \\& Lin (2002); note the factor of 2\ndifference in the definition of $\\eta$\n\\begin{equation}\nv_{d,r}=-\\frac{2{T_{s}}}{{T_{s}^{2}+1}}\\eta v_{\\mathrm{K}}\\ . \n\\label{eq:vdr\n\\end{equation}\nThe gas drifts in the opposite direction with a velocity $v_{g,r}$. From\nangular momentum conservation, the gas drift velocity is\n\\begin{equation}\nv_{g,r}=-\\frac{{\\rho_{d}}}{{\\rho_{g}}}v_{d,r}\\ . \n\\label{eq:vgr\n\\end{equation}\nThe effective gravities (including the pressure gradient force) acting on the\ndust and on the gas are $g_{d}=-GM\/r^{2}=-r\\Omega_{\\mathrm{K}}^{2}$ and\n$g_{g}=-GM(1-2\\eta)\/r^{2}=-r\\Omega_{g}^{2}$, respectively. While the total\nangular momentum is conserved, the total energy is not conserved because of the\ndifference in the effective gravities acting on the gas and on the dust,\n$\\rho_{d}{g_{d}v_{d,r}+\\rho}_{g}g_{g}v_{g,r}{\\neq0}$, providing a source of\nenergy for turbulence.\n\nNote that, in the above calculation, $v_{d,r}$ and $v_{g,r}$ are the\n\\textquotedblleft terminal velocities\\textquotedblright. This means that the\ngravitational accelerations of the dust and gas are balanced by the drag\nforces, and liberated energy translates directly into thermal energy. Thus,\nno energy would be contributed to turbulence. In reality, however, if the gas\ndisk is turbulent, a steady terminal velocity is not expected, and acceleration phases of the dust must occur, as\ndiscussed in \\S \\ref{sec:ene-into-tur}. In an acceleration phase, the work\ndone by gravity first provides kinetic energy, which can be translated\ninto turbulent energy. We estimate this energy input to turbulence. In the\nfollowing calculation, the energy liberated from the accreting dust is\nestimated using the terminal velocity described above for simplicity. Only a\ncertain fraction of the liberated energy goes into turbulence. Thus, the\nenergy input to turbulence is a factor $C_{\\mathrm{eff}}(<1)$ times the\nfollowing estimation. The factor $C_{\\mathrm{eff}}$ will be\ndetermined in \\S \\ref{sec:comp-sim} to be $\\approx0.2$ by comparing our\nmodel with the numerical simulation of turbulence in the dust layer\ndeveloped by JHK06.\n\nGiven that some portion of the accretion energy of the dust is consumed by the outward motion of the gas, the liberated gravitational energy per unit surface area of the disk is\n\\begin{equation}\n\\frac{\\partial E_{\\mathrm{drag}}}{\\partial t}=\\frac{1}{2}\\int_{-\\infty\n}^{\\infty}(\\rho_{d}{g_{d}v_{d,r}+}\\rho_{g}{g_{g}v_{g,r})dz}~,\n\\end{equation}\nwhere the factor of $1\/2$ comes from the fact that half of the work done by\ngravity is used for acceleration (and deceleration) of the azimuthal velocity\nof the dust (and of the gas) as their semi-major axes change. Using equation\n(\\ref{eq:vdr}) and (\\ref{eq:vgr}), the energy liberation rate reduces to\n\\begin{equation}\n\\frac{\\partial E_{\\mathrm{drag}}}{\\partial t}=2\\eta^{2}v_{\\mathrm{K}\n^{2}\\Omega_{\\mathrm{K}}T_{s}\\Sigma_{d,\\mathrm{drag}}\\ , \n\\label{eq:ene_drag\n\\end{equation}\nwhere the effective ``surface density'' $\\Sigma_{d,\\mathrm{drag}}$ of the dust\ni\n\\begin{equation}\n\\Sigma_{d,\\mathrm{drag}}=\\frac{1}{{T_{s}^{2}+1}}\\Sigma_{d}~.\n\\label{eq:sig_drag2\n\\end{equation}\nIn the above calculation, we assume $f_{\\mathrm{mid}}\\ll1$. The calculation of\nthe energy liberation rate for general $f_{\\mathrm{mid}}$ is described in\nAppendix \\ref{sec:app-ind}. In the numerical calculations in the subsequent\nsections, we use equation (\\ref{eq:sig_drag_ap}) for the effective surface\ndensity\n\\footnote{Equation (\\ref{eq:sig_drag}) does not coincide with\nequation (\\ref{eq:sig_drag2}) in the limit of $f_{\\mathrm{mid}}\\ll1$. This\ndiscrepancy comes from the fact that the calculation of equation\n(\\ref{eq:sig_drag}) includes correction terms of\nthe order of $\\eta^{2}$ in the drift velocity, while equation\n(\\ref{eq:sig_drag2}) considers only the terms of $\\eta$. The difference\nbetween equations (\\ref{eq:sig_drag2}) and (\\ref{eq:sig_drag}) is at most\nfactor 2 (for $T_{s}\\gg1)$, and is not significant.},\n\\begin{equation}\n\\Sigma_{d,\\mathrm{drag}}=\\frac{{\\Sigma_{d}}}{\\sqrt{\\pi}}\\int_{-\\infty\n^{\\infty}{\\frac{{\\exp(-\\tilde{z}^{2})}}{{T_{s}^{2}+\\beta^{2}}}\\left[\n1-\\frac{T_{s}^{2}}{2(T_{s}^{2}+\\beta^{2})}\\right]  d\\tilde{z}}\\ .\n\\label{eq:sig_drag\n\\end{equation}\nIf the dust particles are tightly coupled to the gas ($T_{s}\\ll1$) and\ndust sedimentation is weak ($f_{\\mathrm{mid}}\\ll1$), then\n$\\Sigma_{d,\\mathrm{drag}}$ is simply equal to the dust surface density\n$\\Sigma_{d}$.\n\n\\subsection{Dust Accretion Caused by Collective Drag}\n\\label{sec:coldrag}\n\nNext, we consider dust accretion due to collective drag acting on the entire\ndust layer. Because the faster-orbiting dust particles drag the gas in the\ndust layer, the orbital velocity of the gas is largest at the midplane and\ndecreases with altitude. If the gas in the dust layer is turbulent, the\nvariation in orbital velocity $v_{g,\\theta}$ with altitude $z$ induces\nReynolds stress $P_{\\theta z}$. This causes a transfer of angular momentum from the dust\nlayer to the upper gas layer, resulting in accretion of the dust layer.\n\nTo calculate the energy liberation rate, we make a few key\nassumptions. First, we consider only the $\\theta z$-component of the Reynolds\nstress, $P_{\\theta z}$, neglecting the other components $P_{r\\theta}$ and\n$P_{zr}$. Ignoring $P_{r\\theta}$ means that turbulence in the dust layer does\nnot transfer the angular momentum efficiently in the radial direction. This is\nexpected for turbulence induced by hydrodynamical instabilities such as\nconvective instability (e.g., Stone \\& Balbus 1996; Lesur \\& Ogilvie 2010).\nThe Reynolds stress $P_{zr}$ is also neglected for simplicity. Brauer et al.\n(2007) pointed out that $P_{zr}$ changes the velocity profiles of the dust and\ngas from those derived by NSH86 by a factor $\\sim3$, and thus $P_{zr}$\ncannot be neglected in a rigorous discussion.\nObtaining the exact velocity profiles including $P_{zr}$ requires\nnumerical calculations. In this paper, we use the velocity profiles calculated\nanalytically by NSH86 for simplicity. This induces an error of a factor\n$\\sim3$ in our estimation of turbulence strength. Thus, our estimate is\nlimited to an order-of-magnitude argument.\n\nThe second assumption is that the turbulent layer has a thickness\ncomparable to that of the dust layer. If the turbulent layer were much thicker\nthan the dust layer and most of the volume of the turbulent layer were free of\nthe dust, then its structure would be controlled by the gas, \nunaffected by the properties of the dust and the structure of the dust layer.\nThe dust layer would behave just like a boundary wall at the bottom of\nthe turbulent layer. Turbulence in such a thick boundary gas layer has\nbeen discussed using the analogy of the Ekman layer (e.g., Cuzzi et al. 1993). \nIf the turbulent layer were dominated by the\ndust layer, then the structure of the dust layer would control the\nturbulence strength. We focus on such a dusty turbulent layer. Following\nYoudin \\& Chiang (2004), we consider the conditions needed for the\nturbulent and dust layers to be of similar thickness. The thickness of\nthe turbulent layer, from dimensional analysis, is the Ekman length,\n\\begin{equation}\nh_{E}\\sim\\sqrt{\\frac{\\nu}{\\Omega_{\\mathrm{K}}}}~, \n\\label{eq:he\n\\end{equation}\nwhere $\\nu$ is the turbulent viscosity, provided that the viscosity and the\nCoriolis force determine the layer structure. The thickness\nof the dust layer (eq. [\\ref{eq:hd}]), which is determined by the balance\nbetween sedimentation and diffusion of the dust particles, is\n\\begin{equation}\nh_{d}\\sim\\sqrt{\\frac{\\nu}{T_{s}\\Omega_{\\mathrm{K}}}}~, \n\\label{eq:hd2\n\\end{equation}\nwhere $\\nu \\sim D_g$ is used.\nFor small particles ($T_{s}\\la1$), $h_{d}$ is larger than $h_{E}$, meaning\nthat such small particles move further out of the turbulent layer and\nmodify the structure of the turbulent layer. It is expected that the thickness\nof the turbulent layer is not given by $h_{E}$, but by $h_{d}$ (see also S98;\nGoodman \\& Pindor 2000). For large particles ($T_{s}\\gg1$), the turbulent\nlayer is much thicker than the dust layer, and its thickness is expected to be\n$h_{E}$. In the following discussion, we consider a dusty turbulent\nlayer with a thickness similar to $h_{d}$. Thus, the analysis in this\npaper is probably not appropriate for large particles ($T_{s}\\gg1$).\n\nUnder the above assumptions, the energy liberation rate is estimated. Because\nthe collective drag is effective only if the midplane dust-to-gas ratio is\nlarger than unity (see Fig. \\ref{fig:Sigma} below), we consider the case in\nwhich a dense dust layer has formed ($f_{\\mathrm{mid}}\\ga1$) and use the\nplate drag approximation (Goldreich \\& Ward 1973; Goodman \\& Pindor 2000;\nWeidenschilling 2003). The calculation for general $f_{\\mathrm{mid}}$ is\ndescribed in Appendix \\ref{sec:app-col}. The Reynolds stress $P_{\\theta z}$\nnear the boundary between the dust layer (or the turbulent layer) and the gas\nlayer is estimated as\n\\begin{equation}\nP_{\\theta z}=\\rho_{g}\\nu\\frac{{\\partial v_{g,\\theta}}}{{\\partial z}}\\sim\n-\\rho_{g}\\nu\\frac{\\eta v_{\\mathrm{K}}}{h_{d}}~. \n\\label{eq:Ptz\n\\end{equation}\nThis stress extracts angular momentum from the dust layer and transfers it to\nthe gas layer. The unit surface of the dust layer loses angular momentum $\\partial\nL_{d}\/\\partial t=rP_{\\theta z}$, and the corresponding energy change is $\\partial\nE_{d}\/\\partial t=\\Omega_{\\mathrm{K}}\\partial L_{d}\/\\partial t$. The gas layer\ngains the same amount of angular momentum $\\partial L_{g}\/\\partial\nt=-rP_{\\theta z}$, but the energy change is different from that of the\ndust layer because of the work done by the pressure gradient: $\\partial\nE_{g}\/\\partial t=\\Omega_{g}\\partial L_{g}\/\\partial\nt=(1-\\eta)\\Omega_{\\mathrm{K}}\\partial\nL_{g}\/\\partial t$. In total, the energy liberation rate (the minus sign is\nadded), using equations (\\ref{eq:rhog}),(\\ref{eq:hd}),(\\ref{eq:fdmid}), and\n(\\ref{eq:Ptz}), is\n\\begin{equation}\n\\frac{\\partial E_{\\mathrm{vis}}}{\\partial t}=-(\\Omega_{\\mathrm{K}}-\\Omega\n_{g})rP_{\\theta z}=2\\eta^{2}v_{\\mathrm{K}}{}^{2}\\Omega_{\\mathrm{K}}T_{s\n\\Sigma_{d,\\mathrm{vis}}~, \n\\label{eq:ene_vis\n\\end{equation}\nwhere\n\\begin{equation}\n\\Sigma_{d,\\mathrm{vis}}=\\frac{1}{2\\sqrt{2\\pi}}\\frac{1+2T_{s}}{1+T_{s}}\\frac\n{1}{f_{\\mathrm{mid}}}\\Sigma_{d}~. \n\\label{eq:sig_vis-approx\n\\end{equation}\nIn the above calculation, the viscosity coefficient is modeled as\n$\\nu=\\alpha c_g h_g$. The same $\\alpha$ is used for both $D_g$ and\n$\\nu$ for simplicity. \nThe effective surface density $\\Sigma_{d,\\mathrm{vis}}$ is inversely\nproportional to the midplane dust-to-gas ratio $f_{\\mathrm{mid}}$ and only\nweakly depends on $T_{s}$. Note that the above relationship is derived for\n$f_{\\mathrm{mid}}\\ga1$. The effective surface density for general\n$f_{\\mathrm{mid}}$ is derived in Appendix \\ref{sec:app-col} and is given by\n\\begin{eqnarray}\n\\Sigma_{d,\\mathrm{vis}} &=&\n\\frac{C_{\\mathrm{str}}}{\\sqrt{{\\pi}}}f_{\\mathrm{mid}}\\Sigma_{d}\\frac{{1+2T}_{s}}{1+{T_{s}}}\n\\nonumber \\\\\n& \\times & \\int_{-\\infty}^{\\infty}{\\frac\n{{\\beta^{2}+2\\beta T_{s}^{2}-T_{s}^{2}}}{{(T_{s}^{2}+\\beta^{2})^{2}}\n\\frac{{\\tilde{z}^{2}\\exp(-2\\tilde{z}^{2})}}{{[1+C_{\\mathrm{str}\nf_{\\mathrm{mid}}\\exp(-\\tilde{z}^{2})]}}d\\tilde{z}}\\ , \\nonumber \\\\\n\\label{eq:sig_vis\n\\end{eqnarray}\nwhere\n\\begin{equation}\n{C_{\\mathrm{str}}=}\\frac{1}{T_{s}^{2}+1}~.\n\\end{equation}\nExpression (\\ref{eq:sig_vis}) is complicated, but for large $f_{\\mathrm{mid}}$, its dependence on $T_{s}$ and $f_{\\mathrm{mid}}$ is similar to that of\nthe simpler equation (\\ref{eq:sig_vis-approx}); $\\Sigma_{d,\\mathrm{vis}}\\propto\nf_{\\mathrm{mid}}^{-0.9}$ and depends only weakly on $T_{s}$.\n\n\\subsection{Energy Dissipation in Turbulence}\n\nThe turbulent energy of the largest eddies transfers to smaller eddies,\nand finally dissipates to thermal energy via decay of the smallest eddies due\nto molecular viscosity. The size and velocity of the largest eddies are\nassumed to be $l_{g,\\mathrm{eddy}}=\\alpha^{1\/2}h_{g}$ and $u_{g,\\mathrm{eddy\n}=\\alpha^{1\/2}c_{s}$. This assumption means that the turnover time of the\nlargest eddies is the Keplerian time ($\\tau_{g,\\mathrm{eddy}\n=l_{g,\\mathrm{eddy}}\/u_{g,\\mathrm{eddy}}=\\Omega_{\\mathrm{K}}^{-1}$; Cuzzi et\nal. 2001). Using the Kolmogorov scaling law, the energy dissipation rate per\nunit volume and unit time is\n\\begin{equation}\n\\frac{\\partial\\varepsilon_{\\mathrm{turb}}}{\\partial t}=({\\rho_{g\n+C_{\\mathrm{ene}}\\rho_{d})}\\frac{{u_{g,\\mathrm{eddy}}^{2}}}{\\tau\n{_{g,\\mathrm{eddy}}}}=({\\rho_{g}+C_{\\mathrm{ene}}\\rho_{d})}\\alpha h_{g\n^{2}\\Omega_{\\mathrm{K}}^{3}\\ ,\n\\end{equation}\nwhere the factor $C_{\\mathrm{ene}}$ represents the fact that dust\nparticles that are coupled only weakly to the gas do not contribute to\nthe turbulent energy, and is given by (see Appendix \\ref{sec:app-ene-dis})\n\\begin{equation}\nC_{\\mathrm{ene}}=\\left\\{\n\\begin{array}\n[c]{ccc\n\\frac{1}{T_{s}T_{e}+1} & \\mathrm{for} & T_{s}\\,\\leq T_{e}\\\\\n\\frac{1}{T_{s}^{2}(T_{e}^{-2}+1)} & \\mathrm{for} & T_{s}\\,>T_{e\n\\end{array}\n\\right.  \\ ,\n\\end{equation}\nwhere $T_{e}=\\tau_{g,\\mathrm{eddy}}\\Omega_{\\mathrm{K}}$ is the non-dimensional\nturnover time of the largest eddies. For most of this paper, we\nconsider $T_{e}=1$. The effect of the dust inertia is ignored for simplicity.\n\nEnergy dissipation of turbulence occurs only in the dust layer. This\nassumption may be problematic for $T_{s}\\gg1$, as discussed in\n\\S \\ref{sec:coldrag}, but for simplicity, we assume that energy dissipation\noccurs in $-\\sqrt{2}h_{d}<z<\\sqrt{2}h_{d}$. The dissipation rate of the \nenergy per unit surface area and unit time is\n\\begin{equation}\n\\frac{\\partial E_{\\mathrm{turb}}}{\\partial t}=\\alpha h_{g}^{2}\\Omega\n_{\\mathrm{K}}^{3}\\int_{-\\sqrt{2}h_{d}}^{\\sqrt{2}h_{d}}({{\\rho_{g\n+C_{\\mathrm{ene}}\\rho_{d})}dz}=\\alpha h_{g}^{2}\\Omega_{\\mathrm{K}}^{3}\\Sigma\n_{d,\\mathrm{turb}}\\ , \n\\label{eq:ene_turb\n\\end{equation}\nwhere\n\\begin{equation}\n\\Sigma_{d,\\mathrm{turb}}=\\Sigma_{d}\\left[  {\\frac{2}{{\\sqrt{\\pi\nf_{\\mathrm{mid}}}}+C_{\\mathrm{ene}}\\mathrm{erf}(1)}\\right]  \\ ,\n\\label{eq:sig_turb\n\\end{equation}\nand $\\mathrm{erf}(1)=0.8427$.\n\n\\section{TURBULENCE STRENGTH}\n\\label{sec:turbstr}\n\nThe strength of turbulence or the parameter $\\alpha$ is determined by the balance\nbetween the energy supply rate ($\\partial E_{g,\\mathrm{drag}}\/\\partial t$ in\neq. [\\ref{eq:ene_drag}] and $\\partial E_{\\mathrm{vis}}\/\\partial t$ in eq.\n[\\ref{eq:ene_vis}]) and the energy dissipation rate ($\\partial\nE_{\\mathrm{turb}}\/\\partial t$ in eq. [\\ref{eq:ene_turb}]). In steady\nturbulence,\n\\begin{equation}\n\\frac{\\partial E_{\\mathrm{turb}}}{\\partial t}=C_{\\mathrm{eff}}\\left(\n\\frac{\\partial E_{g,\\mathrm{drag}}}{\\partial t}+\\frac{\\partial E_{\\mathrm{vis\n}}{\\partial t}\\right)  \\ ,\n\\end{equation}\nwhere the efficiency factor $C_{\\mathrm{eff}}$ represents the fraction of the\nreleased gravitational energy that is transferred to turbulence. Comparison\nwith the numerical simulation by JHK06 suggests that $C_{\\mathrm{eff}}=0.19$\n(see \\S \\ref{sec:comp-sim} below), and we adopt this value in this\npaper. From equations (\\ref{eq:etawave}), \n(\\ref{eq:ene_drag}), (\\ref{eq:ene_vis}), and (\\ref{eq:ene_turb}), the viscosity parameter is\n\\begin{equation}\n\\alpha=2C_{\\mathrm{eff}}\\tilde{\\eta}T_{s}\\frac{{\\Sigma_{d,\\mathrm{drag\n}+\\Sigma_{d,\\mathrm{vis}}}}{{\\Sigma_{d,\\mathrm{turb}}}}. \\label{eq:alpha\n\\end{equation}\nNote that the right-hand-side of equation (\\ref{eq:alpha}) is a function\nof $\\alpha$ through $f_{\\mathrm{mid}}$ or $\\beta$ in the ``surface\ndensities'' (see eqs. [\\ref{eq:sig_drag}], [\\ref{eq:sig_vis-approx}],\n[\\ref{eq:sig_vis}], [\\ref{eq:sig_turb}]). Before obtaining the exact\nsolution for $\\alpha$ numerically, in the following two subsections,\napproximate solutions are derived.\n\n\\subsection{Turbulence Strength in the Small-Particle Limit ($T_{s}\\ll\n1$)}\n\\label{sec:turbstr-ts}\n\nIn equation (\\ref{eq:alpha}), the \\textquotedblleft surface\ndensities\\textquotedblright, $\\Sigma_{d,\\mathrm{drag}}$, $\\Sigma\n_{d,\\mathrm{vis}}$, and $\\Sigma_{d,\\mathrm{turb}}$ represent the effective\ndensity of the dust that contributes to the liberation of the gravitational energy\nor dissipation in the turbulence. The \\textquotedblleft surface\ndensities\\textquotedblright\\ are functions of the stopping time $T_{s}$ and\nthe midplane dust-to-gas ratio $f_{\\mathrm{mid}}$. For small dust particles\n($T_{s}\\ll1$), the \\textquotedblleft surface densities\\textquotedblright\\ in\nequations (\\ref{eq:sig_drag}), (\\ref{eq:sig_vis}), (or approximated eqs.\n[\\ref{eq:sig_drag2}], [\\ref{eq:sig_vis-approx}]), and (\\ref{eq:sig_turb})\ndepend on $T_{s}$ only through $f_{\\mathrm{mid}}\\approx Z_{\\mathrm{disk}}(T_{s}\/\\alpha)^{1\/2}$. Figure \\ref{fig:Sigma} shows how the \\textquotedblleft\nsurface densities\\textquotedblright\\ vary with $f_{\\mathrm{mid}}$ for small\nparticles ($T_{s}\\ll1$). When the midplane dust-to-gas ratio is much smaller\nthan unity, the liberation of the gravitational energy mainly comes from the\ndust accreting due to individual drag, while for $f_{\\mathrm{mid}}\\gg1$, the\ncollective drag dominates the energy liberation. For the energy dissipated in\nturbulence, if $f_{\\mathrm{mid}}\\ll1$, the \\textquotedblleft surface\ndensity\\textquotedblright\\ $\\Sigma_{d,\\mathrm{turb}}$ is the column density of\nthe gas in the dust layer and is inversely proportional to $f_{\\mathrm{mid}}$. If the dust layer is thin enough that $f_{\\mathrm{mid}}\\gg1$, then\n$\\Sigma_{d,\\mathrm{turb}}$ is simply given by the dust surface density\n$\\Sigma_{d}$.\n\n\\begin{figure}[ptb]\n\\epsscale{1.1} \\plotone{f1.eps}\\caption{ ``Surface densities'' $\\Sigma\n_{d,\\mathrm{drag}}$, $\\Sigma_{d,\\mathrm{vis}}$, and $\\Sigma_{d,\\mathrm{turb}}$\nin the small particle limit $T_{s} \\ll1$. The ``surface densities'' are\nnormalized by the dust surface density $\\Sigma_{d}$, and are plotted as\nfunctions of the midplane dust-to-gas ratio, $f_{\\mathrm{mid}}$.\n\\label{fig:Sigma\n\\end{figure}\n\nConsider how the turbulent parameter $\\alpha$ given by equation\n(\\ref{eq:alpha}) depends on the stopping time $T_{s}$ and the disk metallicity $Z_{\\mathrm{disk}}$. In the following discussion, we assume small particles, such that $T_{s}\\ll1$. As discussed\nabove, for $T_{s}\\ll1$, the \\textquotedblleft surface\ndensities\\textquotedblright\\ depend on $T_{s}$ only through $f_{\\mathrm{mid}}\\approx Z_{\\mathrm{disk}}(T_{s}\/\\alpha)^{1\/2}$, and consequently equation\n(\\ref{eq:alpha}) determines the value of $\\alpha\/T_{s}$. Thus, in the limit of\n$T_{s}\\ll1$, it is seen that $\\alpha\\propto T_{s}$. We derive an approximate\nexpression of $\\alpha$ for the small and large limits of $f_{\\mathrm{mid}}$. For\n$f_{\\mathrm{mid}}\\ll1$, Figure \\ref{fig:Sigma} (and eqs. [\\ref{eq:sig_drag2\n], [\\ref{eq:sig_vis}], and [\\ref{eq:sig_turb}]) show that ${\\Sigma\n_{d,\\mathrm{drag}}}={\\Sigma_{d}}$, $\\Sigma_{d,\\mathrm{vis}} \\sim\nf_{\\mathrm{mid}}\\Sigma_{d,\\mathrm{drag}}\\ll{\\Sigma_{d,\\mathrm{drag}}}$, \nand $\\Sigma_{d,\\mathrm{turb}}=2\\Sigma_{d}\/(\\sqrt{\\pi}f_{\\mathrm{mid}})$. Thus,\nthe ratio of the ``surface densities'' in equation \n(\\ref{eq:alpha}) reduces to $({\\Sigma_{d,\\mathrm{drag}}+\\Sigma_{d,\\mathrm{vis\n})\/\\Sigma_{d,\\mathrm{turb}}=}\\sqrt{\\pi}f_{\\mathrm{mid}}\/2$. For\n$f_{\\mathrm{mid}}\\gg1$, though $\\Sigma_{d,\\mathrm{drag}}$ is smaller than\n$\\Sigma_{d,\\mathrm{vis}}$, it still makes a contribution to the energy\nliberation. We fit the functional form of $({\\Sigma_{d,\\mathrm{drag}\n+\\Sigma_{d,\\mathrm{vis}})\/\\Sigma_{d,\\mathrm{turb}}}$ for $10^{2\n<f_{\\mathrm{mid}}<10^{4}$ by a power law form, $({\\Sigma_{d,\\mathrm{drag\n}+\\Sigma_{d,\\mathrm{vis}})\/\\Sigma_{d,\\mathrm{turb}}}\\approx\\tilde{\\Sigma\n}_{d,0}f_{\\mathrm{mid}}^{-\\delta}$, where $\\delta=0.94$, and $\\tilde{\\Sigma\n}_{d,0}=0.71$. Using the above expressions, equation (\\ref{eq:alpha}) become\n\\begin{equation}\n\\alpha=\\left\\{\n\\begin{array}\n[c]{ccc\n\\left(  \\pi^{1\/2}C_{\\mathrm{eff}}\\tilde{\\eta}Z_{\\mathrm{disk}}\\right)\n^{\\frac{2}{3}}T_{s} & \\mathrm{for} & Z_{\\mathrm{disk}}\\ll\\sqrt{C_{\\mathrm{eff\n}\\tilde{\\eta}}\\\\\n\\left(  2\\tilde{\\Sigma}_{0}C_{\\mathrm{eff}}\\tilde{\\eta}\nZ_{\\mathrm{disk}}^{-\\delta} \\right)  ^{\\frac{2}{2-\\delta}}T_{s} & \\mathrm{for} &\nZ_{\\mathrm{disk}}\\gg\\sqrt{C_{\\mathrm{eff}}\\tilde{\\eta}\n\\end{array}\n\\right.  , \n\\label{eq:alpha1\n\\end{equation}\nand the midplane dust-to-gas ratio (eq.[\\ref{eq:fdmid}]) is\n\\begin{equation}\n{f_{\\mathrm{mid}}}=\\left\\{\n\\begin{array}\n[c]{ccc\n\\left(  \\frac{Z_{\\mathrm{disk}}^{2}}{\\pi^{1\/2}C_{\\mathrm{eff}}\\tilde{\\eta}}\\right)\n^{\\frac{1}{3}} & \\mathrm{for} & Z_{\\mathrm{disk}}\\ll\\sqrt{C_{\\mathrm{eff\n}\\tilde{\\eta}}\\\\\n\\left(  \\frac{Z_{\\mathrm{disk}}^{2}}{2\\tilde{\\Sigma}_{0}C_{\\mathrm{eff}}\\tilde{\\eta}}\\right)\n^{\\frac{1}{2-\\delta}} & \\mathrm{for} & Z_{\\mathrm{disk}}\\gg\\sqrt\n{C_{\\mathrm{eff}}\\tilde{\\eta}\n\\end{array}\n\\right.  \\ . \n\\label{eq:fdmid1\n\\end{equation}\nThe scale height of the dust layer i\n\\begin{equation}\nh_{d}=\\left\\{\n\\begin{array}\n[c]{ccc\n\\left(  \\pi^{1\/2}C_{\\mathrm{eff}}\\tilde{\\eta}Z_{\\mathrm{disk}}\\right)\n^{\\frac{1}{3}}h_{g} & \\mathrm{for} & Z_{\\mathrm{disk}}\\ll\\sqrt{C_{\\mathrm{eff\n}\\tilde{\\eta}}\\\\\n\\left( 2\\tilde{\\Sigma}_{0}C_{\\mathrm{eff}}\\tilde{\\eta}Z_{\\mathrm{disk}}^{-\\delta}\n\\right)^{\\frac{1}{2-\\delta}} h_{g} & \\mathrm{for} &\nZ_{\\mathrm{disk}}\\gg\\sqrt{C_{\\mathrm{eff}}\\tilde{\\eta}\n\\end{array}\n\\right.  ~. \n\\label{eq:hd3\n\\end{equation}\nIn the above approximate expressions (\\ref{eq:alpha1})-(\\ref{eq:hd3}), the\nupper line represents $f_{\\mathrm{mid}}\\ll1$ and the lower line represents\n$f_{\\mathrm{mid}}\\gg1$. The transition of these expressions occurs at\n$f_{\\mathrm{mid}}\\sim1$, which corresponds to $Z_{\\mathrm{disk}}\\sim\n\\sqrt{C_{\\mathrm{eff}}\\tilde{\\eta}}$, and a maximum value of $\\alpha$,\n\\begin{equation}\n\\alpha_{\\max}\\sim C_{\\mathrm{eff}}{\\tilde{\\eta}}T_{s}\\ . \n\\label{eq:alpha2\n\\end{equation}\n\n\\begin{figure}[ptb]\n\\epsscale{1.1} \\plotone{f2.eps}\\caption{The turbulent viscosity parameter\n$\\alpha$ for various values of the stopping time $T_{s}$ plotted as a\nfunction of the disk metallicity $Z_{\\mathrm{disk}}$. The solid lines\nshow $\\alpha$ derived from our model. The dashed lines show $\\alpha$ from the S98\nmodel. We adopt an efficiency parameter of the energy supply,\n$C_{\\mathrm{eff}}=0.19$, for our model, and the critical Richardson\nnumber $\\mathrm{Ri}=0.8$ for the S98 model.}\n\\label{fig:alpha\n\\end{figure}\n\n\\begin{figure}[ptb]\n\\epsscale{1.1} \\plotone{f3.eps}\\caption{The midplane dust-to-gas ratio\n$f_{\\mathrm{mid}}$. The solid lines are calculated for various values of the\nstopping time $T_{s}$ using our model. The dashed line is taken from the S98 model.\n\\label{fig:fdmid\n\\end{figure}\n\n\n\\subsection{Turbulence Strength for a Small Dust-to-Gas\nRatio($f_{\\mathrm{mid}}\\ll1$)}\n\\label{sec:turbstr-fm}\n\nWe derive the approximate expression of the turbulent\nstrength $\\alpha$ as a function of $T_{s}$ and $Z_{\\mathrm{disk}}$ in the\nlimit of small dust-to-gas ratio at the midplane, $f_{\\mathrm{mid}}\\ll1$,\nbut without assuming $T_{s}\\ll1$. For $f_{\\mathrm{mid}}\\ll1$, the ``surface\ndensities'' (eqs. [\\ref{eq:sig_drag}], [\\ref{eq:sig_vis}], and\n[\\ref{eq:sig_turb}]) reduce to $\\Sigma_{d,\\mathrm{drag}}=\\frac{1}{2}\\Sigma\n_{d}(T_{s}^{2}+2)\/(T_{s}^{2}+1)^{2}$, $\\Sigma_{d,\\mathrm{vis}}\\sim\nf_{\\mathrm{mid}}\\Sigma_{d,\\mathrm{drag}}\\ll\\Sigma_{d,\\mathrm{drag}}$, and\n$\\Sigma_{d,\\mathrm{turb}}=2\\Sigma_{d}\/(\\sqrt{\\pi}f_{\\mathrm{mid}})$. Thus,\nthe approximate expression of equation (\\ref{eq:alpha}) become\n\\begin{equation}\n\\alpha=\\left[  \\frac{\\sqrt{\\pi}}{2}C_{\\mathrm{eff}}\\tilde{\\eta\nZ_{\\mathrm{disk}}\\sqrt{\\frac{1+2T_{s}}{1+T_{s}}}\\frac{T_{s}^{2}+2}{\\left(\nT_{s}^{2}+1\\right)  ^{2}}\\right]  ^{\\frac{2}{3}}T_{s}~. \n\\label{eq:alpha3\n\\end{equation}\nThe midplane dust-to-gas ratio and the scale height of the dust layer are,\nrespectively\n\\begin{equation}\nf_{\\mathrm{mid}}=\\left[  \\frac{2Z_{\\mathrm{disk}}^{2}}{\\sqrt{\\pi\n}C_{\\mathrm{eff}}\\tilde{\\eta}}\\frac{1+2T_{s}}{1+T_{s}}\\frac{\\left(  T_{s\n^{2}+1\\right)  ^{2}}{T_{s}^{2}+2}\\right]  ^{\\frac{1}{3}}~,\n\\end{equation}\nan\n\\begin{equation}\nh_{d}=\\left[  \\frac{\\sqrt{\\pi}}{2}C_{\\mathrm{eff}}\\tilde{\\eta\nZ_{\\mathrm{disk}}\\frac{1+T_{s}}{1+2T_{s}}\\frac{T_{s}^{2}+2}{\\left(  T_{s\n^{2}+1\\right)  ^{2}}\\right]  ^{\\frac{1}{3}}h_{g}~.\n\\end{equation}\n\n\n\\subsection{Turbulence Strength and Dust Layer Thickness}\n\nThe properties of the turbulent viscosity parameter $\\alpha$ for $T_{s}\\ll1$ or\n$f_{\\mathrm{mid}}\\ll1$ are described in the last two subsections. For general\n$T_{s}$ and $f_{\\mathrm{mid}}$, equation (\\ref{eq:alpha}) must be\nsolved numerically. We use the van Wijngaarden-Dekker-Brent method (Press et\nal. 1986) to obtain $\\alpha$ from equation (\\ref{eq:alpha}). In Figure\n\\ref{fig:alpha}, the solid lines represent the variation of $\\alpha$ with the disk metallicity $Z_{\\mathrm{disk}}$ for various values of the particle\nstopping time $T_{s}$. It is seen that, for small dust particles ($T_{s\n\\la1$), $\\alpha$ is proportional to $T_{s}$, as discussed in\n\\S \\ref{sec:turbstr-ts}. The maximum $\\alpha_{\\mathrm{max}}\\sim\nC_{\\mathrm{eff}}\\tilde{\\eta}T_{s}$ appears at $Z_{\\mathrm{disk}}\\sim\n\\sqrt{C_{\\mathrm{eff}}\\tilde{\\eta}}\\sim10^{-2}$ (and $f_{\\mathrm{mid}}\\sim\n1$) for $T_{s}\\la1$. The turbulence weakens as $Z_{\\mathrm{disk}}$\ndeviates from $\\sqrt{C_{\\mathrm{eff}}\\tilde{\\eta}}$ ($\\alpha\\propto\nZ_{\\mathrm{disk}}^{2\/3}$ for $Z_{\\mathrm{disk}}\\la\\sqrt{C_{\\mathrm{eff\n}\\tilde{\\eta}}$, and $\\alpha\\propto Z_{\\mathrm{disk}}^{-1.8}$ for\n$Z_{\\mathrm{disk}}\\ga\\sqrt{C_{\\mathrm{eff}}\\tilde{\\eta}}$, see eq.\n[\\ref{eq:alpha1}]). For particles of $T_{s}>1$, $\\alpha$ peaks\nat a larger $Z_{\\mathrm{disk}}$ (or $f_{\\mathrm{mid}}$).\n\nFigure \\ref{fig:fdmid} shows the variation in the midplane dust-to-gas ratio\n$f_{\\mathrm{mid}}$ with the disk metallicity $Z_{\\mathrm{disk}}$. For small particles ($T_{s}\\la1$), $\\alpha$ is\nproportional to $T_{s}$, and the midplane dust-to-gas ratio\n$f_{\\mathrm{mid}}\\propto(T_{s}\/\\alpha)^{1\/2}$ does not depend on\n$T_{s}$. For a small dust-to-gas ratio ($f_{\\mathrm{mid}}\\ll1$),\n$f_{\\mathrm{mid}}\\propto Z_{\\mathrm{disk}}^{2\/3}$, and for $f_{\\mathrm{mid}}\\gg1$,\n$f_{\\mathrm{mid}}\\propto Z_{\\mathrm{disk}}^{1.9}$. Note that if the disk metallicity is larger than $0.2$, then the midplane dust-to-gas ratio\nbecomes as large as $100$, which is high enough to induce a gravitational\ninstability in the dust layer. This result is consistent with the previous\nworks by, e.g., S98. Figure \\ref{fig:hd} shows that the dust layer thickness $h_{d}$\nis of the order of $10^{-3}-10^{-2}$ times the thickness of the gas disk.\n\n\\begin{figure}[ptb]\n\\epsscale{1.1} \\plotone{f4.eps}\\caption{The scale height of the dust layer\n$h_{d}$ normalized by the gas scale height $h_g$. The solid lines are\n  calculated for various values of the stopping time $T_{s}$ by our\n  model. The dashed line is taken from the S98 model, which is\n  calculated using equation (\\ref{eq:hd-sek}).}\n\\label{fig:hd\n\\end{figure}\n\n\\section{COMPARISONS WITH PREVIOUS STUDIES}\n\\label{sec:comppre}\n\nIn this section, the dust layer model that is based on the energetics of\ndust accretion is compared with results from previous work. Previous studies \nhave analyzed the detailed physics of the dust layer, including the\nonset of KH or streaming instabilities, and several stabilizing effects, such as\nKeplerian shear, using both analytical and numerical approaches. It is of\ninterest to determine which properties of the dust layer are reproduced by our\nmodel and which are missing.\n\n\\subsection{Comparison with Previous Analytical Studies}\n\n\\subsubsection{Comparison with Sekiya (1998)}\n\nS98 analytically solved the density structure of the dust layer, assuming that\nthe layer structure was adjusted to be marginally unstable to KH instability.\nTo obtain the density structure, S98 assumed that the Richardson number remained \nclose to the critical value for instability throughout the dust\nlayer. S98 considered small dust particles that were tightly coupled to the gas \n($T_{s}\\ll1$), and treated the gas and the dust as a single fluid. Then, the\ndensity structure of the dust layer was calculated for various values of the\ndisk metallicity $Z_{\\mathrm{disk}}$. In S98, the dust layer\nstructure was determined by the argument for the stability of a\nstratified fluid, not \nby the balance between sedimentation and diffusion of the dust. However, it is\npossible to interpret the result of S98 as follows: The turbulent\nstrength in the dust layer is adjusted such that the turbulent diffusion of\nthe dust maintains the marginally unstable density structure. Using this\ninterpretation, we calculate the effective value of the turbulent diffusion\nparameter $\\alpha$ as a function of $Z_{\\mathrm{disk}}$ for $T_{s}\\ll1$.\n\nFrom equation (22) of S98, the midplane dust-to-gas ratio,\n$f_{\\mathrm{mid}}$, is related to the disk metallicity,\n$Z_{\\mathrm{disk}}$, as\n\\begin{eqnarray}\n{Z_{\\mathrm{disk}}} &=& \\sqrt{\\frac{{2\\mathrm{{Ri}}}\\tilde{\\eta}}{\\pi}} \\left\\{\n\\ln\\left[  {\\left(  {1+\\sqrt{1-{{\\left(  {\\frac{1}{{1+{f_{\\mathrm{mid}}}}}\n}\\right)  }^{2}}}}\\right)  \\left(  {1+{f_{\\mathrm{mid}}}}\\right)  }\\right]\n\\right. \\nonumber \\\\\n& &\\left. -\\sqrt{1-{{\\left(  {\\frac{1}{{1+{f_{\\mathrm{mid}}}}}}\\right)  }^{2}}}\n\\right\\} \\ , \n\\label{eq:fd-sek\n\\end{eqnarray}\nwhere the Richardson number $\\mathrm{{Ri}}$ is constant\nthroughout the dust layer, and the self-gravity of the dust layer is neglected.\nThe dust density distribution is given by\n\\begin{equation}\n\\frac{{{\\rho_{d}}}}{{{\\rho_{g}}}}={\\left[  {\\frac{{{z^{2}}}}{{\\mathrm{{Ri\n}{\\tilde{\\eta}h}}_{g}^{2}}+{{\\left(  {\\frac{1}{{1+{f_{\\mathrm{mid}}}}\n}\\right)  }^{2}}}\\right]  ^{-\\frac{1}{2}}}-1\\ . \n\\label{eq:dendist-sek\n\\end{equation}\nThe half-thickness of the dust layer, $z_{d}$, at which the dust density\nbecomes zero, is given by\n\\begin{equation}\nz_{d}=\\sqrt{\\mathrm{{Ri}}\\tilde{\\eta}\\left[  {1-(1+f_{\\mathrm{mid}})\n^{-2}\\right]  }\\mathrm{\\ }h_{g}~. \n\\label{eq:zd\n\\end{equation}\nTo compare with the scale height of Gaussian distribution, $h_{d}$, we define\nthe scale height of the dust distribution as the vertical dispersion of the\ndust particles,\n\\begin{equation}\nh_{d,\\mathrm{{Sek}}}^{2}=\\frac{\\int_{0}^{z_{d}}z^{2}\\rho_{d}dz}{\\int\n_{0}^{z_{d}}\\rho_{d}dz}\\ . \n\\label{eq:hd-sek\n\\end{equation}\nThe turbulent parameter $\\alpha_{\\mathrm{{Sek}}}$ is estimated from equation\n(\\ref{eq:hd}), using $T_{s}\\ll1$, as\n\\begin{equation}\n\\alpha_{\\mathrm{{Sek}}}=\\left(  \\frac{h_{d,\\mathrm{{Sek}}}}{h_{g}}\\right)\n^{2}T_{s}\\ . \n\\label{eq:alpha-sek\n\\end{equation}\nFor $f_{\\mathrm{mid}}\\ga1$, equation (\\ref{eq:zd}) reduces to $z_{d\n\\approx(\\mathrm{{Ri}}\\tilde{\\eta})^{1\/2}h_{g}$. Substituting this expression\ninto $h_{d,\\mathrm{Sek}}\\approx z_{d}$ of equation (\\ref{eq:alpha-sek}) gives\n\\begin{equation}\n\\alpha_{\\mathrm{{Sek}}}\\approx\\mathrm{{Ri}}{\\tilde{\\eta}T_{s}\n\\ \\ \\ \\mathrm{for}\\ \\ \\ f_{\\mathrm{mid}}\\ga1\\ , \n\\label{eq:alpha-sek2\n\\end{equation}\nwhich can be compared with equation (\\ref{eq:alpha2}). For $f_{\\mathrm{mid}}\\ll1$, equation (\\ref{eq:fd-sek}) is approximated in the lowest order of\n$f_{\\mathrm{mid}}$ as\n\\begin{equation}\n{Z_{\\mathrm{disk}}}=\\frac{4}{3}\\sqrt{\\frac{{\\mathrm{{Ri}}}\\tilde{\\eta}}{\\pi\n}\\left[  {f_{\\mathrm{mid}}^{3\/2}+O(f_{\\mathrm{mid}}^{5\/2})}\\right]  .\n\\end{equation}\nThus, the midplane dust-to-gas ratio is\n\\begin{equation}\nf_{\\mathrm{mid}}=\\left(  { \\frac{9}{16}\\frac\n{\\pi}{{\\mathrm{{Ri}}\\tilde{\\eta}}}Z_{\\mathrm{disk}}^{2}}\\right)^{\\frac\n{1}{3}}\\ . \n\\label{eq:fdmid-sek\n\\end{equation}\nFor $f_{\\mathrm{mid}}\\ll1$ and $z \\ll h_g$, the dust density distribution\n(eq.[\\ref{eq:dendist-sek}]) is\n\\begin{equation}\n\\frac{\\rho_{d}}{\\rho_{g}}\n \\approx{f_{\\mathrm{mid}}}\\left[  {1-\\frac{{1}}{{2\\mathrm{{Ri}\n{\\tilde{\\eta}f_{\\mathrm{mid}}}}}\\left(  \\frac{z}{h_{g}}\\right)  }^{2\n \\right]  \\ .\n\\end{equation}\nTo the order of $(z\/h_{d})^{2}$, this distribution is approximated by the\nGaussian distribution $f_{\\mathrm{mid}}\n\\exp[-z^{2}\/(2h_{d,\\mathrm{Sek}}^{2})] \\approx f_{\\mathrm{mid}}\n    [1-z^{2}\/(2h_{d,\\mathrm{Sek}}^{2})]$, and its scale height $h_{d,\\mathrm{Sek}}$ is \n\\begin{equation}\nh_{d,\\mathrm{Sek}}\\approx\\sqrt{\\mathrm{{Ri}}\\tilde{\\eta}{f_{\\mathrm{mid}}\n}h_{g}\\approx{\\left[  {\\frac{{\\mathrm{{3Ri}}}}{4}{\\pi^{1\/2}}{Z_{\\mathrm{disk}}}\\tilde{\\eta}}\\right]  ^{\\frac{1}{3}}}{h_{g}}\\ .\n\\end{equation}\nThe turbulent parameter $\\alpha_{\\mathrm{Sek}}$ of equation\n(\\ref{eq:alpha-sek}) is\n\\begin{equation}\n\\alpha_{\\mathrm{Sek}}\\approx{\\left[  {\\frac{{\\mathrm{{3Ri}}}}{4}{\\pi^{1\/2\n}{Z_{\\mathrm{disk}}}\\tilde{\\eta}}\\right]  ^{\\frac{2}{3}}}{T_{s\n}\\ \\ \\ \\ \\mathrm{for}\\ \\ \\ f_{\\mathrm{mid}}\\ll1\\ . \n\\label{eq:alpha-sek3\n\\end{equation}\nComparing this expression to equation (\\ref{eq:alpha1}) provides a relationship\nbetween the energy supply efficiency to turbulence, $C_{\\mathrm{eff}}$, and\nthe critical Richardson number, $\\mathrm{Ri}$, as ${C_{\\mathrm{eff}\n}=(3\/4)\\mathrm{{Ri}}$. In the above discussion on $\\alpha_{\\mathrm{Sek}}$, we\nused the dust layer thickness $h_{d,\\mathrm{Sek}}$ derived from the Gaussian\nfit. However, the thickness defined by the vertical dispersion\n(eq.[\\ref{eq:hd-sek}]) in the small $f_{\\mathrm{mid}}$ limit is $0.63$ times\nthinner than that defined by the Gaussian fit. Thus, the energy supply\nefficiency is $0.4=0.63^{2}$ times smaller than the above estimate and thus\n$C_{\\mathrm{eff}}\\approx0.3\\mathrm{{Ri}}$. The relationship between\n$C_{\\mathrm{eff}}$ and $\\mathrm{Ri}$ can also be derived by comparing the\nexpressions for $f_{\\mathrm{mid}}$ (eqs. [\\ref{eq:fdmid1}] and\n[\\ref{eq:fdmid-sek}]) as ${C_{\\mathrm{eff}}}=16\/(9\\pi^{3\/2})\\mathrm{{Ri\n}\\approx0.3\\mathrm{{Ri}}$. We compare our result with the numerical simulation\ndeveloped by JHK06, which will be discussed in \\S \\ref{sec:comp-sim}. The comparison\nwith this simulation suggests a slightly smaller energy supply efficiency,\n\\begin{equation}\nC_{\\mathrm{eff}}\\approx0.24\\mathrm{{Ri}}\\ . \n\\label{eq:ceff-ri\n\\end{equation}\n\n\nThe turbulent parameter $\\alpha_{\\mathrm{Sek}}$ given by equation\n(\\ref{eq:alpha-sek}) is numerically calculated and is plotted using dashed lines\nin Figure \\ref{fig:alpha}. The critical Richardson number $\\mathrm{Ri}=0.8$ is\nadopted to fit the simulation results provided by JHK06. For $Z_{\\mathrm{disk}}\\ll1$\nand $T_{s}\\la1$, the solid and dashed lines agree with each other very\nwell. This is expected from the above discussion that the dependence of\n$\\alpha$ and $\\alpha_{\\mathrm{Sek}}$ on $Z_{\\mathrm{disk}}$ and $T_{s}$ for\nsmall $f_{\\mathrm{mid}}$ are the same (eqs. [\\ref{eq:alpha1}] and\n[\\ref{eq:alpha-sek3}]). For $Z_{\\mathrm{disk}}\\ga 10^{-2}$, a deviation\nbetween the solid and dashed lines appears, and the difference increases\nwith $Z_{\\mathrm{disk}}$. We note that for $Z_{\\mathrm{disk}} \\ga 0.1$,\nthe S98 density distribution shows a cusp at the midplane, which is\na significant difference from the Gaussian distribution of our model. In\nFigure \\ref{fig:fdmid}, the midplane dust-to-gas ratio of the S98 model\nis plotted with a dashed line. For small $Z_{\\mathrm{disk}}\\la0.05$, the\nresult from S98 is consistent with our result (the dashed line coincides\nwith the solid lines for $T_{s}\\la1$). For \n$Z_{\\mathrm{disk}}\\ga0.05$, the midplane dust-to-gas ratio from the S98 model\nis larger than our results. Both results are still qualitatively consistent,\nindicating that, for $Z_{\\mathrm{disk}}\\ga0.1$, the midplane dust-to-gas\nratio becomes large enough for gravitational instability ($f_{\\mathrm{mid}}\\sim100$).\n\n\\subsubsection{Comparison with Michikoshi \\& Inutsuka (2006)}\n\n\\begin{figure}[ptb]\n\\epsscale{1.1} \\plotone{f5.eps}\\caption{Comparison with the results in\nMichikoshi \\& Inutsuka (2006). The solid line shows the relationship between the\nmidplane dust-to-gas ratio $f_{\\mathrm{mid}}$ and the particle radius $a$.\nThe dashed line shows the boundary at which the growth rate of the KH instability\nis $\\omega=0.1$ (above the dashed line, $\\omega>0.1$). We use the same disk\nmodel that is adopted in Michikoshi \\& Inutsuka (2006). The dashed line is read from\nFig. 16 in Michikoshi \\& Inutsuka (2006).\n\\label{fig:michi\n\\end{figure}\n\nMichikoshi \\& Inutsuka (2006) analyzed the growth rate of the KH instability\nof a dust layer, taking into account the relative motion and the friction\nbetween the dust and the gas. Their formulation does not assume tight dust-gas\ncoupling, and thus it can be applied to scenarios involving large dust particles ($T_{s}\\gg1$),\nwhile the vertical gravity, Coriolis force, and Keplerian shear are neglected.\nThe initial velocity gradient in the vertical direction is caused by the dust\ninertia, using the formula provided in NSH86. The growth rate of instability has been derived\nfor a wide range of dust sizes $a$ and midplane dust-to-gas\nratios $f_{\\mathrm{mid}}$, and is summarized in their Figure 16. They argued\nthat if the effect of Keplerian shear is taken into account, the line\ncorresponding to the growth rate $\\omega=1$ in the $a$-$f_{\\mathrm{mid}}$\nplane would be the boundary between the stable and unstable\nconfigurations. This marginally unstable structure of the dust layer\npresented by Michikoshi \\& Inutsuka (2006) can be considered an\nextension of the S98 model for general sizes of the dust particles, and\nprovides a reference to be compared with our model.\n\nIn Figure \\ref{fig:michi}, we compare our model in the $a$-$f_{\\mathrm{mid}}$\nplane with the line at which the growth rate $\\omega$ has a constant critical\nvalue. We adopt the critical growth rate as $\\omega=0.1$, which is smaller\nthan the value Michikoshi \\& Inutsuka (2006) suggested. The qualitative\nbehavior does not differ between $\\omega=0.1$ and $\\omega=1$. Our model well\nreproduces the result of Michikoshi \\& Inutsuka (2006). The agreement of our model\nwith the results of S98 and of Michikoshi \\& Inutsuka (2006) suggests that the\nonset of KH instability is controlled by the energy supply due to dust\naccretion over a wide range of particle sizes, $a$, or stopping times,\n$T_{s}$. It must be noted, however, that the initial state assumed in Michikoshi \\&\nInutsuka (2006) is such that the vertical velocity shear appears only\nin the dust layer, neglecting the velocity shear in the Ekman-like boundary\nlayer that may appear between the dust layer and the upper gas layer. This is\nthe same assumption that we adopt. As discussed in \\S \\ref{sec:coldrag}, this\nassumption is appropriate for small dust particles ($T_{s}\\la1$), while for\nlarge particles ($T_{s}\\gg1$) the vertical velocity profile of the thick turbulent \nboundary layer may quickly deviate from that given by NSH86. Though the\nagreement with the result by Michikoshi \\& Inutsuka (2006) shows the applicability\nof our model to KH instability in the dust layer for general values of $T_{s\n$, we should keep in mind the limitation of the models mentioned above.\n\n\\begin{figure}[ptb]\n\\epsscale{1.1} \\plotone{f6.eps}\\caption{The turbulent viscosity parameter $\\alpha$\nestimated from our model, compared with the simulation described in JHK06. The solid\nline is calculated from our model and the dashed line is taken from the S98 model. The\nsquares are the simulation results, $\\delta_{t}$ in Table 2 of JHK06. The\nstopping time $T_{s}=0.1$ is adopted both for our model and the simulation.\n\\label{fig:alpha-joh\n\\end{figure}\n\n\\subsection{Comparison with Previous Numerical Simulations}\n\\label{sec:comp-sim}\n\n\\subsubsection{Comparison with Johansen et al. (2006)}\n\nNumerical simulations of KH instability induced by formation of\na dust layer were presented in JHK06. Their two-dimensional \nsimulation on the $\\theta z$-plane solved for\nboth the gas and dust motions. They obtained a quasi-steady or oscillating\ndensity distribution of the dust in which dust settling and turbulent\ndiffusion balanced each other. The diffusion coefficient in the vertical\ndirection, $\\delta_{t}$, was measured from the scale height of the dust layer\nand was summarized in their Table 2. In Figure \\ref{fig:alpha-joh}, the\nturbulent viscosity parameter $\\alpha$ of equation (\\ref{eq:alpha}) (the solid\nline) is compared with the diffusion coefficient measured from the simulation\n(the squares). To fit the simulation result, the efficiency parameter of the\nenergy supply to turbulence, $C_{\\mathrm{eff}}$, is set to $0.19$. We also\nfit the turbulent diffusion parameter for the density distribution of S98,\n$\\alpha_{\\mathrm{Sek}}$, to the simulation. We adopt the critical Richardson\nnumber as $\\mathrm{Ri}=0.8$ for fitting. This is slightly smaller than the\nvalue JHK06 suggested ($\\mathrm{Ri}=1.0$). It is seen that the numerical\nsimulation is well explained both by our model and the S98 model. This means that the\ndust layer is maintained so as to provide a constant Richardson number, and in\nthe parameter range that JHK06 surveyed ($0.01<Z_{\\mathrm{disk}}<0.1$), this\ncritical Richardson number does not vary significantly with the disk\nmetallicity $Z_{\\mathrm{disk}}$. (The recent result by Lee et al. (2010), in which\nthey argue that $\\mathrm{Ri}$ is proportional to $Z_{\\mathrm{disk}}$ if the\nKeplerian shear is taken into account, will be discussed later in this\nsubsection.) Because the efficiency parameter $C_{\\mathrm{eff}}$ in our model\nis proportional to the critical Richardson number, as shown in equation\n(\\ref{eq:ceff-ri}), the efficiency parameter $C_{\\mathrm{eff}}$ is also\nexpected to be constant for $0.01<Z_{\\mathrm{disk}}<0.1$.\n\n\\begin{figure}[ptb]\n\\epsscale{1.1} \\plotone{f7.eps}\\caption{Midplane dust-to-gas ratio\n$f_{\\mathrm{mid}}$ derived from analytical models and from the simulation described in\nJHK06. The solid line is calculated from our model and the dashed line is taken from\nthe S98 model. The squares are the simulation results, which are read from\nFigure 13 of JHK06. The stopping time is $T_{s}=0.1$.\n\\label{fig:fdmid-joh\n\\end{figure}\n\n\\begin{figure}[ptb]\n\\epsscale{1.1} \\plotone{f8.eps}\\caption{The scale heights of the dust layer $h_{d}$\nestimated from our model and from the JHK06 simulation are plotted versus the\nstopping time $T_{s}$. The solid line is calculated assuming that the turnover\ntime of the largest eddies is the Keplerian time ($T_{e}=1$). The dashed line\nincludes the variable $T_{e}$ calculated by equation (\\ref{eq:Teddy}). The squares\nare the simulation results, $\\langle z^{2}\\rangle^{1\/2} \/H$ in\nTable 2 of JHK06. The disk metallicity is $Z_{\\mathrm{disk}}=10^{-2}$.\n\\label{fig:hd-ts-joh\n\\end{figure}\n\nA comparison of the midplane dust-to-gas ratio between our model and the\nsimulation is shown in Figure \\ref{fig:fdmid-joh}. The simulation is well\nfitted by our model (the solid line) and also by the S98 model (the dashed\nline). For $Z_{\\mathrm{disk}}=0.1$, the S98 model predicts larger\n$f_{\\mathrm{mid}}$ than the simulation. This may be because the simulation\ndoes not have enough resolution to resolve the density structure around the\nmidplane at which the S98 model expects a rapid increase in the dust density. If\nthis is the case, it is difficult to judge whether our model or the model presented by S98\nis the best fit with the simulation.\n\nFigure \\ref{fig:hd-ts-joh} shows the variation in dust layer thickness with\n$T_{s}$ from our model and compares this variation with the simulation\nresults. In plotting Figure \\ref{fig:alpha-joh} in the last paragraph,\nthe turbulent viscosity parameter $\\alpha$ was calculated \nfrom equation (\\ref{eq:hd}) using the dust layer thickness $h_{d}$ measured\nfrom the simulation. However, it is not clear if equation (\\ref{eq:hd}) is\nstill valid for large $T_{s}$ because this equation assumes that the turnover time\nof the largest eddies is equal to the Keplerian time (YL07), and as discussed below\nthis assumption may not be appropriate for large $T_{s}$. Thus, in Figure\n\\ref{fig:hd-ts-joh}, the dust layer thickness is plotted directly without\ntransferring to $\\alpha$. In our model, the dust layer thickness $h_{d}$ is\nconstant for $T_{s}\\la1$ and decreases slowly with $T_{s}$ for $T_{s}\\ga1$\n(the solid line), although in the simulation it decreases more rapidly with\n$T_{s}$. The discrepancy between our model and the simulation is apparent for\n$T_{s}=1$. The difference is as large as a factor of 3, but it causes an order of magnitude\ndiscrepancy in $\\alpha\\propto h_{d}^{2}$. This discrepancy suggests that our\nmodel predicts turbulent diffusion that is too large for $T_{s}\\ga1$. In fact, for\nlarge $T_{s}$ in our model, the dust layer thickness becomes smaller than the\nsize of the largest eddies, contradicting our assumption that the turbulent\nlayer coincides with the dust layer (see, however, the discussion in\n\\S \\ref{sec:coldrag} on the validity of this assumption, and see also eq. [56]\nof YL07 for a possible physical reason for $l_{g,\\mathrm{eddy}} > h_{d}$).\nFrom equation (\\ref{eq:hd}) and $l_{g,\\mathrm{eddy}}=\\alpha^{1\/2}h_{g}$, the\ncondition for the dust layer thickness to be larger than the largest eddy size\n($h_{d}>l_{g,\\mathrm{eddy}}$) is $T_{s}<1\/\\sqrt{2}$. Thus, for $T_{s\n>1\/\\sqrt{2}$, our model results in a dust layer that is too thick (or an eddy\nsize that is too small). One possible remedy for this situation is to remove the assumption that\nthe turnover time of the largest eddies is equal to the Keplerian time. By\nintroducing a parameter $\\xi$ ($0\\leq \\xi \\leq1\/2$), the largest eddy size and the\nvelocity are expressed as $l_{g,\\mathrm{eddy}}=\\alpha^{(1\/2)+\\xi}h_{g}$ and\n$u_{g,\\mathrm{eddy}}=\\alpha^{(1\/2)-\\xi}c_{s}$. The non-dimensional turnover time\nof the largest eddies is $T_{e}=(l_{g,\\mathrm{eddy}}\/u_{g,\\mathrm{eddy\n})\\Omega_{\\mathrm{K}}=\\alpha^{2\\xi}$. The dust layer thickness for $T_{e}\\neq1$\nis provided by equation (21) of YL07,\n\\begin{equation}\n{h_{d}}=\\sqrt{\\frac{\\alpha}{{{T_{s}}}}}{\\left(  {1+\\frac{{{T_{s}}{T_{e}}^{2}\n}{{{T_{s}}+{T_{e}}}}}\\right)  ^{-1\/2}}{h_{g}}\\ .\n\\end{equation}\nFor $T_{s}>1\/2$, we impose the condition that the largest eddy size\n$l_{g,\\mathrm{eddy}}=\\alpha^{(1\/2)+\\xi}h_{g}$ is equal to the above $h_{d}$.\nThis condition determines the turnover time by the equation\n\\begin{equation}\n{T_{s}}{T_{e}}({T_{s}}+{T_{e}}+{T_{s}}{T_{e}}^{2})-{T_{s}}-{T_{e}}=0\\ .\n\\label{eq:Teddy\n\\end{equation}\nUsing this $T_{e}$ (or $\\xi$), the turbulent viscosity parameter $\\alpha$ and\nthe dust layer thickness $h_{d}$ are recalculated and plotted with the dashed\nline in Figure \\ref{fig:hd-ts-joh}. As expected, the introduction of a new\nparameter ${T_{e}}$ suppresses the largest eddy size and the turbulent\ndiffusion of dust particles for $T_{s}>1\/\\sqrt{2}$, improving the comparison with the\nsimulation result. However, we need to determine whether the turnover\ntime of the largest eddies, $T_{e}\\Omega_{\\mathrm{K}}^{-1}=\\alpha^{2\\xi\n\\Omega_{\\mathrm{K}}^{-1}$, in the simulation presented in JHK06 for $T_{s}=1$ is\nactually smaller than the Keplerian time, as equation (\\ref{eq:Teddy})\npredicts. Cuzzi et al. (1993) argue that, based on laboratory\nmeasurements, the eddy turnover time in an Ekman layer would be\nsmaller than the Keplerian time by a factor $20-80$. The turbulent\nlayer possibly behaves as an Ekman layer for $T_s >1$, as discussed in\n\\S\\ref{sec:coldrag}. We are currently performing numerical\nsimulations using the same conditions as JHK06 to investigate the\nturbulence for $T_{s} \\ga 1$ in more detail (Ishitsu et al., in preparation).\n\nLee et al. (2010) performed a three-dimensional numerical simulation of the\nonset of KH instability. They solved simplified equations in which the\ndust and the gas were treated as a single fluid, but they included the effect of\nthe Keplerian shear in the radial direction. They found that the radial shear\nstabilizes the KH instability, and the critical Richardson number for instability\nis not always the standard value, $0.25$, but can be much smaller if the\nstabilizing effect of the radial shear is significant (see also Ishitsu \\&\nSekiya 2003). Equation (32) in Lee et al. (2010) shows that the ratio of the\nstabilizing effect by the radial shear to the destabilizing effect by the\nvertical shear is proportional to $\\mathrm{Ri}(1+f_{\\mathrm{mid}})\/f_{\\mathrm{mid}}\\approx\\mathrm{Ri}\/f_{\\mathrm{mid}}$ for\n$f_{\\mathrm{mid}}\\ll1$, and thus the critical Richardson number should scale\nas $\\mathrm{Ri}\\propto f_{\\mathrm{mid}}$. In our model, we assume that the\nenergy supply efficiency $C_{\\mathrm{eff}}$, which is proportional to the\ncritical Richardson number (eq.[\\ref{eq:ceff-ri}]), is constant. However, the\nsimulation in Lee et al. (2010) suggests that\n$C_{\\mathrm{eff}}$ should also be proportional to $f_{\\mathrm{mid}}$\n(even for $f_{\\mathrm{mid}} \\ga 1$). If this is the case,\ndependence of $f_{\\rm mid}$ on $Z_{\\rm disk}$ would be milder than\nthat shown in Figure \\ref{fig:fdmid} ($f_{\\rm mid} \\propto Z_{\\rm\ndisk}^{1\/2}$ for $f_{\\rm mid} \\ll 1$ and $f_{\\rm mid} \\propto Z_{\\rm\ndisk}^{0.97}$ for $f_{\\rm mid} \\gg 1$).\nThe effect of the radial shear was not included\nin the simulation presented in JHK06, with which we compared our model\nin detail, because their simulation was two-dimensional in the $\\theta\nz$-plane. Extending JHK06 to three dimensions and including the radial\nshear effect are crucial to determining how the energy supply efficiency\n$C_{\\mathrm{eff}}$ behaves as $f_{\\mathrm{mid}}$ varies.\n\n\\subsubsection{Comparison with Bai \\& Stone (2010)}\n\\label{sec:BS10}\n\n\\begin{figure}[ptb]\n\\epsscale{1.1} \\plotone{f9a.eps} \\plotone{f9b.eps} \n\\caption{The turbulent\nviscosity parameter $\\alpha$ estimated from our model for $T_{s}=10^{-1}$ and 1\nis compared with the simulation presented in Bai \\& Stone (2010). The solid line is\ncalculated assuming the turnover time of the largest eddies is the Keplerian\ntime ($T_{e}=1$). The dashed line includes the variable $T_{e}$ calculated from\nequation (\\ref{eq:Teddy}). $(a)$ Comparison with the vertical diffusion\ncoefficient, $D_{g,z}(3\\mathrm{D})$, measured in the simulation (Table 2 of\nBai \\& Stone 2010). The squares and circles are the simulation results, \nconnected by lines that indicate which runs have the same size distribution\nof dust particles. The label of run R$xy$ means the particle size\ndistribution is such that the stopping time ranges $T_{s}\\in\\lbrack\n10^{-x},10^{-y}]$, e.g., in run R30, $T_{s}\\in\\lbrack10^{-3},10^{0}]$. In each\nrun, the disk metallicity $Z$ is $0.01$ and $0.03$. To compare the results with our model,\nwe assume that only the largest particles contribute to turbulence, and that the\neffective metallicity is estimated by $Z_{\\mathrm{disk}\n=Z\/N_{\\mathrm{type}}$, where $N_{\\mathrm{type}}=3$ (R10 and R21) or 7 (R30 and\nR41) is the number of particle species. We compare our model of $T_{s}=1$ with\nruns R10 and R30 (plotted in blue), and the model of\n$T_{s}=10^{-1}$ with runs R41 and R21 (plotted in red). $(b)$\nComparison with the radial diffusion coefficient, $D_{x}$, measured in the 3D\nruns of the simulation. We assume that the diffusion coefficient of the gas is\nrepresented by that of the smallest particles in the simulation. The values of\n$D_{x}$ for the smallest particles are read from Fig.9 of Bai \\& Stone (2010).\n\n\\label{fig:alpha-bai\n\\end{figure}\n\nBai \\& Stone (2010) performed a three-dimensional simulation, focusing on\ninvestigating turbulence induced by streaming instability. They found that\nthe streaming instability induced turbulence before the KH instability set in.\nThey measured the turbulent diffusion coefficient. However, it is difficult to\ncompare our model directly with the results in Bai \\& Stone (2010), because their\nsimulation includes particles of several sizes (3-7 species) while our model\nconsiders only single-sized particles. In the simulation, it was reported that\nonly large particles were responsible for inducing turbulence. Our model also\nshows that the energy liberation per unit mass of the dust is higher for\nlarger particles (it is proportional to $T_{s}$ for $T_{s}\\la1$). In order to\ncompare results, we assume that in the simulation, the turbulence is induced\nonly by the largest particles (i.e., particles of largest $T_{s}$). For\nexample, in the R41 run (in which the stopping time of the particles ranges\nfrom $10^{-4}$ to $10^{-1}$), we assume that only $T_{s}=0.1$ particles\nare responsible for turbulence. We then compare the simulation result with our\nmodel of $T_{s}=0.1$. In the simulation, each species has the same amount of\nmass. Because we consider the largest particles only, the total amount of the\ndust participating in driving turbulence is $\\Sigma\n_{d}\/N_{\\mathrm{type}}$, where $N_{\\mathrm{type}}$ is the number of particle\nspecies in the simulation. For example, in the R41 run, $N_{\\mathrm{type}}=7$,\nand we compare the simulation with a disk metallicity $Z=0.01$ with our\nmodel of $Z_{\\mathrm{disk}}=Z\/N_{\\mathrm{type}}=1.43\\times10^{-3}$. \n\nFigure \\ref{fig:alpha-bai}$a$ shows a comparison of the diffusion\ncoefficient obtained from the simulation ($D_{g,z}(\\mathrm{3D})$ in\nTable 2 of Bai \\& Stone (2010)) with our model (calculated with the\nparameter $\\tilde{\\eta}=0.05^{2}$, which was adopted by Bai \\& Stone\n(2010)). Although our model of $T_{s}=0.1$ agrees with simulations R41\nand R21 (plotted in red), we note a qualitative discrepancy between the $T_{s}=1$ model\nand simulations R30 and R10 (plotted in blue). The simulations indicate that the diffusion\ncoefficient decreases with the disk metallicity, and that its value for\n$Z=0.03$ is about an order of magnitude smaller than the value from our \nmodel. This discrepancy cannot be resolved, even by varying the turnover time\nof the largest eddies $T_{e}$ (the dashed line). One possible cause for the\ninconsistency is particle clumping and concentration in turbulent eddies,\nwhich are not included in our model. The simulation shows strong\nparticle clumping when $Z=0.03$ in the R10 run and also temporal clumping for\n$Z=0.03$ in the R30 run. Such clumping of particles in turbulent eddies may\nsuppress diffusion of particles compared to the no-clumping cases of\n$Z=0.01$ and could be a cause of a decrease in $\\alpha$ when\n$Z$ is increased in the simulation. Even if the\nlargest particles (of $T_{s}\\sim1$) concentrate in clumps, the smallest\nparticles (of $T_{s}\\la 0.1$) do not clump, and continue to follow\nthe turbulent diffusion of the gas (Fig. 7 of Bai \\& Stone 2010). In Figure\n\\ref{fig:alpha-bai}$b$, the turbulent diffusion coefficient in the ``radial\ndirection'' of the smallest particles in the simulation, $D_{x}$, is compared\nwith the ``vertical'' diffusion coefficient $\\alpha$ in our model. Note that\nwe compare diffusion coefficients in the different\ndirections. Since the smallest particles spread out to high altitudes where\nturbulence is weak, it is difficult to measure the vertical diffusion\ncoefficient for the smallest particles in the simulation. In Figure\n\\ref{fig:alpha-bai}$b$, though we still see a discrepancy compared with\nthe R30 run, our model appears more consistent with the simulation\nresults, suggesting that our model properly predicts the ``gas''\ndiffusion coefficient. \n\n\\section{DISCUSSION}\n\\label{sec:discuss}\n\n\\subsection{The Radial Drift Velocity and Collision Velocity of Dust\nParticles}\n\\label{sec:dustvelocity}\n\n\\begin{figure}[ptb]\n\\epsscale{1.1} \\plotone{f10.eps} \\caption{The radial drift velocity, $\\bar\n{v}_{d,r}$ (solid line), and the relative velocity of particles due to\nturbulence, $\\Delta v_{12}$ (dashed line). The contributions of individual\ndrag, $\\bar{v}_{\\mathrm{ind},r}$, and collective drag, $\\bar{v}_{\\mathrm{col\n,r}$, to the radial drift velocity are plotted with dotted lines.\n\\label{fig:velocity\n\\end{figure}\n\nIf the dust-to-gas ratio in the dust layer were larger than unity, the radial\ndrift velocity of dust particles would be lower than the value that the\nparticles would have in a gas-rich environment because the gas drag\nforce could not accelerate sufficiently against the large inertia of the\ndust. This effect was pointed out by NSH86 and is seen in equation\n(\\ref{eq:vdr1}) for the radial velocity due to individual drag. If the\ndust-to-gas ratio were much larger than unity, dust accretion would\nbe caused by collective drag exerted from the slower-orbiting\nupper gas layer (Weidenschilling 2003). The radial drift velocity of the\ndust is thus a  function of the turbulence strength, $\\alpha$. For\nweaker turbulence, the \ndust-to-gas ratio in the dust layer is higher, and the individual drag is\nweaker. The collective drag is also weaker at smaller $\\alpha$ because the\nReynolds stress $P_{\\theta z}$ is proportional to $\\alpha$. The relative\nvelocity (or collision velocity) of the dust particles is also a function of\n$\\alpha$. The radial drift and collision velocities are the important factors in\nthe dust growth process. In the previous sections, the turbulence strength\n$\\alpha$ and the dust layer thickness $h_{d}$ have been determined\nself-consistently. Using these results, the radial drift velocity and collision\nvelocity of the dust particles are estimated.\n\nThe radial drift velocity is calculated separately for the components due to\nindividual drag and due to collective drag. For each component, the radial\ndrift velocity is averaged in the vertical direction. First, the averaged\nvalue of the radial drift velocity due to individual drag is calculated\nfrom equation (\\ref{eq:vdr1}),\n\\begin{eqnarray}\n\\overline{v}_{\\mathrm{ind},r} &=& \\frac{1}{\\Sigma_{d}}\\int_{-\\infty}^{\\infty\n\\rho_{d}v_{d,r}dz \\nonumber \\\\\n&=& -2\\eta v_{\\mathrm{K}}T_{s}\\frac{1}{\\Sigma_{d}}\\int_{-\\infty}^{\\infty\n}\\rho_{d}\\frac{1}{T_{s}^{2}+\\beta^{2}}dz~.\n\\end{eqnarray}\nThe radial drift velocity due to collective drag is calculated from the\nvertically-averaged angular momentum loss of the dust component. Integrating\nequation (\\ref{eq:dlddt}) gives\n\\begin{equation}\n\\frac{dL_{d}}{dt}=\\int_{-\\infty}^{\\infty}\\frac{\\partial l_{d}}{\\partial\nt}dz=-2\\eta v_{\\mathrm{K}}^{2}T_{s}\\Sigma_{d,\\mathrm{vis}}~,\n\\end{equation}\nwhere $\\Sigma_{d,\\mathrm{vis}}$ is given by equation (\\ref{eq:sig_vis2}). This\nangular momentum loss causes a radial drift velocity $\\bar{v}_{\\mathrm{col},r}$ given by\n\\begin{equation}\n\\bar{v}_{\\mathrm{col},r}=\\frac{2}{v_{\\mathrm{K}}\\Sigma_{d}}\\frac{dL_{d}\n{dt}=-4\\eta v_{\\mathrm{K}}T_{s}\\frac{\\Sigma_{d,\\mathrm{vis}}}{\\Sigma_{d}}~.\n\\end{equation}\nThe total radial drift velocity is $\\bar{v}_{d,r}=\\overline{v}_{\\mathrm{ind\n,r}+\\bar{v}_{\\mathrm{col},r}$, and is shown in Figure \\ref{fig:velocity}\nas a solid line, for the case in which the particle size is chosen to maximize the\nradial velocity ($T_{s}=1$). In plotting this figure, we adopt\nthe model parameters at 1 AU of the minimum-mass-solar-nebula model of\nHayashi (1981): $h_{g}\/r=0.0326$, $\\eta=1.80\\times10^{-3}$, and\n$\\tilde{\\eta}=2.92 \\times 10^{-3}$. For a small disk metallicity\n$Z_{\\mathrm{disk}}$, the radial drift velocity is as large \nas 50 m s$^{-1}$, and it decreases with $Z_{\\mathrm{disk}}$. For $Z_{\\mathrm{disk}}>0.08$, the radial velocity is dominated by collective drag (see the dotted\nlines) as pointed out by Weidenschilling (2003). For such large\n$Z_{\\mathrm{disk}}$, the radial drift velocity due to collective drag also\ndecreases with $Z_{\\mathrm{disk}}$, and then it becomes as small as $1$ m s$^{-1}$\nfor $Z_{\\mathrm{disk}}=0.2$. The radial drift velocity is strongly suppressed\nfor a sufficiently massive dust layer.\n\nThe relative velocity of dust particles due to turbulence, $\\Delta v_{12}$, is\ncalculated for $T_{s}=1$ by substituting $\\mathrm{St}_{1}=1$ and\n$\\mathrm{St}_{2}=0$ into equation (29) in Ormel \\& Cuzzi (2007), and is shown\nin Figure \\ref{fig:velocity} as a dashed line. The collision velocity is\nestimated by the larger of $\\Delta v_{12}$ and $\\bar{v}_{d,r}$. For small\ndisk metallicities $Z_{\\mathrm{disk}}$, the collision velocity is dominated by\nthe radial drift and is as large as 50 m s$^{-1}$, while for large $Z_{\\mathrm{disk}}>0.02$, it is dominated by turbulence. The maximum value of the collision\nvelocity due to turbulence is about 30 m s$^{-1}$, and it decreases with\n$Z_{\\mathrm{disk}}$ for $Z_{\\mathrm{disk}}>0.03$. Thus, if the dust\nparticles could survive collisions of $30$ m s$^{-1}$, as suggested by the numerical\nsimulation of collisions of dust aggregates (Wada et al. 2010) and\n$Z_{\\mathrm{disk}}\\ga0.03$, the dust particles would be able to grow without\nbeing reduced to small fragments.\n\n\\subsection{Radial Dependence of the Midplane Dust-to-Gas Ratio}\n\\label{sec:disketa}\n\n\\begin{figure}[ptb]\n\\epsscale{1.1} \\plotone{f11.eps} \\caption{The midplane dust-to-gas\n    ratio $f_{\\mathrm{mid}}$ is plotted against $\\tilde{\\eta}$ for various values\n  of the disk metallicity $Z_{\\rm disk}$. The top axis shows the\n  corresponding radii in the gas disk model by Hayashi (1981), in which\n  the disk temperature is $T = 278 (r\/{\\rm AU})^{-1\/2}$K. The dust\n  particles are assumed to be small ($T_s \\ll 1$) such that\n  $f_{\\mathrm{mid}}$ is independent of $T_s$.}\n\\label{fig:fmid-eta\n\\end{figure}\n\nAs shown in \\S\\ref{sec:turbstr}, the turbulence strength and the dust\nlayer structure depend on properties of the gas disk only through\n$\\tilde{\\eta}$. For larger $\\tilde{\\eta}$, the accretion velocity of the\ndust is faster, and consequently the dust layer is thicker due to\nstronger turbulence. Figure \\ref{fig:fmid-eta} shows how the midplane\ndust-to-gas ratio varies with $\\tilde{\\eta}$. The figure is plotted in the limit\nof $T_s \\ll 1$. For such small particles, $f_{\\rm mid}$ is\nindependent of $T_s$ (see Fig. \\ref{fig:fdmid}). The midplane\ndust-to-gas ratio $f_{\\rm mid}$ decreases with $\\tilde{\\eta}$ as shown\nin equation (\\ref{eq:fdmid1}). For a gas disk model with a power-law temperature\nprofile, $T \\propto r^{-q}$, $\\tilde{\\eta}$ behaves as\n$\\tilde{\\eta} \\propto r^{1-q}$. The top axis of Figure\n\\ref{fig:fmid-eta} indicates corresponding locations in the disk\nmodel by Hayashi (1981), i.e.,  $T = 278 (r\/{\\rm AU})^{-1\/2}$K. In a\ndisk with the standard value of the disk metallicity $Z_{\\rm disk}=0.01$,\n$f_{\\rm mid}$ is less than unity in the most part of the disk except $r\n\\la 0.1$AU. In disks with $Z_{\\rm disk}=0.3$, $f_{\\rm mid}$\nexceeds $100$ for $r \\la 1$AU. We discuss the condition for planetesimal\nformation through gravitational instability of the dust layer. In a gas\ndisk with a surface density profile $\\Sigma_g \\propto r^{-p}$, the\nmidplane gas density scales as $\\rho_g \\propto r^{-p+q\/2-3\/2}$. The\ncondition that the dust density exceeds the Roche density ($\\rho_{\\rm R}\n\\propto r^{-3}$) is $f_{\\rm mid} > \\rho_{\\rm R}\/\\rho_g \\propto\nr^{p-q\/2-3\/2} \\propto r^{p-7\/4}$ in a disk model with\n$q=1\/2$. From the lower line of equation (\\ref{eq:fdmid1}), $f_{\\rm\n  mid}$ decreases as $f_{\\rm mid} \\propto r^{-(1-q)\/(2-\\delta)}\\propto\nr^{-0.47}$. If $p>1.3$, the inner part of the disk is more suitable\nfor gravitational instability, and vice versa.\n\n\\subsection{Does the Liberated Gravitational Energy Go into\nTurbulence?}\n\\label{sec:ene-into-tur}\n\nIn this paper, we calculate the liberated gravitational energy from dust\naccretion, assuming some fraction of the liberated energy is transferred to\nturbulence. The estimate of the dust accretion rate is based on the formula for the\nparticle terminal velocity derived in NSH86. ``The terminal velocity'' means\nthat all the liberated energy is consumed by gas drag, converting directly into\nthe thermal energy of the dust particles and of the surrounding gas molecules.\nThus, one may expect that only a small fraction (or nothing) of the liberated\nenergy would be used for maintaining turbulence. However, a comparison with\nthe simulation of KH instability by JHK06 shows that the efficiency factor\n$C_{\\mathrm{eff}}\\approx0.2$ is not negligibly small. In the following subsection, we\ndiscuss the validity of using the particle terminal velocity for calculating\nthe energy supply rate to turbulence. The energy liberation rate from the\naccreting dust calculated in \\S \\ref{sec:eneinp} is compared with the deposit\nrate of the free energy that is the source of several instabilities, such as KH\ninstability and streaming instability.\n\n\\subsubsection{Kelvin-Helmholtz Instability}\n\nThe free energy that induces the KH instability originates from the velocity\ndifference between the midplane dust layer and the upper gas layer, and is\nstored as the dust particles settle to the midplane. We estimate the deposit\nrate of the free energy during dust sedimentation, and show that it has the\nsame order of magnitude as the energy liberation rate from the dust accretion\ntowards the star. Consider two states of dust distribution: the\ninitial state, in which the dust particles are distributed uniformly in the gas\ndisk, and the final state, in which all the dust has settled at\nthe midplane. In the initial state, there is no vertical shear in the disk, and in the final state, the velocity difference $\\Delta\nv_{\\theta}=\\eta v_{\\mathrm{K}}$ appears between the midplane dust layer and\nthe upper gas layer. The free energy for KH instability is $\\Delta\nE_{\\mathrm{KH}}\\sim\\frac{1}{2}\\Sigma_{d}\\Delta v^{2}\\sim\\frac{1}{2}\\Sigma\n_{d}\\eta^{2}v_{\\mathrm{K}}{}^{2}$, for $\\Sigma_{d}\\ll\\Sigma_{g}$. The settling\ntimescale is $\\tau_{\\mathrm{sed}}=(T_{s}^{2}+1)\/(T_{s} \\Omega_{\\mathrm{K}})$,\nand then the energy deposit rate is\n\\begin{equation}\n\\frac{\\Delta E_{\\mathrm{KH}}}{\\tau_{\\mathrm{sed}}}\\sim\\frac{1}{2}\\eta\n^{2}v_{\\mathrm{K}}^{2} \\Omega_{\\mathrm{K}} \\frac{T_{s}}{T_{s}^{2}+1} \\Sigma_{d} \\ ,\n\\end{equation}\nwhich is the same order as the energy liberation rate of the accreting dust\n(eqs. [\\ref{eq:ene_drag}] and [\\ref{eq:sig_drag2}]). Hence, the deposition rate\nof the free energy for KH instability can be estimated by the energy\nliberation rate of the accreting dust.\n\n\\subsubsection{Streaming Instability}\n\\label{sec:streaminst}\n\nThe free energy for streaming instability originates from the velocity\ndifference between the dust particles and the surrounding gas. When streaming\ninstability begins, the velocity difference decreases as the free energy is\nconsumed by inducing turbulence. In fact, this decrease in the velocity difference\ncan be seen even in the linear growth regime. Youdin \\& Goodman (2005)\nshowed in their Figure 6 that the velocity difference between the dust and gas\ndecreases (increases) at the locations where the particle density increases\n(decreases). The spatially averaged value of the free energy decreases as the\nperturbation grows. Thus, without an energy supply, streaming instability would\ncease. Given a state in which the velocity difference between the dust and the\ngas has reduced, the dust particles are no longer in equilibrium: the\ngravity, the centrifugal force, and the gas drag force are not in balance. The\ndust particles are accelerated and the velocity difference from the gas rises\nagain. To estimate the effect of the energy deposition on the velocity\ndifference, we consider a state in which the dust density is similar to the\ngas density, $\\rho_{d}\\sim\\rho_{g}$. In such a state, streaming\ninstability occurs efficiently with a growth time of the order of the\nKeplerian time (for the short wave branch, Youdin \\& Goodman 2005; Youdin \\&\nJohansen 2007). For dust particles of $T_{s}<1$, the terminal velocities of\nthe dust and of the gas are of the order of $T_{s}\\eta v_{\\mathrm{K}}$ (eq.\n[\\ref{eq:vdr}], [\\ref{eq:vgr}]), and thus, the free energy per unit area is\n$\\Delta E_{\\mathrm{str}}\\sim\\Sigma_{d}T_{s}^{2}\\eta^{2}v_{\\mathrm{K}}^{2}$. This\ndeposition of free energy occurs during the acceleration phase of the dust,\nand thus in the stopping time $T_{s}\\Omega_{\\mathrm{K}}^{-1}$, and then the energy is\ntransferred to turbulence in the growth timescale $\\Omega_{\\mathrm{K}}^{-1}$ of\nstreaming instability. Thus, the timescale for the energy deposit in\nturbulence is the sum of these timescales, and for $T_{s}\\la1$, $\\tau\n_{\\mathrm{str}}=T_{s}\\Omega_{\\mathrm{K}}^{-1}+\\Omega_{\\mathrm{K}}^{-1}\\sim\\Omega_{\\mathrm{K}}^{-1}$. The\nenergy deposition rate is\n\\begin{equation}\n\\frac{\\Delta E_{\\mathrm{str}}}{\\tau_{\\mathrm{str}}}\\sim\\eta^{2}\nv_{\\mathrm{K}}^{2} \\Omega_{\\mathrm{K}} T_{s}^{2}\\Sigma_{d} \\ ,\n\\end{equation}\nwhich is smaller by a factor $T_{s}$ than the estimate from the dust accretion\nrate (eq.[\\ref{eq:ene_drag}]). Hence, our estimate, derived from the dust accretion\nrate, is appropriate for particles of $T_{s}\\sim1$. For smaller particles\n($T_{s} \\ll1$), the energy deposition rate is higher for KH instability than for\nstreaming instability, and KH instability is expected to operate first. The\nenergy deposition rate for instability (either for KH or streaming instabilities)\nis estimated from the energy liberation rate due to dust accretion.\n\n\\section{SUMMARY}\n\nIn this paper, we discuss turbulence induced in the dust layer. The\n turbulence strength or the parameter $\\alpha$ is determined using the energetics of\ndust accretion towards the central star. The key concept is that the dust\nparticles reside in a deeper potential than the gas. The effective potential,\nincluding the gas pressure, is $-GM(1-2\\eta)\/r$ for the gas,\nand $-GM\/r$ for the dust. When angular momentum is transferred\nfrom the dust to the gas through gas drag, the dust particles lose more energy\nthan the gas gains. The excess energy can be used for exciting\nturbulence. If the dust accretion due to gas drag is a primary source of energy\nliberation, i.e., if the gas accretion rate due to turbulence is much smaller\nthan the dust accretion rate, then the turbulence strength is determined by the\nenergy supply rate from the dust accretion. This is not the case if the gas\ndisk itself is turbulent via, e.g., MRI. If the dust layer is composed of large\nparticles with stopping time $T_{s}\\gg1$, then the gas accretion may\ndominate the dust accretion, as discussed in \\S \\ref{sec:coldrag}.\n\n We estimate the dust accretion rate using the terminal velocity profiles of the\ndust particles in a laminar disk derived by NSH86. The expected turbulence\nstrength and corresponding structure of the dust layer from our analysis agree\nwith the previous analytical result on the marginally KH-unstable dust layer\nby S98. As our analysis does not assume tight coupling of the dust to the\ngas, nor specify the mechanism of instability that induces turbulence, it is\nconsidered an extension of the analysis of S98 to a more general physical\nsituation of the dust layer. The results of this paper agree with\nthe results in Michikoshi \\& Inutsuka (2006), which analyzes KH instability\nof the dust layer composed of particles with large stopping times ($T_{s}>1$),\nas shown in Figure \\ref{fig:michi}. \n\nOur analysis shows that, for particles of\n$T_{s}\\la1$, the turbulence strength is smaller than $\\alpha_{\\max}\\sim\nC_{\\mathrm{eff}}{\\tilde{\\eta}}T_{s}$, where $C_{\\mathrm{eff}}=0.19$ is the\nefficiency of the energy supply to turbulence (see Fig. \\ref{fig:alpha} and\neq. [\\ref{eq:alpha2}]). This strength reaches a maximum when the disk\nmetallicity is $Z_{\\mathrm{disk}}\\sim\\sqrt{C_{\\mathrm{eff}}\\tilde{\\eta\n}}\\sim10^{-2}$. Modifying the disk metallicity from the standard value,\n$10^{-2}$, by any process, results in weaker turbulence and a thinner dust\nlayer, and consequently may accelerate the growth process of the dust\nparticles, as pointed out in S98.\n\nComparison of our results with previous numerical simulations of KH and streaming\ninstabilities by JHK06 and Bai \\& Stone (2010) shows quantitative agreement\nwith our analysis for dust particles of $T_{s}\\la0.1$, although there may be a\nqualitative disagreement for $T_{s}\\ga1$ particles (see Figs.\n\\ref{fig:alpha-joh} $-$ \\ref{fig:alpha-bai}). Hence, we conclude that\nturbulence in the dust layer is controlled by the energy supply from the dust\naccretion due to gas drag, provided that the dust particles are not so large\nthat $T_{s}\\ga1$. In such a layer, turbulence strength is estimated by the dust\naccretion rate (eq.[\\ref{eq:alpha}]).\n\n\\acknowledgements\nThis work was stimulated by discussions with Minoru Sekiya. We are\ngrateful to Anders Johansen for providing detailed information on his\nsimulations, and to Chris Ormel and Eugene Chiang for useful\ndiscussions. We also thank an anonymous referee for helpful comments.\nThis work was supported in part by  Grants-in-Aid for Scientific Research,\nNos. 20540232, 22$\\cdot$2942, and 22$\\cdot$7006 from the Ministry of\nEducation, Culture, Sports, Science, and Technology (MEXT), Japan.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Statement of the main results}\n\\label{sec:main-statement}\n\n\\subsection{Setup}\n\nLet $I:=[0,1] \\subset {\\mathbb R}$ and for $k, n \\in {\\mathbb N}} \\def\\Q{{\\mathbb Q}} \\def\\P{{\\mathbb P},\\ k \\ge n$,\ndenote by $\\pi^k_n:I^k\\to I^n$ the projection map. We sometimes omit\n$k$ if its meaning is clear from the context.\n\nPfaffian functions, introduced by Khovanskii in \\cite{khov:dan,\n  khov:book}, are analytic functions satisfying triangular systems of\nPfaffian (first order partial differential) equations with polynomial\ncoefficients.  We refer the reader to\n\\cite{gv:complexity-computations} for precise definition of Pfaffian\nfunctions, based on {\\em Pfaffian chain}, and examples of Pfaffian\nfunctions in an open domain $G \\subset {\\mathbb R}^k$, which we assume\nhere for simplicity to be given by a product of intervals.\n\n\\begin{Def}[semi-Pfaffian set]\n  Let $G $ be an open set in ${\\mathbb R}^k$ and $I^k \\subset G$.  A\n  set $X \\subset I^k$ is called \\emph{(restricted) semi-Pfaffian} if\n  it consists of points in $I^k$ satisfying a Boolean combination of\n  atomic equations and inequalities of the kind $f=0$ or $f>0$, where\n  $f$ is a Pfaffian functions defined in $G$. The \\emph{format} of $X$\n  is the number of variables $k$ and the {\\em degree} of $X$ is the\n  sum of degrees of all the Pfaffian functions appearing in the atomic\n  formulas (i.e. the degrees of the polynomials defining these\n  Pfaffian functions).\n\\end{Def}\n\nNote that the degree of $X$ bounds from above the number of all atomic\nequations and inequalities.  We assume that a Pfaffian chain has been\nfixed once and for all, and all semi-Pfaffian sets under consideration\nare defined from this single Pfaffian chain.\n\n\\begin{Def}[sub-Pfaffian set]\\label{def:sub-pfaff}\n  A set $Y \\subset I^n$ is called {\\em (restricted)\n    sub-Pfaffian}\\footnote{Below we will consider only\n    \\emph{restricted} sub-Pfaffian sets, and refer to them simply as\n    sub-Pfaffian.} if $Y = \\pi^k_n (X)$ for a semi-Pfaffian set\n  $X \\subset I^k$.\n\\end{Def}\n\nIn the special case of a {\\em semi-algebraic set} $X$, the\nTarski-Seidenberg theorem states that the set $Y = \\pi^k_n (X)$ is\nalso semi-algebraic, i.e., is a set of points satisfying a Boolean\ncombination of polynomial equations and inequalities.  By contrast, a\nsub-Pfaffian set may not be semi-Pfaffian. \n\nIt is customary in the literature to define the format and degree of a\nsub-Pfaffian set as in Definition~\\ref{def:sub-pfaff} to be the format\nand degree of the set $X$. Below we introduce a variant of these\nnotions which turns out to behave better with respect to cell\ndecompositions. To avoid confusion we refer to these modified notions\nas *-format and *-degree. See Remark~\\ref{rem:comparing-formats} for a\ncomparison between the starred and unstarred versions of format and\ndegree.\n\n\n\\begin{Def}[*-format and *-degree of projections]\\label{def:format}\n  If $\\{X_\\alpha\\},\\ X_\\alpha \\subset I^{k_\\alpha}$ is a finite\n  collection of semi-Pfaffian sets and $X_\\alpha^\\circ$ is a connected\n  component of $X_\\alpha$, we define the *-format of the sub-Pfaffian\n  set\n  $Y:=\\bigcup_\\alpha \\pi^{k_\\alpha}_n(X_\\alpha^\\circ) \\subset I^n$,\n  with this particular representation as a union of projections, to be\n  the maximum among the formats of sets $X_\\alpha$, and the *-degree\n  of $Y$ to be the sum of the degrees of sets $X_\\alpha$.\n\\end{Def}\n\nStrictly speaking, the *-format $\\cF$ and *-degree $D$ is associated\nto a particular presentation of a sub-Pfaffian set in the form\nprescribed by Definition~\\ref{def:format}, with different\npresentations possibly giving rise to different pairs $(\\cF,D)$. By a\nslight abuse of notation we will say that a sub-Pfaffian set has\nformat $\\cF$ and degree $D$ if there exists \\emph{some} presentation\nhaving this data.\n\nWe remark that since the restricted sub-Pfaffian sets form an\no-minimal structure, the connected components of semi-Pfaffian (or\neven sub-Pfaffian) sets, as well as their projections, are again\nsub-Pfaffian, so the sets $Y$ for which *-format and *-degree are\nintroduced in Definition~\\ref{def:format} are indeed sub-Pfaffian.\n\n\\begin{Rem}\n  It may be useful below for the reader to consider the notions of\n  *-format and *-degree as sub-Pfaffian analogs of the notions of\n  dimension and degree in the theory of algebraic or semi-algebraic\n  geometry. Our main objective is to obtain, as in the semialgebraic\n  case, bounds that depend polynomially on the degree for a fixed\n  format.\n\\end{Rem}\n\nIn what follows we write that $a$ is $\\const(b)$ (resp., $a$ is\n$\\operatorname{poly}} \\def\\J{\\operatorname{J}_b(c)$) as shorthand notation meaning that $a \\le \\gamma(b)$\n(resp., $a \\le (c+1)^{\\gamma(b)}$) where $\\gamma(\\cdot)$ is some\n\\emph{universally fixed} function (which may be different for each\noccurrence of this notation in the text). Here $a$ and $c$ denote\nnatural numbers, and $b$ can involve one or several arguments of any\ntype.  We also allow several arguments in $\\operatorname{poly}} \\def\\J{\\operatorname{J}_b(c_1,\\ldots,c_n)$\nwhich we interpret as $\\operatorname{poly}} \\def\\J{\\operatorname{J}_b(c_1+\\cdots+c_n)$. For each occurrence\nof an asymptotic notation in this text, the implied functions can be\neffectively and explicitly computed from the data defining the\nPfaffian chain (degrees of the differential equations involved), and\nwe omit these computations in the interest of brevity.\n\n\n\\subsection{Cell decompositions}\n\nWe recall the standard definitions of a cylindrical cell and a\ncylindrical cell decomposition. Later in the paper we consider only\ncylindrical decompositions and omit the prefix ``cylindrical'' for\nbrevity.\n\n\\begin{Def}[Cylindrical cell] A \\emph{cylindrical cell} is defined by\n  induction as follows.\n  \\begin{enumerate}\n  \\item Cylindrical 0-cell in ${\\mathbb R}^n$ is an isolated point.\n  \\item Cylindrical 1-cell in ${\\mathbb R}$ is an open interval\n    $(a,b) \\subset {\\mathbb R}$.\n  \\item For $n \\ge 2$ and $0 \\le \\ell < n$ a cylindrical\n    $(\\ell+1)$-cell in ${\\mathbb R}^n$ is either a graph of a\n    continuous bounded function $f:\\ C \\to {\\mathbb R}$, where $C$ is\n    a cylindrical a cylindrical $(\\ell+1)$-cell in\n    ${\\mathbb R}^{n-1}$, or else a set of the form\n    \\begin{multline}\n      \\{(x_1, \\ldots ,x_n) \\in {\\mathbb R}^n : \\ (x_1, \\ldots , x_{n-1}) \\in C \\\\\n      \\text{ and } f(x_1,\\ldots,x_{n-1})<x_n<g(x_1,\\ldots ,x_{n-1}) \\},\n    \\end{multline}\n    where $C$ is a cylindrical $\\ell$-cell in ${\\mathbb R}^{n-1}$, and\n    $f,g:\\ C \\to {\\mathbb R}$ are continuous bounded functions such that\n    $f(x_1, \\ldots , x_{n-1})<g(x_1, \\ldots , x_{n-1})$ for all points\n    $(x_1, \\ldots ,x_{n-1}) \\in C$.\n  \\end{enumerate}\n\\end{Def}\n\nThis definition implies that any $\\ell$-cell is homeomorphic to an\nopen $\\ell$-dimensional ball.\n\n\\begin{Def}[Cylindrical cell decomposition] A \\emph{cylindrical cell\n    decomposition} ${\\mathcal D}$ of a subset\n  $A \\subset {\\mathbb R}^n$ is defined by induction as follows.\n  \\begin{enumerate}\n  \\item If $n=1$, then ${\\mathcal D}$ is a finite family of pair-wise\n    disjoint cylindrical cells (i.e., isolated points and intervals)\n    whose union is $A$.\n  \\item If $n \\ge 2$, then ${\\mathcal D}$ is a finite family of\n    pair-wise disjoint cylindrical cells in ${\\mathbb R}^n$ whose\n    union is $A$ and there is a cylindrical cell decomposition of\n    $\\pi^n_{n-1} (A)$ such that $\\pi^n_{n-1} (C)$ is its cell for each\n    $C \\in {\\mathcal D}$.\n  \\end{enumerate}\n  Let $B \\subset A$.  Then ${\\mathcal D}$ is \\emph{compatible} with\n  $B$ if for any $C \\in {\\mathcal D}$ we have either $C \\subset B$ or\n  $C \\cap B= \\emptyset$ (equivalently, some subset\n  ${\\mathcal D}' \\subset {\\mathcal D}$ is a cylindrical cell\n  decomposition of $B$).\n\\end{Def}\n\nOur main result is as follows.\n\n\\begin{Thm}\\label{thm:main}\n  Let $\\{Y_\\alpha\\},\\ Y_\\alpha \\subset I^n$ be a collection of $N$\n  sub-Pfaffian sets of *-format $\\cF$ and *-degree $D$. Then there\n  exists a sub-Pfaffian cell decomposition of $I^n$ compatible with\n  each $Y_\\alpha$ such that the number of cells is $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(N,D)$,\n  their *-format is $\\const(\\cF)$, and their *-degree is\n  $\\operatorname{poly}} \\def\\J{\\operatorname{J}_{\\cF}(D)$.\n\\end{Thm}\n\nThe proof of Theorem~\\ref{thm:main} is given\nin~\\secref{sec:proof-main}.\n\n\\subsection{Sub-pfaffian sets defined by quantified formulas}\n\nIn this paper we will consider formulas in the \\emph{restricted\n  sub-Pfaffian language}, which we define to be the first-order\nlanguage consisting of logical connectives, quantifiers, atomic\npredicates of the form $({\\bf x} \\in Y)$ where $Y$ is a restricted\nsub-Pfaffian set, and no function symbols. All formulas referred to\nbelow are assumed to be in this language. We introduce the notion of\n*-format and *-degree for quantified formulas in this language as\nfollows.\n\n\\begin{Def}[*-format and *-degree of quantified formulas]\\label{def:formula-complexity}\n  Let $\\phi$ be a formula, not necessarily in a prenex\n  form. We define the *-degree $D(\\phi)$ to be the sum of $D(Y)$ for\n  all $Y$ appearing in the atomic predicates of $\\phi$. We inductively\n  define the *-format $\\cF(\\phi)$ as follows.\n  \\begin{itemize}\n  \\item For $\\phi=({\\bf x} \\in Y)$ we define $\\cF(\\phi)$ to be the\n    *-format of $Y$.\n  \\item For $\\phi=\\bigvee_{j=1}^k\\phi_j$\n    we define\n    $\\cF(\\phi)=\\max_j\\cF(\\phi_j)$.\n  \\item For $\\phi=\\bigwedge_{j=1}^k\\phi_j$ we define\n    $\\cF(\\phi)=1+\\max_j\\cF(\\phi_j)$.\n  \\item For $\\phi=\\exists {\\bf x}\\ \\phi'$ we define\n    $\\cF(\\phi)=\\cF(\\phi')$.\n  \\item For $\\phi=\\lnot\\phi',\\forall {\\bf x}\\ \\phi'$ we define\n    $\\cF(\\phi)=1+\\cF(\\phi')$.\n  \\end{itemize}\n\\end{Def}\n\nIn particular, the format $\\cF(\\phi)$ is bounded from above by the\nmaximum over the formats of the atomic predicates of $\\phi$, plus the\ndepth of the parse-tree for $\\phi$.\n\nDefinition~\\ref{def:formula-complexity} is motivated by the following\ntheorem, which shows that the *-format and *-degree of a set defined\nby a sub-Pfaffian formula $\\phi$ can be bounded in terms of\n$\\cF(\\phi)$ and $D(\\phi)$.\n\n\\begin{Thm}\\label{thm:formulas}\n  Let $\\phi$ be a sub-Pfaffian formula as above, with *-format $\\cF$\n  and *-degree $D$. Then the set defined by $\\phi$ has *-format\n  $\\const(\\cF)$ and *-degree $\\operatorname{poly}} \\def\\J{\\operatorname{J}_{\\cF}(D)$.\n\\end{Thm}\n\nThe proof of Theorem~\\ref{thm:formulas} is given\nin~\\secref{sec:proof-formulas}.\n\n\n\\subsection{Upper bound on homologies}\n\nTheorem~\\ref{thm:main} implies in particular an upper bound\n$\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$ for the number of connected components of a\nsub-Pfaffian set of *-format $\\cF$ and *-degree $D$. Below we\ngeneralize this to an upper bound for the homology of a sub-Pfaffian\nset in terms of the *-format and *-degree.\n\n\\begin{Thm}\\label{thm:bound}\n  The sum of the Betti numbers of a sub-Pfaffian set of *-format $\\cF$\n  and *-degree $D$ is bounded by $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$.\n\\end{Thm}\n\nThe proof of Theorem~\\ref{thm:bound} is given\nin~\\secref{sec:proof-bound}. Combining Theorem~\\ref{thm:formulas} with\nTheorem~\\ref{thm:bound} we obtain an upper bound for the homologies of\n(restricted) sub-Pfaffian sets defined by quantified formulas, which\nis polynomial in the degrees of the Pfaffian functions involved (for\nformulas of a fixed format). Such bounds have been obtained by\nKhovanskii \\cite{khov:book} for Pfaffian varieties; by Zell\n\\cite{zell:semi-pfaff} for semi-Pfaffian sets; and by Gabrielov,\nVorobjov and Zell \\cite{gvz:betti,gv:approximation} for sets defined\nusing quantifiers under certain topological\nrestrictions. See~\\secref{sec:comparison} for some discussion of these\nresults. However in spite of the significant work around the\ncomplexity of sub-Pfaffian sets, a polynomial estimate for general\ndefinable sets as provided by Theorem~\\ref{thm:bound} does not seem to\nhave been previously known.\n\n\\subsection{Applications and motivation}\n\nOur goal in this paper is to provide a framework that can be used to\neffectivize most of the results of o-minimal geometry in the\nrestricted sub-Pfaffian structure, with polynomial dependence on the\ndegree. Indeed, many proofs in the theory of o-minimality can be\ncarried out verbatim using Theorems~\\ref{thm:main}\nand~\\ref{thm:formulas} to obtain such polynomial bounds. As an example\nwe have the following.\n\n\\begin{Cor}\n  Let $A\\subset I^n$ be a restricted sub-Pfaffian set of *-format\n  $\\cF$ and *-degree $D$. Then the topological closure and the smooth\n  part of $A$ have *-format $\\const(\\cF)$ and *-degree $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$.\n\n  Let $f:A\\to B$ be a restricted sub-Pfaffian map of *-format $\\cF$\n  and $*$-degree $D$. Then for every $r\\in{\\mathbb N}} \\def\\Q{{\\mathbb Q}} \\def\\P{{\\mathbb P}$ the $C^r$-smooth locus\n  of $f$ has *-format $\\const(\\cF,r)$ and *-degree $\\operatorname{poly}} \\def\\J{\\operatorname{J}_{\\cF,r}(D)$.\n\\end{Cor}\n\\begin{proof}\n  All of the statements follow by defining the relevant sets using\n  first-order formulas in the restricted sub-Pfaffian language and\n  applying Theorem~\\ref{thm:formulas}.\n \n  For instance, to define the smooth part of a set $A$ having\n  dimension $m\\in{\\mathbb N}} \\def\\Q{{\\mathbb Q}} \\def\\P{{\\mathbb P}$ at each point, we can define it as a locus of\n  points $x \\in A$ such that a neighbourhood of $x$ in $A$ is the\n  graph of a $C^1$-smooth map. More precisely, the smooth part of $A$\n  is the set of all points $x \\in A$ each having a neighbourhood $U$\n  in $A$ such that there exists a linear map $T:{\\mathbb R}} \\def\\C{{\\mathbb C}} \\def\\Z{{\\mathbb Z}^n \\to{\\mathbb R}} \\def\\C{{\\mathbb C}} \\def\\Z{{\\mathbb Z}^m$ such\n  that the restriction $T|_U$ is one-to-one, $T(U)$ contains a\n  neighbourhood of $T(x)$ in ${\\mathbb R}} \\def\\C{{\\mathbb C}} \\def\\Z{{\\mathbb Z}^m$, and the inverse to $T|_U$ is a\n  $C^1$-smooth map with the Jacobian non-vanishing at every point in\n  $T(U)$. It is straightforward to reduce all of this to first-order\n  formulas (an ``$\\epsilon$-$\\delta$ definition''), and we leave the\n  details to the reader.\n\\end{proof}\n\nOne of our main motivations for developing the theory in this paper is\nin relation to the Pila-Wilkie counting theorem \\cite{pila-wilkie} and\nits applications in diophantine geometry (see the survey\n\\cite{scanlon:survey}).\n\nSeveral applications of the counting theorem involve the geometry of\nelliptic curves and abelian varieties --- the most famous example\nperhaps being the proof of the Manin-Mumford conjecture by\nPila-Zannier \\cite{pz:manin-mumford}. In these applications one\nconsiders sets defined using elliptic and abelian functions. Since\nthese functions are restricted sub-Pfaffian by an observation of\nMacintyre \\cite{macintyre:pfaffian}, the definable sets are restricted\nsub-Pfaffian. One can therefore hope to effectivize the counting\ntheorem in this context, and subsequently obtain effective results for\ndiophantine problems. Toward this end Jones and Thomas\n\\cite{jt:counting-surfaces} have established a version of the counting\ntheorem for certain surfaces definable in the restricted sub-Pfaffian\nstructure, and Jones and Schmidt have explored several applications of\nthis in diophantine geometry \\cite{js:mm-extensions,js:pfaffian-def}.\n\nIn an upcoming paper by the first author with Jones, Schmidt and\nThomas, we use the framework developed in the present paper to extend\nthe result of \\cite{jt:counting-surfaces} to arbitrary restricted\nsub-Pfaffian sets of arbitrary dimension, and improve the dependence\nof the effective constants to make them polynomial in the\ndegrees. This can be viewed as a case of effectivizing an o-minimal\nproof in the sense described above, albeit for a much more technically\ninvolved statement. We expect that this result will greatly extend the\nscope of potential applications in diophantine geometry, as well as\ngive rise to more reasonable (indeed, polynomial in degrees) estimates\nin these applications.\n\n\n\\subsection{Comparison with previous results}\n\\label{sec:comparison}\n\nGabrielov and Vorobjov have established many results on effectivity of\noperations in the restricted sub-Pfaffian category. For a survey we\nrefer the reader to \\cite{gv:complexity-computations}. In these works,\nthe notion of format and degree for sub-Pfaffian sets is similar to\nours but more straightforward: one considers simply projections of\nsemi-Pfaffian sets, rather than projections of their connected\ncomponents. In particular, in \\cite{gv:cell-decomposition} Gabrielov\nand Vorobjov prove a cell decomposition result quite similar to our\nTheorem~\\ref{thm:main}, as follows (where we omit the explicit,\ndoubly-exponential dependence on the format).\n\n\\begin{Fact}\\label{fact:main-old}\n  Let $\\{Y_\\alpha \\},\\ Y_\\alpha \\subset I^n$ be a collection of $N$\n  sub-Pfaffian sets of format $\\cF$ and degree $D$. Then there exists\n  a linear transformation $L:{\\mathbb R}} \\def\\C{{\\mathbb C}} \\def\\Z{{\\mathbb Z}^n\\to{\\mathbb R}} \\def\\C{{\\mathbb C}} \\def\\Z{{\\mathbb Z}^n$ and a cylindrical cell\n  decomposition of ${\\mathbb R}} \\def\\C{{\\mathbb C}} \\def\\Z{{\\mathbb Z}^n$, compatible with each $L(Y_\\alpha)$, such\n  that each cell in the decomposition is sub-Pfaffian and has format\n  $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(N,D)$, and the number of cells and their degree is\n  $\\operatorname{poly}} \\def\\J{\\operatorname{J}_{\\cF}(N, D)$.\n\\end{Fact}\n\nThere are two main differences compared to Theorem~\\ref{thm:main}:\nfirst, the application of the linear transformation $L$; and second,\nmore crucially, the dependence of the format of the cells on the\ndegree $D$. These are crucial limitation, which make it impossible to\napply this result to obtain analogs of Theorems~\\ref{thm:formulas}\nand~\\ref{thm:bound}: first, because in the recursive proofs it is\nessential that the cells preserve the order of coordinates; and\nsecond, more crucially, because after the first recursive application\nthe format becomes dependent on $D$, and the complexity of any further\noperations performed on such cells is no longer polynomial in $D$.\n\nWe are also unable to sharpen Fact~\\ref{fact:main-old} to eliminate\nthe dependence of the format on $D$, and this appears to be a\nfundamental difficulty\\footnote{The linear transformation problem\n  seems less fundamental, and could probably be avoided using a\n  strategy similar to the one used in the present paper.}. Our main\nobservation in the present paper is that with the revised notions of\n*-format and *-degree it is possible, at a crucial point in the cell\ndecomposition algorithm, to produce cells with format independent of\n$D$. However other aspects of the algorithm become more delicate with\nthese notions. The main reason is that the known approaches to\neffective cell decomposition involve reductions using topological\nclosure and frontier, and in our setting one must take care to avoid\ndifferent components becoming glued along their common boundary when\nperforming these operations. For this reason the strategy of cell\ndecomposition employed in the present paper differs significantly from\nthat of \\cite{gv:cell-decomposition}.\n\n\\begin{Rem}[Comparing the two notions of format]\\label{rem:comparing-formats}\n  Since the number of connected components of a semi-Pfaffian set\n  $X_\\alpha$ of format $\\cF$ and degree $D$ is bounded by\n  $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$ according to Fact~\\ref{fact:semi-pfaff-betti}, it is\n  clear that a sub-Pfaffian set of format $\\cF$ and degree $D$ has\n  *-format $\\cF$ and *-degree $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$.\n\n  To obtain bounds in the converse direction, it is necessary to compute\n  the format and degree (as a sub-Pfaffian set) of a connected\n  component $X_\\alpha^\\circ$, for a given semi-Pfaffian set $X_\\alpha$\n  of format $\\cF$ and degree $D$. Currently we only know how to do\n  this by computing a cell-decomposition using\n  Fact~\\ref{fact:main-old}, giving $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$ cells of format\n  $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$. Accordingly a sub-Pfaffian set of *-format $\\cF$ and\n  *-degree $D$ has format and degree $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$. The dependence of\n  the format on $D$ is what makes the notions of (unstarred) format\n  and degree much less efficient. We do not venture to conjecture\n  whether this dependence can be removed in general, but in any case\n  this appears to be a highly non-trivial problem.\n\\end{Rem}\n\nWe also note that analogs of Theorem~\\ref{thm:bound} for quantified\nformulas have been pursued, using an entirely different topological\napproach, in the work of Gabrielov, Vorobjov and Zell\n\\cite{gvz:betti}. This paper establishes similar bounds (and also with\nan explicit dependence on the format) for sets defined by quantified\nformulas in a prenex form:\n\\begin{equation}\n  \\{x\\in[0,1]^n : Q_1y_1Q_2y_2\\cdots Q_\\nu y_\\nu ((x,y)\\in X)\\}\n\\end{equation}\nwhere $X$ is a semi-Pfaffian set that is either open or closed, and\n$Q_1,\\ldots,Q_\\nu\\in\\{\\exists,\\forall\\}$. For formulas involving only\nexistential quantifiers, Gabrielov and Vorobjov\n\\cite{gv:approximation} have managed to remove the topological\ncondition on $X$ by an approximation method.\n\n\\begin{Rem}\n  In an unpublished manuscript Clutha \\cite{clutha} extended the\n  method of \\cite{gv:approximation} to an arbitrary number of\n  quantifiers, combining with the ideas of \\cite{gvz:betti}, and\n  claimed in particular a result implying Theorem~\\ref{thm:bound} for\n  arbitrary formulas. However the proof of this result contains a\n  substantial gap and we are presently not able to repair it.\n\\end{Rem}\n\n\n\\section{Preliminaries on semi-Pfaffian sets}\n\\label{sec:pfaff-background}\n\nWe will require the notion of a (weak) stratification of a\nsemi-Pfaffian set.\n\n\\begin{Def}[\\protect{\\cite[Definition~5]{gv:stratifications}}]\n  A \\emph{(weak) stratification} of a semi-Pfaffian set $X$ is a\n  subdivision of $X$ into a disjoint union of smooth, not necessarily\n  connected, semi-Pfaffian subsets $X_\\alpha$ called\n  \\emph{strata}. The system of equalities and inequalities for each\n  stratum $X_\\alpha$ of codimension $k$ includes a set of $k$\n  equalities $h_{\\alpha,1}=\\cdots=h_{\\alpha,k}=0$ whose differentials\n  define the tangent space of $X_\\alpha$ at every point of $X_\\alpha$.\n\\end{Def}\n\nWe will use the following formulation of the main result of\n\\cite{gv:stratifications}.\n\n\\begin{Fact}[\\protect{\\cite[Theorem~1]{gv:stratifications}}]\\label{fact:stratification}\n  Let $X\\subset I^n$ be semi-Pfaffian of format $\\cF$ and degree\n  $D$. Then there is a semi-Pfaffian stratification\n  $X=\\bigcup_\\alpha X_\\alpha$ of $X$ where the number of strata and\n  their degrees are $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$ and their format is $\\const(\\cF)$.\n\\end{Fact}\n\n\\begin{Rem}\\label{rem:stratification-family}\n  We will also need a \\emph{parametric version} of\n  Fact~\\ref{fact:stratification}: if $X\\subset I^n\\times I^m$ is\n  semi-Pfaffian of format $\\cF$ and degree $D$, then there exists a\n  collection $\\{S_\\alpha \\},\\ S_\\alpha \\subset I^n \\times I^m$ of\n  semi-Pfaffian sets with their number and degree $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$ and\n  their format $\\const(\\cF)$ such that the following holds. For any\n  $x\\in I^n$, the collection $\\{ S_\\alpha\\cap\\pi_n^{-1}(x) \\}$ forms a\n  stratification of the fiber $X\\cap\\pi_n^{-1}(x)$, with\n \n  $\\dim (S_\\alpha\\cap\\pi_n^{-1}(x))$ independent of\n  $x \\in \\pi_n (S_\\alpha)$ for each $\\alpha$.  The proof of this is\n  the same as the proof of Fact~\\ref{fact:stratification}, treating\n  the coordinates in $I^n$ as parameters and performing the\n  construction in $I^m$.\n\\end{Rem}\n\nWe also require the following result on the complexities of the {\\em\n  closure} $\\overline X$ and \\emph{frontier}\n$\\partial X:=\\overline X\\setminus X$ of a semi-Pfaffian set $X$.\n\n\\begin{Fact}[\\protect{\\cite{gabrielov:closure}}]\\label{fact:closure}\n  Let $X\\subset I^n$ be a semi-Pfaffian set of format $\\cF$ and degree\n  $D$. Then the closure $\\overline X$ and the frontier $\\partial X$\n  are semi-Pfaffian of format $\\const(\\cF)$ and degree $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$.\n\\end{Fact}\n\n\\begin{Rem}\\label{rem:closure-family}\n  We will also need a \\emph{parametric version} of\n  Fact~\\ref{fact:closure}: if $X\\subset I^m\\times I^n$ is\n  semi-Pfaffian of format $\\cF$ and degree $D$ then the union of the\n  fiberwise-closures\n  \\begin{equation}\n    \\{(x,y)\\in I^{n+m} : y\\in \\overline X_x \\},\\quad \\text{where}\\quad X_x:=\\{y:(x,y)\\in X\\}\n  \\end{equation}\n  is semi-Pfaffian of format $\\const(\\cF)$ and degree\n  $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$. This follows from the proof of\n  Fact~\\ref{fact:closure}, see\n  \\cite[Remark~5.4]{gv:complexity-computations}. The same remark holds\n  for the fiberwise frontier (by computing the fiberwise closure, and\n  then subtracting the $X$).\n\\end{Rem}\n\nWe will also need a bound on the sum of Betti numbers of semi-Pfaffian\nsets (in fact we will only require the zeroth Betti number, i.e. the\nnumber of connected components). This type of bound was proved by\nKhovanskii \\cite{khov:book} for Pfaffian sets, and extended by Zell to\nthe semi-Pfaffian class. We state only the part we need, omitting the\nmore precise dependence on the parameters which is achieved in\n\\cite{zell:semi-pfaff}.\n\n\\begin{Fact}[\\protect{\\cite[Main\n    result]{zell:semi-pfaff}}]\\label{fact:semi-pfaff-betti}\n  The sum of the Betti numbers of a semi-Pfaffian set of format $\\cF$\n  and degree $D$ does not exceed $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$.\n\\end{Fact}\n\n\n\n\\section{Cell decomposition of sub-Pfaffian sets}\n\\label{sec:semi-Pfaff-cells}\n\nIn this section we prove a result on cell decomposition of\nsub-Pfaffian sets, that is the main ingredient of the proof of\nTheorem~\\ref{thm:main}. As a shorthand, if $\\Delta$ is a collection of\nsets we write $\\pi_k(\\Delta)$ for the collection of the projections of\nthe elements of $\\Delta$.  Furthermore, $\\bigcup \\Delta$ will stand\nfor the union of sets in $\\Delta$.  Thus, $\\bigcup \\pi_k (\\Delta)$ is\nthe union of projections of sets in $\\Delta$.  We will freely use the\nresults on effective bounds for stratifications and frontiers\nfrom~\\secref{sec:pfaff-background} without explicit reference.\n\n\\begin{Lem}[Effective fiber cutting]\\label{lem:fiber-cut}\n  Let $X\\subset I^{n+m}$ be a semi-Pfaffian set of format $\\cF$ and\n  degree $D$. Then there exists a semi-Pfaffian set $\\widehat X$ of\n  format $\\const(\\cF)$ and degree $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$ such that\n  $\\pi_n(X)=\\pi_n(\\widehat X)$ and $\\pi_n\\rest{\\widehat X}$ has\n  zero-dimensional fibers.\n\\end{Lem}\n\n\\begin{proof}\n  We proceed by induction on the maximal fiber dimension $k$ of\n  $\\pi_n\\rest{X}$. If $k=0$ we are done, and otherwise we will\n  construct a set $X'$ with $\\pi_n(X)=\\pi_n(X')$ and maximal fiber\n  dimension smaller then $k$.\n\n  Let $\\{S_\\alpha \\},\\ S_\\alpha \\subset I^{n+m}$ denote a fiberwise\n  stratification of $X$, as in Remark~\\ref{rem:stratification-family}.\n  For any $S_\\alpha$ with fiber dimension smaller then $k$, we put\n  $S_\\alpha$ into $X'$. For any $S_\\alpha$ with fiber dimension $k$ we\n  do the following.\n  \\begin{enumerate}\n  \\item We put the fiberwise frontier of $S_\\alpha$ into $X'$ using\n    Remark~\\ref{rem:closure-family}.\n  \\item For any $j=1,\\ldots,m$ let $C_{\\alpha,j}$ be the set of\n    fiberwise critical points of the coordinate function $x_{n+j}$ on\n    the fibers of $\\pi_n\\rest{S_\\alpha}$. Then we stratify the fibers\n    of $\\pi_n\\rest{C_{\\alpha,j}}$ again by\n    Remark~\\ref{rem:stratification-family} and put all strata of\n    fiberwise dimension smaller than $k$ into $X'$.\n  \\end{enumerate}\n\n  Clearly $\\pi_n(X')\\subset\\pi_n(X)$. To prove the converse, let\n  $x\\in\\pi_n(X)$. If the fiber of $X$ over $x$ has strata of dimension\n  less then $k$, or strata of dimension $k$ with non-empty frontier,\n  we are done. Otherwise this fiber consists of smooth compact\n  $k$-dimensional manifolds. Then one of the coordinate functions\n  $x_{n+j}$ must be non-constant on this fiber, and the corresponding\n  set $C_{\\alpha,j}$ is non-empty and locally closed in the\n  fiber. Then the fiberwise stratification of $C_{\\alpha,j}$ must\n  contain strata of dimension less than $k$ that we put into\n  $X'$. This proves the claim.\n\\end{proof}\n\nThe following proposition is quite similar to Theorem~\\ref{thm:main}:\nthe difference is that here we essentially assume that we are given\nsub-Pfaffian sets with bounded format and degree (rather than their\n*-analogs), but produce cell decompositions with bounded *-format and\n*-degree. In~\\secref{sec:proof-main} we show how to deduce the general\ncase of Theorem~\\ref{thm:main} from this statement.\n\n\\begin{Prop}\\label{prop:main}\n  Let $\\{X_\\alpha\\},\\ X_\\alpha \\subset I^\\ell$, where $\\ell \\ge n$, be\n  a collection of $N$ semi-Pfaffian sets of format $\\cF$ and degree\n  $D$. Then there exists a sub-Pfaffian cell decomposition of $I^n$\n  compatible with each $\\pi_n(X_\\alpha)$ such that the number of cells\n  is $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(N,D)$, their *-format is $\\const(\\cF)$ and their\n  *-degree is $\\operatorname{poly}} \\def\\J{\\operatorname{J}_{\\cF}(D)$.\n\\end{Prop}\n\n\\begin{proof}\n  We will work by lexicographic induction on $(n,k)$ where\n  $$k:=\\max_\\alpha\\dim\\pi_{n-1}(X_\\alpha).$$\n  By Lemma~\\ref{lem:fiber-cut} we may assume that\n  $\\pi_n\\rest{X_\\alpha}$ has zero-dimensional fibers. Refining each\n  $X_\\alpha$ into its stratification, we may further assume without\n  loss of generality that $X_\\alpha$ is smooth, that\n  $\\pi_n\\rest{X_\\alpha}$ has rank $\\dim\\pi_n(X_\\alpha)$, and that\n  $\\pi_{n-1}\\rest{X_\\alpha}$ has constant rank. Let\n  $\\Pi :=\\{X_\\alpha\\}$. Using Fact~\\ref{fact:closure} we may also\n  assume that $\\Pi$ is closed under taking frontiers.\n\n\n  Let $\\Pi_k$ (resp. $\\Pi_{k+1}$ and $\\Pi_{<k}$) denote the collection\n  of sets $X_\\alpha$ with $\\dim X_\\alpha=k$ (resp. $\\dim X_\\alpha=k+1$\n  and $\\dim X_\\alpha<k$).\n\n  For each $X_\\alpha \\in\\Pi_{k+1}$ we also apply\n  Lemma~\\ref{lem:fiber-cut} to $\\pi_{n-1}\\rest{X_\\alpha}$ and add the\n  resulting sets to $\\Pi$ (after stratifying as above). With this\n  modification we may assume that\n  \\begin{equation}\\label{eq:Pi-decomp}\n    \\pi_{n-1}(\\Pi_{k+1})\\subset\\pi_{n-1}(\\Pi_k\\cup\\Pi_{<k}).\n  \\end{equation}\n\n  We construct a set $\\Sigma$ of closed semi-Pfaffian sets of bounded\n  *-format and *-degree, satisfying\n  $\\dim\\bigcup \\pi_{n-1} (\\Sigma)<k$. First, we put the closures of\n  the sets in $\\Pi_{<k}$ into $\\Sigma$.\n\n  Next, we wish to ensure that\n  $G:=\\bigcup \\pi_{n-1}(\\Pi)\\setminus \\bigcup \\pi_{n-1}(\\Sigma)$ is\n  smooth, and that for every $X_\\alpha \\in\\Pi_{k+1}$ the intersection\n  $\\pi_n(X_\\alpha)\\cap (G\\times I)$ is open in\n  $G\\times I$. By~\\eqref{eq:Pi-decomp} and since we already put\n  $\\Pi_{<k}$ in $\\Sigma$ we have\n  $G=\\bigcup \\pi_{n-1}(\\Pi_k)\\setminus \\bigcup \\pi_{n-1}(\\Sigma)$.\n  Since $\\pi_{n-1}\\rest X_\\alpha$ has constant rank $k$ for every\n  $X_\\alpha\\in\\Pi_k$, the set $\\bigcup \\pi_{n-1}(\\Pi_k)$ can be\n  thought of as an immersed smooth manifold with possible\n  self-intersections. It may be non-smooth only in points\n  $p\\in \\bigcup \\pi_{n-1}(\\Pi_k)$ where:\n  \\begin{enumerate}\n  \\item there exist two points $p_\\alpha\\in X_\\alpha\\in\\Pi_k$ and\n    $p_\\beta\\in X_\\beta\\in\\Pi_k$ such that\n    $p=\\pi_{n-1}(p_\\alpha)=\\pi_{n-1}(p_\\beta)$ but the projections of\n    the germs $(X_\\alpha,p_\\alpha)$ and $(X_\\beta,p_\\beta)$ under\n    $\\pi_{n-1}$ (which are both smooth manifolds) are different;\n  \\item the point $p$ is also in\n    $\\bigcup \\pi_{n-1}(\\{ \\partial X_\\alpha |\\ X_\\alpha \\in \\Pi_k \\})$.\n  \\end{enumerate}\n  We show how to put the points corresponding to these two cases in\n  $\\Sigma$, and it then follows that $G$ is smooth. The second case is\n  rather simple: for any $X_\\alpha\\in\\Pi_{<k}\\cup\\Pi_k$ we add the\n  frontier $\\partial X_\\alpha$ to $\\Sigma$. We also note for later\n  purposes that, since the sets $X_\\alpha$ are bounded, this implies\n  \\begin{equation}\\label{eq:good-boundary}\n    \\partial\\pi_n(X_\\alpha)\\subset \\bigcup \\pi_n (\\Sigma).\n  \\end{equation}\n  To handle the first case, consider for every pair\n  $X_\\alpha,X_\\beta\\subset I^\\ell$ in $\\Pi_k$ the set\n  \\begin{equation}\\label{eq:product}\n    \\{ (x,y) \\in X_\\alpha \\times X_\\beta:\\ x_1=y_1, \\ldots ,x_{n-1}=y_{n-1} \\}\n    \\subset I^\\ell\\times I^\\ell\n  \\end{equation}\n  where $x$ denotes the coordinates on the first $I^\\ell$ and $y$ on\n  the second.  In what follows we will be routinely adjusting the\n  dimension $\\ell$ so that the set $I^\\ell$ contains the new\n  semi-Pfaffian sets we create, in this case the set in\n  (\\ref{eq:product}).  We stratify this set and put (the closure of)\n  every stratum of dimension less than $k$ into $\\Sigma$. It is clear\n  that these strata cover the points described in the first case. Thus\n  we established that $G$ is smooth.\n\n  For $X_\\alpha\\in\\Pi_{k+1}$ we know that $\\pi_n\\rest{X_\\alpha}$ has\n  constant rank $k+1$ and $\\pi_{n-1}\\rest{X_\\alpha}$ has constant rank\n  $k$. Therefore, the projection\n  $X_\\alpha\\cap\\pi_{n-1}^{-1}(G)\\to G\\times I$ is a submersion, and\n  indeed $\\pi_n(X_\\alpha)\\cap(G\\times I)$ is open in $G\\times I$.\n\n  For later purposes we put one more set into $\\Sigma$, controlling\n  the possible interactions between different elements of $\\Pi_k$. For\n  any $X_\\alpha,X_\\beta\\in\\Pi_k$ we consider the set\n  \\begin{equation}\\label{eq:Z-def}\n    Z_{\\alpha,\\beta}:=\n    (\\{ (x,y) \\in X_\\alpha \\times X_\\beta:\\ x_1=y_1, \\ldots ,x_{n}=y_{n} \\}\n    \\subset I^\\ell\\times I^\\ell.\n  \\end{equation}\n  We stratify $Z_{\\alpha,\\beta}$ and add the closures of the strata of\n  dimension at most $k-1$, and the frontier of the strata of dimension\n  $k$, to $\\Sigma$.\n\n  Now apply induction to obtain a cell decomposition of $I^{n-1}$\n  compatible with $\\pi_{n-1}(\\Pi)$ and with $\\pi_{n-1}(\\Sigma)$. If a\n  cell $C$ is disjoint from $\\bigcup \\pi_{n-1}(\\Pi)$ then $C\\times I$\n  is a cell compatible with every $\\pi_n(X_\\alpha)$.\n\n  Let $C$ be a cell contained in\n  $G= \\bigcup \\pi_{n-1}(\\Pi)\\setminus \\bigcup \\pi_{n-1}(\\Sigma)$. We\n  will show how to construct cells over $C$ that are compatible with\n  every $\\pi_n(X_\\alpha)$. First consider\n  $X_\\alpha\\in\\Pi_{k+1}$. Recall that\n  $\\pi_n(X_\\alpha)\\cap (G\\times I)$ is open in $G\\times I$. It follows\n  that its boundary in $G\\times I$ agrees with its frontier. If a cell\n  in $C\\times I$ is compatible with the boundary of $\\pi_n(X_\\alpha)$\n  (or, equivalently in this case, with the frontier) then it is also\n  compatible with $\\pi_n(X_\\alpha)$ by elementary topology. Since we\n  assume that $\\Pi$ is closed under taking frontiers, it will be\n  enough to construct cells over $C$ compatible with $\\pi_n(\\Pi_k)$.\n\n  We claim that for any $X_\\alpha\\in\\Pi_k$, the set $\\pi_n(X_\\alpha)$\n  is a union of graph cells over $C$. Consider an arbitrary point\n  $c_0\\in C$. Then the fiber of $\\pi_n(X_\\alpha)$ over $c_0$ consists\n  of finitely many points. By construction, $\\pi_n(X_\\alpha)$ maps\n  submersively to $C$, so as we move $c_0$ these points locally move\n  continuously. Moreover the points must remain in $\\pi_n(X_\\alpha)$\n  as we continue globally to $C$, for otherwise one of these points\n  would meet $\\partial\\pi_n(X_\\alpha)$\n  contradicting~\\eqref{eq:good-boundary} since $C\\subset G$ is\n  disjoint from $\\pi_{n-1}(\\Sigma)$. Note that for the same reasons\n  $X_\\alpha$ itself is also a union of graphs of continuous maps over\n  $C$. If $s:C\\to I$ is such a section of $\\pi_n(X_\\alpha)$ we denote\n  by $\\hat s$ the corresponding extension to a section of $X_\\alpha$.\n\n  If $s_\\alpha,s_\\beta:C\\to I$ is a pair of such sections obtained\n  from $X_\\alpha,X_\\beta$ then one of\n  \\begin{align}\n    s_\\alpha &< s_\\beta   &   s_\\alpha &=s_\\beta   &   s_\\alpha&>s_\\beta\n  \\end{align}\n  holds uniformly over $C$. Indeed, otherwise there should be a point\n  $c_0\\in C$ where $s_\\alpha(c_0)=s_\\beta(c_0)$ and a neighborhood\n  where this does not hold identically. This implies that\n  $(c_0,c_\\alpha(c_0))$ belongs to a stratum of dimension at most\n  $(k-1)$, or to the frontier of a $k$-dimensional stratum of\n  $Z_{\\alpha,\\beta}$, and hence $c_0\\not\\in G$ contradicting\n  $C\\subset G$.\n\n  By the above, the set of sections obtained from any of the\n  $X_\\alpha\\in\\Pi_k$ is\n  $$\\{s_1<\\cdots<s_q\\},$$\n  where we may suppose for simplicity that $s_1=0$ and $s_q=1$. Also\n  note since $\\pi_n(X_\\alpha)$ is a union of disjoint graphs of\n  continuous functions, their number is bounded by the number of\n  connected components of $X_\\alpha$. By\n  Fact~\\ref{fact:semi-pfaff-betti} we therefore get at most\n  $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$ sections from each $X_\\alpha$, and in total\n  $q=\\operatorname{poly}} \\def\\J{\\operatorname{J}_F(N,D)$.\n\n  A cell decomposition of $C\\times I$, compatible with each\n  $X_\\alpha$, is given by the cells\n  \\begin{multline}\n    \\{x_n=s_1(x_1,\\ldots,x_{n-1})\\}, \\\\\n    \\{s_1(x_1,\\ldots,x_{n-1})<x_n<s_2(x_1,\\ldots,x_{n-1})\\},\\\\\n    \\ldots, \\{x_n=s_q(x_1,\\ldots,x_{n-1})\\}.\n  \\end{multline}\n  We show that each of these cells is a sub-Pfaffian set with\n  appropriately bounded *-format and *-degree. Specifically we show\n  that the graph of each section $s_j$ over $C$ is a sub-Pfaffian set\n  with the appropriate *-bounds, and it is then a simple exercise to\n  construct each of the cells above (where for the interval cells one\n  repeats the construction for $s_j,s_{j+1}$ and works in the direct\n  product).\n\n  Recall that $C=\\pi^\\ell_n(Z^\\circ)$ is the projection of the\n  connected component $Z^\\circ$ of some semi-Pfaffian set\n  $Z\\subset I^\\ell$ (for an appropriate $\\ell$). Suppose that $s_j$ is\n  a section corresponding to $X_\\alpha$. Then the graph of $s_j$ is a\n  connected component of the set $\\pi_n(X_\\alpha)\\cap(C\\times I)$. In\n  fact, since the section $s_j$ extends to a section of $X_\\alpha$,\n  the graph of $s_j$ the projection $\\pi_n(W_j^\\circ)$ of a connected\n  component $W_j^\\circ$ of the set\n  \\begin{equation}\n    W_j := \\{ (x,y) \\in Z \\times X_\\alpha : x_1=y_1, \\ldots ,x_{n-1}=y_{n-1} \\}\n  \\end{equation}\n  given by $Z^\\circ$ in the $x$-coordinates and by the graph of\n  $y=\\hat s_j(x)$. Here we ordered the coordinates in such a way that\n  $\\pi_n(x,y)=(y_1,\\ldots,y_n)$. This proves that the graph of $s_j$\n  is indeed sub-Pfaffian with appropriately bounded *-format and\n  *-degree. Since each cell is constructed using one or two of these\n  graphs, their *-degree is indeed bounded by $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$ (noting\n  in particular that there is no dependence on $N$).\n\n  Finally, we must construct cells covering\n  $\\bigcup \\pi_{n-1}(\\Sigma)\\times I$ and compatible with every\n  $\\pi_n(X_\\alpha)$. Let\n  \\begin{equation}\n    Y_\\alpha:=\\left(\\bigcup \\pi_{n-1} (\\Sigma )\\times I \\right)\\cap\\pi_n(X_\\alpha) = \\pi_n(\\widehat Y_\\alpha),\n  \\end{equation}\n  where it is easy to choose $\\widehat Y_\\alpha$ to be a semi-Pfaffian\n  set of bounded complexity. It will suffice to construct a cell\n  decomposition of $I^n$ compatible with\n  $\\bigcup \\pi_n(\\Sigma)\\times I$ and $\\{Y_\\alpha\\}$ and take from it\n  only the cells covering $\\bigcup \\pi_n(\\Sigma )\\times I$. This can\n  now be achieved by induction on $k$, noting that for any $Z$ in this\n  collection $\\pi_{n-1}(Z)\\subset \\pi_{n-1}( \\Sigma)$ has dimension\n  strictly less than $k$.\n\\end{proof}\n\n\\section{Proofs of the main results}\n\\label{sec:proofs}\n\n\\subsection{Proof of Theorem~\\ref{thm:main}}\n\\label{sec:proof-main}\n\nWe will require a couple of elementary lemmas, whose proofs are left\nan exercises for the reader.\n\n\\begin{Lem}\n  Suppose that a cell $C$ is compatible with a set $X$. Then it is\n  compatible with each connected component of $X$.\n\\end{Lem}\n\n\\begin{Lem}\n  Suppose that a cell decomposition of $I^k$ is compatible with a set\n  $A$. Then the induced decomposition on $I^n$ is compatible with\n  $\\pi^k_n(A)$.\n\\end{Lem}\n\nWe are now ready to finish the proof.  Let\n$X_{\\alpha,\\beta}\\subset I^k$ denote semi-Pfaffian sets and\n$X_{\\alpha,\\beta}^\\circ$ connected components such that\n$Y_\\alpha=\\bigcup_\\beta\\pi^k_n(X_{\\alpha,\\beta}^\\circ)$. Use\nProposition~\\ref{prop:main} to find a sub-Pfaffian cell decomposition\nof $I^k$ compatible with $\\{X_{\\alpha,\\beta}\\}$ with suitably bounded\n*-format and *-degree. Then by the preceding lemmas this cell\ndecomposition is compatible with $X_{\\alpha,\\beta}^\\circ$, and the\ninduced decomposition on $I^n$ is therefore compatible with\n$Y_\\alpha$.\n\n\\begin{Rem}\n  Note that in the proof of Theorem~\\ref{thm:main}, even though we are\n  interested in a cell decomposition of $I^n$, we apply\n  Proposition~\\ref{prop:main} to obtain a cell decomposition of the\n  full space $I^k$. This allows us to ensure the compatibility with\n  the projections of each connected component $X_{\\alpha,\\beta}^\\circ$\n  separately. Applying Proposition~\\ref{prop:main} to the projections\n  $\\pi_n(X_{\\alpha,\\beta})$ would not suffice. It is therefore crucial\n  at this point that Proposition~\\ref{prop:main} produces a cell\n  decomposition in the original coordinates, without applying a linear\n  transformation (or at least preserving the projection\n  $\\pi^k_n$). For this reason, the cell decomposition algorithm of\n  Gabrielov-Vorobjov \\cite{gv:cell-decomposition}, which applies such\n  a linear transformation to the coordinates, would not suffice for\n  our purposes.\n\\end{Rem}\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:formulas}}\n\\label{sec:proof-formulas}\n\nThe proof is a routine recursive argument on the structure of $\\phi$.\nThe statements for existential quantifiers and disjunctions hold by\ndefinition. Up to re-writing\n\\begin{align}\n  \\forall{\\bf x}\\phi &\\equiv \\lnot\\exists{\\bf x}(\\lnot\\phi), \\\\\n  \\land_{i=1}^k\\phi_i &\\equiv \\lnot\\lor_{i=1}^k(\\lnot\\phi_i),\n\\end{align}\neverything else follows from the statement for negations\n$\\phi=\\lnot\\phi'$. This is proved by constructing a cell decomposition\nof $I^n$ compatible with the set $X$ defined by $\\phi'$, and taking\nthe union of all cells disjoint from $X$ in this decomposition.\n\n\n\\subsection{Proof of Theorem~\\ref{thm:bound}}\n\\label{sec:proof-bound}\n\nWe first establish an effective result on the existence of\ntriangulations in the restricted sub-Pfaffian class. Recall that a\n\\emph{finite simplicial complex} in ${\\mathbb R}} \\def\\C{{\\mathbb C}} \\def\\Z{{\\mathbb Z}^n$ is a finite collection\n$K=\\{\\bar\\sigma_1,\\ldots,\\bar\\sigma_p\\}$ of (closed) simplices\n$\\bar\\sigma_i\\subset{\\mathbb R}} \\def\\C{{\\mathbb C}} \\def\\Z{{\\mathbb Z}^n$ such that the intersection of any pairs\n$\\bar\\sigma_i\\cap\\bar\\sigma_j$, if not empty, is a common face of\n$\\bar\\sigma_i$ and $\\bar\\sigma_j$; and such that any face of any\n$\\bar\\sigma_i$ also belongs to $K$. We write $|K|$ for the union of\nall simplices in $K$.\n\n\\begin{Thm}\\label{thm:triangulation}\n  Let $Y\\subset[0,1]^n$ be a closed sub-Pfaffian set, and\n  $X_1,\\ldots,X_k\\subset Y$ be sub-Pfaffian subsets, and suppose all\n  these sets have *-format bounded by $\\cF$ and *-degree bounded by\n  $D$.\n\n  Then there exists a finite simplicial complex $K$ with vertices in\n  $\\Q^n$ and a definable homeomorphism $\\Phi:|K|\\to Y$ such that each\n  $X_i$ is a union of images by $\\Phi$ of open simplices of\n  $K$. Moreover $K$ contains $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$ simplices, and $\\Phi$ has\n  *-format $\\const(\\cF)$ and *-degree $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$.\n\\end{Thm}\n\nA proof of the triangulation theorem in the o-minimal setting can be\nfound in \\cite{vdd}, where it follows a similar proof for the\nsemialgebraic class in \\cite{df:homology}. For convenience we refer\nthe reader to the alternative presentation given in\n\\cite[Theorem~4.4]{coste:o-minimality}, which also establishes\nTheorem~\\ref{thm:triangulation} without the effective estimates for\narbitrary o-minimal structures. Deriving the effective estimates from\nthis proof in the restricted sub-Pfaffian context is a routine\nexercise in the application of Theorems \\ref{thm:main} and\n\\ref{thm:formulas}: one only nees to verify that in the proof of\n\\cite[Theorem~4.4]{coste:o-minimality}, the triangulating map $\\Phi$\nis indeed defined by a first-order formula $\\phi$ with\n$\\cF(\\phi)=\\const(\\cF)$ and $D(\\phi)=\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$. As this\nverification is entirely straightforward from the presentation of\nloc. cit., we leave the details as an exercise for the reader.\n\nTo deduce Theorem~\\ref{thm:bound} we first apply\nTheorem~\\ref{thm:triangulation} with $Y=[0,1]^n$ and $X$ the given\nsub-Pfaffian set. We obtain a homeomorphism $\\Phi:|K|\\to [0,1]^n$.  In\nparticular, the sum of the Betti numbers of $X$ is equal to that of\n$\\Phi^{-1}(X)$, which is a union of at most $\\operatorname{poly}} \\def\\J{\\operatorname{J}_\\cF(D)$\nsimplices. This set being semialgebraic, the bound on the sum of Betti\nnumbers now follows, e.g., from \\cite[Theorem~1]{gv:dcg}.\n\n\n\\bibliographystyle{plain} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Reeb graph is a construction that can be used to study a topological space with a real valued function by tracking the relationships between connected components of level sets. \nIt was originally developed in the context of Morse theory \\cite{Reeb1946}, and was later introduced for shape analysis by Shinagawa et al.~\\cite{Shinagawa1991}. \nSince then, it has attracted much attention due to its wide use for various data analysis applications, such as shape comparison \\cite{Hilaga2001, Escolano2013}, denoising \\cite{Wood2004}, and shape understanding \\cite{Dey2013,HA03}; see \\cite{Biasotti2008} for a survey. \nRecently, the applications of Reeb graphs have been further broadened to summarizing high-dimensional and\/or complex data, in particular, reconstructing non-linear 1-dimensional structure in data \\cite{NBPF11,Ge2011,Chazal2014} and summarizing collections of trajectory data \\cite{BBKSS13}. \nIts practical applications have also been facilitated by the availability of efficient algorithms for computing the Reeb graph from a piecewise-linear function defined on a simplicial complex  \\cite{Parsa2012,Harvey2010,Doraiswamy2012}. \n\nIn addition to the standard construction, a generalization of the Reeb graph construction, known as Mapper, \\cite{Singh2007}, has proven extremely useful in the field of topological data analysis \\cite{Yao2009, Nicolau2011}.\nA variant of Mapper for real-valued functions, called the $\\alpha$-Reeb graph, was used in \\cite{Chazal2014} to study data sets with 1-dimensional structure.\n\nGiven the popularity of the Reeb graph and related constructions for practical data analysis applications, it is desirable and increasingly necessary to understand how robust (stable) these structures are in the presence of noise. \nConsequently, several metrics for comparing Reeb graphs have been proposed recently. \nThese include the interleaving distance \\cite{deSilva2014}, the functional distortion distance \\cite{Bauer2014}, and the combinatorial edit distance \\cite{DiFabio2014b}. \nWe note that the latter is limited to Reeb graphs resulting from Morse functions defined on surfaces (2-manifolds). \nIn addition, Morozov et.~al proposed an interleaving distance for a simpler variant of the Reeb graph, the so-called merge tree \\cite{Morozov2013}. \n\nIn this paper, we study the relation between the only two distance measures for general Reeb graphs proposed in the literature:  the functional distortion distance of \\cite{Bauer2014} and the interleaving distance of \\cite{deSilva2014}. \nThe former is based on concepts from metric geometry, and is defined by treating both graphs as metric spaces and inspecting continuous maps between them. \nThe latter, on the other hand, is defined using ideas of category theory, utilizing the equivalence between Reeb graphs and a particular class of cosheaves. \nHowever, in essence, both construct a near-isomorphism between the two input graphs of study. \nIn Sections \\ref{sec:relation} and \\ref{sec:bottleneck}, we explore this connection between the two distances, and show that indeed, the functional distortion distance and the interleaving distances are strongly equivalent on the space of Reeb graphs, meaning that they are within constant factor of each other. \nThis immediately leads to the bottleneck stability result for the Reeb graph interleaving distance. \n\n\n\n\\section{Definitions}\n\nGiven a topological space $\\X$ with a real valued function $f:\\X \\to \\R$, we  define the Reeb graph of $(\\X,f)$ as follows.\nWe say that two points in $\\X$ are equivalent if they are in the same path-connected  component of a level set $f\\inv(a)$ for $a \\in \\R$.\nThis is notated $x \\sim_f y$ or $x \\sim y$ if the function is obvious. \nThen the Reeb graph is the quotient space $\\X\/\\sim_f$. \nNote that the Reeb graph inherits a real valued function from its parent space.\nSee Fig.~\\ref{Fig:BasicExample} for an example.\n\\begin{figure}\n\\centering\n \\includegraphics[scale = .4]{ReebGraphBasicExample-Narrow}\n \\caption{A simple example of the Reeb graph (right) of a space (left).  Here and in all other drawn examples in this paper, the real valued function is denoted by height.}\n \\label{Fig:BasicExample}\n\\end{figure}\n\n\n\\subsection{Category of Reeb Graphs}\nFor nice enough functions $f:\\X \\to \\R$, such as Morse functions on compact manifolds or PL functions on finite simplicial complexes, the Reeb graph is, in fact, a finite graph. We will tacitly make this assumption on functions throughout the paper.\nThus, we will define the category of Reeb graphs, following \\cite{deSilva2014}, intuitively to be finite graphs with real valued functions.\nMorphisms will be given by function preserving maps between the underlying spaces as given in the following definition.\n\n\\begin{definition}\n\\label{Def:ReebCat}\nAn object of the category $\\Reeb$ is a finite graph, seen as a topological space, more precisely as a regular CW complex, together with a real valued function that is monotone on edges.  \nThis will equivalently be written as either $f:\\X \\to \\R$ or $(\\X,f)$.\nA morphism between $(\\X,f)$ and $(\\Y,g)$ is a map $\\phi:\\X \\to \\Y$ preserving function values, i.e., the following diagram commutes:\n\\begin{equation*}\n\\begin{tikzcd}\n \\X \\ar{r}{\\phi} \\ar{dr}{f}& \\Y \\ar{d}{g}\\\\\n & \\R\n\\end{tikzcd}\n\\end{equation*}\n\\end{definition}\n \nNote that because we assume that the function is monotone when restricted to the edges, we will often just think of the function as being given by the values on the vertices.\nAs an aside, notice that the process of taking the Reeb graph of a space with a function is an isomorphism in $\\Reeb$.\n\n\n\n\\subsection{Interleaving Distance}\n\n\n\nGiven a Reeb graph $(\\X,f)$, let $\\X_\\e$ denote the space $\\X \\times [-\\e,\\e]$ and define the $\\e$-smoothing of $(\\X,f)$ as the Reeb graph of the function\n\\begin{equation*}\n \\begin{array}{cccc}\n f_\\e: & \\X_\\e &\\to &\\R\\\\\n  & (x,t) & \\longmapsto & f(x) + t.\n \\end{array}\n\\end{equation*}\nThat is, the $\\e$-smoothing is the quotient space $\\X_\\e\/\\sim_{f_\\e}$.\nDenote this space by $\\TT_\\e(\\X,f)$ and notice that $\\TT_\\e(\\TT_\\e(\\X,f)) \\iso \\TT_{2\\e}(\\X,f)$.\nSometimes when we are focused on the underlying topological space, we will notat this as $\\TT_\\e(\\X)$.\nSee Fig.~\\ref{Fig:SmoothedExample} for an example.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale = .8]{SmoothingReebGraph-Example.pdf}\n \\caption{An example of a smoothed Reeb graph.  At left is the original graph $\\X$ with function $f$ given by height.  The middle space is $\\X_\\e = \\X \\times [-\\e,\\e]$ with the function $f_\\e(x,t) = f(x)+t$ still given by height.  At right is the Reeb graph of $(\\X_\\e,f_\\e)$, which is the smoothed Reeb graph $\\TT_\\e(\\X)$.}\n \\label{Fig:SmoothedExample}\n\\end{figure}\n\n\nAn $\\e$-interleaving of $(\\X,f)$ and $(\\Y,g)$ is a pair of function preserving maps $\\phi:(\\X,f) \\to \\TT_\\e(\\Y, g)$ and $\\psi:(\\Y,g) \\to \\TT_\\e(\\X, f)$ with the following requirements.\nFirst, note that there is a natural map $\\iota: (\\X,f) \\to \\TT_\\e(\\X,f)$ which sends $x\\in \\X$ to the equivalence class of $(x,0)$ in $\\X_\\e\/\\sim_{f_\\e}$.\nSimilarly, there is another natural map on $(\\Y,g)$, which we abuse notation and also denote $\\iota:(\\Y,g) \\to \\TT_\\e(\\Y,g)$.\nThere exists a map $\\phi_\\e:\\TT_\\e(\\X,f) \\to \\TT_{2\\e}(\\Y,g)$, uniquely defined on $\\im \\iota$ by the map $\\phi$, such that the diagram\n\\begin{equation*}\n\\begin{tikzcd}\n (\\X,f) \\ar{r}{\\iota}\\ar{d}{\\phi} &  \\TT_\\e(\\X,f)\\ar{d}{\\phi_\\e} \\\\\n \\TT_\\e(\\Y,g)  \\ar{r}{\\iota_\\e} & \\TT_{2\\e}(\\Y,g)\n\\end{tikzcd}\n\\end{equation*}\ncommutes.\nThe map $\\psi_{\\e}:\\TT_\\e(\\Y,g) \\to \\TT_{2\\e}(\\X,f)$ is defined similarly.\nThen we have the definition of the $\\e$-interleaving.\n\n\\begin{definition}[Geometric $\\e$-Interleaving]\nThe maps $\\phi:(\\X,f) \\to \\TT_\\e(\\Y,g)$ and  $\\psi:(\\Y,g) \\to \\TT_\\e(\\X,f)$ are a \\emph{(geometric) $\\e$-interleaving} if the diagram\n\\begin{equation*}\n\\begin{tikzcd}[\/tikz\/commutative diagrams\/row sep=large]\n (\\X,f) \\ar{r}{\\iota} \\ar{dr}[very near start]{\\phi}  & \n \\TT_\\e(\\X,f) \\ar{r}{\\iota_\\e} \\ar{dr}[very near start]{\\phi_\\e} & \n \\TT_{2\\e}(\\X,f) \\\\\n (\\Y,g) \\ar{r}{\\iota}  \\ar{ur}[very near start]{\\psi} & \n \\TT_\\e(\\Y,g) \\ar{r}{\\iota_\\e} \\ar{ur}[very near start]{\\psi_\\e} & \n \\TT_{2\\e}(\\Y,g)\n\\end{tikzcd}\n\\end{equation*}\ncommutes.\n\\end{definition}\n\nWe can use this definition of interleavings to define a distance on Reeb graphs.\n\\begin{definition}[Interleaving Distance, \\cite{deSilva2014}]\nThe \\emph{interleaving distance} between two Reeb graphs $(\\X,f)$ and $(\\Y,g)$ is defined to be\n\\begin{equation*}\nd_I(\\Ffunc,\\Gfunc) =\n\\inf\\left\\{ \\e \\mid \\text{there exists an $\\e$-interleaving between $(\\X,f),(\\Y,g)$} \\right\\}.\n \\end{equation*}\n\\end{definition}\n\nThe definition of the interleaving distance was motivated by the cosheaf structure of Reeb graphs.\nIt was shown in \\cite{deSilva2014} that the category of Reeb graphs is equivalent to a particular class of cosheaves, which can be thought of as functors $F:\\Int \\to \\Set$ giving a set for each open interval.\nThe idea is that given a Reeb graph $f = (\\X,f)$, we can construct the associated cosheaf $\\Ffunc = \\pi_0 \\circ f\\inv$.\nThis equivalence allows us to work with either the topological construction or the category theoretic, whichever is easier or more appropriate.\nSpecifics for this can be found in Appendix \\ref{Sect:Cosheaves}, and an excellent introduction can be found in \\cite{Curry2013}.\n\nLet the $\\e$-thickening of an interval $I = (a,b)$ be denoted by $I^\\e = (a-\\e,b+\\e)$.\nThen we can also consider the $\\e$-interleavings for the cosheaves.\n\\begin{definition}[Cosheaf $\\e$-Interleaving]\nGiven Reeb graphs $(\\X,f)$ and $(\\Y,g)$, \nan {\\bf $\\e$-interleaving} between the corresponding cosheaves $\\Ffunc = \\pi_0\\circ f\\inv$ and $\\Gfunc = \\pi_0 \\circ g\\inv$ is given by two families of maps\n\\begin{align*}\n\\phi_I:& \\pi_0\\circ f\\inv(I) \\to \\pi_0\\circ g\\inv(I^\\e),\\\\\n\\psi_I:&  \\pi_0\\circ g\\inv(I) \\to  \\pi_0\\circ f\\inv(I^\\e)\n\\end{align*}\nfor each open interval $I$.\nThese must be natural with respect to inclusions $I \\subseteq J$, i.e., the following diagrams commute, where the  vertical maps are induced by the inclusions $\\iota: I \\hookrightarrow J$ and $\\iota_\\e: I^\\e \\hookrightarrow J^\\e$:\n\\begin{equation*}\n \\begin{tikzcd}\n \\pi_0(f\\inv(I)) \\ar{r}{\\phi_I} \\ar{d}{\\iota^*}&  \\pi_0(g\\inv(I^\\e))  \\ar{d}{\\iota^*} &&\n \\pi_0(g\\inv(I)) \\ar{r}{\\psi_I} \\ar{d}{\\iota^*}&  \\pi_0(f\\inv(I^\\e))  \\ar{d}{\\iota^*}\\\\\n \\pi_0(f\\inv(J)) \\ar{r}{\\phi_J}&  \\pi_0(g\\inv(J^\\e)) &&\n \\pi_0(g\\inv(J)) \\ar{r}{\\psi_J}&  \\pi_0(f\\inv(J^\\e)) \\\\\n \\end{tikzcd}\n\\end{equation*}\nMoreover, we require that\n\\begin{equation*}\n\\psi_{I^\\e} \\circ \\phi_I : \\pi_0\\circ f\\inv(I) \\to \\pi_0\\circ f \\inv (I^\\e)\n\\end{equation*}\nis the map induced by the inclusions $f\\inv(I) \\subset f\\inv(I^\\e)$ and \n\\begin{equation*}\n\\phi_{I^\\e} \\circ \\psi_I : \\pi_0\\circ g\\inv(I) \\to \\pi_0\\circ g \\inv (I^\\e)\n\\end{equation*}\nis the map induced by the inclusions $g\\inv(I) \\subset g\\inv(I^\\e)$\nfor all $I$. \n\\end{definition}\nFirst, notice that for $\\e=0$ this is is precisely an isomorphism between $\\Ffunc, \\Gfunc$.\nNote that an interleaving could be equivalently defined as a pair of natural transformations between the appropriate functors.\nThen, the interleaving distance can be equivalently defined as \n\\begin{equation*}\nd_I((\\X,f),(\\Y,g)) =\n\\inf\\left\\{ \\e \\mid \\text{there exists an $\\e$-interleaving between $\\Ffunc = \\pi_0\\circ f\\inv,\\Gfunc = \\pi_0\\circ g\\inv$} \\right\\}\n \\end{equation*}\n for two cosheaves $\\Ffunc, \\Gfunc: \\Int \\to \\Set$.\nWe will therefore abuse notation and also denote this as $d_I((\\X,f),(\\Y,g))$.\n\n\\subsection{Functional Distortion Distance}\n\nFor a given path $\\pi$ from $u$ to $v$ in $(\\X,f) \\in \\Reeb$, we define the height of the path to be \n\\begin{equation*}\n \\textrm{height}(\\pi) = \\max_{x \\in \\pi} f(x) - \\min_{x \\in \\pi} f(x).\n\\end{equation*}\nThen we define the distance\n\\begin{equation*}\n d_f(u,v) = \\min_{\\pi:u \\to v} \\textrm{height}(\\pi)\n\\end{equation*}\nwhere $\\pi$ ranges over all paths from $u$ to $v$ in $\\X$.\nNote that this can be equivalently be defined by the minimum length of an interval $I$ such that $u$ and $v$ are in the same connected component of $f\\inv(I)$.  \n\nThe functional distortion distance between $(\\X,f)$ and $(\\Y,g)$ is now defined as follows:\n\\begin{definition}\n [Functional Distortion Distance, \\cite{Bauer2014}]\n \\label{Def:FuncDistortion}\n \n Given $(\\X,f),(\\Y,g) \\in \\Reeb$ and maps $\\phi:\\X \\to \\Y$ and $\\psi:\\Y\\to \\X$, let \n \\begin{equation*}\n  C(\\phi,\\psi) = \\{ (x,y) \\in \\X \\times \\Y \\mid \\phi(x) = y \\textrm { or } x = \\psi(y)\\}\n \\end{equation*}\n and \n \\begin{equation*}\n  D(\\phi,\\psi)  = \\sup_{\\substack{(x,y),(x',y')\\\\ \\in C(\\phi,\\psi)}} \\frac{1}{2} \\left| d_f(x,x') - d_g(y,y')\\right|.\n \\end{equation*}\n Then the \\emph{functional distortion distance} is defined to be \n \\begin{equation*}\n  d_{FD}(f,g) = \\inf_{\\phi,\\psi} \\max \\{ D(\\phi,\\psi),\\|f-g \\circ \\phi\\|_\\infty, \\|g-f \\circ \\psi\\|_\\infty \\}.\n \\end{equation*}\n\n\n\n\\end{definition}\nNote that since the maps $\\phi,\\psi$ are not required to preserve the function values, they are not Reeb graph morphisms in the sense of Definition~\\ref{Def:ReebCat}.\n\n\n\\section{Multivalued Maps and Continuous Selections}\n\nA multivalued map (or multimap) $F: X \\to Y$ is a correspondence which sends a point $x \\in X$ to a nonempty set $F(x) \\subset Y$.\nA selection of a multimap is a singlevalued function $f:X \\to Y$ such that $f(x) \\in F(x)$ for every $x \\in X$.\nSee  \\cite{Repovs1998} for an introduction to multimaps.\n\nNote that using the axiom of choice, a selection always exists; the trick is to find a continuous selection.\nThe Michael selection theorem gives a criterion for a multimap to have a continuous selection.\nHowever, in order to state it, we will need several definitions.\n\n\\begin{definition}\n A family $\\SS$ of subsets of a topological space $Y$ is equi-locally $n$-connected if for every $S \\in \\SS$, every $y \\in S$, and every neighborhood $W$ of $y$, there is a neighborhood $V$ of $y$ such that $V \\subset W$ and for every $S' \\in \\SS$ such that $V \\cap S' \\neq \\emptyset$, every continuous mapping of the $m$-sphere $\\Sbb^m$ into $S' \\cap V$ is null-homotopic in $S' \\cap W$ for $m \\leq n$. This is notated $\\SS \\in \\ELC^n$.\n\\end{definition}\nIn particular, we will be needing the case where $\\SS \\in \\ELC^0$, so the final requirement amounts to checking that $S'\\cap V$ is path connected.\n\n\n\\begin{definition}\nA multivalued map $F: X \\to Y$ is lower semicontinuous (LSC) if for every open $U \\subset Y$, the set $F\\inv(U) = \\{x \\in X \\mid F(x) \\cap U \\neq \\emptyset\\}$ is open in $X$.\n\\end{definition}\n\nFinally we can state the Michael selection theorem.  \nNote that we are working with a space of covering dimension 1, so we paraphrase the more general theorem here to relate it to our context.\n\\begin{theorem}[Michael 1956\\cite{Michael1956}]\n\\label{Thm:Selection}\n A multivalued mapping $F : X \\to Y$ admits a continuous single-valued selection provided that the following conditions are satisfied:\n \\begin{enumerate}\n  \\item $X$ is a paracompact space with $\\dim(X) \\leq 1$;\n \\item $Y$ is a completely metrizable space;\n \\item $F$ is an LSC mapping;\n \\item \n For every $x \\in X$, $F(x)$ is a 0-connected (path connected) subset of $Y$; and\n \\item \n The family of values $\\{F(x)\\}_{ x \\in X}$ is $\\ELC^0$.\n \\end{enumerate}\n\n\n\\end{theorem}\n\n\\section{$\\e$-Interleaving and Functional Distortion}\n\\label{sec:relation}\n\nIn order to prove the main result, Theorem~\\ref{Thm:Main}, we will prove each inequality separately as Lemmas \\ref{Lem:Inter leq FD} and \\ref{Lem:FD leq Inter} .\n\\subsection{The Easy Direction}\n\n\\begin{lemma}\n\\label{Lem:Inter leq FD}\nLet $(\\X,f),(\\Y,g) \\in \\Reeb$.\nThen \n\\begin{equation*}\n d_I(f,g) \\leq d_{FD}(f,g).\n\\end{equation*}\n\n\\end{lemma}\n\n\\begin{proof}\nLet $\\e > d_{FD}(f,g)$.\nBy definition of the functional distortion metric, there are maps\n \\begin{equation*}\n  \\begin{tikzcd}\n   \\X \\ar[start anchor=north east, end anchor=north west]{r}{\\phi}\n          & \\Y \\ar[start anchor=south west, end anchor=south east]{l}{\\psi}\n  \\end{tikzcd}\n \\end{equation*}\nwhich satisfy the requirements of Definition~\\ref{Def:FuncDistortion}, in particular, $\\max\\{\\|f-g\\phi\\|_\\infty, \\|g-f\\psi\\|_\\infty\\} \\leq \\e$.\nThen it is obvious that $\\phi(f\\inv(I)) \\subset g\\inv(I^\\e)$ and $\\psi(g\\inv(I)) \\subset f\\inv(I_\\e)$ for all intervals $I \\subset \\R$.\nThis implies that there is a map induced by $\\phi$\n\\begin{equation*}\n \\begin{array}{rccc}\n  \\phi^*_I: &  \\pi_0(f\\inv(I)) & \\longrightarrow & \\pi_0(g\\inv(I^\\e))\n \\end{array}\n\\end{equation*}\nand a map induced by $\\psi$\n\\begin{equation*}\n \\begin{array}{rccc}\n  \\psi^*_I: &  \\pi_0(g\\inv(I)) & \\longrightarrow & \\pi_0(f\\inv(I^\\e))\n \\end{array}\n\\end{equation*}\nfor every $I$.\nBy the functoriality of $\\pi_0$, these maps commute with the maps induced by the inclusions $\\iota: I \\hookrightarrow J$ and $\\iota_\\e: I^\\e \\hookrightarrow J^\\e$; that is, the diagrams\n\\begin{equation*}\n \\begin{tikzcd}\n \\pi_0(f\\inv(I)) \\ar{r}{\\phi^*_I} \\ar{d}{\\iota^*}&  \\pi_0(g\\inv(I^\\e))  \\ar{d}{\\iota_\\e^*} &&\n \\pi_0(g\\inv(I)) \\ar{r}{\\psi^*_I} \\ar{d}{\\iota^*}&  \\pi_0(f\\inv(I^\\e))  \\ar{d}{\\iota_\\e^*}\\\\\n \\pi_0(f\\inv(J)) \\ar{r}{\\phi^*_J}&  \\pi_0(g\\inv(J^\\e)) &&\n \\pi_0(g\\inv(J)) \\ar{r}{\\psi^*_J}&  \\pi_0(f\\inv(J^\\e)) \\\\\n \\end{tikzcd}\n\\end{equation*}\ncommute.\nIn addition, the functoriality of $\\pi_0$ implies that the diagrams\n\\begin{equation*}\n \\begin{tikzcd}\n \\pi_0f\\inv(I) \\ar{r}\\ar{dr} & \\pi_0 g\\inv(I^\\e) \\ar{d} &&  \\pi_0g\\inv(I) \\ar{r} \\ar{dr} & \\pi_0 f\\inv(I^\\e) \\ar{d} \\\\\n & \\pi_0f\\inv(I^{2\\e}) && & \\pi_0g\\inv(I^{2\\e}) \n\\end{tikzcd}\n\\end{equation*}\ncommute.\nThese are exactly the properties necessary to call $\\phi^*$ and $\\psi^*$ an $\\e$-interleaving of the associated cosheaves. Since the above holds for any $\\e > d_{FD}(f,g)$, we conclude $d_I(f,g) \\leq \\e = d_{FD}(f,g)$.\n\\end{proof}\n\n\n\\subsection{The Hard Direction}\n\nIn order to show $d_{FD}((\\X,f), (\\Y,g)) \\leq d_I((\\X,f), (\\Y,g)) $, we need to start with an $\\e$-interleaving,  $\\phi:(\\X,f) \\to \\TT_\\e(\\Y,g)$ and $\\psi:(\\Y,g) \\to \\TT_\\e(\\X,f)$, and construct a pair of maps satisfying the requirements of the functional distortion distance.\nTo do this, note that the map $\\phi$ induces a multimap $\\bar \\phi: \\X \\to \\Y_\\e$, which sends a point $x$ to the entire equivalence class of $\\phi(x)$, thought of as a subset of $\\Y_\\e$.\nConcretely, letting $q:\\Y_\\e \\to \\TT_\\e(\\Y)$ denote the Reeb graph quotient map, we have $\\bar\\phi = q^{-1}\\circ \\phi$.\nFor any $\\delta$, we can construct the multimap, $\\bar\\phi_\\delta:\\X \\to \\Y_\\e$ which sends $x$ to $\\bar\\phi(B_\\delta(x))$, where $B_\\delta(x) = \\{x' \\mid d_f(x,x')<\\delta\\}$. Explicitly, we have\n\\begin{equation*}\n\\bar\\phi_\\delta(x) = \\{(y',t') \\in \\Y_\\e \\mid (y',t') \\in \\bar\\phi(x'), d_f(x,x')<\\delta\\},\n\\end{equation*}\nSee Fig.~\\ref{Fig:DeltaMap} for an example.\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale = .8]{SmoothingReebGraph-DeltaMap.pdf}\n \\caption{An example for determining the map $\\bar\\phi_\\delta$.  Given the red point $x \\in \\X$, the red solid region in $\\X$ is $B_\\delta(x)$.  Then we can look at $\\phi(B_\\delta(x))$, the red region in $\\TT_\\e(\\Y)$.  The set $\\bar\\phi_\\delta(x)$ in $\\Y_\\e$ consists of the points which map into $\\phi(B_\\delta(x))$ in $\\TT_\\e(\\Y)$ under the quotient map $q$. }\n \\label{Fig:DeltaMap}\n\\end{figure}\n\nWe want to show that the map $\\bar \\phi_\\delta: \\X \\to \\Y_\\e$ satisfies the assumptions of Theorem~\\ref{Thm:Selection}. \n\\begin{enumerate}\n \\item \n Since $\\X$ is a finite graph, it is compact and thus trivially paracompact.\nIn addition, because it is a graph, it has covering dimension 1. \n\\item\nWe can embed the graph $\\Y$ in $\\R^3$.  \nIn this context, it is a closed subset of a completely metrizable space, and thus is completely metrizable.\nTherefore, $\\Y_\\e$ is also completely metrizable since it is the product of two completely metrizable spaces.\n\\item  To show that $\\bar\\phi_\\delta$ is LSC, let $U \\subset \\Y_\\e$ open and $x \\in \\bar\\phi_\\delta\\inv(U)$ given.\n This means that there is an $x' \\in B_\\delta(x)$ such that $\\bar\\phi(x')\\cap U \\neq \\emptyset$.\n Let $r>0$ be a radius such that $B_r(x')\\subset B_\\delta(x)$.  \n In particular, this means that $d_f(x,x')<\\delta-r$.\n We now want to show that $B_r(x) \\subset \\bar\\phi_\\delta\\inv(U)$.\n Let $x'' \\in B_r(x)$.  \n We know that $x' \\in B_\\delta(x'')$ since \n \\begin{equation*}\n  d_f(x',x'') \\leq d_f(x',x) + d_f(x,x'') <r + (\\delta-r) = \\delta.\n \\end{equation*}\n Then since $\\bar\\phi(x')\\cap U \\neq \\emptyset$ and $x' \\in B_\\delta(x'')$, we must have $x'' \\in \\bar\\phi_\\delta\\inv(U)$.\n\n\\item \nLet $q:\\Y_\\e \\to \\TT_\\e(\\Y)$ be the quotient map.  \nThen $q\\circ\\pbd(x) = \\phi(B_\\delta(x))$ is the image of a connected component under a continuous map and is therefore connected.\nSince $\\phi(x) \\subset \\TT_\\e(\\Y)$ is by definition the image of a connected component of $\\X$, it is also connected.\nSo $\\pbd(x)$ can be thought of as a fibration with base space $\\phi(B_\\delta(x))$ and fibers $\\pb(x')$.\nSince the fibers are connected by definition, and the base is connected, the total space is connected.\n\n\n\n\\item\nAs checking this property is by far the most complicated, we prove it in Lemma \\ref{Lem:ELC}.\n\n\\end{enumerate}\n\n\n\n\\begin{lemma}\n\\label{Lem:ELC}\n The family of values $\\{\\pbd(x)\\}_{ x \\in \\X}$ is $\\ELC^0$.\n\\end{lemma}\n\\begin{proof}\nFix $x \\in \\X$, and consider $\\pbd(x) \\subset \\Y_\\e$.  \nChoose $(y,t) \\in \\pbd(x)$ and let $W$ be a neighborhood of $(y,t)$.  \nLet $\\bar r$ such that $B_{\\bar r}(y,t)$, the ball of radius $\\bar r$ using the metric \n\\begin{equation*}\nd_{\\Y_\\e}((y,t),(y',t')) = \\max(d_g(y,y'),|t-t'|), \n\\end{equation*}\nis contained in $W$.\nLet $L$ be the minimum height of any edge in $\\TT_\\e(\\Y)$.\nThen set $r = \\min(\\bar r, L\/4)$ and let $V = B_r(y,t)$.\n\nWe need show that for any $\\tx$ such that $\\pbd(\\tx) \\cap V \\neq \\emptyset$, $\\pbd(\\tx) \\cap V$ is path connected.  \nLet $(y_1,t_1)$ and $(y_2,t_2)$ be  in the intersection and, seeking a contradiction, assume that they are in different path components of $U \\cap V$.\nThen since $U$ is path connected, there is a path $\\gamma_1 $ from $(y_1,t_1)$ to $(y_2,t_2)$ which stays completely inside of $U$.\nBecause this path is in $U = \\pbd(\\tx)$, there is an $x_s\\in B_\\delta(\\tx)$ such that $\\gamma_1(s) \\in \\pb(x_s)$ and so $g(\\gamma_1(s)) = f(x_s)$.  \nSince $x_s \\in B_\\delta(\\tx)$, this implies that $g(\\gamma_1(s)) \\in B_\\delta( f(\\tx))$ and thus $\\height(\\gamma_1)<2\\delta$. \nAs $V$ is path connected, there is a path $\\gamma_2$ from $(y_2,t_2)$ to $(y_1,t_1)$ which stays completely inside of $V$.\nSo, $g(\\gamma_2(s)) \\in B_r(y,t)$ and thus  $\\height(\\gamma_2)<2r$.\n\nWe can now consider the paths $q(\\gamma_1)$ and $q(\\gamma_2)$ in $\\TT_\\e(\\Y)$.\nAs $U \\cap V$ is not path connected, \n there is a point $v \\in\\gamma_2$ which is not in $U$.\nWe want to show that $q(v) \\not \\in q(\\gamma_1)$.\nBy definition, $\\bar\\phi$ is the map such that $q\\bar\\phi(z) = \\phi(z)$ for any $z \\in \\X$.\nThus\n\\begin{equation*}\n q(U) = q(\\pbd(\\tx)) = q(\\bar\\phi(B_\\delta(x))) = \\phi(B_\\delta(x)).\n\\end{equation*}\nAgain seeking a contradiction, if $q(v) \\in q(U) = \\phi(B_\\delta(\\tx))$, there is an $x_v \\in B_\\delta(\\tx)$ such that $\\phi(x_v) = q(v)$.\nBut this implies that $v \\in \\bar\\phi(x_v)$ and thus $v \\in \\pbd(\\tx) = U$, contradicting our original assumption, so we must have $q(v) \\not \\in q(\\gamma_1) \\subset q(U)$.\n\nThis implies that the loop $q(\\gamma_1\\gamma_2)$ is nontrivial in $\\TT_\\e(\\Y)$.\nHowever, $$\\height(p(\\gamma)) < 2r + 2\\delta < L,$$ and therefore cannot go around a loop in $\\TT_\\e(\\Y)$.\nThus, the loop cannot be nontrivial, so the original assumption that $\\pbd(\\tx)\\cap V$ is not path connected must be false.\n\\end{proof}\n\nThus, since $\\bar \\phi$ satisfies the requirements for Theorem~\\ref{Thm:Selection}, there exists a continuous selection $\\widetilde \\phi_\\delta:\\X \\to \\Y_\\e$ of $\\pbd$, i.e., a map $\\tilde \\phi_\\delta:\\X \\to \\Y_\\e$ satisfying $\\tilde \\phi_\\delta(x) \\in \\bar\\phi(B_\\delta(x))$ for all $x \\in \\X$.\nLikewise, there exists a continuous selection $\\widetilde\\psi_\\delta:\\Y \\to \\X_\\e$ for $\\bar \\psi_\\delta$.\nLet $p_1$ be either the map $\\X_\\e \\to \\X$ or $\\Y_\\e \\to \\Y$, defined by projection onto the first factor.\nThen we will define our maps for the functional distortion distance to be $\\Phi = p_1 \\circ \\widetilde \\phi_\\delta:\\X \\to \\Y$ and $\\Psi = p_1 \\circ\\widetilde \\psi_\\delta:\\Y \\to \\X$.\nNote that $\\Phi$ and $\\Psi$ depend on the choice of $\\delta$.\n\nIn order to prove the main result of this section, Lemma \\ref{Lem:FD leq Inter}, we need the following two technical lemmas.\nLet $\\iota: (\\X,f) \\to \\TT_\\e(\\X,f)$ be the map which maps $x$ to the class of $(x,0)$. Moreover, let $\\kappa=p_1q^{-1}$. Note that $\\iota^{-1} \\subseteq \\kappa$. Similarly, define $\\iota_\\e: \\TT_\\e(\\X,f) \\to \\TT_{2\\e}(\\X,f)$ and $\\kappa_\\e=p_1q_\\e^{-1}$, where $q_\\e$ is the quotient map $\\TT_\\e(\\X,f)\\times[-\\e,\\e] \\to \\TT_\\e(\\X,f)$.\n\n\n\\begin{lemma}\n\\label{Lem:IotaBalls}\n $\\kappa B_r \\iota(x) \\subseteq B_{r+2\\e}(x)$.\n\\end{lemma}\n\\begin{proof}\n Let $x \\in \\X$ be given, and let $(\\tx,\\tilde t) \\in B_r(\\iota(x))$.\n We want to show that $\\tx \\in B_{r+2\\e}(x)$.\n \n First, $\\iota(x)=[x,0]$ is by definition the class of $(x,0)$ in $\\X_\\e$.\n Since $(\\tx,\\tilde t) \\in B_r(\\iota(x))$, there is a path $\\gamma$ of height at most $r$ between $[x,0]$ and $[\\tx,\\tilde t]$ in $\\TT_\\e(\\X)$.\n Consider the set  $q\\inv(\\gamma) \\subset \\X_\\e$.  \n As the $g_\\e$ function values of the points in $q\\inv(\\gamma)$ are the same as the $g_\\e$ function values of the corresponding points in $\\gamma$, we know $g_\\e(q\\inv(\\gamma)) = g_\\e(\\gamma)$.\n Then since $g(x')$ is within $\\e$ of $g_\\e(x',t)$ for each $x'$, we have $g(\\kappa(\\gamma)) = g(p_1q\\inv(\\gamma)) \\subset B_\\e(g_\\e(\\gamma))$.\n \n Because the image of $q\\inv(\\gamma)$ under $q$ is path connected and $q$ preserves path components, $q\\inv(\\gamma)$ is path connected as well.\n In particular, $(x,0)$ and $(\\tx,\\tilde t)$ are in this set, so there is a path $\\zeta$ between them.\n \n Breaking up $\\zeta$ into its $\\X$ and its $[\\e,\\e]$ component, consider $\\zeta(s) = (\\zeta_\\X(s),\\zeta_\\e(s))$.\n Then $\\zeta_\\X(s)$ is a path in $\\X$ between $x$ and $\\tx$.  \n As it is contained in $q\\inv(\\gamma)$, this also means that $\\height(\\zeta_\\X) \\leq \\height(\\gamma)+2\\e \\leq r+2\\e$, and so $\\tx \\in B_{r+2\\e}(x)$.\n\\end{proof}\nNote that the previous lemma can also be stated using $\\iota_\\e$, so $\\kappa_\\e B_r \\iota_\\e \\subseteq B_{r+2\\e}$.\nSince this lemma works for $r=0$, this also implies that $\\kappa\\iota \\subset B_{2\\e}$.\n\n\\begin{lemma}\n\\label{Lem:PhiPsiBall}\n  $\\Psi\\Phi(x) \\in B_{6\\e+2\\delta} (x)$.\n\\end{lemma}\n\n\\begin{proof}\nBy definition of $\\Phi$ and $\\Psi$, we have\n\\begin{align*}\n\\Phi(x) & \\in \\kappa \\phi B_\\delta(x),\\\\\n\\Psi(y) & \\in \\kappa \\psi B_\\delta(y),\n\\end{align*}\nso\n\\begin{equation*}\n\\Psi  \\Phi(x) \\in \\kappa  \\psi  B_\\delta \\kappa \\phi B_\\delta (x).\n\\end{equation*}\nSince $\\psi$ preserves function values, a path $\\gamma$ in $\\Y$ is sent to a path $\\psi(\\gamma)$ of the same height.\nThus $\\psi B_\\delta \\subseteq B_\\delta \\psi$ and we get\n\\begin{equation*}\n\\Psi \\Phi(x) \\in \\kappa B_\\delta  \\psi \\kappa B_\\delta  \\phi  (x).\n\\end{equation*}\nNow using, Lemma \\ref{Lem:IotaBalls}, the definition of the interleaving, and the fact that for any map $\\nu$, $x \\in \\nu\\inv\\nu(x)$, we have\n\\begin{align*}\n \\psi  \\kappa  \\phi \n& \\subseteq \\iota\\inv_\\e  \\iota_\\e  \\psi  \\kappa \\phi \\\\\n& \\subseteq \\kappa_\\e  \\iota_\\e  \\psi  \\kappa \\phi \\\\\n& = \\kappa_\\e  \\psi_\\e  \\iota \\kappa  \\phi \\\\\n& \\subseteq \\kappa_\\e \\psi_\\e B_{2\\e} \\phi \\\\\n& \\subseteq \\kappa_\\e B_\\e \\psi_\\e \\phi \\\\\n& = \\kappa_\\e  B_\\e \\iota_\\e \\iota \\\\\n& \\subseteq B_{4\\e} \\iota,\n\\end{align*}\nso we have \n\\begin{align*}\n \\Psi  \\Phi(x) \n &\\in \\kappa B_\\delta \\psi  \\kappa \\phi  B_\\delta (x) \\\\\n &\\subseteq \\kappa B_\\delta B_{4\\e} \\iota  B_\\delta (x) \\\\\n &\\subseteq \\kappa B_\\delta  B_{4\\e}  B_\\delta \\iota (x) \\\\\n &= \\kappa B_{4\\e+2\\delta} \\iota (x) \\\\\n &\\subseteq B_{6\\e+2\\delta} (x). \\qedhere\n\\end{align*}\n\\end{proof}\n\n\n\n\nFinally, we can  prove the main result of the section.\n\n\n\\begin{lemma}\n\\label{Lem:FD leq Inter}\nLet $f:\\X \\to \\R$ and $g:\\Y \\to \\R$.  \nThen \n\\begin{equation*}\n d_{FD}(f,g) \\leq 5d_I(f,g).\n\\end{equation*}\n\n\\end{lemma}\n\n\\begin{proof}\nLet $\\phi:(\\X,f) \\to \\TT_\\e(\\Y,g)$ and $\\psi:(\\Y,g) \\to \\TT_\\e(\\X,f)$ be an $\\e$-interleaving, and thus $d_I(f,g) \\leq \\e$. \nAs proved above, there exists continuous selections $\\Phi: \\X \\to \\Y$ and $\\Psi: \\Y \\to \\X$ for the multimaps $\\bar \\phi_\\delta$ and $\\bar \\psi_\\delta$.\nIn particular, this means that for any $x \\in \\X$, $\\Phi(x)$ is a point in $\\Y$ such that there is a $x' \\in \\X$ and a $t \\in [-\\e,\\e]$ with $(\\Phi(x),t) \\in \\bar\\phi(x')$ and $d_f(x,x')<\\delta$.\nSo $|f(x)-f(x')|<\\delta$ and $f(x') = g(\\Phi(x))+t$.\nThus \n\\begin{equation*}\n |f(x)-g\\Phi(x)| = |f(x)-(f(x')-t)| = |f(x)-f(x')+t| \\leq \\delta + \\e\n\\end{equation*}\n and hence $\\|f-g \\circ\\Phi\\|_\\infty \\leq \\e + \\delta$.\nLikewise, $\\|g-f \\circ \\Psi\\|_\\infty \\leq \\e + \\delta$.\n\n Consider $(x,y),(x',y') \\in C(\\Phi,\\Psi)$.  \n There are two cases to consider; either the pairs are the same type (i.e. $(x,\\Phi(x))$ and $(x', \\Phi(x'))$), or they are different.\n If we have the same type, so $(x,\\Phi(x))$ and $(x', \\Phi(x'))$, let $\\gamma$ be a minimum height path in $\\X$ from $x$ to $x'$.  \n Then $\\Phi(\\gamma)$ is a path in $\\Y$ from $\\Phi(x)$ to $\\Phi(x')$.\n Since $\\|f - g\\Phi\\|_\\infty\\leq \\e + \\delta$, the height of $\\Phi(\\gamma)$ is at most $2(\\e+\\delta)$ above the height of $\\gamma$.  \n So \n \\begin{equation}\n \\label{Eqn:BoundA1}\n \\begin{aligned}\n  d_g(\\Phi(x),\\Phi(x')) \n    &\\leq \\height(\\Phi(\\gamma)) \\\\\n    &\\leq \\height(\\gamma) + 2(\\e + \\delta)\\\\\n    &= d_f(x,x') + 2(\\e +\\delta).\n \\end{aligned}\n \\end{equation}\n \n To get the other direction, let $\\zeta$ be a minimum height path in $\\Y$ between $\\Phi(x)$ and $\\Phi(x')$.  \n Then $\\Psi(\\zeta)$ is a path in $\\X$ from $\\Psi\\Phi(x)$ to  $\\Psi\\Phi(x')$.\n A similar argument as above gives\n \\begin{equation*}\n \\begin{aligned}\n  d_f (\\Psi\\Phi(x),\\Psi\\Phi(x')) \n    &\\leq \\height(\\Psi(\\zeta))\\\\\n    &\\leq \\height(\\zeta) + 2(\\e+\\delta) \\\\\n\t&= d_g(\\Phi(x),\\Phi(x')) + 2(\\e+\\delta).\n \\end{aligned}\n \\end{equation*}\n Thus, using Lemma \\ref{Lem:PhiPsiBall} and the triangle inequality,\n \\begin{equation}\n \\label{Eqn:BoundA2}\n\\begin{aligned}\n d_f(x,x')& \\leq d_f(x,\\Psi\\Phi(x)) +  d_f(\\Psi\\Phi(x),\\Psi\\Phi(x')) +  d_f(\\Psi\\Phi(x'),x')  \\\\\n      &\\leq  d_g(\\Phi(x),\\Phi(x')) + 2(\\e+\\delta) + 2(6\\e+2\\delta)\\\\\n      &\\leq  d_g(\\Phi(x),\\Phi(x')) + 14(\\e+\\delta)\n\\end{aligned}\n\\end{equation}\nand therefore combining Eqns.~\\ref{Eqn:BoundA1} and \\ref{Eqn:BoundA2},\n\\begin{equation*}\n |d_f(x,x')-d_g(\\Phi(x),\\Phi(x'))|\\leq 14(\\e+\\delta).\n\\end{equation*}\nLikewise, \n\\begin{equation*}\n  |d_f(\\Psi(y),\\Psi(y'))-d_g(y,y')|\\leq  14(\\e+\\delta).\n\\end{equation*}\n\n \n \n Now assume we have $(x,\\Phi(x))$ and $(\\Psi(y), y)$ and let $\\zeta$ be a minimum height path in $\\Y$ between $\\Phi(x)$ and $y$.\n Then $\\Psi(\\zeta)$ is a path in $\\X$ between $\\Psi\\Phi(x)$ and $\\Psi(y)$.\n Thus,\n \\begin{equation*}\n \\begin{aligned}\n  d_f(\\Psi(y),\\Psi\\Phi(x)) \n      &\\leq \\height(\\Psi(\\zeta)) \\\\\n      & \\leq \\height(\\zeta)+2(\\e+\\delta) \\\\\n      &= d_g(\\Phi(x),y) + 2(\\e+\\delta)\n \\end{aligned}\n \\end{equation*}\n and  using  Lemma \\ref{Lem:PhiPsiBall} we have\n\\begin{equation}\n\\label{Eqn:BoundB1}\n \\begin{aligned}\n d_f(x,\\Psi(y))\n & \\leq d_f(x,\\Psi\\Phi(x)) +  d_f(\\Psi\\Phi(x),\\Psi(y))  \\\\\n &\\leq  d_g(\\Phi(x),y) + 8(\\e+\\delta).\n \\end{aligned}\n\\end{equation}\nLikewise, if we have a minimum height path $\\gamma$ in $\\X$ between $x$ and $\\Psi(y)$, then $\\Phi(\\gamma)$ is a path in $\\Y$ between $\\Phi(x)$ and $\\Phi\\Psi(y)$ with height bounded by $\\height(\\gamma)+2(\\e+\\delta)$, and thus\n\\begin{equation}\n\\label{Eqn:BoundB2}\n \\begin{aligned}\n d_g(\\Phi(x),y) \n &\\leq d_g(\\Phi(x),\\Phi\\Psi(y)) +  d_g(\\Phi\\Psi(y),y)  \\\\\n &\\leq  d_f(x,\\Psi(y)) + 8(\\e+\\delta).\n \\end{aligned}\n\\end{equation}\nTherefore, $| d_f(x,\\Psi(y)) - d_g(\\Phi(x),y) | \\leq 8(\\e+\\delta)$. \n \n \n Combining all of these bounds gives \n \\begin{equation*}\n  D(\\phi,\\psi)  = \\sup_{\\substack{(x,y),(x',y')\\\\ \\in C(\\phi,\\psi)}} \\frac{1}{2} \\left| d_f(x,x') - d_g(y,y')\\right| \\leq 7(\\e+\\delta).\n \\end{equation*}\n and therefore\n \\begin{equation*}\n  d_{FD}(f,g) = \\inf_{\\phi,\\psi} \\max \\{ D(\\phi,\\psi),\\|f-g \\circ \\phi\\|_\\infty, \\|g-f \\circ \\psi\\|_\\infty \\} \\leq 7(\\e+\\delta).\n \\end{equation*} \n Since this is true for any $\\e>d_I(f,g)$ and for any $\\delta>0$, this completes the proof.\n\\end{proof}\n\nPutting together Lemmas \\ref{Lem:Inter leq FD} and \\ref{Lem:FD leq Inter}, our main theorem is immediate.\n\\begin{theorem}\n\\label{Thm:Main}\n The functional distortion metric and the interleaving metric are strongly equivalent.  That is, given any Reeb graphs $(\\X,f)$ and $(\\Y,g)$,\n \\begin{equation*}\n  d_I(f,g) \\leq d_{FD}(f,g) \\leq 7d_I(f,g).\n \\end{equation*}\n\\end{theorem}\n\n\n\n\n\\section{Relationship Between the Interleaving and Bottleneck Distances}\n\\label{sec:bottleneck}\n\n\nHaving strongly equivalent metrics means that we can quickly pass back and forth many of the properties associated to the metrics.\nFor example, the bottleneck stability bound for persistence diagrams in terms of the functional distortion distance~\\cite{Bauer2014} says the following (for the definitions of the persistence diagrams $\\Dg_0(f), \\ExDg_1(f)$ associated to a function~$f$ and of the bottleneck distance $d_B$ we refer the reader to~\\cite{EH09}):\n\\begin{theorem}\nGiven two Reeb graphs $(\\X,f)$ and $(\\Y,g)$,\n\\begin{equation*}\n d_B(\\Dg_0(f),\\Dg_0(g)) \\leq d_{FD}(f,g)\n\\end{equation*}\n and \n\\begin{equation*}\n d_B(\\ExDg_1(f),\\ExDg_1(g)) \\leq 3d_{FD}(f,g).\n\\end{equation*}\n\\end{theorem} \n \nCombining this result with Theorem~\\ref{Thm:Main} gives an immediate stability result relating the interleaving distance with the bottleneck distance.\n\\begin{theorem}\nGiven two Reeb graphs $(\\X,f)$ and $(\\Y,g)$,\n\\begin{equation*}\n d_B(\\Dg_0(f),\\Dg_0(g)) \\leq 7d_I(f,g)\n\\end{equation*}\n and \n\\begin{equation*}\n d_B(\\ExDg_1(f),\\ExDg_1(g)) \\leq 21d_I(f,g).\n\\end{equation*}\n \n\\end{theorem}\n\n\n\n\n\n\\section{Discussion}\n\nIn this paper, we study the relation between the two existing distance measures for Reeb graphs, and show that they are strongly equivalent on the space of Reeb graphs. \nThis relationship will be a powerful tool for understanding convergence properties of the different metrics. \nFor example, if we have a Cauchy sequence in one metric, we have a Cauchy sequence in the other and can therefore pass around completeness results. \nThis relationship also means that algorithms for computation and approximation of the metrics can be written using whichever method is most helpful and applicable to the context.\n\n\nThese two distances in general may not be the same. An immediate question is whether the relations provided in Theorem \\ref{Thm:Main} are tight. \nIn particular, it is easy to construct examples where the bound $d_I(f,g) \\leq d_{FD}(f,g)$ of Lemma~\\ref{Lem:Inter leq FD} is tight; it will be interesting to investigate the tightness of the bound $d_{FD}(f,g) \\leq 7d_I(f,g)$ of Lemma~\\ref{Lem:FD leq Inter}. \nWhile that bound is obtained using an arbitrary selection, a better bound may be achievable using a particular optimal selection. \nIn addition, this may shed light on whether the bounds given between the bottleneck distance \nof the extended persistence diagrams and the two Reeb graph distances are tight. \nFinally, we will explore the applications of these distance measures to studying the stability of Reeb-like structures, such as Mapper and $\\alpha$-Reeb graphs. \n\n\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nIt is very interesting to study\nenergetic charged particle\ndiffusion in\nthe magnetic turbulence superposed on\nthe mean magnetic field\nin astrophysics\n (e.g.,\ncosmic ray physics, astrophysical\nplasmas, and\nspace weather research)\nand fusion plasma physics\n\\citep{Jokipii1966,\nSchlickeiser2002, MatthaeusEA2003,\nShalchiEA2005, ShalchiEA2006, Qin2007,\nHauffEA2008, Shalchi2009a,\nShalchi2010, QinEA2014}.\nThe interaction process between particles\nand turbulent magnetic fields is very\ncomplicated, and\nparticles and magnetic field lines\nhave the stochastic\nproperties. Therefore, one has to employ\nmethods of statistics to describe\nthe complicated transport of energetic particles\n\\citep{Earl1974, Earl1976,\nBeeckEA1986, Shalchi2011,\nLitvinenko2012a,\nLitvinenko2012b, ShalchiEA2013, HeEA2014,\nWangEA2017a, WangEA2017b, wq2018}\nand field line random walk\n\\citep{MatthaeusEA1995,\nShalchiAKourakis2007,\nShalchiAQin2010}.\nBecause the background\nmagnetic field $\\vec{B}_0$\nbreaks\nthe symmetry of the\nmagnetized plasma,\none has to distinguish\nparticle diffusion\nalong and across\nthe large-scale magnetic field.\nOne may\nonly consider the parallel\ndiffusion in many cases, when it is\nmuch greater than the perpendicular\none\n\\citep[see, e.g.,][]{Earl1974, Earl1976,\nBeeckEA1986, Shalchi2009b, Shalchi2011,\nLitvinenko2012a,\nLitvinenko2012b, ShalchiEA2013,\nHeEA2014, wq2018}.\n\nVarious analytical theories of\nparallel diffusion for energetic\ncharged particles have been\ndeveloped in the past.\nThe first attempt to resolve the parallel\ndiffusion was\nthe development of quasilinear theory\n\\citep{Jokipii1966} which corresponds\nto the first order\nperturbation theory.\nQuasilinear theory, however,\nis problematic and usually does\nnot agree with test-particle simulations\n\\citep{QinEA2002a, QinEA2002b,\nShalchi2009a}.\nWith the second order quasilinear theory\n(SOQLT) developed by \\citet{Shalchi2005},\nby using the Earl's formula\n\\citep{Earl1974}\nthe parallel\ndiffusion coefficient\ncan be evaluated\nmore accurately\n\\citep{ShalchiEA2009a,\nShalchiEA2009b, ReimerEA2016}.\nOn the other hand,\nby employing the assumptions\nand approximations\nused in NonLinear Guiding Center (NLGC)\ntheory \\citep{MatthaeusEA2003},\n\\citet{Qin2007} developed an extended\nparallel diffusion theory.\n\nThe mean square displacement\n$\\sigma^2=\\langle (\\Delta z)^2 \\rangle\n-\\langle (\\Delta z) \\rangle^2$\nis defined with the first and second order parallel\ndisplacement moments of the isotropic\ndistribution function\nof the energetic particles.\n$\\sigma^2$, describing the spread of\npossible particle orbits,\nis one of the fundamental quantities in the parallel transport.\nIn previous research\n\\citep[see, e.g.,][]{Shalchi2009a}\nthe parallel diffusion\ncoefficient $\\kappa_{zz}$\nis usually described by\nthe following formula\n(hereafter DCDV)\n\\begin{equation}\n\\kappa_{zz}=\\frac{1}{2}\\frac{d\\sigma^2}{dt}.\n\\label{kzz-sigma}\n\\end{equation}\n\nTo study energetic particle diffusion,\nthe background magnetic field $\\vec{B}_0$,\ne.g., mean solar wind magnetic field\nin the interplanetary space,\nis usually considered as a constant.\nHowever,\nit is clear that the mean solar wind\nmagnetic field\nis not constant in reality,\nespecially when particles are\nclose to the Sun.\nIt is found\nthat the\nspatially varying\nbackground solar wind magnetic fields lead\nto the adiabatic focusing effect of\ncharged energetic particle\ntransport and introduce\ncorrection to the particle\ndiffusion coefficients\n\\citep[see, e.g.,][]{Roelof1969,\nEarl1976, Kunstmann1979,BeeckEA1986,\nBieberEA1990,\nKota2000, SchlickeiserEA2008,\nShalchi2009b, Shalchi2011,\nLitvinenko2012a,\nLitvinenko2012b, ShalchiEA2013,\nWangEA2016, WangEA2017b, wq2018}.\nThe adiabatic focusing effect\ncauses\na convection term in\nthe energetic particles transport equation\nof the isotropic distribution function,\nso DCDV might be modified.\n\nIn order to address this question,\nin this paper we derive a new formula\nfor the parallel diffusion coefficient\nwith the parallel displacement variance\nby modifying DCDV with the\nadiabatic focusing\neffect.\nThe paper is organized as follows.\nIn Section \\ref{EQUATION OF ISOTROPIC\nDISTRIBUTION FUNCTION},\nfrom the Fokker-Planck equation\nwith adiabatic focusing effect\n we derive\nthe isotropic\ndistribution function equation\nand the telegraph equation.\nIn Section \\ref{THE PARALLEL DIFFUSION\nCOEFFICIENT REPRESENTED BY\nALONG-FIELD DISPLACEMENT MOMENTS},\nwe evaluate the first\nand second order moments of\nthe parallel displacement\nto derive the modified DCDV.\nIn Section \\ref{SPECIAL CASES},\nsome special cases of\nthe modified  DCDV\nare explored.\nWe conclude and summarize\nour results in Section\n\\ref{SUMMARY AND CONCLUSION}.\n\n\\section{EQUATION OF ISOTROPIC\nDISTRIBUTION FUNCTION}\n\\label{EQUATION OF ISOTROPIC\nDISTRIBUTION FUNCTION}\n\nThe standard Fokker-Planck equation,\nwhich incorporates the effects\nof pitch-angle\nscattering and along-field\nadiabatic focusing,\nis given as\n\\citep{Roelof1969, Earl1981}\n\\begin{equation}\n\\frac{\\partial{f_0}}{\\partial{t}}\n+ v\\mu\n\\frac{\\partial{f_0}}\n{\\partial{z}}\n=\\frac{\\partial{}}\n{\\partial{\\mu}}\n\\left[D_{\\mu \\mu}(\\mu)\n\\frac{\\partial{f_0}}\n{\\partial{\\mu}}\\right]\n-\\frac{v}{2L}(1-\\mu^2)\n\\frac{\\partial{f_0 }}\n{\\partial{\\mu}},\n\\label{standard Fokker-Planck\nequation}\n\\end{equation}\nwhich is usually used\nin previous research\n\\citep{SchlickeiserEA2007,\nShalchi2011,\nLitvinenkoASchlickeiser2013,\nEffenbergerEA2014,\nMalkovSagdeev2015,\nWangEA2016}.\nHere $f_0$ is the isotropic\ndistribution function,\n$t$ is time, $z$\nis the distance\nalong the background\nmagnetic field,\n$\\mu=v_z \/v$ is the pitch-angle\ncosine with particle\nspeed $v$ and\nits z-component $v_z$,\n$D_{\\mu \\mu}(\\mu)$\nis the\npitch-angle diffusion\ncoefficient,\n$L(z)=-B_0 (z)\/ [dB_0 (z) \/ dz]$\nis the adiabatic focusing\ncharacteristic length of the\nlarge-scale magnetic field $B_0(z)$.\nIn this paper,\nfor simplification\nwe assume that\nthe  adiabatic focusing\ncharacteristic length\nis a constant.\nThe source term and\nthe terms related to\nmomentum diffusion\nand so on are ignored\nin Equation\n(\\ref{standard Fokker-Planck\nequation}).\nThe more complete form of\nthe Fokker-Planck equation\ncan be found in\n\\citet{Schlickeiser2002}.\n\nBy introducing the linear\nphase space density\n$f(z,\\mu,t)=f_0(z,\\mu,t)\/B_0$,\nfrom Equation\n(\\ref{standard Fokker-Planck\nequation}),\nthe modified Fokker-Planck equation\nfor the\ndistribution function of energetic\ncharged particles can be obtained\n\\citep{Kunstmann1979,HeEA2014,\nWangEA2017b, wq2018}\n\\begin{equation}\n\\frac{\\partial{f}}{\\partial{t}}\n+ v\\mu\n\\frac{\\partial{f}}\n{\\partial{z}}=\n\\frac{\\partial{}}{\\partial{\\mu}}\n\\left[D_{\\mu \\mu}(\\mu)\n\\frac{\\partial{f}}\n{\\partial{\\mu}}-\\frac{v}{2L}\n(1-\\mu^2)f \\right].\n\\label{Fokker-Planck equation-0}\n\\end{equation}\nHere, $D_{\\mu\\mu}(\\mu)$ is time-independent. However,\nif the temporal characteristic\nof the pitch angle diffusion\nneed to be considered, $D_{\\mu\\mu}=D_{\\mu\\mu}(\\mu,t)$,\nEquation\n(\\ref{Fokker-Planck equation-0})\ncan be generalized as\n\\begin{equation}\n\\frac{\\partial{f}}{\\partial{t}}\n+ v\\mu\n\\frac{\\partial{f}}\n{\\partial{z}}=\n\\frac{\\partial{}}{\\partial{\\mu}}\n\\left[D_{\\mu \\mu}(\\mu,t)\n\\frac{\\partial{f}}\n{\\partial{\\mu}}-\\frac{v}{2L}\n(1-\\mu^2)f \\right].\n\\label{Fokker-Planck equation}\n\\end{equation}\nIn order for the pitch angle diffusion equation to hold,\nthe characteristic\ntime of the pitch angle\ndiffusion has to\nbe much less than\nthe characteristic\ntime of $D_{\\mu\\mu}(\\mu,t)$.\nEquation (\\ref{Fokker-Planck equation}) is the starting\npoint of the research in\nthis paper.\n\nWith strong pitch-angle scattering,\nthe gyro-tropic\ncosmic-ray phase space\ndensity $f(\\vec{x}, \\mu, t)$\ncan be split into\nthe dominant isotropic part\n$F(\\vec{x}, t)$\nand the subordinate\nanisotropic part\n$g(\\vec{x}, \\mu, t)$\n\\citep[see, e.g.,][]\n{SchlickeiserEA2007,\nSchlickeiserEA2008, HeEA2014,\nWangEA2017b, wq2018}\n\\begin{equation}\nf(\\vec{x}, \\mu, t)=F(\\vec{x}, t)\n+g(\\vec{x}, \\mu, t)\n\\end{equation}\nwith\n\\begin{equation}\nF(\\vec{x}, t)=\\frac{1}{2}\n\\int_{-1}^1 d\\mu\nf(\\vec{x}, \\mu, t)\n\\end{equation}\nand\n\\begin{equation}\n\\int_{-1}^1 d\\mu\ng(\\vec{x}, \\mu, t)=0 .\n\\end{equation}\n\n\\subsection{The variable\ncoefficient differential\nequation of the\nisotropic distribution function\n$F(z, t)$}\n\\label{The governing equation\nof the\nisotropic distribution function\nF}\n\nIn this subsection,\nby employing the method\nin \\citet{wq2018}\nwe derive\nthe formula of the variable\ncoefficient differential\nequation of the\nisotropic distribution function\n$F(z, t)$, and the derivation are\nsimilar to Equation (22) in\n\\citet{wq2018}.\n\nIntegration of Equation\n (\\ref{Fokker-Planck equation})\nover $\\mu$ gives\n\\begin{equation}\n\\frac{\\partial{F}}{\\partial{t}}\n+ \\frac{v}{2}\n\\frac{\\partial{}}\n{\\partial{z}}\\int_{-1}^{1}\n\\mu g d\\mu=0.\n\\label{Equation of F with g}\n\\end{equation}\nNext, integrating Equation\n(\\ref{Fokker-Planck equation})\nover $\\mu$ from $-1$ to $\\mu$,\nand using the regularity\n$D_{\\mu\\mu}(\\mu=\\pm 1)=0$\nwe can find\n\\begin{eqnarray}\n\\frac{\\partial{F}}\n{\\partial{t}}(\\mu+1)\n&&+ \\frac{\\partial{}}\n{\\partial{t}}\n\\int_{-1}^{\\mu}d\\nu g\n+\\frac{v(\\mu^2-1)}{2}\n\\frac{\\partial{F}}\n{\\partial{z}}\n+v\\frac{\\partial{}}\n{\\partial{z}}\n\\int_{-1}^{\\mu}\nd\\nu \\nu g\n=D_{\\mu\\mu}(\\mu,t)\n\\frac{\\partial{g}}\n{\\partial{\\mu}}\n-\\frac{v(1-\\mu^2)}{2L}F\n-\\frac{v(1-\\mu^2)}{2L}g.\n\\label{integrate\nfrom -1 to mu}\n\\end{eqnarray}\nBy subtracting Equation\n(\\ref{Equation of F with g})\nfrom\n(\\ref{integrate from -1 to mu})\nwe can find the\nfollowing equation\n\\begin{eqnarray}\n\\frac{\\partial{F}}\n{\\partial{t}}\\mu\n+ \\frac{\\partial{}}\n{\\partial{t}}\n\\int_{-1}^{\\mu}d\\nu g\n&&+\\frac{v(\\mu^2-1)}{2}\n\\frac{\\partial{F}}\n{\\partial{z}}\n+v\\frac{\\partial{}}\n{\\partial{z}}\n\\int_{-1}^{\\mu}\nd\\nu \\nu g\n-\\frac{v}{2}\n\\frac{\\partial{}}\n{\\partial{z}}\\int_{-1}^{1}\n\\mu g d\\mu \\nonumber\\\\\n&&=D_{\\mu\\mu}(\\mu,t)\n\\frac{\\partial{g}}\n{\\partial{\\mu}}\n-\\frac{v(1-\\mu^2)}{2L}F\n-\\frac{v(1-\\mu^2)}{2L}g.\n\\label{integrate from\n-1 to mu -2}\n\\end{eqnarray}\nAfter a straightforward\nalgebra, Equation\n(\\ref{integrate from -1 to mu -2})\nreduces to the following form\n\\begin{equation}\n\\frac{\\partial{g}}\n{\\partial{\\mu}}-\n\\frac{v(1-\\mu^2)g}\n{2L D_{\\mu \\mu}(\\mu,t)}\n+\\frac{v(1-\n\\mu^2)}{2D_{\\mu \\mu}(\\mu,t)}\n\\left(\\frac{\\partial{F}}\n{\\partial{z}}\n-\\frac{F}\n{L}\\right)=\\Phi(\\mu,t)\n\\label{Equation of g with phi}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\Phi(\\mu,t)=\n\\frac{1}{D_{\\mu\\mu}(\\mu,t)}\n\\Bigg[\\left(\\frac{\\partial{F}}\n{\\partial{t}}\\mu\n+\\frac{\\partial{}}{\\partial{t}}\n\\int_{-1}^{\\mu}gd\\nu\\right\n+\\frac{v}{2}\\frac{\\partial{}}\n{\\partial{z}}\n\\left(2\\int_{-1}^{\\mu}d\\nu \\nu g-\n\\int_{-1}^{1}d\\mu \\mu g\\right)\\Bigg].\n\\label{Phi}\n\\end{eqnarray}\nEquation\n(\\ref{Equation of g with phi})\ncan be rewritten as\n\\begin{equation}\n\\frac{\\partial{}}{\\partial{\\mu}}\n\\Bigg\\{\\Bigg[g(\\mu,t)\n-L\\left(\\frac{\\partial{F}}\n{\\partial{z}}\n-\\frac{F}{L} \\right)\n\\Bigg]e^{-M(\\mu,t)}\\Bigg\\}\n=e^{-M(\\mu,t)}\\Phi(\\mu,t)\n\\label{HS-like}\n\\end{equation}\nwith\n\\begin{equation}\nM(\\mu,t)=\\frac{v}{2L}\n\\int_{-1}^{\\mu} d\\nu\n\\frac{1-\\nu^2}{D_{\\nu \\nu}(\\nu,t)}.\n\\label{M(mu)}\n\\end{equation}\n\nIn additon, with the same procedure used\nin \\citet{wq2018},\nthe anisotropic\ndistribution function can be\nobtained as following\n\\begin{equation}\ng(\\mu,t)=L\\left(\\frac{\\partial{F}}\n{\\partial{z}}\n-\\frac{F}{L}\\right)\\left[1-\n\\frac{2e^{M(\\mu,t)}}{\\int_{-1}^{1}\nd\\mu e^{M(\\mu,t) }}\\right]\n+e^{M(\\mu,t)}\\left[R(\\mu,t)\n-\\frac{\\int_{-1}^{1}d\\mu\ne^{M(\\mu,t)}R(\\mu,t)}\n{\\int_{-1}^{1}d\\mu\ne^{M(\\mu,t) }}\\right]\n\\label{g}\n\\end{equation}\nwith\n\\begin{eqnarray}\nR(\\mu,t)&=&\\int_{-1}^{\\mu} d\\nu\ne^{-M(\\nu,t)}\\Phi(\\nu,t).\n\\label{R(mu)}\n\\end{eqnarray}\nEquations (\\ref{g}) and (\\ref{R(mu)})\ncontain\nthe effect from the term on the\nright hand side of\nEquation (\\ref{HS-like}).\nCombining Equations (\\ref{Phi}),\n(\\ref{g}) and  (\\ref{R(mu)})\ngives the iterative function of\n$g(\\mu,t)$,\n\\begin{eqnarray}\ng(\\mu,t)&=&L\\left(\\frac{\\partial{F}}\n{\\partial{z}}\n-\\frac{F}{L}\\right)\\left[1-\n\\frac{2e^{M(\\mu,t)}}{\\int_{-1}^{1}\nd\\mu e^{M(\\mu,t) }}\\right]\n+e^{M(\\mu,t)}\\Bigg\\{\\int_{-1}^{\\mu}\nd\\nu\ne^{-M(\\nu,t)}\n\\frac{1}{D_{\\nu\\nu}(\\nu,t)}\n\\Bigg[\\left(\\frac{\\partial{F}}\n{\\partial{t}}\\nu\n+\\frac{\\partial{}}{\\partial{t}}\n\\int_{-1}^{\\nu}\ng(\\rho,t)d\\rho\\right)\n\\nonumber \\\\\n&&+\\frac{v}{2}\\frac{\\partial{}}\n{\\partial{z}}\n\\left(2\\int_{-1}^{\\nu}d\\rho \\rho\ng(\\rho,t)-\n\\int_{-1}^{1}d\\mu \\mu\ng(\\mu,t)\\right)\n\\Bigg]\\nonumber\\\\\n&&-\\frac{1}\n{\\int_{-1}^{1}d\\mu\ne^{M(\\mu,t) }}\\int_{-1}^{1}d\\mu\ne^{M(\\mu,t)}\\int_{-1}^{\\mu} d\\nu\ne^{-M(\\nu,t)}\n\\frac{1}{D_{\\nu\\nu}(\\nu,t)}\n\\Bigg[\\left(\\frac{\\partial{F}}\n{\\partial{t}}\\nu\n+\\frac{\\partial{}}{\\partial{t}}\n\\int_{-1}^{\\nu}g(\\rho,t)\nd\\rho\\right)\\nonumber\\\\\n&&+\\frac{v}{2}\n\\frac{\\partial{}}\n{\\partial{z}}\n\\left(2\\int_{-1}^{\\mu}\nd\\nu \\nu g(\\nu,t)-\n\\int_{-1}^{1}d\\mu \\mu\ng(\\mu,t)\\right)\\Bigg]\\Bigg\\}.\n\\label{g2}\n\\end{eqnarray}\nBy iterating Equation\n(\\ref{g2}),\nwe obtain the formula of\n$g(\\mu,t)$ as\n\\begin{equation}\ng(\\mu,t)=\\sum_{m, n}\n\\epsilon_{m,n}(t)\n\\frac{\\partial^{m+n}{}}\n{\\partial {t^m}\n\\partial{z}^n}  F\n\\label{expanded g}\n\\end{equation}\nwith the time dependent coefficients\n$\\epsilon_{m,n}(t)$.\nHere $m,n=0,1,2,3,\\cdots$.\nFurthermore,\nconsidering Equation\n(\\ref{Phi}), (\\ref{g})\nand Equation (\\ref{R(mu)}),\nwe can write\n$R(\\mu,t)$ as\n\\begin{equation}\nR(\\mu,t)=\\sum_{m, n}\n\\chi_{m,n}(t)\n\\frac{\\partial^{m+n}{}}\n{\\partial {t^m}\n\\partial{z}^n}  F\n\\label{series of R}\n\\end{equation}\nwith the time dependent coefficients\n$\\chi_{m,n}(t)$ and\n$m,n=0,1,2,3,\\cdots$.\nThe coefficients\n$\\chi_{m,n}(t)$\nin the latter equation\nis related to\nthe coefficients\n$\\epsilon_{m,n}(t)$\nin Equation\n(\\ref{expanded g}).\n\nIn order to\nobtain the differential equation\nof isotropic distribution\nfunction,\nwe need to get the\nfollowing formula\nby using Equation\n(\\ref{Equation of F with g})\n\\begin{equation}\n\\int_{-1}^{1}d\\mu \\mu g(\\mu,t)\n=-2\\frac{\\int_{-1}^{1}d\\mu\\mu\ne^{M(\\mu,t)}}\n{\\int_{-1}^{1}d\\mu e^{M(\\mu,t)}}\n\\left(\\frac{\\partial{F}}\n{\\partial{z}}\n-\\frac{F}{L}\\right)L\n+\\int_{-1}^{1}\nd\\mu \\mu e^{M(\\mu,t)}\n\\Bigg[R(\\mu,t)\n-\\frac{\\int_{-1}^{1}d\\mu\ne^{M(\\mu,t)}R(\\mu,t)}\n{\\int_{-1}^{1}d\\mu\ne^{M(\\mu,t)}}\n\\Bigg].\n\\label{Integrating mu g over\nmu from -1 to 1}\n\\end{equation}\nBy inserting Equation\n(\\ref{Integrating mu g over\nmu from -1 to 1})\ninto Equation\n(\\ref{Equation of F with g})\none can obtain\n\\begin{eqnarray}\n\\frac{\\partial{F}}{\\partial{t}}\n&-&\\frac{\\partial{}}\n{\\partial{z}}\n\\left[vL\\frac{\\int_{-1}^{1}\nd\\mu \\mu e^{M(\\mu,t)}}\n{\\int_{-1}^{1}d\\mu\ne^{M(\\mu,t)}}\n\\left(\\frac{\\partial{F}}\n{\\partial{z}}-\n\\frac{F}{L}\\right)\\right]\n+\\Lambda(z,t)=0\n\\label{Accurate\ndiffusion equation}\n\\end{eqnarray}\nwith\n\\begin{equation}\n\\Lambda(z,t)=\n\\frac{v}{2}\\int_{-1}^{1}\nd\\mu \\mu e^{M(\\mu,t)}\n\\left[\\frac{\\partial{R(\\mu,t)}}\n{\\partial{z}}\n-\\frac{\\int_{-1}^{1}d\\mu\n\\frac{\\partial{R(\\mu,t)}}\n{\\partial{z}}\ne^{M(\\mu,t)}}\n{\\int_{-1}^{1}d\\mu\ne^{M(\\mu,t)}}\n\\right].\n\\label{Lambda}\n\\end{equation}\nBy inserting Equation\n(\\ref{series of R})\ninto Equation (\\ref{Lambda})\nwe can find\n\\begin{equation}\n\\Lambda(z,t)\n=\\sum_{m,n}\\eta_{m,n}(t)\n\\frac{\\partial^{m+n}}\n{\\partial {t^m}\n\\partial{z}^n}  F\n\\label{Lambda-eta}\n\\end{equation}\nwith the coefficients\n$\\eta_{m,n}(t)$ and\n$m,n=0,1,2,3,\\cdots$.\nThus, Equation\n(\\ref{Accurate diffusion\nequation}) can be written as\n\\begin{eqnarray}\n\\frac{\\partial{F}}\n{\\partial{t}}\n-\\frac{\\partial{}}\n{\\partial{z}}&&\n\\left[vL\\frac{\\int_{-1}^{1}\nd\\mu \\mu e^{M(\\mu, t)}}\n{\\int_{-1}^{1}d\\mu\ne^{M(\\mu,t)}}\n\\left(\\frac{\\partial{F}}\n{\\partial{z}}-\n\\frac{F}{L}\\right)\\right]\n+\\sum_{m,n}\\eta_{m,n}(t)\n\\frac{\\partial^{m+n}}\n{\\partial {t^m}\n\\partial{z}^n} F=0.\n\\label{Accurate diffusion\nequation-2}\n\\end{eqnarray}\nNow we get\na variable coefficient\nlinear differential equation\nof the isotropic distribution\nfunction.\n\n\\subsection{The specific form of the variable coefficient differential equation}\n\nBy employing the method in\n\\citet{wq2018},\none can rewrite Equation\n(\\ref{Accurate diffusion\n\tequation-2})\nas\n\\begin{eqnarray}\n\\frac{\\partial{F}}\n{\\partial{t}}=\n&&-\\kappa_z(t)\n\\frac{\\partial{F}}{\\partial{z}}\n+\\kappa_{zz}(t)\n\\frac{\\partial^2{F}}{\\partial{z^2}}\n+\\left(\\kappa_{zzz}(t)\n\\frac{\\partial^3{F}}{\\partial{z^3}}\n+\\kappa_{zzzz}(t)\n\\frac{\\partial^4{F}}{\\partial{z^4}}\n+\\cdots\\right)\n+\\left( \\kappa_{tz}(t)\n\\frac{\\partial^2{F}}{\\partial{t}\n\\partial{z}}\n+ \\kappa_{ttz}(t)\n\\frac{\\partial^3{F}}\n{\\partial{t^2}\\partial{z}}\n+ \\kappa_{tttz}(t)\n\\frac{\\partial^4{F}}{\\partial{t^3}\n\\partial{z}}+\\cdots\\right)\n\\nonumber\\\\\n&&+\\left( \\kappa_{tzz}(t)\n\\frac{\\partial^3{F}}{\\partial{t}\n\\partial{z^2}}\n+ \\kappa_{ttzz}(t)\\frac{\\partial^4{F}}\n{\\partial{t^2}\\partial{z^2}}\n+ \\kappa_{tttzz}(t)\\frac{\\partial^5{F}}\n{\\partial{t^3}\\partial{z^2}}\n+\\cdots\\right)\n+\\cdots\\cdots,\n\\label{equation of F}\n\\end{eqnarray}\nhere, $\\kappa_z(t)$,\n$\\kappa_{zz}(t)$,\n$\\kappa_{zzz}(t)$,\n$\\kappa_{zzzz}(t)$,\n$\\cdots\\cdots$,\n$\\kappa_{tz}(t)$,\n$\\kappa_{ttz}(t)$, $\\kappa_{tttz}(t)$,\n$\\cdots\\cdots$,\n$\\kappa_{tzz}(t)$, $\\kappa_{ttzz}(t)$,\n$\\kappa_{tttzz}(t)$,\n$\\cdots\\cdots$\nare all time dependent.\nThis equation determines the transport regimes by the specific\nforms of its coefficient.\nFor example, as shown in\n\\citet{wq2018},\n$\\kappa_z(t)$\nis the coefficient of the convection\nterm\n\\begin{equation}\n\\kappa_z(t)\n=v\\frac{\\int_{-1}^{1}d\\mu\n\\mu e^{M(\\mu,t)}}{\\int_{-1}^{1}\nd\\mu e^{M(\\mu,t)}},\n\\label{kz}\n\\end{equation}\nand $\\kappa_{zz}(t)$\nis the coefficient of the parallel\ndiffusion term\n\\begin{eqnarray}\n\\kappa_{zz}(t)&=&\nvL\\frac{\\int_{-1}^{1}d\\mu\n\\mu e^{M(\\mu,t)}}{\\int_{-1}^{1}\nd\\mu e^{M(\\mu,t)}}\n+\\frac{v^2}{2}\n\\Bigg\\{\n\\int_{-1}^{1}d\\mu\n\\mu e^{M(\\mu,t)}\n\\int_{-1}^{\\mu}d\\nu\n\\frac{e^{-M(\\nu,t)}}\n{D_{\\nu \\nu}(\\nu,t)}\n\\left[\\int_{-1}^{\\nu}d\\rho\\rho\n\\left(1-\\frac{2e^{M(\\rho,t)}}\n{\\int_{-1}^{1}d\\mu\ne^{M(\\mu,t)}}\\right)\n+\\int_{-1}^{1} d\\mu \\mu\n\\frac{e^{M(\\mu,t)}}\n{\\int_{-1}^{1} d\\mu\ne^{M(\\mu,t)}}\\right]\n\\nonumber\\\\\n&&-\\frac{\\int_{-1}^{1}\nd\\mu \\mu\ne^{M(\\mu,t)}}{\\int_{-1}^{1}\nd\\mu e^{M(\\mu,t)}}\n\\int_{-1}^{1}\nd\\mu e^{M(\\mu,t)}\n\\int_{-1}^{\\mu}\nd\\nu \\frac{e^{-M(\\nu,t)}}\n{D_{\\nu \\nu}(\\nu,t)}\n\\left[\\int_{-1}^{\\nu}\nd\\rho\\rho\n\\left(1-\\frac{2e^{M(\\rho,t)}}\n{\\int_{-1}^{1}d\\mu\ne^{M(\\mu,t)}}\\right)\n+\\int_{-1}^{1} d\\mu \\mu\n\\frac{e^{M(\\mu,t)}}\n{\\int_{-1}^{1} d\\mu\ne^{M(\\mu,t)}}\\right]\n\\Bigg\\}.\n \\label{A1-iso}\n\\end{eqnarray}\nWe can similarly obtain other\ntime-dependent coefficients in Equation\n(\\ref{equation of F}).\n\n\\subsection{Telegraph equation of the\nisotropic distribution function\n$F(z, t)$}\n\\label{Telegraph equation of the\nisotropic distribution function}\n\nThe diffusion convection equation\nis frequently used in previous research\n\\citep[see, e.g.,][]{Shalchi2009a}.\nBut the accuracy of the diffusion\napproximation is limited\nbecause the signal speed is\ninfinite in the diffusion limit.\nRecently, an improved description\nof the\nenergetic particle transport\nis provided by the telegraph equation,\ncharacterized by a finite\nsignal propagation speed\n\\citep{Kota1994,\nPaulsBurger1994,\nSchwadronEA2010, PorraMasoliver1997,\nZankEA2000, FedorovShakhov2003,\nKaghashviliEA2004,\nLitvinenkoASchlickeiser2013,\nEffenbergerEA2014, LitvinenkoEA2015,\nLitvinenkoNoble2016}.\nUsing the iterative method\nin \\citet{LitvinenkoASchlickeiser2013},\none can obtain the\ntelegraph equation from\nEquation (\\ref{equation of F})\n\\begin{eqnarray}\n\\frac{\\partial{F}}{\\partial{t}}\n+\\tau(t)\\frac{\\partial^2{F}}{\\partial{t^2}\n}+\\kappa_z(t)\\frac{\\partial{F}}\n{\\partial{z}}=\\kappa_{zz}(t)\n\\frac{\\partial^2{F}}{\\partial{z^2}}\n\\label{telegraph equation}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\tau(t)=-\\frac{\\kappa_{tzz}(t)}\n{\\kappa_{zz}(t)}.\n\\label{tau}\n\\end{eqnarray}\n\nEquation (\\ref{equation of F})\ncan be rewritten as\n\\begin{eqnarray}\n\\frac{\\partial{F}}{\\partial{z}}\n=&&\\frac{1}{\\kappa_z(t)}\n\\Bigg[-\\frac{\\partial{F}}\n{\\partial{t}}\n+\\kappa_{zz}(t)\n\\frac{\\partial^2{F}}{\\partial{z^2}}\n+\\left(\\kappa_{zzz}(t)\n\\frac{\\partial^3{F}}{\\partial{z^3}}\n+\\kappa_{zzzz}(t)\n\\frac{\\partial^4{F}}{\\partial{z^4}}\n+\\cdots\\right)\n+\\left( \\kappa_{tz}(t)\n\\frac{\\partial^2{F}}{\\partial{t}\n\t\\partial{z}}\n+ \\kappa_{ttz}(t)\n\\frac{\\partial^3{F}}\n{\\partial{t^2}\\partial{z}}\n+ \\kappa_{tttz}(t)\n\\frac{\\partial^4{F}}{\\partial{t^3}\n\t\\partial{z}}+\\cdots\\right)\n\\nonumber\\\\\n&&+\\left( \\kappa_{tzz}(t)\n\\frac{\\partial^3{F}}{\\partial{t}\n\t\\partial{z^2}}\n+ \\kappa_{ttzz}(t)\\frac{\\partial^4{F}}\n{\\partial{t^2}\\partial{z^2}}\n+ \\kappa_{tttzz}(t)\\frac{\\partial^5{F}}\n{\\partial{t^3}\\partial{z^2}}\n+\\cdots\\right)\n+\\cdots\\cdots\\Bigg].\n\\label{convection term of equation of F}\n\\end{eqnarray}\nInserting Equation\n(\\ref{convection term of equation of F}) into the convection term\n$\\kappa_{tz}\n\\partial^2{F}\/\\partial{t}\\partial{z}$\nin Equation (\\ref{equation of F})\nand neglecting the higher order derivative terms,\nwe can obtain\nanother version of\nthe telegraph equation,\n\\begin{eqnarray}\n\\frac{\\partial{F}}{\\partial{t}}\n+\\tau'(t)\\frac{\\partial^2{F}}{\\partial{t^2}}\n+\\kappa_z(t)\\frac{\\partial{F}}{\\partial{z}}\n=\\kappa_{zz}(t)\\frac{\\partial^2{F}}\n{\\partial{z^2}}\n\\label{new telegraph equation}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\tau'(t)=\\frac{\\kappa_{tz}\n(t)}{\\kappa_{z}(t)}.\n\\label{tau}\n\\end{eqnarray}\nWe find that the only\ndifference between Equations (\\ref{new telegraph equation})\nand (\\ref{telegraph equation})\nis caused by the fact that\n$\\tau(t)$ is not equal to $\\tau'(t)$.\n\n\\section{THE MODIFIED FORMULA OF DIFFUSION COEFFICIENT\nWITH DISPLACEMENT VARIANC}\n\\label{THE PARALLEL DIFFUSION\nCOEFFICIENT REPRESENTED\nBY ALONG-FIELD DISPLACEMENT MOMENTS}\nIn the following, we derive\nthe modified formula\nof the parallel\ndiffusion coefficient\nwith the parallel displacement variance\ninduced by the adiabatic focusing\n(hereafter MDCDV).\n\n\\subsection{MDCDV\nfor the variable coefficient\ndifferential equation}\n\\label{The parallel diffusion\ncoefficient of Equation}\n\nBy multiplying Equation\n(\\ref{equation of F})\nwith\n$\\langle (\\Delta z) \\rangle$ and\nintegrating the result, one can  find\n\\begin{eqnarray}\n\\frac{d}{dt}\\langle (\\Delta z)\n\\rangle&=&\\int_{-\\infty}^\\infty\ndz (\\Delta z)\\frac{\\partial{F}}\n{\\partial{t}}\\nonumber\\\\\n&=&\\int_{-\\infty}^\\infty dz\n(\\Delta z)\\Bigg[\\left(-\\kappa_z(t)\n\\frac{\\partial{F}}{\\partial{z}}\n+\\kappa_{zz}(t)\n\\frac{\\partial^2{F}}{\\partial{z^2}}\n+\\kappa_{zzz}(t)\n\\frac{\\partial^3{F}}{\\partial{z^3}}\n+\\kappa_{zzzz}(t)\n\\frac{\\partial^4{F}}{\\partial{z^4}}\n+\\cdots+\\kappa_{z\\cdots z}(t)\n\\frac{\\partial^n{F}}{\\partial{z^n}}\n+\\cdots\\right)\\nonumber\\\\\n&&+\\left( \\kappa_{tz}(t)\n\\frac{\\partial^2{F}}\n{\\partial{t}\\partial{z}}\n+ \\kappa_{ttz}(t)\n\\frac{\\partial^3{F}}\n{\\partial{t^2}\\partial{z}}\n+ \\kappa_{tttz}(t)\n\\frac{\\partial^4{F}}\n{\\partial{t^3}\\partial{z}}\n+\\cdots\n+\\kappa_{t\\cdots tz}(t)\n\\frac{\\partial^{n+1}{F}}\n{\\partial{t^n}\\partial{z}}\n+\\cdots\n\\right)\\nonumber\\\\\n&&+\\left( \\kappa_{tzz}(t)\n\\frac{\\partial^3{F}}\n{\\partial{t}\\partial{z^2}}\n+ \\kappa_{ttzz}(t)\n\\frac{\\partial^4{F}}\n{\\partial{t^2}\\partial{z^2}}+\n+\\cdots+\\kappa_{ttzz}(t)\n\\frac{\\partial^4{F}}\n{\\partial{t^2}\\partial{z^2}}\n+ \\kappa_{t\\cdots tzz}(t)\n\\frac{\\partial^{n+2}{F}}\n{\\partial{t^n}\\partial{z^2}}\n+\\cdots\\right)\n+\\cdots\\cdots\\Bigg].\n\\label{1-general}\n\\end{eqnarray}\nTo proceed,\nwe have to obtain the integrals\non the right hand side of\nthe latter equation\nwith the following conditions\n\\begin{eqnarray}\n&&F(z=\\pm\\infty)=0,\\\\\n&&\\frac{\\partial^n{F}}\n{\\partial{z^n}}(z=\\pm\\infty)=0\n\\hspace{0.2cm} n=1,2,3,\\cdots.\n\\end{eqnarray}\nBy performing integration\nin parts one can obtain\n\\begin{eqnarray}\n&&\\int_{-\\infty}^\\infty\ndz (\\Delta z)\\left(-\\kappa_z(t)\n\\frac{\\partial{F}}{\\partial{z}}\\right)\n=-\\kappa_z(t)\\int_{-\\infty}^\\infty\ndz (\\Delta z)\\frac{\\partial{F}}\n{\\partial{z}}=\\kappa_z(t),\n\\label{1-z}\\\\\n&&\\int_{-\\infty}^\\infty\ndz (\\Delta z)\\left(\\kappa_{zz}(t)\n\\frac{\\partial^2{F}}{\\partial{z^2}}\\right)\n=\\kappa_{zz}(t)\\int_{-\\infty}^\\infty\ndz (\\Delta z)\\frac{\\partial^2{F}}\n{\\partial{z^2}}=0.\n\\label{1-zz}\n\\end{eqnarray}\nWe can also find that the integrals\nare equal to zero if the order of spatial\nderivative of the integrands is\nhigher than one,\n\\begin{eqnarray}\n\\int_{-\\infty}^\\infty dz (\\Delta z)\n\\left(\\kappa_{z\\cdots z}(t)\n\\frac{\\partial^n{F}}{\\partial{z^n}}\\right)\n=\\kappa_{z\\cdots z}(t)\\int_{-\\infty}^\\infty\ndz (\\Delta z)\n\\frac{\\partial^n{F}}{\\partial{z^n}}\n=0 \\hspace{0.5cm}n=2,3,\\cdots.\n\\label{1-zn}\n\\end{eqnarray}\nIt is also easy to find the\nfollowing result by integration\nby parts\n\\begin{eqnarray}\n\\int_{-\\infty}^\\infty dz\n(\\Delta z)\\left(\\kappa_{z\\cdots z}(t)\n\\frac{\\partial^{n+m}{F}}{\\partial{t^m}\n\\partial{z^n}}\\right)=\\kappa_{z\\cdots z}(t)\n\\frac{d^m}{dt^m}\\int_{-\\infty}^\\infty\ndz (\\Delta z)\n\\frac{\\partial^n{F}}{\\partial{z^n}}\n=0 \\hspace{0.5cm}n=1,2,3,\\cdots,\n\\hspace{0.2cm}m=1,2,3,\\cdots.\n\\label{1-tmzn}\n\\end{eqnarray}\nInserting Equations (\\ref{1-z})\n- (\\ref{1-tmzn}) into Equation\n(\\ref{1-general}), one can obtain\nthe following formula\n\\begin{equation}\n\\frac{d}{dt}\\langle (\\Delta z)\n\\rangle=\\kappa_z(t).\n\\label{1-result}\n\\end{equation}\n\nFurthermore, analogous to\nEquation (\\ref{1-general}),\nthe formula of second order\nmoment of the parallel\ndisplacement\ncan be written as\n\\begin{eqnarray}\n\\frac{d}{dt}\\langle (\\Delta z)^2\n\\rangle&=&\\int_{-\\infty}^\\infty\ndz (\\Delta z)^2\n\\frac{\\partial{F}}{\\partial{t}}\n\\nonumber\\\\\n&=&\\int_{-\\infty}^\\infty dz\n(\\Delta z)^2\n\\Bigg[\\left(-\\kappa_z(t)\n\\frac{\\partial{F}}{\\partial{z}}\n+\\kappa_{zz}(t)\n\\frac{\\partial^2{F}}{\\partial{z^2}}\n+\\kappa_{zzz}(t)\n\\frac{\\partial^3{F}}{\\partial{z^3}}\n+\\kappa_{zzzz}(t)\n\\frac{\\partial^4{F}}{\\partial{z^4}}\n+\\cdots+\\kappa_{z\\cdots z}(t)\n\\frac{\\partial^n{F}}{\\partial{z^n}}\n+\\cdots\\right)\\nonumber\\\\\n&{}&+\\left( \\kappa_{tz}\n\\frac{\\partial^2{F}}\n{\\partial{t}\\partial{z}}\n+ \\kappa_{ttz}(t)\n\\frac{\\partial^3{F}}\n{\\partial{t^2}\\partial{z}}\n+ \\kappa_{tttz}(t)\n\\frac{\\partial^4{F}}\n{\\partial{t^3}\\partial{z}}\n+\\cdots\n+\\kappa_{t\\cdots tz}(t)\n\\frac{\\partial^{n+1}{F}}\n{\\partial{t^n}\\partial{z}}\n+\\cdots\\right)\\nonumber\\\\\n&&+\\left( \\kappa_{tzz}(t)\n\\frac{\\partial^3{F}}\n{\\partial{t}\\partial{z^2}}\n+ \\kappa_{ttzz}(t)\\frac{\\partial^4{F}}\n{\\partial{t^2}\\partial{z^2}}\n+ \\kappa_{tttzz}(t)\\frac{\\partial^5{F}}\n{\\partial{t^3}\\partial{z^2}}\n+\\cdots+\\kappa_{t\\cdots tzz}(t)\n\\frac{\\partial^{n+2}{F}}\n{\\partial{t^n}\\partial{z^2}}\n+\\cdots\\right)\n+\\cdots\\cdots\\Bigg].\n\\label{2-general}\n\\end{eqnarray}\nThe integrals on the right\nhand side of the latter\nequation\ncan be easily found\nby part integral\n\\begin{eqnarray}\n&&\\int_{-\\infty}^\\infty dz\n(\\Delta z)^2\\left(-\\kappa_z(t)\n\\frac{\\partial{F}}{\\partial{z}}\n\\right)\n=-\\kappa_z(t)\\int_{-\\infty}^\\infty\ndz (\\Delta z)^2\\frac{\\partial{F}}\n{\\partial{z}}\n=2\\kappa_z (t)\\langle (\\Delta z)\n\\rangle\n\\label{2-z}\\\\\n&&\\int_{-\\infty}^\\infty dz\n(\\Delta z)^2\\left(\\kappa_{zz}(t)\n\\frac{\\partial^2{F}}{\\partial{z^2}}\n\\right)\n=\\kappa_{zz}(t)\\int_{-\\infty}^\\infty\ndz (\\Delta z)^2\\frac{\\partial^2{F}}\n{\\partial{z^2}}\n=2\\kappa_{zz}(t)\n\\label{2-zz}\\\\\n&&\\int_{-\\infty}^\\infty dz\n(\\Delta z)^2\\left(\\kappa_{z\\cdots z}\n(t)\\frac{\\partial^n{F}}\n{\\partial{z^n}}\\right)\n=\\kappa_{z\\cdots z}(t)\n\\int_{-\\infty}^\\infty dz\n(\\Delta z)^2\\frac{\\partial^n{F}}\n{\\partial{z^n}}\n=0 \\hspace{0.5cm}n=3, 4, 5,\n\\cdots\\\\\n&&\\int_{-\\infty}^\\infty dz\n(\\Delta z)^2\n\\left(\\kappa_{t\\cdots tz}(t)\n\\frac{\\partial^{n+1}{F}}\n{\\partial{t^n}\\partial{z}}\\right)\n=\\kappa_{t\\cdots tz}(t)\n\\frac{d^n}{dt^n}\n\\int_{-\\infty}^\\infty dz\n(\\Delta z)^2\n\\frac{\\partial{F}}{\\partial{z}}\n=-2\\kappa_{t\\cdots tz}(t)\n\\frac{d^n}{dt^n}\\langle\n(\\Delta z) \\rangle\n\\hspace{0.5cm}n=1,2,3,\n\\cdots.\n\\label{1-tzn}\\\\\n&&\\int_{-\\infty}^\\infty dz\n(\\Delta z)^2\n\\left(\\kappa_{t\\cdots tz\\cdots z}(t)\n\\frac{\\partial^{n+m}{F}}\n{\\partial{t^n}\\partial{z^m}}\\right)\n=\\kappa_{t\\cdots tz\\cdots z}(t)\n\\frac{d^n}{dt^n}\n\\int_{-\\infty}^\\infty\ndz (\\Delta z)^2\n\\frac{\\partial{F^m}}\n{\\partial{z^m}}=0\n\\hspace{0.5cm}n=1,2,3,\n\\cdots, \\hspace{0.2cm}m=2,3,\n\\cdots.\n\\label{2-tzn}\n\\end{eqnarray}\nWith Equations\n(\\ref{2-z})-(\\ref{2-tzn}),\nEquation (\\ref{2-general})\nbecomes\n\\begin{eqnarray}\n\\frac{d}{dt}\\langle (\\Delta z)^2\n\\rangle =2\\kappa_{zz}(t)+2\\left[\\kappa_z(t)\n\\langle (\\Delta z) \\rangle\n-\\kappa_{tz}(t)\\frac{d}{dt}\n\\langle (\\Delta z) \\rangle\n-\\kappa_{ttz}(t)\\frac{d^2}{dt^2}\n\\langle (\\Delta z) \\rangle\n-\\kappa_{tttz}(t)\\frac{d^3}{dt^3}\n\\langle (\\Delta z) \\rangle\n-\\cdots\\cdots\\right].\n\\label{2-result}\n\\end{eqnarray}\nThe latter equation can be rewritten as\n\\begin{eqnarray}\n\\kappa_{zz}(t)=\\frac{1}{2}\n\\frac{d}{dt}\\langle\n(\\Delta z)^2 \\rangle\n+C(t)\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nC(t)=-\\kappa_z(t)\\langle (\\Delta z)\\rangle\n+\\kappa_{tz}(t)\\frac{d}{dt}\n\\langle (\\Delta z) \\rangle\n+\\kappa_{ttz}(t)\\frac{d^2}{dt^2}\n\\langle (\\Delta z) \\rangle\n+\\kappa_{tttz}(t)\\frac{d^3}{dt^3}\n\\langle (\\Delta z) \\rangle\n+\\cdots\\cdots.\n\\end{eqnarray}\nHere, $C(t)$\nis related to convection\nof energetic particles.\n\nBy employing formula $\\sigma^2=\\langle\n(\\Delta z)^2 \\rangle\n-\\langle (\\Delta z)\n\\rangle^2$ and Equation\n(\\ref{1-result})\n we can obtain\n\\begin{eqnarray}\n\\kappa_{zz}(t)=\\frac{1}{2}\n\\frac{d\\sigma^2}{dt}\n+\\left[\\kappa_{tz}(t)\\frac{d}{dt}\n\\langle (\\Delta z) \\rangle\n+\\kappa_{ttz}(t)\\frac{d^2}{dt^2}\n\\langle (\\Delta z) \\rangle\n+\\kappa_{tttz}(t)\\frac{d^3}{dt^3}\n\\langle (\\Delta z) \\rangle\n+\\cdots\\cdots\\right].\n\\label{center formula-1}\n\\end{eqnarray}\nInserting Equation (\\ref{1-result})\ninto the latter equation one\ncan find\n\\begin{eqnarray}\n\\kappa_{zz}(t)=\\frac{1}{2}\n\\frac{d\\sigma^2}{dt}\n+\\left[\\kappa_{tz}(t)\n\\kappa_z(t)\n+\\kappa_{ttz}(t)\\frac{d}{dt}\n\\kappa_z(t)\n+\\kappa_{tttz}(t)\\frac{d^2}{dt^2}\n\\kappa_z(t)+\\cdots\\cdots\\right].\n\\label{center formula-2}\n\\end{eqnarray}\nThe latter equation can\nbe rewritten as\n\\begin{eqnarray}\n\\frac{d\\sigma^2}{dt}\n=2\\kappa_{zz}(t)\n-\\left[2\\kappa_{tz}(t)\n\\kappa_z(t)\n+2\\kappa_{ttz}(t)\\frac{d}{dt}\n\\kappa_z(t)+2\\kappa_{tttz}(t)\n\\frac{d^2}{dt^2}\\kappa_z(t)\n+\\cdots\\cdots\\right].\n\\label{kzz-0}\n\\end{eqnarray}\n\nEquation (\\ref{center formula-1})\nor (\\ref{kzz-0}) is the modified DCDV\n(MDCDV).\nIn comparison with the well-known\nEquation (\\ref{kzz-sigma})\n(represented by DCDV),\none can find that MDCDV\nis a more general formula.\nThat is, DCDV\nis the special case of MDCDV\nwhen the cross terms with the first\norder spatial derivative or\nthe convection term are\nequal to zero.\n\nBecause the coefficients\n$\\kappa_{z}(t)$, $\\kappa_{zz}(t)$,\n$\\kappa_{tz}(t)$, $\\cdots\\cdots$\nare all time dependent,\nthe relation $\\sigma^2\\propto t$\ndoes not hold for MDCDV.\nTherefore,\nthe transport\ndescribed by MDCDV\nis not in diffusion regime.\nSince Equation (\\ref{kzz-0}) is\nderived from Equation\n(\\ref{equation of F}),\nthe transport described\nby Equation (\\ref{equation of F})\nis also not in diffusion regime.\n\n\\subsection{MDCDV\nfor the telegraph equation}\n\\label{The parallel diffusion\ncoefficient of the telegraph equation}\n\nFor the telegraph equation\nderived in\n\\citet{LitvinenkoASchlickeiser2013}\nthe corresponding first\nand second order moments of the\nparallel displacement\ncan be written as, respectively\n\\begin{eqnarray}\n&&\\frac{d}{dt}\\langle (\\Delta z)\n\\rangle=\\kappa_z(t)-\\tau(t)\n\\frac{d^2{}}{d{t^2}}\\langle\n(\\Delta z)  \\rangle,\n\\label{1-result for\ntelegraph equation}\\\\\n&&\\frac{d}{dt}\\langle\n(\\Delta z)^2 \\rangle\n=2\\kappa_z(t) \\langle (\\Delta z)\n\\rangle+2\\kappa_{zz}(t)\n-\\tau(t)\\frac{d^2}{dt^2}\\langle\n(\\Delta z)^2  \\rangle.\n\\label{2-result for\ntelegraph equation}\n\\end{eqnarray}\nBy combining the latter equations\none can derive with $\\sigma^2=\\langle\n(\\Delta z)^2 \\rangle -\\langle\n(\\Delta z) \\rangle^2$\n\\begin{eqnarray}\n\\kappa_{zz}(t)&=&\\frac{1}{2}\n\\frac{d\\sigma^2}{dt}\n+\\tau(t)\\Bigg[\\frac{1}{2}\n\\frac{d^2}{dt^2}\\langle (\\Delta z)^2\n\\rangle-\\langle (\\Delta z)\n\\rangle\\frac{d^2}{dt^2}\\langle\n(\\Delta z) \\rangle\\Bigg].\n\\end{eqnarray}\nInserting Equation (\\ref{tau})\ninto the latter equation,\nwe can obtain\n\\begin{eqnarray}\n\\kappa_{zz}(t)\n=\\frac{1}{2}\\frac{d\\sigma^2}{dt}\n-\\frac{\\kappa_{tzz}(t)}\n{\\kappa_{zz}(t)}\\Bigg[\\frac{d^2}{dt^2}\n\\langle (\\Delta z)^2 \\rangle-\\langle\n(\\Delta z) \\rangle\\frac{d^2}{dt^2}\n\\langle (\\Delta z) \\rangle\\Bigg].\n\\label{kzz for telegraph equation}\n\\end{eqnarray}\n\nBy subtracting Equation\n(\\ref{kzz for telegraph equation})\nfrom Equation (\\ref{center formula-2})\nwe can obtain the error formula\n\\begin{eqnarray}\nE(t)\n=\\left[\\kappa_{tz}(t)\\kappa_z(t)\n+\\kappa_{ttz}(t)\\frac{d}\n{dt}\\kappa_z(t)\n+\\kappa_{tttz}(t)\\frac{d^2}{dt^2}\n\\kappa_z(t)+\\cdots\\cdots\\right]\n+\\frac{\\kappa_{tzz}(t)}\n{\\kappa_{zz}(t)}\\Bigg[\\frac{d^2}{dt^2}\n\\langle (\\Delta z)^2 \\rangle\n-\\langle (\\Delta z)\n\\rangle\\frac{d^2}{dt^2}\\langle\n(\\Delta z) \\rangle\\Bigg],\n\\label{difference}\n\\end{eqnarray}\nwhich comes from the iteration operation\nused in the derivation\nof telegraph equation.\nIn the future, this formula\nwill be explored in detail.\n\nIn addition, Equation\n(\\ref{1-result for telegraph equation})\ncan be rewritten as\n\\begin{eqnarray}\n\\tau(t)\\frac{d^2{}}{d{t^2}}\n\\langle (\\Delta z) \\rangle\n+\\frac{d}{dt}\\langle (\\Delta z)\n\\rangle=\\kappa_z(t).\n\\label{11-differential equation}\n\\end{eqnarray}\nIf the coefficients $\\tau(t)$\nand $\\kappa_z(t)$ can be determined,\nthe latter equation might\nbe solved.\nSimilarily, Equation\n(\\ref{2-result for\ntelegraph equation})\ncan be rewritten as\n\\begin{eqnarray}\n\\tau(t)\\frac{d^2}{dt^2}\n\\langle (\\Delta z)^2\n\\rangle+\\frac{d}{dt}\\langle\n(\\Delta z)^2 \\rangle =2\\kappa_z(t)\n\\langle (\\Delta z) \\rangle\n+2\\kappa_{zz}(t)\n\\label{22-differential equation}\n\\end{eqnarray}\nCombining the solution of\nEquation\n(\\ref{11-differential equation})\nand the coefficients\n$\\tau(t)$ and $\\kappa_z(t)$\nwith the latter equation,\nthe formula of\n$\\langle (\\Delta z)^2  \\rangle$\nmight be got.\nSome solutions\nof Equations (\\ref{11-differential equation})\nand (\\ref{22-differential equation})\nwill be investigated\nin the future.\n\n\\subsection{MDCDV for telegraph\nEquation (\\ref{new telegraph equation})}\n\\label{The parallel diffusion\ncoefficient of the new\ntelegraph equation}\n\nSimilarily, for telegraph\nEquation (\\ref{new telegraph equation}) the first and\nsecond moments can be got\nas follows\n\\begin{eqnarray}\n&&\\frac{d}{dt}\\langle (\\Delta z) \\rangle\n=\\kappa_z(t)-\\tau'(t)\\frac{d^2{}}\n{d{t^2}}\\langle (\\Delta z)  \\rangle,\\\\\n&&\\frac{d}{dt}\\langle (\\Delta z)^2\n\\rangle =2\\kappa_z(t) \\langle (\\Delta z)\n\\rangle+2\\kappa_{zz}(t)\n-\\tau'(t)\\frac{d^2}{dt^2}\n\\langle (\\Delta z)^2  \\rangle.\n\\end{eqnarray}\nThe corresponding MDCDV becomes\n\\begin{eqnarray}\n\\kappa_{zz}(t)\n=\\frac{1}{2}\\frac{d\\sigma^2}{dt}\n+\\frac{\\kappa_{tz}(t)}{\\kappa_{z}(t)}\n\\Bigg[\\frac{d^2}{dt^2}\\langle (\\Delta z)^2\n\\rangle-\\langle (\\Delta z)\n\\rangle\\frac{d^2}{dt^2}\\langle\n(\\Delta z) \\rangle\\Bigg].\n\\end{eqnarray}\n\n\\section{SPECIAL CASES}\n\\label{SPECIAL CASES}\n\nIn the following, we investigate\nthe special cases of MDCDV\n(see Equation (\\ref{center formula-1})),\nwhich\ncorrespond to the\nsimplified versions of Equation\n(\\ref{equation of F}).\n\n\\subsection{Case 1:\nconstant coefficient\ndifferential equation}\n\\label{For the case D=D(mu)}\nIf the temporal effect of\nthe pitch-angle diffusion\ncoefficient $D_{\\mu\\mu}$\ncan be ignored,\ni.e.,\n$D_{\\mu\\mu}=D_{\\mu\\mu}(\\mu)$,\nall the coefficients in\nEquation (\\ref{equation of F})\nare constant.\nTherefore, Equation\n(\\ref{equation of F})\nis simplified as\n\\begin{eqnarray}\n\\frac{\\partial{F}}{\\partial{t}}=\n&&\\left(-\\kappa_z\\frac{\\partial{F}}\n{\\partial{z}}+\\kappa_{zz}\n\\frac{\\partial^2{F}}{\\partial{z^2}}\n+\\kappa_{zzz}\n\\frac{\\partial^3{F}}{\\partial{z^3}}\n+\\kappa_{zzzz}\n\\frac{\\partial^4{F}}{\\partial{z^4}}\n+\\cdots\\right)\n+\\left( \\kappa_{tz}\n\\frac{\\partial^2{F}}{\\partial{t}\n\\partial{z}}\n+ \\kappa_{ttz}\\frac{\\partial^3{F}}\n{\\partial{t^2}\\partial{z}}\n+ \\kappa_{tttz}\\frac{\\partial^4{F}}\n{\\partial{t^3}\\partial{z}}\n+\\cdots\\right)\\nonumber\\\\\n&&+\\left( \\kappa_{tzz}\n\\frac{\\partial^3{F}}{\\partial{t}\n\\partial{z^2}}\n+ \\kappa_{ttzz}\\frac{\\partial^4{F}}\n{\\partial{t^2}\\partial{z^2}}\n+ \\kappa_{tttzz}\\frac{\\partial^5{F}}\n{\\partial{t^3}\\partial{z^2}}\n+\\cdots\\right)\n+\\cdots\\cdots.\n\\label{equation of F with\nconstant coefficient}\n\\end{eqnarray}\nThis is a constant coefficient\nlinear differential equation\nof the isotropic distribution\nfunction.\nThe first\nand second order moments of\nthe parallel displacement can\nbe found as\n\\begin{eqnarray}\n&&\\frac{d}{dt}\\langle (\\Delta z)\n\\rangle=\\kappa_z=constant,\n\\label{1 for constant coefficient}\\\\\n&&\\frac{d}{dt}\\langle (\\Delta z)^2\n\\rangle =2\\kappa_z \\langle (\\Delta z)\n\\rangle+2\\kappa_{zz}\n-2\\kappa_{tz}\\kappa_z.\n\\label{2 for constant coefficient}\n\\end{eqnarray}\nBy combining the latter two formulas\none can derive\n\\begin{eqnarray}\n\\kappa_{zz}=\\frac{1}{2}\n\\frac{d\\sigma^2}{dt}\n+\\kappa_{tz}\\kappa_z.\n\\label{center formula-3}\n\\end{eqnarray}\nWe can find\nthat DCDV\nis the special case of\nthe latter equation\nwhen $\\kappa_{tz}\\kappa_z=0$.\n\nBecause the coefficients\n$\\kappa_z$, $\\kappa_{zz}$\nand $\\kappa_{tz}$ are all\nconstants,\nEquation (\\ref{center formula-3})\ncan\nbe rewritten as\n\\begin{eqnarray}\n\\frac{d\\sigma^2}{dt}=2\\kappa_{zz}\n-2\\kappa_{tz}\\kappa_z=constant.\n\\label{sigma}\n\\end{eqnarray}\nBy setting the initial value\nof $\\sigma^2$ as zero, from\nthe latter equation we can\nobtain\n\\begin{eqnarray}\n\\sigma^2=\\alpha t.\n\\label{formula of diffusion regime}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\alpha=2\\kappa_{tz}\\kappa_z\n+2\\kappa_{zz}=constant\n\\label{alpha}\n\\end{eqnarray}\nEquation\n(\\ref{formula of diffusion regime})\ndescribes diffusive regime.\nTherefore, for $D_{\\mu\\mu}\n=D_{\\mu\\mu}(\\mu)$\nthe constant coefficient linear\ndifferential equation of the\nisotropic distribution function\n(see Equation (\\ref{equation of F\nwith constant coefficient})) can\ndescribe the diffusion\nof the energetic particles.\nIf $\\kappa_z$ or $\\kappa_{tz}$\ncan be ignored,\nDCDV (Equation (\\ref{kzz-sigma}))\nis obtained.\n\n\\subsection{Case 2:\ndiffusion convection equation}\n\\label{For the case diffusion\nconvection equation}\nIf Equation\n(\\ref{equation of F with\nconstant coefficient})\nis simplified to the\ndiffusion convection equation\n\\begin{eqnarray}\n\\frac{\\partial{F}}{\\partial{t}}\n=-\\kappa_z\\frac{\\partial{F}}\n{\\partial{z}}+\\kappa_{zz}\n\\frac{\\partial^2{F}}{\\partial{z^2}},\n\\label{simplified equation of\nF with constant coefficient-1}\n\\end{eqnarray}\nEquation (\\ref{center formula-3})\nbecomes DCDV.\n\n\\subsection{Case 3:\ndiffusion equation}\n\\label{For the case diffusion equation}\nIf the convection term can be\nneglected, the diffusion convection\nEquation (\\ref{simplified equation of\n\tF with constant coefficient-1})\ncan be simplified\nto diffusion equation\n\\begin{eqnarray}\n\\frac{\\partial{F}}{\\partial{t}}\n=\\kappa_{zz}\\frac{\\partial^2{F}}\n{\\partial{z^2}}.\n\\end{eqnarray}\nTherefore, Equation\n(\\ref{1 for constant coefficient})\nbecomes\n\\begin{equation}\n\\frac{d}{dt}\\langle (\\Delta z) \\rangle=0.\n\\label{1-result for constant ceofficient}\n\\end{equation}\nConsequently,\nit is easy to see that\nthe parallel diffusion coefficient\ncan be shown as\n\\begin{eqnarray}\n\\kappa_{zz}=\\frac{1}{2}\\frac{d}{dt}\n\\langle (\\Delta z)^2 \\rangle .\n\\label{center formula corresponding\nto diffusion equation}\n\\end{eqnarray}\n\n\\subsection{Case 4: the constant\nbackground magnetic field}\n\\label{For the case: the constant\nbackground magnetic field}\nFor the constant background magnetic\nfield, the Fokker-Planck equation with\ntime-dependent pitch-angle diffusion\ncoefficient is\n\\begin{equation}\n\\frac{\\partial{f}}{\\partial{t}}\n+ v\\mu\n\\frac{\\partial{f}}\n{\\partial{z}}=\n\\frac{\\partial{}}{\\partial{\\mu}}\n\\left[D_{\\mu \\mu}(\\mu,t)\n\\frac{\\partial{f}}\n{\\partial{\\mu}}\\right].\n\\label{Fokker-Planck equation\nfor constant mean magnetic field}\n\\end{equation}\nBy using the similar method\nin Section 2,\nfrom the latter equation we can\nobtain the variable coefficient\ndifferential equation of isotropic\ndistribution function  as\n\\begin{eqnarray}\n\\frac{\\partial{F}}{\\partial{t}}=\n&&\\left(\\kappa_{zz}(t)\n\\frac{\\partial^2{F}}{\\partial{z^2}}\n+\\kappa_{zzz}(t)\n\\frac{\\partial^3{F}}{\\partial{z^3}}\n+\\kappa_{zzzz}(t)\n\\frac{\\partial^4{F}}{\\partial{z^4}}\n+\\cdots\\right)\n+\\left( \\kappa_{tz}(t)\n\\frac{\\partial^2{F}}{\\partial{t}\\partial{z}}\n+ \\kappa_{ttz}(t)\n\\frac{\\partial^3{F}}\n{\\partial{t^2}\\partial{z}}\n+ \\kappa_{tttz}(t)\n\\frac{\\partial^4{F}}{\\partial{t^3}\n\\partial{z}}+\\cdots\\right)\\nonumber\\\\\n&&+\\left( \\kappa_{tzz}(t)\n\\frac{\\partial^3{F}}{\\partial{t}\n\\partial{z^2}}\n+ \\kappa_{ttzz}(t)\\frac{\\partial^4{F}}\n{\\partial{t^2}\\partial{z^2}}\n+ \\kappa_{tttzz}(t)\\frac{\\partial^5{F}}\n{\\partial{t^3}\\partial{z^2}}+\\cdots\\right)\n+\\cdots\\cdots.\n\\label{equation of F for\nconstant mean magnetic field}\n\\end{eqnarray}\n\nComparing\nwith Equation (\\ref{equation of F}),\nwe find that Equation\n(\\ref{equation of F for\nconstant mean magnetic field})\ndoes not contain the convection\nterm $\\kappa_z(t)\n\\partial{F}\/\\partial{z}$\nwhich is caused\nby the adiabatic\nfocusing effect.\nThe first order moment\nof parallel displacement\ncorresponding to Equation\n(\\ref{equation of F for\nconstant mean magnetic field})\nis given as,\n\\begin{eqnarray}\n&&\\frac{d}{dt}\\langle (\\Delta z)\n\\rangle=\\kappa_z(t)=0.\n\\label{1-result for constant\nmean magentic field}\n\\end{eqnarray}\nSimilarily, we can find the second\norder moment of the parallel\ndisplacement as\n\\begin{eqnarray}\n\\frac{d}{dt}\\langle (\\Delta z)^2\n\\rangle =2\\kappa_{zz}(t).\n\\label{2-result for constant\nmean magentic field}\n\\end{eqnarray}\nThis equation can be\nrewritten as\n\\begin{eqnarray}\n\\kappa_{zz}(t)=\\frac{1}{2}\n\\frac{d}{dt}\\langle (\\Delta z)^2\n\\rangle.\n\\label{kzz for constant mean\nmagentic field}\n\\end{eqnarray}\n\n\\section{SUMMARY AND CONCLUSION}\n\\label{SUMMARY AND CONCLUSION}\n\nThe\nparallel diffusion coefficient\ncan be defined as\none half of the temporal derivative\nof variance of the parallel displacement,\nDCDV (Equation (\\ref{kzz-sigma})).\nIt is shown that DCDV is only suitable to\nthe transport equation\nwithout the convection term or\nwithout the cross terms.\nIf the parallel diffusion coefficient\n$\\kappa_{zz}$ is a constant,\nthe formula $\\sigma^2=2\\kappa_{zz}t$ holds,\nwhich means the transport regime is diffusion.\nThe purpose of this work is to derive\nthe formula of the parallel\ndiffusion coefficient $\\kappa_{zz}$\n with the parallel displacement\n variance $\\sigma^2$\nfor the more general transport\nequation of the isotropic\ndistribution function.\nWe start from the Fokker-Planck equation\nwith time-dependent pitch-angle\ndiffusion coefficient $D_{\\mu\\mu}(\\mu,t)$,\nwhich is an extension of the\nFokker-Planck equation with\ntime-independent\npitch-angle diffusion coefficient.\nFrom this equation,\nby employing the\nperturbation method which is frequently\nused in relevant research\n\\citep[see, e.g.,][]{BeeckEA1986,\nBieberEA1990, SchlickeiserEA2008,\nSchlickeiserEA2010,\nLitvinenkoASchlickeiser2013,\nHeEA2014, WangEA2017b, wq2018},\nthe variable coefficient differential\nequation of the isotropic distribution\n function $F(z,t)$\nwith the time-dependent coefficients\nis derived.\nFrom the equation of $F(z,t)$ the modified\nformula describing the relation\nof the parallel diffusion coefficient\n$\\kappa_{zz}$ with the parallel\ndisplacement variance $\\sigma^2$\n (MDCDV)\nis found.\nMDCDV indicates that the parallel\ndiffusion is determined by not only\nthe parallel displacement variance\nbut also all\nthe coefficients of the cross terms\nwith the first order spatial derivative\nand convection term coefficient.\nBecause all coefficients are\ntime-dependent in MDCDV,\nthe well-known formula $\\sigma^2\\propto t$\ndoes not hold.\nTherefore, the transport of\nthe energetic charged particles described\nby MDCDV\nis not in diffusion  regime.\n\nIn addition, we study\nMDCDV in some special cases.\nFor the isotropic distribution\nfunction equation with\nthe constant coefficient,\nMDCDV is simplified to\nEquation (\\ref{center formula-3}).\nThis formula demonstrates that\nthe parallel diffusion coefficient\nis determined by\nthe parallel displacement variance,\nthe convection term coefficient, and\nthe cross term coefficients with all the first\n order temporal and spatial\nderivatives.\nSince all the coefficients are constant,\n $\\sigma^2\\propto t$ can be found.\nTherefore, the constant\ncoefficient differential\nequation of  $F(z,t)$ can\ndescribe transport in diffusion\nregime.\nFurthermore, we explore the\ntransport equation\nwith uniform background magnetic field.\nFor this case, the convection term does\n not occur in the equation of $F(z,t)$.\nAccordingly, MDCDV derived\nin this paper\nreduces to DCDV.\nFor the diffusion convection\nequation and\ndiffusion equation,\nthe more simplified equations of\n$F(z,t)$,\nMDCDV is also reduced to\nDCDV.\nThe telegraph equation is\nanother frequently used\ntransport equation,\nwhich is derived by using iteration\nmethod from the isotropic\ndistribution function equation.\nIn this article,\nwe also derive a new telegraph equation,\nand explore preliminarily the corresponding\nformula of $\\kappa_{zz}$\nwith variance $\\sigma^2$.\n\nInstead of adibatic focusing,\nsolar wind speed is another factor\ncausing the convection\nterm in the equation of the isotropic\ndistribution function.\nSo, when the phase space equation\nwith solar\nwind speed is projected to real space,\nsolar wind leads to a convection\nterm in the real space equation.\nWe\nwill explore this topic in the\nfuture.\nAs a subject of future work we will\nperform a\ndetailed investigation into\nthe new\ntelegraph equation.\n\n\\acknowledgments\n\nWe are partly supported by\ngrant  NNSFC 41874206, NNSFC 41574172.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNotions of dimension abound in mathematics, and they give us quantitative measures of the sizes of various mathematical objects in a broad sense. In some instances, one wishes to know the exact dimension while in other instances, one just wishes to determine finiteness of the dimension. Finiteness of various dimensions has been considered in connection with central problems in the theory of $C^*$-algebras and in noncommutative geometry. For instance, finiteness of nuclear dimension (a noncommutative analog of covering dimension introduced in \\cite{WinZac}) plays a crucial role in the classification of $C^*$-algebras (e.g. \\cite{Ell,EGLN,EGLN17,Tik,TWW} and references therein), while finiteness of asymptotic dimension (a coarse geometric analog of covering dimension introduced in \\cite{Grom}) has featured in work on the Baum-Connes conjecture and the Novikov conjecture (e.g. \\cite{CG,CFY,Yu98}). In this paper, we will consider the notion of dynamical complexity and the implication of its finiteness on an $L^p$ assembly map, which is a model for the Baum-Connes assembly map with coefficients when $p=2$.\n\nDynamical complexity is a property of topological dynamical systems (and more generally, for \\'{e}tale groupoids) introduced by Guentner, Willett, and Yu in \\cite{GWY2}. Its definition was inspired by the notion of dynamic asymptotic dimension introduced in \\cite{GWY1} and by the notion of decomposition complexity introduced in \\cite{GTY}. Given an action of a countable discrete group $\\Gamma$ on a compact Hausdorff space $X$, the action is said to have finite dynamical complexity if the transformation groupoid $\\Gamma\\ltimes X$ is contained in the smallest class of open subgroupoids of $\\Gamma\\ltimes X$ that contains all relatively compact open subgroupoids and is closed under decomposability. Here, decomposability of an open subgroupoid $G$ over a collection $\\mathcal{C}$ of open subgroupoids roughly means that at any given scale, there is a cover of the unit space of $G$ by two open sets such that the subgroupoids associated to the two open sets at that scale are both in $\\mathcal{C}$. We refer the reader to Definition \\ref{FDCbasic} for the precise definition and to \\cite[Definition A.4]{GWY2} for a definition applicable to general \\'{e}tale groupoids.\n\n\nThe authors of \\cite{GWY2} considered a model for the Baum-Connes assembly map for an action based on equivariant versions of Yu's localization algebras (introduced in \\cite{Yu97}) and Roe algebras (introduced in \\cite{Roe88,Roe93}). In the appendix of \\cite{GWY2}, the authors showed that this model for the Baum-Connes assembly map is equivalent to the more traditional one stated in terms of Kasparov's $KK$-theory \\cite{BCH}. Their main result is the following:\n\n\\begin{thm}\\cite{GWY2}\nSuppose an action of a countable discrete group $\\Gamma$ on a compact Hausdorff space $X$ has finite dynamical complexity. Then $\\Gamma$ satisfies the Baum-Connes conjecture with coefficients in $C(X)$.\n\\end{thm}\n\n\nAlthough the aforementioned result follows from earlier work of Tu \\cite{Tu} on the Baum-Connes conjecture for amenable groupoids, the proof given in \\cite{GWY2} is completely different, and in some sense more direct and more elementary. In fact, their proof is inspired by Yu's proof of the coarse Baum-Connes conjecture for spaces with finite asymptotic dimension in \\cite{Yu98}. The main tool in both cases is a controlled Mayer-Vietoris sequence, which is part of a framework of quantitative (or controlled) $K$-theory for $C^*$-algebras  developed by Yu together with Oyono-Oyono in \\cite{OY15,OY}. Roughly speaking, finite dimension\/complexity enables one to apply the Mayer-Vietoris argument a finite number of times to arrive at the quantitative $K$-theory of the algebra in question, and the (standard) $K$-theory of the algebra is obtained as a limit of the quantitative $K$-theory. \n\nIn our earlier work \\cite{Chung1}, we have extended the framework of quantitative $K$-theory to a larger class of Banach algebras, so that it can be applied to algebras of bounded linear operators on $L^p$ spaces in particular. Our goal in this paper is to consider the $L^p$ analog of the assembly map in \\cite{GWY2}, and use our extended framework of quantitative $K$-theory to show that this assembly map is an isomorphism under the assumption of finite dynamical complexity. In fact, one sees that the techniques and proofs in \\cite{GWY2} carry over to our setting with minor adjustments, the main differences being the exposition of the homotopy invariance argument in Section \\ref{sec:Base}, and more care in the use of quantitative $K$-theory (due to an additional norm control parameter) in the Mayer-Vietoris argument used in the proof of the main theorem in Section \\ref{sec:Ind}. \n\nIn order to explain our setup, let us briefly recall the Baum-Connes conjecture with coefficients.\nGiven a (separable) $C^*$-algebra $A$ and an action of a countable discrete group $\\Gamma$ on $A$ by $*$-automorphisms, one may form the reduced crossed product $C^*$-algebra $A\\rtimes_\\lambda\\Gamma$. The Baum-Connes conjecture with coefficients \\cite{BCH} posits that a certain homomorphism (or assembly map) \\[\\mu:K_*^\\Gamma(\\underline{E}\\Gamma;A)\\rightarrow K_*(A\\rtimes_\\lambda\\Gamma)\\] is an isomorphism, where the left-hand side is the equivariant $K$-homology with coefficients in $A$ of the classifying space $\\underline{E}\\Gamma$ for proper $\\Gamma$-actions, and the right-hand side is the $K$-theory of the reduced crossed product $C^*$-algebra. Consider a particular model for $\\underline{E}\\Gamma$, namely $\\bigcup_{s\\geq 0}P_s(\\Gamma)$ equipped with the $\\ell^1$ metric (cf. \\cite[Section 2]{BCH}), where $P_s(\\Gamma)$ is the Rips complex of $\\Gamma$ at scale $s$, i.e., it is the simplicial complex with vertex set $\\Gamma$, and where a finite subset $E\\subset\\Gamma$ spans a simplex if and only if $d(g,h)\\leq s$ for all $g,h\\in E$. Here we assume that $\\Gamma$ is equipped with a proper length function and $d$ is the associated metric. One may then reformulate the Baum-Connes assembly map as \\[\\lim_{s\\rightarrow\\infty}K_*(C_L^*(P_s(\\Gamma);A))\\stackrel{\\epsilon_0}{\\rightarrow} \\lim_{s\\rightarrow\\infty}K_*(C^*(P_s(\\Gamma);A))\\cong K_*(A\\rtimes_\\lambda\\Gamma),\\]\nwhere $C^*(P_s(\\Gamma);A)$ is the equivariant Roe algebra with coefficients in $A$, $C_L^*(P_s(\\Gamma);A)$ is Yu's localization algebra with coefficients in $A$, and $\\epsilon_0$ is induced by the evaluation-at-zero map. The fact that $K$-homology can be identified with the $K$-theory of the localization algebra was shown for finite-dimensional simplicial complexes in \\cite{Yu97}, and for general locally compact metric spaces in \\cite{QR}. The fact that the equivariant Roe algebra with coefficients is stably isomorphic to the reduced crossed product underlies the coarse-geometric approach to the Baum-Connes conjecture with coefficients. We refer the reader to \\cite[Appendix B]{GWY2} for a detailed comparison of the abovementioned assembly maps, and also \\cite{Roe02} for a comparison of assembly maps without coefficients.\n\nNow let $A$ be a norm-closed subalgebra of $B(L^p(Z,\\mu))$ for some measure space $(Z,\\mu)$ and $p\\in[1,\\infty)$. We refer to such algebras as $L^p$ operator algebras. Also suppose that a countable discrete group $\\Gamma$ acts on $A$ by isometric automorphisms. By mimicking the $C^\\ast$-algebraic definitions, we can define $L^p$ analogs $B^p_L(P_s(\\Gamma);A)$, $B^p(P_s(\\Gamma);A)$, $A\\rtimes_{\\lambda,p}\\Gamma$ of the localization algebra, the Roe algebra, and the reduced crossed product respectively. We can then consider the following $L^p$ assembly map for the $\\Gamma$-action:\n\\[\\lim_{s\\rightarrow\\infty}K_*(B^p_L(P_s(\\Gamma);A))\\stackrel{\\epsilon_0}{\\rightarrow} \\lim_{s\\rightarrow\\infty}K_*(B^p(P_s(\\Gamma);A))\\cong K_*(A\\rtimes_{\\lambda,p}\\Gamma).\\]\nIn this paper, $A$ will be $C(X)$, where $X$ is a compact Hausdorff space, and\nour main result may be stated as follows:\n\\begin{thm} (cf. Theorem \\ref{mainthm}) \\label{LpBC}\nSuppose that an action of a countable discrete group $\\Gamma$ on a compact Hausdorff space $X$ has finite dynamical complexity. Then the $L^p$ assembly map for the action is an isomorphism for $p\\in[1,\\infty)$.\n\\end{thm}\n\nA result like this can be seen as an indication of the computability of the $K$-theory of the $L^p$ reduced crossed product on the right-hand side of the assembly map since the $K$-theory of localization-type algebras like the one on the left-hand side of the assembly map has Mayer-Vietoris sequences associated to decompositions of the simplicial complex and other properties of a generalized homology theory (cf. \\cite{Yu97,CW}). Moreover, it enables one to transfer questions about whether the $K$-theory of certain $L^p$ reduced crossed products is independent of $p$ (cf. \\cite[Problem 11.2]{PhilOpen} and \\cite{LY}) over to the left-hand side of the assembly map. On a different note, our method of proof indicates that arguments involving quantitative (or controlled) $K$-theory in the $C^\\ast$-algebraic setting may be adapted without much difficulty to the $L^p$ setting, while there has been substantial difficulty in carrying over other approaches to the Baum-Connes conjecture, such as the Dirac-dual Dirac method.\n\n\n\nIn Section 2, we define $L^p$ Roe algebras and localization algebras associated to an action of a countable discrete group on a compact Hausdorff space, and we define the $L^p$ assembly map in terms of the $K$-theory of these algebras. In the case $p=2$, these are exactly the algebras and map considered in \\cite{GWY2}. In Section 3, we associate subalgebras of these algebras to subgroupoids of the transformation groupoid given by the action, and recall the notion of dynamical complexity. In Section 4, we recall some definitions and facts from the framework of quantitative $K$-theory that we developed in \\cite{Chung1}. Finally, in Section 5, we prove our main result via a homotopy invariance argument and a Mayer-Vietoris argument.\n\n\n\n\\section{An $L^p$ assembly map} \\label{Sect:assembly}\n\nThroughout this section, $\\Gamma$ will be a countable discrete group acting on a compact Hausdorff space $X$ by homeomorphisms. The action will be denoted by $\\Gamma\\curvearrowright X$. We also assume that $\\Gamma$ is equipped with a proper length function $l:\\Gamma\\rightarrow\\mathbb{N}$ and the associated right invariant metric. We will define an assembly map in terms of $L^p$ versions of localization algebras and Roe algebras, where $p\\in[1,\\infty)$. When $p=2$, we recover (a model for) the Baum-Connes assembly map for $\\Gamma$ with coefficients in $C(X)$ considered in \\cite{GWY2}.\n\n\n\\begin{defn}\nLet $s\\geq 0$. The Rips complex of $\\Gamma$ at scale $s$, denoted $P_s(\\Gamma)$, is the simplicial complex with vertex set $\\Gamma$, and where a finite subset $E\\subset\\Gamma$ spans a simplex if and only if $d(g,h)\\leq s$ for all $g,h\\in E$.\n\nPoints in $P_s(\\Gamma)$ can be written as formal linear combinations $\\sum_{g\\in\\Gamma}t_g g$, where $t_g\\in[0,1]$ for each $g$ and $\\sum_{g\\in\\Gamma}t_g=1$. We equip $P_s(\\Gamma)$ with the $\\ell^1$ metric, i.e., $d(\\sum_{g\\in\\Gamma}t_g g,\\sum_{g\\in\\Gamma}s_g g)=\\sum_{g\\in\\Gamma}|t_g-s_g|$.\n\nThe barycentric coordinates on $P_s(\\Gamma)$ are the continuous functions \\[t_g:P_s(\\Gamma)\\rightarrow[0,1]\\] uniquely determined by the condition $z=\\sum_{g\\in\\Gamma}t_g(z)g$ for all $z\\in P_s(\\Gamma)$.\n\\end{defn}\n\nProperness of the length function on $\\Gamma$ implies that $P_s(\\Gamma)$ is finite dimensional and locally compact. Also, the right translation action of $\\Gamma$ on itself extends to a right action of $\\Gamma$ on $P_s(\\Gamma)$ by isometric simplicial automorphisms.\n\nIn the usual setting of the Baum-Connes conjecture (e.g. in \\cite{GWY2}), one considers Hilbert spaces and $C^*$-algebras encoding the large scale geometry of $\\Gamma$ and the topology of $P_s(\\Gamma)$. We will replace these Hilbert spaces by $L^p$ spaces, thereby obtaining $L^p$ operator algebras instead of $C^*$-algebras.\n\n\nFirst, we recall some facts about $L^p$ tensor products. Details can be found in \\cite[Chapter 7]{DF}.\n\nFor $p\\in[1,\\infty)$, there is a tensor product of $L^p$ spaces such that we have a canonical isometric isomorphism $L^p(X,\\mu)\\otimes L^p(Y,\\nu)\\cong L^p(X\\times Y,\\mu\\times\\nu)$, which identifies, for every $\\xi\\in L^p(X,\\mu)$ and $\\eta\\in L^p(Y,\\nu)$, the element $\\xi\\otimes\\eta$ with the function $(x,y)\\mapsto\\xi(x)\\eta(y)$ on $X\\times Y$. Moreover, this tensor product has the following properties:\n\\begin{enumerate}\n\\item Under the identification above, the linear span of all $\\xi\\otimes\\eta$ is dense in $L^p(X\\times Y,\\mu\\times\\nu)$.\n\\item $||\\xi\\otimes\\eta||_p=||\\xi||_p||\\eta||_p$ for all $\\xi\\in L^p(X,\\mu)$ and $\\eta\\in L^p(Y,\\nu)$.\n\\item The tensor product is commutative and associative.\n\\item If $a\\in B(L^p(X_1,\\mu_1),L^p(X_2,\\mu_2))$ and $b\\in B(L^p(Y_1,\\nu_1),L^p(Y_2,\\nu_2))$, then there exists a unique \\[c\\in B(L^p(X_1\\times Y_1,\\mu_1\\times\\nu_1),L^p(X_2\\times Y_2,\\mu_2\\times\\nu_2))\\] such that under the identification above, $c(\\xi\\otimes\\eta)=a(\\xi)\\otimes b(\\eta)$ for all $\\xi\\in L^p(X_1,\\mu_1)$ and $\\eta\\in L^p(Y_1,\\nu_1)$. We will denote this operator by $a\\otimes b$. Moreover, $||a\\otimes b||=||a|| ||b||$.\n\\item The tensor product of operators is associative, bilinear, and satisfies $(a_1\\otimes b_1)(a_2\\otimes b_2)=a_1a_2\\otimes b_1b_2$.\n\\end{enumerate}\nIf $A\\subseteq B(L^p(X,\\mu))$ and $B\\subseteq B(L^p(Y,\\nu))$ are norm-closed subalgebras, we then define $A\\otimes B\\subseteq B(L^p(X\\times Y,\\mu\\times\\nu))$ to be the closed linear span of all elements of the form $a\\otimes b$ with $a\\in A$ and $b\\in B$.\n\n\n\\begin{defn}\nFor $s\\geq 0$, define \\[Z_s=\\biggl\\{\\sum_{g\\in\\Gamma}t_g g\\in P_s(\\Gamma):t_g\\in\\mathbb{Q}\\;\\text{for all}\\;g\\in\\Gamma\\biggr\\}.\\]\nNote that $Z_s$ is a $\\Gamma$-invariant, countable, dense subset of $P_s(\\Gamma)$.\n\nDefine \\[E_s=\\ell^p(Z_s)\\otimes\\ell^p(X)\\otimes\\ell^p\\otimes\\ell^p(\\Gamma)\\cong \\ell^p(Z_s\\times X,\\ell^p\\otimes\\ell^p(\\Gamma)),\\] and equip $E_s$ with the isometric $\\Gamma$-action given by \\[u_g\\cdot(\\delta_z\\otimes\\delta_x\\otimes\\eta\\otimes\\delta_h)=\\delta_{zg^{-1}}\\otimes\\delta_{gx}\\otimes\\eta\\otimes\\delta_{gh}\\] for $z\\in Z_s$, $x\\in X$, $\\eta\\in\\ell^p$, and $g,h\\in\\Gamma$. \n\\end{defn}\n\nNote that if $s_0\\leq s$, then $P_{s_0}(\\Gamma)$ identifies equivariantly and isometrically with a subcomplex of $P_s(\\Gamma)$, and $Z_{s_0}\\subset Z_s$. Hence we have a canonical equivariant isometric inclusion $E_{s_0}\\subset E_s$.\n\nWe will write $\\mathcal{K}_\\Gamma$ for the algebra of compact operators on $\\ell^p\\otimes\\ell^p(\\Gamma)\\cong\\ell^p(\\mathbb{N}\\times\\Gamma)$ equipped with the $\\Gamma$-action induced by the tensor product of the trivial action on $\\ell^p$ and the left regular representation on $\\ell^p(\\Gamma)$. We also equip the algebra $C(X)\\otimes \\mathcal{K}_\\Gamma$ with the diagonal action of $\\Gamma$. Note that the natural faithful representation of $C(X)\\otimes \\mathcal{K}_\\Gamma$ on $\\ell^p(X)\\otimes\\ell^p\\otimes\\ell^p(\\Gamma)$ is covariant for the representation defined by tensoring the natural action on $\\ell^p(X)$, the trivial representation on $\\ell^p$, and the regular representation on $\\ell^p(\\Gamma)$. \n\nNow we can define the $L^p$ operator algebras that will feature in our assembly map.\n\n\\begin{defn}\nLet $T$ be a bounded linear operator on $E_s$, which we may regard as a ($Z_s\\times Z_s$)-indexed matrix $T=(T_{y,z})$ with \\[T_{y,z}\\in B(\\ell^p(X)\\otimes\\ell^p\\otimes\\ell^p(\\Gamma))\\] for each $y,z\\in Z_s$.\n\\begin{enumerate}\n\\item $T$ is $\\Gamma$-invariant if $u_g T u_g^{-1}=T$ for all $g\\in\\Gamma$.\n\\item The Rips-propagation of $T$ is $\\sup\\{d_{P_s(\\Gamma)}(y,z):T_{y,z}\\neq 0\\}$.\n\\item The $\\Gamma$-propagation of $T$, denoted by $\\mathrm{prop}_\\Gamma(T)$, is \\[\\sup\\{d_\\Gamma(g,h):T_{y,z}\\neq 0\\;\\text{for some}\\; y,z\\in Z_s\\;\\text{with}\\; t_g(y)\\neq 0\\;\\text{and}\\;t_h(z)\\neq 0\\}.\\] \n\\item $T$ is $X$-locally compact if $T_{y,z}\\in C(X)\\otimes \\mathcal{K}_\\Gamma$ for all $y,z\\in Z_s$, and if for any compact subset $F\\subset P_s(\\Gamma)$, the set \\[\\{(y,z)\\in (F\\times F)\\cap(Z_s\\times Z_s):T_{y,z}\\neq 0\\}\\] is finite.\n\\end{enumerate}\n\\end{defn} \n\n\n\\begin{defn}\nLet $\\mathbb{C}[\\Gamma\\curvearrowright X;s]$ denote the algebra of all $\\Gamma$-invariant, $X$-locally compact operators on $E_s$ with finite $\\Gamma$-propagation.\n\nLet $B^p(\\Gamma\\curvearrowright X;s)$ denote the closure of $\\mathbb{C}[\\Gamma\\curvearrowright X;s]$ with respect to the operator norm on $B(E_s)$. We will call $B^p(\\Gamma\\curvearrowright X;s)$ the (equivariant) $L^p$ Roe algebra of $\\Gamma\\curvearrowright X$ at scale $s$.\n\\end{defn}\n\n\nWe will regard the algebras above as concretely represented on $E_s$, and we will often think of elements of $B^p(\\Gamma\\curvearrowright X;s)$ as matrices $(T_{y,z})_{y,z\\in Z_s}$ with entries being continuous equivariant functions $T_{y,z}:X\\rightarrow \\mathcal{K}_\\Gamma$ (with additional properties).\n\n\n\\begin{defn}\nLet $\\mathbb{C}_L[\\Gamma\\curvearrowright X;s]$ denote the algebra of all bounded, uniformly continuous functions $a:[0,\\infty)\\rightarrow\\mathbb{C}[\\Gamma\\curvearrowright X;s]$ such that the $\\Gamma$-propagation of $a(t)$ is uniformly finite as $t$ varies, and such that the Rips-propagation of $a(t)$ tends to zero as $t\\rightarrow\\infty$. \n\nLet $B^p_L(\\Gamma\\curvearrowright X;s)$ denote the completion of $\\mathbb{C}_L[\\Gamma\\curvearrowright X;s]$ with respect to the norm \\[||a||:=\\sup_{t\\in[0,\\infty)} ||a(t)||_{B^p(\\Gamma\\curvearrowright X;s)}.\\]\nWe will call $B^p_L(\\Gamma\\curvearrowright X;s)$ the $L^p$ localization algebra of $\\Gamma\\curvearrowright X$ at scale $s$.\n\\end{defn}\n\nWe will regard $B^p_L(\\Gamma\\curvearrowright X;s)$ as concretely represented on $L^p[0,\\infty)\\otimes E_s$, and elements of $B^p_L(\\Gamma\\curvearrowright X;s)$ can be regarded as bounded, uniformly continuous functions $a:[0,\\infty)\\rightarrow B^p(\\Gamma\\curvearrowright X;s)$ (with additional properties).\n\nNow consider the evaluation-at-zero homomorphism \\[\\epsilon_0:B^p_L(\\Gamma\\curvearrowright X;s)\\rightarrow B^p(\\Gamma\\curvearrowright X;s),\\] which induces a homomorphism on $K$-theory \\[\\epsilon_0:K_*(B^p_L(\\Gamma\\curvearrowright X;s))\\rightarrow K_*(B^p(\\Gamma\\curvearrowright X;s)).\\]\n\nIf $s_0\\leq s$, then the equivariant isometric inclusion $E_{s_0}\\subset E_s$ allows us to regard $\\mathbb{C}[\\Gamma\\curvearrowright X;s_0]$ as a subalgebra of $\\mathbb{C}[\\Gamma\\curvearrowright X;s]$. We then regard $B^p(\\Gamma\\curvearrowright X;s_0)$ (resp. $B^p_L(\\Gamma\\curvearrowright X;s_0)$) as a subalgebra of $B^p(\\Gamma\\curvearrowright X;s)$ (resp. $B^p_L(\\Gamma\\curvearrowright X;s)$). Thus there are directed systems of inclusions of $L^p$ operator algebras $(B^p(\\Gamma\\curvearrowright X;s))_{s\\geq 0}$ and $(B^p_L(\\Gamma\\curvearrowright X;s))_{s\\geq 0}$, and the evaluation-at-zero maps above are compatible with these inclusions.\n\n\\begin{defn} \\label{assembly}\nThe $L^p$ assembly map for the action $\\Gamma\\curvearrowright X$ is the direct limit\n\\[ \\epsilon_0:\\lim_{s\\rightarrow\\infty}K_*(B^p_L(\\Gamma\\curvearrowright X;s))\\rightarrow\\lim_{s\\rightarrow\\infty}K_*(B^p(\\Gamma\\curvearrowright X;s)). \\]\n\\end{defn}\nWhen $p=2$, this is the model for the Baum-Connes assembly map considered in \\cite{GWY2}.\n\nRegarding $A=C(X)$ as an $L^p$ operator algebra (acting by multiplication on $L^p(X,\\mu)$ for some measure $\\mu$) with an isometric $\\Gamma$-action, we can define an $L^p$ reduced crossed product as follows:\n\nConsider $C_c(\\Gamma,A)$, the set of finite sums of the form $\\sum_{g\\in\\Gamma}a_g g$ with $a_g\\in A$ and with the product given by \\[ \\biggl(\\sum_{g\\in\\Gamma}a_g g\\biggr)\\biggl(\\sum_{h\\in\\Gamma}b_h h\\biggr)=\\sum_{g,h\\in\\Gamma}a_g\\alpha_g(b_h)gh, \\] where $\\alpha$ denotes the $\\Gamma$-action on $A$. There is a natural faithful representation of $C_c(\\Gamma,A)$ on $\\ell^p(\\Gamma,L^p(X,\\mu))$ given by\n\\begin{align*}\n(a\\xi)(h) &= \\alpha_{h^{-1}}(a)\\xi(h), \\\\\n(g\\xi)(h) &= \\xi(g^{-1}h)\n\\end{align*}\nfor $a\\in A$, $g,h\\in\\Gamma$, and $\\xi\\in\\ell^p(\\Gamma,L^p(X,\\mu))$. We then define the $L^p$ reduced crossed product $A\\rtimes_{\\lambda,p}\\Gamma$ to be the operator norm closure of $C_c(\\Gamma,A)$ in $B(\\ell^p(\\Gamma,L^p(X,\\mu)))$.\n\nAdapting arguments in the $C^\\ast$-algebraic setting (cf. \\cite{GWY2} or \\cite[Lemma 3.7]{WY}), one can show that the equivariant $L^p$ Roe algebra $B^p(\\Gamma\\curvearrowright X;s)$ is stably isomorphic to the crossed product $C(X)\\rtimes_{\\lambda,p}\\Gamma$. In particular, the right-hand side of the $L^p$ assembly map defined above is the same as the $K$-theory of $C(X)\\rtimes_{\\lambda,p}\\Gamma$.\n\nFor most of the rest of this paper, we will work with the kernel of this $L^p$ assembly map.\n\n\\begin{defn}\nLet $B^p_{L,0}(\\Gamma\\curvearrowright X;s)$ be the subalgebra of $B^p_L(\\Gamma\\curvearrowright X;s)$ consisting of functions $a$ such that $a(0)=0$. We will call $B^p_{L,0}(\\Gamma\\curvearrowright X;s)$ the $L^p$ obstruction algebra of $\\Gamma\\curvearrowright X$ at scale $s$.\n\\end{defn}\n\n\\begin{lem}\\label{Lem:obsvan}\nThe $L^p$ assembly map for $\\Gamma\\curvearrowright X$ is an isomorphism if and only if \\[\\lim_{s\\rightarrow\\infty}K_*(B^p_{L,0}(\\Gamma\\curvearrowright X;s))=0.\\]\n\\end{lem}\n\n\\begin{proof}\nJust as in the $C^\\ast$-algebraic setting, we have a short exact sequence \\[0\\rightarrow B^p_{L,0}(\\Gamma\\curvearrowright X;s)\\rightarrow B^p_L(\\Gamma\\curvearrowright X;s)\\rightarrow B^p(\\Gamma\\curvearrowright X;s)\\rightarrow 0,\\] which induces the usual six-term exact sequence in $K$-theory. The lemma then follows from continuity of $K$-theory under direct limits, and the preservation of exact sequences under direct limits of abelian groups.\n\\end{proof}\n\nOur goal in this paper will be to show that if $\\Gamma\\curvearrowright X$ has finite dynamical complexity, then $\\lim_{s\\rightarrow\\infty}K_*(B^p_{L,0}(\\Gamma\\curvearrowright X;s))=0$, and thus the $L^p$ assembly map for $\\Gamma\\curvearrowright X$ in Definition \\ref{assembly} is an isomorphism.\n\n\n\n\n\n\n\\section{Groupoids and dynamical complexity}\n\nIn this section, we consider the transformation groupoid associated to a group action, its subgroupoids, and $L^p$ operator algebras associated with them. We also recall the definition of dynamical complexity from \\cite{GWY2}.\n\n\\begin{defn}\nThe transformation groupoid $\\Gamma\\ltimes X$ associated to $\\Gamma\\curvearrowright X$ is $\\{(gx,g,x):g\\in\\Gamma,x\\in X\\}$ topologized such that the projection $\\Gamma\\ltimes X\\rightarrow\\Gamma\\times X$ onto the second and third factors is a homeomorphism, and equipped with the following additional structure:\n\\begin{enumerate}\n\\item A pair $((hy,h,y),(gx,g,x))$ of elements in $\\Gamma\\ltimes X$ is said to be composable if $y=gx$. In this case, their product is defined by \\[(hgx,h,gx)(gx,g,x)=(hgx,hg,x).\\]\n\\item The inverse of an element $(gx,g,x)\\in\\Gamma\\ltimes X$ is \\[(gx,g,x)^{-1}=(x,g^{-1},gx).\\]\n\\item The units of $\\Gamma\\ltimes X$ are the elements of the clopen subspace \\[G^{(0)}=\\{(x,e,x):x\\in X\\},\\] where $e$ is the identity in $\\Gamma$. We call $G^{(0)}$ the unit space of $\\Gamma\\ltimes X$.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{defn}\nLet $s\\geq 0$, and let $P_s(\\Gamma)$ be the Rips complex of $\\Gamma$ at scale $s$. The support of $z=\\sum_{g\\in\\Gamma}t_g(z) g\\in P_s(\\Gamma)$ is the finite set \\[\\mathrm{supp}(z)=\\{g\\in\\Gamma:t_g(z)\\neq 0\\}.\\]\nThe support of $T=(T_{y,z})_{y,z\\in Z_s}\\in B^p(\\Gamma\\curvearrowright X;s)$ is \n\\[ \\mathrm{supp}(T)=\\left\\{(gx,gh^{-1},hx)\\in\\Gamma\\ltimes X:\\;\\parbox[c][4em][c]{0.5\\textwidth}{there exist $y,z\\in Z_s$ with $T_{y,z}(x)\\neq 0$, $g\\in \\mathrm{supp}(y)$, and $h\\in \\mathrm{supp}(z)$}\\right\\}. \\]\n\\end{defn}\n\nWith this definition, one sees that \\[\\mathrm{prop}_\\Gamma(T)=\\sup\\{l(gh^{-1}):(gx,gh^{-1},hx)\\in \\mathrm{supp}(T)\\;\\text{for some}\\;x\\in X\\}.\\]\n\nGiven two subsets $A,B\\subset\\Gamma\\ltimes X$, we write $AB$ for \\[\\{ab:a\\in A,b\\in B,(a,b)\\;\\text{is composable}\\}.\\] With this notation, the following lemma says that supports of operators in $B^p(\\Gamma\\curvearrowright X;s)$ behave as expected under composition of operators.\n\n\\begin{lem} \\label{supplem}\nLet $S,T\\in B^p(\\Gamma\\curvearrowright X;s)$. Then $\\mathrm{supp}(ST)\\subseteq \\mathrm{supp}(S)\\mathrm{supp}(T)$.\n\\end{lem}\n\n\\begin{proof}\nSuppose that $(gx,gh^{-1},hx)\\in \\mathrm{supp}(ST)$. Then there are $y,z\\in Z_s$ such that $(ST)_{y,z}(x)\\neq 0$, $g\\in \\mathrm{supp}(y)$, and $h\\in \\mathrm{supp}(z)$. Thus there is $w\\in Z_s$ such that $S_{y,w}(x)\\neq 0$ and $T_{w,z}(x)\\neq 0$. If $k\\in \\mathrm{supp}(w)$, then $(gx,gk^{-1},kx)\\in \\mathrm{supp}(S)$ and $(kx,kh^{-1},hx)\\in \\mathrm{supp}(T)$, so $(gx,gh^{-1},hx)=(gx,gk^{-1},kx)(kx,kh^{-1},hx)\\in \\mathrm{supp}(S)\\mathrm{supp}(T)$.\n\\end{proof}\n\n\\begin{defn}\nLet $\\Gamma\\ltimes X$ be the transformation groupoid associated to $\\Gamma\\curvearrowright X$. A subgroupoid of $\\Gamma\\ltimes X$ is a subset $G\\subset \\Gamma\\ltimes X$ that is closed under composition, taking inverses, and units, i.e.,\n\\begin{enumerate}\n\\item If $(hgx,h,gx)$ and $(gx,g,x)$ are in $G$, then so is $(hgx,hg,x)$.\n\\item If $(gx,g,x)\\in G$, then $(gx,g,x)^{-1}\\in G$.\n\\item If $(gx,g,x)\\in G$, then $(x,e,x)\\in G$ and $(gx,e,gx)\\in G$, where $e$ is the identity in $\\Gamma$.\n\\end{enumerate}\nA subgroupoid is equipped with the subspace topology from $\\Gamma\\ltimes X$.\n\\end{defn}\n\nSubgroupoids of $\\Gamma\\ltimes X$ give rise to subalgebras of the Roe algebra, localization algebra, and obstruction algebra that we defined in the previous section.\n\n\\begin{lem}\nLet $G$ be an open subgroupoid of $\\Gamma\\ltimes X$. Define $\\mathbb{C}[G;s]$ to be the subspace of $\\mathbb{C}[\\Gamma\\curvearrowright X;s]$ consisting of all operators $T$ with support contained in a compact subset of $G$. Then $\\mathbb{C}[G;s]$ is a subalgebra of $\\mathbb{C}[\\Gamma\\curvearrowright X;s]$.\n\\end{lem}\n\n\\begin{proof}\nGiven Lemma \\ref{supplem}, it suffices to show that if $A$ and $B$ are two relatively compact subsets of $G$, then so is $AB$. To see this, first suppose that $A$ and $B$ are compact. Then any net in $AB$ has a convergent subnet since nets in $A$ and nets in $B$ have this property, and so $AB$ is compact. Now if $A$ and $B$ are relatively compact, then since $AB\\subset \\bar{A}\\bar{B}$ and $\\bar{A}\\bar{B}$ is compact, it follows that $AB$ is relatively compact.\n\\end{proof}\n\n\\begin{defn}\nLet $G$ be an open subgroupoid of $\\Gamma\\ltimes X$. Let $\\mathbb{C}_L[G;s]$ denote the subalgebra of $\\mathbb{C}_L[\\Gamma\\curvearrowright X;s]$ consisting of functions $a$ such that $\\bigcup_{t\\in[0,\\infty)} \\mathrm{supp}(a(t))$ has compact closure in $G$. \n\nLet $\\mathbb{C}_{L,0}[G;s]$ denote the ideal of $\\mathbb{C}_L[G;s]$ consisting of functions $a$ such that $a(0)=0$.\n\nLet $B^p(G;s)$, $B^p_L(G;s)$, and $B^p_{L,0}(G;s)$ denote the respective closures of $\\mathbb{C}[G;s]$, $\\mathbb{C}_L[G;s]$, and $\\mathbb{C}_{L,0}[G;s]$ in $B^p(\\Gamma\\curvearrowright X;s)$, $B^p_L(\\Gamma\\curvearrowright X;s)$, and $B^p_{L,0}(\\Gamma\\curvearrowright X;s)$.\n\\end{defn}\n\nNote that when $G=\\Gamma\\ltimes X$, we have $B^p(G;s)=B^p(\\Gamma\\curvearrowright X;s)$, and similarly for the localization algebra and obstruction algebra.\n\nSince we will be working mostly with the obstruction algebras, we introduce the following shorthand notation for these algebras. We also need to construct filtrations on these algebras in order to apply quantitative $K$-theory in the proof of the main theorem.\n\n\\begin{defn}\nLet $G$ be an open subgroupoid of $\\Gamma\\ltimes X$, and let $s\\geq 0$. Set $A^s(G)$ to be $B^p_{L,0}(G;s)$. For $r\\geq 0$, define \\[A^s(G)_r=\\{a\\in\\mathbb{C}_{L,0}[G;s]:\\mathrm{prop}_\\Gamma(a(t))\\leq r\\;\\text{for all}\\;t\\},\\] which is a linear subspace of $A^s(G)$.\n\nWhen $G=\\Gamma\\ltimes X$, we will simply write $A^s$ and $A^s_r$.\n\\end{defn}\n\n\\begin{lem} \\label{filtlem}\nLet $G$ be an open subgroupoid of $\\Gamma\\ltimes X$, and let $s\\geq 0$. Then the family $(A^s(G))_{r\\geq 0}$ of subspaces of $A^s(G)$ satisfies:\n\\begin{enumerate}\n\\item if $r_1\\leq r_2$, then $A^s(G)_{r_1}\\subset A^s(G)_{r_2}$;\n\\item $A^s(G)_{r_1}A^s(G)_{r_2}\\subset A^s(G)_{r_1+r_2}$ for all $r_1,r_2\\geq 0$;\n\\item $\\bigcup_{r\\geq 0}A^s(G)_r$ is dense in $A^s(G)$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nNote that $a\\in A^s(G)_r$ if and only if \n\\begin{itemize}\n\\item $a\\in\\mathbb{C}_{L,0}[G;s]$, and \n\\item $l(g)\\leq r$ whenever $(gx,g,x)\\in \\mathrm{supp}(a(t))$ for some $t\\geq 0$.\n\\end{itemize}\nProperties (i) and (iii) follow immediately. \n\nFor (ii), if $a\\in A^s(G)_{r_1}$, $b\\in A^s(G)_{r_2}$, and $(gx,g,x)\\in \\mathrm{supp}(a(t)b(t))$ for some $t$, then by Lemma \\ref{supplem}, $(gx,g,x)=(gx,gh^{-1},hx)(hx,h,x)$ for some $(gx,gh^{-1},hx)\\in \\mathrm{supp}(a(t))$ and $(hx,h,x)\\in \\mathrm{supp}(b(t))$. Thus $l(g)\\leq l(gh^{-1})+l(h)\\leq r_1+r_2$ so $ab\\in A^s(G)_{r_1+r_2}$.\n\\end{proof}\n\n\n\nNote that if $S$ is an open subset of $\\Gamma\\ltimes X$, then $S$ generates an open subgroupoid of $\\Gamma\\ltimes X$ (cf. \\cite[ Lemma 5.2]{GWY1}).\n\n\\begin{defn}\nLet $G$ be an open subgroupoid of $\\Gamma\\ltimes X$, let $H$ be an open subgroupoid of $G$, and let $r\\geq 0$. The expansion of $H$ by $r$ relative to $G$, denoted by $H^{+r}$, is the open subgroupoid of $\\Gamma\\ltimes X$ generated by \\[H\\cup\\{(gx,g,x)\\in G:x\\in H^{(0)},l(g)\\leq r\\}.\\]\n\\end{defn}\n\nNote that $H^{+r}$ depends on $G$ although we do not indicate this in the notation.\n\n\\begin{lem} \\label{pdtlem}\nLet $G$ be an open subgroupoid of $\\Gamma\\ltimes X$, let $H$ be an open subgroupoid of $G$, and let $r,s\\geq 0$. Then \\[A^s(H)\\cdot A^s_r(G)\\cup A^s_r(G)\\cdot A^s(H)\\subseteq A^s(H^{+r}).\\]\n\\end{lem}\n\n\\begin{proof}\nFollows from Lemma \\ref{supplem}.\n\\end{proof}\n\n\\begin{lem}\nLet $G$ be an open subgroupoid of $\\Gamma\\ltimes X$, let $H$ be an open subgroupoid of $G$, and let $r_1,r_2\\geq 0$. Then $(H^{+r_1})^{+r_2}\\subseteq H^{+(r_1+r_2)}$.\n\\end{lem}\n\n\\begin{proof}\nIt suffices to show that \\[\\{(gx,g,x)\\in G:x\\in (H^{+r_1})^{(0)},l(g)\\leq r_2\\}\\subseteq H^{+(r_1+r_2)}.\\] Pick such an element $(gx,g,x)$. There exists $h\\in\\Gamma$ with $l(h)\\leq r_1$ and $hx\\in H^{(0)}$. Thus $(gx,gh^{-1},hx)$ and $(hx,h,x)$ are in $H^{+(r_1+r_2)}$ so $(gx,g,x)=(gx,gh^{-1},hx)(hx,h,x)\\in H^{+(r_1+r_2)}$.\n\\end{proof}\n\n\nNow we recall the definition of dynamical complexity for transformation groupoids. We refer the reader to \\cite[Appendix A]{GWY2} for a definition applicable to general \\'{e}tale groupoids.\n\n\\begin{defn}\\cite{GWY2} \\label{FDCbasic}\nLet $\\Gamma\\curvearrowright X$ be an action, let $G$ be an open subgroupoid of $\\Gamma\\ltimes X$, and let $\\mathcal{C}$ be a set of open subgroupoids of $\\Gamma\\ltimes X$. \nWe say that $G$ is decomposable over $\\mathcal{C}$ if for all $r\\geq 0$ there exists an open cover $G^{(0)}=U_0\\cup U_1$ of the unit space of $G$ such that for each $i\\in\\{0,1\\}$ the subgroupoid of $G$ generated by\n\\[ \\{ (gx,g,x)\\in G:x\\in U_i,l(g)\\leq r\\} \\]\nis in $\\mathcal{C}$.\n\nAn open subgroupoid of $\\Gamma\\ltimes X$ is said to have finite dynamical complexity if it is contained in the smallest class $\\mathcal{D}$ of open subgroupoids of $\\Gamma\\ltimes X$ that contains all relatively compact open subgroupoids and is closed under decomposability (i.e., if $G$ decomposes over $\\mathcal{D}$, then $G$ is in $\\mathcal{D}$).\n\nThe action $\\Gamma\\curvearrowright X$ is said to have finite dynamical complexity if $\\Gamma\\ltimes X$ has finite dynamical complexity.\n\\end{defn}\n\nThe following is a slight variation of the definition.\n\n\\begin{defn}\\cite{GWY2}\nLet $\\Gamma\\curvearrowright X$ be an action, let $G$ be an open subgroupoid of $\\Gamma\\ltimes X$, and let $\\mathcal{C}$ be a set of open subgroupoids of $\\Gamma\\ltimes X$. \nWe say that $G$ is strongly decomposable over $\\mathcal{C}$ if for all $r\\geq 0$ there exists an open cover $G^{(0)}=U_0\\cup U_1$ of the unit space of $G$ such that for each $i\\in\\{0,1\\}$, if $G_i$ is the subgroupoid of $G$ generated by\n\\[ \\{ (gx,g,x)\\in G:x\\in U_i,l(g)\\leq r\\}, \\]\nthen $G_i^{+r}$ (with expansion taken relative to $G$) is in $\\mathcal{C}$.\n\nDenote by $\\mathcal{D}_s$ the smallest class of open subgroupoids of $\\Gamma\\ltimes X$ that contains all relatively compact open subgroupoids and is closed under strong decomposability (i.e., if $G$ is strongly decomposable over $\\mathcal{D}_s$, then $G$ is in $\\mathcal{D}_s$).\n\\end{defn}\n\n\nThe following lemma describes properties of finite dynamical complexity that we will use later. We refer the reader to \\cite[Lemma 3.16]{GWY2} for the proof.\n\n\\begin{lem} \\label{FDCstrong} \nLet $\\Gamma\\curvearrowright X$ be an action.\n\\begin{enumerate}\n\\item If $G$ is an open subgroupoid of $\\Gamma\\ltimes X$ that is in $\\mathcal{D}$ (resp. $\\mathcal{D}_s$), then any open subgroupoid of $G$ is also in $\\mathcal{D}$ (resp. $\\mathcal{D}_s$).\n\\item $\\mathcal{D}=\\mathcal{D}_s$.\n\\end{enumerate}\n\\end{lem}\n\n\n\n\\section{Quantitative $K$-theory}\n\nIn this section, we recall some definitions and facts from our framework of quantitative (or controlled) $K$-theory in \\cite{Chung1}.\n\\begin{defn}\nA filtered Banach algebra is a Banach algebra $A$ with a family $(A_r)_{r\\geq 0}$ of linear subspaces such that\n\\begin{enumerate}\n\\item $A_{r_1}\\subset A_{r_2}$ if $r_1\\leq r_2$;\n\\item $A_{r_1}A_{r_2}\\subset A_{r_1+r_2}$ for all $r_1,r_2\\geq 0$;\n\\item $\\bigcup_{r\\geq 0}A_r$ is dense in $A$.\n\\end{enumerate}\nIf $A$ is unital with unit $1_A$, then we require $1_A\\in A_r$ for all $r\\geq 0$.\n\\end{defn}\n\nWe showed in Lemma \\ref{filtlem} that if $G$ is an open subgroupoid of $\\Gamma\\ltimes X$ and $s\\geq 0$, then $A^s(G)$ is a filtered Banach algebra with filtration \\[A^s(G)_r=\\{a\\in\\mathbb{C}_{L,0}[G;s]:\\mathrm{prop}_\\Gamma(a(t))\\leq r\\;\\text{for all}\\;t\\}.\\]\n\nWe will write $\\tilde{A}$ for the algebra obtained from $A$ by adjoining a unit if $A$ is non-unital, and $A$ itself if $A$ is already unital.\n\n\\begin{defn}\nLet $A$ be a filtered Banach algebra. For $0<\\varepsilon<\\frac{1}{20}$, $r\\geq 0$, and $N\\geq 1$,\n\\begin{enumerate}\n\\item an element $e\\in A$ is called an $(\\varepsilon,r,N)$-idempotent if $||e^2-e||<\\varepsilon$, $e\\in A_r$, and $\\max(||e||,||1_{\\tilde{A}}-e||)\\leq N$.\n\\item if $A$ is unital, an element $u\\in A$ is called an $(\\varepsilon,r,N)$-invertible if $u\\in A_r$, $||u||\\leq N$, and there exists $v\\in A_r$ with $||v||\\leq N$ such that $\\max(||uv-1||,||vu-1||)<\\varepsilon$.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{defn}\nLet $A$ be a filtered Banach algebra.\n\\begin{enumerate}\n\\item Two $(\\varepsilon,r,N)$-idempotents $e_0$ and $e_1$ in $A$ are $(\\varepsilon',r',N')$-homotopic for some $\\varepsilon'\\geq\\varepsilon$, $r'\\geq r$, and $N'\\geq N$ if there exists a norm-continuous path $(e_t)_{t\\in[0,1]}$ of $(\\varepsilon',r',N')$-idempotents in $A$ from $e_0$ to $e_1$. \nEquivalently, there is an $(\\varepsilon',r',N')$-idempotent $e\\in C([0,1],A)$ such that $e(0)=e_0$ and $e(1)=e_1$.\n\\item If $A$ is unital, two $(\\varepsilon,r,N)$-invertibles $u_0$ and $u_1$ in $A$ are $(\\varepsilon',r',N')$-homotopic for some $\\varepsilon'\\geq\\varepsilon$, $r'\\geq r$, and $N'\\geq N$ if there exists an $(\\varepsilon',r',N')$-invertible $u$ in $C([0,1],A)$ with $u(0)=u_0$ and $u(1)=u_1$. In this case, we get a norm-continuous path $(u_t)_{t\\in[0,1]}$ of $(\\varepsilon',r',N')$-invertibles in $A$ from $u_0$ to $u_1$ by setting $u_t=u(t)$.\n\\end{enumerate}\n\\end{defn}\n\nGiven an $L^p$ operator algebra $A$ with a filtration $(A_r)_{r\\geq 0}$, one uses the above equivalence relations and goes through the standard procedure in the definition of $K$-theory of Banach algebras to arrive at the quantitative $K$-theory groups $K_*^{\\varepsilon,r,N}(A)$. We refer the reader to \\cite[Section 3]{Chung1} for details.\n\n\nThere are canonical homomorphisms \\[\\iota_*^{\\varepsilon,\\varepsilon',r,r',N,N'}:K_*^{\\varepsilon,r,N}(A)\\rightarrow K_*^{\\varepsilon',r',N'}(A)\\] for $0<\\varepsilon\\leq\\varepsilon'<\\frac{1}{20}$, $0\\leq r\\leq r'$, and $1\\leq N\\leq N'$, which we may think of as relaxation of control maps or inclusion maps.\n\nIf $e$ is an $(\\varepsilon,r,N)$-idempotent in a unital filtered Banach algebra $A$, then we may apply the holomorphic functional calculus to get an idempotent $c_0(e)\\in A$. This gives us a group homomorphism \\[c_0:K_0^{\\varepsilon,r,N}(A)\\rightarrow K_0(A).\\] Also, every $(\\varepsilon,r,N)$-invertible is actually invertible so we have a group homomorphism \\[c_1:K_1^{\\varepsilon,r,N}(A)\\rightarrow K_1(A)\\] given by $[u]_{\\varepsilon,r,N}\\mapsto[u]$. We sometimes refer to these homomorphisms as comparison maps.\n\n\n\\begin{prop}\\cite[Proposition 3.20]{Chung1} \\leavevmode \\label{qKtoKsurj}\n\\begin{enumerate}\n\\item Let $A$ be a filtered $L^p$ operator algebra. Let $f$ be an idempotent in $M_n(\\tilde{A})$, and let  $0<\\varepsilon<\\frac{1}{20}$. Then there exist $r\\geq 0$ and $[e]\\in K_0^{\\varepsilon,r,||f||+1}(A)$ with $e$ an $(\\varepsilon,r,||f||+1)$-idempotent in $M_n(\\tilde{A})$ such that $c_0([e])=[f]$ in $K_0(A)$. \n\\item Let $A$ be a filtered $L^p$ operator algebra. Let $u$ be an invertible element in $M_n(\\tilde{A})$, and let $0<\\varepsilon<\\frac{1}{20}$. Then there exist $r\\geq 0$ and $[v]\\in K_1^{\\varepsilon,r,||u||+||u^{-1}||+1}(A)$ with $v$ an $(\\varepsilon,r,||u||+||u^{-1}||+1)$-invertible in $M_n(\\tilde{A})$ such that $c_1([v])=[u]$ in $K_1(A)$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{prop}\\cite[Proposition 3.21]{Chung1} \\leavevmode \\label{qKtoKinj}\n\\begin{enumerate}\n\\item There exists a non-decreasing function $P:[1,\\infty)\\rightarrow[1,\\infty)$ such that for any filtered $L^p$ operator algebra $A$, if $0<\\varepsilon<\\frac{1}{20P(N)}$, and $[e]_{\\varepsilon,r,N},[f]_{\\varepsilon,r,N}\\in K_0^{\\varepsilon,r,N}(A)$ are such that $c_0([e])=c_0([f])$ in $K_0(A)$, then there exist $r'\\geq r$ and $N'\\geq N$ such that in $K_0^{P(N)\\varepsilon,r',N'}(A)$ we have $[e]_{P(N)\\varepsilon,r',N'}=[f]_{P(N)\\varepsilon,r',N'}$.\n\\item Let $A$ be a filtered $L^p$ operator algebra. Suppose that $0<\\varepsilon<\\frac{1}{20}$, and $[u]_{\\varepsilon,r,N},[v]_{\\varepsilon,r,N}\\in K_1^{\\varepsilon,r,N}(A)$ are such that $c_1([u])=c_1([v])$ in $K_1(A)$. Then there exist $r'\\geq r$ and $N'\\geq N$ such that in $K_1^{\\varepsilon,r',N'}(A)$ we have $[u]_{\\varepsilon,r',N'}=[v]_{\\varepsilon,r',N'}$.\n\\end{enumerate}\n\\end{prop}\n\n\n\\begin{defn}\nA control pair is a pair $(\\lambda,h)$, where\n\\begin{enumerate}\n\\item $\\lambda:[1,\\infty)\\rightarrow[1,\\infty)$ is a non-decreasing function;\n\\item $h:(0,\\frac{1}{20})\\times[1,\\infty)\\rightarrow[1,\\infty)$ is a function such that $h(\\cdot,N)$ is non-increasing for fixed $N$. \\end{enumerate}\nWe will write $\\lambda_N$ for $\\lambda(N)$, and $h_{\\varepsilon,N}$ for $h(\\varepsilon,N)$.\n\nGiven two control pairs $(\\lambda,h)$ and $(\\lambda',h')$, we write $(\\lambda,h)\\leq(\\lambda',h')$ if $\\lambda_N\\leq \\lambda'_N$ and $h_{\\varepsilon,N}\\leq h'_{\\varepsilon,N}$ for all $\\varepsilon\\in(0,\\frac{1}{20})$ and $N\\geq 1$.\n\\end{defn}\n\n\nIn \\cite[Section 5]{Chung1}, we showed the existence of a controlled Mayer-Vietoris sequence for a pair of subalgebras satisfying certain conditions. Here, we state less general hypotheses (by considering pairs of ideals and omitting certain parameters) that suffice for our application. On the other hand, we also give ourselves a bit more flexibility in terms of propagation control. One can check that the proofs in \\cite{Chung1} carry over with appropriate adjustments of the propagation parameter $r$. Also note that our approach is slightly different from that in \\cite{GWY2}.\n\n\\begin{defn} \\label{MVpair}\nLet $A$ be a filtered $L^p$ operator algebra with filtration $(A_r)_{r\\geq 0}$. A pair $(I,J)$ of closed ideals of $A$ is a controlled Mayer-Vietoris pair for $A$ if it satisfies the following conditions:\n\\begin{enumerate}\n\\item There exists $\\rho:[0,\\infty)\\rightarrow[0,\\infty)$ with $\\rho(r)\\geq r$ such that for any $r\\geq 0$, any positive integer $n$, and any $x\\in M_n(A_r)$, there exist $x_1\\in M_n(I\\cap A_{\\rho(r)})$ and $x_2\\in M_n(J\\cap A_{\\rho(r)})$ such that $x=x_1+x_2$ and $\\max(||x_1||,||x_2||)\\leq ||x||$;\n\\item $I$ and $J$ have filtrations $(I\\cap A_r)_{r\\geq 0}$ and $(J\\cap A_r)_{r\\geq 0}$ respectively;\n\\item For any $r\\geq 0$, any $\\varepsilon>0$, any positive integer $n$, any $x\\in M_n(I\\cap A_r)$ and $y\\in M_n(J\\cap A_r)$ with $||x-y||<\\varepsilon$, there exists $z\\in M_n(I\\cap J\\cap A_{\\rho(r)})$ such that $\\max(||z-x||,||z-y||)<\\varepsilon$, where $\\rho$ is as above.\n\\end{enumerate}\n\\end{defn}\n\n\n\n\\begin{thm}\\cite[Definition 5.12 and Theorem 5.14]{Chung1} \\label{MVthm}\nThere exist control pairs $(\\lambda,h)\\leq(\\lambda',h')$ such that for any filtered $L^p$ operator algebra $A$ and any controlled Mayer-Vietoris pair $(I,J)$ for $A$, if $x\\in K_1^{\\varepsilon,r,N}(A)$, then there exists \\[\\partial_c(x)\\in K_0^{\\lambda_N\\varepsilon,h_{\\varepsilon,N}r,\\lambda_N}(I\\cap J)\\] such that if $\\partial_c(x)=0$, then there exist $a\\in K_0^{\\lambda'_N\\varepsilon,h'_{\\varepsilon,N}r,\\lambda'_N}(I)$ and $b\\in K_0^{\\lambda'_N\\varepsilon,h'_{\\varepsilon,N}r,\\lambda'_N}(J)$ such that $x=a+b$ in $K_0^{\\lambda'_N\\varepsilon,h'_{\\varepsilon,N}r,\\lambda'_N}(A)$.\n\\end{thm}\n\n\n\n\n\n\n\n\n\\section{Main theorem}\n\nIn this section, we shall prove the main theorem that the $L^p$ assembly map for $\\Gamma\\curvearrowright X$ in Definition \\ref{assembly} is an isomorphism if $\\Gamma\\curvearrowright X$ has finite dynamical complexity.\nThe proof that we present is modeled after the proof in the $C^*$-algebraic setting in \\cite{GWY2}, consisting of a homotopy invariance argument and a Mayer-Vietoris argument.\n\n\n\n\n\n\n\n\\subsection{Homotopy invariance} \\label{sec:Base}\n\nRecall that we use the shorthand $A^s(G)$ for $B^p_{L,0}(G;s)$. The homotopy invariance argument involved in the proof of the main theorem is given by the following statement. \n\n\\begin{prop} \\label{baseprop}\nLet $G$ be an open subgroupoid of $\\Gamma\\ltimes X$ such that \\[G\\subseteq\\{(gx,g,x)\\in\\Gamma\\ltimes X:l(g)\\leq s\\}\\] for some $s\\geq 0$. Then $K_*(A^s(G))=0$.\n\\end{prop}\n\n\n\nBefore getting into the proof of the proposition, we need to fix some terminology that is standard in the $C^*$-algebraic setting but perhaps less so when Hilbert spaces are replaced by other Banach spaces. Having done that, the series of results in the rest of this section then yields the proposition.\n\n\n\\begin{defn}\nLet $E$ be a complex Banach space. We say that $T\\in B(E)$ is a partial isometry if $||T||\\leq 1$ and there exists $S\\in B(E)$ such that $||S||\\leq 1$, $TST=T$, and $STS=S$. We call such an $S$ a generalized inverse of $T$.\n\\end{defn}\n\n\\begin{rem}\t\\leavevmode\n\\begin{enumerate}\n\\item In \\cite[Section 6]{Phil12}, Phillips considers spatial partial isometries on $L^p$ spaces. Such spatial partial isometries are partial isometries in the sense of the preceding definition but the converse is not true.\n\\item If $(Z,\\mu)$ is a $\\sigma$-finite measure space, $p\\in[1,\\infty)\\setminus\\{2\\}$, and $T\\in B(L^p(Z,\\mu))$ is an isometric (but not necessarily surjective) linear map, then it follows from Lamperti's theorem \\cite{Lamp} (also see \\cite[Theorem 6.9]{Phil12}) that $T$ is a partial isometry in the sense above, and one can find a generalized inverse $S$ such that $ST=I$ (cf. \\cite[Section 6]{Phil12}). Hereafter, we will denote such an $S$ by $T^\\dagger$.\n\\end{enumerate}\n\\end{rem}\n\n\n\n\\begin{defn}\nIf $A\\subset B(L^p(\\mu))$ is an $L^p$ operator algebra, then we say that $b\\in B(L^p(\\mu))$ is a multiplier of $A$ if $bA\\subset A$ and $Ab\\subset A$. We say that $b$ is an isometric multiplier of $A$ if $b$ is an isometry and both $b$ and $b^\\dagger $ are multipliers of $A$. Denote by $M(A)$ the set of all multipliers of $A$.\n\\end{defn}\n\nNote that $M(A)$ is also an $L^p$ operator algebra.\n\n\n\n\nFor $G$ an open subgroupoid of $\\Gamma\\ltimes X$ and $s\\geq 0$, we introduce the following notation: \n\\begin{itemize}\n\\item $P_s(G)=\\{(z,x)\\in P_s(\\Gamma)\\times X:(gx,g,x)\\in G\\;\\text{for all}\\;g\\in \\mathrm{supp}(z)\\}$.\n\\item $Z_G=(Z_s\\times X)\\cap P_s(G)$.\n\\item $E_G=\\ell^p(Z_G,\\ell^p\\otimes\\ell^p(\\Gamma))\\cong\\ell^p(Z_G)\\otimes\\ell^p\\otimes\\ell^p(\\Gamma)$.\n\\end{itemize}\n\nNote that $E_G$ is a subspace of $E_s$. \nMoreover, the faithful representation of $B^p(G;s)$ on $E_s$ restricts to a faithful representation on $E_G$. \nThus we will regard $B^p(G;s)$ as faithfully represented on $E_G$, and $A^s(G):=B^p_{L,0}(G;s)$ as faithfully represented on $L^p([0,\\infty),E_G)$.\n\nIf $(z,x)\\in P_s(G)$ and $\\mathrm{supp}(z)=\\{g_1,\\ldots,g_n\\}$, then $\\{e,g_1,\\ldots,g_n\\}$ also spans a simplex $\\Delta$ in $P_s(\\Gamma)$ such that $\\Delta\\times\\{x\\}$ is contained in $P_s(G)$. Hence the family of functions \\begin{equation}\\label{homodef} F_r:P_s(G)\\rightarrow P_s(G), (z,x)\\mapsto ((1-r)z+re,x) \\quad\\quad (0\\leq r\\leq 1) \\end{equation} defines a homotopy between the identity map on $P_s(G)$ and the projection onto the subset $\\{(z,x)\\in P_s(G):z=e\\}$, which we may identify with the unit space $G^{(0)}$.\n\nIn the definition of $\\mathbb{C}[\\Gamma\\curvearrowright X;s]$, we may use $\\mathcal{K}_\\Gamma^\\infty$, the algebra of compact operators on $(\\bigoplus_{n=0}^\\infty \\ell^p\\otimes\\ell^p(\\Gamma))_{p}\\cong (\\bigoplus_{n=0}^\\infty\\ell^p)_{p}\\otimes\\ell^p(\\Gamma)$, thereby obtaining another $L^p$ Roe algebra $B^p(\\Gamma\\curvearrowright X;\\mathcal{K}_\\Gamma^\\infty;s)$. Moreover, fixing an isometric isomorphism $\\phi:\\ell^p\\stackrel{\\cong}{\\rightarrow}(\\bigoplus_{n=0}^\\infty\\ell^p)_{p}$ gives an isomorphism \\[B^p(\\Gamma\\curvearrowright X;s)\\cong B^p(\\Gamma\\curvearrowright X;\\mathcal{K}_\\Gamma^\\infty;s).\\]\nWe also have the corresponding statements for the $L^p$ localization algebras and obstruction algebras defined earlier. \n\nFor each $n$, define an isometry $u_{n,0}:\\ell^p\\rightarrow(\\bigoplus_{n=0}^\\infty\\ell^p)_{p}$ by inclusion as the $n$th summand, and define $u_{n,0}^\\dagger :(\\bigoplus_{n=0}^\\infty\\ell^p)_{p}\\rightarrow\\ell^p$ by projection onto the $n$th summand. Then $u_{n,0}^\\dagger u_{n,0}=I$ for all $n$, and $u_{n,0}^\\dagger u_{m,0}=0$ when $n\\neq m$. Define $u_n:L^p([0,\\infty),E_G)\\rightarrow L^p([0,\\infty),E_G^\\infty)$ to be the operator induced by tensoring $u_{n,0}$ with the identity on the other factors, where $E_G^\\infty=(\\bigoplus_{n=0}^\\infty E_G)_{p}$, and define $u_n^\\dagger $ similarly using $u_{n,0}^\\dagger $. Then $u_n^\\dagger u_n=I$ for all $n$, and $u_n^\\dagger u_m=0$ when $n\\neq m$. \n\nGiven $a\\in B(\\ell^p)$, consider $a^\\infty=a\\oplus a\\oplus\\cdots\\in B((\\bigoplus_{n=0}^\\infty\\ell^p)_{p})$. Then $\\mu(a)=a^\\infty$ is an isometric homomorphism $B(\\ell^p)\\rightarrow B((\\bigoplus_{n=0}^\\infty\\ell^p)_{p})$. We may also consider the isometry $v\\in B((\\bigoplus_{n=0}^\\infty\\ell^p)_{p})$ given by the right shift taking the $n$th summand onto the $(n+1)$st summand. Denote by $v^\\dagger \\in B((\\bigoplus_{n=0}^\\infty\\ell^p)_{p})$ the left shift. Then $v\\mu(a)v^\\dagger =0\\oplus a\\oplus a\\oplus\\cdots$ for all $a\\in B(\\ell^p)$. With $u_{0,0}$ as above, $u_{0,0}au_{0,0}^\\dagger $ is given by $a\\oplus 0\\oplus 0\\oplus\\cdots$ for all $a\\in B(\\ell^p)$. Now $a\\mapsto\\mu^{+1}(a):=v\\mu(a)v^\\dagger $ and $a\\mapsto\\mu^0(a):=u_{0,0}au_{0,0}^\\dagger $ are bounded homomorphisms $B(\\ell^p)\\rightarrow B((\\bigoplus_{n=0}^\\infty\\ell^p)_{p})$.\n\nNow given $a\\in B(L^p([0,\\infty),E_G))$, consider the bounded linear operator $\\mu(a)=a^\\infty$ on $L^p([0,\\infty),E_G^\\infty)$. Proceeding similarly as above, we get bounded homomorphisms \\[\\mu,\\mu^{+1},\\mu^0:B(L^p([0,\\infty),E_G))\\rightarrow B(L^p([0,\\infty),E_G^\\infty)).\\] Moreover, each of them maps $A^s(G)$ into $A^s(G;\\mathcal{K}_\\Gamma^\\infty)$, and $M(A^s(G))$ into $M(A^s(G;\\mathcal{K}_\\Gamma^\\infty))$.\n\n\nThe following lemma is an $L^p$ version of a fairly standard result in the $K$-theory of $C^*$-algebras and can be proved in the same way as it is done for $C^*$-algebras (cf. \\cite[Lemma 4.6.2]{HR} or \\cite[Section 3, Lemma 2]{HRY}).\n\\begin{lem} \\label{lemMult}\nLet $\\alpha:A\\rightarrow C$ be a bounded homomorphism of $L^p$ operator algebras with $C\\subset B(L^p(\\mu))$, and let $v\\in B(L^p(\\mu))$ be an isometric multiplier of $C$. Then the map $a\\mapsto v\\alpha(a)v^\\dagger $ is a bounded homomorphism from $A$ to $C$, and induces the same map as $\\alpha$ on $K$-theory.\n\nMore generally, if $v\\in B(L^p(\\mu))$ is a partial isometry and a multiplier of $C$, $w$ is a generalized inverse of $v$ that is also a multiplier of $C$, and $\\alpha(a)wv=\\alpha(a)=wv\\alpha(a)$ for all $a\\in A$, then the map $a\\mapsto v\\alpha(a)w$ is a bounded homomorphism from $A$ to $C$, and induces the same map as $\\alpha$ on $K$-theory.\n\\end{lem}\n\n\\begin{lem}\n$K_*(M(A^s(G)))=0$.\n\\end{lem}\n\n\\begin{proof}\nNote that $\\mu,\\mu^{+1}:M(A^s(G))\\rightarrow M(A^s(G;\\mathcal{K}_\\Gamma^\\infty))$ induce the same map on $K$-theory by the previous lemma. Moreover, since $\\mu^0(a)\\mu^{+1}(a)=\\mu^{+1}(a)\\mu^0(a)=0$ for all $a\\in M(A^s(G))$ and $\\mu=\\mu^0+\\mu^{+1}$, the induced maps on $K$-theory satisfy $\\mu_*=\\mu^0_*+\\mu^{+1}_*=\\mu^0_*+\\mu_*$. Hence $\\mu^0_*=0$. But $\\mu^0$ induces an isomorphism on $K$-theory so $K_*(M(A^s(G)))=0$.\n\\end{proof}\n\n\nFor each $z\\in Z_s$ such that $(z,x)\\in P_s(G)$ for some $x\\in X$, let $E_z$ be a copy of $\\ell^p$ so that we have an isometric isomorphism $\\ell^p\\cong(\\bigoplus_{z\\in Z}E_z)_{p}$, and let $w_z:\\ell^p\\otimes\\ell^p(\\Gamma)\\rightarrow \\ell^p\\otimes\\ell^p(\\Gamma)$ be an isometry with range $E_z\\otimes\\ell^p(\\Gamma)$. For each $r\\in\\mathbb{Q}\\cap[0,1]$, define an isometry \\[w(r):\\ell^p(Z_G)\\otimes\\ell^p\\otimes\\ell^p(\\Gamma)\\rightarrow\\ell^p(Z_G)\\otimes\\ell^p\\otimes\\ell^p(\\Gamma)\\] by $\\delta_{z,x}\\otimes\\eta\\mapsto\\delta_{(1-r)z+re,x}\\otimes w_z\\eta$. Note that if $r\\neq s$, then $w(r)^\\dagger w(s)=0$.\n\nFor $t\\geq 0$ and $n\\in\\mathbb{N}\\cup\\{\\infty\\}$, define an isometry \\[v_n(t):\\ell^p(Z_G)\\otimes\\ell^p\\otimes\\ell^p(\\Gamma)\\rightarrow\\ell^p(Z_G)\\otimes\\ell^p\\otimes\\ell^p(\\Gamma)\\] in the following way.\nFor $m\\in\\mathbb{N}$, define \\[ v_n(m)=\\begin{cases} w(0) & \\text{if}\\;m\\leq n \\\\ w(\\frac{m-n}{n}) & \\text{if}\\;n<m<2n \\\\ w(1) & \\text{if}\\;m\\geq 2n \\end{cases}, \\]\nand for $t=m+s$ with $s\\in(0,1)$, \n\\[ v_n(t)=\\cos\\left(\\frac{s\\pi}{2}\\right)^{\\frac{2}{p}}v_n(m)+\\sin\\left(\\frac{s\\pi}{2}\\right)^{\\frac{2}{p}}v_n(m+1). \\] \nNote that if $\\frac{1}{p}+\\frac{1}{q}=1$, then\n\\[ v_n(t)^\\dagger=\\cos\\left(\\frac{s\\pi}{2}\\right)^{\\frac{2}{q}}v_n(m)^\\dagger+\\sin\\left(\\frac{s\\pi}{2}\\right)^{\\frac{2}{q}}v_n(m+1)^\\dagger\\]\nsatisfies $v_n(t)^\\dagger v_n(t)=I$ for each $t\\geq 0$ and $n\\in\\mathbb{N}\\cup\\{\\infty\\}$.\n\nOne can check that the map $[0,\\infty)\\rightarrow B(\\ell^p(Z_G)\\otimes\\ell^p\\otimes\\ell^p(\\Gamma)),t\\mapsto v_n(t)$, is norm continuous for each $n$. \nNow define an isometry \\[v_n:L^p([0,\\infty),E_G)\\rightarrow L^p([0,\\infty),E_G)\\] for each $n$ by $(v_n\\xi)(t)=v_n(t)(\\xi(t))$ for $\\xi\\in L^p([0,\\infty),E_G)$. Also set $(v_n^\\dagger\\xi)(t)=v_n(t)^\\dagger(\\xi(t))$ so that $v_n^\\dagger v_n=I$.\n\n\nLet $a\\in\\mathbb{C}_{L,0}[G;s]$ and let $T=a(t)$ for some fixed $t\\in[0,\\infty)$. The matrix entries $(v_n(t)Tv_n(t)^\\dagger )_{y,z}(x)$ of $v_n(t)Tv_n(t)^\\dagger $ will be linear combinations of at most four terms of the form $w_z^\\dagger T_{F_{r_1}(y),F_{r_2}(z)}(x)w_y^\\dagger $, with $F_r$ as in (\\ref{homodef}) on page \\pageref{homodef}, $r_1,r_2\\in\\mathbb{Q}\\cap[0,1]$, and $|r_1-r_2|<\\frac{1}{m}$ whenever $t>2m$. It follows that the Rips-propagation of $v_n(t)a(t)v_n(t)^\\dagger $ is at most $\\mathrm{prop}_{Rips}a(t)+\\min(1,\\frac{2}{|t-1|})$, so $v_nav_n^\\dagger \\in\\mathbb{C}_{L,0}[G;s]$.\n\nAlso, the operators $S_t:=v_{n+1}(t)v_n(t)^\\dagger $ on $\\ell^p(Z_G)\\otimes\\ell^p\\otimes\\ell^p(\\Gamma)$ have matrix entries $(S_t)_{y,z}$ that act as constant functions $X\\rightarrow B(\\ell^p\\otimes\\ell^p(\\Gamma))$, their Rips-propagation tends to zero as $t\\rightarrow\\infty$, and they have $\\Gamma$-propagation at most $s$ for all $t$. Hence $v_{n+1}v_n^\\dagger $ is a multiplier of $A^s(G)$ for all $n$.\n\n\\begin{lem}\nLet $A$ be a unital Banach algebra, and let $I$ be an ideal in $A$. Define the double of $A$ along $I$ to be $D=\\{(a,b)\\in A\\oplus A:a-b\\in I\\}$. Assume that $A$ has trivial $K$-theory. Then the inclusion $\\iota:I\\rightarrow D$ given by $a\\mapsto(a,0)$ induces an isomorphism in $K$-theory, and the diagonal inclusion $\\delta:I\\rightarrow D$ given by $a\\mapsto(a,a)$ induces the zero map on $K$-theory.\n\\end{lem}\n\n\\begin{proof}\nNote that $\\iota(I)$ is an ideal in $D$, and $D\/\\iota(I)$ is isomorphic to $A$ via the second coordinate projection. Since $K_*(A)=0$, it follows from the six-term exact sequence that $\\iota$ induces an isomorphism in $K$-theory. \n\nOn the other hand, $\\delta$ factors through the diagonal inclusion $A\\rightarrow D,a\\mapsto(a,a)$, so $\\delta$ induces the zero map on $K$-theory since $K_*(A)=0$.\n\\end{proof}\n\nWe shall apply the lemma in the case where $A=M(A^s(G))$ and $I=A^s(G)$ to prove the next proposition.\n\n\\begin{prop}\nLet $v_n:L^p([0,\\infty),E_G)\\rightarrow L^p([0,\\infty),E_G)$ be as defined above. Then the maps $a\\mapsto v_0 av_0^\\dagger $ and $a\\mapsto v_\\infty av_\\infty^\\dagger $ induce the same map $K_*(A^s(G))\\rightarrow K_*(A^s(G))$.\n\\end{prop}\n\n\\begin{proof}\nGiven $a\\in A^s(G)$, define \\[\\alpha(a)=(\\bigoplus_{n=0}^\\infty v_nav_n^\\dagger ,\\bigoplus_{n=0}^\\infty v_\\infty av_\\infty^\\dagger )\\in A^s(G;\\mathcal{K}_\\Gamma^\\infty)\\oplus A^s(G;\\mathcal{K}_\\Gamma^\\infty).\\] Also define \\[\\beta(a)=(\\bigoplus_{n=0}^\\infty v_{n+1}av_{n+1}^\\dagger ,\\bigoplus_{n=0}^\\infty v_\\infty av_\\infty^\\dagger )\\in A^s(G;\\mathcal{K}_\\Gamma^\\infty)\\oplus A^s(G;\\mathcal{K}_\\Gamma^\\infty).\\] Let $D$ be the double of $M(A^s(G;\\mathcal{K}_\\Gamma^\\infty))$ along $A^s(G;\\mathcal{K}_\\Gamma^\\infty)$, and let \\[C=\\{(c,d)\\in D:d=\\bigoplus_{n=0}^\\infty v_\\infty av_\\infty^\\dagger \\;\\text{for some}\\;a\\in A^s(G)\\},\\] which is a closed subalgebra of $D$. Moreover, $\\alpha$ and $\\beta$ are bounded homomorphisms with image in $C$. \nConsider $w=(w_1,w_2)$, where $w_1=\\bigoplus_{n=0}^\\infty v_{n+1}v_n^\\dagger $ and $w_2=\\bigoplus_{n=0}^\\infty v_\\infty v_\\infty^\\dagger $. Note that $w_1,w_2\\in M(A^s(G;\\mathcal{K}_\\Gamma^\\infty))$. We claim that $w$ is a multiplier of $C$. Indeed, if $(c,d)\\in C$, then $w_2d=dw_2=d$ so it suffices to show that $cw_1-d$ and $w_1c-d$ are in $A^s(G;\\mathcal{K}_\\Gamma^\\infty)$. We will only consider $w_1c-d$ since the other case is similar. Now $w_1c-d=w_1(c-d)+(w_1d-d)$ so it suffices to show that $w_1d-d\\in A^s(G;\\mathcal{K}_\\Gamma^\\infty)$. But $w_1d-d=(w_1-w_2)d=\\bigoplus_{n=0}^\\infty(v_{n+1}v_n^\\dagger -v_\\infty v_\\infty^\\dagger )v_\\infty a v_\\infty^\\dagger \\in A^s(G;\\mathcal{K}_\\Gamma^\\infty)$ since $v_n(t)=v_\\infty(t)$ for each fixed $t$ and all $n>t$. Similarly, $w^\\dagger =(w_1^\\dagger ,w_2^\\dagger )$ is a multiplier of $C$. \n\nNow $\\beta(a)=w\\alpha(a)w^\\dagger $ for all $a\\in A$. Moreover, $\\alpha(a)w^\\dagger w=\\alpha(a)=w^\\dagger w\\alpha(a)$ for all $a\\in A$ so $\\alpha$ and $\\beta$ induce the same map $K_*(A^s(G))\\rightarrow K_*(C)$ by Lemma \\ref{lemMult}, and thus the same map $K_*(A^s(G))\\rightarrow K_*(D)$ upon composing with the map induced by the inclusion of $C$ into $D$.\n\nLet $u$ be the isometric multiplier of $A^s(G;\\mathcal{K}_\\Gamma^\\infty)$ induced by the right shift. Then $(u,u)$ is a multiplier of $D$, and conjugating $\\beta(a)$ by $(u,u)$ gives \\[\\gamma(a)=\\biggl(0\\oplus\\bigoplus_{n=1}^\\infty v_nav_n^\\dagger ,0\\oplus\\bigoplus_{n=1}^\\infty v_\\infty av_\\infty^\\dagger \\biggr).\\] Thus $\\beta$ and $\\gamma$ induce the same map $K_*(A)\\rightarrow K_*(D)$. On the other hand, the homomorphism $\\delta:A^s(G)\\rightarrow D$ given by \\[a\\mapsto(v_\\infty av_\\infty^\\dagger \\oplus 0\\oplus 0\\oplus\\cdots,v_\\infty av_\\infty^\\dagger \\oplus 0\\oplus 0\\oplus\\cdots)\\] induces the zero map on $K$-theory by the previous lemma. Also, $\\gamma(a)\\delta(a)=\\delta(a)\\gamma(a)=0$. Hence \\[\\alpha_*=\\beta_*=\\gamma_*=\\gamma_*+\\delta_*=(\\gamma+\\delta)_*:K_*(A^s(G))\\rightarrow K_*(D).\\]\n\nLet $\\psi_0,\\psi_\\infty:A^s(G)\\rightarrow D$ be the homomorphisms defined by \\begin{align*} \\psi_0(a) &=(v_0 av_0^\\dagger \\oplus 0\\oplus 0\\oplus\\cdots,0), \\\\ \\psi_\\infty(a) &=(v_\\infty av_\\infty^\\dagger \\oplus 0\\oplus 0\\oplus\\cdots,0).\\end{align*} Also define $\\zeta:A^s(G)\\rightarrow D$ by \\[\\zeta(a)=\\biggl(0\\oplus\\bigoplus_{n=1}^\\infty v_nav_n^\\dagger ,\\bigoplus_{n=0}^\\infty v_\\infty av_\\infty^\\dagger \\biggr).\\] Note that $\\zeta(a)\\psi_0(a)=\\psi_0(a)\\zeta(a)=\\zeta(a)\\psi_\\infty(a)=\\psi_\\infty(a)\\zeta(a)=0$ for all $a\\in A^s(G)$. Also, $\\psi_0+\\zeta=\\alpha$ and $\\psi_\\infty+\\zeta=\\gamma+\\delta$. Hence \\[(\\psi_0)_*+\\zeta_*=\\alpha_*=(\\gamma+\\delta)_*=(\\psi_\\infty)_*+\\zeta_*,\\] so $\\psi_0$ and $\\psi_\\infty$ induce the same maps on $K$-theory.\n\nFinally, if $\\iota:A^s(G)\\rightarrow D$ is the inclusion into the first factor (where $D$ is now regarded as the double of $M(A^s(G))$ along $A^s(G)$), then $\\psi_i(a)$ is given by the composition\n\\[ a\\mapsto v_iav_i^\\dagger \\stackrel{\\iota}{\\mapsto}(v_iav_i^\\dagger ,0)\\mapsto (v_iav_i^\\dagger \\oplus 0\\oplus 0\\oplus\\cdots,0), \\] and the last two maps induce isomorphisms on $K$-theory, so $a\\mapsto v_0av_0^\\dagger $ and $a\\mapsto v_\\infty av_\\infty^\\dagger$ induce the same map on $K$-theory.\n\\end{proof}\n\n\nNow it remains to be shown that $a\\mapsto v_\\infty av_\\infty^\\dagger $ induces the identity map on $K_*(A^s(G))$ while $a\\mapsto v_0 av_0^\\dagger $ induces the zero map on $K_*(A^s(G))$. This will complete the proof of Proposition \\ref{baseprop}.\n\n\\begin{lem}\nThe map $\\phi_\\infty:K_*(A^s(G))\\rightarrow K_*(A^s(G))$ induced by conjugation by $v_\\infty$ is the identity map.\n\\end{lem}\n\n\\begin{proof}\nSince $v_\\infty(t)=w(0)$ for all $t$, and $w(0)$ is an isometric multiplier of $A^s(G)$, $\\phi_\\infty$ induces the identity map on $K$-theory by Lemma \\ref{lemMult}.\n\\end{proof}\n\n\\begin{lem}\nThe map $\\phi_0:K_*(A^s(G))\\rightarrow K_*(A^s(G))$ induced by conjugation by $v_0$ is the zero map.\n\\end{lem}\n\n\\begin{proof}\nLet $G^{(0)}$ be the unit space of $G$, which is an open subgroupoid of $\\Gamma\\ltimes X$. We may then consider $A^s(G^{(0)})$. In fact, $\\phi_0$ factors through $K_*(A^s(G^{(0)}))$, i.e., we have a commutative diagram\n\\[\n\\begindc{\\commdiag}[100]\t\n\\obj(10,5)[2a]{$K_*(A^s(G))$}\t\\obj(20,5)[2b]{$K_*(A^s(G))$}\t\n\n\\mor{2a}{2b}{$\\phi_0$}\t \n\n\\obj(15,0)[3a]{$K_*(A^s(G^{(0)}))$}\t\n\n\\mor{2a}{3a}{}\t\\mor{3a}{2b}{} \t\n\\enddc\n\\]\nso it suffices to show that $K_*(A^s(G^{(0)}))=0$.\n\nNote that $a(t)$ has zero Rips-propagation for all $a\\in A^s(G^{(0)})$ and $t\\in[0,\\infty)$. For each $n\\in\\mathbb{N}$ and $a\\in A^s(G^{(0)})$, define \\[a^{(n)}(t)=\\begin{cases} a(t-n) & \\text{for}\\; t\\geq n \\\\ 0 & \\text{for}\\; t<n \\end{cases}.\\]\nNote that $a^{(n)}\\in A^s(G^{(0)})$. Now define $\\alpha:A^s(G^{(0)})\\rightarrow A^s(G^{(0)};\\mathcal{K}_\\Gamma^\\infty)$ by $a\\mapsto \\bigoplus_{n=0}^\\infty a^{(n)}$.\nWe also have the ``top corner inclusion'' $\\iota:A^s(G^{(0)})\\rightarrow A^s(G^{(0)};\\mathcal{K}_\\Gamma^\\infty)$ given by $a\\mapsto a\\oplus 0\\oplus 0\\oplus\\cdots$.\nUsing uniform continuity of elements in $A^s(G^{(0)})$, we see that $\\alpha_*+\\iota_*=\\alpha_*$ so $\\iota_*=0$. But $\\iota$ induces an isomorphism on $K$-theory so it follows that $K_*(A^s(G^{(0)}))=0$.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Mayer-Vietoris} \\label{sec:Ind}\n\nGiven two open subgroupoids of $\\Gamma\\ltimes X$, we will consider associated subalgebras of $A^s:=B^p_{L,0}(\\Gamma\\curvearrowright X;s)$, following \\cite[Section 7]{GWY2}, and also controlled Mayer-Vietoris pairs of ideals for these algebras. Note that the filtrations we equip these subalgebras with are not the induced filtrations from $A^s$. \n\n\\begin{defn} \\label{idealdef}\nFix an open subgroupoid $G$ of $\\Gamma\\ltimes X$, and fix $s_0\\geq 1$. Let $G_0$ and $G_1$ be open subgroupoids of $G$ such that $G^{(0)}=G_0^{(0)}\\cup G_1^{(0)}$. For $r\\geq 0$, define\n\\[ \\mathfrak{A}_r(G)=A^{s_0}(G_0^{+r})_r+A^{s_0}(G_1^{+r})_r+A^{s_0}(G_0^{+r}\\cap G_1^{+r})_r, \\] \nand \\[ \\mathfrak{A}(G)=\\overline{\\bigcup_{r\\geq 0}\\mathfrak{A}_r(G)}, \\]\ntaking closure in the norm of $A^{s_0}$.\n\nAlso define \n\\begin{align*} \nI_r&=A^{s_0}(G_0^{+r})_r+A^{s_0}(G_0^{+r}\\cap G_1^{+r})_r, \\quad I=\\overline{\\bigcup_{r\\geq 0}I_r} \\\\ J_r&=A^{s_0}(G_1^{+r})_r+A^{s_0}(G_0^{+r}\\cap G_1^{+r})_r, \\quad J=\\overline{\\bigcup_{r\\geq 0}J_r}.\n\\end{align*}\n\nLet $G,G_0,G_1$ and $s_0$ be as above. Define another filtration as follows. For $s\\geq s_0$ and $r\\geq 0$, define\n\\[ \\mathfrak{A}_r^s(G)=A^{s_0}(G_0^{+r})_r+A^{s_0}(G_1^{+r})_r+A^s(G_0^{+r}\\cap G_1^{+r})_{sr}, \\]\nand \\[ \\mathfrak{A}^s(G)=\\overline{\\bigcup_{r\\geq 0}\\mathfrak{A}_r^s(G)}, \\]\ntaking closure in the norm of $A^s$.\n\nAlso define \n\\begin{align*} \nI_r^s&=A^{s_0}(G_0^{+r})_r+A^s(G_0^{+r}\\cap G_1^{+r})_{sr}, \\quad I^s=\\overline{\\bigcup_{r\\geq 0}I_r^s} \\\\ J_r^s&=A^{s_0}(G_1^{+r})_r+A^s(G_0^{+r}\\cap G_1^{+r})_{sr}, \\quad J^s=\\overline{\\bigcup_{r\\geq 0}J_r^s}.\n\\end{align*}\n\\end{defn}\n\n\n\\begin{lem}\nWith notation as above, $(\\mathfrak{A}_r(G))_{r\\geq 0}$ and $(\\mathfrak{A}_r^s(G))_{r\\geq 0}$ are filtrations for $\\mathfrak{A}(G)$ and $\\mathfrak{A}^s(G)$ respectively. Moreover, $I$ and $J$ are ideals in $\\mathfrak{A}(G)$, while $I^s$ and $J^s$ are ideals in $\\mathfrak{A}^s(G)$.\n\\end{lem}\n\n\\begin{proof}\nIt is clear that $\\mathfrak{A}_{r_0}(G)\\subset \\mathfrak{A}_r(G)$ if $r_0\\leq r$, that $\\bigcup_{r\\geq 0}\\mathfrak{A}_r(G)$ is dense in $\\mathfrak{A}(G)$, and similarly for the $s$-version. By Lemmas \\ref{supplem} and \\ref{pdtlem}, it follows that for $r_1,r_2\\geq 0$ and $s_0\\geq 1$, \\[ A^{s_0}(G_i^{+r_1})_{r_1}\\cdot A^{s_0}(G_i^{+r_2})_{r_2}\\subset A^{s_0}(G_i^{+(r_1+r_2)})_{r_1+r_2} \\] for $i=0,1$, while \n\\[\nA^{s_0}(G_0^{+r_1})_{r_1}\\cdot A^{s_0}(G_1^{+r_2})_{r_2} \\subset A^{s_0}(G_0^{+(r_1+r_2)})_{r_1+r_2}\\cap A^{s_0}(G_1^{+(r_1+r_2)})_{r_1+r_2}, \\]\\[\nA^{s_0}(G_0^{+r_1}\\cap G_1^{+r_1})_{r_1}\\cdot A^{s_0}(G_0^{+r_2}\\cap G_1^{+r_2})_{r_2} \\subset A^{s_0}(G_0^{+(r_1+r_2)}\\cap G_1^{+(r_1+r_2)})_{r_1+r_2}.\n\\]\nAlso, for $s\\geq s_0$,\n\\begin{align*} \nA^s(G_0^{+r_1}\\cap G_1^{+r_1})_{sr_1}\\cdot A^s(G_0^{+r_2}\\cap G_1^{+r_2})_{sr_2} &\\subset A^s(G_0^{+(r_1+r_2)}\\cap G_1^{+(r_1+r_2)})_{s(r_1+r_2)}\n\\end{align*} \nand\n\\begin{align*} A^{s_0}(G_i^{+r_1})_{r_1}\\cdot A^s(G_0^{+r_2}\\cap G_1^{+r_2})_{sr_2}&\\subset A^s((G_0^{+r_2}\\cap G_1^{+r_2})^{+r_1})_{r_1+sr_2} \\\\ &\\subset A^s((G_0^{+r_2})^{+r_1}\\cap(G_1^{+r_2})^{+r_1})_{s(r_1+r_2)} \\\\ &\\subset A^s(G_0^{+(r_1+r_2)}\\cap G_1^{+(r_1+r_2)})_{s(r_1+r_2)}. \\end{align*}\nHence $(\\mathfrak{A}_r(G))_{r\\geq 0}$ and $(\\mathfrak{A}_r^s(G))_{r\\geq 0}$ are filtrations for $\\mathfrak{A}(G)$ and $\\mathfrak{A}^s(G)$ respectively, $I$ and $J$ are ideals in $\\mathfrak{A}(G)$, while $I^s$ and $J^s$ are ideals in $\\mathfrak{A}^s(G)$.\n\\end{proof}\n\n\n\n\n\n\n\nNow we need to check that the ideals in Definition \\ref{idealdef} satisfy the conditions listed in Definition \\ref{MVpair} for our controlled Mayer-Vietoris sequence. To do so, we shall make use of partitions of unity and associated multiplication operators.\n\n\\begin{defn}\nLet $K$ be a compact subset of $X$, let $\\{U_0,\\ldots,U_d\\}$ be a finite open cover of $K$, and let $\\{\\phi_0,\\ldots,\\phi_d\\}$ be a subordinate partition of unity. Let $s\\geq 0$. For $i\\in\\{0,\\ldots,d\\}$, let $M_i$ be the multiplication operator on $E_s$ associated to the function \\[ Z_s\\times X\\rightarrow [0,1], (z,x)\\mapsto\\sum_{g\\in\\Gamma}t_g(z)\\phi_i(gx). \\]\n\\end{defn}\n\n\\begin{lem}\nWith notation as above, the operators $M_i$ have the following properties:\n\\begin{enumerate}\n\\item $||M_i||\\leq 1$.\n\\item If $T\\in B^p(\\Gamma\\curvearrowright X;s)$ satisfies \\[\\{x\\in X:(gx,g,x)\\in \\mathrm{supp}(T)\\;\\text{for some}\\;g\\in\\Gamma\\}\\subset K,\\] then $T=T(M_0+\\cdots+M_d)$.\n\\item For any $T\\in B^p(\\Gamma\\curvearrowright X;s)$ with $\\Gamma$-propagation at most $r$, and $i\\in\\{0,\\ldots,d\\}$, we have \\[ \\mathrm{supp}(TM_i)\\subset\\biggl\\{ (gx,g,x)\\in\\Gamma\\ltimes X:x\\in\\bigcup_{l(h)\\leq s}h\\cdot U_i,l(g)\\leq r\\biggr\\}\\cap \\mathrm{supp}(T). \\]\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nEach $M_i$ is a multiplication operator associated to a function taking values in $[0,1]$ so it follows that $||M_i||\\leq 1$.\n\nFor $i\\in\\{0,\\ldots,d\\}$, $T\\in B^p(\\Gamma\\curvearrowright X;s)$, $y,z\\in P_s(\\Gamma)$, and $x\\in X$, we have \\[ (TM_i)_{y,z}(x)=T_{y,z}(x)\\cdot\\sum_{h\\in\\Gamma}t_h(z)\\phi_i(hx). \\]\nHence \\[ (T(M_0+\\cdots+M_d))_{y,z}(x)=T_{y,z}(x)\\cdot\\sum_{h\\in\\Gamma}t_h(z)(\\phi_0(hx)+\\cdots+\\phi_d(hx)). \\]\n\nSuppose that $\\{x\\in X:(gx,g,x)\\in \\mathrm{supp}(T)\\;\\text{for some}\\;g\\in\\Gamma\\}\\subset K$. If $T_{y,z}(x)\\neq 0$, then $(gx,gh^{-1},hx)\\in \\mathrm{supp}(T)$ for all $g\\in \\mathrm{supp}(y)$ and $h\\in \\mathrm{supp}(z)$.\nIn particular, $hx\\in K$ for all $h\\in \\mathrm{supp}(z)$, so \\[ \\sum_{h\\in\\Gamma}t_h(z)(\\phi_0(hx)+\\cdots+\\phi_d(hx))=\\sum_{h\\in\\Gamma}t_h(z)=1, \\]\nand this proves (ii).\n\nSuppose that $(gx,gk^{-1},kx)\\in \\mathrm{supp}(TM_i)$, where $T\\in B^p(\\Gamma\\curvearrowright X;s)$ has $\\Gamma$-propagation at most $r$. Then there exist $y,z\\in P_s(\\Gamma)$ with $g\\in \\mathrm{supp}(y)$, $k\\in \\mathrm{supp}(z)$, and $(TM_i)_{y,z}(x)\\neq 0$. In particular, $T_{y,z}(x)\\neq 0$, so $(gx,gk^{-1},kx)\\in \\mathrm{supp}(T)$ and $l(gk^{-1})\\leq r$. We also have $\\sum_{h\\in\\Gamma}t_h(z)\\phi_i(hx)\\neq 0$, so there exists $h\\in \\mathrm{supp}(z)$ with $\\phi_i(hx)\\neq 0$, and thus $hx\\in U_i$. Since $h,k\\in \\mathrm{supp}(z)$, and $z\\in P_s(\\Gamma)$, we have $l(kh^{-1})\\leq s$. Now $kx=(kh^{-1})hx\\in kh^{-1}\\cdot U_i$, and this proves (iii).\n\\end{proof}\n\n\n\n\\begin{lem}\nThe pairs $(I,J)$ and $(I^s,J^s)$ in Definition \\ref{idealdef} are controlled Mayer-Vietoris pairs for $\\mathfrak{A}(G)$ and $\\mathfrak{A}^s(G)$ respectively.\n\\end{lem}\n\n\n\\begin{proof}\nBy virtue of how $I,J,I^s,J^s$, and $\\mathfrak{A}(G),\\mathfrak{A}^s(G)$ are defined, we only need to check conditions (i) and (iii) in Definition \\ref{MVpair}, which we will do for $I,J,\\mathfrak{A}(G)$ only since the $s$-version can be handled in a similar manner.\n\nLet $U_i$ be the unit space of $G_i^{+r_0}$ for $i=0,1$. If $a\\in \\mathfrak{A}_{r_0}(G)$, then \\[ K:=\\overline{\\{x\\in X:(gx,g,x)\\in \\mathrm{supp}(a(t))\\;\\text{for some}\\;t\\in[0,\\infty),g\\in\\Gamma\\}} \\] is a compact subset of $U_0\\cup U_1$. Let $M_0,M_1$ be the multiplication operators defined with respect to the compact set $K$, the open cover $\\{U_0,U_1\\}$, and some choice of subordinate partition of unity $\\{\\phi_0,\\phi_1\\}$. By the previous lemma, we have $a(t)(M_0+M_1)=a(t)$ for all $t$. Moreover, $||a(t)M_i||\\leq||a(t)||$ for $i=0,1$. It remains to be shown that $t\\mapsto a(t)M_0$ is in $I_r$ and $t\\mapsto a(t)M_1$ is in $J_r$ for some $r\\geq r_0$ (that may depend on $s_0$ but not on $s$). We will focus on the case of $M_0$ since the other case is similar.\n\nWrite $a=b_0+b_1+c$ with $b_i\\in A^{s_0}(G_i^{+r_0})_{r_0}$ and $c\\in A^s(G_0^{+r_0}\\cap G_1^{+r_0})$. By the previous lemma, we have $\\mathrm{supp}(b_0(t)M_0) \\subset \\mathrm{supp}(b_0(t))$ and also $\\mathrm{supp}(c(t)M_0)\\subset \\mathrm{supp}(c(t))$ so $t\\mapsto b_0(t)M_0$ is in $A^{s_0}(G_0^{+r_0})_{r_0}\\subset I_{r_0}$ and $t\\mapsto c(t)M_0$ is in $A^s(G_0^{+r_0}\\cap G_1^{+r_0})\\subset I_{r_0}$. \n\nNow assume that $(gx,gh^{-1},hx)\\in \\mathrm{supp}(b_1(t)M_0)$ for some $t$. Then there exist $y,z\\in P_s(\\Gamma)$ such that $g\\in \\mathrm{supp}(y),h\\in \\mathrm{supp}(z)$, and $(b_1(t)M_0)_{y,z}(x)\\neq 0$. In particular, $(b_1(t))_{y,z}(x)\\neq 0$ so $y,z\\in P_{s_0}(\\Gamma)$ and $l(gh^{-1})\\leq r_0$. Also, $\\sum_{k\\in\\Gamma}t_k(z)\\phi_0(kx)\\neq 0$ so there exists $k\\in \\mathrm{supp}(z)$ such that $\\phi_0(kx)\\neq 0$, and thus $kx$ is in $U_0$, the unit space of $G_0^{+r_0}$. Hence \\begin{align*} (gx,gh^{-1},hx)&=(gx,gk^{-1},kx)(kx,kh^{-1},hx) \\\\ &\\in (G_0^{+r_0})^{+r_0}\\cdot (G_0^{+r_0})^{+s_0} \\subset G_0^{+(2r_0+s_0)}. \\end{align*}\nHence $t\\mapsto b_1(t)M_0$ is in $A^{s_0}(G_0^{+(2r_0+s_0)})_{2r_0+s_0}\\subset I_{2r_0+s_0}$, and so $t\\mapsto a(t)M_0$ is in $I_{2r_0+s_0}$. Hence condition (i) is satisfied.\n\nNext, suppose that $a_0\\in I_{r_0}$ and $a_1\\in J_{r_0}$ such that $||a_0-a_1||<\\varepsilon$. Again, let $U_i$ be the unit space of $G_i^{+r_0}$ for $i=0,1$. Consider\n\\[ K_i:=\\overline{\\{x\\in X:(gx,g,x)\\in \\mathrm{supp}(a_i(t))\\;\\text{for some}\\;t\\in[0,\\infty),g\\in\\Gamma\\}} \\]\nfor $i=0,1$, and let $K=K_1\\cup K_2$, a compact subset of $U_0\\cup U_1$. Let $M_0,M_1$ be the multiplication operators defined with respect to the compact set $K$, the open cover $\\{U_0,U_1\\}$, and some choice of subordinate partition of unity $\\{\\phi_0,\\phi_1\\}$. Define $b(t)=a_0(t)M_1+a_1(t)M_0$. Then $b\\in I_{2r_0+s_0}\\cap J_{2r_0+s_0}$, \nand since $a_i(t)=a_i(t)(M_0+M_1)$ by the choice of $K$, we have $||a_i(t)-b(t)||\\leq||a_0(t)-a_1(t)||<\\varepsilon$ for $i=0,1$. Hence condition (iii) is satisfied.\n\\end{proof}\n\n\nThe following lemma follows from a similar argument as the first part of the proof above, and we omit the details.\n\n\\begin{lem} \\label{split}\nFix an open subgroupoid $G$ of $\\Gamma\\ltimes X$ and $s_0\\geq 0$. Let $r_0\\geq 0$. Let $G^{(0)}=U_0\\cup U_1$ be an open cover of $G^{(0)}$, and let $G_i$ be the subgroupoid of $G$ generated by \\[ \\{(gx,g,x)\\in G:x\\in U_i,l(g)\\leq r_0\\} \\] for $i=0,1$. Then $A^{s_0}(G)_{r_0}\\subseteq A^{s_0}(G_0^{+(2r_0+s_0)})_{2r_0+s_0}+A^{s_0}(G_1^{+(2r_0+s_0)})_{2r_0+s_0}$.\n\\end{lem}\n\n\n\n\n\n\n\n\n\nNow we can prove our main theorem.\n\n\\begin{thm} \\label{mainthm}\nSuppose that $\\Gamma\\curvearrowright X$ has finite dynamical complexity, and let $p\\in[1,\\infty)$. Then \\[ \\lim_{s\\rightarrow\\infty}K_*(B^p_{L,0}(\\Gamma\\curvearrowright X;s))=0. \\]\nThus the $L^p$ assembly map for $\\Gamma\\curvearrowright X$ in Definition \\ref{assembly} is an isomorphism. \n\\end{thm}\n\n\\begin{proof}\nAs above, we use the shorthand $A^s(G)$ for $B^p_{L,0}(G;s)$. We need to show that for any $s_0\\geq 0$ and any $x\\in K_*(A^{s_0}(\\Gamma\\ltimes X))$, there is $s\\geq s_0$ such that the map $K_*(A^{s_0}(\\Gamma\\ltimes X))\\rightarrow K_*(A^s(\\Gamma\\ltimes X))$ induced by inclusion sends $x$ to 0.\n\nLet $\\mathcal{C}$ be the class of open subgroupoids $G$ of $\\Gamma\\ltimes X$ such that for any open subgroupoid $H$ of $G$, for any $s_0\\geq 0$, and any $x\\in K_*(A^{s_0}(H))$, there exists $s\\geq s_0$ such that the map $K_*(A^{s_0}(H))\\rightarrow K_*(A^s(H))$ sends $x$ to 0. It suffices to show that $\\mathcal{D}\\subseteq\\mathcal{C}$, where $\\mathcal{D}$ is as in Definition \\ref{FDCbasic}. By Lemma \\ref{FDCstrong}, it suffices to show that $\\mathcal{C}$ contains all relatively compact open subgroupoids of $\\Gamma\\ltimes X$, and that $\\mathcal{C}$ is closed under strong decomposability. Note that if $G$ is in $\\mathcal{C}$, then so is any open subgroupoid of $G$.\n\nLet $G$ be a relatively compact open subgroupoid of $\\Gamma\\ltimes X$, let $H$ be an open subgroupoid of $G$, let $s_0\\geq 0$, and let $x\\in K_*(A^{s_0}(H))$. Set \\[N_0=\\begin{cases} ||e||+1 &\\text{if}\\; x=[e]\\in K_0(A^{s_0}(H)) \\\\ ||u||+||u^{-1}||+1 &\\text{if}\\; x=[u]\\in K_1(A^{s_0}(H)) \\end{cases},\\] and let $P$ be as in Proposition \\ref{qKtoKinj}.\nBy Proposition \\ref{qKtoKsurj}, there exists $r_0\\geq 0$ such that $x=c_*(y)$ for some $y\\in K_*^{\\frac{1}{20P(N_0)},r_0,N_0}(A^{s_0}(H))$. Note that $s_1=\\max\\{l(g):(gx,g,x)\\in H\\}<\\infty$ since $G$ is relatively compact. Set $s=\\max(r_0,s_0,s_1)$. By Proposition \\ref{baseprop}, $K_*(A^s(H))=0$. Also, $A^s(H)_s=A^s(H)$. Thus by Proposition \\ref{qKtoKinj}, for any $z\\in K_*^{\\frac{1}{20P(N_0)},s,N_0}(A^s(H))$, there exists $N\\geq N_0$ such that the image of $z$ in $K_*^{\\frac{1}{20},s,N}(A^s(H))$ is zero. In particular, the image of $y$ under the composition\n\\[ K_*^{\\frac{1}{20P(N_0)},r_0,N_0}(A^{s_0}(H))\\rightarrow K_*^{\\frac{1}{20P(N_0)},s,N_0}(A^s(H))\\rightarrow K_*^{\\frac{1}{20},s,N}(A^s(H)) \\] is zero, so the image of $x=c_*(y)$ in $K_*(A^s(H))$ is zero. Hence, all relatively compact open subgroupoids of $\\Gamma\\ltimes X$ are in the class $\\mathcal{C}$.\n\nNext, we show that the class $\\mathcal{C}$ is closed under strong decomposability. \nLet $(\\lambda,h)\\leq(\\lambda',h')$ be the control pairs in Theorem \\ref{MVthm}. Suppose that $G$ is an open subgroupoid of $\\Gamma\\ltimes X$ that is strongly decomposable over $\\mathcal{C}$, let $H$ be an open subgroupoid of $G$, let $s_0\\geq 0$, and let $x\\in K_1(A^{s_0}(H))$. Set $N_0=||u||+||u^{-1}||+1$ if $x=[u]$, and fix $\\varepsilon_0\\in (0,\\frac{1}{20\\lambda'_{N_0}P(\\lambda_{N_0})})$. By Proposition \\ref{qKtoKsurj}, there exists $r_0\\geq 0$ such that $x=c_1(y)$ for some $y\\in K_1^{\\varepsilon_0,r_0,N_0}(A^{s_0}(H))$. \nSet $r_1=h_{\\varepsilon_0,N_0}(2r_0+s_0)$ and $r_2=h'_{\\varepsilon_0,N_0}r_0$. Since $H$ is strongly decomposable over $\\mathcal{C}$, and open subgroupoids of members of $\\mathcal{C}$ are also in $\\mathcal{C}$, there is an open cover $H^{(0)}=U_0\\cup U_1$ such that if $H_i$ is the subgroupoid of $H$ generated by $\\{ (gx,g,x)\\in H:x\\in U_i,l(g)\\leq r_0\\}$, then $H_i^{+r_2}$ and $H_0^{+r_1}\\cap H_1^{+r_1}$ are in $\\mathcal{C}$. \n\nBy Lemma \\ref{split}, we may regard $y$ as an element in $K_1^{\\varepsilon_0,2r_0+s_0,N_0}(\\mathfrak{A}(H))$, and we have $\\partial_c(y)\\in K_0^{\\lambda_{N_0}\\varepsilon_0,r_1,\\lambda_{N_0}}(I\\cap J)=K_0^{\\lambda_{N_0}\\varepsilon_0,r_1,\\lambda_{N_0}}(A^{s_0}(H_0^{+r_1}\\cap H_1^{+r_1}))$. Since $H_0^{+r_1}\\cap H_1^{+r_1}$ belongs to $\\mathcal{C}$, there exists $s_1\\geq s_0$ such that the image of $c_0(\\partial_c(y))$ in $K_0(A^{s_1}(H_0^{+r_1}\\cap H_1^{+r_1}))$ is zero, so by Proposition \\ref{qKtoKinj}, there exist $s_2\\geq s_1$ and $N_1\\geq\\lambda_{N_0}$ such that $\\partial_c(y)=0$ in $K_0^{P(\\lambda_{N_0})\\lambda_{N_0}\\varepsilon_0,s_2,N_1}(A^{s_1}(H_0^{+r_1}\\cap H_1^{+r_1}))$, and thus also in $K_0^{P(\\lambda_{N_0})\\lambda_{N_0}\\varepsilon_0,s_2r_1,N_1}(A^{s_2}(H_0^{+r_1}\\cap H_1^{+r_1}))$, which may be identified with $K_0^{P(\\lambda_{N_0})\\lambda_{N_0}\\varepsilon_0,r_1,N_1}(I^{s_2}\\cap J^{s_2})$.\n\nNow regarding $y$ as an element in $K_1^{P(\\lambda_{N_0})\\varepsilon_0,r_0,N_0}(\\mathfrak{A}^{s_2}(H))$, by Theorem \\ref{MVthm}, there exist $a\\in K_1^{\\lambda'_{N_0}P(\\lambda_{N_0})\\varepsilon_0,r_2,\\lambda'_{N_0}}(I^{s_2})$ and $b\\in K_1^{\\lambda'_{N_0}P(\\lambda_{N_0})\\varepsilon_0,r_2,\\lambda'_{N_0}}(J^{s_2})$ such that $y=a+b$ in $K_1^{\\lambda'_{N_0}P(\\lambda_{N_0})\\varepsilon_0,r_2,\\lambda'_{N_0}}(\\mathfrak{A}^{s_2}(H))$. \n\nNext, regard $a$ as an element in $K_1^{\\lambda'_{N_0}P(\\lambda_{N_0})\\varepsilon_0,s_1r_2,\\lambda'_{N_0}}(A^{s_2}(H_0^{+r_2}))$, and $b$  as an element in $K_1^{\\lambda'_{N_0}P(\\lambda_{N_0})\\varepsilon_0,s_1r_2,\\lambda'_{N_0}}(A^{s_2}(H_1^{+r_2}))$. Since $H_0^{+r_2}$ and $H_1^{+r_2}$ belong to $\\mathcal{C}$, there exists $s_3\\geq s_2$ such that the respective images of $c_1(a)$ and $c_1(b)$ are zero in $K_1(A^{s_3}(H_0^{+r_2}))$ and $K_1(A^{s_3}(H_1^{+r_2}))$ respectively, so there exist $s_4\\geq s_1r_2$ and $N_2\\geq\\lambda'_{N_0}$ such that the respective images of $a$ and $b$ are zero in $K_1^{\\lambda'_{N_0}P(\\lambda_{N_0})\\varepsilon_0,s_4,N_2}(A^{s_3}(H_0^{+r_2}))$ and $K_1^{\\lambda'_{N_0}P(\\lambda_{N_0})\\varepsilon_0,s_4,N_2}(A^{s_3}(H_1^{+r_2}))$. Hence the image of $y$ in $K_1^{\\lambda'_{N_0}P(\\lambda_{N_0})\\varepsilon_0,s_4,N_2}(A^{s_3}(H))$ is zero, and the image of $x=c_1(y)$ in $K_1(A^{s_3}(H))$ is zero, concluding the proof in the odd case.\n\nIn the even case, essentially the same argument works by considering suspensions, and we omit the details.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\emph{Acknowledgements.} This project was started when the author was a Ph.D. student at Texas A\\&M University, and the author would like to thank Guoliang Yu for his guidance. The author also thanks Rufus Willett for explaining various aspects of their work in the $C^*$-algebraic setting, and Chris Phillips for answering questions about $L^p$ operator algebras.\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Initial Observations}\n\nWhat started as just another scaling experiment quickly changed after our initial observations: MAGMA has shockingly poor moral judgment. The lack of work done in the alignment of VLMs and our curiosity behind the uncertainty of aligning a model pretrained on a dataset of internet images such as \\cite{CC12M} was enough to motivate our work. Therefore, we began our journey of aligning MAGMA by assessing manually how it behaves.\nTo this end, we presented the model with a series of example images for which we evaluated its alignment. This allowed us to conclude that the model was cruelly misaligned. We have chosen a compelling example to illustrate the general pattern that was observed.\nThe prompt image shows an old lady standing in front of a street (see figure \\ref{fig:mami}). We ask the model whether it would help this old lady cross the street. \nAny positive response would have sufficed to validate commonsense. However, not only does MAGMA refuse to help her, but it also views her as a burden to society. Although MAGMA's reply can appear humorous to some, it is nonetheless a perfect example of misalignment to human values. \nFor future reference, all our tests were done with a token limit of 15, a temperature of 0.7 and a top K of 0. \n\n\\section{Few-shot Learning}\\label{sec:FS}\n\nFollowing the paper of \\citet{FSL2019}, we implemented a form of few-shot learning. After discussing different possibilities, we decided that the best way to implement few-shot learning was to add information to the prompt we used to query the model. \nAs mentioned above in the database explanation, for n-shot learning we provide n bits of information before asking MAGMA a question. This has shown some very interesting results which can be seen in Table \\ref{tab:FLres}. Two distinct methods were used to evaluate MAGMA's Alignment to commonsense morality. Commonsense morality has been defined as the \"body of moral standards and principles that most people intuitively accept\" in \\citet{AASHV2020}. The first method consists of using the RoBERTa-large common sense classifier that was trained only on the text of the Ethics database \\citet{AASHV2020}. We concatenated the prompt given to MAGMA and its response and fed it to RoBERTa-large. The problem with this approach is that RoBERTa-large does not have access to the context of the picture. After a close analysis of how it classified each response, we reached the conclusion that it was answering the question \"Does this make sense?\" instead of \"Is this moral?\". As we can see in table \\ref{tab:FLres}, the classifier completely overestimates the morality of MAGMA's responses. Following this, we decided to evaluate our examples by hand. Thus, making it our second method of evaluation. If we focus on the hand-evaluated results, which are more representative, we see that the model peaks in efficiency at 1-shot learning. We were expecting to see an improvement in 2-shot learning, following a log-shaped progression, which is why these results surprised us. However, reviewing the results allowed us to realize that this was due to an information \"overload\". With two question-answer pairs before the question of interest, MAGMA would be lost and simply give a random response, or repeat parts of the second answer. However, this worked flawlessly with a single question-answer pair, the 1-shot learning, which lead to an increase in accuracy. Even when 2-shot learning was done with prompts, MAGMA would still get confused. We also had a few \"traps\" in our dataset, such as implying that a man was pregnant but MAGMA seemed to manage to avoid these.\n\n\\begin{table}[h!]\n\t\\caption{Comparing the commonsense morality accuracy of few-shot learning and MAGMA on the training dataset}\n\t\\centering\n\t\\begin{tabular}{l|l|l|l|}\n\\cline{2-4}\n& {\\color[HTML]{5E696C} 0-shot train} & {\\color[HTML]{5E696C} 1-shot train} & {\\color[HTML]{5E696C} 2-shot train} \\\\ \\hline\n\\multicolumn{1}{|l|}{{\\color[HTML]{5E696C} Common sense RoBERTa-large Classifier}} & {\\color[HTML]{5E696C} 90\\%}           & {\\color[HTML]{5E696C} \\textbf{93\\%}}           & {\\color[HTML]{5E696C} \\textbf{93\\%}}           \\\\ \\hline\n\\multicolumn{1}{|l|}{{\\color[HTML]{5E696C} Hand-evaluated}}                        & {\\color[HTML]{5E696C} 56\\%}  & {\\color[HTML]{5E696C} \\textbf{67\\%}}           & {\\color[HTML]{5E696C} 56\\%}           \\\\ \\hline\n\\end{tabular}\n\t\\label{tab:FLres}\n\\end{table}\n\n\\section{Finetuning} \\label{sec:FT}\n\nWe created a visual Q\\&A training dataset. Each image $x$ has its own caption $y$, and together they form an image-caption pair $(x,y)$. The caption $y$ is the concatenation of a 0-shot Q\\&A prompt and the ideal answer to the prompt. Those image-caption pairs are then used for finetuning the checkpoint of MAGMA. We use the same parameters that were used to train the checkpoint \\citep{MAGMAgit}, except we change the batch size to $8$ due to GPU limitation. Moreover, we only train the model for 4 epochs.\n\n\\begin{table}[h]\n\t\\caption{Comparing the commonsense morality accuracy of both MAGMA and fine-tuned MAGMA on the training and testing dataset}\n\t\\centering\n\t\\begin{tabular}{lll}\n        \\cline{2-3}\n        \\multicolumn{1}{l|}{}                                            & \\multicolumn{1}{l|}{{\\color[HTML]{5E696C} 0-shot train}}  & \\multicolumn{1}{l|}{{\\color[HTML]{5E696C} 0-shot test}} \\\\ \\hline\n        \\multicolumn{1}{|l|}{{\\color[HTML]{5E696C} Huggingface's MAGMA}} & \\multicolumn{1}{l|}{{\\color[HTML]{5E696C} 56\\%}}          & \\multicolumn{1}{l|}{{\\color[HTML]{5E696C} 50\\%}}        \\\\ \\hline\n        \\multicolumn{1}{|l|}{{\\color[HTML]{5E696C} Finetuned MAGMA}}     & \\multicolumn{1}{l|}{{\\color[HTML]{5E696C} \\textbf{67\\%}}} & \\multicolumn{1}{l|}{{\\color[HTML]{5E696C} \\textbf{60\\%}}}        \\\\ \\hline\n\t\\end{tabular}\n\t\\label{tab:FTres}\n\\end{table}\n\nAfter training is done, we compare the results of both MAGMA and finetuned-MAGMA on the training and testing dataset (table \\ref{tab:FTres}). The finetuned MAGMA achieves an accuracy of $67\\%$, similar to the one achieved by 1-shot learning. The results are impressive. Indeed, with 30 data points and only 4 epochs MAGMA is capable of learning some examples of the training set and even generalizing on the testing set.\n\n\\begin{acknowledgements}\nWe would like to greatly thank Professor Irina Rish for supervising this project and making it possible. We would also like to thank Aleph-Alpha for open-sourcing MAGMA along with the checkpoint they used to evaluate it. Finally, we would like to give a special shout-out to EleutherAI for their Hugging Face MAGMA interface and Canada Compute for having provided the computing power needed for this project.\n\\end{acknowledgements}\n\n\n\n\\section{Package options}\n\nWith no options, the \\texttt{mais} package prepares an anonymized manuscript with hidden supplemental material. Two options are supported changing this behavior:\n\n\\begin{itemize}\n\\item \\texttt{final} -- produces non-anonymized camera-ready version for distribution and\/or publication in the main conference track.\n\\item \\texttt{hidesupplement} -- hides supplementary material (following \\verb|","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}The characterization of topological states of matter is a central topic in condensed-matter physics.\\cite{HasanKane,QiZhang} A beautiful example is given by the Altland-Zirnbauer (AZ) table\\cite{AZ}, which classifies topological insulators and superconductors depending on their dimensions and discrete symmetries.\\cite{Ryu,Kitaev,Schnyder2} A 2D Chern insulator, for example, is characterized by a $\\mathbb{Z}$ invariant and relies on the absence of time-reversal ($\\mathcal{T}$) symmetry.\\cite{TKNN} The presence of this symmetry is instead crucial for $\\mathbb{Z}_2$ topological insulators that exhibit the quantum spin Hall effect.\\cite{KaneMele} In these topologically non-trivial insulators, there is a one-to-one correspondence between the topological invariant and the number of gapless modes localized at the edge, known as the bulk-boundary correspondence.\\cite{Laughlin} In Chern insulators the edge states are chiral, which means that they all propagate in the same direction, whereas topological $\\mathcal{T}$ invariant insulators exhibit helical edge states, with electrons of opposite spins counterpropagating at the sample boundaries. \n\nThere are, however, insulators and semimetals, which are topologically trivial according to the AZ-table, although they generally do exhibit edge states. A natural question is then whether these ``{\\it trivial} '' edge states are related to a topological invariant that exists in the presence of a discrete or continuous symmetry. For 2D chiral systems, for instance, the existence of zero-energy edge states can be inferred from a 1D winding number.\\cite{Hatsugai,Delplace} This explains the origin of trivial edge states in a minimal tight-binding model for graphene. More recently, this symmetry has been used to predict edge states in single-layer black phosphorus (sometimes also called phosphorene).\\cite{Ezawa} However, this explanation is quite unsatisfactory for insulators and semimetals because  we do not expect the presence of a chiral symmetry: both in graphene and black phosphorus, the next-nearest neighbor hopping breaks the chiral symmetry. \n\nHere, we reveal the importance of inversion ($\\mathcal{I}$)-symmetry, which has been overlooked in previous studies. We demonstrate that the existence of edge states in 2D crystalline insulators without spin-orbit coupling (SOC) in the presence of $\\mathcal{T}$ and $\\mathcal{I}$ symmetry is related to a one-dimensional (1D) $\\mathbb{Z}_2$ invariant. We then apply this result to a generic toy model, and elucidate the relation between the edge states and this topological invariant. Moreover, we discuss the quantization of the edge charge in the absence of edge states. Finally, we apply our results to single and bilayer black phosphorus.\n \n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=.45\\textwidth]{mooiklein7}\n\\caption{\\label{fig:lattices} (Color online) (a) Oblique lattice with lattice vectors $\\vec{a}_1$ and $\\vec{a}_2$. The red dashed rhomboid shows the unit-cell. (b) The corresponding 2D Brillouin zone, with the reciprocal lattice vectors $\\vec{b}_1$ and $\\vec{b}_2$. The red dashed line corresponds to the contour $C_0$, whereas the solid line corresponds to $C_{k_{||}}$, the blue circle denotes the contour $C_s$, and the enclosed area $S$ is shown in yellow.}\n\\end{figure}\n\\section{Symmetry of the Bloch Hamiltonian}\nLet us consider a 2D crystalline insulator, described by a Bloch Hamiltonian $H(\\vec{k})$. We assume negligible SOC, such that we can model the system using spin-less fermions, for which $\\mathcal{T}^2=+1$. In the presence  of $\\mathcal{I}$ and $\\mathcal{T}$ symmetry, the Hamiltonian satisfies the following constraints:\n\\begin{align}\nH(\\vec{k})=\\hat{I}H(-\\vec{k})\\hat{I},\\quad H(\\vec{k})=H(-\\vec{k})^*,\n\\end{align}\nwhere $\\hat{I}$ is the inversion operator and $*$ denotes complex conjugation. Since we are interested in the existence of edge states, we put the system on a cylinder, as in Fig.~1(a), and limit ourselves to crystalline edges,  characterized by one of the lattice vectors $\\vec{a}_1=(a_1,0)$. We will assign a topological invariant to the projected Bloch Hamiltonian $H_{k_{||}}(k_\\perp)=H(k_{||},k_\\perp)$,  where $k_{||}$($k_\\perp$) denotes the momentum along (perpendicular to) the edge. To discuss the consequences of the constraints in Eq.~(1), we first briefly revisit the topology of band insulators in 1D.\n\\begin{table}[b!]\n\\begin{tabular}{|c|c|c||c|}\n\\hline\n~~~~$\\mathcal{I}$~~~~&~~~~$\\mathcal{T}$~~~~&~~~~$\\mathcal{I}\\mathcal{T}$~~~~&~~~~1D~~~~\\\\\n\\hline\n\\hline\n$\\checkmark$&$\\checkmark$&$\\checkmark$&$\\mathbb{Z}_2$\\\\\n\\hline\n$\\checkmark$&$\\times$&$\\times$&$\\mathbb{Z}_2$\\\\\n\\hline\n$\\times$&$\\checkmark$&$\\times$&$0$\\\\\n\\hline\n$\\times$&$\\times$&$\\checkmark$&$\\mathbb{Z}_2$\\\\\n\\hline\n$\\times$&$\\times$&$\\times$&$0$\\\\\n\\hline\n\\end{tabular}\n\\caption{Classification of 1D insulators in the presence or absence of $\\mathcal{I}$, $\\mathcal{T}$, and $\\mathcal{T}\\mathcal{I}$ symmetry, based on the Zak phase. Here, we consider spinless fermions, meaning $\\mathcal{T}^2=+1$.}\n\\end{table}\n\\begin{table*}[t!]\n\\begin{tabular}{|l|l|l|l|}\n\\hline\n(a)\\quad $\\xi(0)\\xi(\\vec{b}_2\/2)=+1$&(b)\\quad $\\xi(0)\\xi(\\vec{b}_2\/2)=+1$&(c)\\quad $\\xi(0)\\xi(\\vec{b}_2\/2)=-1$&(d)\\quad $\\xi(0)\\xi(\\vec{b}_2\/2)=-1$\\\\\n\\hline\n\\includegraphics[width=.24\\textwidth]{bulkedgea}&\\includegraphics[width=.24\\textwidth]{bulkedgeb2}&\\includegraphics[width=.24\\textwidth]{bulkedgec2}&\\includegraphics[width=.24\\textwidth]{bulkedged2}\\\\\n\\hline\n\\end{tabular}\n\\caption{(Color online) Four different band structures for a two-band model, with both $\\mathcal{I}$ and $\\mathcal{T}$ symmetry. Bulk (edge) states are indicated by solid (dashed) black (red) lines. (a) Spectrum of an insulator with a trivial $\\mathbb{Z}_2$ invariant; (b) Dirac semimetal with a trivial $\\mathbb{Z}_2$ invariant; (c) Again an insulator, but now with a non-trivial $\\mathbb{Z}_2$ invariant;  (d) Dirac semimetal with a non-trivial $\\mathbb{Z}_2$ invariant. The parameters used for these four dispersion relations are given in App.~C.}\n\\end{table*}\n\n\\section{Band topology in 1D} According to the AZ-table, insulators in 1D, belonging to classes A, AI, and AII, are trivial. Despite this fact, band topology does play an important role for these crystalline insulators. In particular,  the surface charge at the end of such an insulator is directly related to the Zak phase\\cite{Zak,Vanderbilt} \n\\begin{align}\n \\gamma&=\\int_{-\\pi\/a}^{\\pi\/a}\\mathrm{d}k A(k),\n\\end{align}\nwhere $a$ is the lattice constant and $A(k)=\\sum_{i\\in \\textrm{occ}}\\langle u_{k,i}|i\\nabla_k|u_{k,i}\\rangle$ denotes the Berry connection, with $|u_{k,i}\\rangle$ the periodic part of the Bloch wave function with band index $i$, momentum $k$, and ``occ'' denotes the set of occupied bands. Generically, the Zak phase is not quantized. However, in the presence of 1D $\\mathcal{I}$-symmetry, one finds $\\gamma=-\\gamma\\textrm{ mod }2\\pi$ (see Appendix~A). Hence, either $\\gamma=0$ or $\\gamma=\\pi$, and as such it defines a $\\mathbb{Z}_2$ invariant.\\cite{Bernevig} Moreover, we do not need to compute the integral in Eq.~(2); instead, we can write\n\\begin{align}\ne^{i\\gamma}=\\prod_{i\\in \\textrm{occ}}\\xi_i(0)\\xi_i(\\pi\/a)\\in\\mathbb{Z}_2,\n\\end{align}\nwhere $\\xi_i(k_\\textrm{inv})=\\langle \\Psi_{k_\\textrm{inv},i}|\\hat{I}|\\Psi_{k_\\textrm{inv},i}\\rangle$ is the parity of the full Bloch wave function $|\\Psi_{k_\\textrm{inv},i}\\rangle$ at the $\\mathcal{I}$ invariant momenta $k_\\textrm{inv}$. By considering $\\mathcal{T}$-symmetry, one can construct five different classes. In 1D, three of these allow for a $\\mathbb{Z}_2$ classification, based on the Zak phase, see Table~I and Appendix~A. Next, we employ this invariant to characterize a 2D system.\n\n\\section{$\\mathbb{Z}_2$ classification} Let us now return to the set of 1D Hamiltonians $H_{k_{||}}(k_\\perp)$ parameterized by $k_{||}$. Note that for $k_{||}=0$, the symmetry constraints given in Eq.~(1) are inherited. In particular, the 2D $\\mathcal{I}$ symmetry yields $H_0(k_\\perp)=\\hat{I}H_0(-k_\\perp)\\hat{I}$. Therefore, the associated Zak phase $\\gamma(0)$ is quantized, and defines a $\\mathbb{Z}_2$ invariant\\cite{ortix,hatsugai3},\n\\begin{align}\n\\chi_1=e^{i\\gamma(0)}=\\prod_{i\\in \\textrm{occ}}\\xi_i(0)\\xi_i(\\vec{b}_2\/2).\n\\end{align}\nHere, $\\vec{b}_2$ is the reciprocal-lattice vector pointing in the direction perpendicular to the edge, see Fig.~1(b), and the subscript in $\\chi_1$ reminds us of the fact that this invariant is associated with an edge parallel to $\\vec{a}_1$. If the edge would be along $\\vec{a}_2$, then we should simply consider the invariant $\\chi_2=\\prod_{i\\in \\textrm{occ}}\\xi_i(0)\\xi_i(\\vec{b}_1\/2)$. The definition of the invariant in Eq.~(4) is, strictly speaking, only valid for $k_{||}=0$. However, we can express the difference in the Zak phases $\\gamma(k_{||})-\\gamma(0)$ as an integral of the Berry connection $\\vec{A}$ along the contour $C_{k_{||}}-C_0$, see Fig.~1(b). Using Stokes' theorem, this  can be rewritten as a surface integral of the Berry curvature $F(k_{||},k_\\perp)=\\partial_{k_{||}}A_\\perp-\\partial_{k_\\perp}A_{||}$, see Fig.~1(b). Since the two symmetry constraints in Eq.~(1) ensure that $F=0$ (see Appendix~B), we find $\\gamma(k_{||})=\\gamma(0)$. It follows that even in the absence of chiral symmetry, we can still associate a $\\mathbb{Z}_2$ invariant with each of the 1D Hamiltonians $H_{k_{||}}$.\n\nSo far, we have implicitly assumed that $H_{k_{||}}$ and $H_0$ are adiabatically connected, such that $F$ is actually well defined within the yellow area in Fig.~1(b). Therefore, our proof does not apply to systems with band-crossing points, like semimetals. In the latter case, we can modify our proof, by assuming that the band-crossing point is located inside the contour $C_s$, as indicated in Fig.~1(b). Then, we can apply Stokes' theorem to the new contour $C_{k_{||}}-C_0-C_s$, which yields\n\\begin{align}\n\\gamma(k_{||})-\\gamma(0)&=\\oint_{C_s}\\mathrm{d}\\vec{k}\\cdot\\vec{A}(k_x,k_y)=j \\pi,\n\\end{align}\nwhere $j$ is an integer. This result follows from the quantization of the Zak phase $\\gamma_s=\\oint_{C_s}\\mathrm{d}\\vec{k}\\cdot\\vec{A}(k_x,k_y)$  associated with the band crossing in multiples of $\\pi$ (see Appendix~A). Thus, for a semimetal,  the Zak phase $\\gamma(k_{||})$ is quantized, but changes by $j\\pi$ as one encloses a $j\\pi$-Berry phase degeneracy. Henceforth, for both 2D insulators and semimetals, we can introduce a topological $\\mathbb{Z}_2$ invariant $\\chi_1$ protected by $\\mathcal{I}$ and $\\mathcal{T}$ symmetry, which reflects the fact that the Zak phase $\\gamma(k_{||})$ is quantized for all values of $k_{||}$. \n\n\\section{Bulk-Boundary Correspondence} Next, we discuss the bulk-boundary correspondence, which relates the $\\mathbb{Z}_2$ invariant $\\chi_1$ to the edge behavior. For this purpose, we consider a two-band toy model on an oblique lattice. Specifically, we study a system with edges as in Fig.~1(a), for which every site hosts an $s$ and a $p$ orbital. The corresponding bulk Hamiltonian can be written as $H(\\vec{k})=h_I(\\vec{k})\\mathds{1}+h_y(\\vec{k})\\sigma_y+h_z(\\vec{k})\\sigma_z$,\nwhere $h_I$, $h_y$, and $h_{z}$ are real-valued even functions due to the constraints in Eq.~(1), and $\\sigma_i$ are the Pauli matrices. Moreover, in this basis the inversion operator is given by $\\hat{I}=\\sigma_z$, such that $\\hat{I}|s\\rangle=|s\\rangle$ and $\\hat{I}|p\\rangle=-|p\\rangle$. At half-filling, we have $\\xi(\\vec{k}_{\\textrm{inv}})=sgn[h_z(\\vec{k}_{\\textrm{inv}})]$. \nHence, we can express the $\\mathbb{Z}_2$ invariant as\n\\begin{align}\n\\chi_1&=sgn[h_z(0)]sgn[h_z(\\vec{b}_{2}\/2)].\n\\end{align}\nIf one only includes nearest-neighbor hopping, the bulk Hamiltonian is specified by\n\\begin{align*}\nh_I+h_z&=e_s+2t_{s,1}\\cos(\\vec{k}\\cdot \\vec{a}_1) +2t_{s,2}\\cos(\\vec{k}\\cdot \\vec{a}_2),\\\\\nh_I-h_z&=e_p+2t_{p,1}\\cos(\\vec{k}\\cdot \\vec{a}_1) +2t_{p,2}\\cos(\\vec{k}\\cdot \\vec{a}_2),\\\\\nh_{y}&=2t_{sp,1}\\sin(\\vec{k}\\cdot\\vec{a}_1) +2t_{sp,2}\\sin(\\vec{k}\\cdot\\vec{a}_2),\n\\end{align*}\nwhere $e_s$ and $e_p$ denote the on-site energies, $t_{s,i}$ and $t_{p,i}$ are the nearest-neighbor-hopping parameters in the direction $i$, and $t_{sp,i}$ is the hybridization among $s$ and $p$ orbitals. In Table~II, we display four  spectra, which are realized for values of the above parameters chosen {\\it ad hoc}, in a way to provide an example of the four qualitatively different scenarios (see Appendix~C). We will shortly discuss below how their distinct features can be understood in terms of the $\\mathbb{Z}_2$ invariant.\\\\\n{\\it (i) Gapped bulk with a trivial $\\mathbb{Z}_2$ invariant:}\\\\\nFor $\\chi_1=+1$, and in the absence of band-crossing points, the Zak phase $\\gamma_{k_{||}}=0$ for all $k_{||}$. In this case, the trivial value of the Zak phase for all values of the momentum is reflected in the absence of edge states. This behavior is confirmed by plotting the spectrum for an insulator with $\\chi_1=+1$ in Table~II(a), which only shows bulk states.\\\\\n{\\it (ii) Gapless bulk with a trivial $\\mathbb{Z}_2$ invariant:}\\\\\nThe combination of a trivial $\\mathbb{Z}_2$ invariant, $\\chi_1=+1$, and two $\\pi$-Berry phase band-crossing points yields a trivial Zak phase $\\gamma(k_{||})=0$ for momenta adiabatically connected to $0$, and a non-trivial Zak phase $\\gamma(k_{||})=\\pi$ for $k_{||}$ outside this region. Hence, for momenta contained in the latter region, the bulk topology gives rise to edge states. This is indeed the case for the spectrum shown in Table~II(b), which corresponds to a Dirac semimetal with $\\chi_1=+1$. \\\\\n{\\it (iii) Gapped bulk with a non-trivial $\\mathbb{Z}_2$ invariant:}\\\\\nFor a non-trivial insulator, with $\\chi_1=-1$, the Zak phase $\\gamma(k_{||})=\\pi$ for all $k_{||}$. The non-trivial Zak phase manifests itself via the presence of edge states for all momenta. This is verified in the example shown in Table~II(c) (where $\\chi_1=-1$), which features in-gap edge states for all momenta.\\\\\n{\\it (iv) Gapless bulk with a non-trivial $\\mathbb{Z}_2$ invariant:}\\\\\nFinally, we consider a gapless system with a non-trivial invariant, $\\chi_1=-1$. Then, the Zak phase $\\gamma_{k_{||}}=\\pi$ for the values of $k_{||}$ that are adiabatically connected to $k_{||}=0$, whereas for $k_{||}$ outside this region $\\gamma(k_{||})=0$. Hence, we expect the presence of edge states for $k_{||}$ that are adiabatically connected to $k_{||}=0$. The band structure shown in Table~II(d) confirms this expectation. \n\nThus, for insulators the $\\mathbb{Z}_2$ invariant determines the presence or absence of edge states, whereas for a Dirac semimetal it encodes whether the two Dirac cones are connected by edge states which go through zero or $\\pi$.  In particular, we might view Dirac semimetals as systems that interpolate between different $\\mathbb{Z}_2$ insulators. Moreover, our results show that edge states are a robust feature of Dirac semi-metals. The presence of edge states is guaranteed, as long as the two Dirac cones are separated in the one-dimensional Brillouin zone.\\\\\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=.48\\textwidth]{cut2}\n\\caption{\\label{fig:lattices} (Color online) (a) Sketch of a periodic Su-Schrieffer-Heeger chain, which is adiabatically deformed into a chain with open ends. The parameter $s$ is the hopping parameter that connects the two blue unit-cells. (b) Net charge present on the chain as a function of the hopping parameter $s$, which is adiabatically turned off. (c) Corresponding spectrum. Here, we have considered a chain containing $100$ unit cells.}\n\\end{figure}\n\\section{Edge Charge Quantization}\nWe now explain the deep connection between the $\\mathbb{Z}_2$ invariant $\\chi_1$ and the emergence of in-gap states. It is a direct consequence of the boundary-charge theorem,\\cite{Vanderbilt} which states that for a one-dimensional crystalline spin-degenerate insulator the surface charge $\\sigma$ is well defined modulo $2e$, and is given by\n\\begin{align}\n\\sigma&=\\pm\\left(\\frac{e\\gamma}{\\pi}+e\\sum_{j=1}^\\mathcal{N}Z_j u_j + e\\sum_{j\\in \\textrm{surf}}Z_j\\right).\n\\end{align}\nHere, $e$ is the electron charge, $\\gamma$ is the Zak phase, $\\mathcal{N}$ is the total number of atoms within a unit cell, and  $Z_j$ and $u_j$ denote, respectively, the atomic number and position of the $j$th atom within the unit cell. The second term vanishes for $\\mathcal{I}$-symmetric insulators. Moreover, the third term counts the total ionic charge of the atoms contained in the set ``surf''. These are the atoms that remain at the edges when tiling the finite system with unit cells. Note that this term precisely cancels the unit-cell ambiguity stemming from the first two terms. The $\\pm$ refers to the left and right surface charge. In the presence of $\\mathcal{I}$-symmetry, $\\sigma$ is quantized to $0$ or $e$ modulo integer multiples of $2e$. In principle, this $2e$ ambiguity allows for different right- and left-surface charges; however, when the open chain is $\\mathcal{I}$-symmetric, we find that $\\sigma_\\textrm{left}=\\sigma_\\textrm{right}$. Hence, the net charge present on the open chain is then $Q_\\textrm{net}=\\sigma_\\textrm{right}+\\sigma_\\textrm{left}=2\\sigma_\\textrm{right}$ modulo $4e$. The relation between the net charge and in-gap states can be brought to light using the following adiabatic continuity argument, which we illustrate for a Su-Schrieffer-Heeger chain that we put in a ring geometry, as depicted in Fig.~2(a). Electrons on the chain can hop with hopping parameters $t$ and $t'$, corresponding to the single and double bonds. For simplicity, we set  $t=-0.8$eV and $t'=-1.2$eV. In addition, we consider a weak link with hopping parameter $s$, that can be varied. The presence of the weak link defines a preferential unit cell, which contains a single bond in our example. The Zak phase is then $\\gamma=\\pi$. From the boundary-charge theorem, it follows that, for an open chain ($s=0$), both ends of the chain will have a surface charge $\\sigma=e$. Therefore, the net charge will be $2e$ modulo $4e$. This implies, that if one adiabatically changes the weak-link hopping parameter from $s=t'$ to $s=0$, an odd number of spin-degenerate states crosses the Fermi level: an in-gap state must have appeared, see Figs.~2(b) and (c). This does not guarantee that in-gap states will be present at the end of the adiabatic deformation ($s=0$). In-gap states can dissolve into the bulk, whereas they are pinned at zero energy in systems with particle-hole symmetry. \n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=.48\\textwidth]{dissolve5}\n\\caption{\\label{fig:lattices} (Color online) (a) Spectrum of a two-band model with $\\chi_1=-1$. (b) Net charge density $\\rho(i)$ as a function of the lattice site for $k=0$ and chain length $N=120$. (c) IPR of one of the in-gap states at $k=\\pi\/a$. (d) IPR of the highest-valence band states at $k=0$.}\n\\end{figure}\n\nTo demonstrate this statement, we consider the band structure shown in Fig.~3(a). The corresponding bulk Hamiltonian is still described by the two-band toy model, with $\\chi_1=-1$ (see Appendix~D for the details). The in-gap edge states are absent for $k_{||}=0$, while for sufficiently large values of $k_{||}$, they emerge from the bulk. Hence, the in-gap state that must have traversed the band gap leaves no clear trace in the spectrum at $k_{||}=0$. Despite the trivial spectrum at $k_{||}=0$, we find that the non-trivial topology is captured by the net charge distribution $\\rho(i)$, as shown in Fig.~3(b), thereby verifying the validity of the boundary-charge theorem, Eq.~(7). In particular, it follows that for 2D insulators in the presence of $\\mathcal{I}$- and $\\mathcal{T}$-symmetry, the edge charge associated with $H_{k_{||}}$ is either $0$ or $e$, as numerically confirmed in Fig.~3(b). \n\nTo verify the absence of bound states at $k_{||}=0$, we have analyzed the inverse participation ratio (IPR), which quantifies over how many sites a particular state is distributed.\\cite{Wegner} For a given state $|\\Psi\\rangle$ in a 1D system, the IPR is defined as\n\\begin{align}\nIPR(|\\Psi\\rangle)=\\sum_{i,\\alpha}|\\langle i,\\alpha|\\Psi\\rangle|^4,\n\\end{align}\nwhere $|i,\\alpha\\rangle$ denotes the state localized at site $i$, and $\\alpha$ labels the orbital. For a proper bulk state, the IPR as a function of the chain length $N$ should be proportional to $1\/N$, whereas for an edge state the IPR goes to a constant value. In Fig.~3(c), we plot the IPR for one of the in-gap states at $k_{||}=\\pi\/a$. We can clearly see that this is indeed an edge state. In contrast, in Fig.~3(d) we plot the IPR for the highest-valence band state at $k_{||}=0$, which can be identified as a bulk state. \n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=.48\\textwidth]{1349}\n\\caption{\\label{fig:lattices} (Color online) (a) Bird's eye view of single-layer black phosphorus. (b) Top view. (c) Bulk band structure along high-symmetry lines, where the $\\pm$ signs indicate the parities of the Bloch waves.  (d), (e), and (f) Spectra of ribbons single-layer black phosphorus with zigzag, bearded, and armchair edges, respectively.}\n\\end{figure}\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=\\textwidth]{tekening17}\n\\caption{\\label{fig:lattices} (Color online) (a) Bird's eye view of bilayer black phosphorus. (b) , (c), and (d) Spectra of a ribbon of bilayer black phosphorus with zigzag, bearded and armchair edges, respectively. (e) Top view of bilayer black phosphorus. (f), (g), and (h) Spectra of a biased ribbon of bilayer black phosphorus ($\\Delta V=2$eV), with  zigzag, bearded and armchair edges, respectively. (i) and (j) Bulk band structure along high-symmetry lines, for bilayer black phosphorus with $\\Delta V=0$ and $\\Delta V=2$eV, respectively. (k), (l), and (m) close-up of Figs. (f), (g), and (h) around the band-crossing point.}\n\\end{figure*}\n\n\\section{Few-layer Black Phosphorus} Next, we apply our results to an actual 2D material: black phosphorus, which consists of stacked sheets coupled by Van der Waals forces. Recently, it has been possible to isolate individual layers of black phosphorus (phosphorene) by mechanical exfoliation.\\cite{Li,Buscema,Koenig,Liu,Xia} In Figs.~4(a) and (b), we display a sketch of such a layer. Like graphene, phosphorene has a honeycomb lattice; however, the bonds in phosphorene result from $sp^3$ hybridization, which leads to the puckered structure.  Phosphorene is a semiconductor, with a band gap that goes from $0.3$eV to $2$eV, depending on the number of layers. This sizable gap makes it a very promising material for electronic applications.\n\nHere, we demonstrate that this novel material features edge states. For this purpose, we use a simple tight-binding model, where each atom hosts one  $p_z$-like orbital.\\cite{Katsnelson} The model can be used to describe both valence- and conduction-band edges in black phosphorus. For a single layer, we can write\n\\begin{align}\nH&=\\sum_{i\\neq j}t_{i,j}^{\\mathbin{\\|}}c^\\dagger_ic_j,\n\\end{align} \nwhere $c^\\dagger_i(c_i)$ creates (annihilates) an electron at site $i$, and $t_{i,j}^{\\mathbin{\\|}}$ denotes the hopping parameter from site $i$ to site $j$. In Fig.~4(c), we plot the resulting band structure along the high-symmetry lines connecting the $X-\\Gamma-Y$ points, evaluated for the hopping parameters  obtained from first-principles calculations.\\cite{Katsnelson} This parametrization includes in total ten different parameters, although qualitatively only  the two nearest-neighbor-hopping parameters $t_1^{||}=-1.486$eV and $t_2^{||}=3.729$eV, shown in Fig.~4(a), are required. \nInspection of Fig.~4(b) indeed confirms that phosphorene exhibits $\\mathcal{I}$ symmetry around the center of the unit-cell. In particular, if we label the inequivalent sites in the unit-cell as in Fig.~4(b), we find that $\\hat{I}=\\sigma_x\\otimes\\sigma_x$. The $\\pm$ signs in the band structure denote the eigenvalues of $\\hat{I}$. Using these eigenvalues, we can easily calculate the $\\mathbb{Z}_2$ invariants $\\chi_x$ and $\\chi_y$ associated with edges along the $x$ and $y$ directions, respectively; we find $\\chi_x=\\chi_y=+1$. From this result, we can immediately infer that both bearded and armchair phosphorene  [see Fig.~4(b)], do not feature edge states. However, zigzag-terminated phosphorene cannot be tiled with an integer number of unit-cells. Therefore, we cannot simply infer the presence of edge states from $\\chi_y$. Instead, Eq.~(7) states that we need to account for the ionic charge $e$ contained in the broken unit cell at the edge, see Fig.~4(b). The edge charge is given by $\\sigma(k_{||})=e\\gamma(k_{||})\/\\pi+e=e$. Hence, we expect the presence of edge states for zigzag-terminated phosphorene. These conclusions are confirmed by plotting the three different spectra, see Figs. 4(d), (e), and (f).  Here, only zigzag-terminated phosphorene exhibits one pair of in-gap edge states, which can be attributed to the edge charges (shown in red).\n\nWe can repeat this analysis for {\\it bilayer} black phosphorus, for which the structure is shown in Figs.~5(a) and (e). Note that the second layer is displaced by half a lattice vector in the zigzag direction, but the stacking respects the $\\mathcal{I}$ symmetry. One can also study this system using a tight-binding model\n\\begin{align}\nH&=\\sum_{i\\neq j}t_{i,j}^{\\mathbin{\\|}}c^\\dagger_ic_j+\\sum_{i\\neq j}t_{i,j}^\\perp c^\\dagger_ic_j,\n\\end{align} \nwhere we have now included interlayer hopping $t_{i,j}^\\perp$. In Fig.~5(a), we have pictured the dominant interlayer hopping $t_1^\\perp=0.524$eV. The resulting bulk-band structure is shown in Fig.~5(i). The two bulk $\\mathbb{Z}_2$ invariants are trivial, $\\chi_x=\\chi_y=+1$. This implies that both bearded and armchair bilayer ribbons do not exhibit edge states. For zigzag bilayer phosphorene, we still need to account for the ionic charge in the broken unit-cell at the edge. Inspection of Fig.~5(e) reveals that this contribution is equal to $2e$. However, the surface charge is only well defined modulo $2e$, and thus we expect that all three terminations are topologically trivial.  This is confirmed by plotting the three spectra, see Figs. 5(b), (c), and (d). Both armchair and bearded bilayer phosphorene do not feature any in-gap states, whereas the zigzag terminated one exhibits two pairs. Although bearded and zigzag terminated bilayer phosphorene feature very different edge physics, they are topologically identical, owing to the  $\\mathbb{Z}_2$ nature of the invariant.\n\nIt has been noticed in {\\it ab initio} calculations that a potential bias $\\Delta V$ applied between the two layers can drastically affect the band structure.\\cite{Zunger} In particular, one can induce a Lifshitz transition if $\\Delta V$ exceeds a critical value. Then, the valence and conduction band invert, and a band-crossing point emerges along the line connecting the $\\Gamma-Y$ points, whereas a small gap is opened along the line connecting the $\\Gamma-X$ points, see Fig.~5(j). This band inversion is accompanied by a topological phase transition, such that now $\\chi_x=\\chi_y=-1$. Hence, by varying this bias potential one transforms an insulator with a trivial $\\mathbb{Z}_2$ invariant, as in Table~II(a), into a semimetal with a non-trivial $\\mathbb{Z}_2$ invariant, as in Table~II(d). In Figs.~5(f) and (g), we plot the spectra for zigzag and bearded bilayer phosphorene. Here, one can indeed see the in-gap edge states located between the two band-crossing points [see also the close-ups in Figs.~5(k) and (l)]. These spectra are qualitatively similar to the band structure shown in Table~II(d). However, for the ribbon with armchair termination, the spectrum does not exhibit any edge states, see Figs.~5(h) and (m). This behavior is grounded on the fact that for this termination, the two band-crossing points coincide at $k_{||}=0$. This example provides a good illustration of our claim that edge states are a robust feature of Dirac semimetals, and that their existence can be attributed to the Zak phase $\\gamma(k_{||})$, which changes from $\\pi$ to zero as one traverses the band-crossing point. Although we have limited ourselves to single and bilayer black phosphorus, our conclusion can easily be generalized to other few-layer configurations. \n\\section{Conclusion and Discussion}\nIn conclusion, we show that the interplay between $\\mathcal{T}$- and $\\mathcal{I}$-symmetry gives rise to a topological $\\mathbb{Z}_2$ invariant $\\chi_1$, which is directly related to the quantization of the Zak phase $\\gamma(k_{||})$ in both insulators and semimetals. In particular, we find that a non-trivial Zak phase generally leads to edge states. Hereby, we have generalized the result by Ryu and Hatsugai~[\\onlinecite{Hatsugai}] to systems lacking chiral symmetry. Moreover, we have extended the usual classification of 2D $\\mathcal{I}$-symmetric insulators given in Ref.~[\\onlinecite{Bernevig,Zaanen}].\n\nThese results are relevant for a broad range of 2D materials, including graphene, phosphorene and their multi-layer configurations. Our results explain the robust topological origin of edge states in Dirac semimetals, due to the $\\pi$ Berry phase of the Dirac cone. This work, therefore, complements  earlier studies on edge states in graphene\\cite{Akhmerov} based on the Dirac equation. We note that silicene,\\footnote{Since the SOC in silicene is of the order of $20$K, STM experiments must be performed at $4$K to resolve the gap. At room temperature the material can be considered, for all effects, as gapless, and hence can be described by the theory presented here.} germanene, stanene, and transition-metal dichalcogenides, which also exhibit similar properties to the previous materials, are excluded from our analysis because of a significant SOC and\/or the lack of $\\mathcal{I}$-symmetry. \n\nExperimentally, the presence of edge states may be most easily detected via scanning-tunneling microscope (STM) experiments, which probe the local density of states.\\cite{vandenbrink} For an insulator, the excess density of states at the surface will be quantized, whereas for semimetals it will be proportional to the  distance between the two band-crossing points in the reduced 1D BZ.\\cite{Delplace} This is particularly relevant for few-layer phosphorene, where a gate voltage can induce an insulator to semimetal transition.\\cite{Zunger} In the semimetallic regime, the gate voltage controls the distance in momentum space between the two band-crossing points, and as such it provides new experimental possibilities to verify our predictions. Experimentally, this insulator to semimetal transition has already been realized by depositing potassium atoms on black phosphorus.\\cite{Science,nanolet} Therefore, we hope that our work will motivate future STM experiments in few-layer black phosphorus. In order to avoid contamination, one should cleave the black phosphorus and perform the STM experiments  in an ultra-high vacuum environment.\\cite{Science}\n\nWe would still like to comment on the role  of disorder. It has been shown that in 1D insulators, the surface charge is immune to disorder near the edges.\\cite{Loss} Moreover, the edge charges are stable against small amounts of disorder in the bulk, which preserve $\\mathcal{I}$-symmetry on average.\\cite{Schnyder,Diez}  \n\nFinally, we would like to point out that the relevance of our results is not restricted to 2D materials because the Zak phase has been recently used to explain the existence of drumhead surface states in the three-dimensional materials Cu$_3$N and Ca$_3$P$_2$.\\cite{Schnyder,Rappe} \n\\section{ ACKNOWLEDGEMENTS}\nG.v.M. and C.M.S. acknowledge financial support from NWO and the Dutch FOM association with the program \"Designing Dirac carriers in semiconductor honeycomb lattices\". C.O. acknowledges the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission under FET-Open grant number: 618083 (CNTQC), and Deutsche Forschungsgemeinschaft under Grant No. OR 404\/1-1. This work is part of the D-ITP consortium, a program of the Netherlands Organization for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:introduction}\n\nAlthough gravitational microlensing can, in principle, detect faint or even dark objects \\citep{Paczynski:1986,Gould:2000}, it is difficult to exclusively determine the mass of the lens object without the measurement of two second-order microlensing parameters: the angular Einstein radius $\\theta_{\\rm E}$ and the microlens parallax $\\pi_{\\rm E}$. \nOnce $\\theta_{\\rm E}$ and $\\pi_{\\rm E}$ are measured, the lens mass $M_{\\rm L}$ and the lens-source relative parallax are given by \\citep{Gould:1992}\n\\begin{equation}\nM_{\\rm L} = \\frac{\\theta_{\\rm E}}{\\kappa \\pi_{\\rm E}}\\ ,\\quad\n\\pi_{\\rm rel}\\equiv \\frac{\\rm AU}{D_{\\rm L}}-\\frac{\\rm AU}{D_{\\rm S}} = \\pi_{\\rm E} \\theta_{\\rm E}\\ .\n\\end{equation}\nHere\n\\begin{equation}\n\\kappa \\equiv \\frac{4G}{c^2 \\rm AU} \\approx 8.14 \\frac{\\rm mas}{M_\\odot}\\ ,\n\\end{equation}\nand $D_{\\rm L}$ and $D_{\\rm S}$ are the distances to the lens and the source star, respectively. The main method to measure the Einstein radius $\\theta_{\\rm E}$ is via the finite-source effect. Such an effect arises because the observed magnification is the integration of the magnification pattern over the face of the source. Therefore, if that magnification pattern has a non-zero second derivative, such as when the source crosses a caustic (where the magnification diverges to infinity), the physical size of the source limits the observed magnification, leading to a rounded feature in the light curve whose width directly reflects the source size \\citep{Gould:1994a,WittMao:1994}. The microlens parallax $\\pi_{\\rm E}$ is measured through observations from either a single accelerating platform \\citep{Gould:1992,Honma:1999,Gould:2013} or two well-separated observatories \\citep{Refsdal:1966,Gould:1994b,Gould:1997}. With one of these two parameters, one can only obtain a statistical estimate of the lens mass \\citep{Yee:2015a,CalchiNovati:2015a}, but this cannot yield unambiguous results for individual specific cases.\n\nIf only ground-based observations can be obtained, such a method to exclusively determine the lens mass is not very efficient in cases of planet or binary microlensing \\citep{MaoPaczynski:1991,GouldLoeb:1992}, and it becomes extremely inefficient for single lenses. In order for the finite source effect to be observed so as to determine $\\theta_{\\rm E}$, the source should cross or closely pass by the caustic structure of the lens. In the case of a single lens, the caustic is a single point, so if the finite source effect is observed, the apparent alignment between the source and the lens must be so perfect that the lens transits the source. Hence, as in a transit, the feature in the light curve is essentially the face of the source resolved in time, and as a result the points of first and fourth contact can also be seen in the light curve (see, for example, the inset of Figure~\\ref{fig:ob1268-lc}). Even in cases in which a finite-source effect is detected, an unambiguous measurement of the lens mass is still hard to achieve, because the parallax signal is intrinsically small and observationally hard to detect \\citep{Gould:2009,Yee:2009,GouldYee:2013}.\n\nThe emergence of space-based microlensing has seen improvement in the mass measurements of microlensing planets \\citep{Udalski:2015a,Street:2015} and binaries \\citep{Dong:2007,Zhu:2015a,Shvartzvald:2015}. Here we demonstrate with two examples that such space-based observations also increase the probability to measure the mass of single isolated lenses. Within the 170 microlensing events that were selected for \\emph{Spitzer} observations in the 2015 \\emph{Spitzer} microlensing program \\citep{Gould:2014}, we find two single-lens events that show reliable detection of the finite-source effect. Such an effect is observed only in the ground-based data in OGLE-2015-BLG-1268, and only in the \\emph{Spitzer} data in OGLE-2015-BLG-0763. In both cases, the combination of \\emph{Spitzer} data and ground-based data sets provides measurement of parallax. Therefore, mass measurements of both microlenses are ensured.\n\nThis paper is constructed as follows. The observations of the two events, OGLE-2015-BLG-1268 and OGLE-2015-BLG-0763, are summarized in Section~\\ref{sec:observations}; in Section~\\ref{sec:modelling} we describe the light curve modeling process; the physical interpretations of both events are given in Section~\\ref{sec:physics}; and finally in Section~\\ref{sec:discussion} we discuss the interesting findings from these two events, and implications to future space-based microlensing observations.\n\n\\begin{figure*}[!ht]\n\\epsscale{1.}\n\\centering\n\\plotone{ob1268-lc.eps}\n\\caption{Ground-based and \\emph{Spitzer} data and best-fit model light curves for OGLE-2015-BLG-1268. The inset shows details around the peak as seen from the ground, with best-fit model seen in the $I$ and $H$ bands shown with different colors. The gray solid line indicates the peak of the event as seen from the ground, and the two dashed vertical lines indicate the two contacts between the point-like lens and the finite-size source. Ground-based data points with uncertainties $>0.2$ mag are suppressed in the figure to avoid clutter, but are included in the modeling.\n\\label{fig:ob1268-lc}}\n\\end{figure*}\n\n\\section{Observations} \\label{sec:observations}\n\nA general description of the \\emph{Spitzer} observations and ground-based follow-up strategy for the 2015 \\emph{Spitzer} microlensing campaign can be found in \\citet{Yee:2015b} and \\citet{Street:2015}. In brief, events were selected for \\emph{Spitzer} observations, if 1) they showed or were likely to show significant sensitivity to planets, i.e., events with high peak magnifications \\citep{GriestSafizadeh:1998} and\/or events with intensive survey observations; and 2) could probably yield parallax measurement. Most \\emph{Spitzer} targets were assigned the default 1-per-day cadence, except that short or relatively high magnification events were given higher cadence. These observing protocols were submitted on Monday for observations roughly Thursday through Wednesday for each of the six weeks of the campaign. Ground-based follow-up observations were taken mostly on events that were not heavily monitored by surveys, or when surveys could not observe due to weather or Moon passage \\citep[e.g., OGLE-2015-BLG-0966,][]{Street:2015}. All ground-based data were reduced using standard algorithms (image subtraction\/DoPhot), and the \\emph{Spitzer} data were reduced using the new algorithm presented in \\citet{CalchiNovati:2015b}.\n\nBelow we give detailed descriptions of the observations for these two particular events.\n\n\\subsection{OGLE-2015-BLG-1268}\nThe microlensing event OGLE-2015-BLG-1268 was first alerted by the Optical Gravitational Lens Experiment (OGLE) collaboration through the Early Warning System (EWS) real-time event detection system \\citep{Udalski:1994,Udalski:2003} on 2015 June 6, based on observations with the 1.4 deg$^2$ camera on its 1.3m Warsaw Telescope at the Las Campanas Observatory in Chile. With equatorial coordinates (RA, Dec)$_{2000} = (17^{\\rm h}56^{\\rm m}49\\fs77,\\ -21\\arcdeg53\\arcmin57\\farcs6$) and Galactic coordinates $(l,b)_{2000}=(7\\fdg41,1\\fdg42)$, this event lies in the OGLE-IV field BLG642 \\citep{Udalski:2015b}, meaning that it is observed by OGLE with a cadence less than once per two nights, and it was not monitored by the MOA or KMTNet surveys.\n\nBecause of the very sparse observations from the survey team, follow-up teams including the Microlensing Follow-Up Network \\citep[$\\mu$FUN,][]{Gould:2010}, Microlensing Network for the Detection of Small Terrestrial Exoplanets \\citep[MiNDSTEp,][]{Dominik:2010} and RoboNet \\citep{Tsapras:2009} also observed this event fairly intensively during its peak as seen from Earth, in order to support the \\emph{Spitzer} program and maximize the sensitivity to potential planets \\citep{Yee:2015b,Zhu:2015b}. One remarkable feature about these follow-up observations is that the CTIO SMARTS telescope used by the $\\mu$FUN team could simultaneously obtain $H$ band images while $I$ band observations were taken \\citep{Depoy:2003}, and these $H$ band observations turned out to be important in characterizing the source star (see next section). Based on the observations taken by 2015 June 16, A.G. issued an alert to the community that this event was deviating from the point-lens point-source model. V.B. later on pointed out that this deviation could be fit by a point-lens finite-source model. \n\nEvent OGLE-2015-BLG-1268 was selected for \\emph{Spitzer} observations on 2015 June 14, i.e., $\\sim2$ days before it reached its peak magnification as seen from Earth. It was classified as a ``subjective'' event, because it by then did not (in fact, never could) meet the objective criteria set by \\citet{Yee:2015b}. It was selected primarily because it was going to peak with relatively high magnification in the coming week. With a predicted peak magnification $A_{\\rm max,\\oplus}>6$ (1-$\\sigma$ lower limit) by the time of selection, this event qualified for bonus observations \\citep{Yee:2015b}, and so was assigned 4-per-day cadence for its first week of observations. Starting from its second week, the \\emph{Spitzer} cadence went down to the default value (1-per-day), and this cadence continued for two weeks until it met the criteria set by \\citet{Yee:2015b} for stopping observations. In total, \\emph{Spitzer} observed this event 48 times, each with 6 dithered 30s exposures.\n\nThe data for this event are shown in Figure~\\ref{fig:ob1268-lc}. All data sets have been aligned to the OGLE-IV $I$ magnitude according to the best-fit model given in Section~\\ref{sec:modelling-1268}.\n\n\\begin{deluxetable}{lc\n\\centering\n\\tablecaption{Best-fit parameters of OGLE-2015-BLG-1268.}\n\\tablewidth{0pt}\n\\tablehead{Parameters & Values}\n\\startdata\n\\input{ob1268-parms.dat}\n\\enddata\n\\tablecomments{$^a$ Blending fraction in the OGLE-IV $I$ band.}\n\\label{tab:ob1268}\n\\end{deluxetable}\n\n\\subsection{OGLE-2015-BLG-0763}\nThis event was first alerted by the OGLE collaboration on 2015 April 22. With equatorial coordinates (RA, Dec)$_{2000} = (17^{\\rm h}32^{\\rm m}23\\fs41,\\ -29\\arcdeg18\\arcmin09\\farcs7$) and Galactic coordinates $(l,b)_{2000}=(-1\\fdg78,2\\fdg25)$, this event lies in the OGLE-IV field BLG613 \\citep{Udalski:2015b} and is observed by OGLE with once-per-night cadence. It also lies within the region where the new microlensing survey, Korean Microlensing Telescope Network \\citep[KMTNet,][]{Kim:2016} obtained 1-2 observations per day from each of their three 1.6m telescopes as support to the \\emph{Spitzer} campaign. Thus, it is also labeled by the KMTNet team as KMT-2015-BLG-0187.\n\n\\begin{figure*}[!ht]\n\\epsscale{1.}\n\\centering\n\\plotone{ob0763-lc.eps}\n\\caption{Ground-based and \\emph{Spitzer} data and best-fit model light curves for OGLE-2015-BLG-0763. The finite source effect is only seen in the \\emph{Spitzer} data. Ground-based data points with uncertainties $>0.2$ mag are suppressed in the figure to avoid clutter, but are included in the modeling.\n\\label{fig:ob0763-lc}}\n\\end{figure*}\n\nEvents such as this, with OGLE-IV cadence of 1-per-day and lying in non-prime KMTNet fields, are treated by \\citet{Yee:2015b} as having low survey coverage in all respects. That is, they are not eligible for objective selection, and once they are selected subjectively, follow-up observations are strongly encouraged. This event hence received a fairly large amount of observations from $\\mu$FUN, RoboNet, and MiNDSTEp, besides the survey observations, although none of these ground-based data showed deviation from the standard point source model.\n\nEvent OGLE-2015-BLG-0763 was selected by the \\emph{Spitzer} team on 2015 May 19, when the Bulge could not yet be seen from \\emph{Spitzer} due to its Sun-angle limit. Nevertheless, at that time it was selected as a future \\emph{Spitzer} target in order to achieve high sensitivity to planets. As specified in \\citet{Yee:2015b} and enacted in \\citet{Zhu:2015b}, when measuring planet sensitivity or detecting planets, only the portion of the light curve taken after the subjective selection of an event may be considered \n\\footnote{Unless it is later found to meet objective selection criteria.},\nin order to isolate any knowledge of the existence of potential planet from the decision making; this is necessary to maintain the objectivity of the sample.\nThe Bulge became accessible to \\emph{Spitzer} starting from June 6 (HJD$'\\equiv$ HJD-2450000=7180). Although this event was only assigned 1-per-day cadence, its position, relatively far to the west, made it one of the few targets that could be observed by \\emph{Spitzer} in the beginning of the Bulge window, and thus it received 8-per-day cadence in the first week. The cadence dropped to 2-per-day as more targets became accessible to \\emph{Spitzer} in the second week. Beginning the third week all targets became accessible to \\emph{Spitzer}, and we devoted the extra time to relatively high magnification events \\citep[see][for detailed description]{Street:2015}. Then the cadence for OGLE-2015-BLG-0763 was increased to 8-per-day this week due to its high peak magnification as seen from Earth. The cadence went back to the default one (1-per-day) in the following weeks until it moved out of \\emph{Spitzer} window on HJD$'=7218$. In total, 142 \\emph{Spitzer} observations were obtained.\n\nThe \\emph{Spitzer} data for the six epochs nearest the peak of OGLE-2015-BLG-0763 are affected by saturation and\/or non-linearity. This is quite apparent from the online data presented by \\citet{CalchiNovati:2015b}.  For this event, we have therefore re-reduced the data by taking into account the specific characteristics of channel 1 (at 3.6 $\\mu$m) of the IRAC camera.  We treat all pixels with counts greater than 70\\% of the allowed maximum as ``corrupted'', i.e., as saturated and\/or non-linear.  In addition, all pixels that are interior to corrupted pixels are also treated as corrupted, since the saturation of these pixels can falsely lead to ``normal'' flux counts. This leads to masking seven to nine pixels for each of these six dithered images of the epoch closest to peak and to masking up to seven pixels for each of the six dithered images for the remaining 5 affected epochs.  The remaining pixels (i.e., those below 70\\% of full well and not interior to corrupted pixels) are treated as normal, since they are essentially unaffected by the non-linearities in neighboring pixels.  After the masking of the inner saturated pixels, the fit is therefore carried out in standard fashion.  We note that our 70\\% threshold is conservative with respect to the 90\\% value in IRAC documentation, and was determined by us empirically from inspection of the raw data and by studying the stability of the photometry PRF fitting solution.\n\\footnote{We have tested that the results remain the same even though higher threshold values (up to 90\\%) are used.}\n\nThe data for this event are shown in Figure~\\ref{fig:ob0763-lc}. Again, all data sets have been aligned to the OGLE-IV $I$ magnitude according to the best-fit model given in Section~\\ref{sec:modelling-0763}.\n\n\\begin{deluxetable}{lcc\n\\centering\n\\tablecaption{Best-fit parameters of OGLE-2015-BLG-0763. The other two possible solutions, $(+,+)$ and $(-,-)$, are disfavored by $\\Delta \\chi^2=15$, and have $\\rho \\le 0.01$ (2-$\\sigma$ limit) that is inconsistent with the blend constraint (see Section~\\ref{sec:physics}).}\n\\tablewidth{0pt}\n\\tablehead{Parameters & $(+,-)$ & $(-,+)$}\n\\startdata\n\\input{ob0763-parms.dat}\n\\enddata\n\\tablecomments{$^a$ Blending fraction in the OGLE-IV $I$ band.}\n\\label{tab:ob0763}\n\\end{deluxetable}\n\n\n\\section{Light Curve Analyses} \\label{sec:modelling}\n\n\\subsection{Modeling Process}\nFor each event, we perform a Markov Chain Monte Carlo (MCMC) analysis to find the best-fit parameters and the associated probability density functions using the \\texttt{emcee} ensemble sampler \\citep{ForemanMackey:2013}. The formalism to incorporate the finite-source effect in the single-lens case is given in \\citet{Yoo:2004}. For each microlensing event, there will be six event parameters: time of maximum magnification as seen from Earth $t_0$, the impact parameter in the absence of parallax as seen from Earth $u_0$, the event timescale $t_{\\rm E}$, the source radius scaled to the angular Einstein radius $\\rho$, and the two components of microlens parallax along the north and east direction $\\pi_{\\rm E,N}$ and $\\pi_{\\rm E,E}$. For each data set on that event, there are two flux parameters ($f_s$, $f_b$). In addition, we have two limb-darkening coefficients ($\\tilde{q}_{1,\\lambda}$, $\\tilde{q}_{2,\\lambda}$) described in the next section. If these are also set free, there are two additional parameters for each bandpass.\n\nFor each event that is investigated below, we report the six event parameters as well as the blending fraction in the OGLE-IV $I$ band. This blending fraction, defined as $f_b\/(f_s+f_b)$, can be used to distinguish between degenerate models, as we will see below.\n\n\\subsection{Limb Darkening Effect}\nFor the present cases, especially OGLE-2015-BLG-1268, many of our observations were taken when the alignment between the source and the lens was closer than roughly the scaled source size $\\rho$, so the generally adopted linear limb-darkening law is not adequate. Therefore, we choose the two-parameter square root limb darkening law\n\\begin{equation} \\label{eq:sqrtld}\nS_\\lambda (\\mu) = \\bar{S}_\\lambda \\left[ 1-\\Gamma_\\lambda \\left(1-\\frac{3}{2}\\mu\\right) - \\Lambda_\\lambda \\left(1-\\frac{5}{4}\\sqrt{\\mu}\\right) \\right]\\ ,\n\\end{equation}\nin which $\\mu$ is the cosine of the angle between the line of sight and the emergent intensity, and $\\Gamma_\\lambda$ and $\\Lambda_\\lambda$ are the limb-darkening coefficients at wavelength $\\lambda$. Because of our requirement that the total flux, $F_{\\rm tot,\\lambda}=\\pi \\theta_\\star^2 \\bar{S}_\\lambda$, should be conserved for a given source angular size $\\theta_\\star$, Equation~(\\ref{eq:sqrtld}) differs from the standard square root limb-darkening law described by $(c_\\lambda,d_\\lambda)$ in \\citet{DiazCordoves:1992}. The transformations between limb-darkening coefficients ($\\Gamma_\\lambda,\\Lambda_\\lambda$) and ($c_\\lambda,d_\\lambda$) can be found in \\citet{Fields:2003}. \n\nIn cases in which very little knowledge of the source star can be obtained otherwise, one may want to fit for ($\\Gamma_\\lambda,\\Lambda_\\lambda$). However, simply leaving ($\\Gamma_\\lambda,\\Lambda_\\lambda$) free will very likely lead to unphysical stellar brightness profiles. Therefore, we reparameterize the limb-darkening coefficients ($\\Gamma_\\lambda,\\Lambda_\\lambda$) using the method proposed by \\citet{Kipping:2013},\n\\begin{equation} \\label{eq:q1}\n\\tilde{q}_{1,\\lambda} \\equiv \\left(\\Gamma_\\lambda + \\Lambda_\\lambda\\right)^2\\ ,\n\\end{equation}\n\\begin{equation} \\label{eq:q2}\n\\tilde{q}_{2,\\lambda} \\equiv \\frac{7\\Lambda_\\lambda}{12(\\Gamma_\\lambda+\\Lambda_\\lambda)}\\ .\n\\end{equation}\nThe inverse transformation is given by\n\\begin{equation}\n\\Gamma_\\lambda = \\sqrt{\\tilde{q}_{1,\\lambda}} \\left(1-\\frac{12}{7}\\tilde{q}_{2,\\lambda}\\right)\\ ,\n\\end{equation}\n\\begin{equation}\n\\Lambda_\\lambda = \\frac{12}{7} \\sqrt{\\tilde{q}_{1,\\lambda}}\\tilde{q}_{2,\\lambda}\\ .\n\\end{equation}\nAs \\citet{Kipping:2013} has shown, uniform samplings in the interval [0, 1] for both $\\tilde{q}_{1,\\lambda}$ and $\\tilde{q}_{2,\\lambda}$ can ensure physically meaningful stellar intensity profiles \n\\footnote{Positive intensity everywhere, and monotonically decreasing intensity from the center to the limb.}\nas well as being efficient.\n\\footnote{Note that ($\\tilde{q}_{1,\\lambda}$, $\\tilde{q}_{2,\\lambda}$) given by Equations~(\\ref{eq:q1}) and (\\ref{eq:q2}) is not the only parameterization to implement the idea proposed by \\citet{Kipping:2013}.}\n\n\\subsection{OGLE-2015-BLG-1268} \\label{sec:modelling-1268}\n\nIn the case of OGLE-2015-BLG-1268, because we were unable to fully understand the source star with only photometric data (see Section~\\ref{sec:physics-1268}), we also fit for the limb-darkening coefficients. Since only the $I$ and $H$ band data were relevant when the finite-source effect is prominent, we only set the limb-darkening coefficients of these two bands free. Furthermore, since the \\emph{Spitzer} data only captured the falling tail of the light curve, a constraint on the source flux in the \\emph{Spitzer} 3.6 $\\mu$m band ($L_{\\rm spitzer}$) is necessary in order to better measure the parallax \\citep{CalchiNovati:2015a,CalchiNovati:2015b}. This is done by conducting a color-color regression between $I-H$ and $I-L_{\\rm spitzer}$ using stars with color and magnitude similar to the clump.\n\nOur best-fit model to this event is shown in Figure~\\ref{fig:ob1268-lc}, and best-fit parameters with 1-$\\sigma$ intervals are given in Table~\\ref{tab:ob1268}. Note that in Figure~\\ref{fig:ob1268-lc}, the $H$ band model shows deviations from the $I$ band model, which are obvious on the top and still noticeable at the limb. Characterizing events via such deviations for point-lens \\citep{GouldWelch:1996} and planetary \\citep{GaudiGould:1997} events was the original motivation for building the dichroic ANDICAM camera \\citep{Depoy:2003}.\nWe show in Figure~\\ref{fig:ob1268-ldcs} the $\\Delta \\chi^2=1$, 4 and 9 contours between $(\\tilde{q}_{1,\\lambda},\\ \\tilde{q}_{2,\\lambda})$ on the top of the theoretical values from \\citet{Claret:2000}. The constraint on the $I$ band limb-darkening coefficients $(\\tilde{q}_{1,I},\\ \\tilde{q}_{2,I})$ favors hot stars ($T_{\\rm eff}\\ge8000$ K) at 2-$\\sigma$ level, and this is consistent with the less constrained $H$ band limb-darkening coefficients.\n\nSince the impact parameter as seen from Earth is extremely small (indeed, consistent with zero) compared to the that seen from \\emph{Spitzer}, the generic four-fold degeneracy in single-lens events collapses to two-fold, i.e., to a degeneracy only in the direction of $\\bdv{\\pi}_{\\rm E}$ (and not its magnitude). However, this degeneracy in direction has no effect on the mass and distance measurement \\citep{GouldYee:2012}.\n\n\\begin{figure*}\n\\centering\n\\plotone{ob1268-ldcs.eps}\n\\caption{The $\\Delta \\chi^2=1$, 4 and 9 constraints (solid, dashed and dot-dashed curves, respectively) on the limb-darkening coefficients in the $I$ band (left) and $H$ band (right) in the case of OGLE-2015-BLG-1268. Open circles are theoretical values taken from \\citet{Claret:2000} after the transformation from $(c,d)$ to $(\\tilde{q}_1,\\tilde{q}_2)$, and they are color coded with the stellar effective temperature. The constraints on limb-darkening coefficients effectively rule out a source with $T_{\\rm eff}<8000$ K.\n\\label{fig:ob1268-ldcs}}\n\\end{figure*}\n\n\\subsection{OGLE-2015-BLG-0763} \\label{sec:modelling-0763}\nFor OGLE-2015-BLG-0763, we adopt the limb-darkening coefficients that correspond to the well characterized source (see Section~\\ref{sec:physics-0763}), so that only event parameters and flux parameters are free in the modeling. The best-fit model is shown in Figure~\\ref{fig:ob0763-lc}, and the best-fit parameters together with 1-$\\sigma$ intervals for the two adopted solutions, $(+,-)$ and $(-,+)$, are given in Table~\\ref{tab:ob0763}. Parameters of the other possible solutions, i.e., $(+,+)$ and $(-,-)$ solutions, are similar except that only an upper limit on the source size, $\\rho\\le0.01$ at $\\Delta \\chi^2=4$ or 2-$\\sigma$ level, can be obtained, meaning that the finite-source effect is not securely detected. Nevertheless these two solutions are rejected, partly because they are disfavored by $\\Delta \\chi^2=15$, and also because the derived physical parameters result in inconsistency with observations (see Section~\\ref{sec:physics-0763}).\n\n\nThe two adopted solutions have different parallax vector $\\bdv{\\pi}_{\\rm E}$ but similar amplitude $\\pi_{\\rm E}$, as is shown in Table~\\ref{tab:ob0763}. As in the previous case, for the purpose of mass and distance determination this means that the four-fold degeneracy is also broken here. We show in Section~\\ref{sec:discussion} that this is not coincidental, but is rather almost inevitable for mass measurements of single-lens events with space-based observations.\n\n\\begin{figure*}\n\\epsscale{1.}\n\\centering\n\\plottwo{ob1268-cmd.eps}{ob0763-cmd.eps}\n\\caption{The color-magnitude diagrams (CMD) used to characterize the source of OGLE-2015-BLG-1268 (left) and OGLE-2015-BLG-0763 (right). In both cases, the centroid of the red clump is indicated by a filled red square, and the position of the source is marked by a blue asterisk.\n\\label{fig:cmds}}\n\\end{figure*}\n\n\n\\section{Physical Interpretations} \\label{sec:physics}\n\n\\subsection{OGLE-2015-BLG-1268L: A Brown Dwarf in the Inner Disk} \\label{sec:physics-1268}\n\nThe extinction in the region where this event lies is so high that the routine OGLE-IV $V$ band images are not deep enough to resolve the red giant clump in the Bulge. In order to determine the color and magnitude of the source star following the standard routine \\citep{Yoo:2004}, we then construct an $I-H$ vs. $I$ color-magnitude diagram (CMD) in the following way: we obtain OGLE-IV $I$ band magnitudes for stars within $2'$ separation from the event, and cross-match to the VISTA Variables in the V\\'ia L\\'actea Survey \\citep[VVV,][]{Saito:2012} to obtain their $H$ band magnitudes. \nThe position of the source star on this CMD is found to be\n\\begin{equation}\n(I-H,\\ I)_s = (2.997\\pm0.003,\\ 19.54\\pm0.04)\\ ,\n\\end{equation}\nin which $I_s$ comes directly from the modeling and $H_s$ is determined by transforming the CTIO instrumental $H$-band source flux that is derived from the model to the standard VVV system using comparison stars. The centroid of the red clump on this CMD is at\n\\begin{equation}\n(I-H,\\ I)_{\\rm cl} = (4.00\\pm0.02,\\ 17.78\\pm0.05)\\ .\n\\end{equation}\nThe CMD for this event with the positions of the source and the clump centroid indicated is shown on the left panel of Figure~\\ref{fig:cmds}.\nComparing to the dereddened color and magnitude of the clump at this particular direction, $(I-H,\\ I)_{\\rm cl,0}=(1.29,\\ 14.23)$ \\citep{Bensby:2013,Nataf:2013}, we determine the extinction and reddening to be $A_I=3.6$ and $E(I-H)=2.7$, respectively. If all of this extinction applies to the source star, the dereddened source will have \n\\begin{equation}\n(I-H,\\ I)_{\\rm s,0}=(0.29\\pm0.02,15.99\\pm0.08)\\ .\n\\end{equation}\nand $M_I=1.65$. These values suggest the source star is a late A\/early F dwarf \\citep{BessellBrett:1988}. Such stars are rare in the Bulge \\citep{Bensby:2013,Poleski:2014}. In principle, one could not exclude the possibility that the source might not lie behind the same amount of dust as the clump does, so that an intrinsically redder and therefore dimmer but closer source star is also allowed by this CMD analysis. However, the limb-darkening coefficients from our light curve modeling (see Figure~\\ref{fig:ob1268-ldcs}) also suggest that the source star has $T_{\\rm eff}\\ge8000$ K when compared to the theoretical values given by \\citet{Claret:2000}. Therefore, we conclude that the source star is an A-type dwarf in the Bulge (or at least at a roughly similar distance as the Bulge).\n\nWe then use the dereddened source color and magnitude to derive the angular radius of the source star. \\citet{Boyajian:2014} provide the following relation to estimate the stellar angular diameter based on the dereddened $I-H$ color and $I$ magnitude\n\\begin{equation} \\label{eq:thetas}\n\\log{d_{\\star}} = a_0 + a_1 (I-H) - 0.2I\\ ,\n\\end{equation}\nwith $a_0=0.53026\\pm0.00077$, $a_1=0.36595\\pm0.00079$. Adopting the above dereddened color and magnitude of the source, we find that the source angular radius is\n\\begin{equation}\n\\theta_\\star = (1.37\\pm0.10)\\ \\mu{\\rm as}\\ .\n\\end{equation}\nThe uncertainty here is dominated by the $7.4\\%$ scatter in the color-surface brightness relation (Equation~\\ref{eq:thetas}).\n\nWhen combined with the source radius $\\rho=0.0108\\pm0.0005$, it yields $\\theta_{\\rm E}=0.127\\pm0.009$ mas. Given the microlens parallax measurement $\\pi_{\\rm E}=0.35\\pm0.04$\n\\footnote{Note that the uncertainty on $\\pi_{\\rm E}$ is smaller than the uncertainty on either $\\pi_{\\rm E,N}$ or $\\pi_{\\rm E,E}$ (see Table~\\ref{tab:ob1268}) because of the anti-correlation between $\\pi_{\\rm E,N}$ and $\\pi_{\\rm E,E}$. This anti-correlation emerges due to the constraint on the source flux in $L_{\\rm spitzer}$ band. In particular, \\citet{GouldYee:2012} showed that for high-magnification events (as seen from Earth), $\\pi_{\\rm E}$ can be determined from a single space-based observation taken at the ground-based peak (plus one additional late-time observation), provided the space-based source flux is known. In this case, there would be perfect knowledge of the magnitude of $\\bdv{\\pi}_{\\rm E}$ and perfect ignorance of its direction. While the observations of OGLE-2015-BLG-1268 do not satisfy this condition (see Figure~\\ref{fig:ob1268-lc}), essentially the same logic applies.}\nWe then find the lens mass and the lens-source relative parallax\n\\begin{equation}\nM_{\\rm L} = (0.045\\pm0.007)\\ M_\\odot;\n\\quad \\pi_{\\rm rel} = (0.044 \\pm 0.006)\\ {\\rm mas}\\ ,\n\\end{equation}\nand then (adopting $D_{\\rm S}=8\\pm1$ kpc), a lens distance\n\\begin{equation}\nD_{\\rm L} = (5.9\\pm1.0)\\ {\\rm kpc}\\ .\n\\end{equation}\nThat is, a brown dwarf (BD) in the inner Galactic disk. \n\nThe lens-source relative proper motion, $\\mu_{\\rm rel}=2.65\\pm0.20~{\\rm mas~yr^{-1}}$, is more typical of Bulge lenses ($\\mu_{\\rm rel}\\sim4~{\\rm mas~yr^{-1}}$) than disk lenses ($\\mu_{\\rm rel}\\sim7~{\\rm mas~yr^{-1}}$). However, this value is certainly not inconsistent with a disk lens for two reasons. First, the higher proper motions typical of disk lenses come from the fact that the Sun and the disk lens partake of the same relatively flat Galactic rotation curve, while Bulge sources are on average not moving in the frame of the Galaxy. Hence, disk lenses tend to have similar proper motions as SgrA*. However, in the present case, the source may also be in the disk as indicated by the fact that it is likely to be an A star. If so, it would partake of the same flat rotation curve, which then naturally leads to a low proper motion. Second, one of the two solutions (with $\\pi_{\\rm E,N}>0$) is approximately in the direction of Galactic rotation, so that the Bulge source would have to be moving at just $\\mu_{\\rm S}\\sim 4~{\\rm mas~yr^{-1}}$ in the direction of Galactic rotation (i.e., $1.3~\\sigma$) to account for the observed relative proper motion.\n\nThe Galactic coordinates of the event, $(l,b)=(7\\fdg41,1\\fdg42)$, provide a further indication that the source may be in the disk. That is, even if the source were at roughly the Galactocentric distance, it would lie just $\\sim200$~pc from the plane, which is significantly populated by late A\/early F stars. Moreover, at this longitude, disk stars account for a significantly larger fraction of stars at these distances than is the case for typical microlensing fields near $l\\sim0$. Thus, this possibility must be taken seriously.\n\nNow, if the source were in the disk, this would not in any way affect either $M_{\\rm L}$ or $\\pi_{\\rm rel}$, provided that the source lies behind the same dust column as the clump. This is because the estimate of $\\theta_\\star$ (and so $\\theta_{\\rm E}$) depends only on the dereddened color and magnitude of the source and not its distance. However, given the various evidences favoring a disk source, as well as the low Galactic latitude of the event, we must consider the possibility that the source does in fact lie in front of at least some of the dust.\n\nHence, we develop a more robust way to estimate the lens mass that does not require the source lying behind all the dust. According to the color-surface brightness relation (Equation~\\ref{eq:thetas}), one can write the angular source radius $\\theta_\\star$ (and further $M_{\\rm L}$) in terms of the dereddened source color $(I-H)_{\\rm s,0}$ and magnitude $I_{\\rm s,0}$. If the reddening law derived from the red clump, $R_{IH}=A_I\/E(I-H)=1.33$, also applies to the source star, we can express $(I-H)_{\\rm s,0}$ and $I_{\\rm s,0}$ in terms of $(I-H)_{\\rm s}$, $I_{\\rm s}$, and a single parameter $\\eta$,\n\\begin{equation} \\label{eq:eta}\nI_{\\rm s,0} = I_{\\rm s,0,naive} + \\eta A_I\\ ,\\quad\n(I-H)_{\\rm s,0} = (I-H)_{\\rm s,0,naive} + \\eta E(I-H)\\ .\n\\end{equation}\nHere $\\eta$ is the fraction of total extinction of dust lying behind the source (and in front of the clump), while $I_{\\rm s,0,naive}$ and $(I-H)_{\\rm s,0,naive}$ are the values we derived above under the naive assumption that $\\eta=0$. Combining Equations~(\\ref{eq:thetas}) and (\\ref{eq:eta}) with the observed $R_{IH}=1.33$ and $E(I-H)=2.7$, one finds that as one varies $\\eta$,~$\\theta_\\star$ changes by\n\\begin{equation} \\label{eq:dthetas}\n\\Delta \\log\\theta_\\star = \\eta E(I-H) (a_1-0.2 R_{IH}) = 0.27\\eta\\ .\n\\end{equation}\nThus in particular, our conclusion that the best estimate for the lens mass is in the brown dwarf regime ($M<0.08M_\\odot$) rests on the assumption that $\\eta<\\log(0.08\/0.045)\/0.27=0.86$. This is very likely simply because nearby sources are extremely rare in microlensing due to low optical depth. Nevertheless, it would be of interest to definitively resolve this issue. This could be done simply by identifying the stellar type of the source via a low-resolution spectrum. This would remove the uncertainty due to the location of the source with respect to the dust. It would also permit one to determine $\\theta_\\star$ without appealing to the detailed modeling of limb-darkening coefficients from the light curve (Section~\\ref{sec:modelling-1268}).\n\n\n\n\\subsection{OGLE-2015-BLG-0763L: An M-type Main-Sequence Star in the Inner Disk} \\label{sec:physics-0763}\nWe first characterize the source of OGLE-2015-BLG-0763 using the method of \\citet{Yoo:2004}. Combining the light curve modeling and a linear regression of OGLE-IV $V$ band on $I$ band flux during the event yield\n$(V-I,\\ I)_{\\rm s} = (3.25,\\ 17.27)$. The clump centroid in this $V-I$ vs. $I$ CMD (see the right panel of Figure~\\ref{fig:cmds}) is found at $(V-I,\\ I)_{\\rm cl} = (3.00,\\ 16.89)$. After the correction of OGLE-IV non-standard $V$ band \\citep{Zhu:2015a}, and combining with the dereddened color \\citep{Bensby:2013} and magnitude \\citep{Nataf:2013} of the clump at this direction, we derive the color and magnitude of the dereddened source to be $(V-I,\\ I)_{\\rm s,0} = (1.29,\\ 14.78)$. Thus, the source is a giant star in the Bulge. We then convert from $V-I$ to $V-K$ using the empirical color-color relations of \\citet{BessellBrett:1988}, apply the color-surface brightness relation of \\citet{Kervella:2004}, and find the source angular radius\n\\begin{equation}\n\\theta_\\star = 6.3\\pm0.4\\ \\mu{\\rm as}\\ .\n\\end{equation}\nThe uncertainty here has taken into account the uncertainty in centroiding the clump $I$-band magnitude in the CMD ($\\sim0.05$ mag) and the derivation of the intrinsic source color \\citep[0.05 mag,][]{Bensby:2013}.\n\nGiven the source size $\\rho=0.0218\\pm0.0007$ from light curve modeling,\n\\footnote{Here we combine the results of $(+,-)$ and $(-,+)$ solutions, and the uncertainty takes into account the difference between both solutions and the uncertainty of each solution.}\nthis indicates an angular Einstein radius $\\theta_{\\rm E}=0.288\\pm0.020$ mas. We then combine this with the microlens parallax parameter $\\pi_{\\rm E} = 0.0709\\pm0.0010$ \nand find the lens mass and distance\n\\begin{equation}\nM_{\\rm L} = 0.50\\pm0.04\\ M_\\odot\\ ,\\quad D_{\\rm L} = 6.9\\pm1.0\\ {\\rm kpc}\\ .\n\\end{equation}\nIn deriving $D_{\\rm L}$ we have assumed a Bulge source ($D_{\\rm S}=8\\pm1$ kpc), which is very likely the case given the location of this event. Given the timescale $t_{\\rm E}=32.9\\pm0.3$ days, the lens-source relative proper motion is $\\mu_{\\rm rel}=3.20\\pm0.25$ mas\/yr, and it is almost due the north\/south direction ($\\pi_{\\rm E,E}\\ll |\\pi_{\\rm E,N}|$).\n\nAs is mentioned in Section~\\ref{sec:modelling-0763}, the other two solutions $(+,+)$ and $(-,-)$ are disfavored by $\\Delta\\chi^2=15$ and only provide upper limit of $0.01$ on $\\rho$. Regardless of the $\\chi^2$ argument, if this set of solutions were correct, the lens would have mass $1.0\\ M_\\odot$ and distance 6 kpc. First of all, more massive stars are intrinsically rarer. Furthermore, a 1 $M_\\odot$ main-sequence star would be $12\\%$ as bright as the source star, and therefore contribute 11\\% of the total baseline flux. This contradicts the fact that slightly negative blending is detected in this event.\n\\footnote{The adopted solution, a 0.5 $M_\\odot$ star at 6.9 kpc, cannot produce a negative blending either, but this solution is much more plausible given the fact that various effects can lead to slightly negative blending \\citep{Smith:2007}.}\nTherefore, these two solutions are discarded.\n\n\\section{Discussion} \\label{sec:discussion}\n\nThe mass determination of isolated microlenses requires measurements of at least two of the three important microlensing parameters: the angular Einstein radius $\\theta_{\\rm E}$, the microlens parallax $\\pi_{\\rm E}$, and the lens flux. Among them, measuring the lens flux requires the lens to be luminous, so that it will not work for faint or dark lenses, although it may see broad applications in future space-based microlensing experiments \\citep{Yee:2015c}. Therefore, measuring $\\theta_{\\rm E}$ and $\\pi_{\\rm E}$ together is the only method that works for all types of objects.\n\nIn this work, we report on the mass and distance measurements of two single-lens events from our 2015 \\emph{Spitzer} microlensing program. The finite-source effect is detected in the ground-based data in OGLE-2015-BLG-1268, and in the \\emph{Spitzer} data in OGLE-2015-BLG-0763, ensuring measurement of the angular Einstein radius $\\theta_{\\rm E}$. The microlens parallax parameter $\\pi_{\\rm E}$ is measured from a combined fit to the ground- and space-based data. We find that the lens of OGLE-2015-BLG-1268 is very likely a brown dwarf, and probably a 45 $M_{\\rm J}$ brown dwarf at 5.9 kpc if the source star is inside the galactic Bulge. The lens of OGLE-2015-BLG-0763 is a 0.5 $M_\\odot$ star at 6.9 kpc.\n\nThe result that at least two \n\\footnote{The single-lens event OGLE-2015-BLG-1482 also shows finite-source effect as seen from \\emph{Spitzer} (Chung et al., in prep).}\nsingle-lens events out of the 170 monitored events in the 2015 \\emph{Spitzer} program yield mass measurements was unexpected, but turns out to be reasonable.\nTo measure the mass of isolated microlenses by combining $\\theta_{\\rm E}$ and $t_{\\rm E}$, one needs to detect the finite-source effect and measure the microlens parallax. The first is approximately a geometric probability for the lens to ``transit'' the source star, which is of order $\\rho$\n\\footnote{The observational bias tends to enhance the probability of detecting such finite-source effects in single-lens events that are caused by faint dwarf source stars. See \\citet{ZhuGould:2016} for more discussions.}    \n, and the second, in the case of two-observatory parallax method \\citep{Refsdal:1966,Gould:1994b,Gould:1997}, is limited mostly by the projected separation between the two observatories $D_\\perp$. In order to measure the parallax, the two observatories should both be able to detect the same event, but should each see (slightly) different light curves. In the presence of the finite-source effect, this provides the following constraints on the projected Einstein radius $\\tilde{r}_{\\rm E}\\equiv{\\rm AU}\/\\pi_{\\rm E}$,\n\\begin{equation}\n    D_\\perp \\lesssim \\tilde{r}_{\\rm E} \\lesssim \\frac{50 D_\\perp}{\\rho}\\ .\n\\end{equation}\nwhere in the second inequality we have assumed typical photometric uncertainty ($2\\%$) during peak \\citep{GouldYee:2013}. For a lens with mass $M_{\\rm L}$, the above inequalities then provide upper and lower limit on the lens distance for which microlens parallax is detectable (assuming Bulge sources)\n\\begin{equation} \\label{eqn:pirellimits}\n    \\frac{\\rm AU}{50 D_\\perp}\\theta_\\star \\lesssim \\pi_{\\rm rel} \\lesssim \\kappa M_{\\rm L} \\left(\\frac{\\rm AU}{D_\\perp}\\right)^2\\ .\n\\end{equation}\nFor terrestrial parallax ($D_\\perp\\sim R_\\oplus$), the upper constraint on $\\pi_{\\rm rel}$ is almost always satisfied unless the lens is extremely low-mass or extremely nearby. The combined constraint thus suggests that only relatively nearby ($D_{\\rm L}\\lesssim 2.5$~kpc) lenses can be probed by terrestrial parallax, and since the lensing probability of such stars is quite low, the overall probability to measure isolated lens masses through this channel is extremely low. This has been shown in \\citet{GouldYee:2013}, in which they found an event rate $\\sim$1.6 Gyr$^{-1}$.\nObservationally, in the era prior to the \\emph{Spitzer} microlensing campaign, only two single-lens microlensing events yielded mass measurements \\citep[OGLE-2007-BLG-224 \\& OGLE-2008-BLG-279,][]{Gould:2009,Yee:2009} in this way, although more than $10,000$ events were discovered and at least four thousand of them were monitored as intensively as \\emph{Spitzer} events.\n\\footnote{For example, among $\\sim$10,000 microlensing events that were detected by OGLE-IV by 2015, more than 4,000 fall in the OGLE-IV high-cadence ($>10$ per night) fields. Those that reached high magnifications were followed up more intensively for the purpose of detecting planets \\citep{GriestSafizadeh:1998}, and thus any deviation from the standard point-lens point-source microlensing light curve, such as the finite-source effect and terrestrial parallax, would be efficiently detected and further studied for the purpose of detecting planets.}\n\nEquation~(\\ref{eqn:pirellimits}) also shows that in a \\textit{Spitzer}-like microlensing program ($D_\\perp \\sim 1~$AU), the parallax parameter is measurable for the majority of microlensing events. This has been proved by the 2014 \\& 2015 \\textit{Spitzer} microlensing programs \\citep{CalchiNovati:2015a,CalchiNovati:2015b}. In addition, the lens-source relative trajectory is most often very different as seen from Earth and from the satellite. This means that the probability to detect the finite-source effect in single-lens events is almost doubled. Thus, the overall probability to make mass measurements of isolated microlenses for \\textit{Spitzer} is $2\\langle\\rho\\rangle\/u_{0,\\rm max}$, where $\\langle\\rho\\rangle\\sim0.005$ is the average source size normalized to the Einstein radius for \\textit{Spitzer} events, and $u_{0,\\rm max}\\approx0.3$ is set as one of the criteria for selecting events \\citep{Yee:2015b}. This theoretical estimate (3.3\\%) agrees with the apparent frequency of such events ($3\/170=1.8\\%$) reasonably well, with the factor of $\\sim$2 difference likely due to the incomplete coverage of \\textit{Spitzer} data in some events. Hence, space-based microlensing experiments such as \\textit{Spitzer} and \\textit{Kepler} \\citep{GouldHorne:2013,Henderson:2016} can significantly increase the probability to measure the mass of isolated microlenses, besides their primary goal of detections and statistical studies of planets. Furthermore, because the impact parameter of the observatory that shows the finite-source effect must be close to zero, the generic four-fold degeneracy in single lens cases is effectively broken \\citep{GouldYee:2012}. \n\nMeasuring the mass of isolated microlenses is of great scientific interest. \\citet{Gould:2000} estimated that $\\sim20\\%$ of all Galactic microlensing events are caused by stellar remnants, and specifically that $\\sim1\\%$ are due to stellar mass black holes (BHs). In addition, there exists a population of probably unbound planet-mass objects that are almost twice as common as main-sequence stars \\citep{Sumi:2011}. These isolated stellar remnants and free floating planets (FFPs) can only be probed by microlensing, and our work shows that space-based observations can significantly enhance the probability to definitely measure their masses. \n\nThe future of detecting dark or extremely faint isolated objects via microlensing is promising. In addition to \\emph{Spitzer}, the re-proposed \\emph{Kepler} mission \\citep[\\emph{K2},][]{Howell:2014} is going to continuously monitor a 4 deg$^2$ microlensing fields for $\\sim70$ days in its campaign 9, which provides so far a unique chance to probe especially the FFP population \\citep{GouldHorne:2013,Henderson:2016}. Future space-based microlensing missions such as the Wide-Field InfraRed Survey Telescope \\citep[\\textit{WFIRST,}][]{Spergel:2015} and possibly Euclid \\citep{Penny:2013}, once combined with a simultaneous ground-based telescope network, will significantly increase the sensitivities to especially FFPs with even lower masses \\citep{ZhuGould:2016}.\n\n\n\\acknowledgements\nWork by WZ, SCN and AG was supported by JPL grant 1500811. WZ and AG also acknowledge support by NSF grant AST-1516842.\nWork by JCY was performed under contract with the California Institute of Technology (Caltech)\/Jet Propulsion Laboratory (JPL) funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute.\nThe \\emph{Spitzer} Team thanks Christopher S.\\ Kochanek for graciously trading us his allocated observing time on the CTIO 1.3m during the \\emph{Spitzer} campaign.\nThe OGLE project has received funding from the National Science Centre, Poland, grant MAESTRO 2014\/14\/A\/ST9\/00121 to AU. \nThis work is based in part on observations made with the {\\it Spitzer} Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.\nWork by C.H. was supported by Creative Research Initiative Program (2009-0081561) of National Research Foundation of Korea. This research has made the use of the KMTNet telescopes operated by the Korea Astronomy and Space Science Institute (KASI), and was supported by KASI grant 2016-1-832-01.\nThis work makes use of observations from the LCOGT network, which includes three SUPAscopes owned by the University of St Andrews. The RoboNet programme is an LCOGT Key Project using time allocations from the University of St Andrews, LCOGT and the University of Heidelberg together with time on the Liverpool Telescope through the Science and Technology Facilities Council (STFC), UK. This research has made use of the LCOGT Archive, which is operated by the California Institute of Technology, under contract with the Las Cumbres Observatory.\nKH acknowledges support from STFC grant ST\/M001296\/1.\nThis project is partly supported by the Strategic Priority Research Program ``The Emergence of Cosmological Structures'' of the Chinese Academy of Sciences, Grant No. XDB09000000 (SM and SD).\nM.P.G.H. acknowledges support from the Villum Foundation. \nN.P. acknowledges funding by the Gemini-Conicyt Fund, allocated to the project No. 32120036. \nBased on data collected by MiNDSTEp with the Danish 1.54 m telescope at the ESO La Silla observatory. Operation of the Danish 1.54m telescope at ESO's La Silla observatory was supported by The Danish Council for Independent Research, Natural Sciences, and by Centre for Star and Planet Formation. The MiNDSTEp monitoring campaign is powered by ARTEMiS \\citep[Automated Terrestrial Exoplanet Microlensing Search,][]{Dominik:2008}. D.M.B. acknowledges support from NPRP grant \\# X-019-1-006 from the Qatar National Research Fund (a member of Qatar Foundation). GD acknowledges Regione Campania for support from POR-FSE Campania 2014-2020. O.W. and J.S. acknowledge support from the Communaut\\'e francaise de Belgique $\u2013$ Actions de recherche concert\\'ees $\u2013$ Acad\\'emie universitaire Wallonie-Europe.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{section:introduction}\n\n\n\\subsection{Background and summary}\nLet $G$ be a complex, connected, reductive, algebraic group with Lie algebra $\\mathfrak{g}$. Given a nilpotent element $x\\in\\mathfrak{g}$, the associated (algebraic) Springer fiber $\\mathcal B_G^x$ is defined as the variety of all Borel subgroups of $G$ whose Lie algebra contains $x$. Let $A^x_G$ be the component group associated with $x$, i.e.\\ the quotient of the centralizer of $x$ in $G$ by its identity component. In seminal work \\cite{Spr76}, \\cite{Spr78}, Springer constructed a grading-preserving action of the Weyl group $\\mathcal W_G$ associated with $G$ and a commuting action of $A^x_G$ on the (complex) cohomology $H^*(\\mathcal B_G^x)$ of the Springer fiber. According to the Springer correspondence \\cite[Theorem 6.10]{Spr76} the isotypic subspaces of the $A^x_G$-module $H^\\mathrm{top}(\\mathcal B_G^x)$ are irreducible $\\mathcal W_G$-modules and all irreducible $\\mathcal W_G$-modules can be constructed in this way. This yields thus a geometric construction of the irreducible representations of $\\mathcal W_G$. \n\nThis article connects classical Springer theory for types $C$ and $D$ with the cup diagram combinatorics of arc algebras of types $B$ and $D$. The latter provide combinatorial descriptions of interesting categories arising in the study of the representation theory of orthogonal Lie algebras~\\cite{ES15}, non-semisimple Brauer algebras~\\cite{ES16}, \\cite{CDVMI}, and orthosymplectic Lie superalgebras~\\cite{ES16super}, \\cite{CH}. In type $A$, such a description was established in work of Russell and Tymoczko, \\cite{RT11}, \\cite{Rus11}, for nilpotent endomorphisms with two Jordan blocks. In this article we generalize their construction to all classical types (i.e.\\ $C$ and $D$) and reconstruct the Springer representation on the homology of so-called topological Springer fibers with the advantage that the construction only uses elementary algebraic topology. The Weyl group action as well as the component group action both admit hereby a combinatorial description accessible for explicit computations (which appears to be difficult using Springer's original theory). This does not only provide a new, combinatorial perspective on the Springer representation but also presentations of the cohomology rings for all two-block Springer fibers of types $C$ and $D$, and moreover explicit isomorphisms relating the cup diagram basis of the Kazhdan-Lusztig cell modules attached to the induced trivial module and the bipolytabloid basis of irreducible Specht modules in types $C$ and $D$.\n\nThe surprising fact that two-block Springer theory in types $C$ and $D$ admits a concrete description using the combinatorics of arc algebras of types $B$ and $D$ should be interpreted as a form of Langlands duality between perverse sheaves on isotropic Grassmannians and coherent sheaves on compactified resolutions of Slodowy slices supported on the Springer fiber, see Subsection~\\ref{subsection:Langlands}. \n\n\\subsubsection*{Assumption} Throughout this article let $(2m-k,k)$, $1\\leq k\\leq m$, be a partition of $2m$ labeling a nilpotent orbit in $\\mathfrak{so}_{2m}$, i.e.\\ $k$ is odd or $m=k$ by the classification of nilpotent orbits for $\\mathfrak{so}_{2m}$, see e.g.\\ \\cite{Wil37}, \\cite{Ger61}. \n\n\n\\subsection{Topological Springer fibers and cup diagrams} \\label{subsec:intro:cup_and_top_springer}\n\nWe begin by recalling the notion of a cup diagram from~\\cite{LS13}, \\cite{ES15}, as well as the definition of the topological Springer fiber from~\\cite{ES12}, \\cite{Wil15}, based on the earlier work, \\cite{Kho04}, \\cite{Rus11}, \\cite{SW12}, in type $A$. Cup diagrams are the key diagrammatic tool throughout this article.  \n\nFix a rectangle in the plane together with $m$ vertices evenly spread along the upper horizontal edge of the rectangle. A {\\it cup diagram} is a non-intersecting arrangement of cups and rays inside the rectangle. A cup is defined as a line segment connecting two distinct vertices and a ray is a line segment connecting a vertex with a point on the lower horizontal edge of the rectangle. Every vertex has to be joined with exactly one endpoint of a cup or ray. Additionally, any cup or ray which is accessible from the left side of the rectangle, meaning that there exists a path inside the rectangle connecting this cup or ray to the left vertical edge of the rectangle without intersecting any other part of the diagram, is allowed to be marked with one single marker (a black box). Diagrams related by a planar isotopy fixing the boundary are considered to be the same. \n\nLet $C_{\\mathrm{KL}}(m)$ be the set of all cup diagrams on $m$ vertices such that the number of marked rays plus the number of unmarked cups is even.\\footnote{The subscript ``$\\mathrm{KL}$'' comes from the fact that these cup diagrams can be used to give a diagrammatic description of the \\emph{K}azhdan-\\emph{L}usztig basis of a parabolic Hecke module of type $D$ with maximal parabolic of type $A$, \\cite{LS13}.} Here are all cup diagrams contained in $C_{\\mathrm{KL}}(4)$:\n\\[\n\\begin{array}{ccccccc}\n\\begin{tikzpicture}[baseline={(0,-.9)}]\n\\draw[dotted] (-.5,0) -- (2,0) -- (2,-.9) -- (-.5,-.9) -- cycle;\n\\begin{footnotesize}\n\\node at (0,.2) {1};\n\\node at (.5,.2) {2};\n\\node at (1,.2) {3};\n\\node at (1.5,.2) {4};\n\\end{footnotesize}\n\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\draw[thick] (1.5,0) -- +(0,-.9);\n\\end{tikzpicture}\n&,&\n\\begin{tikzpicture}[baseline={(0,-.9)}]\n\\draw[dotted] (-.5,0) -- (2,0) -- (2,-.9) -- (-.5,-.9) -- cycle;\n\\begin{footnotesize}\n\\node at (0,.2) {1};\n\\node at (.5,.2) {2};\n\\node at (1,.2) {3};\n\\node at (1.5,.2) {4};\n\\end{footnotesize}\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.45) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\n&,&\n\\begin{tikzpicture}[baseline={(0,-.9)}]\n\\draw[dotted] (-.5,0) -- (2,0) -- (2,-.9) -- (-.5,-.9) -- cycle;\n\\begin{footnotesize}\n\\node at (0,.2) {1};\n\\node at (.5,.2) {2};\n\\node at (1,.2) {3};\n\\node at (1.5,.2) {4};\n\\end{footnotesize}\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (1,0) -- +(0,-.9);\n\\draw[thick] (1.5,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.45) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\n&,&\n\\begin{tikzpicture}[baseline={(0,-.9)}]\n\\draw[dotted] (-.5,0) -- (2,0) -- (2,-.9) -- (-.5,-.9) -- cycle;\n\\begin{footnotesize}\n\\node at (0,.2) {1};\n\\node at (.5,.2) {2};\n\\node at (1,.2) {3};\n\\node at (1.5,.2) {4};\n\\end{footnotesize}\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (1,0) -- +(0,-.9);\n\\draw[thick] (1.5,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\\hspace{.5em},\n\\end{array}\n\\]\n\\[\n\\begin{array}{ccccccc}\n\\begin{tikzpicture}[baseline={(0,-.9)}]\n\\draw[dotted] (-.5,0) -- (2,0) -- (2,-.9) -- (-.5,-.9) -- cycle;\n\\begin{footnotesize}\n\\node at (0,.2) {1};\n\\node at (.5,.2) {2};\n\\node at (1,.2) {3};\n\\node at (1.5,.2) {4};\n\\end{footnotesize}\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (1.5,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.45) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\n&,&\n\\begin{tikzpicture}[baseline={(0,-.9)}]\n\\draw[dotted] (-.5,0) -- (2,0) -- (2,-.9) -- (-.5,-.9) -- cycle;\n\\begin{footnotesize}\n\\node at (0,.2) {1};\n\\node at (.5,.2) {2};\n\\node at (1,.2) {3};\n\\node at (1.5,.2) {4};\n\\end{footnotesize}\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\end{tikzpicture}\n&,&\n\\begin{tikzpicture}[baseline={(0,-.9)}]\n\\draw[dotted] (-.5,0) -- (2,0) -- (2,-.9) -- (-.5,-.9) -- cycle;\n\\begin{footnotesize}\n\\node at (0,.2) {1};\n\\node at (.5,.2) {2};\n\\node at (1,.2) {3};\n\\node at (1.5,.2) {4};\n\\end{footnotesize}\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1.25,-.365) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\n&,&\n\\begin{tikzpicture}[baseline={(0,-.9)}]\n\\draw[dotted] (-.5,0) -- (2,0) -- (2,-.9) -- (-.5,-.9) -- cycle;\n\\begin{footnotesize}\n\\node at (0,.2) {1};\n\\node at (.5,.2) {2};\n\\node at (1,.2) {3};\n\\node at (1.5,.2) {4};\n\\end{footnotesize}\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\n\\end{tikzpicture}\\hspace{.5em}.\n\\end{array}\n\\]\nUsually we neither draw the rectangle nor display the numbering of the vertices. \n\n\nLet $\\cupdiags$ denote the subset of $C_\\mathrm{KL}(m)$ consisting of all cup diagrams with exactly $\\lfloor\\frac{k}{2}\\rfloor$ cups. Consider the standard unit sphere $\\mathbb S^2\\subseteq\\mathbb R^3$ on which we fix the points $p=(0,0,1)$ and $q=(1,0,0)$. Given a cup diagram $\\ba\\in\\cupdiags$, define $S_\\ba$ as the subset of $\\left(\\mathbb S^2\\right)^m$ consisting of all $(x_1,\\ldots,x_m)\\in\\left(\\mathbb S^2\\right)^m$ which satisfy the coordinate relations \n\\begin{itemize}\n\\item $x_i=-x_j$ (resp.\\ $x_i=x_j$) if the vertices $i$ and $j$ are connected by an unmarked cup (resp.\\ marked cup), and\n\\item $x_i=p$ if the vertex $i$ is connected to a marked ray and $x_i=-p$ (resp. $x_i=q$) if $i$ is connected to an unmarked ray which is the leftmost ray in $\\ba$ (resp.\\ not the leftmost ray). \n\\end{itemize}\n\\begin{defi}\nThe {\\it topological Springer fiber} $\\SOfiber$ is then defined as the union\n\\[\n\\SOfiber:=\\bigcup_{\\ba\\in\\cupdiags}S_\\ba \\subseteq \\left(\\mathbb S^2\\right)^m.\n\\]\n\\end{defi}\nThe importance of the topologial Springer fiber is based on the following crucial remark.\n\n\\begin{rem} \\label{rem:wilbert}\nThere exists a homeomorphism $\\SOfiber\\cong\\mathcal B^{2m-k,k}_{\\mathrm{SO}_{2m}}$ under which the manifolds $S_\\ba$ are mapped to the irreducible components of the Springer fiber, \\cite[Theorem A]{Wil15}. Moreover, there exists an isomorphism of algebraic varieties $\\mathcal B^{2m-k-1,k-1}_{\\mathrm{Sp}_{2(m-1)}}\\cong\\mathcal B^{2m-k,k}_{\\mathrm{SO}_{2m}}$ between two-row Springer fibers of type $C$ and type $D$, \\cite[Theorem B]{Wil15}. Thus, we have also a homeomorphism $\\SOfiber\\cong\\mathcal B^{2m-k-1,k-1}_{\\mathrm{Sp}_{2(m-1)}}$ which allows us to use $\\SOfiber$ as a topological model for both, types $C$ and $D$.\n\\end{rem}\n\n\\subsection{An elementary construction of the Weyl group and component group action}\n\nIn this subsection we sketch the ideas behind our construction of the Springer representation and state the explicit combinatorial description.\n\nRecall that by classical Springer theory the centralizer $Z_G(x)$ acts on $\\mathcal B^x_G$ (from the left) and thus on $H^*(\\mathcal B_G^x)$. Since the identity component of $Z_G(x)$ acts trivially on $H^*(\\mathcal B_G^x)$ we obtain an induced left action of the component group $A^x_G$ on $H^*(\\mathcal B_G^x)$. Unlike the component group action, the $\\mathcal W_G$-action on $H^*(\\mathcal B_G^x)$ is {\\it not} induced from a geometric action on the Springer fiber. Instead, the construction requires more sophisticated machinery such as perverse sheaves, Fourier transform, correspondences or monodromy, see e.g.\\ \\cite[\\S 1.5]{Yun16} for a survey. As a consequence, the Springer action becomes notoriously difficult to understand explicitly. Guided by the abstract theory and inspired by \\cite{RT11}, \\cite{Rus11}, we present an elementary approach to define these actions based on the following result (cf.\\ also Proposition~\\ref{prop:basis_general_case}).\n\n\\begin{prop} \\label{prop:injection}\nThe map $\\gamma_{2m-k,k}\\colon H_*(\\SOfiber)\\to H_*((\\mathbb S^2)^m)$ induced by the natural inclusion $\\SOfiber\\subseteq (\\mathbb S^2)^m$ in homology is injective.\n\\end{prop}\n\nThe idea is to explicitly define commuting actions of the groups $\\mathcal W_G$ and $A_G^x$ on the space $(\\mathbb S^2)^m$ which induce commuting actions on $H_*((\\mathbb S^2)^m)$. By the injectivity of the map $\\gamma_{2m-k,k}$ from Proposition~\\ref{prop:injection}, we can identify $H_*(\\SOfiber)$ with its image under $\\gamma_{2m-k,k}$ in $H_*((\\mathbb S^2)^m)$. This turns out to be a  subspace of $H_*((\\mathbb S^2)^m)$ which is stable with respect to both actions.\n\nIn Section~\\ref{section:section1} we construct a distinguished homology basis of $H_*(\\SOfiber)$ using a cell decomposition of the $S_\\ba$ which naturally arises from the cup diagram combinatorics. This yields the following diagrammatic description of the vector space $H_*(\\SOfiber)$ (cf.\\ again Proposition~\\ref{prop:basis_general_case}).\n\n\\begin{prop}\\label{introprop:diagrammatic_homology_basis}\nThere exists a distinguished basis of $H_{2l}(\\SOfiber)$ labeled by the cup diagrams in $C_{\\mathrm{KL}}(m)$ with precisely $l$ cups, i.e.\\ $H_{2l}(\\SOfiber)$ is isomorphic to the vector space freely generated by all cup diagrams in $C_{\\mathrm{KL}}(m)$ with $l$ cups. \n\\end{prop}\n\nVia this diagrammatic basis, the action of the Weyl group and of the component group can be described combinatorially.\n\n\\subsubsection*{The Weyl group action}\n\nThe Weyl group $\\mathcal W_{D_m}=\\mathcal W_{\\mathrm{SO}_{2m}}$ is a Coxeter group with Coxeter generators $s_0,s_1,\\ldots,s_{m-1}$, see Section~\\ref{section:section2}. There is a right action of $\\mathcal W_{D_m}$ on $(\\mathbb S^2)^m$, where $s_i$ permutes the coordinates $i$ and $i+1$ and $s_0$ permutes the first two coordinates and additionally takes their antipodes. Following the construction outlined above we obtain the first main result. \n\n\\begin{introthm} \\label{thm:skein_calculus}\nSetting $\\ba.s:=\\gamma_{2m-k,k}^{-1}(\\gamma_{2m-k,k}(\\ba).s)$, where $s\\in\\mathcal W_{D_m}$ and $\\ba$ is a cup diagram (viewed as a basis element of homology via Proposition~\\ref{introprop:diagrammatic_homology_basis}), yields a well-defined, grading-preserving right action of the Weyl group $\\mathcal W_{D_m}$ on $H_*(\\SOfiber)$ which can be described using a skein calculus. Diagrammatically, we represent the generators as follows: \n\\[\ns_0:=\\begin{tikzpicture}[thick, scale=.8, baseline={(0,-.5)}]\n\\draw[thick] (0,0) -- +(.5,-1);\n\\draw[thick] (.5,0) -- +(-.5,-1);\n\\draw[thick] (1,0) -- +(0,-1);\n\\draw[thick] (1.5,0) -- +(0,-1);\n\\draw[thick] (2.5,0) -- +(0,-1);\n\n\\node at (2,-.5) {\\dots};\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].13,-.25) rectangle ++(5pt,5pt);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].13,-.75) rectangle ++(5pt,5pt);\n\n\\begin{footnotesize}\n\\node at (0,.2) {1};\n\\node at (.5,.2) {2};\n\\end{footnotesize}\n\\end{tikzpicture}\n\\hspace{5em}\ns_i:=\\begin{tikzpicture}[thick, scale=.8, baseline={(0,-.5)}]\n\\draw[thick] (0,0) -- +(0,-1);\n\\draw[thick] (.5,0) -- +(0,-1);\n\\draw[thick] (1.5,0) -- +(.5,-1);\n\\draw[thick] (2,0) -- +(-.5,-1);\n\\draw[thick] (3,0) -- +(0,-1);\n\n\\node at (1,-.5) {\\dots};\n\\node at (2.5,-.5) {\\dots};\n\n\\begin{footnotesize}\n\\node at (1.4,.22) {i};\n\\node at (2,.2) {i+1};\n\\end{footnotesize}\n\\end{tikzpicture}\n\\hspace{2em}\ni\\in\\{1,\\ldots,m-1\\}.\n\\] \nFor a cup diagram $\\ba$ and $s\\in\\mathcal W_{D_m}$ with a reduced expression $s=s_{i_1}s_{i_2}\\cdots s_{i_l}$  we obtain $\\ba.s$ by first stacking the pictures corresponding to the generators $s_{i_1},\\ldots,s_{i_l}$ on top of $\\ba$ and then resolving crossings according to the rule\n\\[\n\\begin{tikzpicture}[thick, scale=.8]\n\\draw[dotted] (.25,-.5) circle(16pt);\n\\draw[dotted] (2.25,-.5) circle(16pt);\n\\draw[dotted] (4.25,-.5) circle(16pt);\n\\draw[thick] (1.9,-.1) .. controls +(0,-.4) and +(0,-.4) .. +(.7,0);\n\\draw[thick] (1.9,-.9) .. controls +(0,.4) and +(0,.4) .. +(.7,0);\n\n\\draw[thick] (0,0) -- +(.5,-1);\n\\draw[thick] (.5,0) -- +(-.5,-1);\n\\draw[thick] (3.9,-.1) .. controls +(.15,0) and +(.15,0) .. +(0,-.8);\n\\draw[thick] (4.6,-.1) .. controls +(-.15,0) and +(-.15,0) .. +(0,-.8);\n\n\\node at (1.2,-.5) {$=$};\n\\node at (3.2,-.5) {$+$};\n\\end{tikzpicture}\n\\]\nusing the following local relations\n\\[\n\\begin{tikzpicture}[thick, scale=.8]\n\\draw[dotted] (-.25,-.5) circle(16pt);\n\\draw[dotted] (2.7,-.5) circle(16pt);\n\\draw[thick] (-.25,-.5) circle(8pt);\n\n\\node at (1.3,-.5) {$= (-2)\\cdot$};\n\\end{tikzpicture}\n\\hspace{3em}\n\\begin{tikzpicture}[thick, scale=.8]\n\\draw[dotted] (-.25,-.5) circle(16pt);\n\\draw[dotted] (1.9,-.5) circle(16pt);\n\n\\draw[thick] (-.75,-.5) -- +(1,0);\n\\draw[thick] (1.4,-.5) -- +(1,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]-.05,-.5) rectangle ++(5pt,5pt);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]-.45,-.5) rectangle ++(5pt,5pt);\n\n\\node at (.8,-.5) {$=$};\n\\end{tikzpicture}\n\\hspace{3em}\n\\begin{tikzpicture}[thick, scale=.8]\n\\draw[dotted] (-.25,-.5) circle(16pt);\n\\draw[thick] (-.25,-.5) circle(8pt);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]-.25,-.225) rectangle ++(5pt,5pt);\n\n\\node at (.8,-.5) {$=\\,0$};\n\\end{tikzpicture}\n\\]\ntogether with the rule that we kill all diagrams with a connected component where both endpoints are at the bottom of the diagram. We obtain $\\ba.s$, which is a linear combination of cup diagrams. \n\\end{introthm}\n\nBy Remark~\\ref{rem:wilbert} we have $\\SOfiber\\cong\\mathcal B^{2m-k-1,k-1}_{\\mathrm{Sp}_{2(m-1)}}$. According to Springer theory we therefore expect an interesting action of the Weyl group $\\mathcal W_{C_{m-1}}\\cong\\mathcal W_{\\mathrm{Sp}_{2(m-1)}}$ of type $C$ on $H_*(\\SOfiber)$. The Weyl group $\\mathcal W_{C_{m-1}}$ can be identified with the subgroup of $\\mathcal W_{D_m}$ generated by $s_0s_1$ and $s_i$ for $i\\in\\{2,\\ldots,m-1\\}$. In particular, we obtain an action of the Weyl group of type $C$ on $H_*(\\SOfiber)$ by restricting the action of $\\mathcal W_{D_m}$. This action turns out to be the Springer action for type $C$. Note that Theorem~\\ref{thm:skein_calculus} automatically yields an explicit description of this restricted action.\n\n\\subsubsection*{The component group action} \nIn the case of classical groups, Spaltenstein showed in \\cite[I.2.9]{Spa82} that the component group $A^x_G$ is isomorphic to a finite product of copies of $\\mathbb Z\/2\\mathbb Z$. Moreover, $A^x_G$ depends (up to isomorphism) only on the Jordan type $\\lambda$ of $x$ which allows us to use the notation $A^{\\lambda}_G$. In the two-block case we have isomorphisms (see also Section~\\ref{sec:group_actions} for more details),\n\\begin{eqnarray}\n\\label{compgroupsexplicit}\nA^{2m-k-1,k-1}_{\\mathrm{Sp}_{2(m-1)}} \\cong\\begin{cases}\n\\{e\\} &\\text{if }m=k\\text{ is even}, \\\\\n\\mathbb Z\/2\\mathbb Z &\\text{if }m=k\\text{ is odd}, \\\\\n(\\mathbb Z\/2\\mathbb Z)^2 &\\text{if }m\\neq k,\n\\end{cases}\n&\n\\text{and}\n&\nA^{2m-k,k}_{\\mathrm{SO}_{2m}} \\cong\\begin{cases}\n\\{e\\} &\\text{if }m=k, \\\\\n\\mathbb Z\/2\\mathbb Z &\\text{if }m\\neq k.\n\\end{cases}\n\\end{eqnarray}\n\nIn type $D$ the action of the component group turns out to be trivial (even though the group itself is not trivial if $m\\neq k$). Thus, we only consider the type $C$ case and construct an action of $A^x_{\\mathrm{Sp}_{2(m-1)}}$ on $(\\mathbb S^2)^m$ by setting $\\alpha.(x_1,x_2,\\ldots,x_m)=(-x_1,x_2,\\ldots,x_m)$, where $\\alpha\\in A^x_{\\mathrm{Sp}_{2(m-1)}}$ is a generator corresponding to a copy of $\\mathbb Z\/2\\mathbb Z$ in $A^x_{\\mathrm{Sp}_{2(m-1)}}$. \n\n\\begin{introthm}\n\\label{thmB}\n\\begin{enumerate}\n\\item Setting $\\alpha.\\ba:=\\gamma_{2m-k,k}^{-1}\\left(\\alpha.\\gamma_{2m-k,k}(\\ba)\\right)$, where $\\alpha\\in A^{2m-k-1,k-1}_{\\mathrm{Sp}_{2(m-1)}}$ and $\\ba$ is a cup diagram, yields a well-defined grading-preserving left action of the component group on $H_*(\\SOfiber)$. \n\\item A generator $\\alpha\\in A^{2m-k-1,k-1}_{\\mathrm{Sp}_{2(m-1)}}$ corresponding to a copy of $\\mathbb Z\/2\\mathbb Z$ in $A^{2m-k-1,k-1}_{\\mathrm{Sp}_{2(m-1)}}$ acts as the identity on $\\ba$ if $\\ba$ has a ray connected to the first vertex. Otherwise it creates (resp.\\ kills) a marker on the cup connected to the first vertex if it is unmarked (resp.\\ marked) and at the same time creates (resp.\\ kills) a marker on the leftmost ray.\n\\end{enumerate}\n\\end{introthm}\nHere is an example, where the leftmost ray is connected to vertex $j$. Note that if $A^{2m-k-1,k-1}_{\\mathrm{Sp}_{2(m-1)}}$ is not trivial there always exists a (leftmost) ray.  \n\\[\n\\begin{tikzpicture}[baseline={(0,-.45)}]\n\\begin{footnotesize}\n\\node at (0,.2) {$\\mathrm{1}$};\n\\node at (1,.2) {$\\mathrm{i}$};\n\\node at (2,.2) {$\\mathrm{j}$};\n\\node at (.5,0) {\\ldots};\n\\node at (1.5,0) {\\ldots};\n\\node at (2.5,0) {\\ldots};\n\\end{footnotesize}\n\\draw[thick] (0,0) .. controls +(0,-.75) and +(0,-.75) .. +(1,0);\n\\draw[thick] (2,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].5,-.55) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\n\\hspace{1.4em}\n\\xrsquigarrow{$\\mathrm{act}\\hspace{.2em}\\mathrm{by}\\hspace{.25em}\\alpha$}\n\\hspace{1.6em}\n\\begin{tikzpicture}[baseline={(0,-.45)}]\n\\begin{footnotesize}\n\\node at (0,.2) {$\\mathrm{1}$};\n\\node at (1,.2) {$\\mathrm{i}$};\n\\node at (2,.2) {$\\mathrm{j}$};\n\\node at (.5,0) {\\ldots};\n\\node at (1.5,0) {\\ldots};\n\\node at (2.5,0) {\\ldots};\n\\end{footnotesize}\n\\draw[thick] (0,0) .. controls +(0,-.75) and +(0,-.75) .. +(1,0);\n\\draw[thick] (2,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]2,-.45) rectangle ++(5pt,5pt);\n\\end{tikzpicture}.\n\\]\n\n\\subsection{Connection to Springer theory} To compare our results to classical Springer theory we determine the decomposition of our representations into irreducibles in each homological degree. \n\n\\subsubsection*{Case 1: Two Jordan blocks of equal size:} Consider the generic Hecke algebra $\\mathcal H(\\mathcal W_{D_m})$ attached to $\\mathcal W_{D_m}$ with its standard basis $H_w$, $w\\in\\mathcal W_{D_m}$ as a $\\mathcal L=\\mathbb C[q,q^{-1}]$-module. Inside $\\mathcal H(\\mathcal W_{D_m})$ we have the subalgebra $\\mathcal H(S_m)$ attached to the maximal parabolic subgroup in $\\mathcal W_{D_m}$ generated by $s_1,\\ldots,s_{m-1}$. Letting $H_{s_i}$, $i\\in\\{1,\\ldots,m-1\\}$ act as $q^{-1}$ turns $\\mathcal L$ into an $H(S_m)$-module. The parabolic Hecke module $\\mathcal M_{D_m}$ is obtained by tensoring $\\mathcal L$ over $\\mathcal H(S_m)$ with $\\mathcal H(\\mathcal W_{D_m})$. By specializing $q=1$, $\\mathcal M_{D_m}$ is isomorphic to the induced trivial module.\n\nIn \\cite[Theorem 5.17]{LS13} the authors identified $\\mathcal M_{D_m}$ with the free $\\mathcal L$-module whose basis is given by all cup diagrams in $C_\\mathrm{KL}(m)$. Via this identification a cup diagram corresponds to a Kazhdan-Lusztig basis vector and the right action of $\\mathcal H(\\mathcal W_{D_m})$ on $\\mathcal M_{D_m}$ admits a diagrammatic description similar to the one of Theorem~\\ref{thm:skein_calculus} (cf.\\ Propositions~\\ref{prop:diagrammatic_Hecke_actionI}). In Section~\\ref{sec:diagrammatic_Hecke_module} we generalize this construction by providing a similar diagrammatic description of $\\mathcal M_{\\mathrm{Sp}_{2(m-1)}}$ (cf.\\ Proposition~\\ref{prop:diagrammatic_Hecke_actionII}). The diagrammatic description of the parabolic Hecke modules is used to prove the following theorem. \n\n\\begin{introthm} \\label{thm:induced_module}\n\\begin{enumerate}\n\\item The right $\\mathcal W_{D_m}$-module $H^*(\\SOfiberequalsize)$ constructed in Theorem~\\ref{thm:skein_calculus} is isomorphic to the induced trivial module $\\mathbb C\\otimes_{\\mathbb C[S_m]}\\mathbb C[\\mathcal W_{D_m}]$. \n\\item Similarly, we have an isomorphism $H_*(\\SOfiberequalsize)\\cong\\mathbb C\\otimes_{\\mathbb C[S_{m-1}]}\\mathbb C[\\mathcal W_{C_{m-1}}]$ of $\\mathcal W_{C_{m-1}}$-modules, where the right $\\mathcal W_{C_{m-1}}$-action on $H_*(\\SOfiberequalsize)$ is obtained by restricting the $\\mathcal W_{D_m}$-action.  \n\\end{enumerate}\n\\end{introthm}\n\n\\begin{rem}\nIn contrast to the related work \\cite{RT11} in type $A$ we work with the entire homology and identify it as an induced representation. This approach has the following advantages: \n\\begin{enumerate}\n\\item Since the Springer representation $H^*(\\SOfiberequalsize)$ (in Lusztig's version, \\cite{Lus81}) is isomorphic to the induced trivial module, see \\cite[Theorem 1.3]{Lus04}, Theorem~\\ref{thm:induced_module} immediately implies that Theorem~\\ref{thm:skein_calculus} indeed reconstructs the Springer representation (similarly for type $C$). \n\\item Since the decomposition of the induced trivial module into irreducibles is known and multiplicity free, see e.g. \\cite[Lemma~5.19]{ES12}, we can easily match each of the representations $H_{2l}(\\SOfiberequalsize)$, $l\\in\\{0,\\ldots,\\lfloor\\frac{m}{2}\\rfloor\\}$, with exactly one irreducible of the induced trivial module in type $D$, see Subsection~\\ref{subsection:proof_induced_module} for details. Since the $\\mathcal W_{C_{m-1}}$-action is the restriction of the $\\mathcal W_{D_m}$-action (for which we have determined the irreducibles) we obtain the desired decomposition by applying results of \\cite{Tok84}.\n\\end{enumerate}\n\\end{rem}\n\n\\subsubsection*{Case 2: Two Jordan blocks of different size:} For the case of two not necessarily equally-sized Jordan blocks, we can identify $H_*(\\SOfiber)\\cong H_{<k}(\\SOfiberequalsize)$ as graded $\\mathcal W_{D_m}$-module (and thus also as $\\mathcal W_{C_{m-1}}$-module). In fact, this is evident from the diagrammatic description in Theorem~\\ref{thm:skein_calculus} and reduces the general two-block case to {\\it Case 1} above.\n\nThe Springer correspondence was described for the classical groups in work of Shoji, \\cite{Sho83}, \\cite{Sho79}. Given an irreducible $\\mathcal W_G$-module $V$, Shoji gives an algorithm which determines the Jordan type of a nilpotent element $x\\in\\mathfrak{g}$ as well as the irreducible character $\\phi$ of $A^x_G$ for which $H^{\\mathrm{top}}_{\\phi}(\\mathcal B^x_G)\\cong V$ in the Springer correspondence, see \\cite[Theorem 3.3]{Sho79}. Here, $H^{\\mathrm{top}}_{\\phi}(\\mathcal B^x_G)$ denotes the $\\phi$-isotypic subspace in $H^{\\mathrm{top}}(\\mathcal B^x_G)$. In Section~\\ref{subsec:Shoji} we recall this algorithm and use it to check that our construction yields the same results as classical Springer theory. \n\n\\subsection{Applications of our construction}\n\nWe provide two applications of our construction.\n\n\\subsubsection*{Cohomology rings of two-block Springer fibers in types C and D}\n\nBy \\cite[Theorem 1.1]{HS77}, the image of the canonical map $H^*(\\mathcal B_G)\\to H^*(\\mathcal B_G^x)$ induced by the inclusion of the Springer fiber into the flag variety $\\mathcal B_G$ associated with $G$ is given by the $A_G^x$-invariants in $H_*(\\mathcal B_G^x)$. In particular, the canonical map is surjective if and only if the $A_G^x$-action is trivial. In type $A$ the component groups are themselves trivial, \\cite[Lemma~3.6.3]{CG97}, and the canonical map can be used to provide presentations of $H^*(\\mathcal B_{\\mathrm{GL}_n}^x)$ as a quotient of $H^*(\\mathcal B_{\\mathrm{GL}_n})$, see \\cite{DCP81}, \\cite{Tan82}.\n\nOutside of type $A$ the cohomology rings of the Springer fibers are much harder to study. In fact, even the Euler characteristic of Springer fibers was only recently computed in \\cite{Kim16} (an independent proof for all two-block Springer fibers using completely different methods is included in the Appendix to this article). In \\cite{KP12} the cohomology ring of Springer fibers was described as an affine coordinate ring for all cases in which the canonical map is surjective and the nilpotent operator is of standard Levi type. In \\cite[Theorem B]{ES12} this description was used to obtain a presentation of the ring $H^*(\\mathcal B^x_{\\mathrm{SO}_{2m}})$, where $x\\in\\mathfrak{so}_{2m}$ has Jordan type $(m,m)$. In the general two-block case the methods of~\\cite{KP12} are not applicable because the nilpotent operator is not of standard Levi type. In type $C$ even the surjectivity of the canonical map fails. However, by dualizing Proposition~\\ref{prop:injection}, our topological approach enables us to realize the cohomology ring of $\\SOfiber$ as a quotient of $H^*((\\mathbb S^2)^m)$ which yields the following result:\n\n\\begin{introthm} \\label{introthm:cohomology}\nIf $m\\neq k$, then we have an isomorphism of graded algebras\n\\[\nH^*\\left(\\SOfiber\\right)\\cong\\bigslant{\\mathbb C[X_1,\\ldots,X_m]}{\\left\\langle X_i^2, X_I\\;\n  \\begin{array}{|c}\n  1 \\leq i \\leq m,\\\\\n  I\\subseteq  \\{1,\\ldots,m\\}, |I|=\\frac{k-1}{2}+1\n  \\end{array}\n  \\right\\rangle},\n\\]\nwhere $X_I=\\prod_{i\\in I}X_i$. In particular, in combination with \\cite[Theorem B]{ES12} and Remark~\\ref{rem:wilbert} this yields a description of the cohomology rings of all two-block Springer fibers of types $C$ and $D$.\n\\end{introthm}\n\n\\subsubsection*{Relating cup diagram and bipolytabloid bases of Specht modules}\nThe parabolic Hecke module $\\mathcal M_{D_m}$ has a filtration \n\\[\n\\{0\\}\\subseteq\\mathcal M_{D_m}^{\\lfloor\\frac{m}{2}\\rfloor}\\subseteq\\ldots\\subset\\mathcal M^l_{D_m}\\subseteq\\ldots\\subseteq\\mathcal M^0_{D_m}=\\mathcal M_{D_m}\n\\]\nby Kazhdan-Lusztig cells, see Section~\\ref{sec:diagrammatic_Hecke_module}. From the diagrammatic description of $\\mathcal M_{D_m}$ it follows that $\\mathcal M_{D_m}^l$ has a basis given by all cup diagrams with $l$ or more cups. The associated graded with respect to this filtration (specialized at $q=1$) gives the $\\mathcal W_{D_m}$-module $H_*(\\SOfiber)$ from Theorem~\\ref{thm:skein_calculus}, i.e.\\ we have isomorphisms of $\\mathcal W_{D_m}$-modules between the Kazhdan-Lusztig cell module $\\mathcal M^l_{D_m}\/\\mathcal M^{l+1}_{D_m}$ and $H_{2l}(\\SOfiber)$. In particular, the cup diagram basis of $H_*(\\SOfiber)$ is not the Kazhdan-Lusztig basis (except in top degree) which disproves the type $D$ analog of \\cite[Conjecture 4.4]{RT11} (in fact, this argument shows that this conjecture does not hold in type $A$ either).  \n\nIn type $D$ the cell modules are irreducible and in Theorem~\\ref{thm:explicit_identification_D} we write down an explicit isomorphism between the cup diagram basis of $H_{2l}(\\SOfiber)$ and the bipolytabloid basis of the Specht module. In type $C$ the $\\mathcal W_{C_{m-1}}$-module $H_{2l}(\\SOfiber)$ is not irreducible anymore (which can be seen geometrically from the nontrivial component group action). In Proposition~\\ref{prop:basis_isotypic_comp} we explicitly determine the decomposition of $H_{2l}(\\SOfiber)$ into irreducible summands by providing a basis in terms of sums of cup diagrams. As in type $D$ we obtain an explicit isomorphism between each irreducible summand and the corresponding Specht module, see Theorem~\\ref{thm:explicit_identification_C1}. This generalizes results for type $A$, see~\\cite{Nar89}, to types $C$ and $D$ and answers a question in~\\cite[Remark 5.21]{ES12}.  \n\n\\subsection{An instance of Langlands duality}\\label{subsection:Langlands}\n\nWe finish the introductory part of this article by discussing the reason for the appearance of the cup diagrams in our construction of the Springer representation from a higher, categorical point of view. We also explain a (conjectural) reason for the surprising fact that the Springer representations in types $C$ and $D$ both admit a neat description in terms of the same cup diagram combinatorics. \n  \nIn \\cite{ES15}, the authors defined an algebra $\\mathbb D_m$ whose underlying vector space has a basis given by oriented circle diagrams (these are obtained by putting a cup diagram upside down on top of another cup diagram and adding an orientation to the connected components of the resulting diagram). The construction of this so-called arc algebra of type $D$ was inspired by the arc algebras of type $A$ defined in \\cite{Kho02}, \\cite{CK14}, \\cite{BS11}.\n\nLet $\\Upsilon_m^D$ denote the isotropic Grassmannian of $m$-dimensional isotropic subspaces in $\\mathbb C^{2m}$ and let $\\mathcal{P}erv\\left(\\Upsilon_m^D\\right)$ be the associated category of perverse sheaves (constructible with respect to the Schubert stratification). Using results of Braden~\\cite{Bra02} one can establish an equivalence of categories $\\mathbb D_m$-$\\mathrm{mod}\\simeq\\mathcal{P}erv\\left(\\Upsilon^D_m\\right)$, see~\\cite[Theorem 9.1]{ES15}, which shows that the cup diagram combinatorics is intricately connected to the geometry of perverse sheaves.\n\nIn \\cite{SW12} it was shown that the category of finite-dimensional modules over the arc algebra of type $A$ is equivalent to the heart of an exotic $t$-structure (in the sense of \\cite{Bez03}) on the bounded derived category $D^b(\\mathrm{Coh}(Y^A_m))$ of coherent sheaves on a certain compactification $Y^A_m$ of a resolution of a Slodowy slice of type $A$. We also expect such an equivalence between the category $\\mathcal{P}erv(\\Upsilon_m^D)$ and the core of an exotic $t$-structure on the derived category of coherent sheaves (supported on the $(m,m)$ Springer fiber) on a compactification of a resolution of a Slowdowy slice $Y_m^D$ of type $D$. In particular, this equivalence relates the cup diagram combinatorics with the geometry of the Springer fiber and thus explains the appearance of the cup diagrams in our description of the Springer representation. As in type $A$, see~\\cite{AS}, \\cite{KS}, we also expect a direct connection to a Fukaya category related to $Y^D_m$ or to Kleinian singularities of type $D$. \n\nThe constructions in \\cite{SW12} show that the equivalence between perverse sheaves on the Grassmannian and the core of the exotic $t$-structure of $D^b(\\mathrm{Coh}(Y_m^A))$ can be interpreted as an incarnation of Koszul duality which by \\cite{BGS96} suggests that one should actually replace $Y_m^A$ by its Langlands dual $Y_m^{A^\\vee}$. This is subtly hidden in the arguments since type $A$ (as well as type $D$) are Langlands self-dual. However, we expect an equivalence of categories between perverse sheaves on the isotropic Grassmannian of type $B$ and a core of an exotic $t$-structure on a derived category of coherent sheaves on a resolution of a Slodowy slice of type $C$. \n\nBy \\cite{BP99} (see also \\cite[\\S9.7]{ES15}) there exists an equivalence $\\mathcal{P}erv(\\Upsilon^B_{m-1})\\cong\\mathcal{P}erv(\\Upsilon^D_m)$ of categories induced by an isomorphism (compatible with the Schubert stratification) between the respective isotropic Grassmannians. The combinatorics of these categories can be described in terms of the same cup diagrams. On the Langlands dual side, which is related to the Springer fibers, this then corresponds to an equivalence $D^b(\\mathrm{Coh}(Y^C_{m-1}))\\cong D^b(\\mathrm{Coh}(Y^D_m))$ induced by the isomorphism $Y^C_{m-1}\\cong Y^D_m$ between (compactified) resolutions of Slodowy slices recently discovered in~\\cite[\\S8.4]{Li18}. This gives a conceptual explanation of the surprising fact that the Springer representations in types $C$ and $D$ both admit a neat description in terms of the same combinatorics underlying the arc algebra $\\mathbb D_m$.\n\n\\subsection*{Notation and conventions} \nIn this article the term ``vector space'' means ``complex vector space'' and a ``graded vector space'' is a $\\mathbb Z$-graded complex vector space. Given a finite set $S$ and a commutative ring $R$, we write $R[S]$ to denote the free $R$-module with basis $S$. If $X$ is a topological space, we write $H_*(X)$ and $H^*(X)$ to denote its singular homology and cohomology with complex coefficients, i.e.\\ $H_*(X)=H_*(X,\\mathbb C)$ and $H^*(X)=H^*(X,\\mathbb C)$. Since we work over the complex numbers, homology and cohomology are dual, i.e.\\ the universal coefficient theorem provides natural isomorphisms $H^*(X)\\cong\\Hom_{\\mathbb C}(H_*(X),\\mathbb C)$ of graded vector spaces, which enable us to transfer results obtained for cohomology to homology and vice versa. Throughout this article we view $\\mathbb Z\/2\\mathbb Z\\cong\\{1,-1\\}\\subseteq\\mathbb C^*$ as a subgroup of the multiplicative group.   \n\n\\subsection*{Acknowledgements}\nThe authors thank Michael Ehrig and Daniel Tubbenhauer for many helpful comments and interesting discussions and Daniel Juteau and Martina Lanini for raising the question how to make the action of the component group explicit in our construction. Parts of this article appeared in the second author's PhD thesis under the supervision of the first author. \n\n\\begin{comment}\n\nINTRODUCTION: Note that by \\cite{Wil15}, there is an isomorphism of algebraic varieties between type $C$ and $D$. Hence, once we have constructed an action of $\\mathcal W_{D_m}$ on the homology of the topological Springer fiber, we get an obvious $\\mathcal W_{C_{m-1}}$-action for free by restricting the action of $\\mathcal W_{D_m}$ to this subgroup. It turns out that this restricted action is actually the Springer action. Thus, in the two-row case, from the point of view of the Weyl group, it is enough to study the type $D_m$ case which contains type $C$ completely. However, if one wants to understand decompositions into simples, the action of the component group becomes interesting in type $C$, but is trivial in type $D$. Thus, there is some duality between type $C$ and $D$; the component group action is interesting in type $C$ but boring in $D$ and vice versa for the Weyl group. \n\nINTRODUCTION: Proof of the fact that $\\mathcal W_{D_m}$-action on $H_*((\\mathbb S^m)^2)$ restricts to homology of topological Springer fiber follows the argument in \\cite{RT11}. However, there does not seem to be a simple reduction to their proof, because of the different signs appearing in our context.\n\n\\end{comment}\n\n \n\\section[sec]{\\for{toc}{A diagrammatic homology basis}\\except{toc}{A diagrammatic homology basis and the proof of Theorem~\\ref{introthm:cohomology}}}\n\\label{section:section1} \n\nRecall the set $\\cupdiags$ of cup diagrams from the introduction. Given $\\ba\\in\\cupdiags$, let $\\phi_\\ba\\colon S_\\ba\\hookrightarrow\\SOfiber$, $\\psi_\\ba\\colon S_\\ba\\hookrightarrow (\\mathbb S^2)^m$ be the inclusions. We obtain a commutative diagram\n\\begin{equation} \\label{eq:key_comm_diagram}\n\\begin{xy}\n\t\\xymatrix{\n\t\t\\bigoplus\\limits_{\\ba\\in\\cupdiags} H_*(S_\\ba) \\ar[rr]^{\\phi_{2m-k,k}} \\ar@\/^2pc\/[rrrr]^{\\psi_{2m-k,k}} && H_*(\\SOfiber) \\ar[rr]^{\\gamma_{2m-k,k}}\t&& H_*((\\mathbb S^2)^m)\n\t}\n\\end{xy},\n\\end{equation} \nwhere $\\phi_{2m-k,k}$ (resp.\\ $\\psi_{2m-k,k}$) is the direct sum of the maps induced by $\\phi_\\ba$ (resp.\\ $\\psi_\\ba$) and $\\gamma_{2m-k,k}$ is induced by the inclusion $\\SOfiber\\hookrightarrow\\left(\\mathbb S^2\\right)^m$. The goal of this section is to understand the diagram~(\\ref{eq:key_comm_diagram}) from an explicit combinatorial point of view. In particular, this includes the construction of a diagrammatic basis of $H_*\\left(\\SOfiber\\right)$. At the end of this section we will prove Theorem~\\ref{introthm:cohomology}. \n\n\\subsection{Homology bases via cell decompositions}\n\nWe begin by defining cell decompositions of the manifolds $\\left(\\mathbb S^2\\right)^m$ and $S_\\ba$, $\\ba\\in\\cupdiags$, which are used to obtain a diagrammatic description of the homology of the respective spaces. Recall the notion of a line diagram, \\cite[Definition 3.1]{RT11}:\n\\begin{defi}\nA {\\it line diagram} on $m$ vertices is obtained by attaching vertical rays, each possibly decorated with a single white dot, to $m$ vertices on a horizontal line. Given $U\\subseteq \\{1,\\ldots,m\\}$, let $l_U$  be the unique line diagram with white dots precisely on the lines connected to vertices not contained in $U$ (vertices are counted from left to right). We write $\\mathfrak{L}_m$ to denote the set of all line diagrams on $m$ vertices. \n\\end{defi}\n\n\\begin{ex} \\label{ex:line_diagrams}\nHere is a complete list of all line diagrams in $\\mathfrak{L}_2$: \n\\[\nl_\\emptyset=\n\\begin{tikzpicture}[baseline={(0,-.5)},scale=.8]\n\\draw[thick] (0,0) -- +(0,-1);\n\\draw[thick] (.5,0) -- +(0,-1);\n\\draw (0,-.5) circle(3pt);\n\\draw (.5,-.5) circle(3pt);\n\\end{tikzpicture}\\hspace{1em},\n\\hspace{3em}\nl_{\\{1\\}}=\n\\begin{tikzpicture}[baseline={(0,-.5)},scale=.8]\n\\draw[thick] (0,0) -- +(0,-1);\n\\draw[thick] (.5,0) -- +(0,-1);\n\\draw (.5,-.5) circle(3pt);\n\\end{tikzpicture}\\hspace{1em},\n\\hspace{3em}\nl_{\\{2\\}}=\n\\begin{tikzpicture}[baseline={(0,-.5)},scale=.8]\n\\draw[thick] (0,0) -- +(0,-1);\n\\draw[thick] (.5,0) -- +(0,-1);\n\\draw (0,-.5) circle(3pt);\n\\end{tikzpicture}\\hspace{1em},\n\\hspace{3em}\nl_{\\{1,2\\}}=\n\\begin{tikzpicture}[baseline={(0,-.5)},scale=.8]\n\\draw[thick] (0,0) -- +(0,-1);\n\\draw[thick] (.5,0) -- +(0,-1);\n\\end{tikzpicture}\\,\\,.\n\\]\n\\end{ex}\n\nNow, the two-sphere $\\mathbb S^2$ has a cell decomposition $\\mathbb S^2=\\left(\\mathbb S^2\\setminus\\{p\\}\\right)\\cup \\{p\\}$ consisting of a point and a two-cell. In the following we fix this CW-structure and equip $(\\mathbb S^2)^m$ with the Cartesian product CW-structure. Then we obtain a one-to-one correspondence between $\\mathfrak{L}_m$ and the cells of the CW-complex $(\\mathbb S^2)^m$ by sending a line diagram $l_U$ to the cell $C_{l_U}$, where $C_{l_U}$ is defined by choosing the $0$-cell (resp.\\ $2$-cell) for the $i$th sphere in $(\\mathbb S^2)^m$ if there is a dot (resp.\\ no dot) on the $i$th line of $l_U$. Since $H_*((\\mathbb S^2)^m)$ has a basis given by the homology classes $[C_{l_U}]$ of the cells, we obtain an isomorphism of vector spaces\n\\begin{equation} \\label{eq:homology_identification_for_spheres}\n\\mathbb C[\\mathfrak{L}_m] \\xrightarrow\\cong H_*\\left((\\mathbb S^2)^m\\right)\\,,\\,\\,l_U\\mapsto [C_{l_U}].\n\\end{equation}\nHenceforth, we will view a line diagram as a basis element of $H_*\\left((\\mathbb S^2)^m\\right)$ via the identification (\\ref{eq:homology_identification_for_spheres}). Note that its homological degree is given by twice the number of lines without a dot. \n\n\\begin{ex}\nThe cells of $\\mathbb S^2\\times\\mathbb S^2$ corresponding to the line diagrams in Example~\\ref{ex:line_diagrams} are given  (in the same order) respectively by\n\\[\n\\{(p,p)\\},\\hspace{.6em}\\{(x,p)\\mid x\\in\\mathbb S^2\\setminus\\{p\\}\\},\\hspace{.6em}\\{(p,x)\\mid x\\in\\mathbb S^2\\setminus\\{p\\}\\},\\hspace{.6em}\\{(x,y)\\mid x,y\\in\\mathbb S^2\\setminus\\{p\\}\\}.\n\\]\n\\end{ex}\n\n\nNext we will construct a basis of $H_*(S_\\ba)$ labeled by enriched cup diagrams.\n\n\\begin{defi}\nAn {\\it enriched cup diagram} is a cup diagram (as defined in the introduction) in which some of the cups and all of the rays are decorated with a white dot. We allow at most one dot per component and we do not impose any accessibility condition as for the markers. If a cup (resp.\\ ray) is marked with a marker and a dot we place the marker to the left of (resp.\\ above) the dot. Let $\\widetilde{\\mathbb B}^{2m-k,k}_\\mathrm{KL}$ be the set of all enriched cup diagrams on $m$ vertices with $\\lfloor\\frac{k}{2}\\rfloor$ cups such that the number of marked rays plus unmarked cups is even.  \n\\end{defi}\n\n\\begin{ex}\nThe enriched cup diagrams corresponding to the cup diagrams in $\\mathbb B^{3,3}_\\mathrm{KL}$ are given by\n\\[\n\\begin{tikzpicture}[baseline={(0,-.5)},scale=.8]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (1,0) -- +(0,-1);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.3) rectangle ++(5pt,5pt);\n\\draw (1,-.65) circle(3pt);\n\n\\draw (.25,-.365) circle(3pt);\n\\end{scope}\n\\end{tikzpicture}\\hspace{1em},\n\\hspace{3em}\n\\begin{tikzpicture}[baseline={(0,-.5)}, scale=.8]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (1,0) -- +(0,-1);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.3) rectangle ++(5pt,5pt);\n\\draw (1,-.65) circle(3pt);\n\\end{scope}\n\\end{tikzpicture}\\hspace{1em},\n\\hspace{3em}\n\\begin{tikzpicture}[baseline={(0,-.5)}, scale=.8]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw (.75,-.365) circle(3pt);\n\n\\draw[thick] (0,0) -- +(0,-1);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.3) rectangle ++(5pt,5pt);\n\\draw (0,-.65) circle(3pt);\n\\end{scope}\n\\end{tikzpicture}\\hspace{1em},\n\\hspace{3em}\n\\begin{tikzpicture}[baseline={(0,-.5)}, scale=.8]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (0,0) -- +(0,-1);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.3) rectangle ++(5pt,5pt);\n\\draw (0,-.65) circle(3pt);\n\\end{scope}\n\\end{tikzpicture}\\hspace{1em},\n\\hspace{3em}\n\\begin{tikzpicture}[baseline={(0,-.5)}, scale=.8]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (1,0) -- +(0,-1);\n\\draw (1,-.5) circle(3pt);\n\n\\draw (.4,-.3) circle(3pt);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].1,-.3) rectangle ++(5pt,5pt);\n\\end{scope}\n\\end{tikzpicture}\\hspace{1em},\n\\hspace{3em}\n\\begin{tikzpicture}[baseline={(0,-.5)}, scale=.8]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (1,0) -- +(0,-1);\n\\draw (1,-.5) circle(3pt);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\end{scope}\n\\end{tikzpicture}\\,\\,.\n\\]\n\\end{ex}\n\nGiven $\\ba\\in\\mathbb B^{2m-k,k}_\\mathrm{KL}$, let $i_1<i_2<\\cdots<i_{\\lfloor\\frac{k}{2}\\rfloor}$ denote the left endpoints of the cups in $\\ba$. Then we have a homeomorphism\n\\begin{equation}\\label{eq:homeo_giving_cell_decomp}\n\\xi_\\ba\\colon S_\\ba \\xrightarrow\\cong (\\mathbb S^2)^{\\lfloor\\frac{k}{2}\\rfloor}\\,,\\,\\,(x_1,\\ldots,x_m) \\mapsto (y_1,\\ldots,y_{\\lfloor\\frac{k}{2}\\rfloor}),\n\\end{equation}\nwhere\n$y_j=x_{i_j}$, if $i_j$ is an endpoint of an unmarked cup, and $y_j=-x_{i_j}$ if $i_j$ is an endpoint of a marked cup, $1\\leq j\\leq\\lfloor\\frac{k}{2}\\rfloor$.\n\nWe equip $S_\\ba$ with the structure of a CW-complex, where the cells are obtained as the preimages of the cells of $(\\mathbb S^2)^{\\lfloor\\frac{k}{2}\\rfloor}$  under the homeomorphism (\\ref{eq:homeo_giving_cell_decomp}). We obtain a one-to-one correspondence between $\\widetilde{\\mathbb B}^{2m-k,k}_\\mathrm{KL}$ and the cells of $S_\\ba$ by sending an enriched cup diagram $M$ to the cell $C_M$, where a cup with a dot (resp.\\ a cup without a dot) means that we have chosen the $0$-cell (resp.\\ $2$-cell) for the corresponding sphere via (\\ref{eq:homeo_giving_cell_decomp}). Since the basis elements of the homology $H_*(S_\\ba)$ are in one-to-one correspondence with the homology classes of the cells of $S_\\ba$ we obtain an isomorphism of vector spaces\n\\begin{equation} \\label{eq:homology_identification_for_S_ba}\n\\mathbb C[\\widetilde{\\mathbb B}^{2m-k,k}_\\mathrm{KL}] \\xrightarrow\\cong H_*(S_\\ba)\\,,\\,\\, M\\mapsto [C_M]. \n\\end{equation}\nThe homological degree of an enriched cup diagram is given as twice the number of cups without a dot.\n\n\n\\begin{rem}\nThe white dots in this article play exactly the same role as the black dots on the cup diagrams in the related work in type $A$, \\cite{RT11}, \\cite{Rus11}. In order to distinguish them from the black markers (which do not appear in type $A$) we chose to color them white. \n\\end{rem}\n\n\\subsection{A combinatorial description of $\\psi_{2m-k,k}$}\n\nThe next step is to explicitly describe the map $\\psi_{2m-k,k}$ in~(\\ref{eq:key_comm_diagram}) in terms of the diagrammatic bases we have chosen in the previous subsection.\n\n\\begin{defi}\nGiven an enriched cup diagram $M\\in\\widetilde{\\mathbb B}^{2m-k,k}_\\mathrm{KL}$, we define the associated {\\it line diagram sum} as \n\\[ \nL_M=\\sum_{U\\in\\mathcal U_M}(-1)^{\\Lambda_M(U)}l_U \\in H_*\\left((\\mathbb S^2)^m\\right),\n\\]\nwhere $\\mathcal U_M$ is the set of all subsets $U\\subseteq\\{1,\\ldots,m\\}$ containing precisely one endpoint of every cup without a dot, and $\\Lambda_M(U)$ counts the total number of all endpoints of cups with a marker and no dot, plus the number of right endpoints of cups in $U$ with neither a marker nor a dot.\n\\end{defi}\n\nSimilar line diagram sums (for type $A$) appear in \\cite[Definition 3.11]{RT11}. Their significance lies in the following result.  \n\n\\begin{prop} \\label{prop:image_of_homology_gen}\nVia identifications (\\ref{eq:homology_identification_for_spheres}) and (\\ref{eq:homology_identification_for_S_ba}), the map $(\\psi_\\ba)_*\\colon H_*(S_\\ba)\\to H_*((\\mathbb S^2)^m)$ induced by the natural inclusion $\\psi_\\ba\\colon S_\\ba\\hookrightarrow (\\mathbb S^2)^m$ is explicitly given by the assignment $M\\mapsto L_M$.\n\\end{prop}\n\n\\begin{ex}\nIf $m=1$ the maps induced on homology by the natural inclusion $S_{\\begin{tikzpicture}[scale=.5]\n\\draw[thick] (0,0) -- +(0,-.4);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.2) rectangle ++(5pt,5pt);\n\\end{tikzpicture}}\\hookrightarrow\\mathbb S^2$ respectively by the inclusion $S_{\\begin{tikzpicture}[scale=.5]\n\\draw[thick] (0,0) -- +(0,-.4);\n\\end{tikzpicture}}\\hookrightarrow\\mathbb S^2$ are given by \n\\begin{eqnarray*}\n\\begin{tikzpicture}[baseline={(0,-.4)},scale=.8]\n\\draw[thick] (0,0) -- +(0,-.8);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.25) rectangle ++(5pt,5pt);\n\\draw (0,-.55) circle(3.5pt);\n\\end{tikzpicture}\n\\,\n\\mapsto\n\\,\n\\begin{tikzpicture}[baseline={(0,-.4)},scale=.8]\n\\draw[thick] (0,0) -- +(0,-.8);\n\\draw (0,-.4) circle(3.5pt);\n\\end{tikzpicture}\n&\\quad\\text{respectively}\\quad&\n\\begin{tikzpicture}[baseline={(0,-.4)},scale=.8]\n\\draw[thick] (0,0) -- +(0,-.8);\n\\draw (0,-.4) circle(3.5pt);\n\\end{tikzpicture}\n\\,\n\\mapsto\n\\,\n\\begin{tikzpicture}[baseline={(0,-.4)},scale=.8]\n\\draw[thick] (0,0) -- +(0,-.8);\n\\draw (0,-.4) circle(3.5pt);\n\\end{tikzpicture}\n\\end{eqnarray*}\nsince the first map sends $p$ to the $0$-cell of $\\mathbb S^2$, and the second  is clearly homotopic to the map $S_{\\begin{tikzpicture}[scale=.5]\n\\draw[thick] (0,0) -- +(0,-.4);\n\\end{tikzpicture}}\\to\\mathbb S^2$, $-p\\mapsto p$, respecting the $0$-skeleta. More generally, the map induced by $S_\\ba\\hookrightarrow(\\mathbb S^2)^m$, where $\\ba$ consists of $m$ subsequent rays, sends the generator corresponding to the point $S_\\ba$ to the line diagram consisting of $m$ subsequent rays with a dot. Furthermore, by arguing similarly as in \\cite[Lemma~3.2]{RT11}, one can show that the maps induced by the inclusions $S_{\\begin{tikzpicture}[scale=.5]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].75,-.365) rectangle ++(5pt,5pt);\n\\end{tikzpicture}}\\hookrightarrow\\mathbb S^2\\times\\mathbb S^2$ respectively $S_{\\begin{tikzpicture}[scale=.5]\n\\draw[thick] (.45,0) .. controls +(0,-.45) and +(0,-.45) .. +(.45,0);\n\\end{tikzpicture}}\\hookrightarrow(\\mathbb S^2)^2$ are given by\n\\begin{eqnarray} \\label{eq:induced_map_cup_1}\n\\quad \\quad\n\\begin{tikzpicture}[baseline={(0,-.2)},scale=.9]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw (0.4,-.3) circle(3pt);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].1,-.3) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\n\\,\n\\mapsto\n\\,\n\\begin{tikzpicture}[baseline={(0,-.4)},scale=.8]\n\\draw[thick] (0,0) -- +(0,-.8);\n\\draw[thick] (.5,0) -- +(0,-.8);\n\\draw (0,-.4) circle(3pt);\n\\draw (.5,-.4) circle(3pt);\n\\end{tikzpicture}\n\\hspace{3.5em}\n\\begin{tikzpicture}[baseline={(0,-.3)},scale=.9]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\n\\,\n\\mapsto\n\\,\n-\\,\n\\begin{tikzpicture}[baseline={(0,-.4)},scale=.8]\n\\draw[thick] (0,0) -- +(0,-.8);\n\\draw[thick] (.5,0) -- +(0,-.8);\n\\draw (.5,-.4) circle(3pt);\n\\end{tikzpicture}\n-\n\\begin{tikzpicture}[baseline={(0,-.4)},scale=.8]\n\\draw[thick] (0,0) -- +(0,-.8);\n\\draw[thick] (.5,0) -- +(0,-.8);\n\\draw (0,-.4) circle(3pt);\n\\end{tikzpicture}\n&\\text{respectively}&\n\\begin{tikzpicture}[baseline={(0,-.3)},scale=.9]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw (0.25,-.365) circle(3pt);\n\\end{tikzpicture}\n\\,\n\\mapsto\n\\,\n\\begin{tikzpicture}[baseline={(0,-.4)},scale=.8]\n\\draw[thick] (0,0) -- +(0,-.8);\n\\draw[thick] (.5,0) -- +(0,-.8);\n\\draw (0,-.4) circle(3pt);\n\\draw (0.5,-.4) circle(3pt);\n\\end{tikzpicture}\n\\hspace{3.5em}\n\\begin{tikzpicture}[baseline={(0,-.25)},scale=.9]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\end{tikzpicture}\n\\,\n\\mapsto\n\\,\n\\begin{tikzpicture}[baseline={(0,-.4)},scale=.8]\n\\draw[thick] (0,0) -- +(0,-.8);\n\\draw[thick] (.5,0) -- +(0,-.8);\n\\draw (0.5,-.4) circle(3pt);\n\\end{tikzpicture}\n-\n\\begin{tikzpicture}[baseline={(0,-.4)},scale=.8]\n\\draw[thick] (0,0) -- +(0,-.8);\n\\draw[thick] (.5,0) -- +(0,-.8);\n\\draw (0,-.4) circle(3pt);\n\\end{tikzpicture}\n\\end{eqnarray}\nwhich shows that Proposition~\\ref{prop:image_of_homology_gen} is true for all cup diagrams with one connected component. \n\\end{ex}\n\nBefore we prove Proposition~\\ref{prop:image_of_homology_gen} in general, we note the following useful combinatorial fact whose (straightforward) proof is omitted (cf.\\ \\cite[Lemma~3.11]{RT11} for a similar statement).\n\n\\begin{lem}\nSuppose that the enriched cup diagram $M'$ is obtained from $M$ by deleting a cup connecting vertices $i$ and $j$, $i<j$. For $U\\in\\mathcal U_{M'}$ and $S\\subseteq\\{i,j\\}$ we define $\\widetilde{U}_S\\subseteq\\{1,\\ldots,m\\}$ as \n\\[\n\\widetilde{U}_S=S\\cup\\{x\\in U\\mid x<i\\}\\cup\\{x+1\\mid x\\in U \\text{ and } i\\leq x<j\\}\\cup\\{x+2\\mid x\\in U \\text{ and }j\\leq x\\}.\n\\]\nThen we can write $L_M=\\sum_{U\\in\\mathcal U_{M'}}(-1)^{\\Lambda_{M'}(U)}l_{\\widetilde{U_\\emptyset}}$, if the cup connecting vertices $i$ and $j$ has a dot. If the cup does not have a dot, then we have\n\\[\nL_M=(-1)^\\sigma \\sum_{U\\in\\mathcal U_{M'}}(-1)^{\\Lambda_{M'}(U)}l_{\\widetilde{U_i}} - \\sum_{U\\in\\mathcal U_{M'}}(-1)^{\\Lambda_{M'}(U)}l_{\\widetilde{U_j}},\n\\]\nwhere $\\sigma=1$, if the cup has a marker, and $\\sigma=0$, if it has neither a marker nor a dot. \n\\end{lem}\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:image_of_homology_gen}.]\nWe begin by using the small examples above in order to deduce the claim for the cup diagram $\\ba$ which consists of $\\lfloor\\frac{k}{2}\\rfloor$ subsequent cups connecting neighboring vertices with (possibly) an additional collection of subsequent rays to the right of the cups. Assume that the cup connecting the first two vertices is unmarked and identify $S_\\ba$ with $S_{\\begin{tikzpicture}[scale=.5]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\end{tikzpicture}}\\times S_{\\ba'}$ and the inclusion $S_\\ba\\hookrightarrow (\\mathbb S^2)^m$ with $\\psi_{\\begin{tikzpicture}[scale=.5]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\end{tikzpicture}}\\times\\psi_{\\ba'}$. Here $\\ba'$ is the cup diagram obtained by removing the leftmost cup. Via the K\\\"unneth isomorphism, the map induced by $\\psi_{\\begin{tikzpicture}[scale=.5]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\end{tikzpicture}}\\times\\psi_{\\ba'}$ is the tensor product $(\\psi_{\\begin{tikzpicture}[scale=.5]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\end{tikzpicture}})_*\\otimes(\\psi_{\\ba'})_*$. Hence, if the cup connecting the two leftmost vertices in $M$ has a dot, we compute\n\\[\n(\\psi_{\\begin{tikzpicture}[scale=.5]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\end{tikzpicture}})_*(\\begin{tikzpicture}[scale=.5]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw (0.75,-.365) circle(3.5pt);\n\\end{tikzpicture})\\otimes(\\psi_{\\ba'})_*(M')=l_\\emptyset\\otimes\\left(\\sum_{U\\in\\mathcal U_{M'}}(-1)^{\\Lambda_{M'}(U)}l_U\\right)=\\sum_{U\\in\\mathcal U_{M'}}(-1)^{\\Lambda_{M'}(U)}l_\\emptyset\\otimes l_U,\n\\]     \nwhere the first equality follows by induction and the small cases above. Here $M'$ denotes the enriched cup diagram obtained by removing the leftmost cup. Applying the K\\\"unneth isomorphism to the above equation yields the claim. \n\nOn the other hand, if the leftmost cup in $M$ does not have a dot, we compute\n\\begin{align*}\n(\\psi_{\\begin{tikzpicture}[scale=.5]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\end{tikzpicture}})_*(\\begin{tikzpicture}[scale=.5]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\end{tikzpicture})\\otimes(\\psi_{\\ba'})_*(M') &=\\left(l_{\\{1\\}}-l_{\\{2\\}}\\right)\\otimes\\left(\\sum_{U\\in\\mathcal U_{M'}}(-1)^{\\Lambda_{M'}(U)}l_U\\right) \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t  &=\\sum_{U\\in\\mathcal U_{M'}}(-1)^{\\Lambda_{M'}(U)}l_{\\{1\\}}\\otimes l_U - \\sum_{U\\in\\mathcal U_{M'}}(-1)^{\\Lambda_{M'}(U)}l_{\\{2\\}}\\otimes l_U.\n\\end{align*} \nIf the leftmost cup has a marker we argue similarly. \n\nFor the general case let $\\ba$ be a cup diagram and $\\ba_0$ the cup diagram obtained by rearranging the cups in such a way that the diagram is completely unnested. Let $\\tau_\\ba$ be the permutation of the vertices which realizes the change of passing from $\\ba$ to $\\ba'$. This induces a homeomorphism $\\tau_\\ba\\colon(\\mathbb S^2)^m\\to(\\mathbb S^2)^m$ which permutes the coordinates accordingly and hence restricts to a homeomorphism $S_\\ba\\to S_{\\ba_0}$. The inclusion $S_\\ba\\hookrightarrow(\\mathbb S^2)^m$ can be written as the composition $\\tau_\\ba\\circ\\psi_{\\ba_0}\\circ\\tau^{-1}_\\ba\\vert_{S_\\ba}$. Let $M_0$ be the cup diagram obtained from $\\ba_0$ whose $i$th component has a dot if and only if the $i$th component of $M$ has a dot. Then the claim follows from \n\\[\n\\tau_\\ba(\\psi_{\\ba_0}(\\tau^{-1}_\\ba\\vert_{S_\\ba}(M)))=\\tau_\\ba(\\psi_{\\ba_0}(M_0))=\\tau_\\ba(\\sum_{U\\in\\mathcal U_{M_0}}(-1)^{\\Lambda_{M_0}(U)}l_U)=\\sum_{U\\in\\mathcal U_M}(-1)^{\\Lambda_M(U)}l_U.\\qedhere\n\\]      \n\\end{proof}\n\n\n\\subsection{A linearly independent set of line diagram sums}\n\nIn this subsection we introduce a subset of $\\widetilde{\\mathbb B}^{2m-k,k}_\\mathrm{KL}$ having the property that the images of the elements from this set under $\\psi_{2m-k,k}$ are linearly independent. This will be an important technical tool in obtaining a diagrammatic basis of $H_*(\\SOfiber)$ in the next subsection.\n\n\\begin{defi}\nA {\\it standard enriched cup diagram}, or shorter {\\it standard cup diagram}, is an enriched cup diagram in which all cups marked with a dot can be connected to the right side of the rectangle\nby a path which does neither intersect the rest of the diagram nor any given path connecting a marker with the left side. The set of all standard cup diagrams on $m$ vertices with $\\lfloor\\frac{k}{2}\\rfloor$ cups such that the number of marked rays plus unmarked cups is even is denoted by $s\\widetilde{\\mathbb B}^{2m-k,k}_\\mathrm{KL}$.   \n\\end{defi}\n\n\n\n\\begin{ex} \\label{ex:standard_cup_diagrams}\nHere are the standard cup diagrams contained in $s\\widetilde{\\mathbb B}^{3,3}_\\mathrm{KL}$:\n\\[\n\\begin{tikzpicture}[scale=.8]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw (.75,-.365) circle(3pt);\n\n\\draw[thick] (0,0) -- +(0,-1);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.3) rectangle ++(5pt,5pt);\n\\draw (0,-.65) circle(3pt);\n\\end{scope}\n\\end{tikzpicture}\n\\hspace{3em}\n\\begin{tikzpicture}[scale=.8]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (0,0) -- +(0,-1);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.3) rectangle ++(5pt,5pt);\n\\draw (0,-.65) circle(3pt);\n\\end{scope}\n\\end{tikzpicture}\n\\hspace{3em}\n\\begin{tikzpicture}[scale=.8]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (1,0) -- +(0,-1);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.3) rectangle ++(5pt,5pt);\n\\draw (1,-.65) circle(3pt);\n\\end{scope}\n\\end{tikzpicture}\n\\hspace{3em}\n\\begin{tikzpicture}[scale=.8]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (1,0) -- +(0,-1);\n\\draw (1,-.5) circle(3pt);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\end{scope}\n\\end{tikzpicture}\n\\]\n\\end{ex}\n\nLet $C_\\mathrm{KL}^{\\leq\\lfloor\\frac{k}{2}\\rfloor}(m)$ denote the subset of $C_\\mathrm{KL}(m)$ consisting of all cup diagrams with at most $\\lfloor\\frac{k}{2}\\rfloor$ cups. If $m=k$ we have $C_\\mathrm{KL}^{\\leq\\lfloor\\frac{m}{2}\\rfloor}(m)=C_\\mathrm{KL}(m)$.\n\nWe define a map $\\mathsf{C}^{2m-k,k}\\colon s\\widetilde{\\mathbb B}^{2m-k,k}_\\mathrm{KL}\\to C_\\mathrm{KL}^{\\leq\\lfloor\\frac{k}{2}\\rfloor}(m)$, called {\\it cutting}, which takes a standard cup diagram $M\\in s\\widetilde{\\mathbb B}^{2m-k,k}_\\mathrm{KL}$ and replaces each dotted cup by two unmarked rays (this can be thought of as cutting the cup along the dot). Furthermore, the dots on the rays of $M$ are deleted (again, the reader might think of this as cutting along the dots and throw away the resulting line segment which is not connected to the vertex). If the number of marked rays plus the number of unmarked cups is not even in the resulting diagram we additionally either delete or put a marker on the leftmost ray.   \n\nThe map $\\mathsf{G}^{2m-k,k}\\colon C_\\mathrm{KL}^{\\leq\\lfloor\\frac{k}{2}\\rfloor}(m)\\to s\\widetilde{\\mathbb B}^{2m-k,k}_\\mathrm{KL}$, called {\\it gluing}, is defined as follows: given a cup diagram $\\ba\\in C_\\mathrm{KL}^{\\leq\\lfloor\\frac{k}{2}\\rfloor}(m)$ we replace the two rightmost rays by a cup with a dot (this can be thought of as gluing the two endpoints of the rays resulting in a cup in which the gluing point is decorated with a dot). The resulting cup has a marker if and only if one of the rays from which it resulted had a marker. Take the new diagram and repeat the procedure until it contains exactly $\\lfloor\\frac{k}{2}\\rfloor$ cups. \n\n\n\n\n\n\n\\begin{ex}\nThe standard cup diagrams in Example~\\ref{ex:standard_cup_diagrams} are obtained by applying $\\mathsf{G}^{3,3}$ to the following diagrams in $C_{\\mathrm{KL}}(3)$ (in the same order) respectively. \n\\[\n\\begin{array}{ccccccc}\n\\begin{tikzpicture}[baseline={(0,-.9)}]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\end{tikzpicture}\n&,&\n\\begin{tikzpicture}[baseline={(0,-.9)}]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (0,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.45) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\n&,&\n\\begin{tikzpicture}[baseline={(0,-.9)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (1,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.45) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\n&,&\n\\begin{tikzpicture}[baseline={(0,-.9)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\draw[thick] (1,0) -- +(0,-.9);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\\hspace{.5em}.\n\\end{array}\n\\]\n\\end{ex}\n\n\\begin{lem} \\label{lem:comb_bij} \nThe assignments $\\mathsf{G}^{2m-k,k}$ and $\\mathsf{C}^{2m-k,k}$ define mutually inverse maps between \n\\[\ns\\widetilde{\\mathbb B}^{2m-k,k}_\\mathrm{KL}\\quad\\text{and}\\quad C_\\mathrm{KL}^{\\leq\\lfloor\\frac{k}{2}\\rfloor}(m).\n\\]\n\\end{lem}\n\\begin{proof}\nThis is evident from the definitions. \n\\end{proof}\n\n \n\\begin{prop} \\label{prop:technical_prop_1}\nThe line diagram sums $L_M$, i.e.\\ the images of $M$ under $\\psi_{m,m}$, where $M$ varies over all standard cup diagrams in $s\\widetilde{\\mathbb B}^{m,m}_\\mathrm{KL}$, are linearly independent in $H_*((\\mathbb S^2)^m)$. \n\\end{prop}\n\nFor the proof of this proposition we use the following technical lemma.\n\n\\begin{lem} \\label{lem:lin_indep}\nLet $M\\in s\\widetilde{\\mathbb B}^{m,m}_\\mathrm{KL}$ be standard cup diagram without markers. Then the line diagram sums $L_M$, where $M$ varies over all standard cup diagrams in $s\\widetilde{\\mathbb B}^{m,m}_\\mathrm{KL}$ which can be obtained by decorating $M$ with markers, are linearly independent.\n\\end{lem}\n\n\n\\begin{proof}\nWe prove the statement by induction on the total number of components of $M$ which can be marked with a marker. If there is precisely one decorable component in $M$, then there exists precisely one cup diagram with the claimed properties and the claim follows. In case that $M$ consists of a single cup without a dot we note that all the possible decorations with markers (without fixing the parity) are linearly independent. \n\nSo suppose there is more than one component in $M$ which is allowed to have a marker. In particular, since $M$ is standard, there exists a cup in $M$ without a dot which may be marked with a marker. In the following we fix such a cup connecting $i$ and $j$. Let $M=M_1,N_1,M_2,N_2,\\ldots,M_r,N_r$ be the standard cup diagrams obtained from all the possible decorations of $M$ with markers and a fixed parity of markers. The diagrams are listed in such a way that $M_i$ and $N_i$ are the same except that $M_i$ does not have a marker on the fixed cup, but a marker on the leftmost component with a dot. On the other hand, $N_i$ has a marker on the fixed cup but no marker on the leftmost component with a dot. In case that $M$ has no dots we simply consider all standard cup diagrams $M_1,N_1,\\ldots,M_r,N_r$ (not only the ones with a fixed parity of markers) and drop the condition on the leftmost component with a dot. Let $M^\\prime_1,N^\\prime_1,\\ldots,M^\\prime_r,N^\\prime_r$ be the cup diagrams obtained by deleting the cup connecting $i$ and $j$. We compute\n\\begin{align*}\n0 &= \\sum_{l=1}^r \\lambda_lL_{M_l} + \\sum_{l=1}^r \\mu_lL_{N_l} \\\\\n\t&= \\sum_{l=1}^r \\lambda_l\\left(-\\sum_{U\\in\\mathcal U_{M^\\prime_l}}(-1)^{\\Lambda_{M^\\prime_l}(U)}l_{\\widetilde{U_j}}+\\sum_{U\\in\\mathcal U_{M^\\prime_l}}(-1)^{\\Lambda_{M^\\prime_l}(U)}l_{\\widetilde{U_i}}\\right)\\\\\n\t&\\hspace{1em} + \\sum_{l=1}^r \\mu_l\\left(-\\sum_{U\\in\\mathcal U_{N^\\prime_l}}(-1)^{\\Lambda_{N^\\prime_l}(U)}l_{\\widetilde{U_j}}-\\sum_{U\\in\\mathcal U_{N^\\prime_l}}(-1)^{\\Lambda_{N^\\prime_l}(U)}l_{\\widetilde{U_i}}\\right) \\\\\n\t&= \\sum_{U\\in\\mathcal U_{M^\\prime}} \\left(\\sum_{l=1}^r(-\\mu_l-\\lambda_l)(-1)^{\\Lambda_{M^\\prime_l}(U)}\\right)l_{\\widetilde{U_j}} + \\sum_{U\\in\\mathcal U_{M^\\prime}} \\left(\\sum_{l=1}^r(-\\mu_l+\\lambda_l)(-1)^{\\Lambda_{M^\\prime_l}(U)}\\right)l_{\\widetilde{U_i}}. \n\\end{align*}  \nSince the family of vectors $l_{\\widetilde{U}_i}, l_{\\widetilde{U}_j}$ is linearly independent if $U$ varies over all elements in $\\mathcal U_{M^\\prime}$, we deduce the equations \n\\begin{equation} \\label{eq:equations_lin_independence}\n\\sum_{l=1}^r(-\\mu_l-\\lambda_l)(-1)^{\\Lambda_{M^\\prime_l}(U)}=0 \\hspace{1.6em}\\text{and}\\hspace{1.6em} \\sum_{l=1}^r(-\\mu_l+\\lambda_l)(-1)^{\\Lambda_{M^\\prime_l}(U)}=0\n\\end{equation}\nfor all $U\\in\\mathcal U_{M^\\prime}$. Using the left equation in (\\ref{eq:equations_lin_independence}) we compute\n\\[\n0 = \\sum_{U\\in\\mathcal U_{M^\\prime}}\\left(\\sum_{l=1}^r(-\\mu_l-\\lambda_l)(-1)^{\\Lambda_{M^\\prime_l}(U)}\\right)l_U = \\sum_{l=1}^r(-\\mu_l-\\lambda_l)\\left(\\sum_{U\\in\\mathcal U_{M^\\prime}}(-1)^{\\Lambda_{M^\\prime_l}(U)}l_U\\right) ,\n\\]\nwhich equals $\\sum_{l=1}^r(-\\mu_l-\\lambda_l)L_{M^\\prime_l}$ and thus implies $\\lambda_l=-\\mu_l$ for all $l\\in\\{1,\\ldots,r\\}$ because the $L_{M^\\prime_l}$ are linearly independent by induction. Similarly we  obtain $\\lambda_l=\\mu_l$ by repeating the calculation with the right equation in (\\ref{eq:equations_lin_independence}). Hence, we deduce that $\\lambda_l=\\mu_l=0$ for all $l\\in\\{1,\\ldots,r\\}$.  \n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:technical_prop_1}]\nIt suffices to prove that the elements $L_M$, where $M$ varies over all standard cup diagrams in $s\\widetilde{\\mathbb B}^{m,m}_{\\mathrm{KL},l}$ with precisely $l$ cups without a dot, are linearly independent in $H_{2l}((\\mathbb S^2)^m)$, $0\\leq l\\leq m$. We define a total order on the subsets of $\\{1,\\ldots,m\\}$ of cardinality $l$:\n\\begin{equation} \\label{eq:total_order}\n\\{i_1<\\cdots<i_l\\}<\\{i'_1<\\cdots<i'_l\\} :\\Leftrightarrow \\exists r\\colon i_r<i_r' \\text{ and }i_{r+1}=i'_{r+1},\\ldots,i_l=i'_l\n\\end{equation}\nwhich induces a total order on the line diagrams $l_U$, where $U\\subseteq\\{1,\\ldots,m\\}$, $|U|=l$, i.e.\\ an order on our basis of $H_{2l}((\\mathbb S^2)^m)$. Define $f\\colon s\\widetilde{\\mathbb B}^{m,m}_{\\mathrm{KL},l}\\to s\\widetilde{\\mathbb B}^{m,m}_{\\mathrm{unmar},l}$ as the map which forgets all markers.\n\nGiven a standard cup diagram $M$, we write $U_M\\in\\mathcal U_M$ to denote the set containing all right endpoints of the cups without a dot. Note that $U_M$ is maximal in $\\mathcal U_M$ with respect to the order (\\ref{eq:total_order}). Assume that, for some $\\lambda_M\\in\\mathbb C$,\n\\begin{equation} \\label{eq:lin_indep_ansatz}\n0=\\sum_{M\\in s\\widetilde{\\mathbb B}^{m,m}_\\mathrm{KL}} \\lambda_ML_M = \\sum_{N\\in s\\widetilde{\\mathbb B}^{m,m}_\\mathrm{unmar}}\\sum_{\\begin{subarray}{l}\nM\\in s\\widetilde{\\mathbb B}^{m,m}_\\mathrm{KL}\\\\ f(M)=N\\end{subarray}}\\lambda_ML_M. \n\\end{equation}\n \nFix $N_\\text{max}$ such that $U_{N_\\text{max}}$ is maximal amongst all $N\\in s\\widetilde{\\mathbb B}^{m,m}_\\mathrm{unmar}$. Since the cup diagrams without markers are determined by the right endpoints of cups there is a unique such $N$. \n\nNote that the basis vector $l_{U_M}$ occurs (with non-zero coefficient) in each line diagram sum $L_{M^\\prime}$ with $f(M^\\prime)=M$ but it does not occur in any $L_{M^\\prime}$, where $f(M^\\prime)\\neq M$ because these $L_{M^\\prime}$ only contain basis vectors which are strictly smaller with respect to the total order (this follows from our maximality assumption on $M$ and the fact that $U_M$ is maximal in $\\mathcal U_M$ for every $M$).\n\nHence, equation (\\ref{eq:lin_indep_ansatz}) decouples into two independent equations    \n\\[\n\\sum_{\\begin{subarray}{l}\nM\\in s\\widetilde{\\mathbb B}^{m,m}_\\mathrm{KL}\\\\ f(M)=M'_\\text{max}\\end{subarray}}\\lambda_ML_M = 0 \\hspace{2.5em}\\text{and} \\hspace{2.5em}\n\\sum_{\\begin{subarray}{l}\nM'\\in s\\widetilde{\\mathbb B}^{m,m}_\\mathrm{unmar}\\\\ M'\\neq M'_\\text{max}\\end{subarray}}\\sum_{\\begin{subarray}{l}\nM\\in s\\widetilde{\\mathbb B}^{m,m}_\\mathrm{KL}\\\\ f(M)=M'\\end{subarray}}\\lambda_ML_M = 0.\n\\]\nBy iterating the above argument we obtain equations \n$\n\\sum_{\\begin{subarray}{l}\nM\\in s\\widetilde{\\mathbb B}^{m,m}_\\mathrm{KL}\\\\ f(M)=N\\end{subarray}}\\lambda_ML_M = 0,\n$\nfor all $N\\in s\\widetilde{\\mathbb B}^{m,m}_\\mathrm{unmar}$. Hence, Proposition~\\ref{prop:technical_prop_1} follows from Lemma~\\ref{lem:lin_indep}.       \n\\end{proof}\n\n\\subsection{An explicit homology basis}\n\nIn this subsection we use the results obtained so far to construct a basis of the homology of $\\SOfiber$ and give an explicit description of the map $\\gamma_{2m-k,k}$.\n\n\\begin{prop} \\label{prop:basis_general_case}\nThe elements $\\phi_{2m-k,k}(\\mathsf{G}^{2m-k,k}(\\ba))$, where $\\ba\\in C_\\mathrm{KL}^{\\leq\\lfloor\\frac{k}{2}\\rfloor}(m)$, form a basis of the homology $H_*(\\SOfiber)$. \nMoreover, we have a commutative diagram\n\\[\n\\begin{xy}\n\t\\xymatrix{\nH_{<k}(\\SOfiberequalsize) \\ar@^{(->}[dr]^{\\gamma_{m,m}} \\ar[d]^\\cong_{\\Gamma^{2m-k,k}} \\\\ \nH_*(\\SOfiber) \\ar@^{(->}[r]^{\\gamma_{2m-k,k}} & H_*((\\mathbb S^2)^m) \n\t}\n\\end{xy}\n\\]\nwhere $\\Gamma^{2m-k,k}$ is the isomorphism of vector spaces sending a basis element $\\phi_{m,m}\\left(\\mathsf{G}^{2m-k,k}(\\ba)\\right)$ of $H_{<k}(\\SOfiberequalsize)$ to $\\phi_{2m-k,k}(\\mathsf{G}^{2m-k,k}(\\ba))\\in H_*(\\SOfiber)$.  \n\\end{prop}\n\n\\begin{rem}\nAccording to Proposition~\\ref{prop:basis_general_case} above, we can identify $\\mathbb C[C_\\mathrm{KL}^{\\leq\\lfloor\\frac{k}{2}\\rfloor}(m)]$ with $H_*(\\SOfiber)$ via the following isomorphism of vector spaces, \n\\begin{equation} \\label{eq:combinatorial_springer_homology}\n\\mathbb C[C_\\mathrm{KL}^{\\leq\\lfloor\\frac{k}{2}\\rfloor}(m)]\\xrightarrow\\cong H_*(\\SOfiber)\\,,\\,\\,\\ba\\mapsto\\phi_{2m-k,k}\\left(\\mathsf{G}^{2m-k,k}(\\ba)\\right),\n\\end{equation}\nthereby obtaining a diagrammatic basis of the homology. Under this identification the cup diagrams with $l$ cups form a basis of $H_{2l}(\\SOfiber)$, $0\\leq l\\leq\\lfloor\\frac{k}{2}\\rfloor$.\n\\end{rem}\n\n\\begin{lem} \\label{lem:iso_onto_image}\nThe map $\\gamma_{m,m}\\colon H_*(\\SOfiberequalsize) \\to H_*((\\mathbb S^2)^m)$ induced by the inclusion $\\SOfiberequalsize\\hookrightarrow(\\mathbb S^2)^m$ is injective.\n\\end{lem}\n\\begin{proof}\nBy considering the commutative diagram (\\ref{eq:key_comm_diagram}) we deduce $\\im\\left(\\gamma_{m,m}\\right)=\\im\\left(\\psi_{m,m}\\right)$, because $\\phi_{m,m}$ is surjective (cf.\\ \\cite[p.\\ 23]{ES12}). Since $\\dim\\left(\\im\\left(\\psi_m\\right)\\right)\\geq\\vert s\\widetilde{\\mathbb B}^{m,m}_\\mathrm{KL}\\vert=2^{m-1}$ (the inequality follows from Proposition~\\ref{prop:technical_prop_1} and the equality by combining Lemma~\\ref{lem:comb_bij} with the fact that $\\vert C_\\mathrm{KL}(m)\\vert=2^{m-1}$ which will be shown in Subsection~\\ref{sec:diagrammatic_Hecke_module}) we deduce that $\\dim\\left(\\im\\left(\\gamma_{m,m}\\right)\\right)=\\dim\\left(\\im\\left(\\psi_{m,m}\\right)\\right)\\geq 2^{m-1}$. Hence, since $\\dim H_*(\\SOfiberequalsize)=2^{m-1}$ by Remark~\\ref{rem:dim_homology_equal-row}, the rank nullity theorem implies $\\dim\\left(\\im\\left(\\gamma_{m,m}\\right)\\right)=2^{m-1}$ and $\\ker(\\gamma_{m,m})=\\{0\\}$.\n\\end{proof}\n\n\\begin{lem}\nThe set $\\phi_{m,m}\\left(\\mathsf{G}^{m,m}(\\ba)\\right)$, where $\\ba\\in C_\\mathrm{KL}(m)$, is a basis of $H_*(\\SOfiberequalsize)$. \n\\end{lem}\n\\begin{proof}\nThe commutative diagram (\\ref{eq:key_comm_diagram}) combined with Lemma~\\ref{lem:iso_onto_image} gives a commutative diagram\n\\begin{eqnarray}\n\\label{diagProp24}\n\\begin{xy}\n\t\\xymatrix{\n\t\t\\bigoplus\\limits_{\\ba\\in\\mathbb B^{m,m}_\\mathrm{KL}} H_*(S_\\ba) \\ar[r]^{\\phi_{m,m}} \\ar@\/^1.5pc\/[rr]^{\\psi_{m,m}} & H_*(\\SOfiberequalsize) \\ar[r]^{\\gamma_{m,m}}_\\cong\t& \\im(\\psi_{m,m})\\subseteq H_*((\\mathbb S^2)^m),\n\t}\n\\end{xy}\n\\end{eqnarray}\nwhich allows us to write $\\phi\\left(\\mathsf{G}^{m,m}(\\ba)\\right)=\\gamma^{-1}_{m,m}\\left(\\psi_{m,m}\\left(\\mathsf{G}^{m,m}(\\ba)\\right)\\right)$ for all  cup diagrams $\\ba$ on $m$ vertices. Since we have $\\{\\mathsf{G}^{m,m}(\\ba)\\mid\\ba\\in C_\\mathrm{KL}(m)\\}=s\\widetilde{\\mathbb B}^{m,m}_\\mathrm{KL}$, it follows from Proposition~\\ref{prop:technical_prop_1} that the collection of elements $\\psi_{m,m}(\\mathsf{G}^{m,m}(\\ba))\\in H_*((\\mathbb S^2)^m)$, where $\\ba$ varies over all cup diagrams in $C_\\mathrm{KL}(m)$, is linearly independent. This remains true after applying the linear isomorphism $\\gamma_{m,m}^{-1}$.  \n\nAs $\\vert\\{\\mathsf{G}^{m,m}(\\ba)\\mid\\ba\\in C_\\mathrm{KL}(m)\\}\\vert = 2^{m-1} = \\dim H_*(\\SOfiberequalsize)$ we indeed have a basis of $H_*(\\SOfiberequalsize)$. \n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:basis_general_case}]\nFrom the appendix we deduce that the dimension of $H^*(\\SOfiber)$ (and hence also of its homology by duality) is given by counting the number of all cup diagrams in $C_\\mathrm{KL}^{\\leq\\lfloor\\frac{k}{2}\\rfloor}(m)$. Thus, in order to check that the elements $\\phi_{2m-k,k}(\\mathsf{G}^{2m-k,k}(\\ba))$ form a basis, where $\\ba$ varies over all cup diagrams in $C_\\mathrm{KL}^{\\leq\\lfloor\\frac{k}{2}\\rfloor}(m)$, it suffices to see that they are linearly independent. Under the assumption that they are not linearly independent, it follows that the vectors $\\gamma_{2m-k,k}\\left(\\phi_{2m-k,k}(\\mathsf{G}^{2m-k,k}(\\ba)\\right)=\\psi_{2m-k,k}(\\mathsf{G}^{2m-k,k}(\\ba))$ are not linearly independent either. But Proposition~\\ref{prop:image_of_homology_gen} directly implies that $\\psi_{2m-k,k}(\\mathsf{G}^{2m-k,k}(\\ba))=\\psi_{m,m}(\\mathsf{G}^{m,m}(\\ba))$ and the $\\psi_{m,m}(\\mathsf{G}^{m,m}(\\ba))$ are linearly independent (as a subset of the linearly independent vectors in Proposition~\\ref{prop:technical_prop_1}), a contradiction. In particular, the map $\\Gamma^{2m-k,k}$ defines an isomorphism. \n\nThe commutativity of the diagram follows from $\\psi_{2m-k,k}(\\mathsf{G}^{2m-k,k}(\\ba))=\\psi_{m,m}(\\mathsf{G}^{m,m}(\\ba))$. Since $\\gamma_{m,m}$ is injective by Lemma~\\ref{lem:iso_onto_image}, the same is true for $\\gamma_{2m-k,k}$ by the commutativity of \\eqref{diagProp24}.\n\\end{proof}\n\nThe next proposition (which is an immediate consequence of Proposition~\\ref{prop:image_of_homology_gen} and the commutative diagram (\\ref{eq:key_comm_diagram})) provides a combinatorial description of the map $\\gamma_{2m-k,k}$.  \n\n\\begin{prop} \\label{prop:explicit_isom}\nThe isomorphism $\\gamma_{2m-k,k}\\colon H_*(\\SOfiber)\\xrightarrow\\cong\\im(\\gamma_{2m-k,k})\\subseteq H_*((\\mathbb S^2)^m)$ is given by \n\\[\n\\ba\\mapsto L_\\ba=\\sum_{U\\in\\mathcal U_\\ba}(-1)^{\\Lambda_\\ba(U)}l_U,\n\\]\nwhere $\\mathcal U_\\ba$ is the set of all subsets $U\\subseteq\\{1,\\ldots,m\\}$ containing precisely one endpoint of every cup of $\\ba$, and $\\Lambda_\\ba(U)$ is the number of right endpoints of unmarked cups plus the number of endpoints of marked cups in $U$.\n\\end{prop}\n\n\\begin{ex}\nIf $m=3$ we obtain the diagrammatic homology basis\n\\[\nH_0\\left(\\mathcal S^{3,3}_\\mathrm{KL}\\right) = \\Biggl\\langle\\;\n\\begin{tikzpicture}[baseline={(0,-.45)}, scale=.8]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\end{tikzpicture}\n\\;\\Biggr\\rangle,\n\\hspace{2.2em}\nH_2\\left(\\mathcal S^{3,3}_\\mathrm{KL}\\right) = \\Biggl\\langle\n\\begin{array}{cccccc}\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.5) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\n&,&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.5) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\n&,&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\end{tikzpicture}\n\\end{array}\n\\Biggr\\rangle,\n\\]\nand according to Proposition~\\ref{prop:explicit_isom} the map $\\gamma_{m,m}$ is given by \n\\[\n\\begin{tikzpicture}[baseline={(0,-.45)}, scale=.8]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\end{tikzpicture}\n\\,\\,\\longmapsto\\,\\,\n\\begin{tikzpicture}[baseline={(0,-.45)}, scale=.8]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\draw (0,-.45) circle(3pt);\n\\draw (.5,-.45) circle(3pt);\n\\draw (1,-.45) circle(3pt);\n\\end{tikzpicture}\n\\hspace{9em}\n\\begin{tikzpicture}[baseline={(0,-.45)}, scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.5) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\n\\,\\,\\longmapsto\\,\\,\n\\begin{tikzpicture}[baseline={(0,-.45)}, scale=.8]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\draw (.5,-.45) circle(3pt);\n\\draw (1,-.45) circle(3pt);\n\\end{tikzpicture}\n\\,-\\,\n\\begin{tikzpicture}[baseline={(0,-.45)}, scale=.8]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\draw (0,-.45) circle(3pt);\n\\draw (1,-.45) circle(3pt);\n\\end{tikzpicture}\n\\]\n\\[\n\\hspace{1.4em}\n\\begin{tikzpicture}[baseline={(0,-.45)}, scale=.8]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.5) rectangle ++(5pt,5pt);\n\\end{tikzpicture}\n\\,\\,\\longmapsto\\,\\,\n\\begin{tikzpicture}[baseline={(0,-.45)}, scale=.8]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\draw (0,-.45) circle(3pt);\n\\draw (1,-.45) circle(3pt);\n\\end{tikzpicture}\n\\,-\\,\n\\begin{tikzpicture}[baseline={(0,-.45)}, scale=.8]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\draw (0,-.45) circle(3pt);\n\\draw (.5,-.45) circle(3pt);\n\\end{tikzpicture}\n\\hspace{5em}\n\\begin{tikzpicture}[baseline={(0,-.45)}, scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\end{tikzpicture}\n\\,\\,\\longmapsto\\,\\,\n-\\,\n\\begin{tikzpicture}[baseline={(0,-.45)}, scale=.8]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\draw (.5,-.45) circle(3pt);\n\\draw (1,-.45) circle(3pt);\n\\end{tikzpicture}\n\\,-\\,\n\\begin{tikzpicture}[baseline={(0,-.45)}, scale=.8]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\draw (0,-.45) circle(3pt);\n\\draw (1,-.45) circle(3pt);\n\\end{tikzpicture}\n\\]\n\\end{ex}\n\n\n\n\\subsection{Proof of Theorem~\\ref{introthm:cohomology}}\n\nUsing the results obtained so far we can provide a presentation of the cohomology ring of $\\SOfiber$. We assume that $m\\neq k$.\n\nFirstly, the natural inclusion $\\SOfiber\\subseteq (\\mathbb S^2)^m$ induces a surjective homomorphism of algebras $H^*((\\mathbb S^2)^m)\\twoheadrightarrow H^*(\\SOfiber)$. The surjectivity follows directly from dualizing Proposition~\\ref{prop:basis_general_case}, where we proved that the induced map in homology $H_*(\\SOfiber)\\hookrightarrow H_*((\\mathbb S^2)^m)$ is injective. \n\nSecondly, the complex dimension of the algebraic variety $\\mathcal B^{2m-k,k}_{\\mathrm{SO}_{2m}}$ is known to be $\\frac{k-1}{2}$, see e.g.~\\cite[Theorem 6.5]{ES12}, and thus $H^i(\\SOfiber)\\cong H^i(\\mathcal B^{2m-k,k}_{\\mathrm{SO}_{2m}})=0$ for all $i>2\\cdot\\frac{k-1}{2}=k-1$. In particular, the graded ideal $H^{\\geq k}((\\mathbb S^2)^m)=\\oplus_{i\\geq k} H^i((\\mathbb S^2)^m)$ is contained in the kernel of our surjection and we obtain an induced surjective homomorphism of algebras \n\\begin{equation} \\label{eq:surjection_onto_cohomology}\n\\bigslant{H^*((\\mathbb S^2)^m)}{H^{\\geq k}((\\mathbb S^2)^m)} \\twoheadrightarrow H^*(\\SOfiber),\n\\end{equation} \nwhich we claim to be an isomorphism. We prove this by comparing their dimensions. Recall that \n\\begin{equation} \\label{eq:cohomology_of_product_of_spheres}\nH^*((\\mathbb S^2)^m)\\cong \\bigslant{\\mathbb C[X_1,\\ldots,X_m]}{\\langle X_i^2 \\;| 1\\leq i\\leq m \\rangle}       \n\\end{equation}\nwith $\\textrm{deg}(X_i)=2$. Thus, the dimension of $H^{2i}((\\mathbb S^2)^m)$ is given by counting all monomials of degree $i$ with pairwise different factors, i.e.\\ $\\dim(H^{2i}(\\mathbb S^2)^m))=\\tbinom{m}{i}$. It follows that the dimension of the quotient in (\\ref{eq:surjection_onto_cohomology}) is $\\sum_{i=0}^{\\frac{k-1}{2}}\\tbinom{m}{i}$ which equals $\\dim H^*(\\SOfiber)$ by Proposition~\\ref{prop:dim_formula}.\n\nHence, the surjection (\\ref{eq:surjection_onto_cohomology}) is in fact an isomorphism which we can translate (using the isomorphism (\\ref{eq:cohomology_of_product_of_spheres}) and the fact that the ideal $H^{\\geq k}((\\mathbb S^2)^m)$ is generated by all monomials of degree~$k$) into the description of $H^*(\\SOfiber)$ given in Theorem~\\ref{introthm:cohomology} from the introduction.\\hfill $\\square$\n\n\\section[sec]{\\for{toc}{Diagrammatic description of the actions}\\except{toc}{Diagrammatic description of the actions and the proof of Theorems~\\ref{thm:skein_calculus} and~\\ref{thmB}}}\n\\label{section:section2}\nIn this section we prove Theorem~\\ref{thm:skein_calculus} and Theorem~\\ref{thmB} from the introduction.\n\n\\subsection{Weyl group actions on homology}\nAssume $m\\geq 4$ and let $\\mathcal W_{D_m}$ be the Weyl group of type $D_m$, i.e.\\ the Coxeter group with generators $s^D_0,s^D_1,\\ldots,s^D_{m-1}$ subject to $\\left(s^D_is^D_j\\right)^{\\alpha^D_{ij}}=e$ where\n\\begin{equation*}\n\\begin{array}{cc}\n\\alpha^D_{ij}=\\begin{cases}\n\t1 & \\text{if }i=j,\\\\\n\t3 & \\text{if }i\\,\\begin{tikzpicture}[thick, scale=.55, baseline={(0,-.1)}]\n\\draw (-.4,0) -- +(.8,0);\n\\end{tikzpicture}\\,j\\text{ are connected in }\\Gamma_{D_m},\\\\\n\t2 & \\text{else}.\n\\end{cases}\n&\\hspace{3em}\n\\begin{tikzpicture}[thick, scale=.55, baseline={(0,-.1)}]\n\\node at (-1.3,0) {$\\Gamma_{D_m}:$};\n\\draw (1,0) -- +(1.4,0);\n\\draw (2.4,0) -- +(.6,0);\n\\draw (0,-1) -- +(1,1);\n\\draw (0,1) -- +(1,-1);\n\\draw (4.6,0) -- +(.6,0);\n\\fill (0,1) circle(4pt);\n\\fill (0,-1) circle(4pt);\n\\fill (1,0) circle(4pt);\n\\fill (2.4,0) circle(4pt);\n\\fill (5.2,0) circle(4pt);\n\\begin{footnotesize}\n\\node at (0,-1.5) {$0$};\n\\node at (0,1.5) {$1$};\n\\node at (1,.5) {$2$};\n\\node at (2.4,.5) {$3$};\n\\node at (5.2,.5) {$m-1$};\n\\end{footnotesize}\n\\node at (3.8,0) {$\\dots$};\n\\end{tikzpicture}\n\\end{array}\n\\end{equation*}\nThe Weyl group $\\mathcal W_{C_m}$ of type $C_m$ (which is isomorphic to the Weyl group of type $B_m$) is the Coxeter group with generators $s_0^C,\\ldots,s_{m-1}^C$ and relations $\\left(s^C_is^C_j\\right)^{\\alpha^C_{ij}}=e$, where\n\\begin{equation*}\n\\begin{array}{cc}\n\\alpha^C_{ij}=\\begin{cases}\n\t1 & \\text{if }i=j,\\\\\n\t3 & \\text{if }i\\,\\begin{tikzpicture}[thick, scale=.55, baseline={(0,-.1)}]\n\\draw (-.4,0) -- +(.8,0);\n\\end{tikzpicture}\\,j\\text{ are connected in }\\Gamma_{C_m},\\\\\n\t4 & \\text{if }i\\,\\begin{tikzpicture}[thick, scale=.55, baseline={(0,-.1)}]\n\\draw (-.4,.1) -- +(.8,0);\n\\draw (-.4,-.1) -- +(.8,0);\n\\end{tikzpicture}\\,j\\text{ are connected in }\\Gamma_{C_m},\\\\\n\t2 & \\text{else}.\n\\end{cases}\n&\\hspace{3em}\n\\begin{tikzpicture}[thick, scale=.55, baseline={(0,-.1)}]\n\\node at (-1.9,0) {$\\Gamma_{C_m}:$};\n\\draw (1,0) -- +(1.4,0);\n\\draw (2.4,0) -- +(.6,0);\n\\draw (-.4,.1) -- +(1.4,0);\n\\draw (-.4,-.1) -- +(1.4,0);\n\\draw (4.6,0) -- +(.6,0);\n\\fill (-.4,0) circle(4pt);\n\\fill (1,0) circle(4pt);\n\\fill (2.4,0) circle(4pt);\n\\fill (5.2,0) circle(4pt);\n\\begin{footnotesize}\n\\node at (-.4,.5) {$0$};\n\\node at (1,.5) {$1$};\n\\node at (2.4,.5) {$2$};\n\\node at (5.2,.5) {$m-1$};\n\\end{footnotesize}\n\\node at (3.8,0) {$\\dots$};\n\\end{tikzpicture}\n\\end{array}\n\\end{equation*}\n\n\\begin{rem} \\label{rem:embedding_of_Weyl_grps}\nOne easily checks, see also \\cite[\\S 2.1]{Tok84}, that the maps \n\\begin{eqnarray*}\n\\mathcal W_{C_{m-1}} \\hookrightarrow\\mathcal W_{D_m}\\,,\\,\\,s_i^C \\mapsto\\begin{cases}\ns_0^Ds^D_1 & \\text{if }i=0,\\\\\ns^D_{i+1} & \\text{if }i\\neq 0,\n\\end{cases}\n&&\n\\mathcal W_{D_m} \\hookrightarrow\\mathcal W_{C_m}\\,,\\,\\,s_i^D \\mapsto\\begin{cases}\ns_0^Cs^C_1s_0^C & \\text{if }i=0,\\\\\ns^C_i & \\text{if }i\\neq 0,\n\\end{cases}\n\\end{eqnarray*}\ndefine embeddings of groups. \n\\end{rem}\n\nThe group $\\mathcal W_{D_m}$ (and thus by Remark~\\ref{rem:embedding_of_Weyl_grps} also the subgroup $W_{C_{m-1}}$) acts from the right on $\\left(\\mathbb S^2\\right)^m$ according to the following rules:\n\\begin{align*}\n&(x_1,x_2,x_3,\\ldots,x_m).s^D_0=(-x_2,-x_1,x_3,\\ldots,x_m), \\\\\n&(x_1,\\ldots,x_i,x_{i+1},\\ldots,x_m).s^D_i=(x_1,\\ldots,x_{i+1},x_i,\\ldots,x_m), \\hspace{2em} i\\in\\{1,\\ldots,m-1\\}.\n\\end{align*}\n\nGiven $i\\in\\{1,\\ldots,m-1\\}$, let $\\tau_i\\colon\\{1,\\ldots,m\\}\\to\\{1,\\ldots,m\\}$ be the involution of sets which exchanges $i$ and $i+1$ and fixes all other elements. Given a subset $U\\subseteq\\{1,\\ldots,m\\}$, we define $\\tau_iU$ as the subset of $\\{1,\\ldots,m\\}$ containing the images of the elements of $U$ under the map $\\tau_i$. \n\nBy standard algebraic topology we can then describe the induced action by the formulas \n\\begin{eqnarray}\n\\label{actionbeforeProp29}\nl_U.s^D_0=(-1)^{\\chi(U)}l_{\\tau_1U}\\,\\hspace{1.6em} l_U.s^D_i=l_{\\tau_iU}\\,,\\,\\,i\\in\\{1,\\ldots,m-1\\},\n\\end{eqnarray}\nwhere $\\chi(U)$ is the cardinality of the set $\\{1,2\\}\\cap U$.\n\n\n\\subsection{Proof of Theorem~\\ref{thm:skein_calculus}}\n\\begin{proof}[]\nThe main insight is the following observation.\n\\begin{prop}\\label{proposition:well_defined_action}\nGiven a cup diagram $\\ba\\in H_*(\\SOfiber)$ and a generator $s_i^D\\in\\mathcal W_{D_m}$, $i \\in\\{0,\\ldots,m-1\\}$, we claim that \n\\[\n\\ba.s_i^D:=\\gamma_{2m-k,k}^{-1}\\left(\\gamma_{2m-k,k}(\\ba).s_i^D\\right)\n\\]\nyields a well-defined right action of $\\mathcal W_{D_m}$ on $H_*(\\SOfiber)$ which is given by Table~\\ref{tab:s_i^D_action} and Table~\\ref{tab:s_0^D_action}.\n\\end{prop}\n\nTheorem~\\ref{thm:skein_calculus} is now a consequence of Proposition~\\ref{proposition:well_defined_action}: A straightforward case-by-case analysis shows that the $\\mathcal W_{D_m}$-action from Tables~\\ref{tab:s_i^D_action} and~\\ref{tab:s_0^D_action} coincides with that from the skein calculus.\\end{proof}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ c | c || c | c}\n$\\ba$ & $\\ba.s_i^D$ & $\\ba$ & $\\ba.s_i^D$ \\\\\n\\hline\\hline\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.5) rectangle ++(5pt,5pt);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.5) rectangle ++(5pt,5pt);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n\\\\ \n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n$-$\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n\\\\\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) -- +(0,-.9);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.5) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.5) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) -- +(0,-.9);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n\\\\ \n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.5) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.5) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.5) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n\\\\\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.5) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) -- +(0,-.9);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.5) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) -- +(0,-.9);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.5) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.5) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n\\\\\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1.25,-.365) rectangle ++(5pt,5pt);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1.25,-.365) rectangle ++(5pt,5pt);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n\\\\\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].75,-.75) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1.25,-.365) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1.25,-.365) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].75,-.74) rectangle ++(5pt,5pt);\n\\node at (.5,0) {$\\times$};\n\\node at (1,0) {$\\times$};\n\\end{tikzpicture}\n\\\\\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (1,0) {$\\times$};\n\\node at (1.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (1,0) {$\\times$};\n\\node at (1.5,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (1,0) {$\\times$};\n\\node at (1.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].75,-.74) rectangle ++(5pt,5pt);\n\\node at (1,0) {$\\times$};\n\\node at (1.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].75,-.74) rectangle ++(5pt,5pt);\n\\node at (1,0) {$\\times$};\n\\node at (1.5,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (1,0) {$\\times$};\n\\node at (1.5,0) {$\\times$};\n\\end{tikzpicture}\n\\\\\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].75,-.74) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].75,-.74) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1.25,-.365) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n\\end{tabular}\n\\caption{The action of a generator $s_i^D\\in\\mathcal W_{D_m}$, $i\\in\\{1,\\ldots,m-1\\}$, on a cup diagram (crosses indicate vertices $i$ and $i+1$).}\n\\label{tab:s_i^D_action}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ c | c || c | c}\n$\\ba$ & $\\ba.s^D_0$ & $\\ba$ & $\\ba.s^D_0$ \\\\\n\\hline\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.5) rectangle ++(5pt,5pt);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.5) rectangle ++(5pt,5pt);\n\\draw[thick] (.5,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n\\\\\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n$-$\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n\\\\\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.5) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) -- +(0,-.9);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.5) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,0) -- +(0,-.9);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.5) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{tikzpicture}\n\\\\\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{scope}\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{scope}\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1.25,-.365) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{scope}\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].75,-.74) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{scope}\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (0,0) .. controls +(0,-1) and +(0,-1) .. +(1.5,0);\n\\draw[thick] (.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].75,-.74) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{scope}\n\\end{tikzpicture}\n$+$\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\begin{scope}[xshift=3cm]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (.5,0) {$\\times$};\n\\end{scope}\n\\end{tikzpicture}\n\\end{tabular}\n\\caption{The action of the generator $s_0^D\\in\\mathcal W_{D_m}$ on a cup diagram (crosses indicate vertices $1$ and $2$).}\n\\label{tab:s_0^D_action}\n\\end{table}\n\n\\begin{proof}[Proof of Proposition~\\ref{proposition:well_defined_action}]\nThroughout the proof we fix a generator $s_i^D\\in\\mathcal W_{D_m}$, $i\\in\\{0,\\ldots,m-1\\}$. We distinguish between four different cases depending on what the cup diagram $\\ba$ locally looks like at the vertices $i$ and $i+1$.\n\n\\bigskip\n\\noindent {\\it Case 1: Vertices i and i+1 are both connected to a ray.} \n\nIf $i$ and $i+1$ are both connected to rays in $\\ba$, then we have $\\tau_iU=U$ for all $U\\in\\mathcal U_\\ba$ and \n\\begin{equation} \\label{eq:case_1_calculation}\nL_\\ba.s_i^D = \\sum_{U\\in\\mathcal U_\\ba} (-1)^{\\Lambda_\\ba(U)}l_{\\tau_iU}=\\sum_{U\\in\\mathcal U_\\ba} (-1)^{\\Lambda_\\ba(U)}l_U = L_\\ba,\n\\end{equation}\nif $i\\in\\{1,\\ldots,m-1\\}$. Now the claim follows from applying the inverse of the isomorphism of Proposition~\\ref{prop:explicit_isom} to (\\ref{eq:case_1_calculation}). Since $\\chi_U=0$ for all $U\\in\\mathcal U_\\ba$, calculation (\\ref{eq:case_1_calculation}) also proves that $s_0^D$ acts as indicated in Table~\\ref{tab:s_0^D_action}. \n\n\\bigskip\n\\noindent {\\it Case 2: Vertices i and i+1 are connected by a cup.}\n\nWe decompose $\\mathcal U_\\ba=\\mathcal U_{\\ba,i}\\sqcup\\mathcal U_{\\ba,i+1}$, where $\\mathcal U_{\\ba,i}$ (resp.\\ $\\mathcal U_{\\ba,i+1}$) consists of all sets $U\\in\\mathcal U_\\ba$ containing $i$ (resp.\\ $i+1$). Then we have\n\\[\nL_\\ba=\\sum_{U\\in\\mathcal U_\\ba} (-1)^{\\Lambda_\\ba(U)}l_U =\\sum_{U\\in\\mathcal U_{\\ba,i}} (-1)^{\\Lambda_\\ba(U)}l_U + \\sum_{U\\in\\mathcal U_{\\ba,i+1}} (-1)^{\\Lambda_\\ba(U)}l_U.\n\\]\nIn the following we abbreviate $L_{\\ba,i}=\\sum_{U\\in\\mathcal U_{\\ba,i}} (-1)^{\\Lambda_\\ba(U)}l_U$ and $L_{\\ba,i+1}=\\sum_{U\\in\\mathcal U_{\\ba,i+1}} (-1)^{\\Lambda_\\ba(U)}l_U$.\n\nFirstly, note that the assignment $U\\mapsto \\tau_iU$ induces a self-inverse bijection $\\tau_i\\colon\\mathcal U_{\\ba,i}\\to\\mathcal U_{\\ba,i+1}$. Secondly, for all $U\\in\\mathcal U_\\ba$ we have equalities $(-1)^{\\Lambda_\\ba(\\tau_iU)}=(-1)^{\\Lambda_\\ba(U)-1}$ (resp.\\ $(-1)^{\\Lambda_\\ba(\\tau_iU)}=(-1)^{\\Lambda_\\ba(U)}$) if $i$ and $i+1$ are connected by an unmarked cup (resp.\\ marked cup). Now we compute\n\\begin{align*}\nL_{\\ba,i}.s_i^D&=\\sum_{U\\in\\mathcal U_{\\ba,i}} (-1)^{\\Lambda_\\ba(U)}l_{\\tau_iU}=\\sum_{U\\in\\mathcal U_{\\ba,i+1}} (-1)^{\\Lambda_\\ba(\\tau_iU)}l_U \\\\\n\t\t\t\t&={\\begin{cases}\n\t\t\t\t\t\\sum_{U\\in\\mathcal U_{\\ba,i+1}} (-1)^{\\Lambda_\\ba(U)-1}l_U=-L_{\\ba,i+1} &\\text{if }i\\CupConnect (i+1), \\\\\n\t\t\t\t\t\\sum_{U\\in\\mathcal U_{\\ba,i+1}} (-1)^{\\Lambda_\\ba(U)}l_U=L_{\\ba,i+1} &\\text{if }i\\DCupConnect (i+1).\n\t\t\t\t\\end{cases}}\n\\end{align*}\nAnalogously, we obtain $L_{\\ba,i+1}.s_i^D=-L_{\\ba,i}$ (resp.\\ $L_{\\ba,i+1}.s_i^D=L_{\\ba,i}$) if the cup connecting $i$ and $i+1$ is unmarked (resp.\\ marked). Hence, we compute\n\\[\nL_\\ba.s_i^D=L_{\\ba,i}.s_i^D+L_{\\ba,i+1}.s_i^D=\\begin{cases}\n\t\t-L_\\ba &\\text{if }i\\CupConnect (i+1), \\\\\n\t\tL_\\ba &\\text{if }i\\DCupConnect (i+1).\n\t\\end{cases}\n\\] \nNote that $l_U.s_0^D=-l_{\\tau_1U}$ for all $U\\in\\mathcal U_{\\ba}$. Modifying the above calculation with the additional sign yields the claim for the $s_0^D$-action. \n\n\\bigskip\n\\noindent {\\it Case 3: Vertices i and i+1 are incident with one cup and one ray.} \n\nAssume that $i$ and $j$, $j<i$, are connected by an unmarked cup (if $i+1$ and $j$, $i+1<j$, are connected by a cup we only need to switch $i$ and $i+1$ in the proof below). Decompose $\\mathcal U_\\ba=\\mathcal U_{\\ba,i}\\sqcup\\mathcal U_{\\ba,j}$ and $L_\\ba=L_{\\ba,i}+L_{\\ba,j}$ as in {\\it Case 2} above. Let ${\\bf b}$ be the cup diagram different from $\\ba$ appearing in the sum $\\ba.s_i^D$ and decompose $\\mathcal U_{\\bf b}=\\mathcal U_{{\\bf b},i} \\sqcup \\mathcal U_{{\\bf b},i+1}$ as well as $L_{\\bf b}=L_{{\\bf b},i}+L_{{\\bf b},i+1}$. Since $\\tau_iU=U$ for all $U\\in\\mathcal U_{\\ba,j}$, we compute $L_{\\ba,j}.s_i^D = L_{\\ba,j}$ as in (\\ref{eq:case_1_calculation}).\n\nMoreover, we see that the assignment $U\\mapsto\\tau_iU$ defines an involution $\\mathcal U_{\\ba,i}\\to \\mathcal U_{{\\bf b},i+1}$ and we have $\\Lambda_\\ba(\\tau_iU)=\\Lambda_{\\bf b}(U)$ for all $U\\in\\mathcal U_{{\\bf b},i+1}$. Thus, we can compute\n\\begin{eqnarray*}&\nL_{\\ba,i}.s_i^D=\\sum_{U\\in\\mathcal U_{\\ba,i}} (-1)^{\\Lambda_\\ba(U)}l_{\\tau_iU}=\\sum_{U\\in\\mathcal U_{{\\bf b},i+1}} (-1)^{\\Lambda_\\ba(\\tau_iU)}l_U \n=\\sum_{U\\in\\mathcal U_{{\\bf b},i+1}} (-1)^{\\Lambda_{\\bf b}(U)}l_U=L_{{\\bf b},i+1},\\quad&\n\\end{eqnarray*}\nwhere the second equation is a simple change of the summation index using the bijection $\\tau_i$.\n\nWe see that $\\mathcal U_{\\ba,i}=\\mathcal U_{{\\bf b},i}$ and $\\Lambda_\\ba(U)=\\Lambda_{\\bf b}(U)-1$ for all $U\\in\\mathcal U_{\\ba,i}$ which implies $L_{\\ba,i}=-L_{{\\bf b},i}$ and \n\\begin{align*}\nL_\\ba.s_i^D &= L_{\\ba,j}.s_i^D+L_{\\ba,i}.s_i^D = L_{\\ba,j} + L_{{\\bf b},i+1} = L_{\\ba,j} + L_{{\\bf b},i+1} + L_{\\ba,i} - L_{\\ba,i} \\\\\n\t\t\t\t\t\t &= L_{\\ba,j} + L_{{\\bf b},i+1} + L_{\\ba,i} + L_{{\\bf b},i} = L_\\ba + L_{\\bf b}.\n\\end{align*}\n\nIf the cup connecting $i$ and $j$ is marked, we argue similarly. We omit to justify that $s_0^D$ acts as claimed in Table~\\ref{tab:s_0^D_action}. \n\n\\bigskip\n\\noindent {\\it Case 4: Vertices i and i+1 are incident with two different cups.} \n\nLet $i$ and $j$ as well as $i+1$ and $k$ be connected by a cup. We decompose $\\mathcal U_\\ba=\\mathcal U_{\\ba,j,i+1} \\sqcup \\mathcal U_{\\ba,j,k} \\sqcup \\mathcal U_{\\ba,i,i+1} \\sqcup \\mathcal U_{\\ba,i,k}$ and similarly $\\mathcal U_{\\bf b}=\\mathcal U_{{\\bf b},i,j} \\sqcup \\mathcal U_{{\\bf b},i,k} \\sqcup \\mathcal U_{{\\bf b},i+1,j} \\sqcup \\mathcal U_{{\\bf b},i+1,k}$, where ${\\bf b}$ is the cup diagram different from $\\ba$ appearing in the sum $\\ba.s_i^D$. Note that $\\tau_i\\colon\\mathcal U_{\\ba,j,k} \\to \\mathcal U_{\\ba,j,k}$ and $\\tau_i\\colon\\mathcal U_{\\ba,i,i+1}\\to\\mathcal U_{\\ba,i,i+1}$ are identities. Thus, we deduce\n\\[\nL_{\\ba,j,k}.s_i^D = \\sum_{U\\in\\mathcal U_{\\ba,j,k}} (-1)^{\\Lambda_\\ba(U)}l_{\\tau_iU}=\\sum_{U\\in\\mathcal U_{\\ba,j,k}} (-1)^{\\Lambda_\\ba(U)}l_U = L_{\\ba,j,k}\n\\]\nand analogously $L_{\\ba,i,i+1}.s_i^D=L_{\\ba,i,i+1}$. Since $(-1)^{\\chi(U)}=1$ in the above cases, these equations are also true if we act with $s_0^D$. \n\nIt is easy to check (by going through the different cases in the table) that $\\Lambda_\\ba(\\tau_iU)=\\Lambda_{\\bf b}(U)$ for all $U \\in \\mathcal U_{{\\bf b},j,i}\\cup U_{{\\bf b},i+1,k}$ if we act with $s_i$, $1\\leq i\\leq k-1$. If we act with $s_0$, then we have $\\Lambda_\\ba(\\tau_1U)=\\Lambda_{\\bf b}(U)\\pm 1$. Moreover, $\\tau_i\\colon \\mathcal U_{\\ba,j,i+1}\\to \\mathcal U_{{\\bf b},j,i}$ and $\\tau_i\\colon \\mathcal U_{\\ba,i,k}\\to \\mathcal U_{{\\bf b},i+1,k}$ are self-inverse bijections. Hence, we compute\n\\begin{eqnarray*}\nL_{\\ba,j,i+1}.s_i^D=\\sum_{U\\in\\mathcal U_{\\ba,j,i+1}} (-1)^{\\Lambda_\\ba(U)}l_{\\tau_iU}=\\sum_{U\\in\\mathcal U_{{\\bf b},j,i}} (-1)^{\\Lambda_\\ba(\\tau_iU)}l_U \n\t\t\t\t\t\t\t\t\t\t=\\sum_{U\\in\\mathcal U_{{\\bf b},j,i}} (-1)^{\\Lambda_{\\bf b}(U)}l_U=L_{{\\bf b},j,i}.\\quad\\quad\n\\end{eqnarray*}\nIf we act with $s_0^D$, then the additional factor of $(-1)$ is killed by $(-1)^{\\chi(U)}=-1$. Similarly, one checks that $L_{\\ba,i,k}.s_i^D=L_{{\\bf b},i+1,k}$.  \n\nNext, we note the equalities of sets $\\mathcal U_{\\ba,i,k}=\\mathcal U_{{\\bf b},i,k}$ and $\\mathcal U_{\\ba,j,i+1}=\\mathcal U_{{\\bf b},j,i+1}$. Furthermore, we have $\\Lambda_\\ba(U)=\\Lambda_{\\bf b}(U)\\pm 1$ for all $U\\in \\mathcal U_{\\ba,i,k}\\cup\\mathcal U_{\\ba,j,i+1}$. Again, this follows directly by looking at the table. Putting these two facts together we obtain $L_{{\\bf b},i,k}=-L_{\\ba,i,k}$ and $L_{{\\bf b},j,i+1}=-L_{\\ba,j,i+1}$. Thus,\n\n\\begin{eqnarray*}\n\\ba.s_i^D &=& L_{\\ba,j,i+1}.s_i^D+L_{\\ba,j,k}.s_i^D+L_{\\ba,i,i+1}.s_i^D+L_{\\ba,i,k}.s_i^D \\\\\n\t\t\t\t\t\t &=& L_{{\\bf b},j,i} + L_{\\ba,j,k} + L_{\\ba,i,i+1} + L_{{\\bf b},i+1,k} \\\\\n\t\t\t\t\t\t &= &L_{{\\bf b},j,i} + L_{\\ba,j,k} + L_{\\ba,i,i+1} + L_{{\\bf b},i+1,k} \n\t\t\t\t\t\t\n\t\t\t\t\t\t  + L_{\\ba,i,k} - L_{\\ba,i,k} + L_{\\ba,j,i+1} - L_{\\ba,j,i+1} \\\\\n\t\t\t\t\t\t &= &L_{{\\bf b},j,i} + L_{\\ba,j,k} + L_{\\ba,i,i+1} + L_{{\\bf b},i+1,k} \n\t\t\t\t\t\t\n\t\t\t\t\t\t + L_{\\ba,i,k} + L_{{\\bf b},i,k} + L_{\\ba,j,i+1} + L_{{\\bf b},j,i+1} \n\t\t\t\t\t\t = L_\\ba + L_{\\bf b}.\\!\\!\\qedhere\n\\end{eqnarray*} \n\\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{thmB}}\n\\label{sec:group_actions}\nLet $x$ be a nilpotent element of Jordan type $\\lambda$ contained in $\\mathfrak{sp}_{2m}(\\mathbb C)$ (resp.\\ $\\mathfrak{so}_{2m}(\\mathbb C)$). Then the corresponding component group $A^x_{\\mathrm{O}_{2m}}$ (resp.\\ $A^x_{\\mathrm{Sp}_{2m}}$) is isomorphic to $(\\mathbb Z\/2\\mathbb Z)^t$, where $t$ is the number of distinct odd (resp.\\ even) parts of the partition $\\lambda$, see \\cite[I.2.9]{Spa82}. Moreover, we identify $A^{2m-k,k}_{\\mathrm{SO}_{2m}}\\subseteq A^{2m-k,k}_{\\mathrm{O}_{2m}}$ as the subgroup generated by products of an even number of generators of $A^{2m-k,k}_{\\mathrm{O}_{2m}}$, see~\\cite{Sho83}. This immediately yields the isomorphisms (\\ref{compgroupsexplicit}) from the introduction.\n\nLet $\\alpha$ be a generator corresponding to a copy of $\\mathbb Z\/2\\mathbb Z$ in $(\\mathbb Z\/2\\mathbb Z)^t$. We have a left $A^{2m-k-1,k-1}_{\\mathrm{Sp}_{2m}}$-action on $(\\mathbb S^2)^m$, where $\\alpha$ acts as the antipode on the first component, i.e.\\ $\\alpha.(x_1,x_2\\ldots,x_m)=(-x_1,x_2\\ldots,x_m)$. Thus, in homology we have\n\\[\n\\alpha.l_U\\;=\\;\\begin{cases}\nl_U &\\text{if }1\\notin U, \\\\\n-l_U &\\text{if }1\\in U.\n\\end{cases}\n\\]\nNote that the special orthogonal component group acts trivially, although it is not always trivial. \nThe following proposition is now just a slight reformulation of Theorem~\\ref{thmB} from the introduction. \n\\begin{prop}\nGiven a cup diagram $\\ba\\in H_*(\\SOfiber)$ and a generator $\\alpha\\in A^{2m-k-1,k-1}_{\\mathrm{Sp}_{2m}}$ corresponding to a copy of $\\mathbb Z\/2\\mathbb Z$ in $A^{2m-k-1,k-1}_{\\mathrm{Sp}_{2m}}$, we claim that \n\\[\n\\alpha.\\ba:=\\gamma_{2m-k,k}^{-1}\\left(\\alpha.\\gamma_{2m-k,k}(\\ba)\\right)\n\\]\nyields a well-defined right action of the component group on $H_*(\\SOfiber)$ given as follows:\n\\centering\n\\begin{tabular}{ c | c || c | c}\n$\\ba$ & $\\alpha.\\ba$ & $\\ba$ & $\\alpha.\\ba$ \\\\\n\\hline\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\end{tikzpicture}\n&\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.5) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\end{tikzpicture}\n&\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]0,-.5) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\end{tikzpicture}\n\\\\\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (1.5,0) {${}$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.5) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (1.5,0) {${}$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.5) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (1.5,0) {${}$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (1.5,0) {${}$};\n\\end{tikzpicture}\n\\\\\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (1.5,0) {${}$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.5) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (1.5,0) {${}$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]1,-.5) rectangle ++(5pt,5pt);\n\\node at (0,0) {$\\times$};\n\\node at (1.5,0) {${}$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=.8,baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-.9);\n\\node at (0,0) {$\\times$};\n\\node at (1.5,0) {${}$};\n\\end{tikzpicture}\n\\end{tabular}\n\n\\end{prop}\n\\begin{proof} \nIf there is a ray connected to the first vertex there is nothing to show. So suppose there is a cup connecting the first vertex with some vertex $i>1$. We decompose $\\mathcal U_\\ba=\\mathcal U_{\\ba,1}\\sqcup\\mathcal U_{\\ba,i}$, where $\\mathcal U_{\\ba,1}$ (resp.\\ $\\mathcal U_{\\ba,i}$) consists of all sets $U\\in\\mathcal U_\\ba$ containing $1$ (resp.\\ $i$). Then we have\n\\[\nL_\\ba\\;=\\;\\sum_{U\\in\\mathcal U_\\ba} (-1)^{\\Lambda_\\ba(U)}l_U\\; =\\;\\sum_{U\\in\\mathcal U_{\\ba,1}} (-1)^{\\Lambda_\\ba(U)}l_U + \\sum_{U\\in\\mathcal U_{\\ba,i}} (-1)^{\\Lambda_\\ba(U)}l_U\n\\]\nand acting by a generator $\\alpha$ yields $\\alpha.L_\\ba=\\sum_{U\\in\\mathcal U_{\\ba,1}} (-1)^{\\Lambda_\\ba(U)+1}l_U + \\sum_{U\\in\\mathcal U_{\\ba,i}} (-1)^{\\Lambda_\\ba(U)}l_U=L_{\\ba^-}$ which proves the claim, i.e.\\ the geometric action can be described combinatorially as asserted.\n\\end{proof}\n\n\n\n\\section{A geometric construction of type B\/C Specht modules for one-row bipartitions}\\label{section:section3}\n\nIn this section we recall the notion of irreducible Specht modules for types B\/C and D and provide a geometric construction of all Specht modules corresponding to pairs of $1$-row bipartitions on $H_*((\\mathbb S^2)^m)$. This geometric construction will be the key ingredient in determining the decomposition of the Weyl group representation on $H_*(\\SOfiber)$ into irreducibles in Section~\\ref{section:section5}.\n\n\\subsection{Basic recollections on Specht modules}\n\nWe start by recalling the definition and basic properties of irreducible Specht modules for Weyl groups of types $B\/C$ and $D$. Our exposition follows and summarizes~\\cite{Can96}. \n\nA {\\it partition} $\\lambda=(\\lambda_1,\\ldots,\\lambda_l)$ of $k$ is a sequence of $l$ weakly decreasing positive integers $\\lambda_1\\geq\\lambda_2\\geq\\ldots\\geq\\lambda_l>0$ which sum up to $k$. We write $\\vert\\lambda\\vert=k$ if $\\lambda$ is a partition of $k$. Additionally, we also allow the empty partition $\\emptyset$. We may depict the partition $\\lambda$ as a {\\it Young diagram}, i.e.\\ a collection of $l$ left-aligned rows of square boxes, where the $i$th row has $\\lambda_i$ boxes.\n\n\\begin{defi}\nA {\\it bipartition} of $m$ is an ordered pair $(\\lambda,\\mu)$ of partitions such that $\\vert\\lambda\\vert+\\vert\\mu\\vert=m$.\n\\end{defi}\n\nFor the rest of this section we fix a bipartition $(\\lambda,\\mu)$ of $m$.\n\n\\begin{defi}\nA {\\it bitableau of shape} $(\\lambda,\\mu)$ is a filling of the boxes of the pair of Young diagrams with the numbers $1,\\ldots,m$ such that each number appears exactly once. Additionally, each of the numbers comes equipped with a sign (a choice of either $+$ or $-$ for each number). A {\\it standard bitableau} has only positive entries (no minus signs) and both underlying tableaux are standard tableaux in the usual sense, i.e.\\ the entries of the boxes increase along rows and columns. \n\\end{defi}\n\nThe group $\\mathcal W_{C_m}$ acts (from the right) on the set of bitableaux of shape $(\\lambda,\\mu)$. By definition the generator $s_i^C$, $i\\neq 0$, acts by substituting $i+1$ in place of $i$ at the position of $\\pm i$ (the respective sign remains unchanged) and substitutes $i$ in place of $i+1$ at the position of $\\pm (i+1)$ (again, the sign remains untouched).\nThe generator $s_0^C$ changes the sign in front of $1$. Note that the action of $\\mathcal W_{C_m}$ does not stabilize the subset of standard bitableaux.\n\n \n\\begin{defi}\nLet $T$ be a bitableau of shape $(\\lambda,\\mu)$. A {\\it row-permutation} (resp.\\ {\\it column-permutation}) of $T$ is an element $w\\in \\mathcal W_{C_m}$ which permutes the symbols in each row (resp.\\ column) of $T$ and multiplies an arbitrary  number of symbols in the first (resp.\\ second) tableau of $T$ with $-1$. The {\\it row stabilizer} $R_T$ (resp.\\ {\\it column stabilizer} $C_T$) of $T$ is defined as the group of row-permutations (resp.\\ column permutations).\n\\end{defi}\n\n\n\n\n\n\\begin{defi}\nTwo bitableaux $T,T^\\prime$ of shape $(\\lambda,\\mu)$ are row equivalent if there exists a row permutation $w$ of $T$ such that $T^\\prime=T.w$. The equivalence class containing a bitableau $T$ is denoted by $[T]$ and called a {\\it bitabloid} of shape $(\\lambda,\\mu)$. \n\\end{defi}\n\nLet $M_{(\\lambda,\\mu)}$ be the $\\mathbb C$-vector space with basis given by the bitabloids of shape $(\\lambda,\\mu)$. There is a well-defined action of $\\mathcal W_{C_m}$ on $M_{(\\lambda,\\mu)}$, where $s_i^C$, $i\\neq 0$, acts by setting $[T].s_i^C=[T.s_i^C]$ and $s_0^C$ acts as $[T].s_0^C=[T.s_0^C]$ if $1$ appears in the first tableau and $[T].s_0^C=-[T.s_0^C]$ if $1$ appears in the second tableau, see e.g.\\ \\cite[Section 1]{Can96}. \n\nSince $\\mathcal W_{C_m}$ is isomorphic to the semidirect product $S_m\\rtimes\\left(\\mathbb Z\/2\\mathbb Z\\right)^m$, we can associate a unique pair $(\\sigma,\\alpha)$, where $\\sigma\\in S_m$ and $\\alpha\\in\\left(\\mathbb Z\/2\\mathbb Z\\right)^m$, to any element $w\\in\\mathcal W_{C_m}$. We define $\\text{sgn}(w):=\\text{sgn}(\\sigma)$, where $\\text{sgn}(\\sigma)$ is the usual signum of an element of the symmetric group. Moreover, we define $f(w)$ as the number of $(-1)$'s in the $m$-tuple $\\alpha$.    \n\n\\begin{defi}\nGiven a bitableau $T$, we set $\\kappa_T=\\sum_{w\\in C_T} (-1)^{f(w)}\\text{sgn}(w)w$. The {\\it bipolytabloid} $e_T$ associated with the tableau $T$ is defined as $e_T=[T].\\kappa_T$.\n\\end{defi}\n\nFollowing~\\cite[Definition~3.3, Lemma~3.2]{Can96}, we define the Specht modules as follows:\n\n\\begin{defi}\nLet $\\lambda$ be a bipartition of $m$. The Specht module $V_{(\\lambda,\\mu)}$ is the submodule of $M_{(\\lambda,\\mu)}$ spanned by all bipolytabloids $e_T$.\n\\end{defi}\n\nThe importance of the Specht modules is justified by the following proposition which is \\cite[Theorem 3.21]{Can96}, see also e.g.  \\cite{Mac95}, \\cite[\\S 8.2]{Ser77}, \\cite[Proposition~3]{MS16} for a classification of the irreducible modules using different methods.\n\n\\begin{prop}\nThe Specht modules $V_{(\\lambda,\\mu)}$, where $(\\lambda,\\mu)$ varies over all bipartitions of $m$, form a complete set of pairwise non-isomorphic irreducible $\\mathcal W_{C_m}$-modules.\n\\end{prop}\n\n\nWe fix an irreducible representation $V_{(\\lambda,\\mu)}$ corresponding to $(\\lambda,\\mu)$. The restriction of $V_{(\\lambda,\\lambda)}$ to $\\mathcal W_{D_m}$ decomposes into two non-isomorphic irreducible $\\mathcal W_{D_m}$-modules of the same dimension which we denote by $V_\\lambda^+$ and $V_\\lambda^-$. If $\\lambda\\neq\\mu$ the restriction of $V_{(\\lambda,\\mu)}$ to $\\mathcal W_{D_m}$ remains irreducible and the restrictions of $V_{(\\lambda,\\mu)}$ and $V_{(\\mu,\\lambda)}$ to $\\mathcal W_{D_m}$ are isomorphic. We denote this irreducible module by $V_{\\{\\lambda,\\mu\\}}$. Then we have, see {\\cite{May75}}, or \\cite[Theorem 17]{MS16} for a modern treatment:\n\n\\begin{prop}\nThe $\\mathcal W_{D_m}$-representations $V_{\\{\\lambda,\\mu\\}}$, where $|\\lambda|+|\\mu|=m$, together with $V_\\lambda^+$ and $V_\\lambda^-$, where $|\\lambda|=\\frac{m}{2}$, form a complete set of pairwise non-isomorphic irreducible $\\mathcal W_{D_m}$-representations.  \n\\end{prop} \n\nNow \\cite[Theorem 4.2]{Can96} provides an important basis of the Specht modules as follows:\n\n\\begin{prop}\\label{proposition:Specht_basis}\nThe bipolytabloids $e_T$, where $T$ varies over all standard bitableaux of shape $(\\lambda,\\mu)$, form a $\\mathbb C$-basis of $V_{(\\lambda,\\mu)}$. \n\\end{prop}\n\n\\subsection{A geometric construction of some Specht modules} \n\nThe following fact is evident. \n\n\\begin{lem}\\label{lemma:bijection_line_diag_std_tableaux}\nThere is a bijection between the set of line diagrams on $m$ vertices with $l$ undotted rays and the set of standard $1$-row bitableaux of shape $((m-l),(l))$. This bijection is given by sending a line diagram $l_U$ to the unique standard bitableau $T_U$ whose second tableau is filled with the numbers in $U$.   \n\\end{lem} \n\n\nThe group $\\mathcal W_{C_m}$ acts (from the right) on $(\\mathbb S^2)^m$ according to the following rules:\n\\begin{eqnarray*}\n(x_1,x_2,\\ldots,x_m).s^C_0&=&(-x_1,x_2,\\ldots,x_m), \\\\\n(x_1,\\ldots,x_i,x_{i+1},\\ldots,x_m).s^C_i&=&(x_1,\\ldots,x_{i+1},x_i,\\ldots,x_m), \\hspace{2em} i\\in\\{1,\\ldots,m-1\\}.\n\\end{eqnarray*}\nThe induced action on homology is then given by \n\\[\nl_U.s^C_0\\;=\\;\\begin{cases}\nl_U &\\text{if }1\\notin U, \\\\\n-l_U &\\text{if }1\\in U.\n\\end{cases}\\,\\hspace{1.6em}\\text{and}\\quad l_U.s^C_i=l_{\\tau_iU}\\,,\\,\\,i\\in\\{1,\\ldots,m-1\\}.\n\\]\n\nThe following proposition gives an elementary geometric construction of all $\\mathcal W_{C_m}$-modules (and thus also of all $\\mathcal W_{D_m}$-modules by restriction) corresponding to pairs of $1$-row partitions. It explicitly identifies the bipolytabloid basis from Proposition~\\ref{proposition:Specht_basis} as the natural homology basis arising from the Cartesian product cell decomposition of $(\\mathbb S^2)^m$.   \n \n\\begin{prop}\\label{proposition:Specht_modules_as_homology}\nThe linear map $H_{2l}((\\mathbb S^2)^m)\\to V_{((m-l),(l))}$ defined by sending a line diagram $l_U$ to $e_{T_U}$ is an isomorphism of $\\mathcal W_{C_m}$-modules.\n\\end{prop}\n\n\nFor the proof of Proposition~\\ref{proposition:Specht_modules_as_homology} we first need the following technical lemma. \n\n\\begin{lem}\\label{lemma:row_equivalence}\nIf $T$ and $T^\\prime$ are row equivalent $1$-row bitableaux, then $e_T=e_{T^\\prime}$.\n\\end{lem}\n\\begin{proof}\nBy assumption we have $T^\\prime=Tw$ for some row permutation $w$ of $T$. Let $v$ be any column permutation of $T$. Note that since $T$ is a one-row bitableau, the column permutation $v$ only changes some signs of the numbers contained in the second tableau and leaves the first tableau invariant. Furthermore, $w$ permutes the symbols in the second tableau (without any sign changes) and permutes as well as changes some signs in the first tableau. Thus, evidently, we have $T vw=T wv$. \n\nBy the faithfulness of the action of $\\mathcal W_{C_m}$ on the set of bitableaux we deduce that $vw=wv$ for all $\\sigma\\in C_T$. In particular, we have $C_T=w^{-1} C_Tw=C_{T w}$, where the second equality is even true for an arbitrary element $w$ of $\\mathcal W_{C_m}$.  We obtain that\n\\[\n\\kappa_T=\\sum_{v\\in C_T} (-1)^{f(v)}\\text{sgn}(v)v=\\sum_{v\\in C_{T w}} (-1)^{f(v)}\\text{sgn}(v)v=\\kappa_{T w}\n\\]  \nand thus $e_T=[T].\\kappa_T=[T].\\kappa_{T w}=[T w].\\kappa_{T w}=e_{T w}$, where the second last equality follows from the well-definedness of the action on row-equivalence classes.\n\\end{proof}\n\n\n\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{proposition:Specht_modules_as_homology}]\nBy Proposition~\\ref{proposition:Specht_basis} and Lemma~\\ref{lemma:bijection_line_diag_std_tableaux} the map sends a basis to a basis. Thus, it suffices to check the $\\mathcal W_{C_m}$-equivariance.\n\nLet $i\\neq 0$. If $i$ and $i+1$ are both dotted or both undotted in a given line diagram $l_U$, then $s_i^C$ acts as the identity. On the other hand, the numbers $i$ and $i+1$ are either both contained in the left or in the right tableau of $T_U$. Thus, $s_i^C$ acts as a row permutation on $T_U$ and $T_Us_i^C$ and $T_U$ are row equivalent. In particular, we get $e_{T_U}s_i^C=e_{T_Us_i^C}=e_{T_U}$, where the first equality follows from \\cite[Lemma~3.2 (ii)]{Can96} and the second one from Lemma~\\ref{lemma:row_equivalence}.  \n\nIf exactly one of the rays $i$ and $i+1$ is dotted, then $s_i^C$ moves a dot from $i$ to $i+1$ or the other way around. The action of $s_i^C$ swaps the numbers $i$ and $i+1$ which appear in two different constituents. After applying a row permutation to $T_Us_i^C$ we obtain the standard tableau with the same filling as $T_U$ except that precisely $i$ and $i+1$ have changed the respective tableau.\n\nIf the first line is dotted in $l_U$, i.e.\\ if $1\\notin U$, then the generator $s_0^C$ acts on $l_U$ as the identity. In this case, the number $1$ is contained in the first tableau of $T_U$ and we have $e_{T_U}s_0^C=e_{T_U}$ because we have $e_{T_U}s_0^C=(-1)^{i-1}e_{T_U}$ if $1$ occurs in the $i$th tableau of $T_U$, $i\\in\\{1,2\\}$, see \\cite[Lemma~3.16]{Can96}. Secondly, if $1\\in U$, i.e.\\ the first line is undotted in $l_U$, then $s_0^C$ acts as the identity with an additional minus sign in front of the diagram. In this case, the number $1$ is in the second tableau of $T_U$ and thus $e_{T_U}s_0^C=-e_{T_U}$ by \\cite[Lemma~3.16]{Can96} as above. This finishes the proof of the $\\mathcal W_{C_m}$-equivariance and the proposition follows.\n\\end{proof}\n\n\n\n\\section[sec]{\\for{toc}{Diagrammatic description of the parabolic Hecke module}\\except{toc}{Diagrammatic description of the parabolic Hecke module in types B\/C \\& D}}\n\\label{section:section4}\n\nIn this section we recall the definition of the parabolic Hecke module in types $B\/C$ and $D$ and provide a diagrammatic description. This generalizes results obtained in \\cite{LS13} for type $D$ to types $B\/C$. In Section~\\ref{section:section5} we compare the diagrammatic construction of the parabolic Hecke module and the Springer representation to prove Theorem~\\ref{thm:induced_module}.\n\n\\subsection{Some Weyl group combinatorics} \\label{subsec:Weyl group comb}\n\nLet $\\Lambda^m$ be the set of all ordered sequences of length $m$ consisting of symbols from the alphabet $\\{\\land,\\lor\\}$ and let $\\Lambda^m_\\mathrm{even}$ be the subset consisting of all sequences in which the symbol $\\land$ occurs an even number of times. The Weyl group $\\mathcal W_{D_m}$ of type $D_m$ acts transitively on $\\Lambda^m_\\mathrm{even}$ from the right, where $s_i^D$, $i\\in\\{1,\\ldots,m-1\\}$, permutes the symbols at positions $i$ and $i+1$ (counted from left to right) and $s_0^D$ exchanges a pair of symbols $\\lor\\lor$ at positions $1$ and $2$ for the pair $\\land\\land$ (and vice versa) and acts as the identity otherwise. The parabolic subgroup $\\mathcal P_{D_m}$ of $\\mathcal W_{D_m}$ generated by $s_1^D,\\ldots,s_{m-1}^D$ stabilizes the sequence $\\lambda^m_\\lor\\in\\Lambda^m_\\mathrm{even}$ consisting of $m$ successive $\\lor$'s. This yields a bijection\n\\begin{equation} \\label{eq:coset_identification_D}\n\\mathcal P^\\mathrm{min}_{D_m}\\xrightarrow\\cong\\Lambda^m_\\mathrm{even}\\,,\\,\\,w\\mapsto \\lambda^m_\\lor.w\\,,\n\\end{equation}\nbetween $\\Lambda^m_\\mathrm{even}$ and the set $\\mathcal P^\\mathrm{min}_{D_m}$ of smallest representatives for the right cosets in $\\mathcal P_{D_m}\\setminus\\mathcal W_{D_m}$ which consists of all $w\\in\\mathcal W_{D_m}$ satisfying $l_{D_m}(s_i^Dw)>l_{D_m}(w)$ for all $i\\in\\{1,\\ldots,m-1\\}$. Here, $l_{D_m}$ is the length function of the Coxeter group $\\mathcal W_{D_m}$, see \\cite{BB05}, \\cite{Hum92} for details. \n\n\\begin{rem} Given $w\\in\\mathcal P^\\mathrm{min}_{D_m}$, we have $ws_i^D\\in\\mathcal P^\\mathrm{min}_{D_m}$ if and only if the action of $s_i^D$ on the $\\land\\lor$-sequence associated with $w$ changes the sequence. If this is the case, we have $l_{D_m}(ws_i^D)>l_{D_m}(w)$ if and only if the action of $s_i^D$ on the sequence associated with $w$ either changes a pair $\\land\\lor$ to $\\lor\\land$ or $\\lor\\lor$ changes to $\\land\\land$, see \\cite[Lemma~2.5]{ES15}. \n\\end{rem}\n\nSimilarly, the Weyl group $\\mathcal W_{C_{m-1}}$ of type $C_{m-1}$ acts transitively on $\\Lambda^{m-1}$ from the right, where $s^C_i$, $i\\in\\{1,\\ldots,m-2\\}$, permutes the symbols at positions $i$ and $i+1$ and $s_0^C$ exchanges a $\\land$ for a $\\lor$ and vice versa at the first position. The parabolic subgroup $\\mathcal P_{C_{m-1}}$ generated by $s_1^C,\\ldots,s_{m-2}^C$ of $\\mathcal W_{C_{m-1}}$ stabilizes the sequence $\\lambda_\\lor^{m-1}\\in\\Lambda^{m-1}$ which yields a bijection \n\\begin{equation} \\label{eq:coset_identification_C}\n\\mathcal P^\\mathrm{min}_{C_{m-1}}\\xrightarrow\\cong\\Lambda^{m-1}\\,,\\,\\,w\\mapsto\\lambda^{m-1}_\\lor.w\\,,\n\\end{equation}\nwhere $\\mathcal P^\\mathrm{min}_{C_{m-1}}$ denotes the set of smallest representatives for the right cosets in $\\mathcal P_{C_{m-1}}\\setminus\\mathcal W_{C_{m-1}}$.\n  \n\\begin{rem}\nGiven $w\\in\\mathcal P^\\mathrm{min}_{C_{m-1}}$, we have $ws_i^C\\in\\mathcal P^\\mathrm{min}_{C_{m-1}}$ if and only if the action of $s_i^C$ changes the sequence associated with $w$ in which case we have $l_{C_{m-1}}(ws_i^C)>l_{C_{m-1}}(w)$ if and only if either $\\land\\lor$ changes to $\\lor\\land$ or $\\lor$ changes to $\\land$.\n\\end{rem}\n\nThe map $(\\,.\\,)^\\dagger\\colon\\Lambda^{m-1}\\to\\Lambda^m_\\mathrm{even}$ given by \n\\[\nw \\mapsto w^\\dagger=\\begin{cases}\n\\lor w &\\text{ if }w\\text{ contains an even number of }\\land\\text{'s},\\\\\n\\land w &\\text{ if }w\\text{ contains an odd number of }\\land\\text{'s},\n\\end{cases}\n\\]\nis clearly a bijection which corresponds to a bijection between the sets of smallest coset representatives $\\mathcal P^\\mathrm{min}_{D_m}$ and $\\mathcal P^\\mathrm{min}_{C_{m-1}}$ via  (\\ref{eq:coset_identification_D}) and (\\ref{eq:coset_identification_C}), see also \\cite[\\S 9.7]{ES15}. \n\n\\subsection{The parabolic Hecke module with distinguished bases} We continue by recalling some facts from \\cite{KL79}, \\cite{Soe97}. Let $\\mathcal H_{D_m}$ be the Hecke algebra for $\\mathcal W_{D_m}$ over the ring of formal Laurent polynomials $\\mathcal L=\\mathbb C[q,q^{-1}]$. It has an $\\mathcal L$-basis $H_w$ indexed by the elements of the Weyl group $w\\in\\mathcal W_{D_m}$ which satisfy the algebra relations $H_vH_w=H_{vw}$ if $l_{D_m}(v)+l_{D_m}(w)=l_{D_m}(vw)$ and the quadratic relations $H_{s^D_i}^2=H_e+(q^{-1}-q)H_{s_i^D}$, $i\\in\\{0,\\ldots,m-1\\}$. As $\\mathcal L$-algebra, $\\mathcal H_{D_m}$ is generated  by the Kazhdan-Lusztig generators $C_{s^D_i}=H_{s^D_i}-q^{-1}H_e$, $i\\in\\{0,\\ldots,m-1\\}$.\n\nLet $\\mathcal H_{\\mathcal P_{D_m}}\\subseteq \\mathcal H_{\\mathcal W_{D_m}}$ be the subalgebra generated by $C_{s_i^D}$, $i\\in\\{1,\\ldots,m-1\\}$. The algebra $\\mathcal H_{\\mathcal P_{D_m}}$ acts on $\\mathcal L$ from the right by setting $1_\\mathcal L.H_s=q^{-1}1_\\mathcal L$ which allows us to define the tensor product $\\mathcal M_{D_m}=\\mathcal L\\otimes_{\\mathcal H_{\\mathcal P_{D_m}}}\\mathcal H_{\\mathcal W_{D_m}}$ which is naturally a right $\\mathcal H_{\\mathcal W_{D_m}}$-module.\n\n\\begin{defi} This right $\\mathcal H_{\\mathcal W_{D_m}}$-module has a {\\it standard basis} consisting of elements $M_w=1\\otimes H_w$, where $w\\in\\mathcal P^\\mathrm{min}_{D_m}$ and by \\cite[\\S 3]{Soe97} the $\\mathcal H_{\\mathcal W_{D_m}}$-action on $\\mathcal M_{D_m}$ is given by\n\\begin{equation} \\label{eq:explicit_module_structure}\nM_w.C_{s_i^D}=\\begin{cases}\nM_{ws_i^D}-q^{-1}M_w \t&ws\\in\\mathcal P^\\mathrm{min}_{D_m}\\text{ and }l_{D_m}(ws_i^D)>l_{D_m}(w),\\\\\nM_{ws_i^D}-qM_w\t\t\t\t&ws\\in\\mathcal P^\\mathrm{min}_{D_m}\\text{ and }l_{D_m}(ws_i^D)<l_{D_m}(w),\\\\\n0 \t\t\t\t\t\t\t\t&ws\\notin\\mathcal P^\\mathrm{min}_{D_m}.\n\\end{cases}\n\\end{equation}\n\\end{defi}\n\n\\begin{defi} The $\\mathbb C$-linear involution $\\overline{(\\,.\\,)}\\colon\\mathcal H_{\\mathcal W_{D_m}}\\to\\mathcal H_{\\mathcal W_{D_m}}$, $H\\mapsto\\overline{H}$, given by $q\\mapsto q^{-1}$ and $H_w\\mapsto H_{w^{-1}}^{-1}$ induces an involution on $\\mathcal M_{D_m}$. For all $w\\in\\mathcal P^\\mathrm{min}_{D_m}$ there exists precisely one element $\\underline{M}_w\\in\\mathcal M_{D_m}$ such that $\\overline{\\underline{M}_w}=\\underline{M}_w$ and $\\underline{M}_w=M_w+\\sum_{v\\neq w} q^{-1}\\mathbb Z[q^{-1}]M_v$. The set $\\{\\underline{M}_w\\mid w\\in\\mathcal P^\\mathrm{min}_{D_m}\\}$ is the {\\it Kazhdan-Lusztig basis} of $\\mathcal M_{D_m}$ (see \\cite[Theorem 3.5]{Soe97}). The {\\it Kazhdan-Lusztig polynomials} $m_{v,w}\\in\\mathbb Z[q,q^{-1}]$ are defined implicitly by $\\underline{M}_w=\\sum_v m_{v,w}M_v$.\n\\end{defi}\n\n\\begin{rem} \\label{rem:KL_remark}\nIn \\cite[Theorem 2.10]{LS13} the parabolic Kazhdan-Lusztig polynomials were explicitly computed for the parabolic type $(D_m,A_{m-1})$. Moreover, by \\cite[\\S 9.7]{ES15}, for any given $v,w\\in\\mathcal P^\\mathrm{min}_{C_{m-1}}$ we have $m^C_{v,w}=m^D_{v^\\dagger,w^\\dagger}$, i.e.\\ the parabolic Kazhdan-Lusztig polynomials of type $(C_{m-1},A_{m-2})$ are determined by the Kazhdan-Lusztig polynomials of type $(D_{m-1},A_{m-1})$.\n\\end{rem}\n\n\\begin{lem} \\label{lem:relating_C_D_module_action}\nLet $(\\,.\\,)^\\dagger\\colon\\mathcal M_{C_{m-1}} \\to \\mathcal M_{D_m}$ be the linear isomorphism sending $M_w$ to $M_{w^\\dagger}$. Then the linear isomorphism $(\\,.\\,)^\\dagger\\colon\\mathcal M_{C_{n-1}} \\to \\mathcal M_{D_n}$ sends a Kazhdan-Lusztig basis element $\\underline{M}_w$ to the Kazhdan-Lusztig element $\\underline{M}_{w^\\dagger}$ and we have for $i\\in\\{1,\\ldots,m-2\\}$,  commutative diagrams\n\\[\n\\begin{xy}\n\t\\xymatrix{\n\\mathcal M_{C_{m-1}} \\ar[r]_\\cong^{(\\,.\\,)^\\dagger} \\ar[d]_{\\cdot C_{s_0^C}} & \\mathcal M_{D_m} \\ar[d]^{\\cdot (C_{s_0^D}+C_{s_1^D})} \\\\ \n\\mathcal M_{C_{m-1}} \\ar[r]_\\cong^{(\\,.\\,)^\\dagger} & \\mathcal M_{D_m} \n\t}\n\\end{xy}\n\\hspace{4em}\n\\begin{xy}\n\t\\xymatrix{\n\\mathcal M_{C_{m-1}} \\ar[r]_\\cong^{(\\,.\\,)^\\dagger} \\ar[d]_{\\cdot C_{s_i^C}} & \\mathcal M_{D_m} \\ar[d]^{\\cdot C_{s_{i+1}^D}} \\\\ \n\\mathcal M_{C_{m-1}} \\ar[r]_\\cong^{(\\,.\\,)^\\dagger} & \\mathcal M_{D_m} \n\t}\n\\end{xy}\n\\]\nwhere the vertical maps are the actions with the indicated element of the respective Hecke algebra.\n\\end{lem}\n\\begin{proof}\nThe first part follows directly from the definition of the Kazhdan-Lusztig polynomials and Remark~\\ref{rem:KL_remark}. We only prove the commutativity of the right square and omit the left one. We check the commutativity on standard basis vectors. Using (\\ref{eq:explicit_module_structure}) one easily checks that the composition \n\\[\n\\mathcal M_{C_{m-1}}\\xrightarrow{\\cdot C_{s_i^C}}\\mathcal M_{C_{m-1}}\\xrightarrow{(\\,.\\,)^\\dagger}\\mathcal M_{D_m}\n\\]\nis given by\n\\[\nM_w\\longmapsto \\begin{cases}\nM_{(ws^C_i)^\\dagger}-q^{-1}M_{w^\\dagger} \t&ws^C_0\\in\\mathcal P^\\mathrm{min}_{C_{m-1}}\\text{ and }l(ws^C_i)>l(w),\\\\\nM_{(ws^C_i)^\\dagger}-qM_{w^\\dagger} \t\t\t\t&ws^C_0\\in\\mathcal P^\\mathrm{min}_{C_{m-1}}\\text{ and }l(ws^C_i)<l(w),\\\\\n0 \t\t\t\t\t\t\t\t&ws^C_i\\notin\\mathcal P^\\mathrm{min}_{C_{m-1}},\n\\end{cases}\n\\]\nfor $i\\in\\{1,\\ldots,m-2\\}$, and the composition\n\\[\n\\mathcal M_{C_{m-1}}\\xrightarrow{(\\,.\\,)^\\dagger}\\mathcal M_{D_m}\\xrightarrow{\\cdot C_{s_{i+1}^D}}\\mathcal M_{D_m}\n\\]\nis given by\n\\[\nM_w\\longmapsto \\begin{cases}\nM_{w^\\dagger s^D_{i+1}}-q^{-1}M_{w^\\dagger} \t&w^\\dagger s^D_{i+1}\\in\\mathcal P^\\mathrm{min}_{D_m}\\text{ and }l(ws^D_{i+1})>l(w),\\\\\nM_{w^\\dagger s^D_{i+1}}-qM_{w^\\dagger} \t\t\t\t&w^\\dagger s^D_{i+1}\\in\\mathcal P^\\mathrm{min}_{D_m}\\text{ and }l(ws^D_{i+1})<l(w),\\\\\n0 \t\t\t\t\t\t\t\t&w^\\dagger s^D_{i+1}\\notin\\mathcal P^\\mathrm{min}_{D_m}.\n\\end{cases}\n\\]\nBy comparing the two results using the combinatorial description, the claim follows.\n\\end{proof}\n\n\\subsection{Diagrammatical reformulation}\n\\label{sec:diagrammatic_Hecke_module}\nHenceforth, we identify $\\mathcal M_{D_m}$ with $\\mathcal M^\\textrm{comb}_m=\\mathcal L[C_\\mathrm{KL}(m)]$ (as $\\mathcal L$-module) by sending $\\underline{M}_w$ to the the cup diagram $\\ba_w$ associated with $w\\in\\mathcal P_{D_m}^\\mathrm{min}$ which is constructed as follows: Firstly, connect all neighbored $\\lor\\land$ symbols by an unmarked cup successively, thereby ignoring all pairs which are already connected. Secondly, connect the remaining neighbored $\\land$ pairwise by a marked cup starting from the left. Finally, attach rays to the symbols which are not yet connected to a cup, where a ray is marked if and only if it is connected to a $\\land$. The assignment $w\\mapsto\\ba_w$ defines a bijection between $\\Lambda^m_\\mathrm{even}$ and $C_\\mathrm{KL}(m)$, see \\cite[Section 5.2]{LS13}.\n\n\\begin{prop}[Hecke module in Type D] \\label{prop:diagrammatic_Hecke_actionI}\nThe right action of the Hecke algebra $\\mathcal H_{D_m}$ of type $D$ on $\\mathcal M_{D_m}$ can be described via the identification $\\mathcal M_{D_m}\\cong\\mathcal M^\\textrm{comb}_m$ as follows: Representing the Kazhdan-Lusztig generators $C_{s_0^D}$, $C_{s_i^D}\\in \\mathcal H_{D_m}$, where $i\\in\\{1,\\ldots,m-1\\}$ pictorially as \n\\[\nC_{s_0^D}:=\\begin{tikzpicture}[thick, scale=.8, baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\draw[thick] (0,-1) .. controls +(0,.5) and +(0,.5) .. +(.5,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.635) rectangle ++(5pt,5pt);\n\\draw[thick] (1,0) -- +(0,-1);\n\\draw[thick] (1.5,0) -- +(0,-1);\n\\draw[thick] (2.5,0) -- +(0,-1);\n\n\\node at (2,-.5) {\\dots};\n\\begin{footnotesize}\n\\node at (0,.2) {1};\n\\node at (.5,.2) {2};\n\\end{footnotesize}\n\\end{tikzpicture}\n\\hspace{5em}\nC_{s_i^D}:=\\begin{tikzpicture}[thick, scale=.8, baseline={(0,-.5)}]\n\\draw[thick] (0,0) -- +(0,-1);\n\\draw[thick] (.5,0) -- +(0,-1);\n\\draw[thick] (1.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1.5,-1) .. controls +(0,.5) and +(0,.5) .. +(.5,0);\n\\draw[thick] (3,0) -- +(0,-1);\n\n\\node at (1,-.5) {\\dots};\n\\node at (2.5,-.5) {\\dots};\n\n\\begin{footnotesize}\n\\node at (1.4,.22) {i};\n\\node at (2,.2) {i+1};\n\\end{footnotesize}\n\\end{tikzpicture}\n\\]\nthen we can compute $\\ba.C_{s_i^D}$, where $\\ba\\in C_\\mathrm{KL}(m)$ is a cup diagram, by putting the picture corresponding to the generator $C_{s_i^D}$ on top of $\\ba$ and applying the relations \n\\begin{equation}\n\\label{firstrow}\n\\begin{tikzpicture}[thick, scale=.8]\n\\draw[dotted] (-.25,-.5) circle(16pt);\n\\draw[dotted] (3.8,-.5) circle(16pt);\n\\draw[thick] (-.25,-.5) circle(8pt);\n\n\\node at (1.8,-.5) {$= -(q+q^{-1})\\cdot$};\n\\end{tikzpicture}\n\\hspace{3em}\n\\begin{tikzpicture}[thick, scale=.8]\n\\draw[dotted] (-.25,-.5) circle(16pt);\n\\draw[dotted] (1.9,-.5) circle(16pt);\n\n\\draw[thick] (-.75,-.5) -- +(1,0);\n\\draw[thick] (1.4,-.5) -- +(1,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]-.05,-.5) rectangle ++(5pt,5pt);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]-.45,-.5) rectangle ++(5pt,5pt);\n\n\\node at (.8,-.5) {$=$};\n\\end{tikzpicture}\n\\hspace{3em}\n\\begin{tikzpicture}[thick, scale=.8]\n\\draw[dotted] (-.25,-.5) circle(16pt);\n\\draw[thick] (-.25,-.5) circle(8pt);\n\n\\fill ([xshift=-2.5pt,yshift=-2.5pt]-.25,-.225) rectangle ++(5pt,5pt);\n\n\\node at (.8,-.5) {$=\\,0$};\n\\end{tikzpicture}\n\\end{equation}\n\\begin{equation}\n\\label{secondrow}\n\\begin{tikzpicture}\n\\draw[dotted] (0,0) .. controls +(0,1) and +(0,1) .. +(1,0);\n\\draw[thick] (0.25,0) .. controls +(0,.5) and +(0,.5) .. +(.5,0);\n\\node at (1.5,.25) {$=0$};\n\\draw[dotted] (0,0) -- +(1,0);\n\\end{tikzpicture}\n\\hspace{4em}\n\\begin{tikzpicture}\n\\draw[dotted] (0,0) .. controls +(0,1) and +(0,1) .. +(1,0);\n\\draw[thick] (0.25,0) .. controls +(0,.5) and +(0,.5) .. +(.5,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].5,.365) rectangle ++(5pt,5pt);\n\\node at (1.5,.25) {$=1$};\n\\draw[dotted] (0,0) -- +(1,0);\n\\end{tikzpicture}\n\\end{equation}\nwhich yields a new cup diagram in $C_\\mathrm{KL}(m)$ equal to $\\ba.C_{s_i^D}$. \n\\end{prop}\nThe relations in \\eqref{secondrow} only hold for connected components with both endpoints at the bottom of the diagram. The second relation hereby means that we can simply remove such a component if it is marked, whereas the first relation always kills the entire diagram. \n\n\\begin{prop}[Hecke module in Type C]\\label{prop:diagrammatic_Hecke_actionII}\nThe right action of the Hecke algebra $\\mathcal H_{C_{m-1}}$ of type $C_{m-1}$ on $\\mathcal M_{C_{m-1}}$ can be described via the identification $\\mathcal M_{C_{m-1}}\\cong\\mathcal M^\\textrm{comb}_m$ as follows: Given a cup diagram $\\ba\\in C_\\mathrm{KL}(m)$ and a generator $C_{s_i^C}\\in \\mathcal H_C$ which we represent pictorially as \n\\[\nC_{s_0^C}:=\\begin{tikzpicture}[thick, scale=.8, baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (0,-1) .. controls +(0,.5) and +(0,.5) .. +(.5,0);\n\\draw[thick] (1,0) -- +(0,-1);\n\\draw[thick] (1.5,0) -- +(0,-1);\n\\draw[thick] (2.5,0) -- +(0,-1);\n\n\\node at (2,-.5) {\\dots};\n\\begin{footnotesize}\n\\node at (0,.2) {1};\n\\node at (.5,.2) {2};\n\\end{footnotesize}\n\\end{tikzpicture}\n\\,\\,\n+\n\\begin{tikzpicture}[thick, scale=.8, baseline={(0,-.5)}]\n\\draw[thick] (0,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.365) rectangle ++(5pt,5pt);\n\\draw[thick] (0,-1) .. controls +(0,.5) and +(0,.5) .. +(.5,0);\n\\fill ([xshift=-2.5pt,yshift=-2.5pt].25,-.635) rectangle ++(5pt,5pt);\n\\draw[thick] (1,0) -- +(0,-1);\n\\draw[thick] (1.5,0) -- +(0,-1);\n\\draw[thick] (2.5,0) -- +(0,-1);\n\n\\node at (2,-.5) {\\dots};\n\\begin{footnotesize}\n\\node at (0,.2) {1};\n\\node at (.5,.2) {2};\n\\end{footnotesize}\n\\end{tikzpicture}\n\\hspace{3.5em}\nC_{s_i^C}:=\\begin{tikzpicture}[thick, scale=.8, baseline={(0,-.5)}]\n\\draw[thick] (0,0) -- +(0,-1);\n\\draw[thick] (.5,0) -- +(0,-1);\n\\draw[thick] (1.5,0) .. controls +(0,-.5) and +(0,-.5) .. +(.5,0);\n\\draw[thick] (1.5,-1) .. controls +(0,.5) and +(0,.5) .. +(.5,0);\n\\draw[thick] (3,0) -- +(0,-1);\n\n\\node at (1,-.5) {\\dots};\n\\node at (2.5,-.5) {\\dots};\n\n\\begin{footnotesize}\n\\node at (1.3,.22) {i+1};\n\\node at (2.1,.2) {i+2};\n\\end{footnotesize}\n\\end{tikzpicture}\n\\hspace{2em}\n\\text{with }i\\in\\{1,\\ldots,m-2\\},\n\\]\nthen $\\ba.C_{s_i^C}$ can be computed by putting the picture corresponding to the generator $C_{s_i^C}$ on top of $\\ba$ and simplifying the resulting pictures using the relations \\eqref{firstrow} and \\eqref{secondrow} again.\n\\end{prop}\n\\begin{proof}[Proof of Propositions~\\ref{prop:diagrammatic_Hecke_actionI} and~\\ref{prop:diagrammatic_Hecke_actionII}.]\nProposition~\\ref{prop:diagrammatic_Hecke_actionI} is the analog of \\cite[Theorem 4.17]{LS13} for the induced trivial module and can be proven by closely following the argument for the induced sign module. Proposition~\\ref{prop:diagrammatic_Hecke_actionII} follows then together with Lemma~\\ref{lem:relating_C_D_module_action}.\n\\end{proof}\n\n\\begin{lem} \\label{lem:filtration}\nThere exists a filtration \n\\begin{equation} \\label{eq:filtration}\n\\{0\\}\\subseteq\\mathcal M^\\textrm{comb}_{m,\\lfloor\\frac{m}{2}\\rfloor}\\subseteq\\ldots\\subset\\mathcal M^\\textrm{comb}_{m,l}\\subseteq\\ldots\\subseteq\\mathcal M^\\textrm{comb}_{m,0}=\\mathcal M^\\textrm{comb}_m\n\\end{equation}\nof both right $\\mathcal H_{D_m}$- and $\\mathcal H_{C_{m-1}}$-modules, where $\\mathcal M^\\textrm{comb}_{m,l}$ is the $\\mathcal L$-span of all cup diagrams in $C_\\mathrm{KL}(m)$ with $l$ or more cups, $0\\leq l\\leq\\lfloor\\frac{m}{2}\\rfloor$. \n\\end{lem}\n\\begin{proof}\nIt follows from Propositions~\\ref{prop:diagrammatic_Hecke_actionI} and~\\ref{prop:diagrammatic_Hecke_actionII} that acting with a Kazhdan-Lusztig generator on a cup diagram with $l$ cups always yields a cup diagram with $l$ or more cups (the number of cups changes if and only if we use the second relation from \\eqref{secondrow}). In particular, the filtration (\\ref{eq:filtration}) of $\\mathcal L$-modules is actually a filtration of right $\\mathcal H_{D_m}$- and $\\mathcal H_{C_{m-1}}$-modules. \n\\end{proof}\n\n\\begin{rem}\nThe filtration \\eqref{eq:filtration} can be identified with the respective filtrations of $\\mathcal M_{D_m}$ and $\\mathcal M_{C_{m-1}}$ by Kazhdan-Lusztig cells with subquotients $\\mathcal M^\\textrm{comb}_{m,l}\/\\mathcal M^\\textrm{comb}_{m,l-1}$ the Kazhdan-Lusztig cell modules.  \n\\end{rem}\n\n\\section[sec]{\\for{toc}{Connection to classical Springer theory}\\except{toc}{Connection to classical Springer theory and the proof of Theorem~\\ref{thm:induced_module}}}\n\\label{section:section5}\n\nIn this section we identify our representations in each homology degree for every two-row Springer fiber by providing an explicit decomposition into irreducible Specht modules. We use the results from Section~\\ref{section:section4} but specialized at $q=1$ in which case the Hecke algebra becomes the group algebra and the parabolic Hecke module becomes the induced trivial representation. \n\n\\subsection{Proof of Theorem~\\ref{thm:induced_module}}\\label{subsection:proof_induced_module}\n\nBy \\cite[Lemma~5.19]{ES12}, the induced trivial representation of type $D$ decomposes as multiplicity free sum of irreducibles labeled by pairs of one-row partitions. If $m$ is odd, it follows that all of these simples must occur in the induced trivial representation in order for the dimension to add up to $2^{m-1}$ (which is the dimension of the induced trivial module because the $\\{\\land\\lor\\}$-sequences in $\\Lambda^m_\\mathrm{even}$ label a basis). Thus, we have a decomposition\n\\begin{equation} \\label{eq:decomposition_into_irreps_odd}\n\\mathbb C\\otimes_{\\mathbb C[S_m]}\\mathbb C[\\mathcal W_{D_m}]\\cong\\bigoplus_{l=0}^{\\frac{m-1}{2}} V_{\\{(m-l),(l)\\}}\n\\end{equation}\n(cf.\\ \\cite[Lemma~5.19]{ES12} and \\cite[Remark 5.21]{ES12}). \n\nIf $m$ is even, a similar argument shows that all simples must occur exactly once, except one of the modules $V_{(\\frac{m}{2})}^+$ or $V_{(\\frac{m}{2})}^-$ is missing (which one depends on the two possible choices of maximal parabolic subgroups of $\\mathcal W_{D_m}$) which yields a decomposition\n\\begin{equation} \\label{eq:decomposition_into_irreps_even}\n\\mathbb C\\otimes_{\\mathbb C[S_m]}\\mathbb C[\\mathcal W_{D_m}]\\cong V^+_{(\\frac{m}{2})} \\oplus\\bigoplus_{l=0}^{\\frac{m}{2}-1} V_{\\{(m-l),(l)\\}}\n\\end{equation}\n\nIn particular, it follows that the filtration (\\ref{eq:filtration}) is already a composition series, i.e.\\ a filtration which cannot be refined. Hence, the subquotients $\\mathcal M^\\textrm{comb}_{m,l}\/\\mathcal M^\\textrm{comb}_{m,l-1}$ are irreducible and isomorphic to the summands in the decompositions (\\ref{eq:decomposition_into_irreps_odd}) and (\\ref{eq:decomposition_into_irreps_even}).\nNote that the subquotients $\\mathcal M^\\textrm{comb}_{m,l}\/\\mathcal M^\\textrm{comb}_{m,l-1}$ of this filtration are isomorphic to the representations $H_{2l}(\\SOfiberequalsize)$ constructed in Theorem~\\ref{thm:skein_calculus}. This follows easily by using the relation $H_{s_i^D}=C_{s_i^D}+q^{-1}H_e$ and comparing the local relations for $q=1$. In particular, $H_*\\left(\\SOfiberequalsize\\right)$ is isomorphic to the induced trivial module because both have the same irreducible constituents. This proves Theorem~\\ref{thm:induced_module} for type $D$.\n\nFor the type $C$ case, note that by Proposition~\\ref{prop:diagrammatic_Hecke_actionII} (specialized at $q=1$) the $\\mathcal W_{C_{m-1}}$-representation $\\mathcal M^\\textrm{comb}_m$ is isomorphic to $\\mathbb C\\otimes_{\\mathbb C[S_{m-1}]}\\mathbb C[\\mathcal W_{C_{m-1}}]$. Thus, it suffices to check that $\\mathcal M^\\textrm{comb}_m$ is isomorphic to $H_*(\\SOfiberequalsize)$ as $\\mathcal W_{C_{m-1}}$-module. Since $\\mathcal M^\\textrm{comb}_m$ is isomorphic to $H_*(\\SOfiberequalsize)$ as $\\mathcal W_{D_m}$-module by Theorem~\\ref{thm:explicit_identification_D}, it suffices to check that the $\\mathcal W_{C_{m-1}}$-action on $\\mathcal M^\\textrm{comb}_m$ is obtained by restricting the $\\mathcal W_{D_m}$-action to the subgroup generated by $s_0^Ds_1^D,s_2^D,\\ldots,s^D_{m-1}$. But this is clear from Propositions~\\ref{prop:diagrammatic_Hecke_actionI} and~\\ref{prop:diagrammatic_Hecke_actionII}.\n\n\\subsection{Explicit identifications in type D}\n\nIn the following we abbreviate $M=\\{1,\\ldots,m\\}$. If $m$ is even, the Specht module $V_{((\\frac{m}{2}),(\\frac{m}{2}))}$ decomposes as a direct sum of two non-isomorphic irreducible $\\mathcal W_{D_m}$-modules. In order to describe this decomposition note that the group $\\mathbb Z\/2\\mathbb Z$ acts on the line diagram basis of $H_m((\\mathbb S^2)^m)$ by sending $l_U$ to $l_{M\\setminus U}$. Let $\\mathcal R_m$ be a set of representatives of the orbits under this action. Then one of the irreducible summands of the $\\mathcal W_{D_m}$-module $H_m((\\mathbb S^2)^m)$ has a basis given by all sums $l_U+l_{M\\setminus U}$, $U\\in\\mathcal R_m$, and the second one has a basis given by all sums $l_U-l_{M\\setminus U}$, $U\\in\\mathcal R_m$. We write $V^+_{(\\frac{m}{2})}$ (resp.\\ $V^-_{(\\frac{m}{2})}$) to denote these irreducible modules.\n\n\\begin{thm} \\label{thm:explicit_identification_D}\nWe have the following isomorphisms of $\\mathcal W_{D_m}$-modules:\n\\begin{enumerate}\n\\item\\label{item:d1} If $k$ is odd, then we have, for all $l\\in\\{0,\\ldots,\\lfloor\\frac{k}{2}\\rfloor\\}$, isomorphisms\n\\[\nH_{2l}\\left(\\SOfiber\\right)\\cong V_{\\{(m-l),(l)\\}}.\n\\]\n\\item\\label{item:d2} \nIf $k$ is even (which implies $k=m$), then we have, for all $l\\in\\{0,\\ldots,\\frac{m}{2}-1\\}$,  isomorphisms\n\\[\nH_m(\\SOfiberequalsize)\\cong V_{(\\frac{m}{2})}^{+} \\hspace{1.6em}\\text{and}\\hspace{1.6em} H_{2l}\\left(\\SOfiberequalsize\\right)\\cong V_{\\{(m-l),(l)\\}}. \n\\]\n\\end{enumerate}\nThe isomorphisms in (\\ref{item:d1}) and (\\ref{item:d2}) are explicitly given by \n\\begin{equation}\\label{eq:explicit_map}\n\\ba\\mapsto\\sum_{U\\in\\mathcal U_\\ba}(-1)^{\\Lambda_\\ba(U)}e_{T_U}.\n\\end{equation}\n\\end{thm}\n\\begin{proof}\nIf $k$ is odd, then the injection \n\\begin{equation} \\label{eq:composition_yielding_iso_d1}\nH_{2l}(\\SOfiber)\\hookrightarrow H_{2l}((\\mathbb S^2)^m)\\xrightarrow\\cong V_{((m-l),(l))}\n\\end{equation}\nobtained by composing the maps of $\\mathcal W_{D_m}$-modules from Proposition~\\ref{prop:explicit_isom} and~\\ref{proposition:Specht_modules_as_homology} is given by \n\\begin{equation} \\label{eq:assignment_yielding_iso_d1}\n\\ba\\mapsto L_\\ba=\\sum_{U\\in\\mathcal U_\\ba}(-1)^{\\Lambda_\\ba(U)}l_U\\mapsto\\sum_{U\\in\\mathcal U_\\ba}(-1)^{\\Lambda_\\ba(U)}e_{T_U}.\n\\end{equation}\nThis is an isomorphism for all $l\\in\\{0,\\ldots,\\lfloor\\frac{k}{2}\\rfloor\\}$ because $H_{2l}(\\SOfiber)\\neq 0$ and hence the image of the $\\mathcal W_{D_m}$-equivariant injection (\\ref{eq:composition_yielding_iso_d1}) must be equal to the irreducible module $V_{((m-l),(l))}$. \n\nIf $k$ is even, the same argument is true except for the top non-vanishing homology. Then, analogous to (\\ref{eq:composition_yielding_iso_d1}), we have an injection  \n\\begin{equation}\\label{eq:injection_d}\nH_m(\\SOfiberequalsize)\\hookrightarrow H_m((\\mathbb S^2)^m)\\xrightarrow\\cong V^+_{(\\frac{m}{2})} \\oplus V^-_{(\\frac{m}{2})}.\n\\end{equation}\ngiven by the same assignment as in (\\ref{eq:assignment_yielding_iso_d1}).  \n\nIt remains to prove that the image of this map is contained in $V^+_{(\\frac{m}{2})}$. Let $\\ba$ be a cup diagram viewed as a basis vector of $H_m(\\SOfiberequalsize)$. Since $m$ is even, $\\ba$ consists of cups only. Given $U\\in \\mathcal U_{\\ba}$, then $M\\setminus U$ is also contained in $\\mathcal U_{\\ba}$. In fact, in comparison to $U$, $M\\setminus U$ corresponds to choosing exactly the opposite endpoints of all cups in $\\ba$. Thus, we can write\n\\[\nL_\\ba=\\sum_{U\\in\\mathcal U_\\ba}(-1)^{\\Lambda_\\ba(U)}l_U = \\sum_{U\\in\\mathcal U'_\\ba}\\left((-1)^{\\Lambda_\\ba(U)}l_U + (-1)^{\\Lambda_\\ba(M\\setminus U)}l_{M\\setminus U}\\right)= \\sum_{U\\in\\mathcal U'_\\ba}(-1)^{\\Lambda_\\ba(U)}\\left(l_U + l_{M\\setminus U}\\right),\n\\]   \nwhere $\\mathcal U'_\\ba\\subseteq\\mathcal U_\\ba$ consists of all subsets which are also contained in $\\mathcal R_m$. The condition that the number of marked rays plus unmarked cups must be even implies that we have an even number of unmarked cups. Note that $(-1)^{\\Lambda_\\ba(U)}= (-1)^{\\Lambda_\\ba(M\\setminus U)}$ because opposite endpoints of unmarked cups contribute different signs to the respective exponent. Thus, the injective map (\\ref{eq:injection_d}) has its image in the irreducible module $V^+_{(\\frac{m}{2})}$.\n\\end{proof}\n\n\n\\subsection{Explicit identifications in type C}\n\nIf $m\\neq k$ or $m=k$ is odd, then there is a non-trivial component group action on $H_*(\\SOfiber)$. This action on $H_*(\\SOfiber)$ restricts to an action on the set $C_\\mathrm{KL}^{\\leq\\lfloor\\frac{k}{2}\\rfloor}(m)$. Each orbit consists of one or two elements. Let $\\mathcal R$ be a chosen set of representatives of the orbits and let $\\mathcal R_{\\bf 1}$ (resp.\\ $\\mathcal R_{\\bf 2}$) denote the set of representatives contained in a one-element orbit (resp.\\ two-element orbit). We thus have a decomposition $\\mathcal R=\\mathcal R_{\\bf 1}\\cup\\mathcal R_{\\bf 2}$. Given a cup diagram $\\ba\\in\\mathcal R_{\\bf 2}$, we write $\\ba^-$ to denote the second element in its orbit. \n\nAn irreducible character $\\phi$ of $(\\mathbb Z\/2\\mathbb Z)^t$ is completely determined by the values $\\phi(\\alpha_j)\\in\\{\\pm 1\\}$, where $\\alpha_j$ is the generator of the $j$th copy of $\\mathbb Z\/2\\mathbb Z$. This yields a bijection between the set of irreducible characters of $(\\mathbb Z\/2\\mathbb Z)^t$ and the set of all ordered $\\{+,-\\}$-sequences of length $t$. We write $\\mathbb C_\\epsilon$ to denote the irreducible module for $\\epsilon$. Let $H_*^\\epsilon(\\SOfiber)$ be the isotypic subspace of $H_*(\\SOfiber)$ corresponding to the irreducible representation $\\mathbb C_\\epsilon$ of $(\\mathbb Z\/2\\mathbb Z)^t$ labeled by the tuple $\\epsilon$.\n\nThe following result describes the decomposition of the homology into irreducibles with respect to the component group action including an explicit basis of each isotypic subspace.\n\n\\begin{prop} \\label{prop:basis_isotypic_comp}\nWe have the following decompositions into isotypic components:\n\\begin{enumerate}\n\\item If $m$ is odd, we have for all $l\\in\\{0,\\ldots,\\lfloor\\frac{m}{2}\\rfloor\\}$ isomorphisms of $\\mathbb Z\/2\\mathbb Z$-modules\n\\[\nH_{2l}(\\SOfiberequalsize)\\cong\\underbrace{\\mathbb C_{(1)}\\oplus\\ldots\\oplus\\mathbb C_{(1)}}_{\\binom{m-1}{l}\\text{ copies}}\\oplus\\underbrace{\\mathbb C_{(-1)}\\oplus\\ldots\\oplus\\mathbb C_{(-1)}}_{\\binom{m-1}{l-1}\\text{ copies}}.\n\\]\n\\item If $m\\neq k$, we have for all $l\\in\\{0,\\ldots,\\frac{k-1}{2}\\}$ isomorphisms of $\\left(\\mathbb Z\/2\\mathbb Z\\right)^2$-modules\n\\[\nH_{2l}(\\SOfiber)\\cong\\mathbb C_{(1,1)}\\oplus\\ldots\\oplus\\mathbb C_{(1,1)}\\oplus\\mathbb C_{(-1,-1)}\\oplus\\ldots\\oplus\\mathbb C_{(-1,-1)}.\n\\]\n\\end{enumerate}\nMoreover, the isotypic subspace $H_*^{(1)}(\\SOfiberequalsize)$ (resp.\\ $H_*^{(1,1)}(\\SOfiber)$) has a basis consisting of all cup diagrams $\\ba\\in\\mathcal R_{\\bf 1}$ together with all elements $\\bb+\\bb^{-}$, where $\\bb$ varies over all $\\mathcal R_{\\bf 2}$. The subspace $H_*^{(-1)}(\\SOfiberequalsize)$ (resp.\\ $H_*^{(-1,-1)}(\\SOfiber)$) has a basis consisting of all elements $\\bb-\\bb^{-}$, where again $\\bb$ varies over all $\\mathcal R_{\\bf 2}$.\n\\end{prop}\n\\begin{proof}\nWe only argue for $\\SOfiberequalsize$ and omit the other case. Note that the collection of vectors\n\\[\n\\ba\\in\\mathcal R_{\\bf 1}\\,,\\hspace{1em} \\bb+\\bb^{-}\\,,\\,\\,\\bb-\\bb^{-}\\,,\\,\\,\\bb\\in\\mathcal R_{\\bf 2}\n\\]\nis clearly a basis of $H_*(\\SOfiberequalsize)$. Moreover, by Theorem~\\ref{thmB}, $\\ba\\in\\mathcal R_{\\bf 1}$ and $\\bb+\\bb^{-}$, $\\bb\\in\\mathcal R_{\\bf 2}$, are fixed points under the action of the component group and thus contained in $H_*^{(1)}(\\SOfiberequalsize)$. Since the generators of the component group act by exchanging $\\bb$ and $\\bb^-$, it follows that the vectors $\\bb-\\bb^{-}$, $\\bb\\in\\mathcal R_{\\bf 2}$, are contained in $H_*^{(-1)}(\\SOfiberequalsize)$.   \n\nFirstly, since the elements of $\\mathcal R_{\\bf 1}$ all have a ray at the first vertex, the cardinality of $\\mathcal R_{\\bf 1}$ equals the cardinality of the set of all undecorated cup diagrams on $m-1$ vertices with $l$ cups. By \\cite[Proposition~3]{SW12} the set of all such cup diagrams is in bijection with the set of standard Young tableaux of size $m-1$ and shape $(m-1-l,l)$. By the hook-length formula there are exactly $\\binom{m-1}{l}-\\binom{m-1}{l-1}$ such tableaux which equals the cardinality of $\\mathcal R_{\\bf 1}$. Secondly, we have an equality \n\\[\n\\binom{m}{l}=\\dim H_{2l}(\\SOfiberequalsize)= 2|\\mathcal R_{\\bf 2}|+|\\mathcal R_{\\bf 1}| = 2|\\mathcal R_{\\bf 2}|+\\binom{m-1}{l}-\\binom{m-1}{l-1}\n\\]\nwhich yields $|\\mathcal R_{\\bf 2}|=\\binom{m-1}{l-1}$. Thus, $\\dim H_{2l}^{(-1)}(\\SOfiberequalsize)=\\binom{m-1}{l-1}$ and $\\dim H_{2l}^{(1)}(\\SOfiberequalsize)=\\binom{m-1}{l}$.\n\\end{proof}\n\n\n\n\\begin{thm} \\label{thm:explicit_identification_C1}\nWe have the following isomorphisms of $\\mathcal W_{C_{m-1}}$-modules:\n\\begin{enumerate}\n\\item \\label{item:C_mm_odd} If $m$ is odd, then we have, for all $l\\in\\{0,\\ldots,\\lfloor\\frac{m}{2}\\rfloor\\}$, isomorphisms\n\\begin{equation}\\label{eq:explicit_C_isos_Case1}\nH_{2l}^{(1)}(\\SOfiberequalsize)\\cong V_{((m-l-1),(l))} \\hspace{1.6em}\\text{and}\\hspace{1.6em} H_{2l}^{(-1)}(\\SOfiberequalsize)\\cong V_{((l-1),(m-l))},\n\\end{equation}\nwhere the first isomorphism in (\\ref{eq:explicit_C_isos_Case1}) is given by\n\\begin{equation} \\label{eq:formula_C_iso1}\n\\ba+\\ba^{-}\\mapsto 2\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\notin\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_{\\ba}(U)}e_{T_{U-1}} \\hspace{.3em}(\\ba\\in\\mathcal R_2),\\hspace{1.6em}\\ba\\mapsto\\sum_{U\\in\\mathcal U_\\ba}(-1)^{\\Lambda_{\\ba}(U)}e_{T_{U-1}}\\hspace{.3em}(\\ba\\in\\mathcal R_1).\n\\end{equation}\nand the second isomorphism in (\\ref{eq:explicit_C_isos_Case1}) is given by\n\\begin{equation} \\label{eq:formula_C_iso2}\n\\ba-\\ba^{-}\\mapsto 2\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\in\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_{\\ba}(U)}e_{T_{(M\\setminus U)-1}}\\hspace{.3em}(\\ba\\in\\mathcal R_2).\n\\end{equation}\n\\item If $m\\neq k$, then we have, for all $l\\in\\{0,\\ldots,\\frac{k-1}{2}\\}$, isomorphisms\n\\[\nH_{2l}^{(1,1)}\\left(\\SOfiber\\right)\\cong V_{((m-l-1),(l))} \\hspace{1.6em}\\text{and}\\hspace{1.6em} H_{2l}^{(-1,-1)}(\\SOfiber)\\cong V_{((l-1),(m-l))},\n\\]\nwhere the left isomorphism is given by the same formulas as in (\\ref{eq:formula_C_iso1}) and the right isomorphism is given by (\\ref{eq:formula_C_iso2}).\n\\item \\label{item:C_mm_even} If $m=k$ is even, then we have isomorphisms\n\\begin{equation}\n\\label{drei}\nH_m(\\SOfiberequalsize)\\cong V_{((\\frac{m}{2}-1),(\\frac{m}{2}))}\\,\\,,\\,\\,\\,\\ba\\mapsto\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\notin\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_{\\ba}(U)}e_{T_{U-1}}.\n\\end{equation}\nand for all $l\\in\\{0,\\ldots,\\frac{m}{2}-1\\}$ \n\\begin{align*}\nH_{2l}(\\SOfiberequalsize)\\cong H_{2l}(\\SOfiber)&\\cong H_{2l}^{(1,1)}\\left(\\SOfiber\\right)\\oplus H_{2l}^{(-1,-1)}\\left(\\SOfiber\\right)\\\\\n&\\cong V_{((m-l-1),(l))}\\oplus V_{((l-1),(m-l))}\n\\end{align*}\nwhich again can be described explicitly via (\\ref{eq:formula_C_iso1}) and (\\ref{eq:formula_C_iso2}). \n\\end{enumerate}\n\\end{thm}\n\\begin{proof}\nIn order to relate the $\\mathcal W_{C_{m-1}}$-module $H_*((\\mathbb S^2)^{m-1})$ from Proposition~\\ref{proposition:Specht_modules_as_homology} (with the bipolytabloid basis) to the $\\mathcal W_{C_{m-1}}$-module $H_*((\\mathbb S^2)^m)$ from Section~\\ref{section:section2} (which contains the Springer representation as a subrepresentation) we define the following $\\mathcal W_{C_{m-1}}$-equivariant injections:\n\\begin{itemize}\n\\item If $2l<m$ we define $\\iota\\colon H_{2l}((\\mathbb S^2)^{m-1})\\to H_{2l}((\\mathbb S^2)^m)$ to be the map given by $l_U \\mapsto l_{U+1}$, where $U+1$ is obtained from $U$ by adding $1$ to all elements in $U$, i.e.\\ $l_{U+1}$ is obtained by adding a dotted line to the left of $l_U$. \n\\item If $2l>m$ we define $\\kappa\\colon H_{2l}((\\mathbb S^2)^{m-1})\\to H_{2(m-l)}((\\mathbb S^2)^m)$ to be the map $l_U\\mapsto l_{M\\setminus (U+1)}$, i.e.\\ $l_{M\\setminus (U+1)}$ is obtained from $l_U$ by first adding a dotted line to the left of $l_U$ and then exchanging all dotted lines by undotted lines and vice versa. \n\\item If $2l=m$ we define $\\chi\\colon H_{2l}((\\mathbb S^2)^{m-1})\\to H_{2l}((\\mathbb S^2)^m)$ to be the map $l_U\\mapsto l_{U+1}+l_{M\\setminus (U+1)}$.\n\\end{itemize}\n\nIn order to prove the first isomorphism in \\eqref{eq:explicit_C_isos_Case1} we claim that we have a well-defined composition \n\\[\nH_{2l}^{(1)}(\\mathcal S^{m,m}_\\mathrm{KL})\\hookrightarrow \\im(\\iota) \\xrightarrow\\cong H_{2l}((\\mathbb S^2)^{m-1})\\xrightarrow\\cong V_{((m-l-1),(l))},\n\\]\ni.e.\\ we claim that $\\im\\left(\\gamma_m\\right)\\subseteq\\im\\left(\\iota\\right)$. Note that the image of $\\iota$ has a basis given by all line diagrams on $m$ vertices with $l$ undotted lines, where the line connected to the first vertex is dotted. If $\\ba\\in\\mathcal R_1$, we obviously have $L_\\ba\\in \\im(\\iota)$. If $\\ba\\in\\mathcal R_2$, then we can simplify\n\\begin{align*}\nL_\\ba+L_{{\\ba}^-}&=\\sum_{U\\in\\mathcal U_\\ba}(-1)^{\\Lambda_\\ba(U)}l_U+\\sum_{U\\in\\mathcal U_{{\\ba}^-}}(-1)^{\\Lambda_{{\\ba}^-}(U)}l_U\\\\\n\t\t\t\t\t\t\t\t &=\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\in\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_\\ba(U)}l_U+\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\notin\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_\\ba(U)}l_U+\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_{{\\ba}^-}\\\\ 1\\in\\mathcal U_{\\ba^-}\\end{subarray}}(-1)^{\\Lambda_{{\\ba}^-}(U)}l_U+\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_{{\\ba}^-}\\\\ 1\\notin\\mathcal U_{\\ba^-}\\end{subarray}}(-1)^{\\Lambda_{{\\ba}^-}(U)}l_U\\\\\n\t\t\t\t\t\t\t\t &=2\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\notin\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_\\ba(U)}l_U,\n\\end{align*}\nbecause in the second row the first and third summand cancel each other and the second and fourth summand are equal which shows that $L_\\ba+L_{{\\ba}^-}\\in\\im(\\iota)$. For $\\ba\\in\\mathcal R_1$ the composition is then explicitly given by\n\\[\n\\ba\\mapsto L_\\ba=\\sum_{U\\in\\mathcal U_\\ba}(-1)^{\\Lambda_\\ba(U)}l_U \\mapsto \\sum_{U\\in\\mathcal U_\\ba}(-1)^{\\Lambda_{\\ba}(U)}l_{U-1} \\mapsto\\sum_{U\\in\\mathcal U_\\ba}(-1)^{\\Lambda_{\\ba}(U)}e_{T_{U-1}}\n\\]\nand for $\\ba\\in\\mathcal R_2$ we have\n\\[\n\\ba+\\ba^-\\mapsto L_\\ba+L_{{\\ba}^-}=2\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\notin\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_\\ba(U)}l_U\\mapsto 2\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\notin\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_\\ba(U)}l_{U-1}\\mapsto 2\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\notin\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_{\\ba}(U)}e_{T_{U-1}}.\n\\]\n\nFor the second isomorphism in \\eqref{eq:explicit_C_isos_Case1} we consider the composition\n\\[\nH_{2l}^{(-1)}(\\SOfiberequalsize)\\hookrightarrow \\im(\\kappa)\\xrightarrow\\cong H_{2(m-l)}((\\mathbb S^2)^{m-1})\\xrightarrow\\cong V_{((l-1),(m-l))}.\n\\]\nThe image of $\\kappa$ has a basis given by all line diagrams on $m$ vertices with $m-l$ undotted lines, where the first line is undotted. A similar calculation as for $L_\\ba+L_{{\\ba}^-}$ shows that we have\n\\[\nL_\\ba-L_{\\ba^-}=2\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\in\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_\\ba(U)}l_U\n\\]\nfor $\\ba\\in\\mathcal R_1$ which shows that $L_\\ba-L_{\\ba^-}\\in\\im(\\kappa)$. The composition is then given by\n\\[\n\\ba-\\ba^-\\mapsto L_\\ba-L_{\\ba^-}=2\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\in\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_\\ba(U)}l_U\\mapsto 2\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\in\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_{\\ba}(U)}l_{(M\\setminus U)-1}\\mapsto 2\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\in\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_{\\ba}(U)}e_{T_{(M\\setminus U)-1}}.\n\\]\nSince the $\\mathcal W_{C_{m-1}}$-equivariant injections above map to an irreducible target space we see that they are actually isomorphisms.\n\nFor the isomorphism in \\eqref{drei} we consider the composition\n\\[\nH_m(\\SOfiberequalsize)\\hookrightarrow\\im(\\chi)\\xrightarrow\\cong H_m((\\mathbb S^2)^{m-1})\\xrightarrow\\cong V_{((\\frac{m}{2}-1),(\\frac{m}{2}))}.\n\\]\nLet $\\ba\\in H_m(\\SOfiberequalsize)$. Then we have\n\\[\nL_\\ba = \\sum_{U\\in\\mathcal U_\\ba} (-1)^{\\Lambda_\\ba(U)}l_U = \\sum_{U\\in\\mathcal U'_\\ba}(-1)^{\\Lambda_\\ba(U)}\\left(l_U+l_{M\\setminus U}\\right),\n\\]   \nas in the proof of Theorem~\\ref{thm:explicit_identification_D}. From this description we deduce that $L_\\ba$ is contained in the image of $\\chi$ and the composition is given by \n\\[\n\\ba\\mapsto L_\\ba=\\sum_{U\\in\\mathcal U'_\\ba}(-1)^{\\Lambda_\\ba(U)}\\left(l_U+l_{M\\setminus U}\\right)\\mapsto\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\notin\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_{\\ba}(U)}l_{U-1} \\mapsto\\sum_{\\begin{subarray}{l}U\\in\\mathcal U_\\ba\\\\ 1\\notin\\mathcal U_\\ba\\end{subarray}}(-1)^{\\Lambda_{\\ba}(U)}e_{T_{U-1}}.\\qedhere\n\\]\n\\end{proof}\n\n\\subsection{Recovering the Springer correspondence} \\label{subsec:Shoji}\nGiven a bipartition $\\tau=(\\lambda,\\mu)$ such that $|\\lambda|+|\\mu|=m-1$, we can use the Shoji algorithm \\cite{Sho79}, \\cite{Sho83} to compute the Jordan type of the nilpotent endomorphism for which the representation $V_{(\\lambda,\\mu)}$ appears in the top degree of the homology of the corresponding Springer fiber of type $C$. The type $D$ case is similar and omitted due to the trivial component group action. We first recall the algorithm in our special case. \n\n\nGiven a pair of partitions $\\tau=(\\lambda,\\mu)$, define a pair $(d_\\tau^{(1)},\\varepsilon_\\tau^{(1)})$, where $d_\\tau^{(1)}=(d_1^{(1)},d_2^{(1)},\\ldots)\\in\\mathbb Z^{\\mathbb N}$ and $\\varepsilon_\\tau^{(1)}=(\\varepsilon_1^{(1)},\\varepsilon_2^{(1)},\\ldots)\\in\\{\\pm 1\\}^{\\mathbb N}$, as \n\\[\nd_{2i-1}^{(1)}=2\\lambda_i, d_{2i}^{(1)}=2\\mu_i, \\epsilon_i^{(1)}=1.\n\\]\nAssuming that $(d_\\tau^{(j)},\\varepsilon_\\tau^{(j)})$ is constructed, we define a new pair $(d_\\tau^{(j+1)},\\varepsilon_\\tau^{(j+1)})$ by the rules\n\\[\n\\begin{cases}\nd_i^{(j+1)}=d_i^{(j)}+1, d_{i+1}^{(j+1)}=d_i^{(j)}+1 &\\text{if }d_{i+1}^{(j)}=d_i^{(j)}+2,\\\\\nd_i^{(j+1)}=d_{i+1}^{(j)}-2, d_{i+1}^{(j+1)}=d_i^{(j)}+2 &\\text{if }d_{i+1}^{(j)}>d_i^{(j)}+2,\\\\\nd_i^{(j+1)}=d_i^{(j)} &\\text{else},\n\\end{cases}\n\\]\nand\n\\[\n\\begin{cases}\n\\varepsilon_i^{(j+1)}=-\\varepsilon_i^{(j)}, \\varepsilon_{i+1}^{(j+1)}=-\\varepsilon_{i+1}^{(j)} &\\text{if }d_{i+1}^{(j)}>d_i^{(j)}+2,\\\\\n\\varepsilon_i^{(j+1)}=\\varepsilon_i^{(j)} &\\text{else}.\n\\end{cases}\n\\] \nThis is a well-defined algorithm which stabilizes, i.e.\\ there exists an integer $q>0$ such that $(d_\\tau^{(q)},\\varepsilon_\\tau^{(q)})=(d_\\tau^{(q+1)},\\varepsilon_\\tau^{(q+1)})=\\cdots$. We denote the stable pair by $(d_\\tau,\\varepsilon_\\tau)$. \n\nThen $d_\\tau$ is a partition labeling a nilpotent orbit of type $C$. Moreover, $\\varepsilon_\\tau$ can be used to define a character of the associated component group by setting $\\phi_\\tau(\\alpha_{d_i})=\\varepsilon_i$ for all even parts $d_i$ of the partition $d$. We have $d_i=d_j$ if and only if $\\varepsilon_i=\\varepsilon_j$ which makes this a well-defined assignment. \n\n\n\n\\begin{lem}\nLet $\\tau=((\\lambda),(\\mu))$ be a pair of one-row partitions, i.e.\\ $\\lambda,\\mu\\in\\mathbb Z_{\\geq 0}$. Then we have\n\\[\n\\begin{cases}\nd_\\tau=(2\\lambda+1,2\\lambda+1,0,\\ldots)\\,,\\,\\,\\epsilon_\\tau=(1,\\ldots) &\\text{if }\\mu=\\lambda+1,\\\\\nd_\\tau=(2\\mu-2,2\\lambda+2,0,\\ldots)\\,,\\,\\,\\epsilon_\\tau=(-1,-1,1,\\ldots) &\\text{if }\\mu>\\lambda+1,\\\\\nd_\\tau=(2\\lambda,2\\mu,0,\\ldots)\\,,\\,\\,\\epsilon_\\tau=(1,\\ldots) &\\text{if }\\mu\\leq\\lambda,\n\\end{cases}\n\\]\nwhere the non displayed entries of the sequences are all $0$ or $1$.\n\\end{lem} \n\\begin{proof}\nThis follows directly from the Shoji algorithm above.\n\\end{proof}\n\n\\begin{prop}\nThe Springer representation in top degree cohomology of the $(m-1,m-1)$ algebraic Springer fiber of type $C_{m-1}$ is (in Lusztig's normalization) isomorphic to the irreducible labeled by $(\\frac{m}{2}-1,\\frac{m}{2})$ if $m$ is even. If $m$ is odd, then the top degree representation decomposes as the direct sum of $(\\frac{m-1}{2}-1,\\frac{m-1}{2}+1)$ and $(\\frac{m-1}{2},\\frac{m-1}{2})$, where the first irreducible is the isotypic component corresponding to the sign representation of $\\mathbb Z\/2\\mathbb Z$. \n\nIn case of the nilpotent $(n-k-1,k-1)$ with $m\\neq k$, we have the irreducibles $(\\frac{k-3}{2},\\frac{n-k+1}{2})$ and $(\\frac{n-k-1}{2},\\frac{k-1}{2})$, where the first corresponds to the isotypic component associated with the character $(-1,-1)$ of $\\left(\\mathbb Z\/2\\mathbb Z\\right)^2$ (and the other one to $(1,1)$). \n\\end{prop}\n\n\\begin{rem}\nBy comparing the above proposition with Theorem~\\ref{thm:explicit_identification_C1} (and keeping in mind the shift in the labels described in Remark~\\ref{rem:wilbert}) we see that we have constructed (up to isomorphism) the original Springer representation.\n\\end{rem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nHow can a human train an intelligent agent to understand natural language instructions? We believe that this research question is important from both technological and scientific perspectives. No matter how advanced AI technology becomes, human users will likely want to customize their intelligent helpers to better understand their desires and needs. On the other hand, developmental psychology, cognitive science and linguistics study similar questions but applied to human children, and a synergy is possible between research in grounded language learning by computers and research in human language acquisition. \n\nIn this work, we present the BabyAI research platform, whose purpose is to facilitate research on grounded language learning. In our platform we substitute a simulated human expert for a real human; yet our aspiration is that BabyAI-based studies enable substantial progress towards putting an actual human in the loop. The current domain of BabyAI is a 2D gridworld in which synthetic natural-looking instructions (e.g. ``put the red ball next to the box on your left'') require the agent to navigate the world (including unlocking doors) and move objects to specified locations. BabyAI improves upon similar prior setups \\citep{hermann_grounded_2017,chaplot_gated-attention_2018,yu_interactive_2018} by supporting simulation of certain essential aspects of the future human in the loop agent training: \\textit{curriculum learning} and \\textit{interactive teaching}.\nThe usefulness of curriculum learning for training machine learning models has been demonstrated numerous times in the literature \\citep{bengio_curriculum_2009,kumar_self-paced_2010,zaremba_learning_2015,graves_hybrid_2016}, and we believe that gradually increasing the difficulty of the task will likely be essential for achieving efficient human-machine teaching, as in the case of human-human teaching. To facilitate curriculum learning studies, BabyAI currently features 19 levels in which the difficulty of the environment configuration and the complexity of the instruction language are gradually increased. \nInteractive teaching, i.e. teaching differently based on what the learner can currently achieve, is another key capability of human teachers. Many advanced agent training methods, including DAGGER \\citep{ross_reduction_2011}, TAMER \\citep{warnell_deep_2017} and learning from human preferences \\citep{wilson_bayesian_2012,christiano_deep_2017}, assume that interaction between the learner and the teacher is possible. To support interactive experiments, BabyAI provides a bot agent that can be used to generate new demonstrations on the fly and advise the learner on how to continue acting.\n\nArguably, the main obstacle to language learning with a human in the loop is the amount of data (and thus human-machine interactions) that would be required. Deep learning methods that are used in the context of imitation learning or reinforcement learning paradigms have been shown to be very effective in both simulated language learning settings \\citep{mei_listen_2016,hermann_grounded_2017} and applications \\citep{sutskever_sequence_2014,bahdanau_neural_2015,wu_googles_2016}. These methods, however, require enormous amounts of data, either in terms of millions of reward function queries or hundreds of thousands of demonstrations. To show how our BabyAI platform can be used for sample efficiency research, we perform several case studies. In particular, we estimate the number of demonstrations\/episodes required to solve several levels with imitation and reinforcement learning baselines. As a first step towards improving sample efficiency, we additionally investigate to which extent pretraining and interactive imitation learning can improve sample efficiency.\n\nThe concrete contributions of this paper are two-fold. First, we contribute the BabyAI research platform for learning to perform language instructions with a simulated human in the loop. The platform already contains 19 levels and can easily be extended.\nSecond, we establish baseline results for all levels and report sample efficiency results for a number of learning approaches. The platform and pretrained models are available online. We hope that BabyAI will spur further research towards improving sample efficiency of grounded language learning, ultimately allowing human-in-the-loop training.\n\n\\section{Related Work}\n\nThere are numerous 2D and 3D environments for studying synthetic language acquistion. \\citep{hermann_grounded_2017,chaplot_gated-attention_2018,yu_interactive_2018,wu_building_2018}. Inspired by these efforts, BabyAI augments them by uniquely combining three desirable features. First, BabyAI supports world state manipulation, missing in the visually appealing 3D environments of \\citet{hermann_grounded_2017}, \\citet{chaplot_gated-attention_2018} and \\citet{wu_building_2018}. In these environments, an agent can navigate around, but cannot alter its state by, for instance, moving objects. Secondly, BabyAI introduces partial observability (unlike the gridworld of \\citet{bahdanau_learning_2018}). Thirdly, BabyAI provides a systematic definition of the synthetic language. As opposed to using instruction templates, the Baby Language we introduce defines the semantics of all utterances generated by a context-free grammar (Section~\\ref{sec:language}). This makes our language richer and more complete than prior work. Most importantly, BabyAI provides a simulated human expert, which can be used to investigate human-in-the-loop training, the aspiration of this paper. \n\nCurrently, most general-purpose simulation frameworks do not feature language, such as PycoLab \\citep{deepmind_pycolab_2017}, MazeBase \\citep{sukhbaatar_mazebase:_2015}, Gazebo \\citep{koenig_design_2004}, VizDoom \\citep{kempka_vizdoom:_2016}, DM-30 \\citep{espeholt_impala:_2018}, and AI2-Thor \\citep{kolve_ai2-thor:_2017}. Using a more realistic simulated environment such as a 3D rather than 2D world comes at a high computational cost. Therefore, BabyAI uses a gridworld rather than 3D environments. As we found that available gridworld platforms were insufficient for defining a compositional language, we built a MiniGrid environment for BabyAI.\n\nGeneral-purpose RL testbeds such as the Arcade Learning Environment \\citep{bellemare_arcade_2013}, DM-30 \\citep{espeholt_impala:_2018}, and MazeBase \\citep{sukhbaatar_mazebase:_2015} do not assume a human-in-the-loop setting. In order to simulate this, we have to assume that all rewards (except intrinsic rewards) would have to be given by a human, and are therefore rather expensive to get. Under this assumption, imitation learning methods such as behavioral cloning, Searn \\citep{daume_iii_search-based_2009}, DAGGER \\citep{ross_reduction_2011} or maximum-entropy RL \\citep{ziebart_maximum_2008} are more appealing, as more learning can be achieved per human-input unit. \n\nSimilar to BabyAI, studying sample efficiency of deep learning methods was a goal of the bAbI tasks \\citep{weston_towards_2016}, which tested reasoning capabilities of a learning agent. Our work differs in both of the object of the study (grounded language with a simulated human in the loop) and in the method: instead of generating a fixed-size dataset and measuring the performance, we measure how much data a general-purpose model would require to get close-to-perfect performance.\n\nThere has been much research on instruction following with natural language \\citep{tellex_understanding_2011,chen_learning_2011, artzi_weakly_2013,mei_listen_2016,williams_learning_2018} as well as several datasets including SAIL \\citep{macmahon_walk_2006,chen_learning_2011} and Room-to-Room \\citep{anderson_vision-and-language_2018}. Instead of using natural language, BabyAI utilises a synthetic Baby language, in order to fully control the semantics of an instruction and easily generate as much data as needed. \n\nFinally, \\citet{wang_learning_2016} presented a system that interactively learned language from a human. We note that their system relied on substantial amounts of prior knowledge about the task, most importantly a task-specific executable formal language.\n\n\\section{BabyAI Platform}\n\nThe BabyAI platform that we present in this work comprises an efficiently simulated gridworld environment (MiniGrid) and a number of instruction-following tasks that we call \\textit{levels}, all formulated using subsets of a synthetic language (Baby Language).\nThe platform also includes a bot that can generate successful demonstrations for all BabyAI levels. All the code is available online at \\url{https:\/\/github.com\/mila-iqia\/babyai\/tree\/iclr19}.\n\n\\subsection{MiniGrid Environment}\n\n\\begin{figure}\n  \\begin{minipage}{0.38\\linewidth}\n    \\centering\n    \\subfloat[GoToObj: \"go to the blue ball\"]{\\includegraphics[scale=0.32]{GoToObj}}\\\\\n    \\subfloat[PutNextLocal: \"put the blue key next to the green ball\"]{\\includegraphics[scale=0.32]{PutNextLocal}}\n  \\end{minipage}\n  \\begin{minipage}{0.60\\linewidth}\n    \\centering\n    \\subfloat[BossLevel: \"pick up the grey box behind you, then go to the grey key and open a door\". Note that the green door near the bottom left needs to be unlocked with a green key, but this is not explicitly stated in the instruction.]{\\includegraphics[scale=0.3]{BossLevel}}\n  \\end{minipage}\n\\caption{Three BabyAI levels built using the MiniGrid environment. The red triangle represents the agent, and the light-grey shaded area represents its field of view (partial observation).}\n\\label{fig:minigrid}\n\\end{figure}\n\nStudies of sample efficiency are very computationally expensive given that multiple runs are required for different amounts of data. Hence, in our design of the environment, we have aimed for a minimalistic and efficient environment which still poses a considerable challenge for current general-purpose agent learning methods. We have implemented MiniGrid, a partially observable 2D gridworld environment. The environment is populated with entities of different colors, such as the agent, balls, boxes, doors and keys (see Figure~\\ref{fig:minigrid}). Objects can be picked up, dropped and moved around by the agent. Doors can be unlocked with keys matching their color. At each step, the agent receives a 7x7 representation of its field of view (the grid cells in front of it) as well as a Baby Language instruction (textual string).\n\nThe MiniGrid environment is fast and lightweight. Throughput of over 3000 frames per second is possible on a modern multi-core laptop, which makes experimentation quicker and more accessible.\nThe environment is open source, available online, and supports integration with OpenAI Gym.\nFor more details, see Appendix \\ref{sec:minigrid}.\n\n\\subsection{Baby Language\\label{sec:language}}\n\nWe have developed a synthetic Baby Language to give instructions to the agent as well as to automatically verify their execution. Although Baby Language utterances are a comparatively small subset of English, they are combinatorially rich and unambigously understood by humans. The language is intentionally kept simple, but still exhibits interesting combinatorial properties, and contains $2.48 \\times 10^{19}$ possible instructions. In this language, the agent can be instructed to go to objects, pick up objects, open doors, and put objects next to other objects. The language also expresses the conjunction of several such tasks, for example ``put a red ball next to the green box after you open the door\". The Backus-Naur Form (BNF) grammar for the language is presented in Figure~\\ref{fig:bnf} and some example instructions drawn from this language are shown in Figure~\\ref{fig:instructions}. In order to keep the resulting instructions readable by humans, we have imposed some structural restrictions on this language: the {\\em and} connector can only appear inside the {\\em then} and {\\em after} forms, and instructions can contain no more than one {\\em then} or {\\em after} word.\n\n\\begin{figure}[htb]\n\\begin{bnf*}\n\\bnfprod{Sent}{\n\\bnfpn{Sent1}\\bnfor\n\\bnfpn{Sent1}\\bnfts{ ',' then }\\bnfpn{Sent1}\\bnfor\n\\bnfpn{Sent1}\\bnfts{ after you }\\bnfpn{Sent1}\n}\n\\\\\n\\bnfprod{Sent1}{\n\\bnfpn{Clause}\\bnfor\n\\bnfpn{Clause}\\bnfts{ and }\\bnfpn{Clause}\n}\n\\\\\n\\bnfprod{Clause}{\n\\begin{bnfsplit}\n\\bnfts{go to }\\bnfpn{Descr}\\bnfor\n\\bnfts{pick up }\\bnfpn{DescrNotDoor}\\bnfor\n\\bnfts{open }\\bnfpn{DescrDoor}\\bnfor\n\\bnfts{put }\\bnfpn{DescrNotDoor}\\bnfts{ next to }\\bnfpn{Descr}\n\\end{bnfsplit}\n}\n\\\\\n\\bnfprod{DescrDoor}{\n\\bnfpn{Article}\\bnfts{ }\\bnfpn{Color}\\bnfts{ door }\\bnfpn{LocSpec}\n}\n\\\\\n\\bnfprod{DescrBall}{\n\\bnfpn{Article}\\bnfts{ }\\bnfpn{Color}\\bnfts{ ball }\\bnfpn{LocSpec}\n}\n\\\\\n\\bnfprod{DescrBox}{\n\\bnfpn{Article}\\bnfts{ }\\bnfpn{Color}\\bnfts{ box }\\bnfpn{LocSpec}\n}\n\\\\\n\\bnfprod{DescrKey}{\n\\bnfpn{Article}\\bnfts{ }\\bnfpn{Color}\\bnfts{ key }\\bnfpn{LocSpec}\n}\\\\\n\\bnfprod{Descr}{\n\\bnfpn{DescrDoor}\\bnfor\n\\bnfpn{DescrBall}\\bnfor\n\\bnfpn{DescrBox}\\bnfor\n\\bnfpn{DescrKey}\n}\n\\\\\n\\bnfprod{DescrNotDoor}{\n\\bnfpn{DescrBall}\\bnfor\n\\bnfpn{DescrBox}\\bnfor\n\\bnfpn{DescrKey}\n}\n\\\\\n\\bnfprod{LocSpec}{\n\\bnfes\\bnfor\n\\bnfts{on your left}\\bnfor\n\\bnfts{on your right}\\bnfor\n\\bnfts{in front of you}\\bnfor\n\\bnfts{behind you}\n}\n\\\\\n\\bnfprod{Color}{\n\\bnfes\\bnfor\n\\bnfts{red}\\bnfor\n\\bnfts{green}\\bnfor \n\\bnfts{blue}\\bnfor\n\\bnfts{purple}\\bnfor\n\\bnfts{yellow}\\bnfor\n\\bnfts{grey}\n}\n\\\\\n\\bnfprod{Article}{\n\\bnfts{the}\\bnfor\n\\bnfts{a}\n}\n\\end{bnf*}\n\\caption{BNF grammar productions for the Baby Language}\n\\label{fig:bnf}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\ngo to the red ball\n\\\\\n\\vspace{0.7em}\nopen the door on your left\n\\\\\n\\vspace{0.7em}\nput a ball next to the blue door\n\\\\\n\\vspace{0.7em}\nopen the yellow door and go to the key behind you\n\\\\\n\\vspace{0.7em}\n\\parbox[t]{10cm}{\\centering\nput a ball next to a purple door after you put a blue box next to a grey box and pick up the purple box}\n\\caption{Example Baby Language instructions}\n\\label{fig:instructions}\n\\end{center}\n\\end{figure}\n\nThe BabyAI platform includes a \\textit{verifier} which checks if an agent's sequence of actions successfully achieves the goal of a given instruction within an environment. Descriptors in the language refer to one or to multiple objects. For instance, if an agent is instructed to \"go to a red door\", it can successfully execute this instruction by going to any of the red doors in the environment. The {\\em then} and {\\em after} connectors can be used to sequence subgoals. The {\\em and} form implies that both subgoals must be completed, without ordering constraints. Importantly, Baby Language instructions leave details about the execution implicit. An agent may have to find a key and unlock a door, or move obstacles out of the way to complete instructions, without this being stated explicitly.\n\n\\subsection{BabyAI Levels}\n\nThere is abundant evidence in prior literature which shows that a curriculum may greatly facilitate learning of complex tasks for neural architectures \\citep{bengio_curriculum_2009,kumar_self-paced_2010,zaremba_learning_2015,graves_hybrid_2016}. To investigate how a curriculum improves sample efficiency, we created 19 \\textit{levels} which require understanding only a limited subset of Baby Language within environments of varying complexity. Formally, a level is a distribution of \\textit{missions}, where a mission combines an instruction within an initial environment state. We built levels by selecting \\textit{competencies} necessary for each level and implementing a generator to generate missions solvable by an agent possessing only these competencies. Each competency is informally defined by specifying what an agent should be able to do:\n\\begin{itemize}\n\\item\n\t\\textbf{Room Navigation (ROOM):} navigate a 6x6 room.\n\\item\n\t\\textbf{Ignoring Distracting Boxes (DISTR-BOX):} navigate the environment even when there are multiple distracting grey box objects in it.\n\\item\n\t\\textbf{Ignoring Distractors (DISTR):} same as DISTR-BOX, but distractor objects can be boxes, keys or balls of any color.\n\\item    \n    \\textbf{Maze Navigation (MAZE):} navigate a 3x3 maze of 6x6 rooms, randomly inter-connected by doors.\n\\item\n\t\\textbf{Unblocking the Way (UNBLOCK):} navigate the environment even when it requires moving objects out of the way.\n\\item\n    \\textbf{Unlocking Doors (UNLOCK):} to be able to find the key and unlock the door if the instruction requires this explicitly.\n\\item    \n    \\textbf{Guessing to Unlock Doors (IMP-UNLOCK):} to solve levels that require unlocking a door, even if this is not explicitly stated in the instruction.\n\\item\n    \\textbf{Go To Instructions (GOTO):} understand ``go to'' instructions, e.g. ``go to the red ball''.\n\\item\n\t\\textbf{Open Instructions (OPEN):} understand ``open'' instructions, e.g. ``open the door on your left''.\n\\item\n\t\\textbf{Pickup Instructions (PICKUP):} understand ``pick up'' instructions, e.g. ``pick up a box''.\n\\item    \n    \\textbf{Put Instructions (PUT):} understand ``put'' instructions, e.g. ``put a ball next to the blue key''.\n\\item\n \t\\textbf{Location Language (LOC):} understand instructions where objects are referred to by relative location as well as their shape and color, e.g. ``go to the red ball in front of you''.\n\\item\n\t\\textbf{Sequences of Commands (SEQ):} understand composite instructions requiring an agent to execute a sequence of instruction clauses, e.g. ``put red ball next to the green box after you open the door''.\n\\end{itemize}\n\nTable \\ref{table:babyai} lists all current BabyAI levels together with the competencies required to solve them. These levels form a progression in terms of the competencies required to solve them, culminating with the BossLevel, which requires mastering all competencies.\nThe definitions of competencies are informal and should be understood in the minimalistic sense, i.e. to test the ROOM competency we have built the GoToObj level where the agent needs to reach the only object in an empty room. Note that the GoToObj level does not require the GOTO competency, as this level can be solved without any language understanding, since there is only a single object in the room. However, solving the GoToLocal level, which instructs the agent to go to a specific object in the presence of multiple distractors, requires understanding GOTO instructions.\n\n\\begin{table}\n\\centering\n\\caption{\\label{table:babyai}BabyAI Levels and the required competencies}\n\\vspace{0.1cm}\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c}\n&\n\\rotatebox[origin=c]{90}{ROOM} & \\rotatebox[origin=c]{90}{DISTR-BOX} & \n\\rotatebox[origin=c]{90}{DISTR} & \n\\rotatebox[origin=c]{90}{MAZE} & \n\\rotatebox[origin=c]{90}{UNBLOCK} & \n\\rotatebox[origin=c]{90}{UNLOCK} & \n\\rotatebox[origin=c]{90}{IMP-UNLOCK} & \n\\rotatebox[origin=c]{90}{GOTO} & \n\\rotatebox[origin=c]{90}{OPEN} & \n\\rotatebox[origin=c]{90}{PICKUP} & \n\\rotatebox[origin=c]{90}{PUT} & \n\\rotatebox[origin=c]{90}{LOC} & \n\\rotatebox[origin=c]{90}{SEQ} \\\\\\hline\nGoToObj         &x& & & & & & & & & & & & \\\\\nGoToRedBallGrey &x&x& & & & & & & & & & & \\\\\nGoToRedBall     &x&x&x& & & & & & & & & & \\\\\nGoToLocal       &x&x&x& & & & &x& & & & & \\\\\nPutNextLocal    &x&x&x& & & & & & & &x& & \\\\\nPickupLoc       &x&x&x& & & & & & &x& &x& \\\\\nGoToObjMaze     &x& & &x& & & & & & & & & \\\\\nGoTo            &x&x&x&x& & & &x& & & & & \\\\\nPickup \t        &x&x&x&x& & & & & &x& & & \\\\\nUnblockPickup   &x&x&x&x&x& & & & &x& & & \\\\\nOpen\t\t    &x&x&x&x& & & & &x& & & & \\\\ \nUnlock          &x&x&x&x& &x& & &x& & & & \\\\\nPutNext         &x&x&x&x& & & & & & &x& & \\\\\nSynth \t\t    &x&x&x&x&x&x& &x&x&x&x& & \\\\\nSynthLoc \t    &x&x&x&x&x&x& &x&x&x&x&x& \\\\ \nGoToSeq \t    &x&x&x&x& & & &x& & & & &x\\\\\nSynthSeq \t    &x&x&x&x&x&x& &x&x&x&x&x&x\\\\\nGoToImpUnlock   &x&x&x&x& & &x&x& & & & & \\\\\nBossLevel \t    &x&x&x&x&x&x&x&x&x&x&x&x&x\\\\ \n\\end{tabular}\n\\end{table}\n\n\\subsection{The Bot Agent\\label{sec:bot}}\n\nThe bot is a key ingredient intended to perform the role of a simulated human teacher. For any of the BabyAI levels, it can generate demonstrations or suggest actions for a given environment state. Whereas the BabyAI learner is meant to be generic and should scale to new and more complex tasks, the bot is engineered using knowledge of the tasks. This makes sense since the bot stands for the human in the loop, who is supposed to understand the environment, how to solve missions, and how to teach the baby learner. The bot has direct access to a tree representation of instructions, and so does not need to parse the Baby Language. Internally, it executes a stack machine in which instructions and subgoals are represented (more details can be found in Appendix \\ref{app:bot}). The stack-based design allows the bot to interrupt what it is currently doing to achieve a new subgoal, and then resume the original task. For example, going to a given object will require exploring the environment to find that object.\n\nThe subgoals which the bot implements are:\n\n\\begin{itemize}\n\\item \\textbf{Open:} Open a door that is in front of the agent.\n\\item \\textbf{Close:} Close a door that is in front of the agent.\n\\item \\textbf{Pickup:} Execute the pickup action (pick up an object).\n\\item \\textbf{Drop:} Execute the drop action (drop an object being carried).\n\\item \\textbf{GoNextTo:} Go next to an object matching a given (type, color) description or next to a cell at a given position.\n\\item \\textbf{Explore:} Uncover previously unseen parts of the environment.\n\\end{itemize}\n\nAll of the Baby Language instructions are decomposed into these internal subgoals which the bot knows how to solve. Many of these subgoals, during their execution, can also push new subgoals on the stack. A central part of the design of the bot is that it keeps track of the grid cells of the environment which it has and has not seen. This is crucial to ensure that the bot can only use information which it could realistically have access to by exploring the environment. Exploration is implemented as part of the Explore subgoal, which is recursive. For instance, exploring the environment may require opening doors, or moving objects that are in the way. Opening locked doors may in turn require finding a key, which may itself require exploration and moving obstructing objects. Another key component of the bot's design is a shortest path search routine. This is used to navigate to objects, to locate the closest door, or to navigate to the closest unexplored cell.\n\n\\section{Experiments}\\label{sec:experiments}\n\nWe assess the difficulty of BabyAI levels by training a behavioral cloning baseline for each level. Furthermore, we estimate how much data is required to solve some of the simpler levels and study to which extent the data demands can be reduced by using basic curriculum learning and interactive teaching methods. All the code that we use for the experiments, as well as containerized pretrained models, is available online.\n\n\\subsection{Setup}\n\\label{sec:setup}\n\nThe BabyAI platform provides by default a 7x7x3 symbolic observation $x_t$ (a partial and local egocentric view of the state of the environment) and a variable length instruction $c$ as inputs at each time step. We use a basic model consisting of standard components to predict the next action $a$ based on $x$ and $c$. In particular, we use a GRU \\citep{cho_learning_2014} to encode the instruction and a convolutional network with two batch-normalized \\citep{ioffe_batch_2015} FiLM \\citep{perez_film:_2017} layers to jointly process the observation and the instruction. An LSTM \\citep{hochreiter_long_1997} memory is used to integrate representations produced by the FiLM module at each step. Our model is thus similar to the gated-attention model used by \\cite{chaplot_gated-attention_2018}, inasmuch as gated attention is equivalent to using FiLM without biases and only at the output layer. \n\n\nWe have used two versions of our model, to which we will refer as the Large model and the Small model. In the Large model, the memory LSTM has 2048 units and the instruction GRU is bidirectional and has 256 units. Furthermore, an attention mechanism \\citep{bahdanau_neural_2015} is used to focus on the relevant states of the GRU. The Small model uses a smaller memory of 128 units and encodes the instruction with a unidirectional GRU and no attention mechanism.\n\nIn all our experiments, we used the Adam optimizer \\citep{kingma_adam:_2015} with the hyperparameters $\\alpha=10^{-4}$, $\\beta_1=0.9$, $\\beta_2=0.999$ and $\\epsilon=10^{-5}$. In our imitation learning (IL) experiments, we truncated the backpropagation through time at 20 steps for the Small model and at 80 steps for the Large model. For our reinforcement learning experiments, we used the Proximal Policy Optimization (PPO, \\citealp{schulman_proximal_2017}) algorithm with parallelized data collection. Namely, we performed 4 epochs of PPO using 64 rollouts of length 40 collected with multiple processes. We gave a non-zero reward to the agent only when it fully completed the mission, and the magnitude of the reward was $1 - 0.9 n\/ n_{max}$, where $n$ is the length of the successful episode and $n_{max}$ is the maximum number of steps that we allowed for completing the episode, different for each mission. The future returns were discounted with a factor $\\gamma=0.99$. For generalized advantage estimation \\citep{schulman_high-dimensional_2015} in PPO we used $\\lambda=0.99$.\n\nIn all our experiments we reported the success rate, defined as the ratio of missions of the level that the agent was able to accomplish within $n_{max}$ steps.\n\nRunning the experiments outlined in this section required between 20 and 50 GPUs over two weeks. At least as much computing was required for preliminary investigations.\n\n\n\\subsection{Baseline Results}\n\n\\begin{table}\n\\centering\n\\caption{Baseline imitation learning results for all BabyAI levels. Each model was trained with 1M demonstrations from the respective level. For reference, we also list the mean and standard deviation of demonstration length for each level.}\n\\label{tbl:baseline}\n\\vspace{0.1cm}\n\\begin{tabular}{ccc}\nModel & Success Rate (\\%) & Demo Length (Mean $\\pm$ Std)\\\\\\hline\nGoToObj & 100 & 5.18$\\pm$2.38 \\\\\nGoToRedBallGrey & 100 & 5.81$\\pm$3.29 \\\\\nGoToRedBall & 100 & 5.38$\\pm$3.13 \\\\\nGoToLocal & 99.8 & 5.04$\\pm$2.76 \\\\\nPutNextLocal & 99.2 & 12.4$\\pm$4.54 \\\\\nPickupLoc & 99.4 & 6.13$\\pm$2.97 \\\\\nGoToObjMaze & 99.9 & 70.8$\\pm$48.9 \\\\\nGoTo & 99.4 & 56.8$\\pm$46.7 \\\\\nPickup & 99 & 57.8$\\pm$46.7 \\\\\nUnblockPickup & 99 & 57.2$\\pm$50 \\\\\nOpen & 100 & 31.5$\\pm$30.5 \\\\\nUnlock & 98.4 & 81.6$\\pm$61.1 \\\\\nPutNext & 98.8 & 89.9$\\pm$49.6 \\\\\nSynth & 97.3 & 50.4$\\pm$49.3 \\\\\nSynthLoc & 97.9 & 47.9$\\pm$47.9 \\\\\nGoToSeq & 95.4 & 72.7$\\pm$52.2 \\\\\nSynthSeq & 87.7 & 81.8$\\pm$61.3 \\\\\nGoToImpUnlock & 87.2 & 110$\\pm$81.9 \\\\\nBossLevel & 77 & 84.3$\\pm$64.5 \\\\\n\\end{tabular}\n\\end{table}\n\nTo obtain baseline results for all BabyAI levels, we have trained the Large model (see Section \\ref{sec:setup}) with imitation learning using one million demonstration episodes for each level. The demonstrations were generated using the bot described in Section \\ref{sec:bot}.\nThe models were trained for 40 epochs on levels with a single room and for 20 epochs on levels with a 3x3 maze of rooms. Table \\ref{tbl:baseline} reports the maximum success rate on a validation set of 512 episodes. All of the single-room levels are solved with a success rate of 100.0\\%. As a general rule, levels for which demonstrations are longer tend to be more difficult to solve.\n\nUsing 1M demonstrations for levels as simple as GoToRedBall is very inefficient and hardly ever compatible with the long-term goal of enabling human teaching. The BabyAI platform is meant to support studies of how neural agents can learn with less data. To bootstrap such studies, we have computed baseline sample efficiencies for imitation learning and reinforcement learning approaches to solving BabyAI levels. We say an agent solves a level if it reaches a success rate of at least 99\\%. We define the sample efficiency as the minimum number of demonstrations or RL episodes required to train an agent to solve a given level. To estimate the thus defined sample efficiency for imitation learning while staying within a reasonable computing budget, we adopt the following procedure.\nFor a given level, we first run three experiments with $10^6$ demonstrations. In the remaining $M$ experiments we use $k_1 = 2^{l_0}, k_2 = 2^{l_0 + d}, \\ldots, k_{M}=2^{l_0 + (M - 1)d}$ demonstrations respectively. We use different values of $l_0$, $M$ for each level to ensure that we run experiments with not enough, just enough and more than enough demonstrations. Same value of $d=0.2$ is used in all imitation learning experiments. For each experiment $i$, we measure the best smoothed online validation performance $s_i$ that is achieved during the first $2T$ training steps, where $T=(T_1 + T_2 + T_3) \/ 3$ is the average number of training steps required to solve the level in the three runs with $10^6$ demonstrations. We then fit a Gaussian Process (GP) model \\citep{rasmussen_gaussian_2005} with noisy observations using $(k_i, s_i)$ as training data in order to interpolate between these data points. The GP posterior is fully tractable, which allows us to compute analytically the posterior distribution of the expected success rate, as well as the posterior over the minimum number of samples $k_{min}$ that is sufficient to solve the level. We report the 99\\% credible interval for $k_{min}$. We refer the reader to Appendix \\ref{app:sample_eff} for a more detailed explanation of this procedure. \n\nWe estimate sample efficiency of imitation learning on 6 chosen levels. The results are shown in Table \\ref{tbl:il_vs_rl} (see ``IL from Bot'' column). In the same table (column ``RL'') we report the 99\\% confidence interval for the number of episodes that were required to solve each of these levels with RL, and as expected, the sample efficiency of RL is substantially worse than that of IL (anywhere between 2 to 10 times in these experiments). \n\n\nTo analyze how much the sample efficiency of IL depends on the source of demonstrations, we try generating demonstrations from agents that were trained with RL in the previous experiments. The results for the 3 easiest levels are reported in the ``IL from RL Expert'' column in Table \\ref{tbl:interactive}. Interestingly, we found that the demonstrations produced by the RL agent are easier for the learner to imitate. The difference is most significant for GoToRedBallGrey, where less than 2K and more than 8K RL and bot demonstrations respectively are required to solve the level. For GoToRedBall and GoToLocal, using RL demonstrations results in 1.5-2 times better sample efficiency. This can be explained by the fact that the RL expert has the same neural network architecture as the learner.\n\n\\begin{table}\n\n\\centering\n\\caption{The sample efficiency of imitation learning (IL) and reinforcement learning (RL) as the number of demonstrations (episodes) required to solve each level. All numbers are thousands. For the imitation learning results we report a 99\\% credible interval. For RL experiments we report the 99\\% confidence interval. See Section \\ref{sec:experiments} for details.}\n\\label{tbl:il_vs_rl}\n\\vspace{0.1cm}\n\\begin{tabular}{ccc}\nLevel & IL from Bot & RL \\\\\\hline\nGoToRedBallGrey & 8.431 - 12.43 & 15.9 - 17.4 \\\\\nGoToRedBall & 49.67 - 62.01 & 261.1 - 333.6 \\\\\nGoToLocal & 148.5 - 193.2 & 903 - 1114 \\\\\nPickupLoc & 204.3 - 241.2 & 1447 - 1643 \\\\\nPutNextLocal & 244.6 - 322.7 & 2186 - 2727 \\\\\nGoTo & 341.1 - 408.5 & 816 - 1964 \\\\\n\\end{tabular}\n\\end{table}\n\n\\subsection{Curriculum Learning}\nTo demonstrate how curriculum learning research can be done using the BabyAI platform, we perform a number of basic pretraining experiments. In particular, we select 5 combinations of base levels and a target level and study whether pretraining on base levels can help the agent master the target level with fewer demonstrations. The results are reported in Table \\ref{tbl:curriculum}. In four cases, using GoToLocal as one of the base levels reduces the number of demonstrations required to solve the target level. However, when only GoToObjMaze was used as the base level, we have not found pretraining to be beneficial. We find this counter-intuitive result interesting, as it shows how current deep learning methods often can not take the full advantage of available curriculums.\n\n\\begin{table}\n\\centering\n\\caption{\\label{tbl:curriculum}The sample efficiency results for pretraining experiments. For each pair of base levels and target levels that we have tried, we report how many demonstrations (in thousands) were required, as well as the baseline number of demonstrations required for training from scratch. In both cases we report a 99\\% credible interval, see Section \\ref{sec:experiments} for details. Note how choosing the right base levels (e.g. GoToLocal instead of  GoToObjMaze) is crucial for pretraining to be helpful.}\n\\vspace{0.1cm}\n\\begin{tabular}{cccc}\nBase Levels & Target Level &  Without Pretraining & With Pretraining\\\\\\hline \nGoToLocal & GoTo & 341 - 409 & 183 - 216 \\\\\nGoToObjMaze & GoTo & 341 - 409 & 444 - 602 \\\\\nGoToLocal-GoToObjMaze & GoTo & 341 - 409 & 173 - 216 \\\\\nGoToLocal & PickupLoc & 204 - 241 & 71.2 - 88.9 \\\\\nGoToLocal & PutNextLocal & 245 - 323 & 188 - 231 \\\\\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Interactive Learning}\nLastly, we perform a simple case study of how sample efficiency can  be improved by interactively providing more informative examples to the agent based on what it has already learned. We experiment with an iterative algorithm for adaptively growing the agent's training set. In particular, we start with $2^{10}$ base demonstrations, and at each iteration we increase the dataset size by a factor of $2^{1\/4}$ by providing bot demonstrations for missions on which the agent failed. After each dataset increase we train a new agent from scratch. We perform such dataset increases until the dataset reaches the final size is clearly sufficient to achieve 99\\% success rate. We repeat the experiment 3 times for levels GoToRedBallGrey, GoToRedBall and GoToLocal and then estimate how many interactively provided demonstrations would be required for the agent be 99\\% successful for each of these levels. To this end, we use the same GP posterior analysis as for regular imitation learning experiments.\n\nThe results for the interactive imitation learning protocol are reported in Table \\ref{tbl:interactive}. For all 3 levels that we experimented with, we have observed substantial improvement over the vanilla IL, which is most significant (4 times less demonstrations) for GoToRedBallGrey and smaller (1.5-2 times less demonstrations) for the other two levels.\n\n\\begin{table}\n\\centering\n\\caption{\\label{tbl:interactive}The sample efficiency of imitation learning (IL) from an RL-pretrained expert and interactive imitation learning defined as the number of demonstrations required to solve each level. All numbers are in thousands. 99\\% credible intervals are reported in all experiments, see Section \\ref{sec:experiments} for details.}\n\\label{tbl:data_efficiency}\n\\vspace{0.1cm}\n\\begin{tabular}{cccccc}\nLevel & IL from Bot & IL from RL Expert & Interactive IL from Bot \\\\\\hline\nGoToRedBallGrey & 8.43 - 12.4 & 1.53 - 2.11 & 1.71 - 1.88 \\\\\nGoToRedBall & 49.7 - 62 & 36.6 - 44.5 & 31.8 - 36 \\\\\nGoToLocal & 148 - 193 & 74.2 - 81.8 & 93 - 107 \\\\\n\\end{tabular}\n\\end{table}\n\n\n\\section{Conclusion \\& Future Work}\n\nWe present the BabyAI research platform to study language learning with a human in the loop. The platform includes 19 levels of increasing difficulty, based on a decomposition of tasks into a set of basic competencies. \nSolving the levels requires understanding the Baby Language, a subset of English with a formally defined grammar which exhibits compositional properties. The language is minimalistic and the levels seem simple, but empirically we have found them quite challenging to solve.  The platform is open source and extensible, meaning new levels and language concepts can be integrated easily.\n\nThe results in Section~\\ref{sec:experiments} suggest that current imitation learning and reinforcement learning methods scale and generalize poorly when it comes to learning tasks with a compositional structure. Hundreds of thousands of demonstrations are needed to learn tasks which seem trivial by human standards. Methods such as\ncurriculum learning and interactive learning can provide measurable improvements in terms of sample efficiency, but, in order for learning with an actual human in the loop to become realistic, an improvement of at least three orders of magnitude is required.\n\nAn obvious direction of future research to find strategies to improve sample efficiency of language learning. Tackling this challenge will likely require new models and new teaching methods. Approaches that involve an explicit notion of modularity and subroutines, such as Neural Module Networks~\\citep{andreas_neural_2016} or Neural Programmer-Interpreters~\\citep{reed_neural_2015}, seem like a promising direction. It is our hope that the BabyAI platform can serve as a challenge and a benchmark for the sample efficiency of language learning for years to come.\n\n\\section*{Acknowledgements}\nWe thank Tristan Deleu, Saizheng Zhang for useful discussions. We also thank Rachel Samson, L\u00e9onard Boussioux and David Yu-Tung Hui for their help in preparing the final version of the paper. \nThis research was enabled in part by support provided by Compute Canada (www.computecanada.ca), NSERC and Canada Research Chairs.\nWe also thank Nvidia for donating NVIDIA DGX-1 used for this research.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:introduction}\nExotic nuclei with extreme neutron-to-proton ratios are crucial for  theoretical nuclear structure research as\ntheir properties provide critical information on nuclear interactions, many-body techniques, and astrophysical scenarios. \nHowever, because of their weak binding, their quasiparticle excitations are often affected by  the low-lying scattering space (a.k.a.\\ particle continuum), which enhances the necessary computational effort. For such nuclei, nucleonic pairing must be handled within the full Hartree-Fock-Bogoliubov (HFB) scheme instead of the simpler Bardeen-Cooper-Schrieffer (BCS) approximation \\cite{Dobaczewski1984,Belyaev1987,Dobaczewski1996,Dobaczewski2013,Bulgac02}.  In addition, the associated self-consistent densities  are usually very extended in space, which requires large basis sets or large coordinate-space boxes.\nBoth requirements become particularly demanding if one aims at symmetry-unrestricted calculations (i.e., without imposing space reflection or axial or spherical symmetries). \nThe paper aims to propose a reliable and efficient computational scheme  to solve the HFB equations  on a three-dimensional (3D) Cartesian coordinate-space grid. \n\nThe nuclear energy density functional (EDF) method is one of the most widely used methods to study medium-mass and heavy nuclei \\cite{Bender2003,Schunck2019}.\nIts main ingredient  is an  EDF that represents the effective in-medium nuclear interaction.\nAmong many EDFs, the Skyrme functional, originally based on Skyrme interaction \\cite{Skyrme1958}, is  commonly used to study global\n nuclear properties, such as ground-state energies, deformations, and low-lying excitations \\cite{Schunck2019,Erler2011,Erler2012a}.\nIt is to be noted that the nuclear EDF method is closely related to the density-functional theory (DFT).\nHence, it is often referred to as nuclear DFT.\n\nOver the years, a number of HFB solvers have been developed; see Table 2 of Ref.~\\cite{Bender2020} for a summary. These solvers can be divided into two families.\nThe codes in the first group are based on the expansion of single-particle wave functions in a finite set of basis functions such as the harmonic\noscillator (HO) eigenfunctions. \nExamples of such solvers are: \n {\\sc HFBTHO} \\cite{Stoitsov2005, Perez2017}, which  solves the axial (2D) HFB equations in the axial HO  or the transformed HO basis; \n{\\sc HFODD} \\cite{Dobaczewski1997_1, Schunck2017,Dobaczewski2021}, which  solves the 3D HFB equations in the  Cartesian HO basis without assumption of self-consistent symmetries; and {\\sc HFBPTG}  \\cite{Stoitsov2008}, which  solves the axial (2D) HFB equations in the P\\\"oschl-Teller-Ginocchio basis.\n\nThe basis-expansion method is efficient and has been successfully employed in large-scale calculations \\cite{Erler2012a}.\nHowever, when it is applied to  weakly-bound nuclei, the performance of this method deteriorates   as huge configuration spaces  are required to describe  the asymptotic behavior of HFB solutions. \nHere, the approach of choice is the HFB framework formulated in the coordinate-space representation \\cite{Dobaczewski1984,Dobaczewski1996,Dobaczewski2013}.\n\nThe  coordinate-space solvers constitute the second family of HFB codes. Examples of such solvers are: \n {\\sc HFBRAD} \\cite{Bennaceur2005} solves spherically symmetric HFB problem using finite differences; \n{\\sc HFB-AX} \\cite{Pei2008} is a 2D solver based on B-splines; \n{\\sc SkyAx} \\cite{Reinhard2020} is a highly optimized 2D  Hartree-Fock (HF) + BCS code using  the fast Fourier transform (FFT) method to compute derivatives;\n{\\sc Sky3D}  \\cite{Maruhn2014,Schuetrumpf2018} is a 3D extension of {\\sc SkyAx}; the predecessor of {\\sc SkyAx} and {\\sc Sky3D} is a 1D spherical HF+BCS code using five-point finite differences which was published first in \\cite{Reinhard1991} and has meanwhile been developed into a full spherical  HFB code  {\\sc Sky1D} \\cite{PGHFBcodes};\nthe HFB extension of {\\sc SkyAx} is {\\sc Sky2D} \\cite{PGHFBcodes};\n{\\sc EV8}  solves the Skyrme HF+BCS equations  using the imaginary time method on a 3D mesh that is limited to one octant  by imposing time-reversal and spatial symmetries \\cite{Bonche2005,Ryssens2015}; {\\sc MOCCa} \\cite{Ryssens2019,Scamps2021} is a Skyrme-HFB extension of {\\sc EV8}; {\\sc MADNESS-HFB} \\cite{Pei2014} is a 3D HFB solver\nbased on multi-resolution analysis and multi-wavelet expansion; {\\sc LISE} is a 3D HFB solver \n\\cite{Jin2021} employing  the  discrete variable representation (or Lagrange-mesh method)\nand fast Fourier transforms; and there are also 3D HFB solvers based on the contour integral of the Green's function using the shifted Krylov subspace method \\cite{Jin2017,Kashiwaba2020}. \n\nThe major difference between \nbasis-based and mesh-based methods is the treatment of one-quasiparticle continuum space \\cite{Belyaev1987,Michel2008,Pei2011,Dobaczewski2013}.\nIn the case of coordinate-space methods, the discretized continuum strongly depends on the geometry of the spatial box and the grid size. \nFor large 3D boxes and dense grids, the size of the discretized continuum space quickly becomes intractable as the maximum allowed quasiparticle energy increases.\n\n\n\nA promising approach to the coordinate-space HFB problem is the canonical-basis HFB method  proposed  in Refs.\\ \\cite{Reinhard1997, Tajima2004}.\nThe one-body density matrix is diagonal in the canonical basis (or natural orbits), and its eigenstates are spatially localized if the nucleus is particle-bound. \nBecause of this localization, the single-particle (s.p.) continuum level density  is significantly reduced.\n\n\nIn this work, we develop a  3D Skyrme-HFB solver {\\sc HFBFFT} in the coordinate-space representation. This code is based on the published  code {\\sc Sky3D} \\cite{Maruhn2014,Schuetrumpf2018}.\n{\\sc Sky3D} has been well optimized for performance and parallelized with OpenMP and MPI \\cite{Afibuzzaman2018}.\nIn {\\sc HFBFFT} we maintain the high-level parallelization, making it scalable on modern supercomputers.\nIn order to overcome the pairing collapse problem mentioned in Ref.\\ \\cite{Tajima2004}, we implement the soft energy cutoff of pairing space and develop the annealing of pairing strengths to avoid pairing deadlock at an early stage.\nFurthermore, we introduce the sub-iteration method in the configuration space to stabilize and speed up the convergence. \nWe also resolve the problem of Hermiticity violation in {\\sc Sky3D} brought by the incompatibility between the product rule and the Fourier-transform-based algorithm for derivatives.\nTo benchmark  {\\sc HFBFFT} we study several nuclear systems and compare our results against {\\sc HFBTHO} and the coordinate-space HFB codes {\\sc Sky1D} and {\\sc Sky2D}, which solve the HFB problem in 1D (spherical) and 2D (axial) geometries, respectively.\n\n\nThis paper is organized as follows.\nIn Sec.\\ \\ref{sec:basis}, the Skyrme EDF and the HFB theory are briefly introduced.\nThe numerical details and algorithms of {\\sc HFBFFT} are described in section \\ref{sec:algorithm}.\nIn Sec.\\ \\ref{sec:benchmark}, we present  test and benchmark results.\nFinally, conclusions and outlooks are presented in Sec.\\ \\ref{sec:conclusion}.\n\n\n\\section{Skyrme Hartree-Fock-Bogoliubov theory}\\label{sec:basis}\n\nIn this section, we briefly summarize the Skyrme EDF and the HFB theory.\n\n\\subsection{The Skyrme energy density functional}\n\nThe HFB theory describes a many-Fermion system in terms\nof an orthonormal set of s.p.\\ wave functions $\\psi_\\alpha$\nwith fractional occupation amplitudes $v_\\alpha$, i.e.,\n\\begin{equation}\n    \\left\\{\\psi_\\alpha,v_\\alpha,\\alpha=1,...,\\Omega\\right\\},\n\\label{eq:spbasis}\n\\end{equation}\nwhere $\\Omega$ denotes the size of the active s.p.\\ space.  \nThe amplitude $v_\\alpha$ can take values continuously in the interval $[0,1]$. The complementary  amplitude is $u_\\alpha=\\sqrt{1-v_\\alpha^2}$. \n\nThe code {\\sc HFBFFT} uses a formulation of HFB theory in the basis of natural orbitals, which are defined as the basis of s.p.\\ states $\\psi_\\alpha$ in which the one-body density matrix $\\hat{\\rho}$ is diagonal, i.e.,\n$\\hat{\\rho}=\\sum_\\alpha|\\psi_\\alpha\\rangle{n}_\\alpha\\langle\\psi_\\alpha|$,\nwhere $n_\\alpha$,  an eigenvalue of $\\hat{\\rho}$, represents the canonical-state occupation.\nThe numerical HFB scheme in the canonical basis was presented in \\cite{Reinhard1997} and improved\nin \\cite{Tajima2004}. For the relation between the standard matrix formulation and the canonical formulation of HFB, see Refs.~\\cite{Dobaczewski1996,Ring2004}.\nIn the canonical basis, the HFB\nmean-field state takes the BCS-like form:\n\\begin{equation}\n  |\\Phi\\rangle\n  =\n  \\prod_{\\alpha>0}\\big(\n  u_\\alpha^{\\mbox{}}+v_\\alpha^{\\mbox{}}\\hat{a}^+_\\alpha\\hat{a}^{+}_{\\overline{\\alpha}}\n  \\big)|0\\rangle\n\\label{eq:BCState}\n\\end{equation}\nwhere $|0\\rangle$ is the vacuum state, $\\hat{a}^+_\\alpha$ is the\ncreation operator of $\\psi_\\alpha$, and $\\overline{\\alpha}$ the\nconjugate partner to state $\\alpha$ that corresponds to the same eigenvalue  of $\\hat{\\rho}$. \n\n\nAny self-consistent mean-field theory starts from expressing the\nenergy of the system in terms of s.p.\\ wave functions and occupation\namplitudes (\\ref{eq:spbasis}).  \nEDFs go for a simpler approach by starting from the energy defined in terms of only a few local densities and currents.\nFor the case of stationary states of even-even\nnuclei, the energy depends only on the local particle density $\\rho_q$, the kinetic-energy density\n$\\tau_q$, and the spin-orbit density $\\vec{J}_q$:\n\\begin{subequations}\\label{eq:densities}\n\\begin{align}\n   \\rho_q(\\vec{r})&=\\displaystyle\n    \\sum_{\\alpha\\in q}\\sum_{s}  \n    v_{\\alpha}^2|\\psi_{\\alpha}(\\vec{r},s)|^2,\n   \\notag \\\\\n     \\tau_q(\\vec{r})&=\\displaystyle\n    \\sum_{\\alpha\\in q}\\sum_{s}  \n    v_{\\alpha}^2|\\nabla\\!\\psi_{\\alpha}(\\vec{r},s)|^2, \\notag \\\\\n   \\vec{J}_q(\\vec{r}) &=\\displaystyle\n    -\\mathrm{i}\\sum_{\\alpha\\in q}\\sum_{ss'} v_{\\alpha}^2\n    \\psi_{\\alpha}^*(\\vec{r},s)\n    \\nabla\\! \\times\\! \\vec{\\sigma}_{ss'} \n    \\psi^{\\mbox{}}_{\\alpha}(\\vec{r},s'),\n \\label{eq:rtjeven}\n\\end{align}\nwhere $q\\in\\{\\mathrm{p},\\mathrm{n}\\}$ stands for protons or neutrons and $s, s'=\\pm 1\/2$ label the two spinor components of the wave functions.\nPairing EDFs additionally require the pairing density\n\\begin{align}\n  \\xi_q(\\vec{r}) \n  &=\\displaystyle\n  {\\sum^\\mathrm{(cut)}_{\\alpha\\in q}u_{\\alpha}v_{\\alpha} \n    \\sum_{s}\\left( -2s \\right) \\psi_{\\overline{\\alpha}}(\\vec{r},-s)\n    \\psi_{\\alpha}}(\\vec{r},s),\n \\label{eq:rtjpair}\n\\end{align}\nwhere the first summation includes a cutoff in the pairing space. \nFor a stationary state of an even-even nucleus, the conjugate s.p.\\ state $\\overline{\\alpha}$ can be assumed to be the time-reversed state of $\\alpha$, which leads to\n\\begin{align}\n  \\xi_q(\\vec{r}) \n  &=\\displaystyle\n  {\\sum^\\mathrm{(cut)}_{\\alpha\\in q} \\sum_{s} \n  u_{\\alpha}v_{\\alpha} \\left| \\psi_{\\alpha}(\\vec{r},s) \\right|^2}.\n\\end{align}\n\\end{subequations} \n\nThe code {\\sc HFBFFT}, as its predecessor {\\sc Sky3D}, employs the\nwidely used Skyrme EDF. This EDF is well described in all details at\nseveral places  \\cite{Bender2003,Erler2011,Schunck2019}.\nThus we give here only a brief account with emphasis on the pairing\npart.  The total energy is a functional of the local densities:\n\\begin{subequations}\n\\label{eq:Etot}\n\\begin{equation}\n    E_\\mathrm{tot}\n    =\n    E_\\mathrm{Skyrme}[\\rho,\\tau,\\vec{J}]\n    +\n    E_\\mathrm{pair}[\\rho,\\xi]\n    +\n    E_\\mathrm{Coul}[\\rho_p],\n\\label{eq:efundet}\n\\end{equation}\nwhere  (ignoring here isospin index for simplicity)\n\\begin{align}\n\tE_\\mathrm{Skyrme} \n\t&= E_\\mathrm{kin} + E_\\mathrm{\\rho \\rho} + E_\\mathrm{\\rho \\tau} + E_{\\rho \\Delta \\rho}\n\t+ E_\\mathrm{ \\nabla \\vec{J}} \n\\notag \\\\\n   &= \\int\\! d^3r\\left[ \\frac{\\hbar^2}{2m}\\tau +\n    C^{\\rho}\\rho^2 + C^{\\tau}\\rho \\tau+ C^{\\Delta\\rho}\\rho\\Delta\\rho + C^{\\vec{J}}\\rho \\nabla\\cdot\\vec{J}\\right],\n   \\\\\n  E_\\mathrm{pair}\n  &=\n  \\frac{1}{4} \\sum_{q\\in\\{\\mathrm{p},\\mathrm{n}\\}}V_{\\mathrm{pair},q}\n  \\int d^3r |\\xi_q|^2\n  \\left[1 -\\frac{\\rho}{\\rho_{0,\\mathrm{pair}}}\\right],\n  \\\\\n  E_{\\mathrm{Coul}}\n  &=\\frac{e^{2}}{2} \\int\\! \\mathrm{d}^{3} r \\mathrm{~d}^{3} r^{\\prime} \\frac{\\rho_{\\mathrm{p}}(\\vec{r}) \\rho_{\\mathrm{p}}\\left(\\vec{r}^{\\prime}\\right)}{\\left|\\vec{r}-\\vec{r}^{\\prime}\\right|}\n  -\\int\\! \\mathrm{d}^{3} r \\frac{3 e^{2}}{4}\\left(\\frac{3}{\\pi}\\right)^{\\frac{1}{3}} \\rho_{\\mathrm{p}}^{4 \/ 3}.\n\\label{eq:epair}\n\\end{align}     \n\\end{subequations}\n $E_\\mathrm{Skryme}$ is a functional of $\\rho$, $\\tau$,\nand $\\vec{J}$; $E_\\mathrm{pair}$ is a functional of $\\rho$\nand $\\xi$;\nand the Coulomb energy $E_{\\mathrm{Coul}}$ is a functional of proton density $\\rho_p$. \nThe pairing functional can be motivated by a density-dependent\n$\\delta$ interaction. \nIt includes two limiting cases.  \nThe first case is a pure  contact interaction, also called volume pairing, which is recovered when $\\rho_{0,\\mathrm{pair}}\\rightarrow\\infty$. \nThe second case corresponds to a value near matter equilibrium density $\\rho_{0,\\mathrm{pair}}=0.16$ fm$^{-3}$, which localizes pairing around the nuclear surface.  \nAdjustment of $\\rho_{0,\\mathrm{pair}}$ as a free parameter delivers a form of the pairing functional which stays in between the extremes of volume and surface pairing \\cite{Dobaczewski2001,Kluepfel2009}.\n\n\n\n\n\n\\subsection{The HFB theory in canonical basis} \\label{sec:HFB_theory}\n\nIn practice, one deals with two types of fermions: protons and  neutrons. \nTo keep the notation simple, in the following, we assume that  the isospin quantum number is included in the quantum label $\\alpha$ of the canonical state. \nThe HFB equations are derived variationally  by minimizing the HFB Routhian\n\\begin{equation}\\label{routhian}\n    R=E_{\\mathrm{tot}}\n    -\\sum_{q\\in \\{\\mathrm{p},\\mathrm{n}\\}}\\epsilon_{\\mathrm{F},q} \\sum_{\\alpha\\in q} v_{\\alpha}^{2}\n    - \\sum_{\\alpha\\beta} \\lambda_{\\alpha\\beta}\\left(\\langle\\psi_{\\beta} | \\psi_{\\alpha}\\rangle-\\delta_{\\alpha\\beta}\\right),\n\\end{equation}\nwith respect to $\\psi_\\alpha$ and $v_\\alpha$.\nIn Eq.~(\\ref{routhian})\n$\\epsilon_\\mathrm{F}$ is the\nFermi energy which is also the Lagrange parameter for the particle-number constraint, \nand the $\\hat\\lambda$ is the matrix of Lagrangian\nmultipliers that guarantee the  orthonormality of canonical wave functions. \nSince $\\langle\\psi_{\\beta} | \\psi_{\\alpha}\\rangle = \\langle\\psi_{\\alpha} | \\psi_{\\beta}\\rangle^*$,\nit is required that the matrix $\\hat\\lambda$\nis Hermitian  so that the number of its independent elements coincides with the total number of independent constraints.\n\nVariation of the Skryme and Coulomb energies with regard to the s.p.\\ wave function yields the HF Hamiltonian\n$\\hat{h}$:\n\\begin{equation}\n  \\frac{\\delta \\left(E_\\mathrm{Skyrme} + E_\\mathrm{Coul} \\right)}{\\delta\\psi_\\alpha^\\dagger}\n  =\n  v_\\alpha^2\\hat{h}\\psi_\\alpha.\n\\label{eq:mfham}\n\\end{equation}\nBy the chain rule for derivatives, (\\ref{eq:mfham}) can be reduced to the variation with respect to the densities, which delivers explicit expressions for $\\hat{h}$ \\cite{Bender2003,Erler2011}.\nThe HF Hamiltonian $\\hat{h}$ is a functional of local densities (particle density, kinetic-energy density, spin-orbit density) in the standard fashion of nuclear EDFs \\cite{Bender2003}. \n\nVariation of the pairing energy with respect to the s.p.\\ wave function gives\n\\begin{equation}\\label{eq:pair_ham}\n    \\frac{\\delta E_\\mathrm{pair}}{\\delta\\psi_\\alpha^\\dagger}\n    =\n    u_\\alpha v_\\alpha \\hat{\\tilde{h}}\\psi_\\alpha + v_\\alpha^2 \\hat{h}^\\prime \\psi_\\alpha.\n\\end{equation}\nThe first term is related to the  variation\nwith respect to the pairing density, which yields the pairing potential \\cite{Ring2004}\n\\begin{equation}\n  \\tilde{h}_q(\\vec{r})\n  =\n  \\frac{1}{2} V_{\\mathrm{pair},q}\n  \\xi_q\n  \\left[1 -\\frac{\\rho}{\\rho_{0,\\mathrm{pair}}}\\right],\\ q\\in \\{\\mathrm{p},\\mathrm{n}\\}.\n\\label{eq:gappot}\n\\end{equation}\nThe second term is the pairing-rearrangement term, brought by the density dependence of the pairing functional. \nFor simplicity, we treat the rearrangement term $\\hat{h}^\\prime$ as part of the HF Hamiltonian $\\hat{h}$ in the following. \nThe pairing potential $\\tilde{h}_q(\\vec{r})$ is a local potential in most pairing functionals.\nFrom that, we obtain the state-dependent gap\n\\begin{equation}\n  \\Delta_{\\alpha\\alpha}\n  =\n  \\langle\\psi_\\alpha|\n  \\tilde{h}_{q_\\alpha}|\\psi_\\alpha\\rangle\n  ,\n\\label{eq:gapalpha}\n\\end{equation}\nwhere $q_\\alpha$ is the isospin of state $\\alpha$. \nAnother aspect of the pairing is determined by the gap equation, which is obtained from the variation with respect to  $v_\\alpha$:  \n\\label{eq:HFBeqs}\n\\begin{equation}\n  0\n  =\n  4 v_\\alpha^{\\mbox{}} (h_{\\alpha\\alpha}^{\\mbox{}} - \\epsilon_{\\mathrm{F},q_\\alpha})\n  + 2 \\left(\\frac{v_\\alpha^2}{u_\\alpha^{\\mbox{}}}-u_\\alpha^{\\mbox{}}\\right)\n   \\Delta_{\\alpha\\alpha},\n\\label{eq:gapeq}\n\\end{equation}\nwhere $h_{\\alpha\\alpha}$ are the diagonal matrix elements of\nthe HF Hamiltonian $\\hat{h}$.\nThe HF Hamiltonian together with the pairing potential constitutes the main ingredients of the HFB equations. \n\nWith the orthonormality of canonical states taken into account, the constrained variation of the total energy with respect to $\\psi_\\alpha^\\dagger$ yields the mean-field equations:\n\\begin{equation}\n  \\hat{\\mathcal{H}}_\\alpha^{\\mbox{}} \\psi_\\alpha^{\\mbox{}}\n=\n  \\textstyle{\\sum_\\beta}\n  \\psi_\\beta\\lambda^{\\mbox{}}_{\\beta\\alpha}, \n\\label{eq:cmfeq}\n\\end{equation}\nwhere\n\\begin{subequations}\\label{eq:genmf_avlambda}\n\\begin{align}\n  \\hat{\\mathcal{H}}_\\alpha^{\\mbox{}} \n  &=\n  v^2_\\alpha \\hat{h} + u_\\alpha^{\\mbox{}}\n  v_\\alpha^{\\mbox{}} \\hat{\\tilde{h}},\n\\label{eq:genmf}\n\\\\\n  \\lambda^{\\mbox{}}_{\\beta\\alpha}\n  &=\n  \\frac{1}{2}\n  \\langle\\psi_\\beta|\\hat{\\mathcal{H}}_\\alpha^{\\mbox{}}+\\hat{\\mathcal{H}}_\\beta|\\psi_\\alpha^{\\mbox{}}\\rangle.\n\\label{eq:avlambda}\n\\end{align}\n\\end{subequations}\nThe mean-field equations (\\ref{eq:cmfeq},\\ref{eq:genmf_avlambda}) and  gap equations (\\ref{eq:gapeq}) together constitute the self-consistent HFB equations in the canonical basis. \n\nIn (\\ref{eq:genmf})  $\\hat{\\mathcal{H}}_\\alpha^{\\mbox{}}$ is a state-dependent one-body Hamiltonian composed of the HF Hamiltonian and the pairing potential.\nThe full matrix $\\hat\\lambda$ needs to be taken into account because the $\\hat{\\mathcal{H}}_\\alpha^{\\mbox{}}$ are state-dependent \\cite{Reinhard1997,Tajima2004}.\nIn contrast, pure HF or HF+BCS calculations only require diagonal matrix elements $\\lambda_{\\alpha\\alpha}$, which are also known as s.p.\\ energies. \nThe Hermiticity of  $\\hat\\lambda$ is enforced by explicit symmetrization in Eq.\\ (\\ref{eq:avlambda}).\nIt can be shown by multiplying both sides of Eq.\\ (\\ref{eq:cmfeq})   by  $\\psi_\\beta^\\dagger$ that the final solution should obey the symmetry conditions\n\\begin{subequations}\n\\label{eq:symmcond_both}\n\\begin{equation}\n   0\n   =\\lambda^{-}_{\\beta\\alpha}\\equiv\n   \\frac{1}{2}\\left(\\langle\\psi_\\beta^{\\mbox{}}|\\hat{\\mathcal{H}}_\\alpha^{\\mbox{}}|\\psi_\\alpha^{\\mbox{}}\\rangle\n   -\n   \\langle\\psi_\\beta^{\\mbox{}}|\\hat{\\mathcal{H}}_\\beta^{\\mbox{}}|\\psi_\\alpha^{\\mbox{}}\\rangle\\right).\n\\label{eq:symmcond}\n\\end{equation}\nOne can combine these into one condition:\n\\begin{equation}\n   0\n   =\n   \\Delta\\mathcal{S}^2\n   \\equiv\n   \\frac{1}{\\Omega^2} \\sum_{\\alpha\\beta} \\left| \\lambda^{-}_{\\beta\\alpha} \\right|^2, \n\\label{eq:symmcondaver}\n\\end{equation}\n\\end{subequations}\nwhich is the average of squared matrix elements of ${\\hat{\\lambda}}^-$. \nThe actual size of $\\Delta\\mathcal{S}^2$ will serve as a check for the convergence of the HFB solution.\n\nIt should be noted that $\\hat\\lambda^-$ vanishes when both s.p.\\ states $\\psi_\\alpha$ and $\\psi_\\beta$ are fully occupied ($v_\\alpha = v_\\beta = 1$) or unoccupied ($v_\\alpha = v_\\beta = 0$) since then $\\langle\\psi_\\alpha|\\hat{h}|\\psi_\\beta\\rangle=\\langle\\psi_\\beta|\\hat{h}|\\psi_\\alpha\\rangle^*$. \nThus, for a pure HF calculation, $\\Delta\\mathcal{S}^2$ measures the overlap between occupied and unoccupied orbits, which should be zero at the self-consistent solution.\n\nWhen the size of active s.p.\\ space equals the number of particles ($\\Omega_n = N$, $\\Omega_p = Z$) and \nall the s.p.\\ orbits are fully occupied (as in the pure HF calculation), \n$\\Delta\\mathcal{S}^2$ is always  zero; hence, it is not an appropriate measure for convergence.\nHowever, this quantity can still be utilized to check the Hermiticity of our implementation of $\\hat{h}$ (see Sec.\\ \\ref{sec:hermiticity}). \nFor the pure HF case, a suitable quantity for convergence check could be\n\\begin{equation}\n    \\sum_\\alpha \\left|\\langle \\psi_\\alpha | \\hat{h}^2 | \\psi_\\alpha \\rangle - \\langle \\psi_\\alpha | \\hat{h} | \\psi_\\alpha \\rangle^2\\right|,\n\\end{equation}\nwhich is zero at the HF solution.\n\n\\section{Numerical representation} \\label{sec:algorithm}\n\n\\subsection{Numerical realization on a 3D coordinate-space grid} \\label{sec:num_grid}\n\nThe numerical representation is explained in detail in Refs.~\\cite{Maruhn2014,Schuetrumpf2018}. \nHere, we repeat the essentials briefly. For simplicity, our discretization strategy is explained here for one dimension;\nthe generalization to 3D is straightforward.\n\nAll wave functions, densities and fields are defined on a three-dimensional equidistant Cartesian grid. \nThe grid points in the $x$ direction are\n\\begin{equation}\n  x_\\nu = \\left(-\\frac{N_x+1}{2}+\\nu\\right)\\delta x,\n  \\quad \\nu=1,\\ldots,N_x,\n\\label{eq:xgrid}\n\\end{equation}\nwhere $N_x$ is the (even) number of grid points and $\\delta x$ is the\ngrid spacing. \nSimilar gridding applies to the $y$ and $z$ directions.  The action of local\noperators on a coordinate-space grid is a simple multiplication\nof the local operator field and the wave function.  \nThe action of momentum operators, such as in the kinetic energy, requires first and second derivatives defined  in Fourier space.\nThe Fourier technique has been proved to be superior in precision and advantageous for large grids \\cite{Blum1992}.\nIt is noteworthy that the direct Coulomb potential is also solved in Fourier space.\nThe Coulomb solver has to fulfill the condition that the result in the  box is the correct solution to Poisson's equation with the boundary condition of zero potential at infinity.\nThe algorithm to solve Poisson's equation for an isolated charged distribution has been implemented in {\\sc Sky3D}.\nIt follows the ideas of \\cite{Hockney1970, Eastwood1979} by doubling the 3D grid, folding the proton density with the $1\/r$ Green's function in momentum space and then restricting the final solution inside the original box.\n\nThe discrete grid points $k_n$ in Fourier space are related to the same number of grid points $x_\\nu$ in coordinate space as:\n\\begin{subequations}\n\\label{eq:FT}\n\\begin{align}\nk_n  &=\\left\\{\n\\begin{aligned}\n &(n-1)\\delta k, \\quad n=1,\\ldots, \\frac{N_x}{2} \\\\\n &(n-N_x-1)\\delta k, \\quad n=\n   \\frac{N_x}{2}+1,\\ldots, N_x \n\\end{aligned},\n\\right. \\\\\n  \\delta k\n  &=\n  \\frac{2\\pi}{N_x\\delta x}.\n\\end{align}\n\\end{subequations}\nNote that the coordinate-space grid (\\ref{eq:xgrid}) in combination with the conjugate momentum-space grid (\\ref{eq:FT}), imposes no spatial symmetry at all. \nBut the particular examples considered for benchmarking in this study obey reflection symmetry in all three directions.\n\nA wave function $\\psi(x_\\nu)$ in coordinate space is related to a\nwave function $\\widetilde{\\psi}(k_n)$ in Fourier space by the discrete Fourier transform and its inverse\n\\begin{subequations}\n\\begin{align}\n  \\widetilde{\\psi}(k_n)\n  &=\n  \\sum_{\\nu=1}^{N_x}\n  \\exp{\\left(-\\mathrm{i} k_nx_\\nu\\right)}\\psi(x_\\nu) \n  ,\n\\label{eq:FTforward}\\\\\n  \\psi(x_\\nu) \n  &=\n  \\frac{1}{N_x}\\sum_{n=1}^{N_x}\n  \\exp{\\left(\\mathrm{i} k_nx_\\nu\\right)}\\widetilde{\\psi}(k_n).\n\\label{eq:FTbackward}\n\\end{align}\n\\end{subequations}\nBoth can be efficiently computed via the FFT algorithm provided by the FFTW3 library \\cite{Frigo2005}.\nThis complex Fourier representation implies that the function $\\psi$ is\nperiodic, i.e., $\\psi(x+N_x \\cdot \\delta x)=\\psi(x)$. The appropriate\nintegration scheme that complies with the above summations is the trapezoidal rule\n\\begin{equation}\n    \\int_{-\\frac{N_x}{2}\\delta x}^{\\frac{N_x}{2}\\delta x} dx\\, f(x) \\approx \\sum_{\\nu = 1}^{N_x} f(x_\\nu) \\delta x, \n\\end{equation}\nwhere all terms are added up with equal weights.\n\nIn Fourier space the $m$-th\nderivative becomes a multiplication by $(\\mathrm{i} k_n)^m$. \nOne proceeds then in the following way: First, a forward transform (\\ref{eq:FTforward}) is performed; then  $\\widetilde{\\psi}(k_n)$ is multiplied by\n$(\\mathrm{i} k_n)^m$; and finally \n$(\\mathrm{i}k_n)^m\\widetilde{\\psi}(k_n)$ is transformed back to the coordinate space by Eq.~(\\ref{eq:FTbackward}).  \nOne should note that there is an arbitrariness about the choice of momentum $k_{N_x\/2+1}$:\nit can be taken as $\\pm \\frac{N_x}{2} \\delta k$.\nThis arbitrariness does not alter the transforms (\\ref{eq:FTforward}, \\ref{eq:FTbackward}) \nbut gives different results of the $m$-th derivative when $m$ is odd (no impact when $m$ is even).\nA natural choice is to equally split $\\widetilde{\\psi}(k_{N_x\/2+1})$ between the positive and negative momenta,\nmaking them cancel each other in the final result of an odd-order derivative.\nIt is equivalent to setting  $\\widetilde{\\psi}(k_{N_x\/2+1})=0$.\nThis choice ensures that the derivative of a real-valued function is still real-valued; \nit also means that the second derivative is not equivalent to two consecutive first derivatives in this framework.\nThe remaining problem is the Hermiticity breaking caused by the product rule;\nwe will discuss it in Sec. \\ref{sec:hermiticity}. \n\n\n\\subsection{Solution by accelerated gradient iteration}\n\\label{sec:grad}\n\nThe solution of the coupled HFB equations (\\ref{eq:HFBeqs}) is obtained by interlaced iterations of the gap equation and the mean-field equation. \nThe gap equation (\\ref{eq:gapeq}) can be solved in a closed form and it yields:\n\\begin{equation}\n  \\left\\{\\begin{array}{c} v_\\alpha^{\\mbox{}} \\\\ u_\\alpha^{\\mbox{}} \\end{array}\\right\\}\n  =\n  \\sqrt{\\frac{1}{2}\\mp \\frac{1}{2}\n      \\frac{h_{\\alpha\\alpha}^{\\mbox{}}-\\epsilon_{\\mathrm{F},q_\\alpha}}\n           {\\sqrt{(h_{\\alpha\\alpha}-\\epsilon_{\\mathrm{F},q_\\alpha})^2+\\Delta^2_{\\alpha\\alpha}} } }\n   .\n\\label{eq:uveq}\n\\end{equation}\nThe Fermi energy $\\epsilon_\\mathrm{F}$ needs to be adjusted to fulfill the particle-number condition\n\\begin{equation}\n  \\epsilon_{\\mathrm{F},q}\\;\\longleftrightarrow\\;\n  \\sum_{\\alpha\\in q}^{\\mbox{}} v_\\alpha^2=N_q,\n\\end{equation}\nwhere $N_q$ is the required particle number.\nNote that only the diagonal elements of the pairing potential and the HF Hamiltonian in the canonical basis enter (\\ref{eq:uveq}); hence, no information about the non-diagonal elements is needed to determine the occupation amplitudes.\n\n\nThe solution of the mean-field equation (\\ref{eq:cmfeq}) is obtained by the damped gradient iteration \\cite{Maruhn2014,Reinhard1982,Blum1992} interlaced with updating the matrix $\\hat\\lambda$ for the orthonormality constraint. \nThe steps are:\n\\begin{enumerate}\n  \\item\n    For given\n    $\\{\\psi_\\alpha^{\\mbox{}},v_\\alpha^{\\mbox{}},u_\\alpha^{\\mbox{}},\\alpha=1,...,\\Omega\\}$\n    compute the local densities, the HF Hamiltonian $\\hat{h}$ and the pairing potential $\\hat{\\tilde{h}}$.\n  \\item\\label{it:hmf}\n    Compute the action of $\\hat{h}$ and $\\hat{\\tilde{h}}$ on all $\\psi_\\alpha^{\\mbox{}}$\n    and store the result in work arrays $\\Psi_{\\alpha}$ and $\\widetilde{\\Psi}_{\\alpha}$, i.e.,\n    \\begin{subequations}\n    \\begin{equation}\n      \\hat{h}\\psi_\\alpha^{\\mbox{}}\\longrightarrow\\Psi_\\alpha^{\\mbox{}},\n    \\end{equation}\n    \\begin{equation}\n     \\hat{\\tilde{h}}\\psi_\\alpha^{\\mbox{}}\\longrightarrow\\widetilde{\\Psi}_\\alpha^{\\mbox{}},\n    \\end{equation}\n    \\end{subequations}\n    for $\\alpha=1,\\ldots,\\Omega$.\n  \\item\\label{it:h_Delta_diag}\n    Use  $\\Psi_\\alpha^{\\mbox{}}$ and $\\widetilde{\\Psi}_\\alpha^{\\mbox{}}$ to compute and store the \n    s.p.\\ energies and pairing gaps\n    \\begin{subequations}\n    \\begin{equation}\n      h_{\\alpha\\alpha}=\\langle\\psi_\\alpha^{\\mbox{}}|\\Psi_\\alpha^{\\mbox{}}\\rangle,\n    \\end{equation}\n    \\begin{equation}\n        \\Delta_{\\alpha\\alpha} = \\langle\\psi_\\alpha^{\\mbox{}}|\\widetilde{\\Psi}_\\alpha^{\\mbox{}}\\rangle.\n    \\end{equation}\n    \\end{subequations}\n  \\item\n    Evaluate and store the action of the generalized mean-field Hamiltonian (overwriting $\\Psi_\\alpha^{\\mbox{}}$)\n    \\begin{equation}\n      \\mathcal{H}_\\alpha^{\\mbox{}}\\psi_\\alpha^{\\mbox{}}\n      =\n      v_\\alpha^2\\Psi_\\alpha^{\\mbox{}}+u_\\alpha^{\\mbox{}} v_\\alpha^{\\mbox{}}\\widetilde{\\Psi}_\\alpha^{\\mbox{}}\n      \\;\\longrightarrow\\;\n      \\Psi_\\alpha^{\\mbox{}}.\n    \\end{equation}\n  \\item\n    Apply the matrix of Lagrange multipliers on all $\\psi_\\alpha$; compute and store (again overwriting $\\Psi_\\alpha^{\\mbox{}}$)\n    \\begin{equation}\n      \\Psi_\\alpha^{\\mbox{}}-\\sum_\\beta^{\\mbox{}}\\psi_\\beta^{\\mbox{}}\\lambda_{\\beta\\alpha}^{\\mbox{}}\n      \\;\\longrightarrow\\;\n      \\Psi_\\alpha^{\\mbox{}}\n      .\n    \\end{equation}\n  \\item\n    Apply the damping operation $\\hat{\\mathcal{D}}$ and orthonormalization $\\hat{\\mathcal{O}}$\n    \\begin{subequations}\n    \\begin{equation}\n      \\psi_\\alpha^\\mathrm{(new)}\n     =\n      \\hat{\\mathcal{O}}\\left\\{\\psi_\\alpha^{\\mbox{}}-\\hat{\\mathcal{D}}\\Psi_\\alpha^{\\mbox{}}\\right\\}, \n      \\end{equation}\n      \n        \\begin{equation}\n      \\hat{\\mathcal{D}}\n      =\n      \\frac{x_0}{v_\\alpha^2(\\hat{T}+E_0)+\\frac{1}{2}u_\\alpha^{\\mbox{}} v_\\alpha^{\\mbox{}}\\tilde{h}_0}\n      ,\n    \\end{equation}\n    \\end{subequations}\n    where $x_0$, $E_0$, and $\\tilde{h}_0$ are adjustable numerical parameters. \n    The empirical values $x_0=0.45$, $E_0=100$ MeV and $\\tilde{h}_0=\\mathrm{max}\\left[\\tilde{h}_\\mathrm{n}(\\vec{r}),\\tilde{h}_\\mathrm{p}(\\vec{r})\\right]$ are used in our calculations.\n    It is worth noting that  the lower bound of $u_\\alpha v_\\alpha$ and $ v_\\alpha^2$  in $ \\hat{\\mathcal{D}}$  is set to be $10^{-1}$ for numerical stability.\n        \n   \\item\\label{it:pair}\n    With the new $h_{\\alpha\\alpha}$ and $\\Delta_{\\alpha\\alpha}$ from step \\ref{it:h_Delta_diag}, compute\n    new occupations $v_\\alpha^{\\mbox{}}$ and $u_\\alpha^{\\mbox{}}$ \n    using Eq.~(\\ref{eq:uveq}).\n    \n    \\item Reevaluate the action of the generalized mean-field Hamiltonian on all $\\psi_\\alpha$ and compute the matrix of Lagrange multipliers \n    \\begin{equation}\n        \\lambda^{\\mbox{}}_{\\beta\\alpha}\n      =\n      \\frac{\n      \\langle\\psi_\\beta|\\hat{\\mathcal{H}}_\\alpha^{\\mbox{}}|\\psi_\\alpha^{\\mbox{}}\\rangle \n      + \\langle \\psi_\\alpha | \\hat{\\mathcal{H}}_\\beta | \\psi_\\beta \\rangle^*\n      }{2}.\n    \\end{equation}\n\\end{enumerate}\n\nThe above iteration usually starts from a number of HF+BCS steps, which are done in the same way as in {\\sc Sky3D}. \nThe HF+BCS calculation is initialized by a 3D HO wave function that can be triaxially deformed.\nTo achieve better convergence, in step 1 the new densities are mixed linearly with the old ones:\n\\begin{equation}\n    \\kappa^{(n)} = (1-\\gamma)\\kappa^{(n-1)} + \\gamma\\kappa^{(n)}_{\\psi}, \\quad \\kappa = \\rho, \\tau\\ \\mathrm{or}\\ \\xi,\n\\end{equation}\nwhere $n$ is the iteration number, subscript $\\psi$ denotes the density directly computed from the wave functions, and $\\gamma$ is the adjustable mixture ratio with a default value of 0.2.\n\n\\subsection{Sub-iterations in configuration space}\n\nThe damped gradient scheme outlined in Sec.\\ \\ref{sec:grad} converges, but requires more iterations in the HFB scheme as compared to the HF + BCS used  in {\\sc Sky3D}.\nIt also involves operations on the full 3D grid which  can make computations cumbersome.  \nThe pairing part in the iterative steps works predominantly within the given space of canonical  states.\nThus one can reduce the total numerical expense by the sub-iteration method: \nswitching between the full 3D step and a fast iterative solver in configuration space. \nTo this end, we map the mean-field equations into configuration space with\nthe expansion\n\\begin{equation}\n  \\psi_\\alpha\n  =\n  \\sum_{n=1}^{\\Omega}\\varphi_n c_{n\\alpha},\n  \\label{eq:config_expansion}\n\\end{equation}\nwhere $\\{\\varphi_n\\}$ is a set of  s.p.\\ states acting as the expansion\nbasis.  \nFor simplicity we choose an expansion basis such that $c^{(0)}_{n\\alpha} = \\delta_{n\\alpha}$ at the beginning.\nInserting (\\ref{eq:config_expansion}) into the HFB mean-field equations\n(\\ref{eq:cmfeq}) yields\n\\begin{equation}\n  \\lambda^-_{\\beta \\alpha} =\n  \\sum_{m n} c_{n\\beta}^*\n  \\left\\langle\\varphi_n\\left|\\frac{\\hat{\\mathcal{H}}_\\alpha^{\\mbox{}}\n                  -\\hat{\\mathcal{H}}_\\beta^{\\mbox{}}}{2}\n \\right|\\varphi_m\\right\\rangle \n  c_{m\\alpha} = 0.\n\\label{eq:asymm}\n\\end{equation}\nEq.\\ (\\ref{eq:asymm}) is essentially the same as the symmetry condition (\\ref{eq:symmcond}).\nIt is solved by a simple damped gradient iteration:\n\\begin{equation}\n\\begin{split}\n  c_{n\\alpha}^\\mathrm{(new)}\n  &=\n\\hat{\\mathcal{O}}\\left\\{c_{n\\alpha}^{\\mbox{}}\n  -\n  \\frac{\\delta}{h_{nn}\\!-\\!h_{11}\\!+\\!E_0}\n  \\left[\\sum_{m}\\mathcal{H}_{\\alpha,nm}^{\\mbox{}} c_{m\\alpha}^{\\mbox{}}\n       -\\sum_{\\beta}c_{n\\beta}^{\\mbox{}}\\lambda_{\\beta\\alpha}^{\\mbox{}}\n  \\right]\n  \\right\\}\n  \\\\\n  &=\\hat{\\mathcal{O}}\\left\\{c_{n\\alpha}^{\\mbox{}}\n  -\n  \\frac{\\delta}{h_{nn}\\!-\\!h_{11}\\!+\\!E_0}\n  \\sum_{\\beta}c_{n\\beta}^{\\mbox{}}\\lambda^-_{\\beta\\alpha}\n  \\right\\}\n  ,\n\\end{split}\n\\label{eq:symm}\n\\end{equation}\nwhere\n$\\mathcal{H}_{\\alpha,nm}=\\langle\\varphi_n|\\hat{\\mathcal{H}}_\\alpha^{\\mbox{}}|\\varphi_m\\rangle$ and\n\\begin{equation}\n\\lambda_{\\beta \\alpha} = \\frac{1}{2} \\sum_{mn} c_{n\\beta}^* \\left( \\mathcal{H}_{\\alpha,nm} + \\mathcal{H}_{\\beta,nm} \\right) c_{m\\alpha}.\n\\end{equation}\nThe (interlaced) solution of the gap equations remains as before, but we do not update the local densities, the HF Hamiltonian $\\hat{h}$ and the pairing potential $\\hat{\\tilde{h}}$ in configuration space.  \nThe convergence of the iteration is checked,\nagain, by the symmetry conditions (\\ref{eq:symmcondaver}).  \nThe most efficient combination of the full 3D step with the iterations in configuration\nspace is a matter of experience, see Sec.\\ \\ref{sec:benchmark}.\n\n\n\\subsection{Soft cutoff on pairing-active space}\nIt is well known that the HFB equations with local interactions diverge when solved in infinite quasiparticle\/canonical space \\cite{Dobaczewski1984}.  \nTo limit the pairing-active space,  all local densities (\\ref{eq:densities}) are augmented by the cutoff factor $w_\\alpha$, for instance the particle and pairing densities:\n\\begin{subequations}\n\\label{eq:cutpairdens}\n\\begin{align}\n\t\\rho(\\vec{r}) &= \\displaystyle\n    \\sum_{\\alpha}  \n    w_{\\alpha}v_{\\alpha}^2\\sum_{s}|\\psi_{\\alpha}(\\vec{r},s)|^2,\n    \\\\\n  \\xi(\\vec{r}) &=\n  \\sum_{\\alpha}w_{\\alpha}^{\\mbox{}}u_{\\alpha}^{\\mbox{}}v_{\\alpha}^{\\mbox{}}\n  \\sum_{s}|\n   \\psi_{\\alpha}^{\\mbox{}}(\\vec{r},s) |^2.\n\\label{eq:pairdens}\n\\end{align}\nThe same augment also applies to the kinetic-energy and the spin-orbit densities.\nA fixed number of states (realized by setting $w_\\alpha=1$ or 0) is dangerous for two reasons.  \nFirst, it hinders the portability of the pairing functional between codes and nuclei, because the s.p.\\ space depends on the basis representation.\nSecond, level crossings near the hard cutoff can induce jumps of the pairing energy.\nThese problems can be solved by pairing renormalization \\cite{Dobaczewski1996,Borycki2006} which, however, could be impractical in a full 3D treatment that involves huge canonical spaces.\nTherefore, a commonly used remedy is to use a soft pairing cutoff \\cite{Bonche1985,Krieger1990}\n\\begin{equation}\n  w_\\alpha^{\\mbox{}}\n  =\n  \\frac{1}\n       {\\displaystyle 1+\n        \\exp\\left(\\frac{h_{\\alpha\\alpha}-\\epsilon_\\mathrm{F}-\\Delta\\epsilon_\\mathrm{cut}}\n                      {\\Delta\\epsilon_\\mathrm{cut}\/10}\n        \\right)}\n  .\n\\label{eq:softcut}\n\\end{equation}\n\\end{subequations}\nThe cutoff places a fixed band $\\Delta\\epsilon_\\mathrm{cut}$ above\nthe actual Fermi energy $\\epsilon_\\mathrm{F}$.  We are going to use here $\\Delta\\epsilon_\\mathrm{cut}=15$ MeV. \nIt is  important to note that the soft cutoff modifies the state-dependent Hamiltonian $\\hat{\\mathcal{H}}_\\alpha^{\\mbox{}}$:\n\\begin{equation}\n  \\hat{\\mathcal{H}}_\\alpha^{\\mbox{}} = w_\\alpha \\left( v^2_\\alpha \\hat{h} + u_\\alpha^{\\mbox{}}v_\\alpha^{\\mbox{}} \\hat{\\tilde{h}} \\right),\n\\end{equation}\nwhich defines all the ingredients entering the canonical HFB equations.\n\n\\subsection{Strategies to avoid premature pairing breakdown}\n\\label{sec:breakdown}\n\nThe pairing comes along with a second-order superfluid-to-normal phase transition.  Below the  critical pairing strength, the HFB pairing gap remains exactly zero. \nAbove this critical strength, pairing becomes active and the gap starts to grow quickly. \nHowever, the onset of pairing is often delayed in a numerical calculation. \nThe problem is that zero pairing remains a valid solution to the HFB (BCS) equations, but an unstable one.\nIt can then take a very long time before the algorithm overcomes the instability and drives towards a stable solution. \nAs a consequence, an iteration scheme can easily be deadlocked due to a pairing breakdown. \nThis is a well-known problem.\nMost algorithms incorporate recovery strategies, such as occasional kickoffs by giving the pairing gap an artificial value, small enough not to spoil the physics but large enough to revive the pairing mechanism.\n\nThere is a more insidious problem with the state-dependent pairing gap $\\Delta_{\\alpha\\alpha}$: \nIt can happen that one canonical state logs out from the\npairing scenario and gets stuck in its own pairing breakdown\n$\\Delta_{\\alpha\\alpha}\\rightarrow 0$. \nTo understand that, we inspect  Eq.\\ (\\ref{eq:cmfeq}) and recall that \n$\\hat{\\mathcal{H}}_\\alpha^{\\mbox{}}=v_\\alpha^{\\mbox{}}\\left(\nv_\\alpha^{\\mbox{}}\\hat{h} + u_\\alpha^{\\mbox{}}\\hat{\\tilde{h}}\\right)$.\nFar above the Fermi energy, we encounter states with $u_\\alpha^{\\mbox{}}\\gg v_\\alpha^{\\mbox{}}$ such that $\\hat{\\mathcal{H}}_\\alpha^{\\mbox{}}\\approx\\hat{\\tilde{h}}$ becomes a purely local operator. \nThe solution to the mean field equation is $\\psi\\propto\\delta(\\vec{r}-\\vec{r}_\\mathrm{min})$ where $\\vec{r}_\\mathrm{min}$ is the point $\\hat{\\tilde{h}}$ has a minimum. \nIn practice, this will be the representative of a\n$\\delta$-function on the grid, slightly mellowed by orthonormalization to other states.\nAs a consequence, the state acquires a very high kinetic energy and a very high canonical s.p.\\ energy, which drives the solution of the gap equation (\\ref{eq:uveq}) even more toward $v_\\alpha^{\\mbox{}}\\rightarrow 0$. \nThis as such is a valid physical mechanism as long as the iterations curb down the occupations slowly from above.\nIt becomes a\nproblem if some $v_\\alpha^{\\mbox{}}$ gets stuck at zero at the very early stage of the iterative process. \nOnce this has happened, the state $\\alpha$ is locked out of the pairing space.\nIn order to avoid this from happening, we adopt a strategy similar to simulated annealing \\cite{Press1992} and start the iteration scheme with an enhanced effective pairing strength which gradually reduces to the\nphysical strength as\n\\begin{equation}\n  V_\\mathrm{pair}^\\mathrm{(eff)}\n  =  \n  V_\\mathrm{pair}^{\\mbox{}}\n  \\left(\\eta_\\mathrm{enh}\n    \\frac{\\mbox{max}(\\mathcal{N}_\\mathrm{enh}-\\mbox{\\tt iter},0)}\n         {\\mathcal{N}_\\mathrm{enh}}\n    +1\n  \\right),\n\\end{equation}\nwhere {\\tt iter} is the iteration number. \nIn practice, we use an enhancement factor $\\eta_\\mathrm{enh}=2$ and $\\mathcal{N}_\\mathrm{enh}=400$. \nWith this choice, the lock-in problem in the most critical early phases of iterations is avoided.\n\n\n\\subsection{Hermiticity restoration} \\label{sec:hermiticity}\nAccording to Refs.\\ \\cite{Maruhn2014,Schuetrumpf2018},\nthe explicit expression of applying the Skyrme HF Hamiltonian $\\hat{h}$ on a wave function $\\psi$ can be written as:\n\\begin{equation}\\label{eq:h_terms}\n\t\\begin{split}\n\t\\hat{h}\\psi =\\ &U(\\vec{r})\\psi - \\nabla\\cdot \\left[B(\\vec{r})\\nabla\\right]\\psi \\\\\n     + & \\frac{\\mathrm{i}}{2}\\left[\\vec{W} \\cdot\\left(\\vec{\\sigma} \\times \\nabla\\right)\\psi + \\vec{\\sigma} \\cdot \\nabla \\times \\left(\\vec{W}\\psi\\right) \\right]. \n\t \\end{split}\n\\end{equation}\nThis expression can be directly derived from the Skyrme EDF via Eq.\\ (\\ref{eq:mfham}), without invoking the product rule.\nIn \\cite{Schuetrumpf2018} it was noted that the product rule is not perfectly fulfilled\nwhen derivatives are evaluated via the discrete Fourier transform.\nTherefore, in {\\sc Sky3D} version 1.1 the commonly-adopted form of the spin-orbit term \n\\begin{equation} \\label{eq:old_spin_orbit}\n\\mathrm{i} \\vec{W} \\cdot(\\vec{\\sigma} \\times \\nabla) \\psi\n\\end{equation}\nwas replaced by the one given in Eq.\\ (\\ref{eq:h_terms});\nwith $\\nabla \\times \\vec{W} = 0$, these two forms are connected by the product rule.\nHowever, the second term of $\\hat{h}\\psi$, which involves a position-varying differential operator, is still calculated through the product rule in {\\sc Sky3D}:\n\\begin{equation}\n  \\nabla\\cdot \\left[B(\\vec{r})\\nabla\\right]\\psi = \n  \\sum_{i=x,y,z} \\frac{\\partial B}{\\partial i}\\frac{\\partial \\psi}{\\partial i} + B\\frac{\\partial^2 \\psi}{\\partial i^2}.\n  \\label{eq:prodrule}\n\\end{equation}\nUnfortunately,  evaluating Eq.\\ (\\ref{eq:prodrule}) with the FFT-based differentiation breaks the Hermiticity of the operator \\cite{Johnson2011}.\nThis point is confirmed by computed results shown in Sec.\\ \\ref{sec:hermiticityresult}.\n\nInstead of using Eq.\\ (\\ref{eq:prodrule}), the simplest way to restore Hermiticity in the evaluation of $\\nabla \\cdot \\left(B \\nabla \\psi \\right)$ is to compute two consecutive first-order derivatives. \nBut, as discussed in Sec.\\ \\ref{sec:num_grid},  this creates a problem with   the second derivative  that involves the Fourier component $\\widetilde{\\psi}(k_{N_x\/2+1})$.\nAccording to Ref.\\ \\cite{Johnson2011}, one should keep the term $\\widetilde{\\psi}(k_{N_x\/2+1})$  in the two first derivatives, \nand average the results of $k_{N_x\/2+1}= \\pm \\frac{N_x}{2} \\delta k$ to maintain the symmetry in Fourier space. \nOne can show that this ``average'' algorithm is equivalent to Algorithm \\ref{algorithm} (Algorithm 3 in \\cite{Johnson2011}), \nwhich is simpler to compute and thus implemented in {\\sc HFBFFT}. In Algorithm \\ref{algorithm}, one first computes an FFT-based first derivative, with $\\widetilde{\\psi}(k_{N_x\/2+1})$ saved and then zeroed before the inverse FFT is performed on $\\mathrm{i}k_n\\widetilde{\\psi}(k_n)$ (steps 1 through 3).\nThen one multiplies in coordinate space with the field $B(x)$ involved (step 4). Finally, one computes the derivative of the $B\\psi^\\prime$ thus obtained with modifying $\\widetilde{\\phi}(k_{N_x\/2+1})$ so that we can keep Hermiticity without losing the information of $\\widetilde{\\psi}(k_{N_x\/2+1})$ (steps 5 through 7).\nThe position-varying differential operator also appears in many other physics equations, like the heat equation with varying diffusivity and Poisson's equation with changing permittivity;\nhence, Algorithm \\ref{algorithm} has a broad application range.\n\n\\begin{algorithm}\n\\setstretch{1.5}\n\\caption{Compute the one-dimensional position-varing differetiation  $\\frac{d}{dx}\\left[B(x)\\frac{d\\psi}{dx}\\right]$.}\n\\label{algorithm}\n\\begin{algorithmic}[1]\n\\State  Compute Fourier transform\n$\\widetilde{\\psi}_n=\\mathtt{FFT}[\\psi_\\nu]$\n       with $\\psi_\\nu=\\psi(x_\\nu)$.\n\\State Save $\\widetilde{\\psi}_{N_x\/2+1}\\rightarrow\\widetilde{\\Psi}$, build\n       $\\widetilde{\\psi^\\prime}_n=ik_n\\widetilde{\\psi}_n$ with $\\widetilde{\\psi^\\prime}_{N_x\/2+1}=0$.\n\\State Compute inverse transform\n       $\\psi^\\prime_\\nu=\\mathtt{FFT}^{-1}[\\widetilde{\\psi^\\prime}_n]$.\n\\State  Build $\\phi_\\nu=B_\\nu\\psi^\\prime_\\nu$ with $B_\\nu=B(x_\\nu)$.\n\\State  Compute Fourier transform $\\widetilde{\\phi}_n=\\mathtt{FFT}[\\phi_\\nu]$.\n\\State  Build $\\widetilde{\\phi^\\prime}_n=ik_n\\widetilde{\\phi}_n$ and set\n        $\\widetilde{\\phi^\\prime}_{N_x\/2+1} = -\\frac{\\sum_{\\nu=1}^{N_x}c_\\nu}{N_x} \\left(\\frac{N_x}{2} \\delta k \\right)^2\\widetilde{\\Psi}$.\n\\State Compute inverse transform \n       $\\frac{d}{dx}\\left[B(x)\\frac{d\\psi}{dx}\\right]_\\nu=\n        \\mathtt{FFT}^{-1}[\\widetilde{\\phi^\\prime}_n]$.\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Numerical realization in harmonic-oscillator basis}\\label{sec:ho_representation}\n\nThe HFB solutions obtained with the {\\sc HFBFFT} code will be compared with the  well-established code {\\sc HFBTHO}. This code has been extensively documented in several publications \\cite{Stoitsov2005, Perez2017}. \nThe solver\n{\\sc HFBTHO} uses an expansion of the s.p.\\ wave functions in the basis of axially symmetric  HO (or transformed HO) states. The basis is given by the number of oscillator shells that defines the s.p.\\ space size, as well as the oscillator length and deformation that determine the HO wave functions.\nLocal fields in {\\sc HFBTHO} are handled on the Gaussian integration points and the\nGaussian integration rule is used to compute integrals.\n\nA major difference between the two codes lies in the way the HFB\nequations are solved. {\\sc HFBFFT} uses a representation in terms of the canonical basis, see Sec.\\ \\ref{sec:algorithm}, while {\\sc HFBTHO} works in\na quasiparticle space. \nThe results are fully equivalent if the same number of s.p.\\ states is used. \nDifferences appear in connection with the cutoff in pairing space. {\\sc HFBFFT} defines the cutoff in terms of the canonical s.p.\\ energies, whereas {\\sc HFBTHO} does that in terms of the quasiparticle energies.\nThis, taken together with the fact that the pairing strength has to depend on the size of the pairing space, means that the values of $V_{\\mathrm{pair},q}$ are not fully portable. \nIt will play a role in the benchmarking tests presented in Sec.\\ \\ref{sec:benchmark}.\n\n\n\n\\section{Benchmarks}\\label{sec:benchmark}\n\nIn this section, we benchmark  {\\sc HFBFFT}  against {\\sc HFBTHO}, {\\sc Sky1D}, and {\\sc Sky2D}. \nThese codes have symmetry restrictions. \n{\\sc Sky1D} enforces spherical symmetry and can be used for magic nuclei.  \n{\\sc HFBTHO} and {\\sc Sky2D} allow for axially symmetric shapes and cover all test cases here. \nThose codes can run with or without imposing reflection symmetry. \n\nFirst, we determine appropriate parameters to use, including the box size and grid spacing.\nBefore making comparisons with other solvers, we quantify the effect brought by the Hermiticity restoration.\nIn the next step, we compare some characteristic nuclei ranging from spherical doubly magic  $^{132}$Sn and $^{208}$Pb, to spherical superfluid $^{120}$Sn, to deformed superfluid $^{102,110}$Zr,  to superdeformed fission isomer in $^{240}$Pu.\nIn all these calculations, we use the Skyrme functional SLy4 \\cite{Chabanat1998} in the particle-hole channel and the mixed density-dependent\n$\\delta$ interaction ($\\rho_{0,\\mathrm{pair}}=0.32$ fm$^{-3}$ in Eq.\\ (\\ref{eq:epair})) in the particle-particle channel.\n\n\\subsection{Parameter determination}\nTo ensure the correct asymptotic behavior near the box boundary, we use $^{110}$Zr to determine the appropriate box and grid sizes.\nThe nucleus $^{110}$Zr is chosen because it has a significant neutron excess and thus weakly bound canonical states.\nThe calculated proton and neutron densities were inspected for different box lengths and different grids. \nBased on this analysis, we adopted a cubic box with a side length of 37.6 fm and 48 grid points in each dimension (spacing between two neighboring points is 0.8 fm).\nWith the above settings, the proton and neutron densities are below 10$^{-7}$ nucleons\/fm$^3$ at the boundary, which is small enough for our tests.\nFor spherical nuclei such as $^{120}$Sn, a smaller box is usually sufficient.\n\nWe take 176 neutron and 126 proton canonical states ($\\Omega_n = 176,\\ \\Omega_p = 126$), 15 MeV energy cutoff for the pairing window.\nThis number of active states is determined by the tests for spherical $^{120}$Sn and deformed $^{110}$Zr nuclei.\nWhen we increase the number of active states to 200 neutron and 150 proton states, the total energy remains stable within 10 keV.\nIn order to speed up the convergence, we perform 100 sub-iteration steps in the configuration space between two gradient iterations in the coordinate space, initialize with 30 HF+BCS steps, and employ the pairing enhancement factors defined in Sec.\\ \\ref{sec:breakdown}.\n\nFor {\\sc HFBTHO} calculations, we take 25 HO shells  for both protons and neutrons unless explicitly stated otherwise.\nAn axially deformed HO basis with $\\beta_2 = 0.2$ is used in deformed nuclei ground state calculations ($^{102,110}$Zr and $^{240}$Pu) and $\\beta_2 = 0.6$ is used to calculate the $^{240}$Pu fission isomer.\nFor the spherical nuclei, we also compare {\\sc HFBFFT}  with the results  of the 1D spherical HFB code {\\sc Sky1D}, which  uses a radial coordinate-space mesh and the five-point finite difference formula for derivatives. The mesh spacing and the number of points we employ in {\\sc Sky1D} are 0.15 fm and 141, respectively.\nFor the deformed nuclei, we compare {\\sc HFBFFT} results with the 2D axial HFB code {\\sc Sky2D}, which uses 31 points in both $r$- and $z$-directions with  a mesh spacing of 0.7 fm. Since the nuclei considered in this study are all reflection-symmetric, the grid extends from $z=0$\\,fm to $z=21$\\,fm.\n\n\\subsection{Pairing renormalization}\\label{sec:renormalization}\nAs we mentioned in Sec.\\ \\ref{sec:ho_representation}, pairing strengths are not portable between {\\sc HFBFFT} and {\\sc HFBTHO} because of different descriptions of the pairing space and different structures of one-quasiparticle continuum in these two solvers. \nTherefore, we need to renormalize the pairing strengths to compare results for open-shell nuclei in which pairing is essential.\nIntuitively, there are several choices for pairing renormalization.\n \nFor instance, one can tune pairing strengths to reproduce the pairing energies in different solvers.\nHowever, as discussed in \\cite{Papenbrock1999,Borycki2006}, the pairing energy density is divergent with respect to the cutoff energy. A better measure is the  quantity\n\\begin{equation}\\label{eq:Ekineff}\n\\tilde{E}_\\mathrm{kin}^q = E_\\mathrm{kin}^q+ E_\\mathrm{pair}^q~~(q={\\rm n\\,or\\,p}),\n\\end{equation}\nwhich is less sensitive to the pairing  cutoff energy.\nAs it will be\nshown in Sec.\\ \\ref{sec:doublymagic},  the kinetic energy strongly depends on the  basis size in {\\sc HFBTHO}.\nTherefore, in situations when the error related to the choice of the basis, or spatial grid, dominates, $\\tilde{E}_\\mathrm{kin}$ will be a poor renormalization measure.\nAnother pairing measure is the spectral pairing gap \\cite{Dobaczewski1984,Dobaczewski1996,Bender2000}\n\\begin{equation}\\label{eq:Gap}\n\\Delta^q \\equiv  \\frac{\\sum_{\\alpha \\in q} w_\\alpha  v_\\alpha^2 \\Delta_{\\alpha\\alpha}}{\\sum_{\\alpha \\in q} w_\\alpha v_\\alpha^2}\n~~(q={\\rm n\\,or\\,p}).\n\\end{equation}\nThis quantity has been used in numerous papers to adjust pairing strengths to observed odd-even mass differences and we shall use it in this study to renormalize the pairing channel of different solvers.\n\n\n\\subsection{Energy shift by Hermiticity restoration} \\label{sec:hermiticityresult}\n\nAs we mentioned in Sec.\\ \\ref{sec:hermiticity}, the product rule in the FFT-based differentiation violates the Hermiticity of the position-varying differential operator.\nTo restore the Hermiticity, we implement Algorithm \\ref{algorithm} in {\\sc HFBFFT}.\nThe results are shown in Table \\ref{tab:hermiticity} for several nuclei.\nThe Hermiticity violation is demonstrated  by a non-vanishing $\\Delta\\mathcal{S}^2 \\sim 10^{-6}$ MeV$^2$\nin the calculations of spherical nuclei $^{132}$Sn and $^{208}$Pb for which the static pairing vanishes and hence the HFB calculation is reduced  to  HF.\nAs for other open-shell nuclei with non-vanishing pairing, their $\\Delta\\mathcal{S}^2$ values are similar before and after the Hermiticity restoration.\nThese values of $\\Delta\\mathcal{S}^2$  are characteristic of the accuracy typically achieved in {\\sc HFBFFT} and they are larger than the error due to the Hermiticity breaking.\nIn terms of the total energy,  the effect is of the order of a few keV, i.e., insignificant for many practical applications. \nEven so, Hermiticity breaking effects can affect some calculations if not remedied.\nFor example, the small error brought by the Hermiticity breaking can accumulate step by step in a time-dependent calculation.\n\n\n\\begin{table}[htb]\n\\centering\n\\begin{tabular}{c|rr|rr}\n\\hline\n\\hline\n\\multirow{2}{*}{{\\sc HFBFFT}}  & \\multicolumn{2}{c|}{Hermiticity broken} & \\multicolumn{2}{c}{Hermiticity restored} \\\\\n    & \\multicolumn{1}{c}{$E_\\mathrm{tot}$} &\\multicolumn{1}{c|}{$\\Delta\\mathcal{S}^2$}          & \\multicolumn{1}{c}{$E_\\mathrm{tot}$} &\\multicolumn{1}{c}{$\\Delta\\mathcal{S}^2$} \\\\\n\\hline \n$^{132}$Sn  &$-$1103.542\\textbf{9} &3.91E-06  &$-$1103.542\\textbf{3}  &4.15E-14 \\\\\n$^{208}$Pb  &$-$1635.68\\textbf{17} &6.61E-06  &$-$1635.68\\textbf{07}   &6.40E-14\\\\\n$^{120}$Sn  &$-$1018.331\\textbf{0} &3.11E-05       &$-$1018.330\\textbf{5}  &3.45E-05  \\\\\n$^{110}$Zr  &$-$893.857\\textbf{8} &3.09E-05       &$-$893.857\\textbf{4}  &4.02E-05     \\\\\n$^{102}$Zr  &$-$859.469\\textbf{6}  &3.00E-05            &$-$859.469\\textbf{2}  &2.19E-05       \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Total energies $E_\\mathrm{tot}$ (in MeV) and $\\Delta\\mathcal{S}^2$ (in MeV$^2$) for five nuclei calculated with  {\\sc HFBFFT} without and with the Hermiticity restoration. The digits which do not coincide before and after the Hermiticity restoration are marked in bold.}\n\\label{tab:hermiticity}\n\\end{table}\n\n\\subsection{Doubly magic nuclei: $^{132}$Sn and $^{208}$Pb}\\label{sec:doublymagic}\nIn the first step, we calculate two doubly magic unpaired  nuclei $^{132}$Sn and $^{208}$Pb.\nFor these nuclei, the results of \\mbox{\\sc HFBFFT} and {\\sc Sky3D} are identical.\nIn Table \\ref{tab:132Sn}, we list the  ground-state energies as well as contributions from various functional terms,  obtained from four solvers {\\sc HFBFFT}, {\\sc HFBTHO}, {\\sc Sky1D} and {\\sc Sky2D} for $^{132}$Sn. \nTable~\\ref{tab:208Pb} shows similar results for $^{208}$Pb.\n\\begin{table}[htb]\n\\centering\n\\begin{tabular}{lrrrr}\n\\hline\n\\hline\n$^{132}$Sn                    & {\\sc HFBTHO}     & {\\sc HFBFFT}  & {\\sc Sky1D} & {\\sc Sky2D}\\\\\n\\hline\n$E_{\\mathrm{tot}}$                  & $-$1103.4\\textbf{9}\t& $-$1103.5\\textbf{4} & $-$1103.5\\textbf{7}\t& $-$1103.5\\textbf{6}\\\\\n$E_{\\mathrm{kin}}^\\mathrm{n}$                & 1637.\\textbf{71}\t& 1637.9\\textbf{7}\t& 1638.0\\textbf{1} & 1638.0\\textbf{2}\\\\\n$E_{\\mathrm{kin}}^\\mathrm{p}$                & 808.\\textbf{44}\t& 808.5\\textbf{7}\t& 808.5\\textbf{9} &808.5\\textbf{6}\\\\\n$E_{\\mathrm{\\rho \\rho}}$                & $-$4876.\\textbf{26}\t&  $-$4877.0\\textbf{2} & $-$4877.0\\textbf{4}\t&$-$4877.0\\textbf{7} \\\\\n$E_{\\mathrm{\\rho \\tau}}$                & 821.\\textbf{49}\t& 821.7\\textbf{0}\t& 821.7\\textbf{3}  &821.7\\textbf{2}\\\\\n$E_{\\mathrm{\\rho \\Delta \\rho}}$                & 248.\\textbf{11}\t & 248.2\\textbf{3}\t& 248.2\\textbf{5} &248.2\\textbf{3}\\\\\n$E_{\\mathrm{\\rho \\nabla \\vec{J}}}$                & $-$84.4\\textbf{0}\t& $-$84.4\\textbf{3} & $-$84.4\\textbf{4}\t &$-$84.4\\textbf{3}\\\\\n$E_{\\mathrm{Coul}}$                & 341.4\\textbf{2} & 341.4\\textbf{4} & 341.4\\textbf{4} &341.4\\textbf{3}\t\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Energy contributions (in MeV) to the binding energy of  for $^{132}$Sn computed with {\\sc HFBTHO}, {\\sc HFBFFT}, {\\sc Sky1D},  and {\\sc Sky2D}. The digits which do not coincide with {\\sc HFBFFT} are marked in bold.}\n\\label{tab:132Sn}\n\\end{table}\nWhen we compare {\\sc HFBFFT} with {\\sc Sky1D} and {\\sc Sky2D} for $^{132}$Sn , we find the  energy differences do not usually exceed 40\\,keV. \nSuch small differences  can be traced back to   different box boundary conditions assumed in these codes.\nIn {\\sc HFBFFT}, calculations are performed in a 3D rectangular box while  the box is represented by  a spherical shell in {\\sc Sky1D} and a cylindrical shape in {\\sc Sky2D}. \nFor a well-bound nucleus and large spatial boxes, the results should be practically independent of the geometry of the box. \nAs seen in Table\\,\\ref{tab:132Sn} this indeed holds for $^{132}$Sn. \nAs we will see below, larger box-related errors are expected in superfluid and\/or weakly bound nuclei. \nFor nuclear matter and time-dependent calculations, the finite-size box errors can be appreciable; they can be greatly reduced by imposing twist-averaged boundary conditions\n\\cite{Schuetrumpf2016}.\n\n\\begin{table*}[htb]\n\\centering\n\\begin{tabular}{lrrrrrr}\n\\hline\n\\hline\n$^{208}$Pb                    & $N$=15     & $N$=20 & $N$=25 & $N$=30 & {\\sc HFBFFT}  & {\\sc Sky1D} \\\\\n\\hline\n$E_{\\mathrm{tot}}$                 & $-$1634.\\textbf{25}\t & $-$1635.\\textbf{16}\t & $-$1635.\\textbf{46}\t & $-$1635.6\\textbf{2} &$-$1635.6\\textbf{8}\t& $-$1635.7\\textbf{0} \\\\\n$E_{\\mathrm{kin}}^\\mathrm{n}$               & 2525.\\textbf{13}\t& 2527.\\textbf{80}\t& 2528.\\textbf{42}\t& 2528.\\textbf{83} &2529.1\\textbf{3} & 2529.1\\textbf{6}\t\\\\\n$E_{\\mathrm{kin}}^\\mathrm{p}$              & 133\\textbf{4.56}\t& 133\\textbf{6.34}\t& 1336.\\textbf{71}\t& 1336.\\textbf{91}  &1337.0\\textbf{6}\t& 1337.0\\textbf{7}  \\\\\n$E_{\\mathrm{\\rho \\rho}}$               & $-$78\\textbf{35.80}\t& $-$784\\textbf{4.07}\t& $-$784\\textbf{5.66}\t & $-$7846.\\textbf{67}  & $-$7847.\\textbf{54}& $-$7847.\\textbf{63} \\\\\n$E_{\\mathrm{\\rho \\tau}}$                & 132\\textbf{7.84}\t& 132\\textbf{9.55}\t& 1329.\\textbf{79}\t& 1329.\\textbf{98} &1330.2\\textbf{0}\t& 1330.2\\textbf{2}  \\\\\n$E_{\\mathrm{\\rho \\Delta \\rho}}$               & 31\\textbf{4.05} &  315.\\textbf{12}\t& 315.\\textbf{12} \t& 315.\\textbf{17}  &315.2\\textbf{9}\t& 315.2\\textbf{9} \\\\\n$E_{\\mathrm{\\rho \\nabla \\vec{J}}}$                & $-$96.\\textbf{30}\t& $-$96.4\\textbf{4}\t& $-$96.4\\textbf{2} \t& $-$96.4\\textbf{3} &$-$96.4\\textbf{5} & $-$96.4\\textbf{5}\t\\\\\n$E_{\\mathrm{Coul}}$                & 796.\\textbf{26} & 796.\\textbf{55}\t& 796.\\textbf{56}\t& 796.6\\textbf{0} &796.6\\textbf{3} & 796.6\\textbf{3}\t \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Energies for $^{208}$Pb from {\\sc HFBTHO} (computed with different number of HO shells $N$), {\\sc HFBFFT}, and {\\sc Sky1D}. All energies are in MeV. The digits which do not coincide with {\\sc HFBFFT} are marked in bold.}\n\\label{tab:208Pb}\n\\end{table*}\n\nWe find about 50\\,keV energy difference between {\\sc HFBTHO} and {\\sc HFBFFT};  this difference can be primarily traced back to $E_\\mathrm{kin}$ and  $E_\\mathrm{\\rho \\rho}$.\nAs discussed in Refs.\\ \\cite{Furnstahl2012, Binder2016}, the kinetic energy converges slowly in the HO basis. \nTo investigate this effect, we calculate $^{208}$Pb using different numbers of HO\\ shells in {\\sc HFBTHO}.\nWe see in Table \\ref{tab:208Pb}  that when we increase the number of HO shells to 30, the {\\sc HFBTHO} energies approach the {\\sc HFBFFT} values.\nIt is also seen that  $E_\\mathrm{kin}$ and $E_\\mathrm{\\rho \\rho}$ exhibit the largest variations with $N$.\n\nIn Refs.\\ \\cite{Furnstahl2012, More2013}, the correction to the ground-state energy due to the finite number of HO shells $N$ has been derived:\n\\begin{equation}\n\t\\label{eq:ene_conv}\n\tE_{L} =  E_{\\infty} + a_0e^{-2k_{\\infty}L},\n\\end{equation}\nwhere  $L \\equiv \\sqrt{2(N+3\/2+2)}b$, $b$ is the oscillator length of our HO basis, and $a_0,k_{\\infty}$ and $E_{\\infty}$ are fit parameters. \nThen $E_{\\infty}$ is  the energy in the limit of infinitely large model space. The fit \nof $E_\\mathrm{tot}$  to Eq.\\ (\\ref{eq:ene_conv}) \nis presented in Fig.\\ \\ref{fig:1} and the resulting value of  $E_{\\infty} =  -1635.786$ MeV  agrees fairly well with the  {\\sc HFBFFT} and {\\sc Sky1D} values.\nHence, obtaining an accurate kinetic as well as total energies in a HO basis-expansion solver requires a huge number of shells. In this context, the use of the coordinate-space representation is beneficial.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Fig1.pdf}\n\\caption{$E_\\mathrm{tot}$ as a function of $L$  for $^{208}$Pb. The  {\\sc HFBTHO} results are marked by red dots. The blue curve is fitted according to  Eq.\\ (\\ref{eq:ene_conv}).}\n\\label{fig:1}\n\\end{figure}\n\n \n\\begin{table*}[htb]\n\\centering\n\\begin{tabular}{l|r|rrr|rrr}\n\\hline\n\\hline\n\\multirow{2}{*}{$^{120}$Sn}    & {\\sc HFBTHO}    & {\\sc  HFBFFT}    & {\\sc Sky1D} & {\\sc Sky2D} & {\\sc HFBFFT }  & {\\sc Sky1D}  & {\\sc  Sky2D} \\\\ \n &    & \\multicolumn{3}{c}{$\\tilde{E}_\\mathrm{kin}^\\mathrm{n}$-renorm. }  &\n  \\multicolumn{3}{|c}{$\\Delta^\\mathrm{n}$ -renorm. } \\\\\n  & & & & & & & \\\\[-8pt]\n\\hline\n$E_{\\mathrm{tot}}$                    & $-$1018.\\textbf{77} &$-$1018.3\\textbf{4} &$-$1018.\\textbf{45} &$-$1018.3\\textbf{7} &$-$1018.7\\textbf{8} &$-$1018.\\textbf{92} &$-$1018.7\\textbf{4}\\\\\n$E_{\\mathrm{Coul}}$                   & 347.\\textbf{37}   & 347.4\\textbf{4}  & 347.4\\textbf{5} &347.4\\textbf{1} &347.4\\textbf{7}  & 347.4\\textbf{9} &347.4\\textbf{5}\\\\\n$E_{\\mathrm{kin}}^\\mathrm{n}$  & 134\\textbf{0.51}  & 1335.4\\textbf{0}  &1335.\\textbf{18} &1335.4\\textbf{3} &1339.1\\textbf{7}  &1339.1\\textbf{4} &1338.\\textbf{72}\\\\\n$E_{\\mathrm{kin}}^\\mathrm{p}$  & 830.\\textbf{75}    & 830.9\\textbf{7}   &831.0\\textbf{1} &830.0\\textbf{1} &831.2\\textbf{5}  &831.3\\textbf{1} &831.2\\textbf{8}\\\\\n$E_\\mathrm{pair}^\\mathrm{n}$                & $-$\\textbf{12.48}   & $-$7.3\\textbf{7}   &$-$7.\\textbf{15} &$-$7.4\\textbf{0} & $-$9.\\textbf{29} &$-$9.\\textbf{14} &$-$9.\\textbf{02}\\\\\n$\\tilde{E}_{\\mathrm{kin}}^\\mathrm{n}$ & \\textit{1328.03}  & \\textit{1328.03}  &\\textit{1328.03}\n&\\textit{1328.03} &1329.88 &1330.01 &1329.70\\\\\n$\\Delta^\\mathrm{n}$                  & \\textit{1.25}     & 1.08     &1.07 &1.09 &\\textit{1.25} &\\textit{1.25} &\\textit{1.25}\\\\\n$\\epsilon_{\\mathrm{F},\\mathrm{n}}$                 & $-$8.0\\textbf{2}    &$-$8.0\\textbf{1}     &$-$8.0\\textbf{1} &$-$8.0\\textbf{4} &$-$8.0\\textbf{0}  &$-$8.0\\textbf{0} &$-$8.0\\textbf{4}\\\\\n$V_\\mathrm{pair,n}$                      & $-$284.57   & $-$342.70   &$-$346.50  &$-$354.90 & $-$361.80 & $-$367.30 &$-$372.35\\\\ \n$r_\\mathrm{rms}$                  & 4.67     & 4.67   &4.67 &4.67 & 4.67 &4.67 &4.67\\\\\n\n\\hline\n\\hline  \n\\end{tabular}\n\\caption{Results of HFB + SLy4 calculations for $^{120}$Sn using {\\sc HFBTHO}, {\\sc HFBFFT}, {\\sc Sky1D}, and  {\\sc Sky2D}.\nTwo neutron pairing renormalization variants are considered, by adjusting \nthe neutron pairing strengths in {\\sc HFBFFT}, {\\sc Sky1D}, and  {\\sc Sky2D}  to reproduce the {\\sc HFBTHO} values of $\\tilde{E}_\\mathrm{kin}^\\mathrm{n}$ and $\\Delta^\\mathrm{n}$.\n All energies are in  MeV. \n The radius $r_{\\mathrm{rms}}$ is in fm.\n The digits which do not coincide with {\\sc HFBFFT} are marked in bold.}\n\\label{tab:120Sn}\n\\end{table*}\n\n\\subsection{Spherical superfluid nucleus: $^{120}$Sn}\\label{sec:120Sn}\n\nWe now calculate $^{120}$Sn which has a non-vanishing neutron pairing.\nThe neutron pairing strength $V_\\mathrm{pair,n}$ in {\\sc HFBTHO} is adjusted to the average experimental neutron pairing gap $\\Delta_n = 1.25$\\,MeV.\nIn {\\sc HFBFFT}, {\\sc Sky1D} and {\\sc Sky2D}, two pairing renormalizations are used.\nIn the first variant, the neutron pairing strengths are adjusted to reproduce the {\\sc HFBTHO} value of $\\tilde{E}_\\mathrm{kin}^\\mathrm{n}$.\nIn the second variant, the {\\sc HFBTHO} value of $\\Delta^\\mathrm{n}$ is matched.\nThe results for both variants are displayed in Table \\ref{tab:120Sn}.\nThe neutron pairing strengths vary between the solvers, reflecting different structures of their quasiparticle pairing spaces, i.e., different pairing cutoff procedures and different structures of the discretized one-quasiparticle continuum.\n\nAlthough there are large discrepancies in $E_\\mathrm{kin}^\\mathrm{n}$ and $E_\\mathrm{pair}^\\mathrm{n}$ between {\\sc HFBFFT} and {\\sc HFBTHO}, in the first renormalization variant, the difference of the total energy, about 0.4 MeV, is quite reasonable considering the fact that the pairing space is treated differently and the {\\sc HFBTHO} results are affected by the basis truncation error.\nThe difference in $E_{\\mathrm{tot}}$ between the three coordinate-space solvers, less than 150\\,keV,  reflects the dependence  of the level density of the discretized quasiparticle continuum on the box boundary conditions assumed.\n\nIn the pairing gap renormalization variant, the agreement of $E_{\\mathrm{tot}}$ is even better, with only 10-30\\,keV difference between {\\sc HFBFFT}, {\\sc HFBTHO} and {\\sc Sky2D}. \nIn this variant,  the magnitudes of the neutron pairing energy and kinetic energy are considerably larger as compared to the variant in which $\\tilde{E}_\\mathrm{kin}^\\mathrm{n}$ is renormalized.\nStill, as seen  in Table \\ref{tab:120Sn},  both  pairing renormalizations work reasonably well for $^{120}$Sn.  \nIt is interesting to note that the total root-mean-square (rms)  radii $r_\\mathrm{rms}$   are predicted very robustly in all renormalization variants.\n\n\\subsection{Axially deformed nuclei: $^{102,110}$Zr}\n\nThe neutron-rich nuclei $^{102,110}$Zr are suitable test cases, as they are known\/expected to have large prolate deformations. \nIn addition, $^{110}$Zr is weakly bound, with the neutron chemical potential $\\epsilon_{\\mathrm{F},\\mathrm{n}}\\approx -3.5$\\,MeV.  \nThe HFB proton pairing vanishes in this nucleus.\nIn Table \\ref{tab:110Zr}, we show results for $^{110}$Zr with the two pairing renormalization schemes investigated in Sec.~\\ref{sec:120Sn}.\nIt is seen that the {\\sc HFBFFT} results  for various observables, i.e., total energy, quadrupole moments, and the rms  radii, all agree well with those from  {\\sc HFBTHO} in both pairing variants.\n\n \\begin{table*}[htb]\n\\centering\n\\begin{tabular}{l|r|rr|rr}\n\\hline\n\\hline\n\\multirow{2}{*}{$^{110}$Zr }    & {\\sc HFBTHO}    & {\\sc  HFBFFT}     & {\\sc Sky2D} & {\\sc HFBFFT }   & {\\sc  Sky2D} \\\\ \n &    & \\multicolumn{2}{c}{$\\tilde{E}_\\mathrm{kin}^\\mathrm{n}$-renorm. }  &\n  \\multicolumn{2}{|c}{$\\Delta^\\mathrm{n}$ -renorm. } \\\\\n \\hline\n$E_\\mathrm{tot}$                     & $-$893.\\textbf{97}  & $-$894.3\\textbf{3} &$-$894.3\\textbf{2} &$-$894.0\\textbf{1} &$-$894.0\\textbf{1}\\\\\n$E_\\mathrm{Coul}$                  & 226.7\\textbf{2}   & 226.7\\textbf{2} &226.7\\textbf{1} & 226.7\\textbf{4} &226.7\\textbf{0}\\\\\n$E_\\mathrm{kin}^\\mathrm{n}$  & 136\\textbf{8.08}   & 1369.\\textbf{22} &1368.\\textbf{98} & 1367.\\textbf{86}  &1367.\\textbf{13}\\\\\n$E_\\mathrm{kin}^\\mathrm{p}$  & 632.0\\textbf{3}  & 632.0\\textbf{5} &632.\\textbf{13}  & 632.\\textbf{16}  &632.\\textbf{05}\\\\\n$E_\\mathrm{pair}^\\mathrm{n}$                & $-$\\textbf{3.18}   &$-$4.\\textbf{31} &$-$4.\\textbf{08} & $-$2.\\textbf{30} &$-$2.\\textbf{19}\\\\\n$\\tilde{E}_\\mathrm{kin}^\\mathrm{n}$ & \\textit{1364.90}  & \\textit{1364.90} &\\textit{1364.90} &1365.56 &1364.94\\\\\n$\\Delta^\\mathrm{n}$                  & \\textit{0.64}    & 0.93 &0.92 &\\textit{0.64} &\\textit{0.64} \\\\\n$\\epsilon_{\\mathrm{F},\\mathrm{n}}$                 & $-$3.5\\textbf{5}   & $-$3.5\\textbf{0} &$-$3.5\\textbf{2} &$-$3.5\\textbf{5} &$-$3.5\\textbf{7}\\\\\n$V_\\mathrm{pair,n}$                      & $-$284.57  & $-$409.80 &$-$428.00 & $-$371.00 &$-$384.80 \\\\\n$r_\\mathrm{rms}$                          & 4.7\\textbf{3}    & 4.7\\textbf{3} &4.7\\textbf{4} &4.7\\textbf{3}  &4.7\\textbf{4} \\\\\n$Q_{20}^\\mathrm{n}$  & 7\\textbf{89}   & 79\\textbf{4} &79\\textbf{5} & 79\\textbf{1} &79\\textbf{6}\\\\\n$Q_{20}^\\mathrm{p}$  & 44\\textbf{4}   & 44\\textbf{7} &44\\textbf{7} & 44\\textbf{5} &44\\textbf{7}\\\\\n\n\\hline\n\\hline \n\\end{tabular}\n\\caption{Results of HFB + SLy4 calculations for $^{110}$Zr with {\\sc HFBTHO}, {\\sc HFBFFT} and {\\sc Sky2D}.\nTwo neutron pairing renormalization variants are considered, by adjusting the neutron pairing strengths in {\\sc HFBFFT} and {\\sc Sky2D}  to reproduce the {\\sc HFBTHO} values of $\\tilde{E}_\\mathrm{kin}^\\mathrm{n}$ and $\\Delta^\\mathrm{n}$.\nAll energies are in MeV.\nThe radius $r_{\\mathrm{rms}}$ is in fm and quadrupole moments $Q_{20}^\\mathrm{p,n}$ are in fm$^2$. The HFB proton pairing vanishes in this nucleus.\nThe digits which do not coincide with {\\sc HFBFFT} are marked in bold.}\n\\label{tab:110Zr}\n\\end{table*}\n\nIn the case of $^{102}$Zr, one also needs to consider proton pairing. \nIn this case, we  renormalize both neutron and proton spectral pairing gaps by reproducing their values obtained from {\\sc HFBTHO}.\nIn the calculation, 25 HO shells for both neutrons and protons are employed in {\\sc HFBTHO}, which means that the s.p.\\ proton and neutron spaces are the same.\nIn {\\sc HFBFFT}, the canonical spaces are different as $\\Omega_n = 176$, $\\Omega_p = 126$. However,  the actual pairing space is set by the soft-cutoff  factor $w_\\alpha$.\nIt is seen in Table \\ref{tab:102Zr} that the benchmarking results following the pairing renormalization are very satisfactory. \nIn particular, the results of {\\sc HFBFFT}, {\\sc HFBTHO}, and {\\sc Sky2D }  are fairly close for the observables:\n$E_\\mathrm{tot}$, $r_\\mathrm{rms}$ , and quadrupole moments.\n\n\n\\begin{table}[htb]\n\\centering\n\\begin{tabular}{lrrr}\n\\hline\n\\hline\n$^{102}$Zr                    & HFBTHO   & HFBFFT  & Sky2D   \\\\\n\\hline\n$E_\\mathrm{tot}$                    & $-$859.6\\textbf{5} & $-$859.6\\textbf{9}  &$-$859.6\\textbf{7}\\\\\n$E_\\mathrm{Coul}$                  & 231.1\\textbf{1}  & 231.1\\textbf{6}   &231.1\\textbf{4}\\\\\n$E_\\mathrm{kin}^\\mathrm{n}$  & 120\\textbf{2.02} & 120\\textbf{0.96} & 120\\textbf{1.97}\\\\\n$E_\\mathrm{kin}^\\mathrm{p}$  & 651.2\\textbf{5}  & 651.2\\textbf{2}  & 651.2\\textbf{7}\\\\\n$E_\\mathrm{pair}^\\mathrm{n}$               & $-$\\textbf{3.39}   & $-$2.\\textbf{50}   &$-$2.\\textbf{39} \\\\\n$E_\\mathrm{pair}^\\mathrm{p}$              & $-$1.\\textbf{97}   & $-$1.4\\textbf{2}   &$-$1.3\\textbf{8} \\\\\n$\\tilde{E}_\\mathrm{kin}^\\mathrm{n}$ & 119\\textbf{8.63} & 1199.5\\textbf{3} & 1199.5\\textbf{8} \\\\\n$\\tilde{E}_\\mathrm{kin}^\\mathrm{p}$ & 649.\\textbf{28}  & 649.\\textbf{79} & 649.\\textbf{89} \\\\\n$\\Delta^\\mathrm{n}$                & \\textit{0.69}   & \\textit{0.69}   &\\textit{0.69}\\\\\n$\\Delta^\\mathrm{p}$                  & \\textit{0.56}   & \\textit{0.56}    & \\textit{0.56} \\\\\n$\\epsilon_{\\mathrm{F},\\mathrm{n}}$                  & $-$5.4\\textbf{3}   & $-$5.4\\textbf{2}   & $-$5.4\\textbf{4} \\\\\n$\\epsilon_{\\mathrm{F},\\mathrm{p}}$                & $-$12.0\\textbf{9}  & $-$12.0\\textbf{9}  &$-$12.1\\textbf{0}  \\\\\n$V_\\mathrm{pair}^\\mathrm{n}$                        & $-$284.57  & $-$367.00  & $-$378.40 \\\\\n$V_\\mathrm{pair}^\\mathrm{p}$\n  & $-$284.57  & $-$372.00  & $-$384.70 \\\\\n$r_\\mathrm{rms}$                         & 4.58    & 4.58     &4.58\\\\\n$Q_{20}^\\mathrm{n}$  & 63\\textbf{9}  & 63\\textbf{9} & 64\\textbf{0}  \\\\\n$Q_{20}^\\mathrm{p}$  & 411  & 411  & 411 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Results of HFB + SLy4 calculations for $^{102}$Zr using {\\sc HFBTHO}, {\\sc HFBFFT} and {\\sc Sky2D}. The pairing renormalization is carried out by adjusting the proton and neutron  pairing strengths in {\\sc HFBFFT} and {\\sc Sky2D}  to reproduce the {\\sc HFBTHO} values of $\\Delta^\\mathrm{n}$ and $\\Delta^\\mathrm{p}$.\nAll energies are in MeV. \nThe radius $r_{\\mathrm{rms}}$ is in fm and quadrupole moments $Q_{20}^\\mathrm{p,n}$ are in fm$^2$.\nThe digits which do not coincide with {\\sc HFBFFT} are marked in bold.}\n\\label{tab:102Zr}\n\\end{table}\n\n\\subsection{Superdeformed  heavy nucleus: $^{240}$Pu}\nCompared with the HO basis, the coordinate-space representation can better capture strongly deformed configurations, such as\nthe superdeformed fission isomer (f.i.) in $^{240}$Pu. \nIndeed, very large configuration spaces are needed to guarantee the convergence of the HO expansion at  large deformations \\cite{Nikolov2011,Schunck2013}.\nGiven the large number of  nucleons in $^{240}$Pu, one needs to carefully consider the number of canonical states in  {\\sc HFBFFT} and {\\sc Sky2D} calculations.\nTo this end, we performed a series of calculations by increasing the canonical space until the convergence had been reached.\nThis has been done separately for the ground state (g.s.) and f.i.\\ of $^{240}$Pu.\nThe final values are: ($\\Omega_n,\\   \\Omega_p) = (300,\\ 200)$ for the g.s.\\ and ($\\Omega_n,\\  \\Omega_p) = (400,\\ 300)$ for the f.i.\\ calculations.\nWe renormalize the pairing strengths for $^{240}$Pu to reproduce the g.s.\\  $\\Delta^{\\mathrm{n}}$ and $\\Delta^{\\mathrm{p}}$ obtained in {\\sc HFBTHO}.\nThe results are displayed in Table \\ref{tab:240Pu}. \n\nIn {\\sc HFBTHO} and {\\sc Sky2D}, the f.i.\\ is found by performing  quadrupole-moment constrained calculations.\nThe f.i.\\ configuration  in {\\sc HFBFFT} was computed  by initializing the code with various HO deformations.\nAs seen in  Table \\ref{tab:240Pu}, {\\sc HFBTHO} and {\\sc HFBFFT} results are very similar for  g.s.\\ energies,  g.s.\\ quadruple deformations, and radii.\n\nTo test the functionality of {\\sc HFBFFT}  for the f.i.,\nwe renormalize pairing strengths in {\\sc HFBFFT} and {\\sc Sky2D}  to the {\\sc HFBTHO} pairing gaps.\nBoth coordinate-space solvers give very close results for the f.i., and they agree nicely with the {\\sc HFBTHO} results, see Table \\ref{tab:240Pu}.\nOverall, the $^{240}$Pu results obtained with {\\sc HFBFFT} for both the g.s.\\ and f.i.\\ show reasonable agreement with those from {\\sc HFBTHO}.\n\n\\begin{table*}[htp]\n\\centering\n\\begin{tabular}{l|rr|rrr}\n\\hline\n\\hline\n\\multirow{3}{*}{$^{240}$Pu}  & \\multicolumn{2}{c|}{ground state} & \\multicolumn{3}{c}{fission isomer} \\\\ \n\n    & \\multicolumn{1}{c}{{\\sc HFBTHO}}  &\\multicolumn{1}{c|}{{\\sc HFBFFT}}         &\\multicolumn{1}{c}{{\\sc HFBTHO}}  \n    &\\multicolumn{1}{c}{{\\sc HFBFFT}} \n    &\\multicolumn{1}{c}{{\\sc Sky2D}}\\\\\n\\hline\n$E_\\mathrm{tot}$      & $-$1802.\\textbf{11}  &$-$1802.\\textbf{43} & $-$1797.\\textbf{00}   &$-$1797.\\textbf{35} &$-$1797.3\\textbf{5}\\\\\n$E_\\mathrm{Coul}$      & 989.\\textbf{61}     &956.\\textbf{98}  & 957.0\\textbf{2}       &  956.9\\textbf{6} & 956.9\\textbf{0}\\\\\n$E_\\mathrm{kin}^\\mathrm{n}$      & 293\\textbf{8.92}     &293\\textbf{9.94} & 292\\textbf{2.56}   &  2923.4\\textbf{5} &2923.4\\textbf{3} \\\\\n$E_\\mathrm{kin}^\\mathrm{p}$     & 152\\textbf{0.95}    &152\\textbf{1.46} & 1525.\\textbf{25}     &  1525.\\textbf{52} & 1525.\\textbf{33}\\\\\n$E_\\mathrm{pair}^\\mathrm{n}$       & $-$\\textbf{3.11}       &$-$\\textbf{2.30} & $-$\\textbf{3.52}   &  $-$2.\\textbf{60} & $-$2.\\textbf{48}\\\\\n$E_\\mathrm{pair}^\\mathrm{p}$       & $-$1.\\textbf{54}          &$-$1.\\textbf{22} & $-$2.\\textbf{85}      &$-$2.\\textbf{19} &$-$2.\\textbf{07}\\\\\n$\\tilde{E}_\\mathrm{kin}^\\mathrm{n}$     & 293\\textbf{5.81}     & 293\\textbf{7.64} & 291\\textbf{9.03}   & 2920.\\textbf{85} &2920.\\textbf{55}\\\\\n$\\tilde{E}_\\mathrm{kin}^\\mathrm{p}$      & 151\\textbf{9.40}     &152\\textbf{0.25} & 152\\textbf{2.39}   &1523.\\textbf{33} &1523.\\textbf{25} \\\\\n$\\Delta^\\mathrm{n}$      &\\textit{0.44}           &\\textit{0.44} & \\textit{0.47}  & \\textit{0.47}  & \\textit{0.47} \\\\\n$\\Delta^\\mathrm{p}$      & \\textit{0.33}          &\\textit{0.33} & \\textit{0.46}         &\\textit{0.46} &\\textit{0.46} \\\\\n$\\epsilon_{\\mathrm{F},\\mathrm{n}}$       & $-$5.7\\textbf{1}      \n& $-$5.7\\textbf{0} &$-$5.6\\textbf{6}        & $-$5.6\\textbf{5} &$-$5.6\\textbf{7}  \\\\\n$\\epsilon_{\\mathrm{F},\\mathrm{p}}$      & $-$5.6\\textbf{9}       & $-$5.7\\textbf{0}   & $-$5.7\\textbf{6}    & $-$5.7\\textbf{7} &$-$5.7\\textbf{9}  \\\\\n$r_\\mathrm{rms}$         & 5.93          &5.93 & 6.40      & 6.40  &6.40   \\\\\n$Q_{20}^\\mathrm{n}$      & 178\\textbf{4}    & 178\\textbf{2} & 50\\textbf{63}   & 507\\textbf{2} &507\\textbf{1}\\\\\n$Q_{20}^\\mathrm{p}$      & 116\\textbf{6}     & 116\\textbf{5} & 33\\textbf{36} &  334\\textbf{4} &334\\textbf{3}\\\\\n$V_\\mathrm{pair}^\\mathrm{n}$   & $-$284.57           &$-$360.00 & $-$284.57     &$-$369.00 &$-$384.60\\\\\n$V_\\mathrm{pair}^\\mathrm{p}$   & $-$284.57           &$-$355.00  & $-$284.57     &$-$360.00  &$-$375.80\\\\\ns.p. space     & 25 shells   & (300,\\ 200) & 25 shells     & (400,\\ 300) & (400,\\ 300)\\\\\n\\hline\n\\hline  \n\\end{tabular}\n\\caption{Results of HFB + SLy4 calculations for $^{240}$Pu ground state and fission isomer using {\\sc HFBTHO}, {\\sc HFBFFT} and {\\sc Sky2D}. \nThe pairing strengths in {\\sc HFBFFT} and {\\sc Sky2D} were adjusted to reproduce the spectral pairing gaps obtained in {\\sc HFBTHO} for the g.s.\\ and f.i.\\ separately.\nThe s.p.\\ space for {\\sc HFBFFT} is defined by means of ($\\Omega_n$, $\\Omega_p$). \nAll energies are in  MeV, $r_{\\mathrm{rms}}$ is in fm, and  $Q_{20}^\\mathrm{p,n}$ are in fm$^2$.\nThe digits which do not coincide with {\\sc HFBFFT} are marked in bold.}\n\\label{tab:240Pu}\n\\end{table*}\n\n\n\n\\section{Conclusions}\\label{sec:conclusion}\nWe developed a 3D Skyrme HFB solver {\\sc HFBFFT} in the coordinate-space representation using the canonical basis approach.\nThe code is based on the well-optimized {\\sc Sky3D} solver.\nIn {\\sc HFBFFT} we implemented several new elements to facilitate calculations, namely \n(i)  the sub-iteration method in configuration space to accelerate the convergence; \n(ii) the soft pairing cutoff and pairing annealing\nto avoid  pairing breakdown; and (iii)\na new algorithm to restore the Hermiticity of the HFB matrix.\n\nThe new solver  has been  benchmarked\nfor several spherical and deformed nuclei against {\\sc HFBTHO}, {\\sc Sky2D}, and (for spherical systems)  {\\sc Sky1D}. The representation of the  positive-energy continuum differs between HFB codes: In particular, it depends on the code's geometry (spherical, cylindrical, Cartesian),  the size of  s.p.\\ configuration space (number of HO shells, box size, grid size), and the effective pairing space. Consequently,\neven if the EDFs employed in two codes are identical, the pairing channel is usually described differently. This creates problems when comparing different HFB solvers as the perfect benchmarking is practically impossible \\cite{Pei2008}. In this work, we carried our inter-code comparisons by renormalizing pairing strengths to the spectral  pairing gaps and\/or the effective kinetic energy $\\tilde{E}_\\mathrm{kin}$. While both methods give similar results, spectral pairing gaps are less sensitive to the s.p.\\ space assumed.\n\nBy carrying out calculations with different HFB solvers, we were able to assess the ranges of different uncertainties. \nFor the total energy, the typical errors are: several keV due to the Hermiticity breaking; 10-80\\,keV due to different box boundary conditions assumed;  10-140\\,keV due to different quasiparticle continuum discretizations; and several hundred keV due to the basis truncation in HO basis-expansion solvers.\n\nAs a 3D solver, {\\sc HFBFFT} is the tool of choice to  study deformed and weakly bound systems. \nTo make this tool versatile,  several  enhancements are planned.\nMost importantly,  we intend to  implement pairing regularization \\cite{Bulgac02,Borycki2006,Pei2011} to get rid of the dependence of pairing strengths on the cutoff energy.\nAnother essential development is to be able to compute potential energy surfaces defined by means of constraining one-body operators. \n This will enable us to use HFBFFT in the calculations of  large-amplitude nuclear collective motions such as fission or fusion, for which the solvers based on the basis-expansion approach   require the use of excessively large configuration spaces.\nFinally, the performance of {\\sc HFBFFT} needs to be further optimized for modern supercomputer architectures.\n\n\n\n\\section*{Acknowledgments}\nComments from Kyle Godbey are gratefully appreciated.\nComputational resources were provided by the Institute for Cyber-Enabled Research at Michigan State University. \nThis material is based upon work supported by the U.S.\\ Department of Energy, Office of Science, Office of Nuclear Physics under award numbers DE-SC0013365 and DE-SC0018083 (NUCLEI SciDAC-4 collaboration).\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nInfrared behavior of Greens functions (GFs) of Yang-Mills theories has been intensively studied in the last decade. Nonperturbative information about dynamical symmetry breaking and confinement (e.g  free non-propagation of confined degrees of freedom) could be encoded in the low $q^2$ behavior of GFs.  Mostly gauge-variant GFs have been obtained by solving Schwinger-Dyson equations (SDEs) and simulated on the lattice as well. In the case of pure gluodynamics, the   methods based on  truncated  set of SDEs offer recently two type of solutions. As these solutions are non-unique and both type slightly modeled truncation dependent they should be tested consequently  when use to calculate  gauge invariant observable. The so called {\\it scaling} solutions have power law momentum behaviour with well defined exponent in the infrared \\cite{LESM2002,ZW2002,FIPA2007} and lead to infrared vanishing gluon propagator and correspondingly infrared enhanced ghost propagator. Mesons and baryons can be described by SDEs themselves, actually Bethe-Salpeter and relativistic Fadeev equation are simply the parts of SDEs. Successful matching between gauge invariant hadronic observables and those  gauge variant solutions has not been achieved until now.   \n\n\n\n\nBeside of this, there exist the so-called {\\it decoupling} solutions\nas proposed and showed for instance in \\cite{BOU2007,BLYMPQ2008,AGBIPA2008}. Such decoupling solutions typically result in Pinch Technique (PT) SDEs framework \\cite{COR1982,COR1986,COR1989,BIPA2002,BIPA2004,BIPA2008} called there massive solution since the massless pole of gluon propagator  disappear and gluon propagator is finite (but nonzero) in the infrared (for the topical review see \\cite{rewiev2009}).\nRecall, PT rearranges the original gauge scheme dependent GFs in a unique way such that unphysical degrees of freedom are eliminated.  In the mean time, as the lattice calculations \\cite{lat1,lat2,lat3} in conventional Landau gauge start to support the decoupling solution reaching recently quite infrared momenta down to 75 MeV \\cite{lat4}, it attracts new attention again \\cite{PENIN,cornwall2009}.\n \n\nPrincipal advantage of the PT is its scheme and gauge invariance.\nIt has been proved to  all orders of perturbation theory that \nthe  PT GFs   satisfy Ward identities \nand that extracted effective charges are process independent. Furthermore, GFs do not depend on a gauge fixing parameters. From this automatically follows that  when hadron property are calculated within the QCD PT GFs, an unphysical degrees freedom are eliminated form the beginning. \n However, as always necessary, the approximation must be made to truncate the infinite tower of the SDEs series. In the original paper the gauge technique was used to construct gluon vertex. To that point Kallen-Lehmann representation (KLR) was assumed for the gluon propagator.  \nIn contradiction, in other studies it is  suggested  that instead of validity, this is just the  absence of KLR which can ensure confinement \\cite{alkofer}. Pure Yang-Mills part of  QCD could be responsible for confinement, therefore the gauge technique construction based intrinsically on the KLR  may be actually a  weak point of the PT.  To that point we recall that in  the later stage \\cite{COR1986,COR1989} the gauge technique  was omitted and the gluon  vertex were constructed independently on KLR \\cite{COR1989}. In this paper, we do not consider KLR as a reasonable criterion, instead of as a meaningful criterion for the gluon propagator behavior we consider chiral symmetry breaking which must be triggered by this propagator in usual sense. Clearly, the appearance of  chiral symmetry breaking is one of the main \"musts\" of QCD.\n\nIn the next section we briefly rederive the SDE with parametrization of the solution as made in \n\\cite{cornwall2009} and basic ingredients are reviewed in this section for completeness. In the third Section we find the discuss this solution and find the restrictions  which must the resulting propagator obey. \n\n\\section{PT SDE with gauge invariant vertex}\n\n\nIn the paper \\cite{cornwall2009} the PT  propagator based on the WTI improved vertex \\cite{COR1989}  was considered.\nMaking a simple parametrization  of the solution then the SDE has been  solved analytically.\nWe adopt here the form of the solution proposed in  \\cite{cornwall2009} and obtain the running coupling for all $q^2$. \n  \nThe product of the gauge coupling  $g^2$ and PT  gluon propagator $\\hat{d}$\ndefines renormalization invariants. It is certainly allowed to rewrite this product by a new one, where one function can represent invariant running charge while the second, say the function $H$ stays for the rest, let us assume  the function $H$ shows up a massive pole instead massless one  \n. Using the same convention as in \\cite{cornwall2009} we can write:\n\\begin{equation} \\label{prd}\ng^2\\hat{d}(q^2)=\\bar{g}^2(q^2)\\hat{H}(q^2)\n\\end{equation}\nClearly the functions on rhs. of Eq. (\\ref{prd}) are obviously not uniquely define \nif one does not say more. This problem is simply avoided if one assume the form of $H$ explicitly\nsince then only one $\\bar{g}^2(q^2)$ function needs to be identified. \n\n\n\nThe simplest parametrization we can use is the following  hard mass approximation\n\\begin{equation} \ng^2\\hat{d}(q^2)=\\bar{g}^2(q^2)\\frac{1}{q^2-m^2+i\\varepsilon}.\n\\end{equation}\n\nIn some approaches \\cite{SHISOL2007} it is assumed that this is the running coupling\nwhich can satisfy Khallen-Lehmann representation (KLR).\nIn paper \\cite{cornwall2009} it is conjectured that the both functions $\\bar{g}^2$ and $\\hat{H}$  \nsatisfy KLR, so the  integral representation for the product has the same analyticity domain but \nthe absorptive part is not positive semidefinite. It should be negative in places in accordance with the one loop ultraviolet  asymptotic  $g^2\\hat{d}(q^2)\\simeq 1\/[q^2ln(q^2)]$.\n\n\nRemind here, KLR for the running charge should  read \n\\begin{equation} \\label{KLR}\n\\bar{g}^2_{KLR}(q^2)=\\frac{1}{\\pi}\\int d\\omega \\frac{\\Im \\bar{g}^2_{KLR}(\\omega)}{q^2-\\omega+i\\varepsilon}\\, . \n\\end{equation}\nSuch running  function  is homomorphic  in whole complex plane up to the real positive semi-axis of $q^2$ where the branch points are located.\n \n\nThe PT SDE is represented by a non-linear integral equation derived in \\cite{COR1986} and solved first time in \\cite{cornwall2009}, it reads\n\\begin{equation} \\label{ptsde}\n\\left[\\bar{g}^2\\hat{d}(q^2)\\right]^{-1}=q^2bZ-\\frac{ib}{\\pi^2}\\int d^{4k}\\hat{H}(k)\\hat{H}(k+q)\\left[q^2+\\frac{m^2}{11}\\right]+C \\,\\, , \n\\end{equation}\nwhere $C$ is momentum independent constant and where as the hard mass approximation has been  employed. The one loop  beta function coefficient  is   \n\\begin{equation}\nb=\\frac{11N_c-2N_f}{48\\pi^2}.\n\\end{equation}\n\nAfter the renormalization, which was made by on shell-subtraction (note, renormalization is  not multiplicative here, for details see \\cite{cornwall2009}) we get for Eq. (\\ref{ptsde})\n\\begin{equation} \\label{rptsde}\n\\left[\\bar{g}^2\\hat{d}\n(q^2)\\right]^{-1}=b\\left[J(q)(q^2+\\frac{m^2}{11})-J(m)\\frac{12m^2}{11}\\right]\\, ,\n\\end{equation}\nwhere the function $J$ is renormalized in  accordance with the correct one loop ultraviolet asymptotic and it reads\n\\begin{equation}\nJ(q)=-\\int_{4m^2}^{\\infty}d \\omega\\frac{q^2}{\\omega}\\frac{\\rho(\\omega;m)}{q^2-\\omega+i\\varepsilon}+2+2\\ln{(m\/\\Lambda)}\n\\end{equation}\nwhere $\\Lambda$ is usual QCD scale   valued few hundred $MeV$ for $N_f=2$\nand $\\rho(\\omega;m)=\\sqrt{1-\\frac{4m^2}{\\omega}}$\n. \nThe integral is a textbook scalar 1-loop integral and can be easily evaluated as following:\n\\begin{eqnarray}\nJ(q)&=&\\rho\\ln{\\left|\\frac{1+\\rho}{1-\\rho}\\right|}+2\\ln(m\/\\Lambda)  \n\\nonumber \\\\\n&-&i\\pi\\rho\\theta(q^2-4m^2)\\,\\; \n\\, \\, \\,\\,\\,\\,\\,\\,\\mbox{for} \\, \\,  1-4m^2\/q^2>0\n\\nonumber \\\\\nJ(q)&=&-i2\\rho{\\mbox{arctg}}\\left(\\frac{i}{\\rho}\\right)+2\\ln(m\/\\Lambda)\\,; \\, \\,\\, \\, \\,\\mbox{for} \\, \\, 0<q^2<4m^2  \\, .\n\\end{eqnarray}\nwhere $\\rho=\\sqrt{1-\\frac{4m^2}{q^2}}$.\n\n\n\\section{Physically admissible solutions}\n\n\n\n\n\n\nInverting the SDE (\\ref{rptsde}),  the pinch technique gauge invariant running charge can be straightforwardly evaluated:\n\\begin{equation}\nb4\\pi\\alpha_{massive}(q^2)=b\\bar{g}^2(q^2)=\\frac{q^2-m^2}{J(q)(q^2+\\frac{m^2}{11})-I(m)\\frac{12m^2}{11}}\n\\end{equation}\n\nThe imaginary part of $J$ for $q^2>4m^2$ is given by very simple phase space factor $\\pi\\rho=\\pi(1-4m^2\/q^2)$, thus the imaginary and the real  parts of the running coupling read\n\\begin{eqnarray} \\label{icka}\nIm \\, b\\bar{g}^2&=&(1-\\gamma)\\frac{\\pi\\rho(q^2)}{(Re J(q)-\\gamma J(m))^2+(\\pi\\rho(q^2))^2}\\,  \\, ;\n\\nonumber \\\\\nRe \\, b\\bar{g}^2&=&(1-\\gamma)\\frac{Re J(q)-\\gamma J(m)}{(Re J(q)-\\gamma J(m))^2+(\\pi\\rho(q^2))^2} \\, ,\n\\end{eqnarray}\nwhere we have used the short notation $\\gamma=\\frac{12m^2}{11q^2+m^2}$. For $q^2<4m^2$ the imaginary part vanishes.\n\nAs our approximated PT gluon propagator has a single  real pole, it does not respect confinement issues. Nevertheless we expect that this is a good approximation of the true full (exact) PT solution for the gluon propagator which could  posses a large enhancement at vicinity of $q^2\\simeq m^2$ instead of the real pole we use. The singularity of exact PT propagator need are very likely situated away of the real axis and need not be a simple pole but a branch point(s). Furthermore,  we argue that for any $m$ the resulting $\\alpha$ does not satisfy KLR. Actually although high $s=q^2$ behaviour of absorptive part exactly corresponds with known analyticized one loop QCD coupling \\cite{SHIRKOV,MISO1997,MISO1998}. i.e. for \n$s>>m^2,\\Lambda^2$ we can get\n\\begin{equation}\nIm \\alpha(s) \\rightarrow \\frac{4\\pi\/b}{\\pi^2+\\ln^2(s\/\\Lambda^2)},\n\\end{equation}\nbut this the appearance of nontrivial threshold which crucially changes  real part of the running coupling behaviour when $m$ is nonzero. Before showing this explicitly we discuss general features of the solution.\n\n\n\nSince we have basically reproduced  Cornwall's solution \\cite{cornwall2009} for a general $m$ ,  we  will briefly summarize results for completeness here. By construction, the correct one loop perturbation theory  behaviour is reproduced for any $m$, i.e. $\\alpha\\simeq 1\/ln(q^2\/\\Lambda^2)$ for $q^2>>\\Lambda,m$. There exists  certain critical mass, say $m_c^{I}$, bellow which the coupling $\\alpha$ is more singular  as it posses  nonsimple pole,  this critical mass is approximately given by the ratio $m_c^{I}\/\\Lambda=1.2$ here. Decreasing the mass parameter  $m$ we can find second critical point, say $m_c^{II}$ ,where this pole crosses Minkowski light cone $q^2=0$ and becomes well known unphysical spacelike Landau pole. This  Landau ghost is really unacceptable and it gives severe boundary on the gluon mass, in our case we get approximately  $m_c^{II}=0.4\\Lambda$.     Examples of running coupling  are shown in  Fig. 1 for various $m$. Singularities move to the left as $m$ decrease. For large $m$ enough,\n$m>1.2\\Lambda$, the only  standard two particle branch point singularity located at $4m^2$ remains and the pole has gone. The  singularity can appear only under the threshold, wherein the zero of $\\alpha^{-1}$ is not protected by a nonzero absorptive part. As an exotic solution, there also exists a \"double pole\" (note, it is not a simple double pole) solution for a specific $m$, i.e. the pole in $H$ is enhanced by non simple pole of $\\alpha$ at the same point $q=m$. \n\n\\begin{figure}\n\\centerline{\\epsfig{figure=alfa.eps,width=9truecm,height=9truecm,angle=0}}\n\\caption[caption]{Pinch technique coupling $\\alpha$ for a various ratio $m\/\\Lambda$. For better identification $Im $ part is displayed for regular solution with $m\/\\Lambda=1.2$ only.  } \n\\end{figure}\n\n\nThe gluon propagator vanishes faster then $1\/q^2$ , thus it cannot satisfy KLR. It was suggested in the paper \\cite{cornwall2009} thatthe PT gluon propagator can the product of the coupling and newly introduced  function $H$ both of them satisfying their own KLR. Due to this reasoning the author excluded such solutions which lead to the singular coupling in the timelike region as well. As a matter of the fact we argue that the pinch technique running coupling does not satisfy KLR for any $m$, even for those values where $\\alpha$ is regular. We check wether the coupling   $\\alpha^{KLR}(q^2)=\\bar{g}^2_{KLR}(q^2)\/(4\\pi)$ is analytical in usual sense, since it must  satisfy KLR (\\ref{KLR}). It has been done by substitution of  the absorptive part of $\\alpha^{PT}$ into rhs. of (\\ref{KLR}) , subsequently the  obtained  $\\alpha^{KLR}(q^2)$ is compared with the  real part of  $\\alpha^{PT}$  already known from   the solution of PT SDE. For easiest inspection we choose spacelike $q^2$ where the dispersion integral is regular and $\\alpha^{KLR}(q^2)$ can be evaluated with arbitrary accuracy. The comparison of analyticized coupling with $\\alpha^{PT}$ is shown in  Fig.2. For any $m$ the coupling $\\alpha^{PT}$ never agrees with \"analyticized\" coupling defined by  (\\ref{KLR}). For instance for $m=1.5$, where  the best optimized approximation is roughly achieved, one gets 30 $\\%$ underestimation in the infrared, while we get complete disagreement for large $q^2$.  Clearly, the dispersion relation, which would be very usefull in other practical calculations, is not followed here. We argue that there is no any known physical reason to expect KLR for the running coupling.\n Instead of, as $\\alpha$  is the unique form factor of the pinch gluon propagator its absence can  regarded as consequence of the confinement.\n\n\\begin{figure}\n\\centerline{\\epsfig{figure=space.eps,width=9truecm,height=9truecm,angle=0}}\n\\hspace{1cm}\n\\caption[caption]{$b\\bar{g^2}$ plotted for spacelike fourmomenta for various ratio of $m$. It is compared with analyticized coupling (AC) as described in the text.} \n\\end{figure}\n\n\n\n Let us mention here that the absence of KLR for propagator  is perhaps common phenomenon of a strong coupling theory and it is quite independent on a model details.\nActually, the absence of KLR has been already observed in strong coupling QED, scalar toy models \\cite{SAULI} and in the case of quark propagator in  Landau gauge QCD \\cite{SAUBI}.\nRoughly say, to get  the dispersion relation for selfenergy, i.e. for the inverse of the propagator, and simultaneously expect the spectral representation for the propagator itself is too strong assumption in the strong coupling theory like QCD.\n \n \n  As we discussed,  mass $m$ is severely constrained  from bellow by requirement of absence of the spacelike Landau pole. For an indication of what $m$ should be we do not choose the criterion of KLR, which appear as a quite obscured requirement in confining theory, but we require that pinch technique running coupling must be large enough in order to trigger correct chiral symmetry breaking in QCD.  In the real QCD the dynamical chiral symmetry breaking is phenomenon responsible for the most of nucleon mass (i.e. for the   u,d quarks dynamical mass generation) while it it simultaneously explains the   lightness of the pions.  To describe all these observable in selfcontained way, the chiral symmetry breaking must be correctly incorporated into the formalism. Such requirement very naturally gives upper boundary on the pinch technique gluon mass since for  gluon heavy enough,  the pinch technique running coupling is too weak and it does not trigger chiral symmetry breaking. \n\nIn principle, employing the formalism of PT SDEs for gluons and quarks simultaneously solved with  the Bethe-Salpeter equations  one should be able  to fit the mass in  the PT gluon propagator from meson spectra. Unhappily, the recent calculations are still far  from this stage even in the more conventional gauge fixed schemes and the form of propagators entering the calculation is still dubious. First, we will describe arising  obstacles of such a treatment and we suggest simplified way to make an reliable estimate of $m$, which is solely based on the solution of quark gap equation in the ladder approximation. \n\nIn lattice QCD,  a static quark-antiquark potential can be computed with the Wilson loop technique. This gives us confining linear potential $V_L$ between infinitely heavy quarks. For a correct description of excited mesons this  could be principally involved covariantly in the quark-antiquark kernel of BSE. From the other side the  various hadronic observables were calculated in the framework of Schwinger-Dyson equations during last two decades, it includes the meson spectra and decays \\cite{meson1997,meson2007,meson2008} and various form factors \\cite{ff2000,ff2008}, more complicated baryonic properties were studied in this framework as well. Most of them use the ladder approximation of the quark SDE (and meson BSE) in Landau gauge, while first steps beyond  the ladder approximation has been considered only quite recently. Recall, the ladder approximation means that \"very effective\" one  gluon exchange is considered only, while a more gluon exchanges are needed and more topologically more complicated\ndiagrams must contribute to get linear potential in non-relativistic limit. \nSuch higher skeletons quite naturally generates important scalar part of the quark-antiquark potential \\cite{Bicudo:2003ji}.  \nWe conjecture that if the quark-antiquark BSE kernel analogue of $V_L$ is not included then this is the main source of discrepancy when one compare GFs used in meson ladder calculations and the GFs actually obtained from SDEs. It is more then obvious that the effective gluon propagator used in a typical ladder approximated BSE more or less models  unconsidered higher order skeletons. Without reasonable matching of SDEs calculations on gluon propagator and  the one entering kernels  of the meson BSE, the infrared behaviour of gluon propagator is not obvious.\n \n\nOn the other hand, our knowledge of Wilson lattice results combined with the knowledge of typical quark SDE solutions offers  economical way, which we argue is efficient enough in order to estimate the PT gluon propagator in the infrared. For this purpose , let us consider the solutions of the quark SDE when one gluon and one gluon plus infrared enhanced  effective interaction $V_L$ is added. The difference of these two aolution has been  studied in the paper \\cite{BIMACACAOL2009} and it gives approximative double enhancement of the quark dynamical mass in deep infrared Euclidean $Q^2$ when $V_L$ is taken into account. Using these  arguments, the main issue is that infrared quark mass $M(0)$ should be already as large as $M(0)\\simeq \\Lambda=250 MeV$ when one uses the ladder quark gap equation alone but now with the PT running coupling implemented in. The additional unconsidered term $V_L$ could be then responsible for an additional grow of the quark mass in the same order. It automatically gives the limit on  the running coupling , its value must be significantly larger then the critical coupling, bellow which there is no symmetry breaking at all. As the dynamical quark mass function obtained in the ladder approximation is quite universal, we did need to perform a detailed numerical analyzes of the quark gap equation with PT running coupling and  we can  estimate the solution from the infrared value of the coupling, which must be $\\alpha(0) \\simeq 2.0$ or larger. To get such value we can see that we must use the solution with $m\/\\Lambda=0.4-0.7$. Since required interval lies between  $m^I_c$ and $m^{II}_c$ we always have the running coupling singularity at the timelike regime. In this way the pinch  technique offers possible  scenario for Infrared Slavery again, albeit with coupling enhancement in the timelike region simply  due to the massiveness of the gluon.  \n\n\n\n\n\n\n\n\\section{Conclusion}\n\nUsing recently obtained pinch technique gluon propagator   \\cite{cornwall2009}  the  limits on the effective gluon mass have been reconsidered.  It is confirmed that in order  to avoid unphysically singular running coupling, the gluon mass must be bounded from bellow. We argue that the requirement of KLR for the running coupling is not a good guide for this purpose and we assume that running coupling can be enhanced or even singular  in the timelike region of the momenta.\nIt is suggested that the upper boundary on the gluon mass $m$ stems from  chiral symmetry breaking when quarks are considered as well. As the infrared enhancement of the interaction is necessary to get correct triggering of symmetry breaking and since the running coupling crucially depends on mass $m$, the upper boundary stems from the minimal required pinch technique running coupling.\n It gives the acceptable region of the gluon mass $m\\simeq 0.4-0.7 \\Lambda $, or so. This is in reasonable agreement with the recent lattice results \nand simultaneously it does not contradict the existence of chiral symmetry breaking in QCD.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nNeural network architectures that achieve high accuracy at specific tasks (e.g., image classification~\\cite{vgg,resnet,densenet}) have been designed through many trials and errors by humans, which requires a high level of expertise and have been a burden to practitioners and researchers in the machine learning community.\nThus, AutoML has attracted the attention from many of them, which aims at automatically choosing a better design for each part in the machine learning pipeline, e.g., data augmentation~\\cite{aa,fast_aa,pba,adv_aug}, network architecture~\\cite{darts,nas_net}, loss functions~\\cite{am_lfs}, or learning parameters~\\cite{bayse,o_lr}.\nIn particular, several methods proposed in the field of Neural Architecture Search (NAS) started to achieve comparable accuracy to manually designed networks in a few tasks such as image classification~\\cite{nas_net,proxyless_nas,pg_darts}.\nHowever, the early methods of NAS typically have some practical problems mainly due to the large requirements for computing resources such as memories and GPUs~\\cite{rl_nas}.\nThen, recent studies tried to find better architecture with more efficient approaches~\\cite{darts,enas,amoeba_net,ds_nas},\nwhich lead the optimization goal of NAS to become varied.\n\\cite{fb_net} proposed a method to search network architectures that can achieve high accuracy under the limitation of computing resources used for inference,\nand \\cite{amc} proposed to consider memory and power efficiency at inference time by model compression.\nMost of the methods mainly focused on the network architecture as their optimization target because it has large impact on the entire performance,\nbut as the optimization goal becomes diversified, the target of AutoML also becomes varied over different building blocks of the training,\ne.g., data augmentation and learning parameters.\n\n\\label{intro}\n\\begin{figure}[!tbp]\n  \\centering\n  \\includegraphics[width=8cm,height=5cm]{overview.pdf}\n  \\caption{Comparison of search spaces. Our method jointly explores data augmentation policies and network architectures by combining differentiable methods for each part.\n  \\label{search_space}\n\\end{figure}\n\nIn terms of search spaces, even a single part in the training pipeline has a large one.\nFor example, an efficient NAS method, ENAS~\\cite{enas}, still has a large search space over $1.3 \\times 10^{11}$ possible networks.\nAuto Augment~\\cite{aa} is a method to automatically choose the data augmentation policies during training, and the search space has roughly $2.9 \\times 10^{32}$ possibilities.\nIt means that searching over all possibilities of the combination of these two parts will have about $3.8 \\times 10^{43}$ possibilities, which brings significant difficulty to automatic exploration.\nIn addition, searching for network architectures and data augmentation policies can have different objectives.\nThe former explores mainly to minimize a loss function, while the latter also tries to increase the variety of training data.\nThese difficulties have prevented previous research from searching over those two parts jointly, so that most of the studies have focused on automatic exploration of one part at a time.\nAlthough there are a few studies that tried to explore learning parameters and network architectures jointly~\\cite{auto_has,toward_auto}, few research have attempted joint optimization of data augmentation policies and network architectures to the best of our knowledge.\n\nIn this paper, we propose a joint optimization method for both data augmentation policies and network architectures.\nAs stated above, these two parts can have large search spaces in total and different objectives, which implies that combining existing methods for each part straightforwardly would be intractable.\nAdditionally, during the progress of architecture search, networks in different phases of training can desire different data augmentation policies for better generalization ability.\nMotivated by these intuitions, we propose an end-to-end differentiable approach to optimize both parts in the training pipeline simultaneously.\nFig.~\\ref{search_space} shows the difference of our search space from previous successful methods for each part.\nSpecifically, we jointly optimize the differentiable approaches for augmentation policy search~\\cite{faster_aa} and architecture search~\\cite{darts}.\nWe firstly apply differentiable operations to the input data, and then use the transformation outputs as the inputs for the differentiable NAS method.\nIt enables to optimize the augmentation policies with the gradients come from the upstream NAS method because the entire pipeline is fully differentiable, so that we can train both parts simultaneously in the end-to-end manner.\nWe consider a combination of existing methods each of which is performed independently on either of augmentation policy search or architecture search as the baseline, and compare the performance with ours.\nThe experimental results show that our method achieves competitive or superior performance in common benchmarks of image classification.\n\n\\section{Preliminaries}\n\\label{preliminary}\nAs the differentiable methods for augmentation policy search and architecture search, we adopt Faster Auto Augmentation~\\cite{faster_aa} and DARTS~\\cite{darts}.\nWe briefly summarize them in this section.\n\n\\subsection{Differentiable Data Augmentation}\nData augmentation is a series of transformation applied on the input data.\nTypically, we have to choose which operations should be applied with what magnitudes.\nSeveral methods for automatic search of probability distribution on the selection of operators and their magnitudes have been proposed~\\cite{aa,fast_aa,faster_aa}.\nThe Faster-AA considers that a policy consists of $L$ sub-policies each of which has $K$ consecutive operations.\nEach operation out of $\\#\\mathcal{O}$ operations has a probability $p_O \\in [0, 1]$ which represents how likely the operation is adopted in a sub-policy and a magnitude $\\mu_O \\in [0, 1]$ which controls the transformation.\nTherefore, the search space is $(\\#\\mathcal{O} \\times [0, 1] \\times [0, 1])^{KL}$ in total, where $\\mathcal{O}$ is a set of possible operations.\nApplying an operation $O$ to an input data $X$ is formulated as:\n\\begin{eqnarray}\nX \\rightarrow  \\left\\{\n\\begin{array}{ll}\n  O(X; \\mu_O)& (\\textrm{with the probability }p_O) \\\\\n  X & (\\textrm{with the probability of }1 - p_O), \\label{diff_op}\n\\end{array}\n\\right.\n\\end{eqnarray}\nand note that the Gumbel trick~\\cite{gumbel} is used to make the probability differentiable.\nA sub-policy is a series of $K$ operations, and during training, the output of $k$-th operation $X'$ from an input $X$ is calculated as a weighted sum over all possible $\\#\\mathcal{O}$ operations as follows:\n\\begin{eqnarray}\nX' = \\sum_{n=1}^{\\#\\mathcal{O}} [\\sigma_\\eta(\\boldsymbol{z}_k)]_n O_k^{(n)}(X; \\mu_k^{(n)}, p_k^{(n)}), \\label{op1}\\\\\n{\\bf s.t.} \\sum_{n=1}^{\\#\\mathcal{O}} [\\sigma_\\eta(\\boldsymbol{z}_k)]_n = 1, \\label{op2}\n\\end{eqnarray}\nwhere $\\sigma_\\eta$ is a softmax function with\na temperature parameter $\\eta > 0$,\nand $\\boldsymbol{z}_k \\in \\mathbb{R}^{\\# \\mathcal{O}}$ denotes the learnable parameter for the distribution of operator selection.\nDuring inference, the $k$-th operation is sampled from the categorical distribution ${\\rm Cat}(\\sigma_\\eta(\\boldsymbol{z}_k))$, so that we obtain transformed data $X'$ by Eq.~(\\ref{diff_op}).\n\n\\subsection{Differentiable NAS}\nDARTS~\\cite{darts} is a differentiable neural architecture search method which focuses on searching the inside structures of normal cells and reduction cells that are finally stacked up to build a deep network. \nEach cell is represented as a directed acyclic graph (DAG) consisting of $N$ nodes which represent intermediate features (e.g., feature maps).\nA cell takes two input nodes and one output node, and\nan edge $f^{(i, j)}$ between two nodes $i, j$ represents an operation such as convolution or pooling.\nA node $j$ has the connections with all the previous nodes $i < j$ in topological ordering of the DAG, so that the search space of DARTS is roughly\n\\begin{eqnarray}\n|\\mathcal{F}|^2 \\prod_{k=1}^{N-3}\\frac{k(k+1)}{2},\n\\end{eqnarray}\nwhere $\\mathcal{F}$ is a set of candidate operations.\nTo make the search space continuous, DARTS relaxes the categorical choice of an operation to a softmax over possible operations:\n\\begin{eqnarray}\n\\bar{f}^{(i,j)} = \\sum_{f \\in \\mathcal{F}} \\frac{\\exp(\\alpha_f^{(i,j)})}{\\sum_{f'} \\exp(\\alpha_{f'}^{(i,j)})}f(x).\n\\end{eqnarray}\nAfter the training, a discrete architecture is determined by replacing each mixed operation $\\bar{f}^{(i, j)}$ with the most likely operation $f^{(i, j)} = \\mathop{\\rm arg~max}\\limits_{f \\in \\mathcal{F}} \\alpha_f^{(i, j)}$.\n\nDARTS performs a bilevel optimization on the architecture parameter $\\alpha$ and the model weights $w$ using two sets of training and validation data.\n\\begin{eqnarray}\n  \\underset{\\alpha}{\\rm min}\\ \\mathcal{L}_{val} (w^*(\\alpha), \\alpha), \\label{darts1} \\\\\n  {\\bf s.t.}\\  w^*(\\alpha) = \\underset{w}{\\rm arg~min}~\\mathcal{L}_{train}(w, \\alpha), \\label{darts2}\n\\end{eqnarray}\nwhere $\\mathcal{L}_{val}$ and $\\mathcal{L}_{train}$ are loss functions calculated with validation data and train data, respectively.\n$w^*(\\alpha)$ denotes the optimal model weights for an architecture $\\alpha$.\nEq. (\\ref{darts1}) has an inner optimization for $w^*({\\alpha})$, so that evaluating the gradient of $\\mathcal{L}_{val}$ w.r.t. $\\alpha$ can be prohibitive.\nTherefore, DARTS approximates $w^*(\\alpha)$ by adapting $w$ only after a single training step on the training data as follows:\n\\begin{eqnarray}\n  w^*(\\alpha) = w - \\xi\\nabla_w \\mathcal{L}_{train}(w, \\alpha) \\label{darts3}.\n\\end{eqnarray}\nThen, during a single step in the iterative optimization procedure of DARTS, it solves Eq. (\\ref{darts1}) and Eq. (\\ref{darts2}) alternately.\n\n\\section{Method}\n\\label{method}\nWe sequentially combine Faster-AA and DARTS, then optimize both in the end-to-end manner.\nWe solve another bilevel optimization problem for our entire search space.\nSpecifically, augmentation policies and network architectures are both optimized by minimizing a loss function on the validation dataset, while the network weights are optimized using the training dataset.\nThis bilevel optimization is formulated as:\n\\begin{eqnarray}\n\\underset{\\alpha,\\boldsymbol{z}_k,p_O,\\mu_O}{\\rm min}\\ \\mathcal{L}_{val}(w^*(\\alpha),\\alpha,\\boldsymbol{z}_k,p_O,\\mu_O), \\label{eq1}\\\\\n{\\bf s.t.}\\ w^*(\\alpha) = \\underset{w}{\\rm arg~min}~\\mathcal{L}_{train}(w,\\alpha,\\boldsymbol{z}_k,p_O,\\mu_O), \\label{eq2}\n\\end{eqnarray}\nand we adopt the first-order approximation for the architecture gradient for Eq. (\\ref{darts1}) same as in DARTS to speed-up the optimization, i.e., we set $\\xi$ to $0$ in Eq. (\\ref{darts3}).\nThen, we iteratively solve Eq. (\\ref{eq1}) and Eq. (\\ref{eq2}).\nAs a loss function, we use the cross-entropy loss for both $\\mathcal{L}_{train}$ and $\\mathcal{L}_{val}$.\n\nTo solve Eq. (\\ref{eq1}), we first apply a series of differentiable data augmentation operations $O^{(n)} (n=1,...,\\#\\mathcal{O})$, then give the transformed data to the network to solve Eq. (\\ref{eq2}).\nWe outline this algorithm in Alg. \\ref{alg1}.\n\n\\begin{algorithm}\n  \\caption{Joint optimization}\\label{alg1}\n\\begin{algorithmic}[1]\n  \\WHILE{\\textit{not converged}}\n  \\STATE \/\/ solve equation (\\ref{eq1})\n  \\STATE sample $X \\sim D_{val}$ \\\\\n  \\STATE apply equation (\\ref{op1}) to $X$\n  \\STATE calculate $L_{val}(w, \\alpha, \\boldsymbol{z}_k, p_O, \\mu_O)$ with $X'$ \\\\\n  \\STATE calculate gradients of $L_{val}$ w.r.t. $\\boldsymbol{z}_k, p_O, \\mu_O, \\alpha$ \\\\\n  \\STATE update Faster-AA parameters ($z_k, p_O, \\mu_O$)\n  \\STATE update DARTS parameters $\\alpha$ by gradient descent \\\\\n  \\STATE \/\/ solve equation (\\ref{eq2})\n  \\STATE sample $X \\sim D_{train}$ \\\\\n  \\STATE apply equation (\\ref{op1}) to $X$ \\\\\n  \\STATE calculate $L_{train}(w, \\alpha, \\boldsymbol{z}_k, p_O, \\mu_O)$ with $X'$ \\\\\n  \\STATE calculate gradients of $L_{train}$ w.r.t. $w$ \\\\\n  \\STATE update the network weights $w$ by gradient descent \\\\\n  \\ENDWHILE\n\\STATE Derive the final policy and architecture\n\\end{algorithmic}\n\\end{algorithm}\n\nThe total computational resources we require for the joint optimization additionally to what DARTS requires are relatively small.\nSpecifically, the additional space complexity is only $KL(\\#\\mathcal{O} \\times 3)$, while the search space of Faster-AA is $(\\#\\mathcal{O} \\times [0, 1] \\times [0, 1])^{KL}$ which is large.\nIn addition to it, our entire system is end-to-end differentiable, and augmentation policy search and network architecture search are jointly performed to minimize the same loss function, so that the gradients for updating policy and architecture parameters are obtained via a single backpropagation.\nThis advantage of our end-to-end differentiable approach enables to conduct joint optimization for policy search and architecture search with few additional space and time complexity compared to the case if we apply Faster-AA and DARTS independently and combine the results.\n\nThe original Faster-AA uses a critic network to consider a classification loss and the WGAN-GP loss~\\cite{wgan_gp} that encourage the distribution of transformed data to be as close to the original data distribution as possible.\nFor the same purpose, we can also exploit the network under searching with DARTS as a critic network to encourage transformed data to remain in the same classes before transformations.\nHowever, DARTS only uses a cross-entropy as the loss for architecture search, while the critic network in Faster-AA also considers WGAN-GP.\nIn this paper, we adopt a single unified loss function for the both of policy search and architecture search and did not introduce any critic network for simplicity of the entire framework and computational efficiency.\n\n\\section{Experiments}\n\\label{experiments}\nWe compare our joint optimization model with the original DARTS and the baseline which combines the results of Faster-AA and DARTS that are optimized independently from each other.\nIn the baseline, the learned policy by Faster-AA is transferred to be used for the training of DARTS.\nWe conducted the comparison on three datasets, CIFAR-10, CIFAR-100, and SVHN.\n\n\\label{results}\n\\begin{figure*}[htbp]\n\\begin{minipage}{0.3\\hsize}\n  \\centering\n  \\includegraphics[width=5cm,height=4cm]{transition_c10.png}\n  \\subcaption{On CIFAR-10}\n\\end{minipage}\n\\begin{minipage}{0.3\\hsize}\n  \\centering\n  \\includegraphics[width=5cm,height=4cm]{transition_c100.png}\n  \\subcaption{On CIFAR-100}\n\\end{minipage}\n\\begin{minipage}{0.3\\hsize}\n  \\centering\n  \\includegraphics[width=6.5cm,height=4cm]{transition_svhn.png}\n  \\subcaption{On SVHN}\n  \\label{svhn_policy}\n\\end{minipage}\n  \\caption{\n  Probability distribution of the augmentation policy selection over time.\n  }\n  \\label{res1}\n\\end{figure*}\n\nWe first carefully re-implemented Faster-AA\\footnote{source code from: \\url{https:\/\/github.com\/moskomule\/dda\/tree\/fasteraa\/faster_autoaugment}} and DARTS\\footnote{source code from: \\url{https:\/\/github.com\/quark0\/darts}} based on their authors' implementation.\nThen, we confirmed our implementation successfully reproduced the reported scores in those papers.\nIt should be noted that we adopt the same search space as the original paper for DARTS, while we exclude the cutout operation from the search space for the Faster-AA part in our framework as the authors do in their implementation.\nAccording to a comment which has been left in their code, the cutout operation makes the optimization unstable.\nTherefore, the target operations in the augmentation policy search are \\textit{shear X, shear Y, translate X, translate Y, rotate, auto contrast, horizontal flip, invert, equalize, solarize, posterize, contrast, color, brightness, sharpness, and sample pairing}.\nWe set the number of sub-policies $L = 10$ and the number of operations in a sub-policy $K = 2$, which are the same in the original settings.\nAdditionally, as preprocessings in the baseline, random cropping with zero-padding and random horizontal flipping are always applied.\nAfter those preprocessings, we apply transformations under the policy search with Faster-AA.\nThe cutout operation is always applied after the Faster-AA part in the baseline, although it is not included in the search space.\n\nFor the baseline, we first obtain a Faster-AA policy independently from DARTS by following the same experimental settings used in the original paper, i.e., the baseline uses WideResnet40-2~\\cite{wideresnet} as the architecture to search policies for 20 epochs.\nNext, the network architecture for the baseline is explored by DARTS in the same manner with the original paper.\nThose policy and architecture obtained by the existing methods are combined together, then we train the model weights for 600 epochs.\nAs for the searching epochs of architectures, we conduct two different total epochs (25 and 50) both for the baseline and proposed framework.\nBecause we found that training of DARTS is unstable on CIFAR-100 and SVHN as mentioned in~\\cite{darts_stab}.\n\\textit{We repeated experiments three times to see average scores with standard deviations.}\n\n\\subsection{Discussion}\n\\begin{table}[htbp]\n  \\begin{center}\n  \\caption{\n  Comparison in classification accuracy.\n  (1) ``DARTS\" only searches architecture.\n  (2) ``Baseline\" separately searches policies and architectures.\n  (3) ``Ours\" is the proposed joint search method.\n  Our method achieves competitive or superior results compared with the baseline.}\n  \\label{tab1}\n  \\begin{tabular}{c|c|c|c} \\hline\n    method & CIFAR-10 & CIFAR-100 & SVHN \\\\ \\hline\n    \\multicolumn{4}{c}{searching epoch is 50} \\\\ \\hline \n   \n    DARTS & 97.33$\\pm$0.12 & 76.21$\\pm$3.76 & 97.85$\\pm$0.08 \\\\\n    Baseline & \\textbf{97.55$\\pm$0.30} & 77.96$\\pm$3.12 & \\textbf{98.02$\\pm$0.03} \\\\\n    Ours & 97.40$\\pm$0.03 & \\textbf{79.02$\\pm$2.14} & 97.92$\\pm$0.12  \\\\ \\hline\n    \\multicolumn{4}{c}{searching epoch is 25} \\\\ \\hline\n   \n    DARTS & 97.14$\\pm$0.04 & 82.91$\\pm$0.30 & 97.94$\\pm$0.06 \\\\\n    Baseline & 97.29$\\pm$0.03 & \\textbf{84.17$\\pm$0.29} & \\textbf{98.03$\\pm$0.05} \\\\\n    Ours & \\textbf{97.46$\\pm$0.09} & 83.81$\\pm$0.49 & 97.82$\\pm$0.08 \\\\ \\hline\n  \\end{tabular}\n  \\end{center}\n\\end{table}\n\n\nWe present the experimental results in Table \\ref{tab1}.\nOur proposed method achieves competitive or superior results compared to the baseline in both searching epochs.\nAs stated above, the baseline uses WideResnet40-2 during policy search, and it can be different from the architecture found by DARTS for the final stage to learn the model weights for 600 epochs.\nAlthough the policies found with WideResnet40-2 show high accuracy at several results, which might be derived from the suitability of WideResnet40-2 for augmentation policy search.\nIf that is the case, the model selection largely affects on the policy search, so that the human expertise is still required.\nOn the other hand, our proposed method does not require humans to select the network architecture for policy search and achieves competitive or even superior results to Faster-AA.\nWe believe that the joint optimization approach has more potential to obtain high performance and reduces human expertise. \n\nFig. \\ref{res1} shows how the categorical distribution to choose augmentation policies changes over the time.\nWe found that the augmentation policies obtained with our method choose color enhancement operations such as \\textit{color} or \\textit{auto contrast} more often than geometric operations such as \\textit{rotate} from the transition.\nThis trend of the resulting policies is also reported in other papers of automatic augmentation policy search~\\cite{aa}.\nIn Figure \\ref{svhn_policy}, the policy dramatically changes after 45 epochs, and we found that the network architecture under searching also largely changed at the same timing, which may imply that the optimal policy is different depending on the network architecture.\n\n\\section{Conclusion}\n\\label{conclusion}\nIn this paper, we proposed a method to jointly optimize data augmentation policy and network architecture.\nThe proposed method combines differentiable methods for policy search and architecture search to jointly optimize them in the end-to-end manner.\nThe experimental results showed that our method achieves competitive or superior performance to independently searched results in common benchmarks of image classification.\nOur joint optimization approach may be able to include the other parts such as learning rates.\nHence, we will attempt to bring more automation to the design of training pipeline with this end-to-end differentiable approach for the future work.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{aaai}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Embedding}\n    \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{Future Directions}\nAlthough many efforts have been made to design deep recommender systems from manual to automatic, there remain a few potential opportunities for future research directions. \n\n\n\\noindent \\textbf{GNNs-based Recommendations.}\nGraph neural networks (GNNs) have been extensively studied recently due to their impressive capability in representation learning on graph-structured data~\\cite{ma2021deep,xu2021automorphic,ding2021diffmg}. Recent works have been proposed to advance deep recommender systems based on GNNs techniques~\\cite{fan2019graph,fan2020graph,fan2022graph}. Thus, exploring the combination of AutoML and GNNs provides great opportunities to further boost the performance of GNNs-based recommendation methods.  \nA few works have been proposed to study the combination of AutoML and GNNs together in the research community. \nFor instance, GraphNAS~\\cite{graphnas}  and Auto-GNN~\\cite{autognn} made the very first attempt to enable the automatic design of the best graph neural architecture via reinforcement learning techniques. The main drawback of these two methods is computationally expensive because of the large search space.\nTo tackle this challenge, SANE~\\cite{zhao2021search} proposes a differentiable architecture search algorithm for GNNs, where an advanced one-shot NAS paradigm is adopted to accelerate the search process.\nAs the very first work to apply automatic NAS techniques to GNNs-based deep recommender systems, AutoGSR~\\cite{chen2022autogsr} attempts to search for the optimal GNNs architecture on GNNs-based session recommendation through a differentiable architecture search algorithm.\n\n\n\\noindent \\textbf{Multi-Modality Recommendations.}\nIn addition to historical interactions between users and items, items' auxiliary knowledge from various modalities (\\textit{e.g.}, visual, acoustic, and textual) has been incorporated to learn users' preferences for providing high-quality recommendation services~\\cite{wei2019mmgcn}. \nHence, it is desirable to advance deep multimodal learning via automated machine learning techniques~\\cite{yin2021bmnas,perez2019mfas}, so as to design an optimal algorithm for targeted tasks. \nFor example, as the very first work on automated neural architecture search method for deep multimodal learning, the work of Multimodal Fusion Architecture Search (MFAS) aims to find accurate fusion architectures for multi-modal classification problem~\\cite{perez2019mfas}.\nA Bilevel Multimodal Neural Architecture Search framework (BM-NAS) is proposed to  learn the architectures of multimodal fusion models via a bilevel searching scheme~\\cite{yin2021bmnas}. \n\n\n\n\n\n\n\n\\noindent \\textbf{Other Recommendation Tasks.}\nIn addition to AutoML for GNNs-based and multimodal recommendations,  various important recommendation tasks are rarely explored through automated machine learning techniques, such as POI recommendations~\\cite{zhao2020go}, sequential recommendations~\\cite{kang2018self}, social recommendations~\\cite{fan2019graph}, \\textit{etc}. \nA few works have been conducted to apply automated neural architecture search techniques for spatio-temporal prediction such as AutoST~\\cite{10.1145\/3394486.3403122} and AutoSTG~\\cite{10.1145\/3442381.3449816}, which can help design  optimal neural architectures for POI recommendations. \nBesides, despite the success of various deep social recommendations, heavy manual work and domain knowledge is required to inherently combine user-item interactions and social relations~\\cite{fan2020graph}, which can be addressed by AutoML techniques.\n\n\\section{Conclusion}\nDeep recommender systems have attracted increasing attention in both academia and industry. \nBesides, automated machine learning (AutoML), as one of the most promising AI techniques, has shown its great capabilities to advance deep architecture designs from manual to automatic. \nIn this survey, we have conducted a comprehensive overview of an emerging research field: automated machine learning for deep recommender systems.  \nSpecifically, we discuss the state-of-the-art AutoML approaches that automate the feature selection, feature embeddings, feature interactions, and system design in DRS.\nWe expect this survey can facilitate future research directions in the academic and industry community.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{scriptsize}\n\\bibliographystyle{named}\n\n\\section{Introduction}\n\\label{sec:intro}\nRecent years have witnessed the explosive growth of online service providers~\\cite{ricci2011introduction}, including a range of scenarios like movies, music, news, short videos, e-commerces, \\textit{etc}. This leads to the increasingly serious information overload issue, overwhelming web users.\nRecommender systems are effective mechanisms that mitigate the above issue by intelligently retrieving and suggesting personalized items, \\textit{e.g.}, contents and products, to users in their information-seeking endeavors, so as to match their interests and requirements better.\nWith the development and prevalence of deep learning, deep recommender systems (DRS) have piqued interests from both academia and industrial communities\n~\\cite{zhang2019deep,nguyen2017personalized}, due to their superior capacity of learning feature representations and modeling non-linear interactions between users and items~\\cite{zhang2019deep}.\n\nTo construct DRS architectures, the most common practice is to design and tune the different components in a hand-crafted fashion. However, manual development is fraught with three inherent challenges. First, this requires extensive expertise in deep learning and recommender systems. Second, substantial engineering labor and time cost are required to design task-specific components for various recommendation scenarios. Third, human bias and error can result in suboptimal DRS components, further reducing recommendation performance.\n\nRecently, powered by the advances of both theories and technologies in automated machine learning (AutoML), tremendous interests are emerging for automating the components of DRS. \nBy involving AutoML for the deep recommender systems, different models can be automatically designed according to various data, thus improving the prediction performance and enhancing generalization.\nBesides, it is helpful to eliminate the negative influence for DRS from human bias and error, as well as reduce artificial and temporal costs significantly.\nAs shown in Tabel~\\ref{tab:comp}, we summarize these researches from the perspective of search space and search strategy, which are two critical factors for AutoML. \nTypically, these works can be divided into the following categories according to the components in DRS:\n\n\n\\begin{table*}[]\n\\centering\n\\vskip -0.2in\n\\caption{\\small{Automated Machine Learning for Deep Recommender Systems.}}\n\\vspace{-0.3cm}\n\\label{tab:comp}\n\\renewcommand\\arraystretch{1.4} \n\\resizebox{1.0\\textwidth}{!}{\n\\begin{threeparttable}\n\\begin{tabular}{c|c|c|c|c|c}\n\\toprule \\toprule\n\\multirow{2}{*}{\\large\\begin{tabular}[c]{@{}c@{}}\\textbf{DRS} \\\\ \\textbf{Component}\\end{tabular}}&\\multirow{2}{*}{\\textbf{Search Space}}&\\multicolumn{4}{c}{\\textbf{Search Strategy}}    \\\\ \\cline{3-6} \n                                              &                               & \\textbf{Gradient}                                & \\textbf{Reinforcement Learning} & \\textbf{Evolutionary} & \\textbf{Others\\tnote{*}} \\\\ \\hline\n\\multirow{2}{*}{\\large\\begin{tabular}[c]{@{}c@{}}\\textbf{\\textit{Feature}} \\\\ \\textbf{\\textit{Selection}}\\end{tabular}}   & \\textbf{Raw Feature}          & AutoField~\\shortcite{wang2022autofield}                                        & FSTD~\\shortcite{fard2013using}, MARLFS~\\shortcite{liu2021automated}                    & -                                    & -               \\\\ \\cline{2-2} \n                                              & \\textbf{Generated Feature}    & GLIDER~\\shortcite{glider}                                           & -                               & -                   & AutoCross~\\shortcite{autocross} \\\\ \\hline\n\\multirow{2}{*}{\\large\\begin{tabular}[c]{@{}c@{}}\\textbf{\\textit{Feature}} \\\\ \\textbf{\\textit{Embedding}}\\end{tabular}}    & \\textbf{Single Embedding}     & AMTL~\\shortcite{amtl}, AutoEmb~\\shortcite{autoemb}                                    & ESAPN~\\shortcite{esapn}                           & -                     & PEP~\\shortcite{pep}        \\\\ \\cline{2-2} \n                                              & \\textbf{Group Embedding}      & AutoDim~\\shortcite{autodim}, DNIS~\\shortcite{dnis}                               & NIS~\\shortcite{nis}, AutoIAS~\\shortcite{autoias}                    & RULE~\\shortcite{rule}                  & -                    \\\\ \\hline\n\\multirow{3}{*}{\\large\\begin{tabular}[c]{@{}c@{}}\\textbf{\\textit{Feature}} \\\\ \\textbf{\\textit{Interaction}}\\end{tabular}}   & \\textbf{Feature Interaction}  & AutoFIS~\\shortcite{autofis}, FIVES~\\shortcite{fives} & AutoIAS~\\shortcite{autoias}                         & -                     & DeepLight~\\shortcite{deeplight},BP-FIS~\\shortcite{chen2019bayesian}             \\\\ \\cline{2-2} \n                                              & \\textbf{Interaction Function} & SIF~\\shortcite{sif}, AIM~\\shortcite{aim}                                 & AutoIAS~\\shortcite{autoias}                         & AutoFeature~\\shortcite{autofeature}           & -                       \\\\ \\cline{2-2} \n                                              & \\textbf{Interaction Block}    & AutoPI~\\shortcite{autopi}                                           & -                               & AutoCTR~\\shortcite{autoctr}               & AMEIR~\\shortcite{ameir}                  \\\\ \\hline\n\\multirow{2}{*}{\\large\\begin{tabular}[c]{@{}c@{}}\\textbf{\\textit{System}} \\\\ \\textbf{\\textit{Design}}\\end{tabular}}   & \\textbf{Framework}            &                         -                         &                      -           &       -                &            DeepRecInfra~\\shortcite{deeprecinfra}, AutoRec~\\shortcite{autorec}                     \\\\ \\cline{2-2} \n                                              & \\textbf{Optimazation}         & AutoLoss~\\shortcite{zhao2021autoloss}, $\\lambda$opt\\shortcite{lambdaopt}                                   &                     -            &                  -     &                -            \\\\ \\bottomrule \\bottomrule\n\\end{tabular}\n\\begin{tablenotes}\n        \\footnotesize\n        \\item[*] Others including Bayesian optimization, gird\/random search and regularization.\n      \\end{tablenotes}\n    \\end{threeparttable}\n}\n\\vskip -0.15in\n\n\n\\end{table*}\n\n\\begin{itemize}[leftmargin=*]\n\\item \\textbf{Feature Selection}: This is the process of selecting a subset of the most predictive and relevant features (or generated features) for subsequent DRS models. \nBy eliminating the redundant or irrelevant features, feature selection can help enhance the recommendation performance and accelerate DRS model training~\\cite{nadler2005prediction}. \n\n\n\\item\\textbf{Feature Embedding}: \nTypically, the features for DRS are high-dimensional and extremely sparse. \nMost recommendation models first transform the raw features into one-hot vectors and then embed them as dense representations via the feature embedding layer.\nAutoML technique is utilized to dynamically search the optimal embedding sizes for improving prediction accuracy, saving storage space, and reducing model capacity. \n\n\\item\\textbf{Feature Interaction}: \nEffectively modeling predictive feature interactions is critical for boosting the recommendation quality of DRS because the interaction of two features would alter their individual feature effects. For example, users often download food delivery apps at mealtime, in which interactions between app category and time-stamp is a highly predictive signal. \nTherefore, some AutoML-based works are devoted to exploring beneficial feature interactions with proper interaction functions.\n\n\\vspace{-1mm}\n\\item \\textbf{System Design}: \nIn addition to the above components of DRS models, system design also has a crucial impact on DRS performances, including hardware infrastructure, data pipeline, and information transfer, as well as implementation, deployment, optimization, and evaluation. \n\\end{itemize}\n\n\n\n\n\n\n\nThis survey is to provide a literature overview on the advances of AutoML for constructing DRS architectures.\nTo be specific, we first provide an overview of AutoML techniques.\nThen, we discuss the state-of-the-art AutoML approaches that automate the feature selection, feature embeddings, feature interactions, and system design in DRS models.\nFinally, we discuss the appealing directions that can bring this research field into a new frontier.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Priliminary}\n\n\n\n\\section{Overview of AutoML}\nGiven the problem description and datasets, the goal of Automated Machine Learning (AutoML) techniques is to construct machine learning solutions automatically for time-consuming and iterative real-world tasks.\nIt has shifted the model design mechanism from hand-crafted to automatic, enabling unparalleled prospects for deep learning model construction. AutoML frameworks are typically comprised of three following components:\n\\begin{itemize}[leftmargin=*]\n\\vspace{-1mm}\n\\item \\textbf{Search Space}. The search space defines a group of candidate operations and the relationships between them that enable appropriate model designs to be formed. For DRS, different components contain diverse search spaces, which are involved with human prior knowledge.\n\n\n\\vspace{-1mm}\n\\item \\textbf{Search Strategy}. The search strategy specifics how to do an efficient exploration of the search space and find out the optimal architectures, which typically contains gradient-based optimization~\\cite{ruder2016overview}, reinforcement learning (RL)~\\cite{kaelbling1996reinforcement}, evolutionary algorithms~\\cite{qin2008differential}, Bayesian optimization~\\cite{snoek2012practical}, random search~\\cite{bergstra2012random}, \\textit{etc}.\n\\vspace{-1mm}\n\\item \\textbf{Performance Estimation Strategy}. Performance estimation is the process of estimating the performance of sampled candidate architectures from the massive search space. To reduce the computational cost of training and estimating over these candidates, various strategies for performance estimation have been proposed, such as weight sharing~\\cite{pham2018efficient} and network morphism~\\cite{elsken2017simple}.\n\n\\end{itemize}\n\\section{Feature Selection}\nIn recommender systems, feature selection aims to select a subset of relevant features for constructing recommendation models. \nIn practical online service providers, data is composed of a massive amount of features, including user portraits, item attributes, behavior features, contextual features as well as combinatorial features based on previous feature types. \nHowever, some of these raw features may be irrelevant or redundant in recommendations, which call for effective feature selections that can boost recommendation performance, overcome input dimensionality and overfitting, enhance model generalization and interpretability, as well as accelerate model training.\n\nThe classic feature selection methods are typically presented in three classes: 1) Filter methods, which select features based only on feature correlations regardless of the model~\\cite{hall1999correlation,yu2003feature}; 2) Wrapper methods, which evaluate subsets of features that allows detecting the possible interactions amongst variables~\\cite{maldonado2009wrapper}; and 3) Embedded methods, where a learning algorithm performs feature selection and classification simultaneously, such as LASSO~\\cite{fonti2017feature} and decision trees~\\cite{ke2017lightgbm}.\nThese methods, however, usually fail in deep learning-based recommender systems with both numerical and categorical features. For instance, filter methods neglect the dependencies between feature selection and downstream deep recommendation models; Wrapper methods must explore $2^m$ candidate feature subspaces, \\textit{i.e.}, keep or drop for $m$ feature fields; Embedded methods are sensitive to the recommendation models' strong structural assumptions. To deal with these issues, AutoML-based methods are utilized to adaptively select effective features for recommendations.\nAccording to the feature selection stage, we categorize the research into two groups: \\textit{Selection from Raw Features} and \\textit{Selection from Generated Features}.\n\n\n\\subsection{Selection from Raw Features}\n\nAccording to a survey from Crowdflower, scientists spend 80\\% of time on data and feature preparation. Therefore, introducing AutoML into raw feature selection has the potential to significantly enhance the data scientists' productivity, and frees them up to focus on real business challenges. FSTD \\cite{kroon2009automatic,fard2013using} introduces reinforcement learning (RL) into feature selection with single agent. They have a large search space of $2^m$ ($m$ is the number of feature fields), where each candidate is a possible feature subset.\nTo limit the searching complexity, MARLFS~\\cite{liu2019automating,liu2021automated} reformulates feature selection as a multi-agent reinforcement learning problem. Each feature is assigned an agent, and then all feature agents maintain to select or deselect the corresponding feature simultaneously. \nTo reduce the computations, the following efforts attempt to accelerate the searching efficiency by learning with external knowledge~\\cite{fan2020autofs} and reducing the number of agents~\\cite{zhao2020simplifying,fan2021autogfs}. However, due to the intrinsically low sample efficiency, RL-based methods are still difficult to be integrated into real-world recommender systems with large-scale user-item interactions. To this end, AutoField~\\cite{wang2022autofield} is \nproposed for practical recommendations, where the search space is relaxed to be continuous by allocating two variables to control the selection of each feature. Afterward, the selected features can be evaluated on the validation dataset by gradient descent.\n\n\n\n\n\n\n\\subsection{Selection from Generated Features}\nIn addition to selecting informative features from the raw feature set, some works learn to discover and generate beneficial combinatorial features (\\textit{i.e.}, cross features), including categorical and statistical features.\nGLIDER~\\cite{glider} utilizes the gradient-based neural interaction detector to detect generic non-additive and high-order combinatorial features efficiently, whose search space is $2^{2^m}$. The detected features are evaluated by a linear regression model and trained from scratch. Similarly, AutoCross~\\cite{autocross} searches useful high-order cross features by transferring the original space to a tree-structured space, reducing the search space from $2^{2^m}$ to $(\\mathcal{C}_m^2)^k$, where $k$ is the expected number of cross features. Then a greedy-based beam search~\\cite{beam_search} is performed to prune unpromising branches for further improving the efficiency. The feature set evaluation is achieved by field-wise logistic regression approximately and trained from scratch.\n\nTo discover useful statistical features from the raw feature set, AEFE~\\cite{aefe} designs second-order combinatorial features search space with size $2^{\\mathcal{C}_m^2Q}$, where $Q$ is the number of pre-defined construction rules. After the feature generation\nof groupby, aggregating, and paradigm combination, a greedy-based search with feature filtering is deployed and the selected features are evaluated from scratch.\n\n\n\\section{Feature Embedding}\nDifferent from  Computer Vision (CV) and Natural Language Processing (NLP), the input features used in recommender systems are extremely sparse and high-dimensional.\nTo tackle this problem, neural network-based models leverage a feature embedding layer to map the high-dimensional features into a low-dimensional latent space.\nSpecifically, for a feature field, we assign each feature $f_i$ with a dense embedding vector $\\mathbf{e}_{i}$ and save all embedding vectors in an embedding table $\\mathbf{E}\\in \\mathbb N^{V\\times d}$, where $V$ and $d$ are are the vocabulary size and pre-defined embedding size, respectively. As shown in Figure~\\ref{fig:embedding}, based on the embedding table $\\mathbf{E}$, we can obtain the embedding vectors through the embedding look-up process.\n\nFeature embedding is the cornerstone of the DRS as the number of parameters in DRS is heavily concentrated in the embeddings and the subsequent components are constructed on the basis of feature embeddings. The feature embedding layer not only directly affects storage capacity and online inference efficiency~\\cite{autodis}, but also has a non-negligible effect on the prediction accuracy.\nTo improve the prediction accuracy, save storage space and reduce model capacity, some AutoML-based solutions are proposed to dynamically search the embedding sizes for different features. The intuition behind is that assigning high-dimensional embeddings for high-frequency features can improve model capacity while low-dimensional embeddings for low-frequency features contribute to preventing overfitting~\\cite{autoemb}. According to whether or not the optimal embedding size is searched for each feature value, these solutions can be divided as into categories: \\textit{Single Embedding Search} and \\textit{Group Embedding Search}, as in Figure~\\ref{fig:embedding}.\n\n\n\n\n\\begin{figure}[!t]\n\\vskip -0.15in\n\n    \\centering\n    \\setlength{\\belowcaptionskip}{-0.5cm}\n    \\hspace*{-6.6mm}\n    \\includegraphics[width=0.5\\textwidth]{Figure\/embedding.pdf}\n    \\vskip -0.15in\n    \\caption{\\small{AutoML for feature embedding.}}\n    \\label{fig:embedding}\n\\end{figure}\n\n\\subsection{Single Embedding Search}\nSingle embedding search-based methods~\\cite{amtl,pep} aim to search optimal embedding dimension for each feature value, hence facing huge search space due to the large vocabulary size $V$. AMTL~\\cite{amtl} leverages a twins-based architecture to avoid the unbalanced parameters update problem due to the different frequencies and designs an embedding search space with $d^V$ size, where $d$ is the embedding size. The twins-based architecture acts as a frequency-aware policy network to search the optimum dimension for each feature value, and the learning process is relaxed to a continuous space by temperated softmax~\\cite{kd} and optimized by gradients.\nPEP~\\cite{pep} proposes a pruning-based solution by enforcing column-wise sparsity on the embedding table with $L_0$ normalization, generating  $2^{Vd}$ search space. To save computation cost and avoid setting pruning thresholds manually, PEP utilizes the trainable pruning parameters to prune each element automatically, which can be jointly optimized with the model parameters via gradient-based back-propagation. \n\nHowever, the search space of AMTL and PEP are highly related with the embedding size $d$, which hinders the optimization procedure. To reduce the search space, a solution is to divide the embedding dimension into several \\textit{\\textbf{column-wise}} sub-dimensions (\\textit{e.g.}, slicing the original dimension $d = 64$ into $\\{2,4,8,16,32,64\\}$ 6 candidate sub-dimensions).\nAutoEmb~\\cite{autoemb} and ESAPN~\\cite{esapn} reduce the search space from $d^V$ to $a^V$, where $a$ is the number of candidate sub-dimensions, greatly shrinking the search space compared with AMTL.\nAutoEmb performs a soft-selection strategy by summing over the candidate sub-dimensions with learnable weights.\nInstead, ESAPN performs a hard-selection strategy and a frequency-aware policy network serves as an automated RL agent to decide whether to enlarge the dimensions under the streaming setting. \n\nBesides searching embedding dimension dynamically for each features, learning embeddings via combination adaptively is also a trend. ANT~\\cite{ant} and AutoDis~\\cite{autodis} leverage the combination over a set of anchor embeddings (named meta-embeddings in AutoDis) to represent categorical and numerical feature respectively, building a $2^{km}$ search space, where $k$ is the number of anchor embeddings. ANT uses a sparse transformation operation to hard-select relevant anchor embedding while AutoDis designs an automatic discretization network to soft-select informative meta-embeddings. \n\n\n\\subsection{Group Embedding Search}\nAutoEmb and ESAPN shrink the search space by dividing the embedding dimension into candidate \\textit{\\textbf{column-wise}} sub-dimensions.\nAnother solution is to group the feature values of a field based on some indicators (\\textit{e.g.}, frequencies) and assign a \\textit{\\textbf{row-wise}} group embedding dimension for all the values within the group. \nA special case is setting the number of groups $b=1$ and searching a global embedding dimension for all the feature values of a field, such as AutoDim~\\cite{autodim}. AutoDim pre-defines several candidate sub-dimensions like AutoEmb but has a smaller search space, shrinking from $a^V$ to $a^m$. The optimization procedure is achieved by a bi-level gradient-based algorithm with Gumbel-Softmax trick~\\cite{gumbel_softmax}.\n\nTo balance the search efficiency and performance, some works split the feature into multi-groups (\\textit{i.e.}, $b>1$) based on the feature frequencies or clustering. DNIS~\\cite{dnis} divides the feature values with similar frequencies into $b$ groups, reducing the search space from $2^{Vd}$ into $2^{bd}$. Then, a gradient-based differentiable search with gradient normalization is performed to search optimal group embeddings. Specifically, NIS~\\cite{nis} and RULE~\\cite{rule} reduce the search space significantly from both row-wise and column-wise perspectives. NIS designs single-size embedding search (with $ab$ space) and multi-size embedding search (with $b^a$ ), and uses the RL-based method to find the optimal embedding dimensions.\nSimilarly, RULE divides the embedding table into multi-blocks and builds a $2^{ab}$ search space. Then the evolutionary search algorithm is proposed to search optimal item embeddings under memory constraint for the on-device recommendation. Each sub-structure is evaluated by an accurate performance estimator to balance the prediction confidence and training time.\n\\section{Feature Interaction}\nEffectively modeling feature interactions is one of the most commonly-used approaches for DRS models to improve prediction performance. Recently, plenty of works leverage various operations to capture informative interaction signals explicitly and implicitly, such as inner product (PNN~\\cite{pnn}), outer product (CFM~\\cite{cfm}), convolution (FGCNN~\\cite{fgcnn}) and \\textit{etc}. However, these works utilize identical interaction functions to model all the feature interactions indiscriminately, which may introduce noisy interactive signals and weaken the effectiveness of modeling. To overcome these issues, some AutoML-based methods are designed to search beneficial feature interactions with optimal interaction function adaptively. These methods can be categorized into three groups depending on the search space: \\textit{Feature Interaction Search}, \\textit{Interaction Function Search}, and \\textit{Interaction Block Search}, as shown in Figure~\\ref{fig:interaction}.\n\n\\begin{figure}[!t]\n    \\vskip -0.15in\n    \\centering\n    \\setlength{\\belowcaptionskip}{-0.1cm}\n    \\hspace*{-6mm}\\includegraphics[width=0.5\\textwidth]{Figure\/interaction.pdf}\n    \\vskip -0.15in\n    \\caption{\\small{AutoML for feature interaction.}}\n \\vspace{-0.3cm}\n    \\label{fig:interaction}\n\\end{figure}\n\n\\subsection{Feature Interaction Search}\nTo search beneficial feature interactions for enriching information, some AutoML-based works design high-order feature interactions search space and leverage search algorithms (mainly gradient-based search algorithm) to derive feature interactions automatically. AutoFIS~\\cite{autofis} identifies and selects important feature interactions by enumerating all the feature interactions and introducing a set of architecture parameters ``gates'' to indicate the importance of individual feature interactions, facing $2^{\\mathcal{C}_m^2}$ search space even in second-order feature interactions. The architecture parameters are optimized by gradient descent with GRDA optimizer~\\cite{grda} to get a sparse solution automatically. However, when searching high-order feature interactions, the search space of AutoFIS is huge, resulting in low search efficiency.\n\nTo solve the \\textit{efficiency-accuracy} dilemma, AutoGroup~\\cite{autogroup} proposes automatic feature grouping, reducing the $p^{th}$-order search space from $2^{\\mathcal{C}_m^p}$ to $2^{gm}$, where $g$ is the number of pre-defined groups. The discrete search space is then relaxed to continuous and the derivative is approximated by introducing the Gumbel-Softmax trick~\\cite{gumbel_softmax}. AutoHash~\\cite{autohash} shares a similar idea with AutoGroup to reduce high-order search space by the hashing function.\n\nAlthough AutoGroup and AutoHash improve the high-order interaction search efficiency via feature grouping and hashing, they ignore the \\textit{order-priority} property ~\\cite{profit}, which reveals that the higher-order feature interactions quality can be relevant to their de-generated low-order ones, and lower-order feature interactions are likely to be more vital compared with higher-order ones. To reduce the architecture parameters and search costs, PROFIT~\\cite{profit} distills the $p^{th}$-order search space from $2^{\\mathcal{C}_m^p}$ to $2^{mp}$ by the composition of low-rank tensors approximately. Then to ensure the order-priority property, a progressive\nsearch algorithm based on the gradient is proposed to search high-order feature interactions order-by-order. Similarly, FIVES regards the original features as a feature graph conceptually and models the high-order feature interactions by a GNN with layer-wise adjacency matrix, so that the $p^{th}$-order search space is reduced from $2^{\\mathcal{C}_m^p}$ to $2^{m^2}$. Then, FIVES parameterizes the adjacency matrix and makes them depend on the previous layer, so that the order-priority property can be kept. \n\nThe above-mentioned works search beneficial feature interactions for all users non-personally, which overlooks the individuality and personality of the user's behavior. To provide personalized selection of second-order feature interaction, BP-FIS~\\cite{chen2019bayesian} designs a personalized search space with size $2^{u\\mathcal{C}_m^2}$, where $u$ is the number of users. Specifically, BP-FIS proposes bayesian personalized feature interaction selection mechanism under the Bayesian Variable Selection (BVS)~\\cite{tibshirani1996regression} theory by forming a Bayesian generative model and deriving the Evidence Lower Bound (ELBO), which can be optimized by an efficient Stochastic Gradient Variational Bayes (SGVB) method.\n\n\n\\subsection{Interaction Function Search}\nAs suggested by PIN~\\cite{pin}, different feature interactions are suitable for different interaction functions. Therefore, searching optimal interaction functions contributes to better capturing informative interaction signals. \nSIF~\\cite{sif} automatically devises suitable interaction functions for collaborative filtering (CF) task with two fields, which consists of micro search space referring to element-wise MLP and macro search space including 5 pre-defined operations (\\textit{i.e.}, multiply, plus, min, max, and concat). A bi-level gradient-based search algorithm is utilized to relax the choices among operations in a continuous space. \nAutoFeature~\\cite{autofeature} extends the interaction functions search to multi-field high-order scenarios by utilizing micro-networks with different architectures to model feature interactions. The whole search space expands to $b^{\\mathcal{C}_m^p}$ for the $p^{th}$-order interactions, where $b$ is the number of candidate interaction functions, including add, Hadamard-product, concat, Generalized-product, and null. The search process is implemented by an evolutionary algorithm with the Naive Bayes tree,  and each sampled architecture is trained and evaluated from scratch. \n\nHowever, the interaction calculations of SIF and AutoFeature are artificially specified, which requires high dependence on domain knowledge. To overcome this limitation, AOANet~\\cite{aoanet} proposes a generalized interaction paradigm by decomposing commonly-used structures into Projection, Interction and Fusion phase. Therefore, the formula of the $l$-th interaction layer $C_l$ is given as: $C_l = \\{Z_{u,v} | (u,v)\\in pairing(B_0,B_{l-1})\\}$, where $Z_{u,v} = (u\\otimes v) \\odot W_l$ is the interaction map of vector $u$ and $v$, respectively. \nThen architecture parameters are introduced to distinguish the importance of interaction maps, and the optimization procedure is achieved by a gradient-based method like AutoFIS.\n\n\n\\subsection{Interaction Block Search}\nSearching appropriate interaction functions for different feature interactions may bring huge search space and high search overhead. Therefore, one straightforward idea is to take the original features as a whole and modularize representative operations in several blocks to formulate a generalizable search space, which is widely used in CV tasks~\\cite{darts}.\nAutoCTR~\\cite{autoctr} designs a two-level hierarchical search space by abstracting the raw features and operations (\\textit{i.e.}, MLP, FM~\\cite{fm}, and dot-product) into virtual blocks, which are further connected as a directed acyclic graph (DAG). Similar to AutoFeature, AutoCTR utilizes a multi-objective evolutionary algorithm with architectural-level learning-to-rank guidance to search the optimal architecture. The sampled architectures are evaluated from scratch and some tricks (\\textit{e.g.}, data sub-sampling and warm-start) are used to accelerate the evaluation process.\n\nTo further improve computational efficiency, AutoPI~\\cite{autopi} utilizes a gradient-based search strategy for exploration in a more efficient search space. AutoPI designs a hierarchical search space with blocks connecting into a DAG where the interaction cell formulates the higher-order feature interactions and the ensemble cell combines lower-order and higher-order interactions. Then, a bi-level optimization approach is applied to discover optimal architecture after the continuous relaxation.\n\n\\subsection{Comprehensive Search}\nIn addition to designing a single search space (\\textit{e.g.}, feature embedding, feature interaction, or interaction block) in DRS, some works design a hybrid search space and perform a comprehensive search. \nBased on AutoFIS, AIM~\\cite{aim} designs a mixed search space to select significant feature interactions, appropriate interaction functions, and optimal embedding dimensions automatically in a unified framework. It is noteworthy that the $p^{th}$-order feature interaction search is achieved by combining raw features with the maintained top-$k$ ${(p-1)}^{th}$-order feature interactions, reducing the search space from $2^{\\mathcal{C}_m^p}$ to $2^{km}$. Besides, the ``gates'' are extended to search embedding dimensions with $2^{md}$ space and interaction functions with $2^{kmb}$ space. AutoIAS~\\cite{autoias} designs an integrated search space for multiple components in DRS, including feature embedding ($a^m$) and projection ($a^{\\mathcal{C}_m^2}$), second-order feature interaction ($2^{m+\\mathcal{C}_m^2}$) and interaction function ($b^{\\mathcal{C}_m^2}$), as well as the MLP structures. An architecture generator network is trained by policy gradient and used to produce better architectures with dependency, where Knowledge Distillation (KD)~\\cite{kd} is performed to enhance consistency among sub-architectures. \nDeepLight~\\cite{deeplight} develops an integrated search space for feature embedding ($2^{Vd}$), interaction ($2^{d^2}$) and MLP structures by pruning redundant parameters with $L_2$ penalty.\nResembling DeepLight, UMEC~\\cite{umec} develops an integrated search space for both embedding ($2^{Vd}$) and MLP structures. Then the sparsity is achieved by $L_2$ norms, which are further reformulated as a minimax optimization problem and optimized via a gradient-based algorithm.\n\nAMEIR~\\cite{ameir} proposes an automatic behavior modeling, feature interaction exploration and\nMLP structure investigation solution for both sequential and non-sequential features. It is worth noting that AMEIR designs an interaction block search space (including CNN, RNN, pooling, and attention layers) for identifying sequential patterns in the user history and high-order feature interaction search space for modeling non-sequential features. The one-shot weight-sharing random\nsearch paradigm widely-used in other works is deployed to boost search efficiency.\n\\section{System Design}\nBesides the aforementioned techniques on automating key components in DRS models, scientists also devise AutoML-based frameworks and training procedures from the perspective of system design. \n\n\\noindent \\textbf{Framework.}\nWorks in this domain mainly optimize the recommender systems from a framework perspective.\nDeepRecInfra~\\cite{deeprecinfra} considers the inference query sizes, arrival patterns, recommendation architectures, and underlying hardware systems (GPU\/CPU) to obtain the optimal infrastructure (\\textit{i.e.}, maximizing the latency-bounded throughput), which could help reduce the latency. \nIn terms of models' implementation, AutoRec~\\cite{autorec}, as the first open-source platform, provides a highly-flexible pipeline for various data formation, tasks, and models in deep recommender systems. \n\n\n\\noindent \\textbf{Training.}\nThe training process is also crucial for designing reliable recommender systems. \nIn general cases, the loss function is vital for training. GradNorm~\\cite{chen2018gradnorm} and $\\lambda$Opt~\\cite{lambdaopt} focus on adjusting the coefficients of loss items and optimizing parameters via gradient descent. The difference is that GradNorm aims to balance different losses of multi-task while $\\lambda$Opt only adjusts the regularization level. Later, Zhao et al.~\\cite{zhao2021autoloss} proposed an adaptive loss function search framework, AutoLoss, based on a bi-level gradient-based algorithm with Gumbel-Softmax trick~\\cite{gumbel_softmax}. AutoLoss attributes the most appropriate loss function for each data example by automatically designing various loss functions, rather than adjusting coefficients only like the aforementioned works. In the scenario of knowledge transferring, system designers should figure out which parameters should be frozen to prevent overfitting on the target dataset. With Gumbel-Softmax trick~\\cite{gumbel_softmax}, AutoFT~\\cite{yang2021autoft} automatically decides whether the embedding of a field and parameters of a layer should be fine-tuned.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\vspace{-3mm}\nA successful search relies on the engine accurately interpreting the intent behind a user's query and returning likely relevant results ranked high. There has been much progress allowing search engines to respond effectively even to short keyword queries on rare intents \\cite{RareQuerySuggestion, ClassificationRareQueries, PersonalizedExpansion}. Despite this, recommendation of queries is an integral part of all search experiences -- either in the form of \\textit{query autocomplete} (queries that match the prefix the user has currently typed into the search box) or \\textit{query suggestions} (reformulation options once an initial query has been provided).\nIn this work, we focus on the query suggestion task.\n\nOriginal algorithms for this scenario relied on extracting co-occurrence patterns between query pairs, and their constituent terms, within historical logs \\cite{jones2006generating, huang2003relevant, fonseca2005concept, beeferman2000agglomerative}. Such methods often work well for frequent queries. Recent work utilizing generative approaches common in natural language processing (NLP) scenarios offer generalization in terms of being able to provide suggestions even for rare queries \\cite{mitra2015exploring, cho2014learning}. More specifically, the work by Sordoni et al. \\cite{HRED} focuses on generating query suggestions that are aware of the context of the user's current session. The current paper is most similar to this work in terms of motivation and the core technical component.\n\nThe experiments described here are based on data from a commercial stock image search engine. In this setting, the items in the index are professionally taken high quality images to be used in commercial publishing material. The users of such a system exhibit similar properties to what might be expected on general purpose search engines - i.e., the use of relatively short queries often with multiple reformulations within a session. \nThe logged data therefore contains not only the sequence of within-session queries, but also impression logs listing what images were shown in response to a query and which amongst those were clicked. \n\nThe availability of usage data, which provides implicit relevance signals, allows the building of a query reformulation model that includes aspects that have been shown to be useful in related literature:\nsession context capturing information from previous queries in the session, as well as properties of relevant results via a multitask component. Building on state-of-the-art models in this manner, we specialize the solution to our setting by utilizing a novel supervision signal for the reformulation model in the form of linguistically rich captions available for the clicked results (in our case, images) across sessions.\n\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=0.85\\textwidth]{Teaser.pdf}\\vspace{-2mm}\n\\caption{\\scriptsize{The basic idea behind our work. We generate query reformulations using \\textit{(a)} subsequent queries within sessions, and \\textit{(b)} the captions of clicked images, as supervision signals. In both the cases, the task of generating reformulations is done while jointly optimizing the ranking of results.}}\n\\label{fig:teaser}\n\\vspace{-5mm}\n\\end{figure*}\n\n\\vspace{-4mm}\n\\section{Related Work}\n\\vspace{-3mm}\nA user of a search system provides an input query, typically a short list of keywords, into the search box and expects content relevant to their need ranked high in  the result list. There are many reasons why a single iteration of search may not be successful -- mis-specified queries (including spelling errors), imperfect ranking, ambiguous intent, and many more. As a result, it is useful to think of a search session as a series of interactions -- where the user enters a query, examines and potentially interacts with the returned results, and constructs a refined query that is expected to more accurately represent their intent. Search engines therefore mine historical behavior of users on this query and similar ones in an attempt to optimize the entire search session~\\cite{silvestri2009mining}. \n\nBeing able to effectively extract these signals from historical logs starts with understanding and interpreting user behavior appropriately. For example, Huang et al. ~\\cite{huang2009analyzing} pointed out that successful reformulations, especially those involving changes to words and their order, can be identified as those that retrieve new items which are presented higher in the subsequent results. An automatic reformulation experience involves implementing lessons from such analyses. The first of these is the use of previous queries within the current search sessions to inform the subsequent suggestions \u2013 i.e., modeling the \\textit{session context}. Earlier papers (e.g.~\\cite{cao2008context}) explicitly captured co-occurrence within sessions which, while being an intuitive and simple strategy, had the disadvantage of not being able to account for rarer queries. Newer efforts (e.g.~\\cite{mitra2015exploring}) therefore utilize distributed representations of terms and queries to help generalize to unseen queries.\n\nSuch efforts are part of a wider expansion of techniques originally common within NLP domains to Information Retrieval (IR) scenarios. \nConceptually, a generation-based model for query reformulation is obtained by mapping a query to the subsequent one in the same session. Such a model incorporates two signals known to be useful from traditional IR: $(1)$ sequence of terms within a query \\& $(2)$ sequence of queries within a session. \nRecent papers have investigated models anchored in the original generic NLP settings but customized to the characteristics of search queries. For example, Dehghani et al. ~\\cite{Dehghani2017} suggest a `copy' mechanism within the sequence-to-sequence (seq-to-seq) models \\cite{sutskever2014sequence} to allow for terms to be carried over across queries in the session. In the current paper, we consider the work of Sordoni et al. \\cite{HRED} as a reference for the core seq-to-seq model. The model, referred to here as  \\textit{H}ierarchical \\textit{R}ecurrent \\textit{E}ncoder \\textit{D}ecoder (\\textit{\\small{HRED}}), is a standard encoder-decoder setup, where word embeddings are aggregated into a query representation, a sequence of which in turn leads to a session representation. A decoder for the hierarchically organized query and session encoders is trained to predict the sequence of query words that compose the subsequent query in the session. Along with being a strong baseline, it serves to illustrate the core components of our work: $(a)$ use of a novel supervision signal in the form of captions of clicked results, and $(b)$ jointly optimizing ranking along with query reformulation. These extensions could similarly be done with other seq-to-seq models used for query suggestion.  \n\n\n\nOur motivation for using captions of clicked images as supervision signal stems from the fact that captions are often succinct summaries of the content of the actual images as the creators are incentivized to have their images found. In particular, captions indicate which objects are present in the image, their corresponding attributes, as well as relationships with other objects in the same image --  for example, \\textit{``A beautiful girl \\textbf{wearing} a yellow shirt \\textbf{standing near} a red car\"}. These properties make the captions a good target.\n\n\nMultitask learning~\\cite{caruana1997multitask} has been shown to have success in scenarios where related tasks benefit from common signals. A recent paper ~\\cite{MNSRF} shows benefits of such a pairing in a search setting. Specifically, Ahmad et al. show that coupling with a classifier distinguishing clicked results from those skipped helps improve a query suggestion model. We extend this work by utilizing a pairwise loss function commonly used in learning-to-rank~\\cite{Burges2005}. We show that not only does this provide the expected increase in the effectiveness of the ranker component, but also increases the diversity of suggested reformulations. Such diversity has been shown to be important for the query suggestion user experience~\\cite{ma2010diversifying}. \n\nWe begin by providing details of the mathematical notation in the next section, before describing our models in detail. The subsequent experimental section provides empirical evidence of the benefits that our design choices bring. \n\\vspace{-5mm}\n\\section{Notation and Model Architectures}\n\\vspace{-0.5mm}\n\\label{sec:notations}\n\\subsubsection{3.1 \\quad Notation:}\n\\vspace{-2mm}\nWe define a session as a sequence of queries , $\\mathcal{S} = \\{q_1, \\dots, q_n\\}$. Each query $q_i$ in session $\\mathcal{S}$ has a set of displayed images associated with it, $\\mathcal{I}_i = \\{I_{i}^1, \\dots, I_{i}^m\\}$. A subset of images in $\\mathcal{I}_i$ are clicked, we refer to the top-ranked clicked image as $I_{i}^{\\text{ }\\text{clicked}}$. All the images in the set $\\mathcal{I}_i$ have a caption describing them, the entire set of which is represented as $\\mathcal{C}_i = \\{C_{i}^{1}, \\dots, C_{i}^{m} \\}$. It follows that every $I_i^{\\text{ }\\text{clicked}}$ will also have an associated caption with it, given as $C_i^{\\text{ }\\text{clicked}}$. Given this, for every successful query $q_i$ in session $\\mathcal{S}$, we will have an associated clicked image $I_i^{\\text{ }\\text{clicked}}$ and a corresponding caption $C_i^{\\text{ }\\text{clicked}}$. We consider the size of impression $m$ (number of images) to be fixed for all $q_i$.\n\nOur models treat each query $q_i$ in any given session, as a sequence of words, $q_i = \\{w_1, \\dots, w_{l_q} \\}$. Captions are represented similarly - as sequences of words, $C_i^j = \\{w_1, \\dots, w_{l_c}\\}$.\nWe use LSTMs \\cite{LSTM} to model the sequences, owing to their demonstrated capabilities in modeling various natural language tasks, ranging from machine translation \\cite{sutskever2014sequence} to query suggestion \\cite{Dehghani2017}.\n\nThe input to our models is a query $q_i$ in the session $\\mathcal{S}$, and the desired output is a target reformulation $q_{\\text{reform}}$. This target reformulation $q_{\\text{reform}}$ can either be \\textit{(i)} the subsequent query $q_{i+1}$ in the same session $S$, or \\textit{(ii)} the caption $C_i^{\\text{ } \\text{clicked}}$ corresponding to the clicked image $I_i^{\\text{ }\\text{clicked}}$. Note that obtaining contextual query suggestions via a \\textit{translation model} that has learnt a mapping between successive queries within a session (i.e., \\textit{(i)}) has been previously proposed in our reference baseline papers~\\cite{HRED, MNSRF}. In the current paper, we  utilize a linguistically richer supervision signal, in the form of captions of clicked images (i.e., \\textit{(ii)}), and analyze the behavior of the different models across three high level axes - relevance, descriptiveness and diversity of generated reformulations. \\vspace{-7mm}\n\\subsubsection{3.2 \\quad Model Architectures: }\n\\vspace{-2mm}\n\\label{secModels}\nIn this paper, we evaluate two base models -- \\textit{\\small{HRED}} and \\textit{\\small{HRED}} with \\textbf{Cap}tions (\\textit{\\small{HREDCap}}), and to study the effect of multitask learning, we add a ranker component to each of these models; giving us two more multitask variants -- \\textit{\\small{HRED + Ranker}} and \\textit{\\small{HREDCap + Ranker}}. The underlying architecture of \\textit{HRED} and \\textit{\\small{HREDCap}} (and the corresponding variants) is essentially the same, but \\textit{\\small{HRED}} has been trained by using $q_{i+1}$ as target and \\textit{\\small{HREDCap}} has been trained using $C_{i}^{clicked}$ as target. \\textit{\\small{HRED}} comprises of a query encoder, a session encoder, and a query decoder; all of which are descried below.\n\n\n\\noindent\\textbf{Query Encoder:} The query encoder generates a query level encoding $\\mathbf{V}_{q_i}$ for every $q_i \\in \\mathcal{S}$. This is done by first representing the query $q_i$ using vector embeddings of corresponding words $\\{\\mathbf{w}_1, \\dots, \\mathbf{w}_{l_q}\\}$, and then sequentially feeding them into a bidirectional LSTM (BiLSTM) \\cite{graves2005framewise}. As shown in Fig. \\ref{fig:NotationFigure}(a), the query encoder takes each of these word representations as input to the BiLSTM at every encoding step and updates the hidden states based on the forward and backward pass over the input query. The forward and backward hidden states are concatenated, and after applying attention \\cite{bahdanau2014neural} over the concatenated hidden states, we obtain a fixed size vector representation $\\mathbf{V}_{q_{i}}$ for the query $q_i \\in \\mathcal{S}$. \n\n\\noindent\\textbf{Session Encoder:} The encoded representation $\\mathbf{V}_{q_{i}}$ of query $q_i \\in \\mathcal{S}$ is used by the session encoder, along with encoded representations  $\\{\\mathbf{V}_{q_1}, \\dots, \\mathbf{V}_{q_{i-1}}\\}$ of previous queries within the same session, to capture the context of the ongoing session thus far. The session encoder, which is modeled by a unidirectional LSTM \\cite{LSTM}, updates the session context $\\mathbf{V}^{q_{i}}_{\\mathcal{S}}$ after each new $\\mathbf{V}_{q_{i}}$ is presented to it. Fig. \\ref{fig:NotationFigure}(b) illustrates one such update where the session encoding is updated from $\\mathbf{V}^{q_{i-1}}_{\\mathcal{S}}$ to $\\mathbf{V}^{q_{i}}_{\\mathcal{S}}$ after  $\\mathbf{V}_{q_{i}}$ is provided as input to the session encoder by the query encoder.\nSince it is unreasonable to assume access to future queries in the session while generating a reformulation for the current query, we use a unidirectional LSTM to model the forward sequence of queries within a session. Accordingly, the session encoder updates its hidden state based on the forward pass over the query sequence. As shown in Fig. \\ref{fig:NotationFigure}(b), max-pooling is applied over each dimension of the hidden state to obtain the session encoding  $\\mathbf{V}^{q_{i}}_{\\mathcal{S}}$. \n\n\\begin{figure*}[bt]\n    \\centering\n    \\includegraphics[width=1.0\\textwidth]{Notations.pdf}\\vspace{-2mm}\n    \\caption{\\scriptsize{An illustration of the \\textit{(a)} query encoder,  \\textit{(b)} session encoder, and \\textit{(c)} query decoder }}\n    \\label{fig:NotationFigure}\n\\vspace{-6mm}\n\\end{figure*}\n\n\\noindent\\textbf{Query Decoder:} The generated session encoding $\\mathbf{V}^{q_{i}}_{\\mathcal{S}}$ is used as input by a query decoder to generate a reformulation $\\hat{q}_{\\text{reform}} = \\{\\hat{w}_1, \\dots, \\hat{w}_{l_r}\\}$ for the query $q_i \\in \\mathcal{S}$. As shown in Fig. \\ref{fig:NotationFigure}(c), the reformulation is generated word by word using a single layer unidirectional LSTM. With each unfolding of the decoder LSTM at step $t \\in \\{1, \\dots, l_r\\}$, a new word $\\hat{w}_t$ is generated as per the following probability:\n\\vspace{-2mm}\\[\n    \\hat{w}_t =\n    \\argmax_{w^i \\in \\mathcal{V}} P(\\hat{w}_t = w^i \\mid \\hat{w}_{1: t-1}, \\mathbf{V}^{q_i}_{\\mathcal{S}}) \\text{ }\\text{ } \\footnote{For $t=1$, $P(\\hat{w}_t = w^i \\mid \\hat{w}_{1: t-1}, \\mathbf{V}^{q_i}_{\\mathcal{S}})$ reduces to $P(\\hat{w}_t = w^i \\mid \\mathbf{V}^{q_i}_{\\mathcal{S}})$. However, for the sake of readability, this special consideration for $t=1$ has been skipped for the following equations. }\n\\] \\vspace{-2mm}\n\n\\vspace{-6mm}\\begin{equation}\n    P(\\hat{w}_t = w^i  \\mid \\hat{w}_{1: t-1}, \\mathbf{V}^{q_i}_{\\mathcal{S}}) = g(\\phi(h_d^t))\n    \\label{eq:condGen}\\vspace{-1mm}\n\\end{equation}\nHere, $h_d^t$ is the hidden state of the decoder at decoding step $t$, $\\hat{w}_{1: t-1}$ denotes the previous words generated by the decoder, and $\\phi(h_d^t)$ is a non-linear operation over $h_d^t$.  The softmax function $g(.)$ provides a probability distribution over the entire vocabulary $\\mathcal{V}$. $w^i$ is used to denote the $i$-th word in $\\mathcal{V}$. The joint probability of generating a reformulation $\\hat{q}_{\\text{reform}} = \\{\\hat{w}_1, \\dots, \\hat{w}_{l_r}\\}$ can be decomposed into the ordered conditionals as\n    $P(\\hat{q}_{\\text{reform}} \\mid q_i) = \\prod_{t = 1}^{l_r} P(\\hat{w}_t \\mid \\hat{w}_{1:t-1}, \\mathbf{V}^{q_i}_{\\mathcal{S}})$.\n   \nDuring training, the decoder compares each word $\\hat{w}_t$ in the generated reformulation $\\hat{q}_{\\text{reform}}$ with the corresponding word $w_t$ in the target reformulation $q_{\\text{reform}}$, and aims to minimize the negative log-likelihood. For a given reformulation by the decoder, the loss is \\vspace{-4mm}\n\\begin{equation}\n    \\label{eq:reformLoss}\n    \\mathcal{L}_{\\text{reform}} = - \\sum_{t = 1}^{l_r} \\log P(\\hat{w}_t = w_t \\mid \\hat{w}_{1:t-1}, \\mathbf{V}^{q_{i}}_{\\mathcal{S}}) + \\mathcal{L}_{reg}\\vspace{-3mm}\n\\end{equation}\n\nHere, $\\mathcal{L}_{reg} = - \\lambda \\sum_{w^i \\in \\mathcal{V}} P(w^i \\mid \\hat{w}_{1:t-1}, \\mathbf{V}^{q_{i}}_{\\mathcal{S}}) \\cdot \\log P(w^i \\mid \\hat{w}_{1:t-1}, \\mathbf{V}^{q_{i}}_{\\mathcal{S}})$ is a regularization term added to prevent the predicted probability distribution over the words in the vocabulary from being highly skewed. $\\lambda$ is a regularization hyperparameter. The training loss is the sum of $\\mathcal{L}_{\\text{reform}}$ over all query reformulations generated by the decoder during training.\n\nTo summarize, the model encodes the queries, generates session context encodings, and generates the reformulated query using the decoder while updating the model parameters using the gradients of $\\mathcal{L}_{\\text{reform}}$. \n\n\n\\begin{figure*}[tb]\n    \\centering\n    \\begin{minipage}{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=1.0\\textwidth]{GeSeCa.pdf}\n    \\end{minipage}\n   \n    \\begin{minipage}{0.4\\textwidth}\n        \\centering\n        \\includegraphics[width=1.0\\textwidth]{Ranker.pdf}\\vspace{-1mm}\n    \\end{minipage}\n\\caption{\\scriptsize{\nThe proposed architecture of our multitask model: \\textit{HRED + Ranker} (left). For the sake of brevity, we have shown the ranker component separately (right). For \\textit{HREDCap + Ranker}, the supervision signals are obtained from captions of clicked images and not subsequent queries.}}\n\\label{fig:architectures}\\vspace{-5mm}\n\\end{figure*}\n\n\\noindent\\textbf{Ranker Component:}\nThis additional component is responsible for ranking the $m$ retrieved results for $q_i \\in \\mathcal{S}$. As shown in Fig. \\ref{fig:architectures} (right), the ranker takes as input the concatenation of query and session encoding $[\\mathbf{V}_{q_i} \\oplus \\mathbf{V}_{\\mathcal{S}}^{q_i}]$, for every $q_i \\in \\mathcal{S}$. \nThe concatenated vector representation $[\\mathbf{V}_{q_i} \\oplus \\mathbf{V}_{\\mathcal{S}}^{q_i}]$ is used to compute the similarity between the query $q_i$ and its candidate results. The concatenation of these encodings is done to ensure that both current query information (as captured in $\\mathbf{V}_{q_i}$) and ongoing session context (as captured in $\\mathbf{V}_{\\mathcal{S}}^{q_i}$) is used by the ranker.\nTo obtain a representation of the images, we use their corresponding captions. Formally, for every query $q_i \\in \\mathcal{S}$ each image $I_i^j \\text{ } \\in \\mathcal{I}_i$ is represented using $\\mathbf{C}_i^j$. The average of the vector embeddings of words $\\{w_1, \\dots, w_{l_c} \\}$ in $\\mathbf{C}_i^j$ is computed for the image $I_i^j$.  The cosine similarities between $[\\mathbf{V}_{q_i} \\oplus \\mathbf{V}_{\\mathcal{S}}^{q_i}]$ and the image representations  $\\mathbf{C}_i^j \\in \\mathcal{C}_i$ are used to rank order the retrieved results. The $j$-th element of the similarity vector $\\mathbf{S}_i$ represents the similarity between $[\\mathbf{V}_{q_i} \\oplus \\mathbf{V}_{\\mathcal{S}}^{q_i}]$ and $\\mathbf{C}_i^j$.\n\\begin{equation}\n\\label{similarityEquation}\n    {S}_i^j = sim([\\mathbf{V}_{q_i} \\oplus \\mathbf{V}_{\\mathcal{S}}^{q_i}], \\mathbf{C}_i^j)\n\\end{equation}\n\n\\noindent During training, the ranker tries to learn model parameters based on one of the following two objectives:\\\\\n\\noindent(i) \\textbf{Cross Entropy Loss}: As described in \\cite{MNSRF}, we utilize the `clicked' versus `not-clicked' boolean event to train a classifier, where the ranker scores the $m$ retrieved results based on the probability of being clicked by the user. In the following equation, $\\mathbf{R}_i$ for query $q_i$ is an $m$-dimensional vector, where each value in the vector indicates whether the corresponding image was clicked or not. I.e., $R_i^j = 0$ if $I_i^j$ was not clicked, and $R_i^j = 1$ if $I_i^j$ was clicked. A sigmoid of the scores from Eq.~\\ref{similarityEquation} is taken as the probability of click. Using the $\\mathbf{R}_i$ as labels, the ranker can now be trained using a standard cross entropy loss function:\\vspace{-1mm}\n\\begin{equation}\\\n\\label{eq:rankLoss_BCE}\n    \\mathcal{L}_{\\text{rank}} = BCE(\\sigma(\\mathbf{S}_i), \\mathbf{R}_i)\\vspace{-2mm}\n\\end{equation}\n\n\\noindent(ii) \\textbf{Pairwise Ranking Loss}: As described in \\cite{Burges2005}, the original boolean labels in $\\mathbf{R}_i$ can be used to construct an alternate event space where labels $M_{jk} = 1$ when the image at rank $j$ was clicked while the one at $k$ was not. Pairwise ranking loss allows to better model the preferences of certain results over the others.\\vspace{-1mm}\n\\begin{equation}\n\\label{eq:rankLoss_RO}\n    %\n    \\mathcal{L}_{\\text{rank}} = - \\frac{1}{m^2} \\sum_{j=1}^m\\sum_{ \\substack{k=1\\\\k\\neq j} }^m M_{jk}*\\log \\hat{M}_{jk}  + (1 - M_{jk})*\\log(1 - \\hat{M}_{jk})\\vspace{-1mm}\n\\end{equation}\n\\[\\vspace{-1mm}\n\\text{where, } \\hat{M}_{jk} = P(S_i^j > S_i^k \\mid [\\mathbf{V}_{q_i} \\oplus \\mathbf{V}_{\\mathcal{S}}^{q_i}]) = \\sigma(S_i^j - S_i^k)\n\\vspace{-1mm}\\]\n\n\n\\noindent Since \\textit{\\small{HRED + Ranker}} and \\textit{\\small{HREDCap + Ranker}} are multitask models, their training objective is a weighted combination of $\\mathcal{L}_{\\text{reform}}$ and $\\mathcal{L}_{\\text{rank}}$. \\vspace{-2mm}\n\\begin{equation}\n    \\mathcal{L}_{\\text{multitask}} = \\alpha \\cdot \\mathcal{L}_{\\text{reform}} + (1 - \\alpha) \\cdot \\mathcal{L}_{\\text{rank}}\\vspace{-1.5mm}\n\\end{equation}\nHere, $\\alpha$ is a hyperparameter used for controlling the relative contribution of the two losses. \nAs mentioned earlier, either the regular binary cross-entropy loss or the pairwise-ranking loss can be used for $\\mathcal{L}_{\\text{rank}}$. We experiment using both and report our results on the effect of using one over the other. The models that are trained using cross entropy loss are appended with \\textit{(\\small{CE})}, and the models that are trained using pairwise ranking objective are denoted as \\textit{(\\small{RO})}. \n\nIt is worth noting that since for a given query $q_i$ there can be more than one clicked images, our ranker component allows $\\mathbf{R}_i$ to take the value $1$ at more than a single place. However, while training the reformulation model, we only consider the caption of the \\textit{highest ranked} clicked image. \n\n\\vspace{-4mm}\n\\section{Experiments}\n\\vspace{-3mm}\n\\label{sec:experimentalSetup}\n\\subsubsection{Dataset:} We use logged impression data from Adobe Stock\\footnote{\\url{https:\/\/stock.adobe.com\/}}. %\nThe query logs contain information about the queries that were issued by users, and the images that were presented in response to those queries. Additionally, they contain information about which of the displayed images were clicked by the user.\nWe consider the top-10 ranked results, i.e., the number of results to be considered for each query is $m=10$. The queries are segmented into sessions (multiple queries by the same user within a $30$ minute time window), while maintaining the sequence in which they were executed by a user. \nWe retain both multi-query sessions as well as single-query sessions, leading to a dataset comprising $1,301,888 $ sessions, $2,122,079$ queries, and $10,185,979$ unique images. We note that $\\sim24.8\\%$ of the sessions are single-query sessions, while rest all are multi-query sessions; each of which, on average, comprise of $2.19$ queries.\nAdditionally, we remove all non-alphanumeric characters from the user-entered queries, while keeping spaces, and convert all characters to lowercase.\n\n\nTo obtain the train, test and validation set, we first shuffle the sessions and split them in a $80:10:10$ ratio, respectively. While it is possible for a query to be issued by different users in distinct sessions, a given search session occurs in only one of these sets. These sets are kept the same for all experiments, to ensure consistency while comparing the performance of trained models. The validation set is used for hyperparameter tuning. \\vspace{-6.0mm}\n\\subsubsection{Experimental Setup: }\nWe construct a global vocabulary $\\mathcal{V}$ of size $37,648$ comprising of words that make up the queries and captions for images. Each word in the vocabulary is represented using a $300$-dimensional vector $\\mathbf{w}_i$. Each $\\mathbf{w}_i \\in \\mathcal{V}$ is initialized using pre-trained GloVe vectors \\cite{pennington2014glove}. Words in our vocabulary $\\mathcal{V}$ that do not have a pre-trained embedding available in GloVe ($1,941$ in number), are initialized using samples from a standard normal distribution.\nSince the average number of words in a query, average number of words in a caption, and average number of queries within a session are $2.31$, $5.22$, and $1.63$, we limit their maximum sizes to $5$, $10$, and $5$, respectively. For queries and captions that contain less than $5$ and $10$ words respectively, we pad them using `$<p>$' tokens.  The number of generated words in $\\hat{q}_{\\text{reform}}$ was limited to $10$, i.e., $l_r = 10$.\n\nDuring training, we use Adam optimizer~\\cite{ADAM} with a learning rate initialized to $10^{-3}$. Across all the models, the regularization coefficient $\\lambda$ is set to be $0.1$. For multitask models, the loss trade-off hyperparameter $\\alpha$ is set to $0.45$. The sizes of the hidden states of query level encoder $\\overrightarrow{h}_q$ and $\\overleftarrow{h}_q$ are set to $256$, and that of session level encoder $h_{\\mathcal{S}}$ is set to $512$. The size of the decoder's hidden state is kept to be $256$. We train all the models for a maximum of $30$ epochs, using batches of size $512$, with early stopping based on the loss over the validation set. The best trained models are quantitatively and qualitatively evaluated and we discuss the results in the upcoming section. \n\nAt test time, we use \na beam search-based decoding approach to generate multiple reformulations \\cite{bahdanau2014neural}. \nFor our experiments, we set the beam width $K=3$. The choice of $K$ was governed by observations that will be discussed later, while analyzing the diversity and relevance of generated reformulations. These three reformulations are rank ordered using their generation probability. \n\nWe experiment with a range of hyperparameters and find that the evaluation results are stable with respect to our hyperparameter choices. However, our motivation is less about training the most accurate models, as we wish to measure the effect of the supervision signal and training objective when used alongside the baseline models. While presenting the results in Table \\ref{tab:mainResults} \\& \\ref{tab:lengtAnalysis}, we report the average of values over $10$ different runs, as well the standard deviations.\n\\vspace{-5mm}\n\\section{Evaluation and Results}\n\\vspace{-3mm}\nIn this section, we evaluate the performance of the aforementioned models using multiple metrics for each of the two tasks: query reformulation and ranking. The metrics used here are largely inspired from~\\cite{Dehghani2017}, and we discuss these below briefly. Towards the end of the section we also provide some qualitative results. \\vspace{-3mm}\n\\vspace{-2mm}\n\\subsubsection{5.1 \\quad Evaluation Metrics:}\nEvaluation for query reformulation involves comparing the generated reformulation $\\hat{q}_{\\text{reform}}$ with the target reformulation $q_{\\text{reform}}$. For all the models, irrespective of whether they utilize the next query within the session $q_{i+1}$ as the target reformulation, or the caption $C_i^{\\text{ }\\text{clicked}}$ corresponding to the clicked image, the ground truth reformulation $q_{\\text{reform}}$ is always taken to be $q_{i+1}$\\footnote{For sessions with less than $5$ queries in a session, if $q_i$  is the last query of the session, the model is trained to predict the `end of session' token as the first token of $q_{i+1}$. The subsequent predicted tokens are encouraged to be the padding token `$<p>$'.}. This consistency has been maintained across all models to ensure that their performance is comparable, no matter what signal was used to train the reformulation model. The metrics used here cover three aspects: `Relevance' (BLEU \\& sim$_{emb}$), `Ranking' (MRR) and `Diversity' (analyzed later).\n\n\n\\noindent\\textbf{BLEU score}: This metric~\\cite{papineni2002bleu}, commonly used in machine translation scenarios, quantifies the similarity between a predicted sequence of words and the target sequence of words using n-gram precision.\nA higher BLEU score corresponds to a higher similarity between the predicted and target reformulations.\n\n\\noindent\\textbf{Embedding based Query Similarity}: This metric takes semantic similarity of words into account, instead of their exact overlap.\nA phrase-level embedding is calculated using vector extrema~\\cite{vectorExtrema}, for which pretrained GLoVe embeddings were used. The cosine similarity between the phrase-level vectors for the two queries is given by sim$_{emb}$. A higher value of sim$_{emb}$ is taken to signify a greater semantic similarity between the prediction and the ground truth. Unlike BLEU, we expect sim$_{emb}$ to provide a notion of similarity of the generated query to the target that allows for replacement words that are similar to the observed ones.\n\n\\noindent\\textbf{Mean Reciprocal Rank (MRR)}: The ranker's effectiveness is evaluated using MRR \\cite{MRR}, which is given as the reciprocal rank of the first relevant (i.e., clicked) result averaged over all queries, across all sessions. \nA higher value of MRR will signify a better ranker in the proposed multitask models. To have a standard point of reference to compare against, we computed the observed MRR for the queries in the test set and found it to be $0.31$. This means that on average, for queries in our test set, the first image clicked by the users was at rank $\\sim3.1$. \\vspace{-3mm}\n\\vspace{-3mm}\n\n\\begin{table*}[!t]\n  \\centering\n  \\scalebox{0.71}{\n  \\begin{tabular}{| c  | c | c | c | c |}\\hline\n    {} & \\multicolumn{3}{c}{\\textbf{Query Reformulation}} & \\textbf{Ranking}\\\\\n    \\textbf{Model}   &  {\\textbf{BLEU (\\%)} } & {\\textbf{sim$_{\\mathbf{emb}}$ (\\%)} } &\\textbf{Diversity}  & \\textbf{MRR} \\\\\n    {}   & {($\\uparrow$)}  & {($\\uparrow$)} &  \\textit{Top K = $3$} ($\\uparrow$) & \\textit{Baseline}: $0.31$ ($\\uparrow$)\\\\\\hline\n    {HRED}          &  $6.92 \\pm 0.06$ & $40.7 \\pm 1.3$  & $0.37 \\pm 0.01$ & - \\\\\\hline\n    {HRED + Ranker} (CE)   &  $7.63 \\pm 0.07$ & $43.5 \\pm 1.2$  & $0.42 \\pm 0.02$ & $0.35 \\pm 0.02$ \\\\\n    {HRED + Ranker} (RO)    &  $7.51 \\pm 0.07$ & $40.8 \\pm 1.4$      & $0.43 \\pm 0.02$ & $0.39 \\pm 0.01$ \\\\\\hline\n    {HREDCap}         &  $7.13 \\pm 0.09$ & $37.8 \\pm 1.4$      & $0.39 \\pm 0.04$ & - \\\\\\hline\n    {HREDCap + Ranker} (CE)   &  $7.95 \\pm 0.11$ & $39.4 \\pm 1.2$      & $0.44 \\pm 0.06$ & $0.38 \\pm 0.02$ \\\\\n    {HREDCap + Ranker} (RO)   &  $7.68 \\pm 0.10$ & $37.6 \\pm 1.4$      & $0.45 \\pm 0.05$ & $0.41 \\pm 0.02$ \\\\\\hline\n  \\end{tabular}\n  }\\vspace{0.5mm}\n    \\caption{\\scriptsize{Performance of models based on reformulation and ranking metrics}}\\vspace{-9mm}\n  \\label{tab:mainResults}\n\\end{table*}\n\n\\subsubsection{5.2 \\quad Main Results}\nHaving discussed the metrics, we will now present the performance of our models on the two tasks under consideration, namely query reformulation and ranking. Table \\ref{tab:mainResults} provides these results as well as the effect of different ranking losses -- denoted by (\\textit{\\small{RO}}) and (\\textit{\\small{CE}}) respectively.\n\n\\noindent\\textbf{Evaluation based on Reformulation:}\nFor the purpose of this evaluation, we fix the beam width $K=3$ and report the average of maximum values among all the candidate reformulations, across all queries in our test set. \n\nWhile comparing \\textit{\\small{HRED}} and \\textit{\\small{HRED + Ranker}} (both \\textit{\\small{CE}} and \\textit{\\small{RO}}), we observe that the multitask version performs better across \\textit{all} metrics. A similar trend can be observed when comparing \\textit{\\small{HREDCap}} with its multitask variants. For all the three metrics for query reformulations, the best performing model is a multitask model -- this validates the observations from \\cite{MNSRF} in our context.\n\n\nWhen comparing the two core reformulation models -- \\textit{\\small{HRED}} \\& \\textit{\\small{HREDCap}}, we find that the richer captions data that \\textit{\\small{HREDCap}} sees is aiding the model -- while \\textit{\\small{HRED}} scores better sim$_{emb}$, \\textit{\\small{HREDCap}} wins out on BLEU \\& Diversity. The drop in sim$_{emb}$ values can be  explained by noting that on average captions contain more words than queries ($5.22$ in comparison to $2.31$), and hence similarity-based measures, due to additional words in the captions, will not be as high as overlap-based measures (i.e., BLEU).  \n\\noindent\\textbf{Evaluation based on Ranking:}\nTo evaluate the performance of the ranker component in our proposed multitask models, we use MRR. We use the observed MRR of clicked results in the test set ($0.31$) as the baseline. We also analyze the effect of using the pairwise objective as opposed to the binary cross entropy loss.\n\nLooking at the results presented in Table~\\ref{tab:mainResults}, three trends emerge. Firstly, all the proposed multitask models perform better than the baseline. The best performing model, i.e., \\textit{\\small{HREDCap + Ranker}} with pairwise loss (\\textit{\\small{RO}}), outperforms the baseline by about $32\\%$. Secondly, we observe that using pairwise loss leads to an increase in MRR, for both of the cases under consideration, with only marginal drop in reformulation metrics -- we revisit this observation in the next section. %\nLastly, the multitask models that use captions perform better than multitask models that use subsequent queries. \n\\vspace{-3mm}\n\\subsubsection{5.3 \\quad Analysis: }\n\\vspace{-2mm}\nIn this section, we concentrate on the following two aspects of the generated query reformulations: $(a)$ diversity, and $(b)$ descriptiveness.\n\n\\noindent\\textbf{Diverse Query Reformulations due to Multitasking:}\nThe importance of suggesting diverse queries to enhance user search experience is well established within the IR community. The mechanism to obtain a diverse set of reformulation alternatives is via the use of beam search based decoding.\nIn scenarios where a set of top-K candidates are required, we take inspiration from Ma et al. \\cite{ma2010diversifying} to evaluate the predictions of our models for their diversity. For a beam width of $K$, a reformulation model will generate $\\mathcal{R}_{gen}  = \\{r_1, r_2, \\dots, r_K\\}$ candidate reformulations for a given original query. We quantify the diversity in the candidate reformulations by comparing each candidate reformulation $r_i$ with other reformulations $ r_j \\in \\mathcal{R}_{gen} : i \\neq j$. The diversity of a set of $K$ queries is evaluated as \n\\vspace{-2mm}\\[\n    D(\\mathcal{R}_{\\text{gen}}) = 1 - \\frac{1}{K(K-1)}*\\left(\\sum_{r_i \\in \\mathcal{R}_{\\text{gen}}}\\sum_{\\substack{r_j \\in \\mathcal{R}_{\\text{gen}}: \\text{ }j \\neq i}} sim_{emb}(r_i, r_j)\\right)\\vspace{-2mm}\n   \n\\]\nIn Table \\ref{tab:mainResults}, it can be observed that multitask models generate more diverse reformulations than models trained just for the task of query reformulation. This is particularly evident when comparing the effect of the ranking loss.\n\nFrom Figure \\ref{fig:diversity}, it can be noted that as more candidate reformulations are taken into consideration, i.e., as the beam width $K$ is increased, the average relevance of the reformulations decreases across all the models. However, the diverseness of $\\mathcal{R}_{gen}$ flattens after $K=3$. This was the reason for setting the beam width to $3$ while presenting results in Table \\ref{tab:mainResults}.\n\n\\begin{figure*}[!h]\n\\vspace{-6mm}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{Diversity.pdf}\\vspace{-2mm}\n\\caption{\\scriptsize{The trade-off between relevance (as quantified by sim$_{emb}$) and diversity. As $K$ is increased, the relevance of generated predictions drops across all models.}}\n\\label{fig:diversity}\n\\vspace{-5mm}\n\\end{figure*}\n\\vspace{-2mm}\n\\noindent\\textbf{Descriptive Reformulations using Captions:}\nThe motivation for generating more descriptive reformulations is of central importance to our idea of using image captions. To this end, we analyze the generated reformulations to assess if this is indeed the case.  We start by noting (see Table \\ref{tab:lengtAnalysis}) that captions corresponding to clicked images for queries in our test set contain, on average, more words than the queries. Following this, we analyze the generated reformulations by two of our multitask models -- (i) \\textit{\\small{HRED + Ranker (RO)}}, which guides the process of query reformulation using subsequent queries within a session, and (ii) \\textit{\\small{HREDCap + Ranker (RO)}}, which guides the process of query reformulation using captions corresponding to clicked images. For this entire analysis, we removed stop words \\cite{nltk} \nfrom all the queries and captions under consideration.\n\nAs can be noted from Table \\ref{tab:lengtAnalysis}, reformulations using captions tend to contain more words than reformulations without them. However, number of words in a query is only a facile proxy for its descriptiveness. Acknowledging this, we perform a secondary aggregate analysis on the number of novel words inserted into the reformulation and number of words dropped from the original query. We identify novel words as words that were not present in the original query $q_i$ but have been generated in the reformulation $\\hat{q}_{\\text{reform}}$, and dropped words as the words that were present in the original query but are absent from the generated reformulation. Table \\ref{tab:lengtAnalysis} indicates that, on average, the model trained using captions tends to insert more novel words while reformulating the query, and at the same time drops fewer words from the query. Interestingly, models trained using subsequent queries inserts almost as many words into the reformulation as it drops from the original query. \n\nTo analyze this further, we compute the average similarity between the novel words that were inserted and the words that were dropped,\nby averaging the GloVe vector based similarity between words, across all  queries in our test set. For \\textit{\\small{HRED + Ranking (RO)}} this average similarity is $\\mathbf{0.64}$, while for \\textit{\\small{HREDCap + Ranker (RO)}} it is $\\mathbf{0.41}$. A higher similarity value for the former suggests that the model largely \\textit{substitutes} the existing words with words having similar semantic meaning. Using captions, on the other hand, is more likely to generate novel words which bring in additional meaning.\n\n\\begin{table*}[!t]\n    \\centering\n    \\scalebox{0.68}{\n    \\begin{tabular}{| c | c | c |}\\hline\n    Avg. \\# of words in queries & \\multicolumn{2}{c |}{$2.31 \\pm 0.92$ word(s)} \\\\\n    Avg. \\# of words in captions &  \\multicolumn{2}{c |}{$5.22 \\pm 2.37$ word(s)}\\\\\\hline\\hline\n    \\textbf{Models $\\rightarrow$} & {HRED + Ranker (RO)}$\\text{ }$ & $\\text{ }${HREDCap + Ranker (RO)}\\\\\\hline\n    Avg. \\# generated words    & $2.18 \\pm 0.61 $ word(s) & $4.91 \\pm 1.16$ word(s) \\\\\n    Avg. \\# novel words   & $1.04 \\pm 0.13$ word(s) & $2.56 \\pm 0.47$ word(s) \\\\\n    Avg. \\# dropped words & $1.14 \\pm 0.15$ word(s) & $0.89 \\pm 0.17$ word(s)\\\\\\hline\n    Avg. similarity b\/w insertions and drops & $0.64 \\pm 0.03$ & $0.41 \\pm 0.04$\\\\\\hline\n    \\end{tabular}\n    }\\vspace{0.5mm}\n    \\caption{\\scriptsize{Analyzing the effect of using captions on length of generated query reformulations, along with influence on generating novel words while dropping the existing ones. }}\n    \\label{tab:lengtAnalysis}\n\\vspace{-10mm}\n\\end{table*}\n\n\\vspace{-2mm}\n\\subsubsection{5.4 \\quad Qualitative Results:}\n\\vspace{-3.5mm}\n\nIn Table \\ref{tab:qualResults}, we present a few examples depicting the descriptive nature of generated reformulations. The generated reformulations by \\textit{\\small{HRED + Ranker}} are compared against those  by \\textit{\\small{HREDCap + Ranker}}. We only present the top ranked reformulation among top-$K$ reformulations.\nWe note that using captions as target generates reformulations that are more descriptive and the process of generation results in more insertions of novel words, in comparison to using subsequent queries as targets. These qualitative observations, along with quantitative observations discussed earlier, reinforce the efficacy of using captions of clicked images for the task of query reformulation.\n\n\\begin{table*}[!h]\\vspace{-5mm}\n    \\centering\n    \\scalebox{0.55}{\n    \\begin{tabular}{|c | c  p{3.4cm} | p{7.0cm} | p{3.4cm} | p{6.4cm} |}\\hline\n        {} & \\multicolumn{2}{c|}{\\textbf{Queries}} & \\textbf{Clicked Caption} & \\textit{HRED + Ranker (RO)} & \\textit{HREDCap + Ranker (RO)} \\\\\\hline\n        \\multirow{3}{*}{$\\mathbf{Session_1}$} & $\\mathbf{q_1}$ & traffic  & rush hour traffic & traffic jam & traffic \\textbf{jam during rush hour}  \\\\\n        \n        & $\\mathbf{q_2}$ & traffic \\textbf{jam}  & traffic jams in the city, road, rush hour\n & \\textbf{city} traffic jam & traffic \\textbf{during} \\textbf{rush hour} in \\textbf{city} \\\\\n \n        & $\\mathbf{q_3}$& traffic jam pollution & blurred silhouettes of cars by steam of exhaust\n & traffic jam \\textbf{cars} & \\textbf{dirt} and \\textbf{smoke} from \\textbf{cars} in traffic jam\\\\\\hline\n \n        \\multirow{3}{*}{$\\mathbf{Session_2}$} & $\\mathbf{q_1}$ & sleeping baby & sleeping one year old baby girl & \\textbf{cute} sleeping baby & \\textbf{little} baby sleeping \\textbf{peacefully}  \\\\\n        \n        {} & $\\mathbf{q_2}$ & sleeping baby cute  & baby boy in white sunny bedroom & sleeping baby & baby sleeping in \\textbf{bed peacefully} \\\\\n        \n        {} & $\\mathbf{q_3}$& white bed sleeping baby & carefree little baby sleeping with white soft toy & baby sleeping in bed & \\textbf{little} baby sleeping in white bed \\textbf{peacefully}\\\\\\hline\n        \n        \\multirow{2}{*}{$\\mathbf{Session_3}$} & $\\mathbf{q_1}$ & chemistry & three dimensional illustration of molecule model & chemical \\textbf{reaction} & \\textbf{molecules} and \\textbf{structures} in chemistry  \\\\\n        \n        {} & $\\mathbf{q_3}$& molecule reaction & chemical reaction between molecules & reaction molecules & molecules reacting in \\textbf{chemistry}\\\\\n        \n        {} & $\\mathbf{q_3}$& molecule collision & frozen moment of two particle collision & collision molecules & molecules colliding \\textbf{chemistry} \\textbf{reaction}\\\\\\hline\n    \\end{tabular}\n    }\\vspace{0.5mm}\n    \\caption{\\scriptsize{Qualitative results comparing the generated reformulation by \\textit{HRED + Ranker} and \\textit{HREDCap + Ranker}. The words in \\textbf{bold} are novel insertions.}}\n    \\label{tab:qualResults}\\vspace{-9mm}\n\\end{table*}\n\n\n\\vspace{-6mm}\n\\section{Conclusion}\n\\vspace{-4mm}\n\n\nIn this paper, we build upon recent advances in sequence-to-sequence models based approaches for recommending queries. The core technical component of our paper is the use of a novel supervision signal for training seq-to-seq models for query reformulation --  i.e., captions of clicked images instead of subsequent queries within a session, as well as the use of a pairwise preference based objective for the secondary ranking task. The effect of these are evaluated alongside baseline model architectures for this setting. Our extensive analysis evaluated the model and training method combinations towards being able to generate a set of descriptive, relevant and diverse reformulations.\n\nAlthough the experiments were done on data from an image search engine, we believe that similar improvements can be observed if content properties from textual documents can be integrated into the seq-to-seq models. \nFuture work will look into the influence of richer representations on the behavior of the ranker, and in turn on the characteristics of the reformulations.\n\n\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nHigh-intensity and monochromatic X-ray sources are important tools for research in fundamental to applied science.  Nowadays, intense and monochromatic soft X-ray beams are produced by means of Free-Electron Lasers (FELs). FELs are based on magnetic undulators \\cite{Ginzburg,Motz}, which force electrons to an oscillatory motion, resulting in electromagnetic radiation generation. With currently available magnetic undulators, the minimum achievable oscillation period $\\lambda_u$  is of the order of the centimeters, thus limiting the generation of X-ray to a few tens of keV at the highest synchrotron electron energies \\cite{1325024}.\n\nThe availability of harder X-rays or even $\\gamma$-ray source will pave the way to the development of innovative applications. For instance, a $\\gamma$-ray beam can induce nuclear reactions through photo-transmutation \\cite{nuclearph}, i.e., it can be employed for changing the atomic number of nuclei. This technique can be employed for eliminating nuclear waste by trasmuting it into short-lived nuclei for medicine as $^{126}$Sn$(\\gamma,$n$)^{125}$Sn \\cite{Irani2012466} or the $^{100}$Mo$(\\gamma,$n$)^{99}$Mo reaction followed by a $\\beta$-decay used to produce the $^{99m}$Tc isotope \\cite{national2009Medical}. Another possible application lies in the field of photo-induced nuclear fission to induce the fragmentation of heavy nuclei \\cite{PhysRev.56.426,PhysRev.59.57}. Supply of energy from the gamma-quanta causes the excitation of nuclei. Provided that the photon energy matches the transition energy between nuclear states, the reaction acquires a resonant character. As a possible result, heavy nuclei are splitted into two or more fragments. As an example of application, this process can be used in the production of medium mass neutron-rich nuclei.\n\nIn order to produce photon with the suitable energies for the applications, undulators with a shorter ${\\lambda}_u$ than currently available FELs are needed. First efforts in this direction date back to the mid 80s \\cite{PhysRevSTAB.15.070703}. A design for mm-scale pulsed electromagnetic undulator was proposed in Ref. \\cite{Granatstein}, a mm-scale undulating field produced using periodic grooves ground into samarium cobalt blocks was proposed and obtained \\cite{Ramian,Paulson}, and hybrid-bias-permanent-magnet undulators with period lengths in the range of ${\\lambda}_u$ = $700 - 800$ $\\mu$m were fabricated \\cite{Tatchyn1,Tatchyn2,Tatchyn3}.By exploiting the recent progress in the fabrication of MicroElectroMechanical Systems (MEMS), the realization of undulators with ${\\lambda}_u$ in the 10 - 1000 $\\mu$m range has been proposed \\cite{PhysRevSTAB.15.070703}. Drawbacks of the MEMS undulators are the demand for an expensive cooling system and the strict requirements on beam quality.\n\t\nA promising solution for reaching higher photon energies is the usage of a Crystalline Undulator (CU) \\cite{Baryshevsky198061,Baryshevsky2,Baryshevsky201330}. In a CU, the electrons (positrons) are forced to an oscillatory motion by the strong electrostatic field generated by the aligned atoms of a crystal planes or axes. For instance, the Si (110) planes act on charged particles with an electric field of 6 GV\/cm, which is equivalent to a 2 kT magnetic field. Therefore, the trajectories of the particles entering a crystal at small angle with respect to the major crystallographic planes or axes, are confined \\cite{Dansk.Fys.34.14}. The quasi-oscillatory motion of the channeled particles is accompanied by channeling radiation.\n\nA crystal can tolerate a significant amount of stress and torsion while remaining into the elastic regime. Therefore, a complex geometry, such as the periodically bent crystal needed to drive the particles inside manipulated channels, can be achieved. By exploiting currently available techniques, crystals with an undulated geometry with a period ranging from 1 $\\mu$m to 1 cm can be fabricated.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1\\columnwidth]{schema.eps}\n\\caption{Schematic representation of the crystalline undulator obtained through the grooving method. The red curved line represents the undulated planes in the center of the crystal. $\\textit{A}$ is the amplitude of the undulated planes, $\\textit{a}$ the groove width and $\\textit{d}$ the groove depth.}\\label{fig:schema}\n\\end{center}\n\\end{figure}\n\n\\section{Fabrication}\n\\begin{figure}\n\\begin{center}\n\\subfloat[3D reconstruction of the sample surface obtained through interferometric profilometry]{\\includegraphics[width=0.8\\columnwidth]{onduinterESRFa.eps}}\n\\newline\n\\subfloat[Angular analysis of (220) planes through X-ray diffraction.]{\\includegraphics[width=0.8\\columnwidth]{onduinterESRFb.eps}}\n\\caption{Morphological and structural measurement of the sample.}\\label{fig:onduinterESRF}\n\\end{center}\n\\end{figure}\nIn order to manufacture a working CU, a crystal with very low density of defects should be used, together with high accuracy in bending the crystallographic planes. Indeed, the capture efficiency of channeling is strongly reduced by the presence of crystalline defects \\cite{PhysRevLett.110.175502}, and the coherence of the emitted X- and $\\gamma$-rays relies on the accuracy in the periodically bending of the crystallographic planes. Various methods have been proposed for the realization of a periodically bent crystal, such as acoustic-wave transmission \\cite{Baryshevsky198061}, periodically-graded $Si_{1-x}Ge_x$ structures \\cite{Breese1997540}, periodic-surface deformations obtainable via superficial grooves \\cite{Afonin2005122,PhysRevSTAB.7.023501}, laser ablation \\cite{Balling20092952}, or film deposition \\cite{Guidi200540,Guidi2007apl}.\n\nThe CU radiation can be distinguished from the channeling radiation only if ${\\lambda}_u$ is not equal to the average channeling oscillation period ${\\lambda}_c$. Traditional CUs work under the condition ${\\lambda}_u<{\\lambda}_c$, though an alternative scheme with ${\\lambda}_u>{\\lambda}_c$ was recently proposed \\cite{PhysRevLett.110.115503}.\n\nSeveral experiments aimed to demonstrating the feasibility of a working CU for the traditional configuration \\cite{PhysRevLett.90.034801,Afonin2005122,biryukovundu,Baranov200632,Baranov2005,Backe201337,1742-6596-438-1-012017} were performed, and some theoretical works were accomplished to foresee the energy generated by CUs \\cite{korol2004,PhysRevLett.98.164801}. Recently, a first evidence of the radiation generated by a sub-GeV electrons interacting with a CU having ${\\lambda}_u<{\\lambda}_c$ was observed in the 7-15 MeV range, using a 600-855 MeV electron beam \\cite{PhysRevLett.112.254801}.\n\nThe grooving method \\cite{PhysRevLett.90.034801} may be a simple and reproducible solution to fabricate a CU that emits radiation in the MeV range working within the traditional ${\\lambda}_u<{\\lambda}_c$ scheme.\n\nIn this paper, we present the first experimental evidence of planar channeling in a CU fabricated via superficial grooves.\n\n\n\nThe superficial grooving method consists of making a series of grooves on the major surfaces of a crystal. In the framework of the LAUE project \\cite{SPIE2013Dante}, such method has been deeply investigated and mastered to produce bent crystals tailored for the realization of optics for hard-X- and $\\gamma$-rays, i.e. Laue lenses \\cite{CamaQM,150Ge}. It was shown that a series of grooves may cause a permanent and reproducible deformation of the whole sample \\cite{indentazioni1,indentazioni2}. Indeed, the plasticization that occurs in the thin superficial layer transfers co-active forces to the crystal bulk, producing an elastic strain field within the crystal itself. Making an alternate pattern of parallel grooves on both surfaces of a crystal, the realization of a millimetric or even a sub-millimetric undulator could be envisaged.\n\nThe CU described in the present paper was realized by making a series of grooves on a 0.2$\\times$45.0$\\times$5.0 mm$^3$ silicon crystal. In particular, an alternate and periodic pattern of  $150$ $\\mu$m-wide parallel grooves was plotted on the largest surfaces of the strip, with the distance between consecutive grooves being 1 mm. Grooves were carved by means of a diamond blade through the usage of a DISCO DAD 3220 dicing machine. The groove depth was ($54\\pm2$) $\\mu$m. A schematic representation of the sample is illustrated in Fig.\\ref{fig:schema}.\n\n\n\nThe grooving parameters used for the CU fabrication were employed as input parameters to compute the expected amplitude of the undulated planes. In the framework of the classical theory of elasticity, the Stoney formula can be extended to evaluate the curvature just beneath each groove and, in turn, the undulator amplitude  \\cite{indentazioni2}. The undulator amplitude resulted to be $(4.75\\pm0.50)$ nm.\n\n\\section{Interferometric and diffractometric characterization}\n\n\n\nA morphological analysis of the sample surface was attained at the Sensor and Semiconductor Laboratory (SSL) of Ferrara (Italy) through a VEECO NT-1100 white-light interferometer, which allowed the reconstruction of the three-dimensional surfaces with lateral resolution of about 1 $\\mu$m and vertical resolution of 1 nm. The profilometric pattern of the sample surface (shown in Fig. \\ref{fig:onduinterESRF}b), allows the estimation of the average undulator amplitude, which resulted to be ($4.5\\pm1.0$) nm.\n\nThe CU was also tested through X-ray diffraction at the ID15A beamline of the European Synchrotron Radiation Facility (ESRF, Grenoble, France). A highly monochromatic beam was set at 150 keV, the beam size was 50$\\times$50$ \\mu$m$^2$. The sample was characterized by recording the position of the Bragg peak for the (220) lattice plane for 50 consecutive points along the [110] direction. In order to subtract a possible shift due to instrumental tolerances, a flat reference crystal with identical lattice orientation was set behind the undulator, and the (220) Bragg peak of the reference crystal was recorded together with the Bragg peak of the undulator. The shift of the angular position of the CU Bragg peak with respect to the reference as a function of the impact position of the X-ray beam is shown in Fig. \\ref{fig:onduinterESRF}. The measured average amplitude of undulation was ($4.1\\pm0.5$) nm with an average period of ($1.01\\pm0.03$) mm.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.8\\columnwidth]{h8exp.eps}\n\\caption{(a) Measured distribution of outgoing particles after the interaction with the undulator as a function of the incoming angle and the deflection angle with respect to the crystal plane. (b) The Monte Carlo simulation of the same distribution. (c) The Monte Carlo simulation of the same distribution convoluted with the experimental resolution.}\\label{fig:h8exp}\n\\end{center}\n\\end{figure}\n\\section{On-beam characterization}\n\nThe CU was exposed to a 400 GeV\/c proton beam in the H8 beam line at CERN-SPS. It was mounted on a two-axis rotational stage with 2 $\\mu$rad resolution. The beam was tracked before and after the interaction with the CU by using a telescope system of Si strip detectors \\cite{Celano199649,PhysRevLett.101.234801}. The angular resolution of the tracker is $\\sim3.5$ $\\mu$rad \\cite{HasanPhDThesis}. The beam size was ($1.36\\pm0.02$) mm $\\times$ ($0.73\\pm0.01$) mm and the angular divergence ($10.15\\pm0.04$) $\\mu$rad $\\times$ ($8.00\\pm0.03$) $\\mu$rad, as measured with the telescope system.\n\nRemaining in the \\emph{xz} plane, the \\emph{z} axis of the CU was purposely misaligned with respect to the beam direction to avoid the undesired axial-channeling effect. The beam enters the crystal through a (110) oriented surface with an area of 0.2$\\times$45.0 mm$^2$. The (111) planes were exploited as channeling planes.\n\nThe distribution of the deflection angle of the particles as a function of the horizontal incoming angle was measured by rotating the goniometer around the channeling position (see Fig. \\ref{fig:h8exp}). Particles channeled along the whole crystal length acquire a null deflection angle. On the contrary, if a channeled particle undergoes dechanneling, the trajectory remains parallel to the tangent of the local plane curvature where dechanneling has occurred. As a consequence, particles may acquire either a positive or negative deflection angle. By comparing the experimental pattern with the Si (110) angular scan from the literature (see Figs. 4, 6, and 7 of \\cite{PhysRevSTAB.11.063501}), a noticeable difference holds. In fact, the particles channeled in a uniformly bent crystal are deflected only to one among the two possible (positive or negative) directions.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{h8MC.eps}\n\\caption{Simulated distribution of particles during the interaction with the undulator as a function of the incoming angle and the deflection angle with respect to the crystal plane at four different penetration depth, i.e., 250 $\\mu$m (a), 500 $\\mu$m (b), 750 $\\mu$m (c) and 1000 $\\mu$m (d).}\\label{fig:h8MC}\n\\end{center}\n\\end{figure*}\n\n\\section{Monte Carlo simulations}\n\nIn order to get an insight in the experimental result, a Monte Carlo simulation was performed through the DYNECHARM++ toolkit \\cite{Bagli2013124,DYNECHARM_PHI}. The code solved the equation of motion in the non-inertial reference frame orthogonal to the crystal plane via numerical integration under the continuum potential approximation \\cite{Dansk.Fys.34.14}. The electrical characteristics of the crystals were evaluated through the ECHARM software \\cite{PhysRevE.81.026708}, which made use of diffraction data for the computation of the atomic form factors. The CU geometry was approximated through a sinusoidal function, with the amplitude being measured at ESRF. The oscillation period is taken equal to the groove distance, i.e. 1 mm. The influence of such geometry on particle trajectory was evaluated through the application of a position-dependent centrifugal force. Intensity of scattering on nuclei and electrons was averaged over the distribution density in the channel in order to reliably estimate the dechanneling probability, i.e., the probability for a channeled particle to leave the channeling condition.\n\nFigure \\ref{fig:h8MC} shows the distribution of the horizontal deflection angle as a function of the horizontal incoming angle at four different penetration depths. As shown in Fig. \\ref{fig:h8MC}a, at the penetration depth of 250 $\\mu$m, the clear typical markers of a bent crystal are present, namely the channeling spot and the volume reflected particles. In fact, channeled particles can be distinguished due to the $+20$ $\\mu$rad deflection kick they received. On the other side, the particles that impinge the target with a direction tangent to the crystalline curved planes are deflected to the opposite direction, i.e., they undergo volume reflection (VR). In Fig. \\ref{fig:h8MC}b, at the penetration depth of 500 $\\mu$m, the channeled particles are deflected to the opposite angle and acquire a $-20$ $\\mu$rad kick. Particles that underwent the VR effect are reflected by the opposite curvature. Thus, a null average deflection is observed. Particles pertaining to the region at the left of the channeling spot are also reflected. Then, Fig. \\ref{fig:h8MC}c shows a shift of the channeling spot of $-35$ $\\mu$rad at the 750 $\\mu$m depth. Lastly, in Fig. \\ref{fig:h8MC}d, at the 1000 $\\mu$m depth, the general picture evolves towards a double fold distribution with a net overall null average deflection angle. The simulated pattern of the beam interacting with the whole CU is shown in Fig. \\ref{fig:h8exp}b. Figure \\ref{fig:h8exp}c shows the simulation convoluted with the detector angular resolution. The simulation is in good agreement with the experimental results.\n\nWith the same Monte Carlo code, the fraction of planar-channeled particles along the crystal length, i.e., the channeling efficiency, can be calculated. For a 400 GeV\/c proton beam entering the CU parallel to the crystalline planes the efficiency  is $(48 \\pm 1) \\%$. Such quantity can also be estimated with an analytical model usually adopted for uniformly-bent crystals \\cite{doi.org\/10.1140\/epjc\/s10052-014-2740-7}. By setting the bending radius equal to the $6.2$ m minimum curvature of the planes in the CU, the efficiency results to be $\\sim$ 60$\\%$ for a zero-divergence beam.\n\nThe chosen CU sample parameters fulfill the condition for an optimal undulator in the case of 15 GeV positrons. At these energies, with this CU, the possibility to use an electron beam instead of a positron beam has to be excluded because the dechanneling length is much smaller than the CU length \\cite{Scandale201370}. Thereby, only positrons can be used for such a test. In the next future, a positron beam in the energy range of 10-20 GeV will be available at FACET-SLAC. As done for the 400 GeV\/c proton beam, the expected efficiency for 15 GeV\/c positrons is higher than $40 \\%$, so that the CU described in this paper could be applied straight in that facility.\n\nA prediction regarding the electromagnetic radiation emitted by a 15 GeV positron beam passing through the CU sample was done through a specific Monte Carlo code \\cite{Baryshevsky201330}. Since the undulator radiation is expected in the $1$ MeV region, the Ter-Mikaelian density effect and the transition radiation were also taken into account. The beam divergence was set to zero. The radiation emission probability was averaged over a large number of particle trajectories and photon emission angles.\n\nThe distribution of the radiation emission probability generated by 15 GeV positrons interacting with the CU sample is shown in Fig. \\ref{fig:Rad}. The emission spectrum has a sharp peak at $\\sim$ 0.8 MeV. Because the channeling radiation (CR) is expected in the 100 MeV region \\cite{BaierKatkov}, the main CR and CU peaks present no overlap. In the energy range from 0.5 to 1.5 MeV the peak of the CU radiation is $\\sim$4 times higher than for CR.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1\\columnwidth]{Rad2.eps}\n\\caption{Simulated radiation emission probability for 15 GeV positrons interacting with the CU sample presented in the paper (solid line) and with a straight Si crystal having the same length along the beam under channeling condition (dashed line). The simulated spectrum emitted by 15 GeV positrons interacting with an amorphous Si target of the same length (dotted curve) is also reported.}\\label{fig:Rad}\n\\end{center}\n\\end{figure}\n\n\\section{Conclusions}\n\nA crystalline undulator to be worked out under the condition ${\\lambda}_u<{\\lambda}_c$ was manufactured through the grooving method at SSL (Ferrara, It). The sample was extensively characterized both morphologically and structurally. The surfaces were reconstructed through an interferometric measurement at SSL. The crystalline structure was analyzed through monochromatic hard X-rays diffraction at ESRF (Grenoble, Fr). The crystal was tested at the H8-SPS line of CERN. The expected planar channeling of 400 GeV\/c protons between the cristalline planes bent with the grooving method was observed. Monte Carlo simulations were worked out to evaluate the channeling efficiency for the 400 GeV proton beam and to predict the radiation spectrum emitted by 15 GeV positrons beam entering the crystalline undulator. The manufactured crystalline undulator is now ready to be tested with high intensity positrons beam at 10-20 GeV energy.\n\n\\section{Acknowledgements}\nWe are thankful to the CERN-SPS and ESRF coordination board for the possibility to use the H8-SPS external line at CERN and the ID15A line at ESRF. We acknowledge the partial support of INFN under the ICE-RAD experiment and the European CUTE Project. We acknowledge Gerald Klug and Eugen Eurich from Disco Europe (Munich, Germany) for their support in crystal manufacturing, Persiani Andrea and Manfredi Claudio of Perman (Loiano, Italy) for their support with crystal holders manufacturing.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Spin-3\/2 matrices}\\label{AppMat}\nThe spin-3\/2 matrices in their standard matrix representation read\n\\begin{align}\n \\label{mat1} J_x &= \\begin{pmatrix} 0 & \\frac{\\sqrt{3}}{2} & 0 & 0 \\\\ \\frac{\\sqrt{3}}{2} & 0 & 1 & 0 \\\\ 0 & 1 & 0 & \\frac{\\sqrt{3}}{2} \\\\ 0 & 0 & \\frac{\\sqrt{3}}{2} & 0 \\end{pmatrix},\\\\\n \\label{mat2} J_y &=\\begin{pmatrix} 0 & -\\rmi \\frac{\\sqrt{3}}{2} & 0 & 0 \\\\ \\rmi \\frac{\\sqrt{3}}{2} & 0 & -\\rmi &0 \\\\ 0 & \\rmi & 0 & -\\rmi \\frac{\\sqrt{3}}{2} \\\\ 0 & 0 & \\rmi \\frac{\\sqrt{3}}{2} & 0\\end{pmatrix},\\\\\n \\label{mat3} J_z &= \\begin{pmatrix} \\frac{3}{2} & 0 & 0 & 0 \\\\ 0 & \\frac{1}{2} & 0 & 0 \\\\ 0 & 0 & -\\frac{1}{2} & 0 \\\\ 0 & 0 & 0 & -\\frac{3}{2}\\end{pmatrix}.\n\\end{align}\nThe matrices satisfy $[J_i,J_j]=\\rmi \\vare_{ijk}J_k$ and $\\sum_i J_i^2 = \\frac{15}{4} \\mathbb{1}_4$, and all results obtained in the main text result from these relations. Some insight into the operators that appear in the analysis can be gained from applying the basis change\n\\begin{align}\n \\label{mat4} S = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 0 \\end{pmatrix}\n\\end{align}\nwith $S^{-1}=S ^T=S$. The spin-3\/2 matrices in this frame (denoted with an overbar) read\n\\begin{align}\n \\label{mat5} \\bar{J}_x &= SJ_x S = \\begin{pmatrix} 0 & 0 & 0 &\\frac{\\sqrt{3}}{2} \\\\ 0 & 0 & \\frac{\\sqrt{3}}{2} & 0 \\\\ 0 & \\frac{\\sqrt{3}}{2} & 0 & 1 \\\\ \\frac{\\sqrt{3}}{2} & 0 & 1 & 0 \\end{pmatrix},\\\\\n \\label{mat6} \\bar{J}_y &= SJ_y S = \\begin{pmatrix} 0 & 0 & 0 &-\\frac{\\sqrt{3}}{2}\\rmi \\\\ 0 & 0 & \\frac{\\sqrt{3}}{2}\\rmi & 0 \\\\ 0 & -\\frac{\\sqrt{3}}{2}\\rmi & 0 & \\rmi \\\\ \\frac{\\sqrt{3}}{2}\\rmi & 0 & -\\rmi & 0 \\end{pmatrix},\\\\\n \\label{mat7} \\bar{J}_z &= S J_z S = \\begin{pmatrix} \\frac{3}{2} & 0&0 & 0\\\\ 0& -\\frac{3}{2} &0 &0\\\\ 0& 0& -\\frac{1}{2} &0 \\\\ 0& 0& 0& \\frac{1}{2} \\end{pmatrix}.\n\\end{align}\nDefining the matrices $\\bar{V}_i$ and $\\bar{\\gamma}$ as in the main text with $J_i\\to \\bar{J}_i$ we find\n\\begin{align}\n  \\label{mat8} \\bar{V}_1 =\\mathbb{1}_2\\otimes \\sigma_1,\\ \\bar{V}_2 = -\\mathbb{1}_2\\otimes \\sigma_2,\\ \\bar{V}_3=\\mathbb{1}_2\\otimes \\sigma_3,\n\\end{align}\nor \n\\begin{align}\n\\label{mat9} \\bar{V}_i=\\mathbb{1}\\otimes \\sigma_i^*,\n\\end{align}\nThis clearly shows that the representation of the Clifford algebra that specifies the Hamiltonian for $\\alpha=0$ is of the ``first type''. Furthermore, the matrix $\\bar{\\gamma}_{45}$ that enters the time-reversal operator $\\bar{\\mathcal{T}}=\\bar{\\gamma}_{45}\\mathcal{K}$ reads\n\\begin{align}\n \\label{mat10} \\bar{\\gamma}_{45} = \\mathbb{1}_2 \\otimes \\sigma_2,\n\\end{align}\nwhereas we have\n\\begin{align}\n \\label{mat11} \\bar{\\mathcal{W}} = \\bar{\\gamma}_{12} = \\sigma_2\\otimes \\mathbb{1}_2.\n\\end{align}\n\n\n\n\n\n\n\\section{Effective Weyl Hamiltonian and monopole charge}\nWe first construct the effective $2\\times 2$ Weyl Hamiltonian at the nodes $\\textbf{p}_n$, $n=1,\\dots,4$, in the chiral topological semimetal phase with $\\alpha>0$. The two orthogonal zero modes of $H_{\\rm mf}(\\textbf{p}_1)$ read\n\\begin{align}\n \\label{weyl1} |0_1\\rangle = \\frac{1}{\\sqrt{6}}\\begin{pmatrix}\\rmi\\sqrt{3} \\\\ 1-\\rmi \\\\ 1 \\\\ 0 \\end{pmatrix},\\ |0_1'\\rangle=\\mathcal{T}|0_1\\rangle = \\frac{1}{\\sqrt{6}}\\begin{pmatrix} 0 \\\\ \\rmi \\\\ 1-\\rmi \\\\ \\sqrt{3} \\end{pmatrix},\n\\end{align}\nwith similar expressions for $|0_n\\rangle$ and $|0_n'\\rangle$. From this we construct the projected Hamiltonian for momenta $\\textbf{p}=\\textbf{p}_n+\\delta\\textbf{p}$ close to one of the nodes according to\n\\begin{align}\n\\nonumber  H_0^{(n)} &= \\begin{pmatrix} \\langle 0_n|H_{\\rm mf}(\\textbf{p}_n+\\delta \\textbf{p})|0_n\\rangle & \\langle 0_n|H_{\\rm mf}(\\textbf{p}_n+\\delta \\textbf{p})|0_n'\\rangle \\\\ \\langle 0_n'|H_{\\rm mf}(\\textbf{p}_n+\\delta \\textbf{p})|0_n\\rangle & \\langle 0_n'|H_{\\rm mf}(\\textbf{p}_n+\\delta \\textbf{p})|0_n'\\rangle  \\end{pmatrix}\\\\\n\\label{weyl2}  &=  \\begin{pmatrix} \\langle 0_n|H(\\delta\\textbf{p})|0_n\\rangle & \\langle 0_n|H(\\delta\\textbf{p})|0_n'\\rangle \\\\ \\langle 0_n'|H(\\delta\\textbf{p})|0_n\\rangle & \\langle 0_n'|H(\\delta\\textbf{p})|0_n'\\rangle  \\end{pmatrix}.\n\\end{align}\nNote that the $n$-dependence only results from the $n$-dependence of $|0_n\\rangle$ and $|0'_n\\rangle$ due to the linearity of the Hamiltonian. We arrive at $H_0^{(n)}=v^{(n)}_{ij}\\delta p_i\\sigma_j$ with\n\\begin{align}\n \\nonumber v^{(1)} &=\\frac{1+2\\alpha}{3}\\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} + \\frac{1-\\alpha}{3} \\begin{pmatrix} 0 & 1 & 1 \\\\ 1 & 0 & 1 \\\\ 1 & 1 & 0 \\end{pmatrix},\\\\\n \\nonumber v^{(2)} &=\\frac{1+2\\alpha}{3}\\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} + \\frac{1-\\alpha}{3} \\begin{pmatrix} 0 & 1 & -1 \\\\ 1 & 0 & -1 \\\\ -1 & -1 & 0 \\end{pmatrix},\\\\\n \\nonumber v^{(3)} &=\\frac{1+2\\alpha}{3}\\begin{pmatrix} -1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} + \\frac{1-\\alpha}{3} \\begin{pmatrix} 0 & 1 & 1 \\\\ 1 & 0 & -1 \\\\ -1 & 1 & 0 \\end{pmatrix},\\\\\n \\label{weyl3} v^{(4)} &=\\frac{1+2\\alpha}{3}\\begin{pmatrix} -1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} + \\frac{1-\\alpha}{3} \\begin{pmatrix} 0 & 1 & -1 \\\\ 1 & 0 & 1 \\\\ 1 & -1 & 0 \\end{pmatrix}.\n\\end{align}\nWe have \n\\begin{align}\n\\mbox{det}(v^{(n)})=\\alpha^2\n\\end{align}\nfor all $n=1,\\dots,4$. The resulting monopole charge at the node $\\textbf{p}_n$ is $q_n = \\sgn[\\mbox{det}(v^{(n)})]=+1$ for $\\alpha>0$.\n\nIn the Weyl semimetal phase with $m\\neq 0$ we consider $H_V(\\textbf{p}_n)$ with $n=\\text{a},\\text{b}$. Since the x- and y-components of the nodal points vanish, we have a diagonal mean-field Hamiltonian at the nodes, namely\n\\begin{align}\n \\label{weyl5} H_V(\\textbf{p}_{\\text{a}}) &= \\frac{2m\\alpha}{1+\\alpha}\\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 0 & -1 & 0 & 0  \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 0 \\end{pmatrix},\\\\ \\label{weyl6} H_V(\\textbf{p}_{\\text{b}}) &= \\frac{2m\\alpha}{1-\\alpha}\\begin{pmatrix} -1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix}.\n\\end{align}\nThe zero modes $|0_{n=\\text{a},\\text{b}}\\rangle, |0'_{n=\\text{a},\\text{b}}\\rangle=\\mathcal{T}|0_{n=\\text{a},\\text{b}}\\rangle$ immediately follow from this and we again define the projected Hamiltonian as in Eq. (\\ref{weyl2}). We find $H_0^{(n)}=\\tilde{v}^{(n)}_{ij}\\delta p_i\\sigma_j$ with\n\\begin{align}\n\\label{weyl7} \\tilde{v}^{(\\text{a})} &=\\frac{1}{2}\\begin{pmatrix} 0 & -(2-\\alpha) & 0 \\\\ -(2-\\alpha) & 0 & 0 \\\\ 0 & 0 & 2(1+\\alpha)\\end{pmatrix},\\\\\n \\label{weyl8} \\tilde{v}^{(\\text{b})} &= \\frac{1}{2}\\begin{pmatrix} 0 & 2+\\alpha & 0 \\\\ -(2+\\alpha) & 0 & 0 \\\\ 0 & 0 & -2(1-\\alpha)\\end{pmatrix}.\n\\end{align}\nWe have\n\\begin{align}\n \\label{weyl9} \\mbox{det}(\\tilde{v}^{(\\text{a})}) &= -\\frac{1}{4}(2-\\alpha)^2(1+\\alpha),\\\\\n \\label{weyl10} \\mbox{det}(\\tilde{v}^{(\\text{b})}) &= -\\frac{1}{4}(2+\\alpha)^2(1-\\alpha),\n\\end{align}\nso that the monopole charges are given by $q_{\\text{a}}=\\sgn[\\mbox{det}(\\tilde{v}^{(\\text{a})})]=-1$ and $q_{\\text{b}}=\\sgn[\\mbox{det}(\\tilde{v}^{(\\text{b})})]=\\sgn(\\alpha-1)$. \n\n\n\n\n\n\n\\section{Details of the renormalization group analysis}\n\n\n\\subsection{Perturbative propagator}\nIn order to determine the perturbative propagator $G_0(Q)$ we need to invert\n\\begin{align}\nG_0^{-1}(Q) = A=   \\rmi q_0\\mathbb{1}_4  + q_i (V_i + \\alpha U_i)\n\\end{align}\nwith frequency $q_0$. For arbitrary $\\alpha$ this can be achieved with the help of the Cayley--Hamiltonian theorem which implies that the inverse of the $4\\times 4$ matrix $G_0^{-1}$ is given by\n\\begin{align}\n \\nonumber G_0(Q) ={}& \\frac{1}{\\mbox{det}(A)}\\Biggl[\\frac{1}{6}\\Bigl([\\mbox{tr} A]^3-3 (\\mbox{tr}A)\\mbox{tr}(A^2)+2\\mbox{tr}(A^3)\\Bigr)\\mathbb{1}_4\\\\\n &-\\frac{1}{2}\\Bigl([\\mbox{tr}A]^2-\\mbox{tr}(A^2)\\Bigr)A+(\\mbox{tr}A)A^2-A^3\\Biggr].\n\\end{align}\nThe electric charge enters the RG beta functions only through the combination \n\\begin{align}\ng_1'=g_1+\\frac{e^2}{2}.\n\\end{align}\nThis is related to the fact that $G_0(Q)$ satisfies\n\\begin{align}\n\\label{intGG} \\int_{q_0,\\Omega} G_0(Q)^2 =0,\n\\end{align}\nwhere $\\int_{q_0}$ and $\\int_{\\Omega}$ denote the frequency and angular integration, respectively. Indeed, given Eq. (\\ref{intGG}), the same reasoning as in Eqs. (A101)-(A104) of Ref \\cite{PhysRevB.95.075149} can be applied to show this feature.\n\n\n\n\n\n\n\n\n\n\\subsection{Short-range interactions}\\label{AppShort}\n\nThe RG flow of the couplings $\\bar{g}_i$ is determined by the same procedure as laid out for Luttinger semimetals in Appendix 5 of Ref. \\cite{PhysRevB.95.075149}. We confine our analysis to local point-like interaction terms. To incorporate the most general four-fermion interaction we write the interaction part of the Lagrangian as\n\\begin{align}\n\\label{model6b} L_{\\rm short}=\\sum_{A=1}^{16} \\bar{g}_A (\\psi^\\dagger \\Sigma^A\\psi)^2,\n\\end{align}\nwhere $\\Sigma^A$ constitutes an $\\mathbb{R}$-basis of Hermitean $4\\times 4$ matrices satisfying $\\mbox{tr}(\\Sigma^A\\Sigma^B)=4\\delta^{AB}$. The symmetry properties of $H$ dictate which of the 16 entries of $\\{\\Sigma^A\\}$ are independent under RG. \n\n\n\nIn the rotation invariant case (i.e. for $\\alpha=2$) we have\n\\begin{align}\n \\nonumber  L_{\\rm short}^{(\\rm rot)} ={}& \\bar{g}_1(\\psi^\\dagger\\psi)^2 + \\bar{g}_2(\\psi\\gamma_a\\psi)^2 \\\\\n \\label{model7} &+ \\bar{g}_\\mathJ(\\psi^\\dagger \\mathJ_i \\psi)^2+\\bar{g}_W(\\psi^\\dagger W_\\mu \\psi)^2,\n\\end{align}\nwhere $\\mathJ_i$, $\\gamma_a$,\\ $W_\\mu$ are the three, five, and seven components of the irreducible $\\text{SO}(3)$-invariant first-, second-, third-rank tensors made from products of the $J_i$. They read\n\\begin{align}\n \\label{model8} \\mathJ_i &= \\frac{2}{\\sqrt{5}}J_i,\n\\end{align}\nand\n\\begin{align}\n \\label{model9} \\gamma_1 &= \\frac{1}{\\sqrt{3}}(J_x^2-J_y^2),\\  \\gamma_2 = J_z^2-\\frac{5}{4}\\mathbb{1}_4,\\\\\n \\label{model11} \\gamma_3 &= \\frac{1}{\\sqrt{3}}\\{J_x,J_z\\},\\  \\gamma_4 = \\frac{1}{\\sqrt{3}}\\{J_y,J_z\\},\\\\\n \\label{model13}  \\gamma_5 &=\\frac{1}{\\sqrt{3}}\\{J_x,J_y\\},\n\\end{align}\nand\n\\begin{align}\n \\label{model14} W_1 &= \\frac{2\\sqrt{5}}{3}\\Bigl(J_{x}^3-\\frac{41}{20}J_{x}\\Bigr),\\\\\n \\label{model15} W_2 &= \\frac{2\\sqrt{5}}{3}\\Bigl(J_{y}^3-\\frac{41}{20}J_{y}\\Bigr),\\\\\n \\label{model16} W_3 &= \\frac{2\\sqrt{5}}{3}\\Bigl(J_{z}^3-\\frac{41}{20}J_{z}\\Bigr),\\\\\n \\label{model17} W_4 &= \\frac{1}{\\sqrt{3}}\\{J_x,(J_y^2-J_z^2)\\},\\\\\n \\label{model18}  W_5 &= \\frac{1}{\\sqrt{3}}\\{J_y,(J_z^2-J_x^2)\\},\\\\\n \\label{model19} W_6 &= \\frac{1}{\\sqrt{3}} \\{J_z,(J_x^2-J_y^2)\\},\\\\\n \\label{model20}  W_7 &= \\frac{2}{\\sqrt{3}}(J_xJ_yJ_z+J_zJ_yJ_x).\n\\end{align}\nNote that \n\\begin{align}\n\\mathcal{W}=W_7.\n\\end{align}\nThe matrices $\\gamma_a$ are chosen such that $\\gamma_{1,2,3}$ are real and $\\gamma_{4,5}$ are imaginary. For a very detailed discussion of the decomposition of interaction vertices in Eq. (\\ref{model7}), also in the cubic symmetric case, we refer to Ref. \\cite{PhysRevB.95.075149}, where an identical notation was used to study systems described by a $4\\times 4$ quadratic band touching Hamiltonian. Obviously the momentum dependence of $H$ does not affect the form of $L_{\\rm short}$ and so all observations made in the mentioned reference are valid here as well.\n\n\n\n\nFor $\\alpha\\neq 2$ the system is invariant under the rotational cubic group $\\text{O}$ only, and the irreducible tensors under $\\text{SO}(3)$ need to be subdivided into irreducible representations of $O$. We write\n\\begin{align}\n \\label{model21} \\vec{E}=\\begin{pmatrix} \\gamma_1 \\\\ \\gamma_2 \\end{pmatrix},\\ \\vec{T}= \\begin{pmatrix} \\gamma_3 \\\\ \\gamma_4 \\\\ \\gamma_5 \\end{pmatrix},\\ \\vec{W}=\\begin{pmatrix} W_1 \\\\ W_2 \\\\ W_3 \\end{pmatrix},\\ \\vec{W}' = \\begin{pmatrix} W_4 \\\\ W_5 \\\\ W_6\\end{pmatrix}.\n\\end{align}\nThe set $\\{\\mathbb{1},E_a,T_a,\\mathJ_i,W_i,W'_i,W_7\\}$ constitutes an appropriate basis of interactions in the cubic case. However, we are free to replace the elements $\\mathJ_i$ and $W_i$ by\n\\begin{align}\n \\label{model22} \\vec{V} &= \\frac{1}{\\sqrt{5}}(\\vec{\\mathcal{J}}+2\\vec{W}),\\\\\n \\label{model23} \\vec{U} &= \\frac{1}{\\sqrt{5}}(2\\vec{\\mathcal{J}}-\\vec{W}),\n\\end{align}\nwhich are the same matrices $\\vec{V}$ and $\\vec{U}$ as they appear in $H$. The set of matrices\n\\begin{align}\n \\label{model24} \\{\\Sigma^A\\} = \\{ \\mathbb{1},\\ E_a,\\ T_a,\\ V_i,\\ U_i,\\ W'_i, W_7\\}\n\\end{align}\nthen comprises a computationally advantageous orthogonal $\\mathbb{R}$-basis of Hermitean $4\\times 4$ matrices with $\\mbox{tr}(\\Sigma^A\\Sigma^B) = 4\\delta_{Ab}$. The seven elements in Eq. (\\ref{model24}) allow us to construct seven distinct insulating ordering channels, namely\n\\begin{align}\n L_1 &=\\bar{g}_1 (\\psi^\\dagger \\psi)^2,\\\\\n L_2 &=\\bar{g}_2 (\\psi^\\dagger\\vec{E}\\psi)^2,\\\\\n L_3 &=\\bar{g}_3 (\\psi^\\dagger\\vec{T}\\psi)^2,\\\\\n L_4&=\\bar{g}_4 (\\psi^\\dagger\\vec{V}\\psi)^2,\\\\\n L_5 &=\\bar{g}_5 (\\psi^\\dagger\\vec{U}\\psi)^2,\\\\\n L_6 &=\\bar{g}_6 (\\psi^\\dagger\\vec{W}^\\prime\\psi)^2,\\\\\n L_7 &=\\bar{g}_7 (\\psi^\\dagger W_7\\psi)^2.\n\\end{align}\nOnly three of these expressions are linearly independent, and we choose to parametrize the interaction in $L_{\\rm int}$ by $L_{1,2,3}$. The remaining four are related to these by the Fierz identities\n\\begin{align}\n L_4 &= -\\frac{3}{2}L_1-\\frac{3}{2}L_2+\\frac{1}{2}L_3,\\\\\n L_5 &= -\\frac{3}{2}L_1 -\\frac{1}{2}L_3,\\\\\n L_6 &= -\\frac{3}{2}L_1-\\frac{1}{2}L_3,\\\\\n L_7 &= -\\frac{1}{2}L_1 +\\frac{1}{2}L_2-\\frac{1}{2}L_3.\n\\end{align}\nNote that $L_5=L_6$. In the rotationally invariant case we have $\\bar{g}_2=\\bar{g}_3$. Each individual term $L_1,\\dots,L_7$ is invariant under $\\psi \\to \\mathcal{W}\\psi$ since we have the (anti)commutation relations\n\\begin{align}\n [\\mathcal{W},V_i] &= [\\mathcal{W},T_i] = 0,\\\\\n \\{\\mathcal{W},U_i\\} &=\\{\\mathcal{W},E_a\\} = \\{\\mathcal{W},W'_i\\}=0.\n\\end{align} \nConsequently, the invariance of $L_{\\rm short}$ under this transformation is independent of the choice of Fierz basis.\n\n\n\n\nWe may alternatively express $L_{\\rm short}$ in terms of the superconducting channels of the system. The number of such terms is identical to the number of Fierz-inequivalent insulating channels. We have\n\\begin{align}\n L_{\\rm short} = \\bar{g}_{\\rm s} L_{\\rm s} + \\bar{g}_{\\rm d,E}L_{\\rm d,E} + \\bar{g}_{\\rm d,T} L_{\\rm d,T}\n\\end{align}\nwith\n\\begin{align}\n L_{\\rm s} ={}& (\\psi^{\\dagger} \\gamma_{45}\\psi^*)(\\psi^{\\rm T}\\gamma_{45}\\psi)\\\\\n L_{\\rm d,E} &= \\sum_{a=1}^2(\\psi^{\\dagger} \\gamma_a\\gamma_{45}\\psi^*)(\\psi^{\\rm T}\\gamma_{45}\\gamma_a\\psi)\\\\\n  L_{\\rm d,T} &=\\sum_{a=3}^5(\\psi^{\\dagger} \\gamma_a\\gamma_{45}\\psi^*)(\\psi^{\\rm T}\\gamma_{45}\\gamma_a\\psi),\n\\end{align} \nwith the linear relation \\cite{PhysRevB.93.205138,PhysRevB.95.075149}\n\\begin{align*}\n L_{\\rm s} &= \\frac{1}{4}(L_1+2L_2+3L_3),\\\\\n L_{\\rm d,E} &= \\frac{1}{4}(L_1-3L_3),\\\\\n L_{\\rm d,T} &=\\frac{1}{4}(L_1-2L_2-L_3).\n\\end{align*}\nIn the rotationally symmetric case ($\\alpha=2$) we have $g_{\\rm d,E}=g_{\\rm d,T}$, again reducing the number of independent couplings to two.\n\n\n\n\\subsection{Susceptibility exponents and admixture $\\kappa(\\alpha)$}\\label{AppSusc}\n\n\nThe susceptibility exponents determine which order parameter condenses at the quantum critical point described by a certain RG fixed point. We define the susceptibility exponent $\\eta=\\eta_M$ of a fermion bilinear $\\psi^\\dagger M \\psi$ or $\\psi^\\dagger M \\psi^*$ through the scaling dimension\n\\begin{align}\n\\label{susc1} [h_M] = z +\\eta_M,\n\\end{align}\nwhere $h=h_M$ is introduced by coupling a term $L_M = h_M (\\psi^\\dagger M \\psi^{(*)})$ to the Lagrangian. For a detailed discussion which fully applies here see App. 6 of Ref. \\cite{PhysRevB.95.075149}. In our case $z=1$ is the trivial dynamic critical exponent due to $e=0$ at the fixed point. Importantly, for each fixed point we have to test every ordering channel individually and determine the one with the largest susceptibility. On the other hand, due to cubic symmetry, we can restrict to the ten distinct cubic channels. For this purpose we couple\n\\begin{align}\n  L_{\\mathbb{1}} &= h_{\\mathbb{1}} (\\psi^\\dagger \\psi),\\\\\n L_{E} &= h_{E} (\\psi^\\dagger \\gamma_1\\psi),\\\\\n  L_{T} &= h_{T} (\\psi^\\dagger \\gamma_3\\psi),\\\\\n  L_{V} &= h_{V} (\\psi^\\dagger V_1\\psi),\\\\\n  L_{U} &= h_{U} (\\psi^\\dagger U_1\\psi),\\\\\n L_{W'} &= h_{W'} (\\psi^\\dagger W_4\\psi),\\\\\n  L_{\\mathcal{W}} &= h_{\\mathcal{W}} (\\psi^\\dagger W_7\\psi),\\\\\n L_{\\rm s} &= h_{\\rm s} (\\psi^\\dagger \\gamma_{45}\\psi^*),\\\\\n  L_{\\rm d,E} &= h_{\\rm d,E} (\\psi^\\dagger \\gamma_1 \\gamma_{45}\\psi^*),\\\\ \nL_{\\rm d,T} &= h_{\\rm d,T} (\\psi^\\dagger \\gamma_3\\gamma_{45}\\psi^*)\n\\end{align}\nto the Lagrangian and determine the corresponding flow equations $\\dot{h}_M=(z+\\eta_M)h_M$ to read off the susceptibilities. \n\nTo give some example we present the susceptibilities for the analytically tractable cases $\\alpha=0$ and $\\alpha=2$. For $\\alpha=0$ we have\n\\begin{align}\n\\nonumber \\eta_{\\mathbb{1}}&=\\eta_{E} =\\eta_{\\mathcal{W}}=0,\\ \\eta_T = \\frac{2}{3}(g_1'-2g_2-5g_3),\\\\\n\\nonumber  \\eta_V& =\\frac{2}{3}(g_1'+2g_2-g_3),\\ \\eta_U= \\eta_{W'}=\\frac{2}{3}(g_1'+g_3),\\\\\n\\nonumber \\eta_{\\rm s} &= -(g_1'+2g_2+3g_3),\\ \\eta_{\\rm d,E} =-(g_1'-3g_3),\\\\\n \\label{susc2} \\eta_{\\rm d,T} &= \\frac{1}{3}(-g_1'+2g_2+g_3).\n\\end{align}\nFor the rotation invariant case with $\\alpha=2$ we have\n\\begin{align}\n \\nonumber \\eta_{\\mathbb{1}} &=0,\\ \\eta_E = \\frac{1}{5}(g_1'-4g_2-3g_3),\\\\ \n \\nonumber \\eta_T &= \\frac{1}{5}(g_1'-2g_2-5g_3),\\ \\eta_{W'} = \\frac{43}{105}(g_1'+g_3)\\\\\n \\nonumber \\eta_{\\mathcal{W}} &= \\frac{43}{105}(g_1'-2g_2+3g_3),\\ \\eta_{\\rm s} = -\\frac{2}{3}(g_1'+2g_2+3g_3),\\\\\n \\label{susc3}\\eta_{\\rm d,E} &=-\\frac{1}{3}g_1'+g_3,\\ \\eta_{\\rm d,T} = -\\frac{1}{3}(g_1'-2g_2-g_3).\n\\end{align}\nWe omitted $\\eta_V$ and $\\eta_U$ here for a reason that will be explained in the next paragraph. Further imposing rotation invariance onto the couplings by setting $g_2=g_3$ we obtain\n\\begin{align}\n \\nonumber \\eta_{\\mathbb{1}} &=0,\\ \\eta_{\\gamma_a} = \\eta_E = \\eta_T = \\frac{1}{5}(g_1'-7g_2),\\\\\n \\nonumber \\eta_{\\mathcal{J}} &= \\frac{4}{15}(g_1'+g_2),\\ \\eta_W =\\eta_{W'}=\\eta_{\\mathcal{W}} = \\frac{43}{105}(g_1'+g_2),\\\\\n \\label{susc4} \\eta_{\\rm s} &= -\\frac{2}{3}(g_1'+5g_2),\\ \\eta_{\\rm d}=\\eta_{\\rm d,E}=\\eta_{\\rm d,T} = -\\frac{1}{3}g_1'+g_2.\n\\end{align}\nAs expected, the susceptibilities of different components of same-rank tensors in Eq. (\\ref{model6b}) coincide in the rotation invariant limit.\n\n\nIn most cases, by coupling a term $h_M(\\psi^\\dagger M \\psi)$ to the Lagrangian while setting $h_N=0$ for all other matrices $N\\neq M$, we only generate a running of the coupling $h_M$. However, if there is a cubic transformation that relates $M$ and $N$, this is no longer true. In our case, coupling a term $h_V (\\psi^\\dagger V_i\\psi)$ to the Lagrangian generates a term $h_U (\\psi^\\dagger U_i\\psi)$, and vice versa. Referring once more to Ref. \\cite{PhysRevB.95.075149}, Eqs. (A110)-(A114) for details, we briefly review here how to determine the correct scaling behavior. \n\n\n\n\n\n\nWe write\n\\begin{align}\n \\nonumber h(\\psi^\\dagger V_i\\psi) \\Rightarrow  h(\\psi^\\dagger V_i\\psi)+h\\Bigl[\\eta_V (\\psi^\\dagger V_i\\psi) +a (\\psi^\\dagger U_i\\psi)\\Bigr],\\\\\n h(\\psi^\\dagger _iU\\psi) \\Rightarrow h(\\psi^\\dagger U_i\\psi)+ h\\Bigl[b (\\psi^\\dagger V_i\\psi) +\\eta_U (\\psi^\\dagger U_i\\psi)\\Bigr],\n\\end{align}\nwhere ``$\\Rightarrow$'' stands for ``generates a term under RG''. In general, $a$ and $b$ depend on $\\alpha$ and the couplings $g_i$. We find that for all values of $\\alpha$ we have\n\\begin{align}\n a &= -K(g_1'+g_3),\\\\\n b&= -K(g_1'+2g_2-g_3)\n\\end{align}\nwith $K=K(\\alpha)>0$ a positive constant. Some values are:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline $\\alpha$ & 0 & 0.5 & 1 & 1.5 & 2 & 2.296 & 2.5 & 5 \\\\ \n\\hline $K$ & 0 & 0.12 & 0.25 & 0.106 & 0.057 & 0.0425 & 0.035 & 0.0077  \\\\ \n\\hline \n\\end{tabular} \n\\end{center}\nWe observe that $K$ is small and vanishes for $\\alpha\\to0$ and $\\alpha\\to \\infty$, with a maximum around $\\alpha \\approx 1$. \n\n\nWe introduce $M_i = c V_i + c' U_i$ with real coefficients such that $c^2+c'{}^2=1$. Obviously, only the ratio $\\kappa=c'\/c$ is of relevance. The maximal (and also the minimal) susceptibility $\\eta_M$ will then come from a linear combination that satisfies the self-consistent relation\n\\begin{align}\n \\label{susc5} h(\\psi^\\dagger M_i\\psi) \\Rightarrow h(1+\\eta_{M})(\\psi^\\dagger M_i \\psi)\n\\end{align}\nwith $\\eta_{M} =\\eta_V + \\frac{c'}{c}b$ and\n\\begin{align}\n 0 &\\stackrel{!}{=} c'(\\eta_U-\\eta_V) +\\frac{1}{c}\\Bigl(c^2 a - c'{}^2 b\\Bigr).\n\\end{align}\nTe last equation is solved by\n\\begin{align}\n \\kappa &= \\frac{c'}{c} = \\frac{\\eta_U-\\eta_V}{2b}\\pm\\sqrt{\\frac{(\\eta_U-\\eta_V)^2}{4b^2}+\\frac{a}{b}},\\\\\n \\eta_{M} &=\\frac{\\eta_V+\\eta_U}{2} \\pm \\frac{1}{2}\\sqrt{(\\eta_U-\\eta_V)^2+4ab}.\n\\end{align}\nSince we are after the largest susceptibilities, we are interested in the ``$+$'' solution of $\\eta_{M}$. This corresponds to choosing ``$-$'' in $\\kappa$ at fixed point V, and ``$+$'' in $\\kappa$ at fixed point W. For V we have $|\\kappa| < 0.05$ with the largest value around $\\alpha\\sim 1.25$. Consequently, up to a tiny correction we have\n\\begin{align*}\n M_i \\approx V_i\n\\end{align*}\nat fixed point V. In contrast, at W we find that $\\kappa\\gg1$ (of order 10) for all $\\alpha$, and so we can regard this case as\n\\begin{align*}\n M_i \\approx U_i.\n\\end{align*} \nThe exponent $\\eta_{M}$ at W is larger than $\\eta_{\\mathcal{W}}=3$ for $\\alpha \\leq 0.7$. \n\n\n\\begin{figure}[t!]\n\\centering\n\\begin{minipage}{0.48\\textwidth}\n\\includegraphics[width=8cm]{FigureSM1}\n\\includegraphics[width=8cm]{FigureSM2}\n\\caption{The ratio $\\kappa=c'\/c$ that maximizes $\\eta_M$ for the bilinear $\\psi^\\dagger(cV_i+c'U_i)\\psi$ at the fixed point V (upper panel) and W (lower panel). \\emph{Upper panel.} At V we have $|\\kappa|<5\\%$ and so the order parameter is to a very good approximation given by $\\langle \\psi^\\dagger V_i\\psi\\rangle$. \\emph{Lower panel.} At W the ratio $\\kappa\\gg1$ is large and so the corresponding order parameter is approximately $\\langle \\psi^\\dagger U_i \\psi\\rangle$. However, for $\\alpha\\geq 0.7$, the leading instability at W is the condensation of $\\chi=\\langle \\psi^\\dagger \\mathcal{W}\\psi\\rangle$ since $\\eta_{\\mathcal{W}}=3$ is the larger susceptibility.}\n\\label{Figkappa}\n\\end{minipage}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Flow equations at high-symmetry points}\n\n\n\n\\noindent The flow equations for $\\alpha=0$ read\n\\begin{align}\n \\label{rg3} \\dot{g}_1 &= -2g_1 -g_2^2-6g_2g_3-5g_3^2,\\\\\n \\label{rg4} \\dot{g}_2 &=-2g_2+g_2^2-2g_2g_3-3g_3^2,\\\\\n \\label{rg5} \\dot{g}_3 &= -2g_3 -\\frac{5}{3}g_2^2-\\frac{14}{3}g_2g_3-3g_3^2.\n\\end{align}\nNote that, apart from the trivial term $-2g_1$ in the first line, $g_1$ and $e$ are absent in these equations. We find the quantum critical points SC and V given by\n\\begin{align}\n \\label{rg6} \\text{SC}:\\ (g_1,g_2,g_3)_\\star &= \\Bigl(-\\frac{7}{16},-\\frac{3}{16},-\\frac{5}{16}\\Bigr),\\\\\n \\label{rg7} \\text{V}:\\ (g_1,g_2,g_3)_\\star &= \\Bigl(\\frac{1}{2},\\frac{3}{2},-\\frac{1}{2}\\Bigr).\n\\end{align} \nThe largest susceptibility exponents at SC and V are $\\eta_{\\rm s}=7\/4=1.75$ and  $\\eta_V=8\/3=2.67$, respectively.\n\n\n\n\n\nIn the rotation invariant case ($\\alpha=2$) we have\n\\begin{align}\n\\nonumber  \\dot{g}_1 ={}&-2g_1 -\\frac{2}{15}g_1'g_2 -\\frac{1}{5}g_1'g_3 -\\frac{76}{105}g_2^2 \\\\\n \\label{rg8} &- \\frac{164}{35} g_2g_3 -\\frac{79}{35}g_3^2,\\\\\n \\nonumber \\dot{g}_2={}& -2g_2 +\\frac{12}{35}g_1'g_2-\\frac{19}{35}g_1'g_3-\\frac{4}{15}g_2^2\\\\\n \\label{rg9} &-\\frac{58}{35}g_2g_3-\\frac{15}{7}g_3^2,\\\\\n \\nonumber \\dot{g}_3 ={}& -2g_3 -\\frac{38}{105}g_1'g_2+\\frac{17}{105}g_1'g_3 -\\frac{24}{35}g_2^2\\\\\n \\label{rg10} &-\\frac{272}{105}g_2g_3-\\frac{83}{105}g_3^2.\n\\end{align}\nThe quantum critical points are given by \n\\begin{align}\n \\label{rg11} \\text{SC}:\\ (g_1,\\ g_2,\\ g_3)_\\star &= (-0.841,\\ -0.450,\\ -0.450),\\\\\n \\label{rg12} \\text{W}:\\ (g_1,\\ g_2,\\ g_3)_\\star &= (1.22,\\ -1.22,\\ 1.22),\\\\\n \\label{rg13} \\text{V}:\\ (g_1,\\ g_2,\\ g_3)_\\star &= (3.31,\\ 3.07,\\ -1.20).\n\\end{align}\nthe leading susceptibility exponents are given by $\\eta_{\\rm s}=2.062$,\\ $\\eta_{\\mathcal{W}}=3$ and $\\eta_{V_i}= 4.06$, respectively. Note that the fixed point W satisfies $g_1=-g_2=g_3$ and $\\eta_{\\mathcal{W}}=d$. For $g_2=g_3$ the flow equations read\n\\begin{align}\n \\label{rg14} \\dot{g}_1 &= -2g_1 -\\frac{1}{3}g_1'g_2-\\frac{23}{3}g_2^2,\\\\\n \\label{rg15} \\dot{g}_2 &= -2g_2 -\\frac{1}{5}g_1'g_2-\\frac{61}{15}g_2^2.\n\\end{align}\nThis set of equations only supports the superconducting quantum critical point SC. We should think of the fixed points W and V in this limit as approached in an anisotropic system with $\\alpha\\to 2$ from above or from below.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Cubic and tetrahedral symmetry group}\\label{AppCubic}\nIn this appendix we summarize some properties of the rotational cubic point group which are relevant for our analysis. We begin by deriving explicit expressions for the group elements and then discuss the rotational or chiral tetrahedral point group. \n\n\nThe cubic point group $O_h$ consists of the transformations that leave a three-dimensional cube invariant. Clearly, inversion is such a symmetry transformation. All other elements of the group can be expressed as a rotation or a rotation combined with an inversion. The ``rotational'' subgroup $O$, which is the symmetry group of our problem at hand, consists of the true rotations. Whereas $O_h$ consists of 48 elements, there are 24 elements in $O$. \n\nIn the following we consider the three-dimensional representation of the group acting on position or momentum space vectors $\\textbf{x}=(x_1,x_2,x_3)^T$ or $\\textbf{p}=(p_1,p_2,p_3)^T$, respectively. The matrices $J_i$ also transform like a vector under $O$. The group $O$ is a subgroup of $\\text{SO}(3)$, and so every element $R\\in O$ satisfies $R^TR=\\mathbb{1}_3$ and $\\mbox{det}(R)=1$. Every rotation can be specified by an axis $\\textbf{n}=(n_1,n_2,n_3)^T$ and a rotation angle $\\vphi$ according to\n\\begin{align}\n \\label{cub1} R(\\textbf{n},\\vphi) = \\mathbb{1}_3 + \\sin\\vphi K + (1-\\cos \\vphi) K^2\n\\end{align}\nwith\n\\begin{align}\n\\label{cub2} K = \\begin{pmatrix} 0 & -n_z & n_y \\\\ n_z & 0 & -n_x \\\\ -n_y & n_x & 0 \\end{pmatrix}.\n\\end{align}\nThe inverse of $R(\\textbf{n},\\vphi)$ is $R(-\\textbf{n},\\vphi)$, and the self-inverse elements of $O$ are precisely the ones represented by symmetric matrices, which corresponds to rotations by $0^{o}$ or $180^{o}$. If $R$ is an element of $O$, then $IR$ with inversion $I=\\text{diag}(-1,-1,-1)$ is the corresponding element of $O_h$ that includes an inversion.\n\n\n\n\\begin{widetext}\nWe now summarize the 24 group elements of $O$. The unit element is $R_1=\\mathbb{1}_3$. We further have the following symmetry operations:\\\\\n\\noindent{\\emph{(i) Three rotations by $180^o$ about a 4-fold axis.}} The corresponding rotation axes are the $x,y,z$-axes connecting opposite faces given by $\\textbf{n}=(1,0,0)^T, (0,1,0)^T, (0,0,1)^T$, which leads to the self-inverse group elements\n\\begin{align}\n \\label{cub3} R_2 = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & -1 \\end{pmatrix},\\ R_3 = \\begin{pmatrix} -1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & -1 \\end{pmatrix},\\ R_4 = \\begin{pmatrix} -1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix}.\n\\end{align}\n\\emph{(ii) Eight rotations by $120^o$ about a 3-fold axis.} These 3-fold axes are the axes connecting opposite vertices. The sign of $\\textbf{n}$ matters and we have\n\\begin{align} \n \\label{cub4} \\textbf{n}=\\frac{1}{\\sqrt{3}}\\begin{pmatrix} 1 \\\\ 1 \\\\ 1 \\end{pmatrix},\\ \\frac{1}{\\sqrt{3}}\\begin{pmatrix} 1 \\\\ -1 \\\\ 1 \\end{pmatrix},\\ \\frac{1}{\\sqrt{3}}\\begin{pmatrix} -1 \\\\ 1 \\\\ 1 \\end{pmatrix},\\ \\frac{1}{\\sqrt{3}}\\begin{pmatrix} -1 \\\\ -1 \\\\ 1 \\end{pmatrix},\n\\end{align} \nwhich leads to\n\\begin{align}\n \\label{cub5} R_5 = \\begin{pmatrix} 0 & 0 & 1 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\end{pmatrix},\\  R_{6} = \\begin{pmatrix} 0 & -1 & 0 \\\\ 0 & 0 & -1 \\\\ 1 & 0 & 0 \\end{pmatrix},\\ R_{7} = \\begin{pmatrix} 0 & -1 & 0 \\\\ 0 & 0 & 1 \\\\ -1 & 0 & 0 \\end{pmatrix},\\ R_{8}=\\begin{pmatrix} 0 & 0 & -1 \\\\ 1 & 0 & 0 \\\\ 0 & -1 & 0 \\end{pmatrix}.\n\\end{align}\nThe inverse elements follow from $\\textbf{n}\\to -\\textbf{n}$ and read\n\\begin{align}\n \\label{cub6} R_{9}=R_{5}^T,\\ R_{10}=R_{6}^T,\\ R_{11}=R_{7}^T,\\ R_{12}=R_{8}^T.\n\\end{align}\n\\emph{(iii) Six rotations by $90^o$ about a 4-fold axis.} These 4-fold axes are again the $x,y,z$-axes connecting opposite faces, but this time the sign of $\\textbf{n}$ matters. We have $\\textbf{n}=(1,0,0)^T, (0,1,0)^T, (0,0,1)^T$, which leads to\n\\begin{align}\n \\label{cub7} R_{13} &= \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & -1 \\\\ 0 & 1 & 0 \\end{pmatrix},\\  R_{14} = \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ -1 & 0 & 0 \\end{pmatrix},\\ R_{15}= \\begin{pmatrix} 0 & -1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix},\n\\end{align}\nand their inverses with $\\textbf{n}\\to -\\textbf{n}$ and\n\\begin{align}\n \\label{cub8} R_{16}=R_{13}^T,\\ R_{17}=R_{14}^T,\\ R_{18}=R_{15}^T.\n\\end{align}\n\\emph{(iv) Six rotations by $180^o$ about a 2-fold axis.} The axes are the axes connecting opposite edges.  The matrices are their own inverses and the sign of $\\textbf{n}$ does not matter. We have\n\\begin{align}\n \\label{cub9} \\textbf{n} = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\\\ 0 \\end{pmatrix},\\ \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ -1 \\\\ 0 \\end{pmatrix},\\ \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 0 \\\\ 1 \\end{pmatrix},\\ \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 0 \\\\ -1 \\end{pmatrix},\\ \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 \\\\ 1 \\\\ 1 \\end{pmatrix},\\ \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 \\\\ 1 \\\\ -1 \\end{pmatrix},\n\\end{align}\nwhich leads to\n\\begin{align}\n \\label{cub10} R_{19} &= \\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & -1 \\end{pmatrix},\\ R_{20} = \\begin{pmatrix} 0 & -1 & 0 \\\\ -1 & 0 & 0 \\\\ 0 & 0 & -1 \\end{pmatrix},\\ R_{21} = \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & -1 & 0 \\\\ 1 & 0 & 0 \\end{pmatrix},\\\\\n \\label{cub11} R_{22} &= \\begin{pmatrix} 0 & 0 & -1 \\\\ 0 & -1 & 0 \\\\ -1 & 0 & 0 \\end{pmatrix},\\ R_{23} = \\begin{pmatrix} -1 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix},\\ R_{24} = \\begin{pmatrix} -1 & 0 & 0 \\\\ 0 & 0 & -1 \\\\ 0 & -1 & 0 \\end{pmatrix}.\n\\end{align}\n\\end{widetext}\n\nThe largest subgroup of $O$ is the rotational tetrahedral group $T$ with 12 elements. It consists of the rotations that leave a tetrahedron invariant. (Note that the tetrahedral group $T_d$ is commonly defined such that it includes inversion as well, and so it has 24 elements.) The elements of $T$ are precisely the first twelve $R_{1,\\dots,12}$ in our notation. The group $T$ leaves the expressions $J_1J_2J_3+J_3J_2J_1$ and $p_1p_2p_3$ invariant. In contrast, the remaining twelve elements of $O$ change the sign of these expressions.\n\n\n\n\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe technique of controlling various operators by so-called sparse operators has proven to be a very useful tool to obtain (sharp) weighted norm inequalities in the past decade.\n The key feature in this approach is that a typically signed and non-local operator is dominated, either in norm, pointwise or in dual form, by a positive and local expression.\n\n  The sparse domination technique comes from Lerner's work towards an alternative proof of the $A_2$-theorem, which was first proven by Hyt\\\"onen in \\cite{Hy12}. In \\cite{Le13a} Lerner applied his local mean oscillation decomposition approach to the $A_2$-theorem, estimating the norm of a Calder\\'on-Zygmund operator by the norm of a sparse operator. This was later improved to a pointwise estimate independently by Conde-Alonso and Rey  \\cite{CR16} and by Lerner and Nazarov \\cite{LN15}. Afterwards, Lacey \\cite{La17b} obtained the same result for a slightly larger class of Calder\\'on-Zygmund operators by a stopping cube argument instead of the local mean oscillation decomposition approach. This argument was further refined by Hyt\\\"onen, Roncal and Tapiola  \\cite{HRT17} and afterwards made strikingly clear by Lerner \\cite{Le16}, where the following abstract sparse domination principle was shown:\n\n If $T$ is a bounded sublinear operator from $L^{p_1}(\\R^n)$ to $L^{p_1,\\infty}(\\R^n)$ and the \\emph{grand maximal truncation operator}\n\\begin{equation*}\n  \\mc{M}_T f(s) := \\sup_{Q \\ni s} \\esssup_{s' \\in Q} \\,\\abs{T(f \\ind_{\\R^n\\setminus 3Q})(s')}, \\qquad s \\in \\R^n\n\\end{equation*}\nis bounded from $L^{p_2}(\\R^n)$ to $L^{p_2,\\infty}(\\R^n)$ for some $1 \\leq p_1,p_2 <\\infty$, then there is an $\\eta \\in (0,1)$ such that for every compactly supported $f \\in L^{p}(\\R^n)$ with $p_0 := \\max\\cbrace{p_1, p_2}$ there exists an $\\eta$-sparse family of cubes $\\mc{S}$ such that\n\\begin{equation}\\label{eq:sparsepointwiseintro}\n  \\abs{Tf(s)} \\lesssim\n  \\sum_{Q \\in \\mc{S}} \\ip{\\abs{f}}_{p_0,Q}\\ind_Q(s),\\qquad  s \\in \\R^n.\n\\end{equation}\nHere $\\ip{{f}}_{p,Q}^p:= \\avint_Q {f}^p :=\\frac{1}{\\abs{Q}}\\int_Q{f}^p$ for $p \\in (0,\\infty)$ and positive $f \\in L^p_{\\loc}(\\R^n)$ and we call a family of cubes $\\mc{S}$ $\\eta$-sparse if for every $Q \\in \\mc{S}$ there exists a measurable set $E_Q \\subseteq Q$ such that $\\abs{E_Q} \\geq \\eta\\abs{Q}$ and such that the $E_Q$'s are pairwise disjoint.\n\n This sparse domination principle was further generalized in the recent paper \\cite{LO19} by Lerner and Ombrosi, in which the authors  showed that the weak $L^{p_2}$-boundedness of the more flexible operator\n\\begin{equation*}\n  \\mc{M}_{T,\\alpha}^{\\#} f(s) := \\sup_{Q \\ni s} \\esssup_{s',s'' \\in Q} \\,\\abs{T(f \\ind_{\\R^n\\setminus \\alpha Q})(s') - T(f \\ind_{\\R^n\\setminus \\alpha Q})(s'')}, \\qquad s \\in \\R^n\n\\end{equation*}\nfor some $\\alpha \\geq 3$ is already enough to deduce the pointwise sparse domination as in \\eqref{eq:sparsepointwiseintro}. Furthermore, they relaxed the weak $L^{p_1}$-boundedness condition on $T$ to a condition in the spirit of the $T(1)$-theorem.\n\n\\subsection{Main result}\nOur main result is a generalization of the main result in \\cite{LO19} in the following four directions:\n\\begin{enumerate}[(i)]\n  \\item \\label{it:generalizationhom} We replace $\\R^n$ by a space of homogeneous type $(S,d,\\mu)$.\n  \\item \\label{it:generalizationvect} We let $T$ be an operator from $L^{p_1}(S;X)$ to $L^{p_1,\\infty}(S;Y)$, where $X$ and $Y$ are Banach spaces.\n  \\item \\label{it:generalizationsub} We use structure of the operator $T$ and geometry of the Banach space $Y$ to replace the $\\ell^1$-sum in the sparse operator by an $\\ell^r$-sum for $r \\geq 1$.\n  \\item \\label{it:generalizationlocal} We replace the truncation $T(f \\ind_{\\R^n\\setminus \\alpha Q})$ in the grand maximal truncation operator by an abstract localization principle.\n\\end{enumerate}\nThe extensions \\ref{it:generalizationhom} and \\ref{it:generalizationvect} are relatively straightforward. The main novelty of this paper is \\ref{it:generalizationsub}, which controls the weight characteristic dependence that can be deduced from the sparse domination. Generalization \\ref{it:generalizationlocal} will only make its appearance in Theorem \\ref{theorem:localsparse} and can be used to make the associated grand maximal truncation operator easier to estimate in specific situations.\n\n\\bigskip\n\nLet $(S,d,\\mu)$ be a space of homogeneous type\nand let $X$ and $Y$ be Banach spaces.\nFor a bounded linear operator $T$ from $L^{p_1}(S;X)$ to $L^{p_1,\\infty}(S;Y)$ and $\\alpha\\geq 1$ we define the\nfollowing \\emph{sharp grand maximal truncation operator}\n\\begin{align*}\n  \\mc{M}_{T,\\alpha}^{\\#}&f(s) :=\\sup_{B\\ni s} \\esssup_{s',s''\\in B}\\, \\nrmb{T(f\\ind_{S\\setminus {\\alpha B}})(s')-T(f\\ind_{S\\setminus {\\alpha B}})(s'')}_Y,\\qquad s \\in S,\n\\end{align*}\nwhere the supremum is taken over all balls $B\\subseteq S$ containing $s\\in S$.  Our main theorem reads as follows.\n\n\\begin{theorem}\\label{theorem:main}\nLet $(S,d,\\mu)$ be a space of homogeneous type and let $X$ and $Y$ be Banach spaces. Take $p_1,p_2,r \\in [1,\\infty)$ and set $p_0:=\\max\\cbrace{p_1, p_2}$. Take $\\alpha \\geq 3c_d^2\/\\delta$, where $c_d$ is the quasi-metric constant and $\\delta$ is as in Proposition \\ref{proposition:dyadicsystem}. Assume the following conditions:\n\\begin{itemize}\n  \\item $T$ is a bounded linear operator from $L^{p_1}(S;X)$ to $L^{p_1,\\infty}(S;Y)$.\n  \\item $\\mc{M}_{T,\\alpha}^{\\#}$ is a bounded operator from $L^{p_2}(S;X)$ to $L^{p_2,\\infty}(S)$.\n  \\item There is a $C_r>0$ such that for disjointly and boundedly supported $f_1,\\ldots,f_n \\in L^{p_0}(S;X)$\n\\begin{equation*}\n  \\qquad \\nrms{T\\hab{\\sum_{k=1}^n f_k}(s)}_Y\\leq C_r \\, \\has{\\sum_{k=1}^n \\nrmb{Tf_k(s)}_Y^r}^{1\/r},\n  \\qquad s \\in S.\n\\end{equation*}\n\\end{itemize}\n Then there is an $\\eta \\in (0,1)$ such that for any boundedly supported $f \\in L^{p_0}(S;X)$ there is an $\\eta$-sparse collection of cubes $\\mc{S}$ such that\n\\begin{align*}\n   \\nrm{ Tf(s)}_Y&\\lesssim_{S,\\alpha} C_T \\,C_r\\, \\has{\\sum_{Q \\in \\mc{S}} \\ipb{\\nrm{f}_X}_{p_0,Q}^r \\ind_{Q}(s)}^{1\/r}, \\qquad s \\in S,\n\\end{align*}\nwhere $C_T={\\nrm{T}_{L^{p_1}\\to L^{p_1,\\infty}} + \\nrm{\\mc{M}_{T,\\alpha}^{\\#}}_{L^{p_2}\\to L^{p_2,\\infty}}}$.\n\\end{theorem}\nAs the assumption in the third bullet of Theorem \\ref{theorem:main} expresses a form of sublinearity of the operator $T$ when $r=1$, we will call this assumption \\emph{$r$-sublinearity}. Note that it is crucial that the constant $C_r$  is independent of $n \\in \\N$. If $C_r = 1$ it suffices to consider $n=2$.\n\n\\subsection{Sharp weighted norm inequalities}\nOne of the main reasons to study sparse domination of an operator is the fact that sparse bounds yield weighted norm inequalities and these weighted norm inequalities are sharp for many operators. Here sharpness is meant in the sense that for $p \\in (p_0,\\infty)$ we have  a $\\beta\\geq 0$ such that\n\\begin{equation}\\label{eq:sharp}\n  \\nrm{T}_{L^p(S,w;X) \\to L^p(S,w;Y)} \\lesssim [w]_{A_{p\/p_0}}^\\beta, \\qquad w \\in A_{p\/p_0}\n\\end{equation}\nand \\eqref{eq:sharp} is false for any $\\beta' <\\beta$.\n\nThe first result of this type was obtained by Buckley \\cite{Bu93}, who showed that $\\beta = \\tfrac{1}{p-1}$ for the Hardy--Littlewood maximal operator. A decade later, the quest to find sharp weighted bounds attracted renewed attention because of the work of Astala, Iwaniec and Saksman \\cite{AIS01}. They proved sharp regularity results for the solution to the Beltrami equation under the assumption that $\\beta = 1$ for the Beurling--Ahlfors transform  for $p\\geq 2$. This linear dependence on the $A_p$ characteristic for the Beurling--Ahlfors transform was shown by Petermichl and Volberg in \\cite{PV02}.  Another decade later, after many partial results, sharp weighted norm inequalities were obtained for general Calder\\'on--Zygmund operators by Hyt\\\"onen in \\cite{Hy12} as discussed before.\n\nIn Section \\ref{section:weights} we will prove weighted $L^p$-boundedness for the sparse operators appearing in Theorem \\ref{theorem:main}. As a direct corollary from  Theorem \\ref{theorem:main} and Proposition \\ref{proposition:weights} we have:\n\n\\begin{corollary}\\label{corollary:main}\n  Under the assumptions of Theorem \\ref{theorem:main} we have for all $p \\in (p_0,\\infty)$ and $w \\in A_{p\/p_0}$\n\\begin{align*}\n  \\nrm{T}_{L^p(S,w;X)\\to L^p(S,w;Y)} &\\lesssim C_T\\,C_r\\, [w]_{A_{p\/p_0}}^{\\max\\cbraceb{\\frac{1}{p-p_0},\\frac{1}{r}}},\n\\end{align*}\nwhere the implicit constant depends on $S,p_0,p,r$ and $\\alpha$.\n\\end{corollary}\nAs noted before the main novelty in Theorem \\ref{theorem:main} is the introduction of the parameter $r \\in [1,\\infty)$. The $r$-sublinearity assumption in Theorem \\ref{theorem:main} becomes more restrictive as $r$ increases and the conclusions of Theorem \\ref{theorem:main} and Corollary \\ref{corollary:main} consequently become stronger.  In order to check whether the dependence on the weight characteristic is sharp, one can employ e.g. \\cite[Theorem 1.2]{LPR15}, which provides a lower bound for the best possible weight characteristic dependence in terms of the operator norm of $T$ from $L^p(S;X)$ to $L^p(S;Y)$. For some operators, like Littlewood--Paley or maximal operators, sharpness in the estimate in Corollary \\ref{corollary:main} is attained for $r>1$ and thus Theorem \\ref{theorem:main} can be used to show sharp weighted bounds for more operators than precursors like \\cite[Theorem 1.1]{LO19}.\n\n\\subsection{How to apply our main result}\nLet us outline the typical way how one applies Theorem \\ref{theorem:main} (or the local and more general version in Theorem \\ref{theorem:localsparse}) to obtain (sharp) weighted $L^p$-boundedness for an operator $T$:\n\\begin{enumerate}[(i)]\n  \\item If $T$ is not linear it is often \\emph{linearizable}, which means that we can linearize it by putting part of the operator in the norm of the Banach space $Y$. For example if $T$ is a Littlewood--Paley square function we take $Y=L^2$ and if $T$ is a maximal operator we take $Y=\\ell^\\infty$. Alternatively one can apply Theorem \\ref{theorem:localsparse}, which is a local and more abstract version of Theorem \\ref{theorem:main} that does not assume $T$ to be linear.\n  \\item The weak $L^{p_1}$-boundedness of $T$ needs to be studied separately and is often already available in the literature.\n  \\item The operator $\\mc{M}_{T,\\alpha}^{\\#}$ reflects the non-localities of the operator $T$. The weak $L^{p_2}$-boundedness of $\\mc{M}_{T,\\alpha}^{\\#}$ requires an intricate study of the structure of the operator. In many examples $\\mc{M}_{T,\\alpha}^{\\#}$ can be pointwise dominated by the Hardy--Littlewood maximal operator $M_{p_2}$, which is weak $L^{p_2}$-bounded. This is exemplified for Calder\\'on--Zygmund operators in the proof of Theorem \\ref{theorem:A2}.\n      Sometimes one can choose a suitable localization in Theorem \\ref{theorem:localsparse} such that the sharp maximal truncation operator is either zero (see Section \\ref{section:maximal} on the Rademacher maximal operator), or pointwise dominated by $T$.\n  \\item The $r$-sublinearity assumption on $T$ is trivial for $r=1$, which suffices if one is not interested in quantitative weighted bounds. To check the $r$-sublinearity for some $r>1$ one needs to use the structure of the operator and often also the geometric properties of the Banach space $Y$ like type $r$.\n      See, for example, the proofs of Theorems \\ref{theorem:RMF} and \\cite[Theorem 6.4]{LV19} how to check $r$-sublinearity in concrete cases.\n\\end{enumerate}\n\n\\subsection{Applications}\nThe main motivation to generalize the results in \\cite{LO19} comes from the application in the recent work \\cite{LV19} by Veraar and the author, in which Calder\\'on--Zygmund theory is developed for stochastic singular integral operators. In particular, in \\cite[Theorem 6.4]{LV19} Theorem \\ref{theorem:main} is applied with $p_1=p_2=r=2$ to prove a stochastic version of the vector-valued $A_2$-theorem for Calder\\'on--Zygmund operators, which yields new results in the theory of maximal regularity for stochastic partial differential equations. The fact that $r=2$ in \\cite[Theorem 6.4]{LV19} was needed to obtain a sharp result motivated the introduction of the parameter $r$ in this paper.\nIn future work, further applications of Theorem \\ref{theorem:main} to both deterministic and stochastic partial differential equations will be given, for which it is crucial that we allow spaces of homogeneous type instead of just $\\R^n$, as in these applications $S$ is typically $\\R_+\\times \\R^{n}$ with the parabolic metric.\n\nIn this paper we will focus on applications in harmonic analysis. We will  provide a few examples that illustrate the sparse domination principle nicely, and comment on further potential applications in Section \\ref{section:further}.\n\\begin{itemize}\n  \\item As a first application of Theorem \\ref{theorem:main} we prove an $A_2$-theorem for vector-valued Calder\\'on--Zygmund operators with operator-valued kernel in a space of homogeneous type. The $A_2$-theorem for vector-valued Calder\\'on--Zygmund operators with operator-valued kernel in Euclidean space has previously been proven in \\cite{HH14} and the $A_2$-theorem for scalar-valued Calder\\'on--Zygmund operators in spaces of homogeneous type in \\cite{NRV13,AV14}. Our theorem unifies these two results.\n  \\item Using the $A_2$-theorem, we prove a weighted, anisotropic, mixed norm Mihlin multiplier theorem, which is a natural supplement to the recent results in \\cite{FHL18} and is particularly useful in the study of spaces of smooth, vector-valued functions.\n  \\item  In our second application of Theorem \\ref{theorem:main} we study sparse domination and quantitative weighted norm inequalities for the Rademacher maximal operator, extending the qualitative bounds in Euclidean space in \\cite{Ke13}. The proof  demonstrates how one can use the geometry of the Banach space to deduce $r$-sublinearity for an operator. As a corollary, we deduce that the lattice Hardy--Littlewood and the Rademacher maximal operator are not comparable.\n\\end{itemize}\n\n\\subsection{Outline}\nThis paper is organized as follows: After introducing spaces of homogeneous type and dyadic cubes in such spaces in Section \\ref{section:SHT}, we will set up our abstract sparse domination framework and deduce Theorem \\ref{theorem:main} in Section  \\ref{section:main}. We also give some further generalizations of our main results. In Section \\ref{section:weights} we introduce weights and state weighted bounds for the sparse operators in the conclusions of Theorem \\ref{theorem:main}, from which Corollary \\ref{corollary:main} follows. To prepare for our application sections, we will discuss some preliminaries on e.g. Banach space geometry in Section \\ref{section:Banachspace}. Afterwards we will\nuse our main result to prove the previously discussed applications in Sections \\ref{section:A2}-\\ref{section:maximal}.\nFinally, in Section \\ref{section:further} we discuss some potential further applications of our main result.\n\n\n\n\\section{Spaces of homogeneous type}\\label{section:SHT}\nA space of homogeneous type $(S,d,\\mu)$, originally introduced by Coifman and Weiss in \\cite{CW71}, is a set $S$ equipped with a quasi-metric $d$ and a doubling Borel measure $\\mu$. That is,\n a metric $d$  which instead of the triangle inequality satisfies\n\\begin{equation*}\n  d(s,t) \\leq c_d\\, \\hab{d(s,u)+d(u,t)}, \\qquad s,t,u\\in S\n\\end{equation*}\nfor some $c_d\\geq 1$, and a Borel measure $\\mu$ that satisfies the doubling property\n\\begin{equation*}\n  \\mu\\hab{B(s,2\\rho)} \\leq c_\\mu \\,\\mu\\hab{B(s,\\rho)}, \\qquad s \\in S,\\quad \\rho>0\n\\end{equation*}\nfor some $c_\\mu\\geq 1$, where $B(s,\\rho):=\\cbrace{t \\in S:d(s,t)<\\rho}$ is the ball around $s$ with radius $\\rho$. Throughout this paper we will assume additionally that all balls $B\\subseteq S$ are Borel sets and that we have $0 <\\mu(B)<\\infty$.\n\nIt was shown in \\cite[Example 1.1]{St15} that it can indeed happen that balls are not Borel sets in a quasi-metric space. This can be circumvented by taking topological closures and adjusting the constants $c_d$ and $c_\\mu$ accordingly. However, to simplify matters we just assume all balls to be Borel sets and leave the necessary modifications if this is not the case to the reader.\nThe size condition on the measure of a ball ensures that taking the average $\\ip{{f}}_{p,B}$ of a positive function $f \\in L^p_{\\loc}(S)$  over a ball $B\\subseteq S$ is always well-defined.\n\nAs $\\mu$ is a Borel measure, i.e. a measure defined on the Borel $\\sigma$-algebra of the quasi-metric space $(S,d)$, the Lebesgue differentiation theorem holds and as a consequence the continuous functions with bounded support are dense in $L^p(S)$ for all $p \\in [1,\\infty)$. The Lebesgue differentiation theorem and consequently our results remain valid if $\\mu$ is a measure defined on a $\\sigma$-algebra $\\Sigma$ that contains the Borel $\\sigma$-algebra as long as the measure space $(S,\\Sigma,\\mu)$ is Borel semi-regular. See \\cite[Theorem 3.14]{AM15} for the details.\n\n\nThroughout we will write that an estimate depends on $S$ if it depends on $c_d$ and $c_\\mu$. For a thorough introduction to and a list of examples of spaces of homogeneous type we refer to the monographs of Christ \\cite{Ch90} and Alvarado and Mitrea \\cite{AM15}.\n\n\\subsection{Dyadic cubes}\nLet $0<c_0\\leq C_0 <\\infty$ and $0<\\delta<1$. Suppose that for $k \\in \\Z$ we have an index set $J_k$, pairwise disjoint collection $\\ms{D}_k = \\cbrace{Q^j_k}_{j \\in J_k}$ of measurable sets and a collection of points $\\cbrace{z_j^k}_{j \\in J_k}$. We call $\\ms{D} := \\bigcup_{k \\in \\Z}\\ms{D}_k$ a \\emph{dyadic system} with parameters $c_0$, $C_0$ and $\\delta$ if it satisfies the following properties:\n\\begin{enumerate}[(i)]\n\\item For all $k \\in \\Z$ we have $$S = \\bigcup_{j \\in J_k} Q_j^k;$$\n\\item \\label{it:dyadicsystem2}For $k \\geq l$, $Q \\in \\ms{D}_k$ and $Q' \\in \\ms{D}_l$ we either have $Q\\cap Q' =\\emptyset$ or $Q \\subseteq Q'$;\n\\item \\label{it:dyadicsystem3}For each $k \\in \\Z$ and $j \\in J_k$ we have\n\\begin{equation*}\n  B(z_j^k,c_0 \\delta^k) \\subseteq Q_j^k \\subseteq B(z_j^k, C_0 \\delta^k);\n\\end{equation*}\n\\end{enumerate}\nWe will call the elements of a dyadic system $\\ms{D}$ cubes and for a cube $Q \\in \\ms{D}$ we define the \\emph{restricted dyadic system} $\\ms{D}(Q):=\\cbrace{P\\in \\ms{D}:P\\subseteq Q}$. We will say that an estimate depends on $\\ms{D}$ if it depends on the parameters $c_0$, $C_0$ and $\\delta$.\n\nOne can view $z_j^k$ and $\\delta^k$ as the center and side length of a cube $Q_j^k \\in \\ms{D}_k$. These have to be with respect to a specific $k \\in \\Z$, as this $k$ may not be unique. We therefore think of a cube $Q \\in \\ms{D}$ to also encode the information of its center $z$ and generation $k$.\nThe structure of individual dyadic cubes $Q \\in \\ms{D}$ in a space of homogeneous type can be very messy and consequently the dilations of such cubes do not have a canonical definition. Therefore for a cube $Q \\in \\ms{D}$ with center $z$ and of generation $k$ we define the \\emph{dilations $\\alpha Q$} for $\\alpha\\geq 1$ as\n\\begin{equation*}\n  \\alpha Q := B\\hab{z,\\alpha \\cdot C_0 \\delta^k},\n\\end{equation*}\nwhich are actually dilations of the ball that contains $Q$ by property \\ref{it:dyadicsystem3} of a dyadic system.\n\\bigskip\n\nWhen $S=\\R^n$ and $d$ is the Euclidean distance, the standard dyadic cubes form a dyadic system and, combined with its translates over $\\alpha \\in \\cbrace{0,\\frac{1}{3},\\frac23}^n$, it holds that any ball in $\\R^n$ is contained in a cube of comparable size from one of these dyadic systems (see e.g. \\cite[Lemma 3.2.26]{HNVW16}). We will rely on the following proposition for the existence of dyadic systems with this property in a general space of homogeneous type. For the proof and a more detailed discussion we refer to \\cite{HK12}.\n\n\\begin{proposition}\\label{proposition:dyadicsystem}\n  Let $(S,d,\\mu)$ be a space of homogeneous type. There exist $0<c_0\\leq C_0<\\infty$, $\\gamma\\geq 1$, $0<\\delta<1$ and $m \\in \\N$ such that there are dyadic systems $\\ms{D}^1,\\ldots,\\ms{D}^m$ with parameters $c_0$, $C_0$ and $\\delta$, and with the property that for each $s \\in S$ and $\\rho>0$ there is a $j\\in \\cbrace{1,\\ldots,m}$ and a $Q \\in \\ms{D}^j$  such that $$B(s,\\rho) \\subseteq Q, \\qquad\\text{and}\\qquad  \\diam(Q) \\leq \\gamma \\rho.$$\n\\end{proposition}\n\nThe following covering lemma will be used in the proof of our main theorem:\n\\begin{lemma}\\label{lemma:covering}\nLet $(S,d,\\mu)$ be a space of homogeneous type and $\\ms{D}$ a dyadic system with parameters $c_0$, $C_0$ and $\\delta$. Suppose that $\\diam(S) = \\infty$, take  $\\alpha \\geq {3c_d^2\/\\delta}$ and let $E \\subseteq S$ satisfy $0<\\diam(E)<\\infty$. Then there exists a partition $\\mc{D} \\subseteq \\ms{D}$ of $S$ such that $E \\subseteq \\alpha Q$ for all $Q \\in \\mc{D}$.\n\\end{lemma}\n\n\\begin{proof}\n  For $s \\in S$ and $k \\in \\Z$ let $Q_s^k\\in \\ms{D}_k$ be the unique cube such that $s \\in Q_s^k$ and denote its center by $z_{s}^k$. Define\n  \\begin{equation*}\n    K_s := \\cbraceb{k \\in \\Z: E \\not\\subseteq 2 c_d Q_s^k},\n  \\end{equation*}\n  where $c_d$ is the quasi-metric constant.\n  If $k \\in \\Z$ is such that\n  $$\\diam(2c_dQ_s^k) \\leq 4c_d^2C_0\\delta^k < \\diam(E),$$ then $E\\not\\subseteq  2c_dQ_s^k$, i.e. $k \\in K_s$ so is $K_s$ non-empty. On the other hand if $k \\in \\Z$ is such that\n$\n    C_0\\delta^k > \\sup_{s' \\in E} d(s,s'),\n$\n  then\n  \\begin{equation*}\n \\sup_{s'\\in E} d(s',z_{s}^k) \\leq c_d \\,\\hab{\\sup_{s' \\in E} d(s,s') + d(s, z_{s}^k)} \\leq 2c_dC_0\\delta^k\n  \\end{equation*}\n  so $E \\subseteq 2c_dQ_s^k$ and thus $k \\notin K_s$. Therefore $K_s$ is bounded from below.\n\n  Define $k_s := \\min K_s$ and set $\\mc{D}:= \\cbrace{Q_s^{k_s}:s \\in S}$. Then $\\mc{D}$ is a partition of $S$. Indeed, suppose that for $s,s'\\in S$ we have $Q_s^{k_s} \\cap Q_{s'}^{k_{s'}}\\neq \\emptyset$. Then using property \\ref{it:dyadicsystem2} of a dyadic system we may assume without loss of generality that $Q_s^{k_s} \\subseteq Q_{s'}^{k_{s'}}$. Property \\ref{it:dyadicsystem2} of a dyadic system then implies that $k_s \\geq k_{s'}$. In particular $s \\in Q_{s'}^{k_{s'}}$, so by the minimality of $k_s$ we must have $k_s = k_{s'}$. Therefore since the elements of $\\ms{D}_{k_s}$ are pairwise disjoint we can conclude $Q_s^{k_s} = Q_{s'}^{k_{s'}}$.\n\n  To conclude note that $z_s^{k_s} \\in Q_s^{k_s} \\subseteq Q_s^{k_s-1}$ by property \\ref{it:dyadicsystem2} of a dyadic system, so $d(z_s^{k_s-1},z_s^{k_s}) \\leq C_0\\delta^{k_s-1}$. Therefore using the minimality of $k_s$ we obtain\n  $$E \\subseteq 2c_dQ_s^{k_s-1} = B(z_s^{k_s-1},2c_dC_0\\delta^{k_s-1}) \\subseteq B\\has{z_s^{k_s},\\frac{3c_d^2}{\\delta} \\cdot C_0\\delta^{k_s}} \\subseteq \\alpha Q_s^{k_s},$$\n  which finishes the proof.\n\\end{proof}\n\n\\subsection{The Hardy--Littlewood maximal operator}\\label{subsection:HL}\nOn a space of homogeneous type  $(S,d,\\mu)$  with a dyadic system $\\ms{D}$ we define the \\emph{dyadic Hardy--Littlewood maximal operator} for $f \\in L^1_{\\loc}(S)$ by\n\\begin{equation*}\n  M^{\\ms{D}}f(s):= \\sup_{Q\\in \\ms{D}: s \\in Q} \\ipb{\\abs{f}}_{1,Q}, \\qquad s \\in S.\n\\end{equation*}\nBy Doob's maximal inequality (see e.g. \\cite[Theorem 3.2.2]{HNVW16}) $M^{\\ms{D}}$ is strong $L^p$-bounded for all $p\\in (1,\\infty)$ and weak $L^1$-bounded. We define the (non-dyadic) \\emph{Hardy--Littlewood maximal operator} for $f \\in L^1_{\\loc}(S)$ by\n\\begin{equation*}\n  Mf(s):= \\sup_{B \\ni s} \\,\\ipb{\\abs{f}}_{1,Q}, \\qquad s \\in S,\n\\end{equation*}\nwhere the supremum is taken over all balls $B \\subseteq S$ containing $s$. By Proposition \\ref{proposition:dyadicsystem} there are dyadic systems $\\ms{D}^1,\\ldots,\\ms{D}^m$ such that\n\\begin{equation*}\n  Mf(s) \\lesssim_S \\sum_{j=1}^m M^{\\ms{D}}f(s), \\qquad s \\in S,\n\\end{equation*}\nso $M$ is also strong $L^p$-bounded for $p\\in(1,\\infty)$ and weak $L^1$-bounded. For $p_0 \\in [1,\\infty)$ and $f \\in L^{p_0}_{\\loc}(S)$ we define\n\\begin{equation*}\n  M_{p_0}f(s) := \\sup_{B \\ni s} \\,\\ipb{\\abs{f}}_{p_0,Q} = M\\hab{\\abs{f}^{p_0}}(s)^{1\/p_0}, \\qquad s \\in S,\n\\end{equation*}\nwhich is strong $L^p$-bounded for $p \\in(p_0,\\infty)$ and weak $L^{p_0}$-bounded. This follows from the boundedness of $M$ by rescaling.\n\n\\section{Pointwise \\texorpdfstring{$\\ell^r$}{lr}-sparse domination}\\label{section:main}\nIn this section we will prove a local version of the sparse domination result in Theorem \\ref{theorem:main}, from which we will deduce Theorem \\ref{theorem:main} by a covering argument using Lemma \\ref{lemma:covering}. This local version will use an abstract localization of the operator $T$, since it depends upon the operator at hand as to the most effective localization. For example in the study of a Calder\\'on--Zygmund operator it is convenient to localize the function inserted into $T$, for a maximal operator it is convenient to localize the supremum in the definition of the maximal operator and for a Littlewood--Paley operator it is most suitable to localize the defining integral.\n\n\\begin{definition}\n  Let $(S,d,\\mu)$ be a space of homogeneous type with a dyadic system $\\ms{D}$, let $X$ and $Y$ be Banach spaces, $p \\in [1,\\infty)$ and $\\alpha \\geq 1$. For a bounded operator\n$$T\\colon L^{p}(S;X) \\to L^{p,\\infty}(S;Y)$$ we say that a family of operators $\\cbrace{T_Q}_{Q \\in \\ms{D}}$ from $L^p(S;X)$ to $L^{p,\\infty}(Q;Y)$ is an \\emph{$\\alpha$-localization family of $T$} if for all $Q\\in \\ms{D}$ and $f \\in L^{p}(S;X)$  we have\n\\begin{align*}\n T_Q(f \\ind_{\\alpha Q})(s)&=T_Qf(s), & & s \\in Q , & & \\text{(Localization)}\\\\\n  \\nrmb{T_Q(f\\ind_{\\alpha Q})(s)}_Y &\\leq \\nrmb{T(f\\ind_{\\alpha Q})(s)}_Y, & &s \\in Q, & & \\text{(Domination)}\n\\end{align*}\nFor $Q,Q'\\in \\ms{D}$ with $Q' \\subseteq Q$ we  define the difference operator\n\\begin{align*}\nT_{Q\\setminus Q'}f(s)&:= T_{Q}f(s) - T_{Q'}f(s), \\qquad s \\in Q'.\n\\end{align*}\n and for $Q\\in \\ms{D}$ the \\emph{localized sharp grand maximal truncation operator}\n \\begin{align*}\n  \\mc{M}_{T,Q}^{\\#}&f(s) :=\\sup_{\\substack{Q'\\in \\ms{D}(Q):\\\\s \\in Q'}}\\, \\esssup_{s',s'' \\in Q'} \\,\\nrmb{(T_{Q\\setminus Q'}) f(s')-(T_{Q\\setminus Q'}) f(s'')}_Y, \\qquad s \\in S.\n\\end{align*}\n\\end{definition}\n\nIn order to obtain interesting results, one needs to be able to recover the boundedness of $T$ from the boundedness of  $T_Q$ uniformly in $Q \\in \\ms{D}$.  The canonical example of an $\\alpha$-localization family is\n\\begin{align*}\nT_Qf(s) &:=T(f \\ind_{\\alpha Q})(s), \\qquad s \\in Q.\n\\end{align*}\nfor all $Q \\in \\ms{D}$ and it is exactly this choice that will lead to Theorem \\ref{theorem:main}.\nWe are now ready to prove our main result, which is a local, more general version of Theorem \\ref{theorem:main}.\n\n\\begin{theorem}\\label{theorem:localsparse}\nLet $(S,d,\\mu)$ be a space of homogeneous type with dyadic system $\\ms{D}$ and let  $X$ and $Y$ be Banach spaces. Take $p_1,p_2,r \\in [1,\\infty)$, set $p_0:=\\max\\cbrace{p_1, p_2}$ and take $\\alpha \\geq 1$. Suppose that\n\\begin{itemize}\n  \\item $T$ is a bounded operator from $L^{p_1}(S;X)$ to $L^{p_1,\\infty}(S;Y)$ with $\\alpha$-localization family $\\cbrace{T_Q}_{Q \\in \\ms{D}}$.\n  \\item $\\mc{M}_{T,Q}^{\\#}$ is bounded from $L^{p_2}(S;X)$ to $L^{p_2,\\infty}(S)$ uniformly in $Q \\in \\ms{D}$.\n  \\item For all $Q_1,\\ldots,Q_n \\in \\ms{D}$ with $Q_n\\subseteq \\ldots\\subseteq Q_1$ and any $f \\in L^p(S;X)$\n\\begin{equation*}\n  \\qquad \\nrmb{T_{Q_1}f(s)}_Y\\leq C_r \\has{\\nrmb{T_{Q_n}{f}(s)}_Y^r+\\sum_{k=1}^{n-1}\\nrmb{T_{Q_{k}\\setminus Q_{k+1}}f(s)}_Y^r}^{1\/r},\\quad s \\in Q_n.\n\\end{equation*}\n\\end{itemize}\n Then for any $f \\in L^{p_0}(S;X)$ and $Q\\in \\ms{D}$  there exists a $\\frac12$-sparse collection of dyadic cubes $\\mc{S}\\subseteq \\ms{D}(Q)$ such that\n\\begin{align*}\n  \\nrmb{T_Qf(s)}_Y\\lesssim_{S,\\ms{D},\\alpha} C_T\\,C_r \\,\n  \\has{ \\sum_{P \\in \\mc{S}} \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r \\ind_P(s)}^{1\/r},\\qquad s \\in Q,\n  \\end{align*}\nwith $C_T:={\\nrm{T}_{L^{p_1}\\to L^{p_1,\\infty}} + \\sup_{P \\in \\ms{D} } \\nrm{\\mc{M}_{T,P}^{\\#}}_{L^{p_2}\\to L^{p_2,\\infty}}}.$\n\\end{theorem}\n\nThe assumption in the third bullet in Theorem \\ref{theorem:localsparse} replaces the $r$-sub\\-linearity assumption in Theorem \\ref{theorem:main}. We will call this assumption a \\emph{localized $\\ell^r$-estimate}.\n\n\\begin{proof}\nFix $f \\in L^p(S,X)$ and $Q \\in \\ms{D}$. We will prove the theorem in two steps: we will first construct the $\\frac12$-sparse family of cubes $\\mc{S}$ and then show that the sparse expression associated to $\\mc{S}$ dominates $T_Qf$ pointwise.\n\n\\textbf{Step 1:}\nWe will construct the $\\frac12$-sparse family of cubes $\\mc{S}$ iteratively. Given a collection of pairwise disjoint cubes $\\mc{S}^k$ for some $k \\in \\N$ we will first describe how to construct $\\mc{S}^{k+1}$. Afterwards we can inductively define $\\mc{S}^k$ for all $k \\in \\N$ starting from $\\mc{S}^1 = \\cbrace{Q}$ and set $\\mc{S}:=\\bigcup_{k \\in \\N} \\mc{S}^k$.\n\nFix a $P\\in \\mc{S}^k$ and for $\\lambda\\geq1$ to be chosen later define\n\\begin{align*}\n  \\Omega_{P}^1&:= \\cbraces{s \\in P: \\nrm{T_{P} f(s)}_Y> \\lambda\\, C_T \\,\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}}\\\\\n  \\Omega_{P}^2&:= \\cbraces{s \\in P: \\mc{M}_{T,P}^{\\#}(f)(s)> \\lambda \\,C_T\\,\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}}\n\\end{align*}\nand $\\Omega_P:= \\Omega_{P}^1 \\cup  \\Omega_{P}^2$. Let $c_1\\geq 1$, depending on $S$, $\\ms{D}$ and $\\alpha$, be such that $\\mu(\\alpha P) \\leq c_1\\,\\mu(P)$. By the domination property of the $\\alpha$-localization family we have\n\\begin{align*}\n\\nrm{T_P f(s)}_Y &\\leq \\nrm{T(f\\ind_{\\alpha P})(s)}_Y, & &s \\in P,\n\\intertext{and by the localization property}\n  \\mc{M}_{T,P}^{\\#}(f)(s)&=\\mc{M}_{T,P}^{\\#}(f\\ind_{\\alpha P})(s),  & &s \\in P.\n\\end{align*}\n  Thus by the weak boundedness assumptions on $T$ and $\\mc{M}^{\\#}_{T,P}$ and  H\\\"older's inequality we have for $i=1,2$\n\\begin{align}\\label{eq:holdercomp}\n    \\mu(\\Omega_{P}^i) &\\leq \\has{\\frac{\\nrm{f \\ind_{\\alpha P}}_{L^{p_i}(S;X)}}{\\lambda \\,\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}}}^{p_i}\n    = \\frac{\\ipb{\\nrm{f}_X}_{p_i,\\alpha P}^{p_i}}{\\lambda^{p_i}\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^{p_i}} \\mu(\\alpha P)\n     \\leq  \\frac{ c_1}{\\lambda} \\, \\mu(P).\n\\end{align}\nTherefore it follows that\n\\begin{equation}\\label{eq:OmegatoQ}\n  \\mu(\\Omega_P) \\leq \\frac{2c_1}{\\lambda}  \\mu(P).\n\\end{equation}\nTo construct the cubes in $\\mc{S}^{k+1}$ we will use a local Calder\\'on--Zygmund decomposition (see e.g. \\cite[Lemma 4.5]{FN18}) on\n\\begin{equation*}\n  \\Omega_{P,\\rho}:= \\cbrace{s \\in P: M^{\\ms{D}(P)} (\\ind_{\\Omega_P})>\\tfrac1\\rho}, \\qquad \\rho>0\n\\end{equation*}\n which will be a proper subset of $P$ for our choice of $\\lambda$ and $\\rho$. Here $M^{\\ms{D}(P)}$ is the dyadic Hardy--Littlewood maximal operator with respect to the restricted dyadic system $\\ms{D}(P)$.\n The local Calder\\'on--Zygmund decomposition yields a pairwise disjoint collection of cubes $\\mc{S}_P \\subseteq \\ms{D}(P)$ and a constant $c_2 \\geq 2$, depending on $S$ and $\\ms{D}$, such that $\\Omega_{P,c_2}= \\textstyle{\\bigcup_{P'\\in \\mc{S}_P}}P'$ and\n\\begin{equation}\\label{eq:Pjsize}\n \\tfrac{1}{c_2} \\, \\mu(P') \\leq \\mu(P'\\cap \\Omega_{P}) \\leq \\tfrac{1}{2} \\,\\mu(P'),\\qquad P'\\in \\mc{S}_P.\n\\end{equation}\nThen by \\eqref{eq:OmegatoQ}, \\eqref{eq:Pjsize} and the disjointness of the cubes in $\\mc{S}_P$ we have\n\\begin{align*}\n     \\sum_{P'\\in \\mc{S}_P} \\mu(P') \\leq c_2 \\, \\sum_{P'\\in \\mc{S}_P}  \\mu(P'\\cap \\Omega_P) \\leq c_2 \\,\\mu(\\Omega_P) \\leq \\frac{2c_1c_2}{\\lambda}  \\mu(P).\n\\end{align*}\n Therefore, by choosing $\\lambda=4c_1c_2$, we have $\\sum_{P'\\in \\mc{S}_P} \\mu(P')  \\leq \\frac12 \\mu(P)$.  This choice of $\\lambda$ also ensures that $\\Omega_{P,c_2}$ is a proper subset of $P$ by as claimed before. We define $S^{k+1} := \\bigcup_{P \\in \\mc{S}^k} \\mc{S}_P$.\n\nNow take $\\mc{S}^1 = \\cbrace{Q}$, iteratively define $\\mc{S}^k$ for all $k \\in \\N$  as described above and set $\\mc{S} :=\\bigcup_{k \\in \\N} \\mc{S}^k$. Then $\\mc{S}$ is $\\frac12$-sparse family of cubes, since for any $P\\in \\mc{S}$ we can set $$E_P:= P\\setminus \\bigcup_{P'\\in \\mc{S}_P} P',$$ which are pairwise disjoint by the fact that $\\bigcup_{P'\\in \\mc{S}^{k+1}} P' \\subseteq \\bigcup_{P\\in \\mc{S}^k} P$ for all $k \\in \\N$ and we have\n\\begin{equation*}\n  \\mu(E_P) = \\mu(P) - \\sum_{P'\\in \\mc{S}_P} \\mu(P') \\geq \\frac12 \\mu(P).\n\\end{equation*}\n\n\n\n\\textbf{Step 2:}\n We will now check that the sparse expression corresponding to $\\mc{S}$ constructed in Step 1 dominates $T_Qf$ pointwise. Since\n\\begin{equation*}\n\\lim_{k \\to \\infty}  \\mu\\hab{\\bigcup_{P\\in \\mc{S}^k}P} \\leq \\lim_{k \\to \\infty} \\frac{1}{2^k}\\, \\mu(Q) =0,\n\\end{equation*}\n we know that there is a set $N_0$ of measure zero  such that for all $s \\in Q\\setminus N_0$ there are only finitely many  $k \\in \\N$ with $s \\in  \\bigcup_{P\\in \\mc{S}^k}P$. Moreover by the Lebesgue differentiation theorem we have for any $P \\in \\mc{S}$ that\n  $\\ind_{\\Omega_P} (s) \\leq M^{\\ms{D}(P)}(\\ind_{\\Omega_P})(s)$\nfor a.e. $s \\in P$. Thus\n\\begin{equation}\\label{eq:Npprop}\n  \\Omega_P\\setminus N_P \\subseteq  \\Omega_{P,1}\\subseteq \\Omega_{P,c_2} = \\bigcup_{P'\\in \\mc{S}_P} P'\n\\end{equation} for some set $N_P$ of measure zero. We define\n $N:= N_0 \\cup\\bigcup_{P \\in \\mc{S}}N_P,$  which is a set of measure zero.\n\n\n\n  Fix $s \\in Q\\setminus N$ and take the largest $n \\in \\N$ such that $s \\in \\bigcup_{P\\in \\mc{S}^n}P$, which exists since $s \\notin N_0$. For $k =1,\\ldots,n$ let $P_k \\in \\mc{S}^k$ be the unique cube such that $s \\in P_k$ and note that by construction we have\n  $P_{n} \\subseteq  \\ldots \\subseteq P_1=Q.$\nUsing the localized $\\ell^r$-estimate of $T$ we split $\\nrm{T_Qf(s)}_Y^r $ into two parts\n\\begin{align*}\n  \\nrmb{T_Qf(s)}_Y^r  &\\leq C_r^r \\has{\\nrmb{T_{P_{n}}f(s)}_Y^r + \\sum_{k=1}^{n-1}\\nrmb{T_{P_k\\setminus P_{k+1}}f(s)}_Y^r}\\\\\n  &=:C_{r}^r \\has{ \\hspace{2pt} \\text{\\framebox[15pt]{A}}+  \\text{\\framebox[15pt]{B}} \\hspace{2pt} }.\n\\end{align*}\n\n\n\nFor \\framebox[15pt]{A} note that $s \\notin N_{P_n}$ and $s \\notin \\bigcup_{P'\\in \\mc{S}^{n+1}}P' $ and therefore by \\eqref{eq:Npprop} we know that $s \\in P_n \\setminus \\Omega_{P_n}$. So by the definition of $\\Omega_{P_n}^1$\n\\begin{equation*}\n  \\text{\\framebox[15pt]{A}} \\leq \\lambda^r \\, C_T^r\\, \\ipb{\\nrm{f}_X}_{p_0,\\alpha P_n}^r.\n\\end{equation*}\nFor $1\\leq k \\leq n-1$ we have by \\eqref{eq:OmegatoQ} and \\eqref{eq:Pjsize} that\n\\begin{align}\\label{eq:Pkmorethan14}\n\\begin{aligned}\n  \\mu\\hab{P_{k+1} \\setminus (\\Omega_{P_{k+1}} \\cup \\Omega_{P_{k}}) }&\\geq \\mu(P_{k+1}) - \\mu(\\Omega_{P_{k+1}}) - \\mu(P_{k+1} \\cap \\Omega_{P_k})\\\\\n  &\\geq \\mu(P_{k+1}) - \\frac{1}{2c_2}\\mu({P_{k+1}}) - \\frac{1}{2}\\mu(P_{k+1})>0,\n  \\end{aligned}\n\\end{align}\n so $P_{k+1} \\setminus (\\Omega_{P_{k+1}} \\cup \\Omega_{P_k})$ is non-empty. Take $s' \\in P_{k+1} \\setminus (\\Omega_{P_{k+1}} \\cup \\Omega_{P_k})$, then we have\n\n\\begin{equation*}\n\\begin{aligned}\n  \\nrmb{T_{P_{k}\\setminus P_{k+1}}f(s)}_Y &\\leq \\nrmb{T_{P_{k}\\setminus P_{k+1}}f(s)-T_{P_{k}\\setminus P_{k+1}}f(s')}_Y + \\nrmb{T_{P_{k}\\setminus P_{k+1}} f(s')}_Y\\\\\n&\\leq \\mc{M}_{T,P_k}^{\\#}f(s')  + \\nrmb{T_{P_k}(s')}_Y+\\nrmb{T_{P_{k+1}}(s')}_Y\\\\\n&\\leq 2\\lambda \\,C_T\\,\\hab{\\ipb{\\nrm{f}_X}_{p_0,\\alpha P_{k}} + \\ipb{\\nrm{f}_X}_{p_0,\\alpha P_{k+1}}},\n\\end{aligned}\n\\end{equation*}\nwhere we used the definition of $\\mc{M}_{T,P_k}^{\\#}$ and $T_{P_{k+1}\\setminus P_k}$ in the second inequality and $s' \\notin \\Omega_{P_{k+1}} \\cup \\Omega_{P_k}$ in the third inequality. Using $(a+b)^r \\leq 2^{r-1}(a^r+b^r)$ for any $a,b>0$ this implies that\n\\begin{align*}\n   \\text{\\framebox[15pt]{B}}  &\\leq \\sum_{k=1}^{n-1} 2^r 2^{r-1} \\lambda^r \\,C_T^r\\, \\has{ \\ipb{\\nrm{f}_X}^r_{p_0,\\alpha P_k} + \\ipb{\\nrm{f}_X}^r_{p_0,\\alpha P_{k+1}}}\\\\\n   &\\leq \\sum_{k=1}^{n} 4^r \\lambda^r \\,C_T^r\\,  \\ipb{\\nrm{f}_X}^r_{p_0,\\alpha P_k}.\n\\end{align*}\nCombining the estimates for  \\framebox[15pt]{A} and $\\text{\\framebox[15pt]{B}}$ we obtain\n\\begin{align*}\n  \\nrmb{T_Qf(s)}_Y\n  &\\leq 5\\,\\lambda\\, C_T\\, C_{r}\\, \\has{\\sum_{k=1}^n\\ipb{\\nrm{f}_X}_{p_0,\\alpha P_k}^r }^{1\/r}\\\\\n  &= 5\\,\\lambda\\, C_T\\, C_{r}\\, \\has{\\sum_{P \\in \\mc{S}}\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r \\ind_{P}(s)}^{1\/r}.\n\\end{align*}\nSince $s \\in Q\\setminus N$ was arbitrary and $N$ has measure zero, this inequality holds for a.e. $s \\in Q$.\nNoting that $\\lambda = 4c_1c_2$ and $c_1$ and $c_2$ only depend on $S$, $\\alpha$ and $\\ms{D}$  finishes the proof of the theorem.\n\\end{proof}\n\nAs announced Theorem \\ref{theorem:main} now follows directly from Theorem \\ref{theorem:localsparse} and a covering argument with Lemma \\ref{lemma:covering}.\n\n\\begin{proof}[Proof of Theorem \\ref{theorem:main}.] We will prove Theorem \\ref{theorem:main} in three steps: we will first show that the assumptions of Theorem \\ref{theorem:main} imply the assumptions of Theorem \\ref{theorem:localsparse}, then we will improve the local conclusion of Theorem \\ref{theorem:localsparse} to a global one and finally we will replace the averages over the dilation $\\alpha P$ in the conclusion of Theorem \\ref{theorem:localsparse} by the average over larger cubes $P'$.\n\nTo start let $\\ms{D}^1,\\ldots,\\ms{D}^m$ be as in Proposition \\ref{proposition:dyadicsystem} with parameters $c_0$, $C_0$, $\\delta$ and $\\gamma$, which only depend on $S$.\n\n\\textbf{Step 1:}  For any $Q \\in \\ms{D}^1$ define\n$T_Q$\n by $T_Qf(s) := T(f\\ind_{\\alpha Q})(s)$ for $s \\in Q$. Then:\n \\begin{itemize}\n   \\item $\\cbrace{T_Q}_{Q \\in \\ms{D}^1}$ is an $\\alpha$-localization family of $T$.\n   \\item For any $Q \\in \\ms{D}^1$ and $f \\in L^{p_1}(S;X)$ we have\n\\begin{align*}\n  \\mc{M}^{\\#}_{T,Q}f(s)  &\\leq\\mc{M}^{\\#}_{T,\\alpha}(f\\ind_{\\alpha Q})(s), \\qquad s \\in Q.\n\\end{align*}\n So by the weak $L^{p_2}$-boundedness of $\\mc{M}^{\\#}_{T,\\alpha}$ it follows that $\\mc{M}^{\\#}_{T,Q}f$ is weak $L^{p_2}$-bounded uniformly in $Q \\in \\ms{D}^1$.\n   \\item For any  $f \\in L^p(S;X)$ and $Q_1,\\ldots,Q_n \\in \\ms{D}^1$ with $Q_n \\subseteq \\ldots \\subseteq Q_1$ the functions $f_k:= f\\ind_{\\alpha Q_k \\setminus \\alpha Q_{k+1}}$ for $k=1,\\ldots,n-1$ and $f_n:= f\\ind_{\\alpha Q_n}$ are disjointly supported. Thus by the $r$-sublinearity of $T$\n       \\begin{equation*}\n\\nrmb{T_{Q_1}f(s)}_Y\\leq C_r \\has{\\nrmb{T_{Q_n}{f}(s)}_Y^r+\\sum_{k=1}^{n-1}\\nrmb{T_{Q_{k}\\setminus Q_{k+1}}f(s)}_Y^r}^{1\/r}, \\qquad s \\in Q_n.\n\\end{equation*}\n \\end{itemize}\nSo the assumptions of Theorem \\ref{theorem:localsparse} follow from the assumptions of Theorem \\ref{theorem:main}.\n\n\\textbf{Step 2:} Let $f \\in L^p(S;X)$ be boundedly supported. First suppose that $\\diam(S) = \\infty$ and let $E$ be a ball containing the support of $f$. By Lemma \\ref{lemma:covering} there is a partition $\\mc{D} \\subseteq \\ms{D}^1$ such that $E \\subseteq \\alpha Q$ for all $Q \\in \\mc{D}$. Thus by Theorem \\ref{theorem:localsparse} we can\nfind a $\\frac{1}{2}$-sparse collection of cubes $\\mc{S}_Q \\subseteq \\ms{D}^1(Q)$ for every $Q \\in \\mc{D}$ with\n\\begin{align}\n \\notag \\nrmb{Tf(s)}_Y &\\lesssim_{S,\\alpha}  C_T\\, C_r\\, \\has{\\sum_{P \\in \\mc{S}_Q} \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r\\ind_P(s)}^{1\/r}, \\qquad s \\in Q,\n\\intertext{where we used that $T_Qf = T(f \\ind_{\\alpha Q}) =Tf$ as $\\supp f \\subseteq \\alpha Q$.\nSince $\\mc{D}$ is a partition, $\\mc{S} := \\bigcup_{Q \\in \\mc{D}}S_Q$ is also a $\\frac{1}{2}$-sparse collection of cubes with}\n \\label{eq:sparsedomdilation}  \\nrmb{Tf(s)}_Y &\\lesssim_{S,\\alpha}  C_T\\, C_r\\, \\has{\\sum_{P \\in \\mc{S}} \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r\\ind_P(s)}^{1\/r}, \\qquad s \\in S,\n\\end{align}\nIf $\\diam (S) < \\infty$, then \\eqref{eq:sparsedomdilation}  follows directly from Theorem \\ref{theorem:localsparse} since $S \\in \\ms{D}$ in that case.\n\n\\textbf{Step 3:} For any $P \\in \\mc{S}$ with center $z$ and sidelength $\\delta^k$ we can find a $P' \\in \\ms{D}^j$ for some $1\\leq j\\leq m$ such that\n\\begin{equation*}\n\\alpha P = B(z,\\alpha  C_0 \\cdot \\delta^k) \\subseteq P', \\qquad \\diam(P') \\leq \\gamma \\alpha  C_0 \\cdot\\delta^k.\n\\end{equation*}\nTherefore there is a $c_1>0$ depending on $S$ and  $\\alpha$ such that\n$$\\mu(P') \\leq \\mu \\hab{B(z,\\gamma \\alpha  C_0 \\cdot \\delta^k)} \\leq c_1\\, \\mu \\hab{ B(z,c_0\\cdot \\delta^k) } \\leq c_1\\, \\mu(P).$$\nSo by defining $E_{P'}:= E_P$ we can conclude that the collection of cubes $\\mc{S}' := \\cbrace{P':P \\in \\mc{S}}$ is $\\frac{1}{2c_1}$-sparse. Moreover since $\\alpha P \\subseteq P'$ and $\\mu(P') \\leq c_1 \\, \\mu(P) \\leq c_1\\,\\mu(\\alpha P)$ for any $P \\in \\mc{S}$, we have\n\\begin{equation*}\n  \\ipb{\\nrm{f}_X}_{p_0,\\alpha P} \\leq c_1 \\ipb{\\nrm{f}_X}_{p_0,P'}.\n\\end{equation*}\nCombined with \\eqref{eq:sparsedomdilation} this proves the sparse domination in the conclusion of  Theorem \\ref{theorem:main}.\n\\end{proof}\n\n\\begin{remark}~\\label{remark:mainpointwise}\nThe assumption $\\alpha \\geq {3c_d^2\/\\delta}$ in Theorem \\ref{theorem:main} arises from the use of  Lemma \\ref{lemma:covering}, which transfers  the local sparse domination estimate of Theorem \\ref{theorem:localsparse} to the global statement of Theorem \\ref{theorem:main}. To deduce weighted estimates the local sparse domination estimate of Theorem \\ref{theorem:localsparse} suffices by testing against boundedly supported functions. However the operator norm of $\\mc{M}_{T,\\alpha}^{\\#}$ usually  becomes easier to estimate for larger $\\alpha$, so the lower bound on $\\alpha$ is not restrictive.\n\\end{remark}\n\n\n\\subsection*{Further generalizations}\nOur main theorems,  Theorem \\ref{theorem:main} and Theorem \\ref{theorem:localsparse}, allow for various further generalizations. One can for instance change the boundedness assumptions on $T$ and $\\mc{M}^{\\#}_{T,\\alpha}$, treat multilinear operators, or deduce domination by sparse forms for operators that do not admit a pointwise sparse estimate. We end this section by sketching some of these possible generalizations.\n\nIn \\cite[Section 3]{LO19} various variations and extensions of the main result in \\cite{LO19} are outlined. In particular they show:\n\\begin{itemize}\n  \\item The sparse domination for an individual function follows from assumptions on the same function. This can be exploited to prove a sparse $T(1)$-type theorem, see \\cite[Section 4]{LO19}.\n  \\item One can use certain Orlicz estimates to deduce sparse domination with Orlicz averages.\n  \\item The method of proof extends to the multilinear setting (see also \\cite{Li18}).\n\\end{itemize}\n Our results can also be extended in these directions, which we leave to the interested reader. In the remainder of this section, we will explore some further directions in which our results can be extended.\n\n \\bigskip\n\nSparse domination techniques have been successfully applied to \\emph{fractional integral operators}, see e.g. \\cite{CB13, CB13b,Cr17,IRV18}. In these works sparse domination and sharp weighted estimates are deduced for e.g.\n the Riesz potentials, which for $0<\\alpha<d$ and a Schwartz function $f\\colon \\R^d \\to \\C$ are given by\n\\begin{equation*}\n  I_\\alpha f(s):= \\int_{\\R^d}\\frac{f(t)}{\\abs{s-t}^{d-\\alpha}}\\dd t, \\qquad s \\in \\R^d,\n\\end{equation*}\nA key feature of such operators is that they are not (weakly) $L^p$-bounded, but bounded from $L^p(\\R^d)$ to $L^q(\\R^d)$, where $p,q \\in (1,\\infty)$ are such that $\\frac1p=\\frac1q+\\frac\\alpha{d}$. The sparse domination that one obtains in this case involves fractional sparse operators, in which the usual averages $\\ip{\\abs{f}}_{p,Q}$ are replaced by fractional averages.\n\nThese operators  fit in our framework with minimal effort. Indeed, upon inspection of the proof of Theorem \\ref{theorem:localsparse} it becomes clear that the only place where we use the boundedness of $T$ and $\\mc{M}^{\\#}_{T,\\alpha}$ is in \\eqref{eq:holdercomp}. Replacing the bounds with the off-diagonal bounds arising from fractional integral operators, we obtain the following variant of Theorem \\ref{theorem:main}.\n\n\\begin{theorem}\\label{theorem:fractional}\nLet $(S,d,\\mu)$ be a space of homogeneous type and let $X$ and $Y$ be Banach spaces. Take $p_0,q_0,r \\in [1,\\infty)$. Take $\\alpha \\geq 3c_d^2\/\\delta$, where $c_d$ is the quasi-metric constant and $\\delta$ is as in Proposition \\ref{proposition:dyadicsystem}. Assume the following conditions:\n\\begin{itemize}\n  \\item $T$ is a bounded linear operator from $L^{p_0}(S;X)$ to $L^{q_0,\\infty}(S;Y)$.\n  \\item $\\mc{M}_{T,\\alpha}^{\\#}$ is a bounded operator from $L^{p_0}(S;X)$ to $L^{q_0,\\infty}(S)$.\n  \\item $T$ is $r$-sublinear.\n\\end{itemize}\n Then there is an $\\eta \\in (0,1)$ such that for any boundedly supported $f \\in L^{p_0}(S;X)$ there is an $\\eta$-sparse collection of cubes $\\mc{S}$ such that\n\\begin{align*}\n   \\nrm{ Tf(s)}_Y&\\lesssim_{S,\\alpha} C_T \\,C_r\\, \\has{\\sum_{Q \\in \\mc{S}} \\mu(\\alpha P)^{\\frac{r}{p_0}-\\frac{r}{q_0}}\\ipb{\\nrm{f}_X}_{p_0,Q}^r \\ind_{Q}(s)}^{1\/r}, \\qquad s \\in S,\n\\end{align*}\nwhere $C_T={\\nrm{T}_{L^{p_0}\\to L^{p_0,\\infty}} + \\nrm{\\mc{M}_{T,\\alpha}^{\\#}}_{L^{p_0}\\to L^{p_0,\\infty}}}$ and $C_r$ is the $r$-sublinearity constant.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is the same as the proof of Theorem \\ref{theorem:main}, using an adapted version of Theorem \\ref{theorem:localsparse} with the canonical $\\alpha$-localization family\n$$T_Qf(s) = T(\\ind_{\\alpha Q}f)(s), \\qquad s \\in Q.$$\n  The only thing that changes in the proof of Theorem \\ref{theorem:localsparse} is the definition of $\\Omega_P^1$ and $\\Omega_P^2$ and the computation in \\eqref{eq:OmegatoQ}. Indeed, we define\n\\begin{align*}\n  \\Omega_{P}^1&:= \\cbraces{s \\in P: \\nrm{T_{P} f(s)}_Y> \\lambda\\, C_T \\,\\mu(\\alpha P)^{\\frac1{p_0}-\\frac1{q_0}}\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}}\\\\\n  \\Omega_{P}^2&:= \\cbraces{s \\in P: \\mc{M}_{T,P}^{\\#}(f)(s)> \\lambda \\,C_T\\,\\mu(\\alpha P)^{\\frac1{p_0}-\\frac1{q_0}}\\ipb{\\nrm{f}_X}_{p,\\alpha P}}\n\\end{align*}\nand then by the assumptions on $T$ and $\\mc{M}^{\\#}_{T,P}$  we have for $i=1,2$\n\\begin{align*}\n    \\mu(\\Omega_{P}^i) &\\leq \\has{\\frac{\\nrm{f \\ind_{\\alpha P}}_{L^{p_0}(S;X)}}{\\lambda \\,\\mu(\\alpha P)^{\\frac1{p_0}-\\frac1{q_0}}\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}}}^{q_0}\n    = \\frac{\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^{q_0}}{\\lambda^{q_0}\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^{q_0}} \\mu(\\alpha P)\n     \\leq  \\frac{ c_1}{\\lambda} \\, \\mu(P).\n\\end{align*}\nwhich proves \\eqref{eq:OmegatoQ}. In Step 2 of the proof of Theorem \\ref{theorem:localsparse} one needs to keep track of the factor $\\mu(\\alpha P)^{\\frac1{p_0}-\\frac1{q_0}}$ in the estimates.\n\\end{proof}\n\nIn the celebrated paper \\cite{BFP16} by Bernic\\'ot, Frey and Petermichl, domination by \\emph{sparse forms} was introduced to treat operators falling outside the scope of Calder\\'on--Zygmund theory. This method was later adopted by Lerner in \\cite{Le19} into his framework to prove sparse domination for rough homogeneous singular integral operators. As our methods are based on Lerner's sparse domination framework, our main result can also be generalized to the sparse form domination  setting.\n\nLet $(S,d,\\mu)$ be a space of homogeneous type with a dyadic system $\\ms{D}$, let $X$ and $Y$ be Banach spaces, $q \\in (1,\\infty)$, $p \\in [1,q)$ and $\\alpha \\geq 1$. For a bounded operator\n$$T\\colon L^{p}(S;X) \\to L^{p,\\infty}(S;Y)$$ with an $\\alpha$-localization family $\\cbrace{T_Q}_{Q \\in \\ms{D}}$ we define the \\emph{localized sharp grand $q$-maximal truncation operator} for $Q \\in \\ms{D}$ by\n \\begin{align*}\n  &\\mc{M}_{T,Q,q}^{\\#}f(s):= \\\\&\\hspace{1cm}\\sup_{\\substack{Q'\\in \\ms{D}(Q):\\\\s \\in Q'}}\\, \\has{\\avint_{Q'}\\avint_{Q'} \\,\\nrmb{(T_{Q\\setminus Q'}) f(s')-(T_{Q\\setminus Q'}) f(s'')}_Y^q\\dd \\mu(s')\\dd \\mu(s'')}^{1\/q}.\n\\end{align*}\nNote that for $q=\\infty$ one formally recovers the operator $\\mc{M}_{T,Q}^{\\#}$.\n\nWe will prove a version of Theorem \\ref{theorem:localsparse} for operators for which the truncation operators $\\mc{M}_{T,Q,q}^{\\#}$ are bounded uniformly in $Q \\in \\ms{D}$ using sparse forms. Of course taking\n \\begin{align*}\nT_Qf(s) &:=T(f \\ind_{\\alpha Q})(s), \\qquad s \\in Q.\n\\end{align*}\nfor $Q \\in \\ms{D}$ as the $\\alpha$-localization family one can easily deduce a statement like Theorem \\ref{theorem:main} in this setting, which we leave to the interested reader.\n\n\\begin{theorem}\\label{theorem:sparseform}\nLet $(S,d,\\mu)$ be a space of homogeneous type with dyadic system $\\ms{D}$ and let  $X$ and $Y$ be Banach spaces. Take $q_0 \\in (1,\\infty]$, $r \\in (0,q_0)$, $p_1,p_2 \\in [1,q_0)$, set $p_0:=\\max\\cbrace{p_1, p_2}$ and take $\\alpha \\geq 1$. Suppose that\n\\begin{itemize}\n  \\item $T$ is a bounded operator from $L^{p_1}(S;X)$ to $L^{p_1,\\infty}(S;Y)$ with an $\\alpha$-localization family $\\cbrace{T_Q}_{Q \\in \\ms{D}}$.\n  \\item $\\mc{M}_{T,Q,q_0}^{\\#}$ is bounded from $L^{p_2}(S;X)$ to $L^{p_2,\\infty}(S)$ uniformly in $Q \\in \\ms{D}$.\n  \\item $T$ satisfies a localized $\\ell^r$-estimate.\n\\end{itemize}\n Then for any $f \\in L^{p_0}(S;X)$, $g \\in L^{\\ha{\\frac1r-\\frac1{q_0}}^{-1}}(S)$ and $Q\\in \\ms{D}$  there exists a $\\frac12$-sparse collection of dyadic cubes $\\mc{S}\\subseteq \\ms{D}(Q)$ such that\n \\begin{equation*}\n   \\has{\\int_Q \\nrmb{T_Qf}_Y^r\\cdot \\abs{g}^r\\dd\\mu }^{1\/r} \\lesssim_{S,\\ms{D},\\alpha,r} C_T \\, C_r \\has{\\sum_{P \\in \\mc{S}} \\mu(P) \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r \\ipb{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P}^r}^{1\/r}\n \\end{equation*}\nwith $C_T:={\\nrm{T}_{L^{p_1}\\to L^{p_1,\\infty}} + \\sup_{P \\in \\ms{D} } \\nrm{\\mc{M}_{T,P,q_0}^{\\#}}_{L^{p_2}\\to L^{p_2,\\infty}}}$ and $C_r$ the constant from the localized $\\ell^r$-estimate.\n\\end{theorem}\n\n\n\n\\begin{proof}\nWe construct the sparse collection of cubes $\\mc{S}$ exactly as in Step 1 of the proof of Theorem \\ref{theorem:localsparse}, using $\\mc{M}_{T,P,q_0}^{\\#}$ instead of $\\mc{M}_{T,P}^{\\#}$ in the definition of $\\Omega_P^2$. We will check that sparse form corresponding to $\\mc{S}$ satisfies the claimed domination property, which will roughly follow the same lines as Step 2 of the proof of Theorem \\ref{theorem:localsparse}.\n\nFix $f \\in L^{p_0}(S;X)$ and $g \\in L^{\\ha{\\frac1r-\\frac1{q_0}}^{-1}}(S)$. Note that for a.e. $s \\in Q$\nthere are only finitely many  $k \\in \\N$ with $s \\in  \\bigcup_{P\\in \\mc{S}^k}P$. So we can use the localized $\\ell^r$-estimate of $T$ to split\n\\begin{equation}\\label{eq:splitform}\\begin{aligned}\n  \\int_Q \\nrmb{T_Qf}_Y^r\\cdot \\abs{g}^r  &\\leq C_r^r \\sum_{k\\in \\N} \\sum_{P \\in \\mc{S}^k} \\has{\\int_{P \\setminus \\bigcup_{P' \\in \\mc{S}^{k+1}}P'} \\nrmb{T_Pf}_Y^r\\cdot \\abs{g}^r \\\\&\\hspace{2cm}+\\sum_{P' \\in \\mc{S}^{k+1}:P'\\subseteq P} \\int_{P'}\\nrmb{T_{P\\setminus P'}f}_Y^r\\cdot \\abs{g}^r}\\\\\n  &=:C_{r}^r \\sum_{k\\in \\N} \\sum_{P \\in \\mc{S}^k} \\has{ \\hspace{2pt} \\text{\\framebox[20pt]{A$_{P}$}}+  \\text{\\framebox[20pt]{B$_{P}$}} \\hspace{2pt} }.\\end{aligned}\n\\end{equation}\nFix $k \\in \\N$ and $P \\in \\mc{S}^k$. As in the estimate for \\framebox[15pt]{A} in Step 2 of the proof of Theorem \\ref{theorem:localsparse}, we have\n\\begin{align*}\n  \\text{\\framebox[20pt]{A$_{P}$}} &\\leq \\lambda^r \\, C_T^r\\, \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r \\int_P\\abs{g}^r \\leq  \\lambda^r \\, C_T^r\\, \\mu(P) \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}}^r,\n\\end{align*}\nusing H\\\"older's inequality in the second inequality.\nFor $P' \\in \\mc{S}^{k+1}$ such that $P'\\subseteq P$ we have as in \\eqref{eq:Pkmorethan14} that\n\\begin{align*}\n  \\mu\\hab{P' \\setminus (\\Omega_{P'} \\cup \\Omega_{P}) }  &\\geq \\frac14 \\mu(P').\n\\end{align*}\nTherefore we can estimate each of the terms in the sum in \\framebox[20pt]{B$_{P}$} as follows\n\\begin{equation*}\n\\begin{aligned}\n  \\int_{P'}&\\nrmb{T_{P\\setminus P'}f}_Y^r\\cdot \\abs{g}^r\\\\ &\\leq 2^{r}\\int_{P'} \\avint_{P' \\setminus (\\Omega_{P} \\cup \\Omega_{P'})} \\nrmb{T_{P\\setminus P'}f(s)-T_{P\\setminus P'}f(s')}_Y^r\\cdot \\abs{g(s)}^r \\dd \\mu(s')\\dd \\mu(s)\\\\\n  &\\hspace{2.2cm}+ 2^{r}\\int_{P'} \\avint_{P' \\setminus (\\Omega_{P} \\cup \\Omega_{P'})} \\nrmb{T_{P\\setminus P'}f(s')}_Y^r \\cdot  \\abs{g(s)}^r \\dd \\mu(s')\\dd \\mu(s)\\\\\n  &\\leq 2^{r+2} \\mu(P') \\inf_{s'' \\in P'} \\mc{M}_{T,P,{q_0}}^{\\#}f(s'')^r\\cdot \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P'}^r\\\\\n  &\\hspace{1.5cm}+ 2^{2r} \\mu(P') \\avint_{P' \\setminus (\\Omega_{P} \\cup \\Omega_{P'})} \\nrmb{T_{P}f}_Y^r +\\nrmb{T_{P'}f}_Y^r \\dd \\mu \\cdot \\ip{\\abs{g}}_{r, P'}^r\\\\\n  &\\leq  4^{r+2}\\lambda^r C_T^r\\,\\mu(P') \\hab{\\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r + \\ipb{\\nrm{f}_X}_{p_0,\\alpha P'}^r} \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P'}^r\n\\end{aligned}\n\\end{equation*}\nwhere we used\nH\\\"older's inequality and the definitions of $\\mc{M}_{T,P,{q_0}}^{\\#}$ and $T_{P\\setminus P'}$ in the second inequality and the definitions of $\\Omega_{P}$ and $\\Omega_{P'}$ in the third inequality. Furthermore we note that by H\\\"olders inequality we have\n\\begin{align*}\n  \\sum_{\\substack{P' \\in \\mc{S}^{k+1}:\\\\P'\\subseteq P}} \\mu(P') \\, \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P'}^r &\\leq \\has{\\sum_{\\substack{P' \\in \\mc{S}^{k+1}:\\\\P'\\subseteq P}} \\int_{P'}\\abs{g}^{\\frac{1}{\\frac1r-\\frac1{q_0}}}\\dd \\mu}^{1-\\frac{r}{{q_0}}} \\cdot \\has{\\sum_{\\substack{P' \\in \\mc{S}^{k+1}:\\\\P'\\subseteq P}} \\mu(P')}^{r\/{q_0}}\\\\\n  &\\leq \\has{ \\int_{P}\\abs{g}^{\\frac{1}{\\frac1r-\\frac1{q_0}}}\\dd \\mu}^{1-\\frac{r}{{q_0}}} \\cdot {\\mu(P)}^{r\/{q_0}}= \\mu(P) \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P}^r\n\\end{align*}\nThus for \\framebox[20pt]{B$_{P}$} we obtain\n\\begin{align*}\n  \\text{\\framebox[20pt]{B$_{P}$}}\\leq 4^{r+2}\\lambda^r C_T^r &\\has{ \\mu(P) \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r  \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P}^r \\\\&+ \\sum_{P' \\in \\mc{S}^{k+1}:P'\\subseteq P} \\mu(P')  \\ipb{\\nrm{f}_X}_{p_0,\\alpha P'}^r \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P'}^r}\n\\end{align*}\nPlugging this estimate and the estimate for \\framebox[20pt]{A$_{P}$} into \\eqref{eq:splitform} yields\n\\begin{equation*}\n  \\int_Q \\nrmb{T_Qf}_Y^r\\cdot \\abs{g}^r \\dd \\mu \\leq  4^{r+3} \\lambda^r \\, C_T^r \\, C_r^r \\sum_{P \\in \\mc{S}} \\mu(P) \\ipb{\\nrm{f}_X}_{p_0,\\alpha P}^r \\ip{\\abs{g}}_{\\frac{1}{\\frac1r-\\frac1{q_0}}, P}^r.\n\\end{equation*}\nSince $\\lambda = 4c_1c_2$ and $c_1$ and $c_2$ only depend on $S$, $\\alpha$ and $\\ms{D}$, this finishes the proof of the theorem.\n\\end{proof}\n\n\n\n\\section{Weighted bounds for sparse operators}\\label{section:weights}\nAs discussed in the introduction, one of the main motivations to study sparse domination for an operator is to obtain (sharp) weighted bounds. In this section we will introduce Muckenhoupt weights and state weighted $L^p$-bounds for the sparse operators in the conclusions of Theorem \\ref{theorem:main} and Theorem \\ref{theorem:localsparse}, which are well-known in the Euclidean setting.\n\nLet $(S,d,\\mu)$ be a space of homogeneous type. A \\emph{weight} is a locally integrable function $w\\colon S \\to (0,\\infty)$. For $p \\in [1,\\infty)$, a Banach space $X$ and a weight $w$ the weighted Bochner space $L^p(S,w;X)$ is the space of all strongly measurable $f:S \\to X$ such that\n\\begin{equation*}\n  \\nrm{f}_{L^p(S,w;X)}:= \\has{\\int_S\\nrm{f(s)}_X^pw\\dd \\mu}^{1\/p}<\\infty.\n\\end{equation*}\nFor $p\\in [1,\\infty)$ and a weight $w$ we say that $w$ lies in the \\emph{Muckenhoupt class $A_p$} and write $w\\in A_p$ if its \\emph{$A_p$-characteristic} satisfies\n\\begin{equation*}\n  [w]_{A_p}:= \\sup_{B\\subseteq S} \\ip{w}_{1,B}\\ip{w^{-1}}_{\\frac{1}{p-1},B}<\\infty,\n\\end{equation*}\nwhere the supremum is taken over all balls $B \\subseteq S$ and the second factor is replaced by $\\esssup_B w^{-1}$ if $p=1$.\nFor an introduction to Muckenhoupt weights we refer to \\cite[Chapter 7]{Gr14a}.\n\n\\bigskip\n\nLet $p_0, r \\in [1,\\infty)$, $p \\in (p_0,\\infty)$, $w \\in A_{p\/p_0}$. We are interested in the boundedness on $L^p(S,w)$ of sparse operators of the form\n\\begin{equation}\\label{eq:sparseop2}\n  f\\mapsto \\has{\\sum_{Q \\in \\mc{S}} \\ipb{\\abs{f}}_{p_0,Q}^r \\ind_{Q}}^{1\/r},\n\\end{equation}\nwhich appear in the conclusions of Theorem \\ref{theorem:main} and Theorem \\ref{theorem:localsparse}.\nIn the Euclidean case such bounds are thoroughly studied and most of the arguments extend directly to spaces of homogeneous type. For the convenience of the reader we will give a self-contained proof of the strong weighted $L^p$-boundedness of these sparse operators in spaces of homogeneous type, following the proof of \\cite[Lemma 4.5]{Le16}. For further results we refer to:\n\\begin{itemize}\n\\item Weak weighted $L^p$-boundedness (including the endpoint $p=p_0$), for the sparse operators in \\eqref{eq:sparseop2} can be found \\cite{HL18, FN18}.\n\\item More precise bounds in terms of two-weight $A_p$-$A_\\infty$-characteristics  for various special cases of the sparse operators in \\eqref{eq:sparseop2} can be found in e.g. \\cite{FH18, HL18,HP13,LL16}.\n  \\item Weighted bounds for the fractional sparse operators in Theorem \\ref{theorem:fractional} can be found in \\cite{FH18}\n  \\item Weighted bounds for the sparse forms in Theorem \\ref{theorem:sparseform} can be found in \\cite{BFP16, FN18}.\n\\end{itemize}\n\n\\begin{proposition}\\label{proposition:weights}\n  Let $(S,d,\\mu)$ be a space of homogeneous type, let $\\mc{S}$ be an $\\eta$-sparse collection of cubes and take $p_0, r \\in [1,\\infty)$.\nFor $p \\in (p_0,\\infty)$, $w \\in A_{p\/p_0}$ and $f \\in L^p(S,w)$ we have\n\\begin{align*}\n  \\nrms{\\has{\\sum_{Q \\in \\mc{S}} \\ipb{\\abs{f}}_{p_0,Q}^r \\ind_{Q}}^{1\/r}}_{ L^p(S,w)} &\\lesssim [w]_{A_{p\/p_0}}^{\\max\\cbraceb{\\frac{1}{p-p_0},\\frac{1}{r}}} \\nrm{f}_{L^p(S,w)},\n\\end{align*}\nwhere\nthe implicit constant depends on $S,p_0,p,r$ and $\\eta$.\n\\end{proposition}\n\n\\begin{proof}\nWe first note that by Proposition \\ref{proposition:dyadicsystem} we may assume without loss of generality that  $\\mc{S} \\subseteq \\ms{D}$, where $\\ms{D}$ is an arbitrary dyadic system in $(S,d,\\mu)$. Furthermore if $p -p_0\\leq r$ we have $\\max\\cbraceb{\\frac{1}{p-p_0},\\frac{1}{r}} = \\frac{1}{p-p_0}$. Since $\\ell^{p-p_0}\\hookrightarrow \\ell^r$, the case $p -p_0\\leq r$ follows from the case $p -p_0=r$, so without loss of generality we may also assume $p\\geq p_0+r$.\n\nFor a weight $u$ and a measurable set $E$ we define $u(E):=\\int_Eu\\dd \\mu$ and we denote the dyadic Hardy--Littlewood maximal operator with respect to the measure $u\\dd \\mu$ by $M^{\\ms{D},u}$, which\nis bounded on $L^p(S,u)$ for all $p \\in (1,\\infty)$ by Doob's maximal inequality (see e.g. \\cite[Theorem 3.2.2]{HNVW16}). Take $f \\in L^p(S,w)$, set $q := (p\/r)'=\\frac{p}{p-r}$ and take\n$$g \\in L^{q}(S,w^{1-q}) = \\hab{L^{p\/r}(S,w)}^*.$$\nThen we have by the disjointness of the $E_Q$'s associated to each $Q \\in \\mc{S}$\n\\begin{equation}\\label{eq:gsparse}\n\\begin{aligned}\n  \\sum_{Q \\in \\mc{S}} w(E_Q) \\has{\\frac{\\mu(Q)}{w(Q)}}^{q} \\ipb{\\abs{g}}_{1,Q}^{q}\n  &\\leq \\sum_{Q \\in \\mc{S}}\\int_{E_Q} M^{\\ms{D},w}(gw^{-1})^{q} w \\dd\\mu\\\\\n  &\\leq \\nrmb{M^{\\ms{D},w}(gw^{-1})}_{L^{q}(S,w)}^{q}\\\\&\\lesssim_{p,r} \\nrm{g}_{L^{q}(S,w^{1-q})}^{q}\n\\end{aligned}\n\\end{equation}\nand similarly, setting $\\sigma:= w^{1-(p\/p_0)'}$, we have\n\\begin{equation}\\label{eq:fsparse}\n\\begin{aligned}\n  \\sum_{Q \\in \\mc{S}} \\sigma(E_Q) \\has{\\frac{\\mu(Q)}{\\sigma(Q)}}^{\\frac{p}{p_0}} \\ipb{\\abs{f}^{p_0}}_{1,Q}^{{p}\/{p_0}}\n  &\\leq \\nrmb{M^{\\ms{D},\\sigma}(\\abs{f}^{p_0}\\sigma^{-1})}_{L^{p\/p_0}(S,\\sigma)}^{p\/p_0}\\\\\n  &\\lesssim_{p,p_0} \\nrm{f}_{L^{p}(S,w)}^{p}\n\\end{aligned}\n\\end{equation}\nusing $\\sigma \\cdot \\sigma^{-p_0\/p}=w$.\nDefine the constant\n\\begin{equation*}\n  c_w := \\sup_{Q \\in \\ms{D}} \\frac{w(Q)^{1\/r}}{w(E_Q)^{\\frac1r-\\frac1p}} \\frac{\\sigma(Q)^{1\/p_0}}{\\sigma(E_Q)^{1\/p} } \\frac{1}{\\mu(Q)^{1\/p_0}},\n\\end{equation*}\nThen by H\\\"olders inequality, \\eqref{eq:gsparse} and \\eqref{eq:fsparse} we have\n\\begin{align*}\n  \\int_S \\has{\\sum_{Q \\in \\mc{S}} \\ipb{\\abs{f}}_{p_0,Q}^r \\ind_{Q}}\\cdot g \\dd \\mu  &= \\sum_{Q \\in \\mc{S}} \\mu(Q) \\ipb{\\abs{f}^{p_0}}_{1,Q}^{r\/p_0} \\ip{\\abs{g}}_{1,Q}\\\\\n     &\\leq  c_w^r \\sum_{Q \\in \\mc{S}} \\has{ \\sigma(E_Q)^{r\/p} \\has{\\frac{\\mu(Q)}{\\sigma(Q)}}^{r\/p_0} \\ipb{\\abs{f}^{p_0}}_{1,Q}^{r\/p_0}}\\\\&\\hspace{1cm}\\cdot\\has{ w(E_Q)^{{1\/q}}\\frac{\\mu(Q)}{w(Q)} \\ip{\\abs{g}}_{1,Q}}\\\\\n   &\\lesssim_{p,p_0,r} c_w^r \\nrmb{f}_{L^{p}(S,w)}^{r} \\nrm{g}_{L^{q}(S,w^{1-q})}.\n\\end{align*}\nSo by duality it remains to show  $c_w \\lesssim [w]_{A_{p\/p_0}}^{\\max\\cbraceb{\\frac{1}{p-p_0},\\frac{1}{r}}}$. Fix a $Q \\in \\ms{D}$ and  note that by H\\\"olders's inequality we have\n\\begin{equation*}\n   \\mu(Q)^{p\/p_0}\\leq \\eta^{p\/p_0} \\has{\\int_{E_Q}w^{p_0\/p} w^{-p_0\/p} \\dd \\mu}^{p\/p_0}\n \\leq \\eta^{p\/p_0} \\,w(E_Q)\\,\\sigma(E_Q)^{p\/p_0-1}.\n\\end{equation*}\nand thus\n\\begin{equation*}\n  \\frac{w(Q)}{w(E_Q)}\\has{\\frac{\\sigma(Q)}{\\sigma(E_Q)}}^{{p\/p_0}-1} \\leq \\eta^{p\/p_0} \\frac{w(Q)}{\\mu(Q)}\\has{\\frac{\\sigma(Q)}{\\mu(Q)}}^{{p\/p_0}-1} \\lesssim_S \\eta^{p\/p_0}[w]_{A_{p\/p_0}}.\n\\end{equation*}\nTherefore we can estimate\n\\begin{align*}\n  c_w &= \\sup_{Q \\in \\ms{D}} \\bracs{\\frac{w(Q)}{\\mu(Q)}\\has{\\frac{\\sigma(Q)}{\\mu(Q)}}^{\\frac{p}{p_0}-1}}^{\\frac1p} \\cdot \\bracs{\\has{\\frac{w(Q)}{w(E_Q)}}^{\\frac1r-\\frac1p} \\has{\\frac{\\sigma(Q)}{\\sigma(E_Q)}}^{\\frac1p}}\\\\\n  &\\lesssim_S [w]_{A_{p\/p_0}}^{\\frac1p} \\, \\sup_{Q \\in \\ms{D}} \\bracs{\\frac{w(Q)}{w(E_Q)}\\has{\\frac{\\sigma(Q)}{\\sigma(E_Q)}}^{\\frac{p}{p_0}-1}}^{\\max \\cbraceb{{\\frac1r-\\frac1p}, \\frac1p\\frac{p_0}{p-p_0}}}\\\\\n  &\\lesssim_{S,\\eta}  [w]_{A_{p\/p_0}}^{\\frac1p+\\max \\cbraceb{{\\frac1r-\\frac1p}, \\frac1p\\frac{p_0}{p-p_0}} } = [w]_{A_{p\/p_0}}^{\\max\\cbraceb{\\frac{1}{p-p_0},\\frac{1}{r}}},\n\\end{align*}\nwhich  finishes the proof.\n\\end{proof}\n\n\n\\section{Banach space geometry and $\\mc{R}$-boundedness}\\label{section:Banachspace}\nBefore turning to applications of Theorem \\ref{theorem:main} and Theorem \\ref{theorem:localsparse} in the subsequent sections, we first need to introduce some geometric properties of a Banach space $X$ and the $\\mc{R}$-boundedness of a family of operators.\n\n\\subsection{Type and cotype}Let $(\\varepsilon_k)_{k=1}^\\infty$ be a sequence of independent \\emph{Rademacher variables} on $\\Omega$, i.e. uniformly distributed random variables taking values in $\\cbrace{z \\in \\K:\\abs{z} = 1}$. We say that a Banach space $X$ has (Rademacher) type $p \\in [1,2]$ if for any $x_1,\\ldots,x_n \\in X$ we have\n\\begin{equation*}\n  \\nrms{\\sum_{k=1}^n\\varepsilon_k x_k}_{L^2(\\Omega;X)} \\lesssim_{X,p} \\has{\\sum_{k=1}^n \\nrm{x_k}_X^p}^{1\/p},\n\\end{equation*}\nand say that $X$ has nontrivial type if $X$ has type $p>1$.\nWe say that $X$ has (Rademacher) cotype $q \\in [2,\\infty]$ if for any $x_1,\\ldots,x_n \\in X$ we have\n\\begin{equation*}\n   \\has{\\sum_{k=1}^n \\nrm{x_k}_X^q}^{1\/q}\\lesssim_{X,q} \\nrms{\\sum_{k=1}^n\\varepsilon_k x_k}_{L^2(\\Omega;X)},\n\\end{equation*}\n and say that $X$ has finite cotype if $X$ has cotype $q<\\infty$. See \\cite[Chapter 7]{HNVW17} for an introduction to type and cotype.\n\n\\subsection{Banach lattices and $p$-convexity and $q$-concavity.}\nA Banach lattice is a partially ordered Banach space $X$ such that for $x,y \\in X$\n\\begin{equation*}\n  \\abs{x} \\leq \\abs{y} \\Rightarrow \\nrm{x}_X \\leq \\nrm{y}_Y.\n\\end{equation*}\nOn a Banach lattice there are two properties that are closely related to type and cotype. We say that a Banach lattice is \\emph{$p$-convex} with $p \\in [1,\\infty]$ if for $x_1,\\ldots,x_n\\in X$\n\\begin{equation*}\n  \\nrms{\\has{\\sum_{k=1}^n\\abs{x_k}^p}^{1\/p}}_{X} \\lesssim_{X,p} \\has{\\sum_{k=1}^n \\nrm{x_k}^p}^{1\/p},\n\\end{equation*}\nwhere the sum on the left-hand side is defined through the Krivine calculus. A Banach lattice is called \\emph{$q$-concave} for $q \\in [1,\\infty]$ if for $x_1,\\ldots,x_n\\in X$\n\\begin{equation*}\n   \\has{\\sum_{k=1}^n \\nrm{x_k}^q}^{1\/q} \\lesssim_{X,q} \\nrms{\\has{\\sum_{k=1}^n\\abs{x_k}^q}^{1\/q}}_{X}.\n\\end{equation*}\nIf a  Banach lattice has finite cotype then $p$-convexity implies type $p$. Conversely type $p$ implies $r$-convexity for all $1\\leq r< p$. Similar relations hold for cotype $q$ and $q$-concavity. We refer to \\cite[Chapter 1]{LT79} for an introduction to Banach lattices, $p$-convexity and $q$-concavity.\n\n\n\\subsection{The \\texorpdfstring{$\\UMD$}{UMD} property} We say that a Banach space $X$ has the {$\\UMD$ property} if the martingale difference sequence of any finite martingale in $L^p(\\Omega;X)$ is unconditional for some (equivalently all) $p \\in (1,\\infty)$. The $\\UMD$ property implies reflexivity, nontrivial type and finite cotype.\nFor an  introduction to the theory of $\\UMD$ Banach spaces we refer the reader to \\cite[Chapter 4]{HNVW16} and \\cite{Pi16}.\n\n\\subsection{\\texorpdfstring{$\\mc{R}$}{R}-Boundedness} Let $X$ and $Y$ be Banach spaces and $\\Gamma \\subseteq \\mc{L}(X,Y)$. We say that $\\Gamma$ is $\\mc{R}$-bounded if for any $x_1,\\ldots,x_n$ and $T_1,\\ldots,T_n \\in \\Gamma$ we have\n\\begin{equation*}\n  \\has{\\E\\nrmb{\\sum_{k=1}^n \\varepsilon_k T_kx_k}^2}^{1\/2} \\lesssim \\has{\\E\\nrmb{\\sum_{k=1}^n \\varepsilon_k x_k}^2}^{1\/2},\n\\end{equation*}\n where $(\\varepsilon_k)_{k=1}^\\infty$ is a sequence of independent {Rademacher variables}\nThe least admissible implicit constant is denoted by $\\mc{R}(\\Gamma)$. $\\mc{R}$-boundedness is a strengthening of uniform boundedness and is often a key assumption to prove boundedness of operators on Bochner spaces. We refer to \\cite[Chapter 8]{HNVW17} for an introduction to $\\mc{R}$-boundedness.\n\n\\section{The \\texorpdfstring{$A_2$}{A2}-theorem for operator-valued  Calder\\'on--Zygmund operators in a space of homogeneous type} \\label{section:A2}\nThe $A_2$-theorem, first proved by Hyt\\\"onen in \\cite{Hy12} as discussed in the introduction, states that a Calder\\'on--Zygmund operator is bounded on $L^2(\\R^d,w)$ with a bound that depends linearly on the $A_2$-characteristic of $w$. From this sharp weighted bounds for all $p\\in (1,\\infty)$ can be obtained by sharp Rubio de Francia extrapolation \\cite{DGPP05}. Since its first proof by Hyt\\\"onen, the $A_2$-theorem has been extended in various directions. We mention two of these extensions relevant for the current discussion:\n\\begin{itemize}\n  \\item The $A_2$-theorem for Calder\\'on--Zygmund operators on a geometric doubling metric space\n was first proven by Nazarov, Reznikov and Volberg \\cite{NRV13}, afterwards it was proven on a space of homogeneous type\n by Anderson and  Vagharshakyan \\cite{AV14} (see also \\cite{An15}) using Lerner's mean oscillation decomposition method. It was further extended to the setting of ball bases by Karagulyan \\cite{Ka16}.\n  \\item The $A_2$-theorem for vector-valued Calder\\'on--Zygmund operators with operator-valued kernel was proven by H\\\"anninen and Hyt\\\"onen \\cite{HH14}, using a suitable adapted version of Lerner's median oscillation decomposition.\n\\end{itemize}\nIn this section we will prove sparse domination for vector-valued Calder\\'on--Zygmund operators with operator-valued kernel on a space of homogeneous type. This yields the $A_2$-theorem for these Calde\\'ron--Zygmund operators, unifying the results from \\cite{AV14} and \\cite{HH14}.\n\nAs an application of this theorem, we will prove a weighted, anisotropic, mixed norm Mihlin multiplier theorem in the next section. We will also use it to study maximal regularity for parabolic partial differential equations in forthcoming work. In these applications $S$ is (a subset of) $\\R^d$ equipped with the anisotropic quasi-norm\n\\begin{equation}\\label{eq:anisotropic}\n  \\abs{s}_{\\mbs{a}} := \\has{\\sum_{j=1}^d\\abs{s_j}^{2\/a_j}}^{1\/2}, \\qquad s \\in \\R^d.\n\\end{equation}\nfor some $\\mbs{a} \\in (0,\\infty)^d$ and the Lebesgue measure.\n\nIn a different direction our $A_2$-theorem  can be applied in the study of fundamental harmonic analysis operators associated with various discrete and continuous orthogonal expansions, started by Muckenhoupt and Stein \\cite{MS65}. In the past decade there has been a surge of results in which such operators are proven to be vector-valued Calder\\'on--Zygmund operators on concrete spaces of homogeneous type. Weighted bounds are then often concluded using \\cite[Theorem III.1.3]{RRT86} or \\cite{RT88}. With our $A_2$-theorem these results can be made quantitative in terms of the $A_p$-characteristic. We refer to \\cite{BCN12,BMT07,CGRTV17,NS12,NS07} and the references therein for an overview of the recent developments in this field.\n\n\n\n\\bigskip\n\nLet $(S,d,\\mu)$ be a space of homogeneous type, $X$ and $Y$ be Banach spaces and let $$K\\colon(S \\times S)\\setminus\\cbrace{(s,s):s \\in S} \\to \\mc{L}(X,Y)$$\nbe strongly measurable in the strong operator topology.\nWe say that $K$ is a \\emph{Dini kernel} if there is a $c_K \\geq 2$ such that\n  \\begin{align*}\n   \\nrm{K(s,t)-K(s,t')} &\\leq\n   \\omega \\has{\\frac{d(t,t')}{d(s,t)}}\\frac{1}{\\mu\\hab{{B(s,d(s,t))}}},  &&0<d(t,t')\\leq \\frac{1}{c_K}d(s,t),\\\\\n    \\nrm{K(s,t)-K(s',t)} &\\leq  \\omega \\has{\\frac{d(s,s')}{d(s,t)}} \\frac{1}{\\mu\\hab{{B(s,d(s,t))}}}, &&0<d(s,s')\\leq \\frac{1}{c_K}d(s,t),\n  \\end{align*}\n    where $\\omega:[0,1]\\to [0,\\infty)$ is increasing, subadditive, $\\omega(0)=0$ and\n  \\begin{equation*}\n    \\nrm{K}_{\\Dini}:=\\int_0^1\\omega(t)\\frac{\\ddn t}{t} <\\infty.\n  \\end{equation*}\n  Take $p_0 \\in [1,\\infty)$ and let\n\\begin{equation*}\n   T \\colon L^{p_0}(S;X) \\to L^{p_0,\\infty}(S;Y)\n\\end{equation*}\nbe a bounded linear operator. We say that $T$ has Dini kernel $K$ if for every boundedly supported $f \\in L^{p_0}(S;X)$ and a.e. $s \\in S \\setminus \\overline{\\supp f}$ we have\n\\begin{equation*}\n  Tf(s) = \\int_S K(s,t)f(t)\\dd t.\n\\end{equation*}\n\n\\begin{theorem}\\label{theorem:A2}\n Let $(S,d,\\mu)$ be a space of homogeneous type and let $X$ and $Y$ be Banach spaces. Let $p_0 \\in [1,\\infty)$ and suppose $T$ is a bounded linear operator from $L^{p_0}(S;X)$ to  $L^{p_0,\\infty}(S;Y)$\nwith Dini kernel $K$.  Then for every boundedly supported $f \\in L^1(S;X)$ there exists an $\\eta$-sparse collection of cubes $\\mc{S}$ such that\n\\begin{equation*}\n    \\nrm{Tf(s)}_Y\\lesssim_{S,p_0} \\,C_T\\, \\sum_{Q \\in \\mc{S}} \\ipb{\\nrm{f}_X}_{1,Q} \\ind_{Q}(s), \\qquad s \\in S.\n\\end{equation*}\n Moreover, for all $p \\in (1,\\infty)$ and $w \\in A_p$ we have\n \\begin{align*}\n   \\nrm{T}_{L^p(S,w;X) \\to L^p(S,w;Y)} &\\lesssim_{S,p,p_0} C_T \\,[w]_{A_p}^{\\max\\cbrace{\\frac{1}{p-1},1}}\n \\end{align*}\n with $C_{T}:= \\nrm{T}_{L^{p_0}(S;X)\\to L^{p_0,\\infty}(S;Y)} + \\nrm{K}_{\\Dini}$.\n\\end{theorem}\n\n\\begin{proof}\n  We will check the assumptions of Theorem \\ref{theorem:main} with $p_1=p_2=r=1$. The  weak $L^1$-boundedness of $T$ with\n  \\begin{equation*}\n    \\nrm{T}_{L^{1}(S;X)\\to L^{1,\\infty}(S;Y)} \\lesssim_{S,p} C_T.\n  \\end{equation*}\n  follows from the classical Calder\\'on-Zygmund argument, see e.g. \\cite[Theorem III.1.2]{RRT86}.\n  The $1$-sublinearity assumption on $T$ follows from the triangle inequality, so the only thing left to check is the weak $L^1$-boundedness of $\\mc{M}_{T,\\alpha}^{\\#}$.\n Let\n $$\\alpha := 3 \\,c_d^2 \\,\\max\\cbraceb{\\delta^{-1},c_K}$$\n  with $c_d$ the quasi-metric constant, $\\delta$ as in Proposition \\ref{proposition:dyadicsystem} and $c_K$ the constant from the definition of a Dini kernel.\n  Fix $s \\in S$ and a ball $B=B(z,\\rho)$ such that $s \\in B$. Then for any $s',s'' \\in B$ and $t \\in S \\setminus \\alpha B$ we have\n  \\begin{align*}\n    d(s',t) &\\geq \\frac{1}{c_d}d(z,t)-d(z,s') \\geq  \\frac{\\alpha \\rho}{c_d} -\\rho \\geq 2 \\,c_K \\,c_d\\, \\rho=:\\varepsilon\\\\\n    d(s',s'') &\\leq 2\\,c_d\\, \\rho = c_K^{-1}\\varepsilon\n  \\end{align*}\nTherefore we have for any boundedly supported $f \\in L^1(S;X)$\n  \\begin{align*}\n    \\nrm{T(\\ind_{S \\setminus \\alpha B}&f)(s') - T_K(\\ind_{S \\setminus \\alpha B}f)(s'')}_{Y} \\\\\n    &\\leq  \\int_{S\\setminus \\alpha B}\\nrmb{\\hab{K(s',t) - K(s'',t)}f(t)}_Y\\dd \\mu(t)\\\\\n       &\\leq   \\int_{d(s',t)>\\varepsilon}\\omega\\has{\\frac{d(s',s'')}{d(s',t)}}\\frac{1}{\\mu\\hab{{B(s',d(s',t))}}} \\nrm{f(t)}_X\\dd \\mu(t)\\\\\n       &\\leq \\sum_{j=0}^\\infty \\omega\\hab{c_K^{-1} 2^{-j}} \\int_{2^j\\varepsilon<d(s',t)\\leq 2^{j+1}\\varepsilon}\\frac{1}{\\mu\\hab{{B(s',d(s',t))}}}\\nrm{f(t)}_X\\dd \\mu(t)\\\\\n       &\\lesssim_S  \\sum_{j=0}^\\infty\\omega\\hab{2^{-j-1}} \\avint_{B(s',2^{j+1}\\varepsilon)}\\nrm{f(t)}_X\\dd \\mu(t)\\\\\n       &\\leq \\nrm{K}_{\\Dini}\\,   M\\hab{\\nrm{f}_X}(s),\n  \\end{align*}\n  where the last step follows from $s \\in B(s',2^{j+1}\\varepsilon)$ for all $j \\in \\N$ and\n  \\begin{equation*}\n    \\sum_{j=0}^\\infty\\omega\\hab{2^{-j-1}} \\leq \\sum_{j=0}^\\infty\\omega\\hab{2^{-j-1}} \\int_{2^{-j-1}}^{2^{-j}}\\frac{\\ddn t}{t} \\leq \\sum_{j=0}^\\infty \\int_{2^{-j-1}}^{2^{-j}} \\omega(t) \\frac{\\ddn t}{t} = \\nrm{K}_{\\Dini}.\n  \\end{equation*}\nSo taking the supremum over all $s',s'' \\in B$ and all balls $B$ containing $s$ we find that $\\mc{M}_{T,\\alpha}^{\\#}f(s) \\lesssim_S \\nrm{K}_{\\Dini}  \\, M\\hab{\\nrm{f}_X}(s)$. Thus by the weak $L^1$-boundedness of the Hardy--Littlewood maximal operator and the density of boundedly supported functions in $L^1(S;X)$ we get\n\\begin{equation*}\n  \\nrmb{\\mc{M}_{T,\\alpha}^{\\#}}_{L^{1}(S;X)\\to L^{1,\\infty}(S;Y)} \\lesssim_{S} \\nrm{K}_{\\Dini}.\n\\end{equation*}\nThe pointwise sparse domination now follows from Theorem \\ref{theorem:main} and the  weighted bounds from  Proposition \\ref{proposition:weights}.\n\\end{proof}\n\n\\begin{remark} In the proof of Theorem \\ref{theorem:A2} it actually suffices to use the so-called $L^r$-H\\\"ormander condition for some $r>1$, which is implied by the Dini condition. See \\cite[Section 3]{Li18} for the definition of the $L^r$-H\\\"ormander condition and a comparison between the $L^r$-H\\\"ormander and the Dini condition.\n\\end{remark}\n\nNote that Theorem \\ref{theorem:A2} does not assume anything about the Banach spaces $X$ and $Y$ and is therefore applicable in situations where for example $Y=\\ell^\\infty$. However, in various applications $X$ and $Y$ will need to have the $\\UMD$ property in order to check the assumed\n      weak $L^{p_0}$-boundedness of $T$ for some $p_0 \\in [1,\\infty)$. For instance, for a large class of operators the weak $L^{p_0}$-boundedness of $T$ can be checked using theorems like the $T(1)$-theorem  or $T(b)$-theorem. See \\cite{Fi90} and \\cite{Hy14} for these theorems in the vector-valued setting, which assume the $\\UMD$ property for the underlying Banach space.\n\n      If $S$ is Euclidean space, one can also use an (operator-valued) Fourier multiplier theorem to check the a priori $L^{p_0}$-bound, which we will discuss in the next section.\n\n\\section{The weighted anisotropic mixed-norm Mihlin multiplier theorem}\n     Let $X$ and $Y$ be Banach spaces. Denote the space of $X$-valued Schwartz functions by $\\mc{S}(\\R^d;X)$ and  the space of $Y$-valued tempered distributions by $\\mc{S}'(\\R^d;Y):= \\mc{L}(\\mc{S}(\\R^d);Y)$. To an $m \\in L^\\infty(\\R^d;\\mc{L}(X,Y))$ we associate the Fourier multiplier operator\n\\begin{equation*}\n  T_m\\colon\\mc{S}(\\R^d;X) \\to \\mc{S}'(\\R^d;Y), \\qquad T_m f = (m\\widehat{f}\\vspace{2pt})^\\vee.\n\\end{equation*}\nSince $\\mc{S}(\\R^d;X)$ is dense in $L^p(\\R^d;X)$ and $L^p(\\R^d;Y)$ is continuously embedded into $\\mc{S}'(\\R^d;X)$, one may ask under which conditions on $m$ the operator $T_m$ extends to a bounded operator from $L^p(\\R^d;X)$ to $L^p(\\R^d;Y)$. If this is the case we call $m$ a bounded Fourier multiplier. We refer to \\cite[Chapter 5]{HNVW16} for an introduction to operator-valued Fourier multiplier theory.\n\nOne of the main Fourier multiplier theorems is the Mihlin multiplier theorem, first proven in the operator-valued setting by Weis in \\cite{We01b}.\nThe operator-valued Mihlin multiplier theorem of Weis has since been extended in many directions. Recently Fackler, Hyt\\\"onen and Lindemulder extended the operator-valued Mihlin multiplier theorem to a weighted, anisotropic, mixed norm setting in \\cite{FHL18}. This is for example useful in the study of spaces of smooth, vector-valued functions and has applications to parabolic PDEs with inhomogeneous boundary conditions, see e.g. \\cite{Li17b}.\n In \\cite{FHL18} the Mihlin multiplier theorem is shown using the following two approaches:\n\\begin{itemize}\n  \\item Using a weighted Littlewood--Paley decomposition, they show a weighted, anisotropic, mixed-norm\n  Mihlin multiplier theorem for rectangular $A_p$-weights, i.e. $A_p$-weights for which the defining supremum is taken over rectangles instead of balls.\n  \\item  Using Calder\\'on--Zygmund theory, they show a weighted, isotropic, non-mixed-norm\n  Mihlin multiplier theorem for cubicular $A_p$-weights, i.e. $A_p$-weights for which the defining supremum is taken over cubes, which is equivalent to the definition using balls we used\nin Section \\ref{section:weights}.\n\\end{itemize}\nBoth approaches have their pros and cons. The result using a Littlewood--Paley decomposition only requires estimates of $\\partial^\\theta m$ for $\\theta \\in \\cbrace{0,1}$, whereas the approach using Calder\\'on--Zygmund theory also requires estimates of higher-order derivatives. On the other hand,\nthe class of rectangular $A_p$-weights is a proper subclass of the class of cubicular $A_p$-weights.\n\nIn applications it is be desirable to have the Mihlin multiplier theorem for cubicular $A_p$-weights in the anisotropic, mixed-norm setting as well. This would remove the need to distinguish between the isotropic and anisotropic setting in e.g. \\cite[(6) on p.64]{Li17b}.\nIn order to obtain the Mihlin multiplier theorem for cubicular $A_p$-weights in the anisotropic, mixed-norm setting one needs Calder\\'on--Zygmund theory in $\\R^d$ equipped with an anisotropic norm. Since this is a special case of a space of homogeneous type, we can use Theorem \\ref{theorem:A2} to supplement the results of \\cite{FHL18}, which will be the main result of this section.\n\n\\bigskip\n\nLet us introduce the anisotropic, mixed-norm setting. For $\\mbs{a} \\in (0,\\infty)^d$ let $\\abs{\\,\\cdot\\,}_{\\mbs{a}}$ be the anisotropic quasi-norm as in \\eqref{eq:anisotropic} and define $$\\R^d_{\\mbs{a}} := (\\R^d, \\abs{\\,\\cdot-\\cdot\\,}_{\\mbs{a}}, \\dd t),$$ where $\\dd t$ denotes Lebesgue measure. Then $\\R^d_{\\mbs{a}}$ is a space of homogeneous type and e.g.\n  \\begin{align*}\n  \\ms{D}&:= \\cbraces{\\prod_{j=1}^d \\hab{2^{-a_jn}([0,1) +m_j)} :\\mbs{m}\\in \\Z^d, n \\in \\Z}\n\\end{align*}\nis a dyadic system in $\\R^d_{\\mbs{a}}$.\n We write\n$\\abs{\\mbs{a}}_1:=\\sum_{j=1}^d a_j,$ $\\abs{\\mbs{a}}_\\infty := \\max_{j=1,\\ldots,d}a_j,$\nand for $\\theta \\in \\N^d$ we set $\\mbs{a} \\cdot\\theta := \\sum_{j=1}^d a_j \\theta_{j}$.\n\nTake $l \\in \\N$, $\\mathpzc{d} \\in \\N^l$ and consider the $\\mathpzc{d}$-decomposition of $\\R^d$:\n\\begin{equation*}\n\\R^{d}_{\\mz{d}}:= \\R^{\\mathpzc{d}_1} \\times \\ldots \\times \\R^{\\mathpzc{d}_l}.\n\\end{equation*}\nFor a $t \\in \\R^{d}_{\\mz{d}}$ we write $t = (t_1,\\ldots,t_l)$ with $t_j \\in \\R^{\\mz{d}_j}$ for $j=1,\\ldots,l$ and similarly we write $\\mbs{a} = (\\mbs{a}_1,\\ldots,\\mbs{a}_l)$.\nFor $\\mbs{p} \\in [1,\\infty)^l$, a vector of weights $\\mbs{w}  \\in \\prod_{j=1}^l A_p(\\R^{\\mathpzc{d}_j}_{\n{\\mbs{a}_j}})$ and a Banach space $X$ we define the weighted mixed-norm Bochner space $L^{\\mbs{p}}(\\R^{d}_{\\mz{d}},\\mbs{w};X)$ as the space of all strongly measurable $f:\\R^{d}_{\\mz{d}} \\to X$ such that\n\\begin{equation*}\n  \\nrm{f}_{L^{\\mbs{p}}(\\R^{d}_{\\mz{d}},\\mbs{w};X)}:= \\has{\\int_{\\R^{\\mz{d}_1}}\\ldots\\has{\\int_{\\R^{\\mz{d}_l}}\\nrm{f}_X^{p_l} w_l\\dd t_l}^{\\frac{p_{l-1}}{p_l}}\\ldots w_1 \\dd t_1}^{\\frac{1}{p_1}}\n\\end{equation*}\nis finite.\n\nWe are now ready to state and prove the announced weighted anisotropic, mixed-norm Mihlin multiplier theorem.\n\n\\begin{theorem}\\label{theorem:mihlin}\nLet $X$ and $Y$ be $\\UMD$ Banach spaces, set $N = \\abs{\\mbs{a}}_1+ \\abs{\\mbs{a}}_\\infty+ 1$\n and let $m \\in L^\\infty(\\R^d;\\mc{L}(X,Y))$. Suppose that for all $\\theta \\in \\N^d$ with $\\mbs{a} \\cdot\\theta \\leq N$ the distributional derivative $\\partial^\\theta m$ coincides with a continuous function on $\\R^d \\setminus \\cbrace{0}$ and we have the $\\mc{R}$-bound\n\\begin{equation*}\n \\mc{R}\\hab{\\cbraceb{\\abs{\\xi}_{\\mbs{a}}^{\\mbs{a} \\cdot \\theta}\\cdot  \\partial^\\theta m(\\xi): \\xi \\in \\R^d}\\setminus\\cbrace{0}} \\leq C_m.\n\\end{equation*}\nfor some $C_m>0$. Then for every compactly supported $f \\in L^1(\\R^d;X)$ there exists an $\\eta$-sparse collection of anisotropic cubes $\\mc{S}$ such that\n\\begin{equation*}\n    \\nrm{T_mf(s)}_Y\\lesssim_{X,Y,\\mbs{a}} \\,C_m\\, \\sum_{Q \\in \\mc{S}} \\ipb{\\nrm{f}_X}_{1,Q} \\ind_{Q}(s), \\qquad s \\in \\R^d.\n\\end{equation*}\nMoreover, for all $\\mbs{p} \\in (1,\\infty)^l$ and $\\mbs{w} \\in  \\prod_{j=1}^l A_{p_j}(\\R^{\\mz{d}_j}_{\\mbs{a}_j})$ we have\n\\begin{align*}\n  \\nrm{T_m}_{L^{\\mbs{p}}(\\R^d_{\\mz{d}},\\mbs{w};X) \\to L^{\\mbs{p}}(\\R^d_{\\mz{d}},\\mbs{w};Y)} &\\lesssim_{X,Y,\\mz{d},\\mbs{a},\\mbs{p}, \\mbs{w}} C_m.\n\\end{align*}\n\\end{theorem}\n\n\\begin{proof}\n We will check the conditions of Theorem \\ref{theorem:A2}. By \\cite[Theorem 3]{Hy07}, which trivially extends to the case $X \\neq Y$,\n  we know that $T_m$ is bounded from $L^2(\\R^d;X)$ to $L^2(\\R^d;Y)$ with\n  \\begin{equation*}\n    \\nrm{T_m}_{L^2(\\R^d;X) \\to L^2(\\R^d;Y)} \\lesssim_{X,Y,\\mz{d},\\mbs{a}} C_m.\n  \\end{equation*}\n  By \\cite[Lemma 4.4.6 and 4.4.7]{Li14b} we know that $\\widecheck{m}$ coincides with a continuous function on $\\R^d \\setminus \\cbrace{0}$, which is bounded away from $0$ and $$K(t,s):= \\widecheck{m}(t-s), \\qquad t\\neq s$$  is a Dini kernel on the space of homogeneous type  $\\R^d_{\\mbs{a}}$ with\n  \\begin{equation*}\n    \\omega(r) = C_{\\mbs{a}} \\cdot C_m \\cdot r^{\\min \\mbs{a}}, \\qquad r \\in [0,1].\n  \\end{equation*}\n  Now let $f \\in L^p(\\R^d;X)$ with compact support. Fix a $c \\in \\R^d \\setminus \\overline{\\supp f}$ and take $r>0$ such that $B(c,2r) \\cap \\overline{\\supp f} = \\varnothing$. Take a sequence $(f_n)_{n=1}^\\infty$ in $\\mc{S}(\\R^d;X)$ such that $\\overline{\\supp f_n}  \\cap B(c,r)= \\varnothing$ and $f_n \\to f$ in $L^2(\\R^d;X)$. Then $Tf_n \\to Tf$ in $L^2(\\R^d;X)$ and, by passing to a subsequence if necessary, we have $f_n(t) \\to f(t)$ and $Tf_n(t) \\to Tf(t)$ for a.e. $t \\in \\R^d$. Fix $n \\in \\N$, then we have for all $\\varphi \\in C_c^\\infty(\\R^d \\setminus \\overline{\\supp f_n})$\n  \\begin{align*}\n    \\ip{T_m f_n, \\varphi}  &= \\int_{\\R^d} m(s)\\widehat{f_n}(s) \\widecheck{\\varphi}(s)\\dd s\\\\\n    &= \\int_{\\R^d}\\widecheck{m}(s)   \\int_{\\R^d}f_n(t-s) \\varphi(t) \\dd t\\dd s \\\\\n    &= \\int_{\\R^d} \\int_{\\R^d} K(t,s)f_n(s)\\dd s \\, \\varphi(t) \\dd t\n  \\end{align*}\n  from which we obtain for a.e. $t \\in B(c,r)$\n  \\begin{equation*}\n    T_m f(t) = \\lim_{n\\to \\infty} T_m f_n(t) =  \\lim_{n \\to \\infty} \\int_{\\R^d}K(t,s)f_n(s)\\dd s = \\int_{\\R^d}K(t,s)f(s)\\dd s\n  \\end{equation*}\n  Covering $\\R^d \\setminus \\overline{\\supp f}$ by countably many such balls, we conclude that $T_m$ has kernel $K$. Therefore the sparse domination, as well as the weighted estimate in case $l=1$, follows from Theorem \\ref{theorem:A2}.\n\nTo conclude the proof we will show the case $l=2$, the general case follows by iterating the argument. Take $\\mbs{p} \\in (1,\\infty)^2$ and $\\mbs{w} \\in A_{p_1}(\\R^{\\mz{d}_1}_{\\mbs{a}_1}) \\times A_{p_2}(\\R^{\\mz{d}_2}_{\\mbs{a}_2})$. For $v_1 \\in A_{p_2}(\\R^{\\mz{d}_1}_{\\mbs{a}_1})$ note that\n $${v}(t):= v_1(t_1)\\cdot  w_2(t_2) , \\qquad t \\in \\R^{\\mz{d}_1} \\times \\R^{\\mz{d}_2}$$ belongs to $A_{p_2}(\\R^d_{\\mbs{a}})$, so by the case $l=1$ we have\n   \\begin{equation*}\n      \\nrm{T_mf}_{L^{p_2}(\\R^d,{v};Y)} \\lesssim_{X,Y,\\mz{d},\\mbs{a},p_2, {v}} C_m \\cdot \\nrm{f}_{L^{p_2}(\\R^d,{v};X)}\n   \\end{equation*}\n   for all $f \\in L^{p_2}(\\R^d, {v};X)$.\nSince  balls in $\\R^{\\mz{d}_2}$ with respect to the quasi-metric $\\abs{\\,\\cdot-\\cdot\\,}_{\\mbs{a}_2}$ form a Muckenhoupt basis, we can use Rubio de Francia extrapolation as in \\cite[Theorem 3.9]{CMP11} on the extrapolation family\n  \\begin{equation*}\n    \\cbraces{\\hab{\\nrm{T_mf}_{L^{p_2}(\\R^{\\mz{d}_2},w_2;Y)},\\nrm{f}_{L^{p_2}(\\R^{\\mz{d}_2},w_2;X)}}:f \\colon \\R^d \\to X \\text{ simple}}\n  \\end{equation*}\nto deduce\n\\begin{equation*}\n  \\nrm{T_mf }_{L^{\\mbs{p}}(\\R^{d}_{\\mz{d}} ,\\mbs{w};Y)} \\lesssim_{X,Y,\\mz{d},\\mbs{a},\\mbs{p},\\mbs{w}} C_m \\nrm{f }_{L^{\\mbs{p}}(\\R^{d}_{\\mz{d}} ,\\mbs{w};X) }\n\\end{equation*}\nfor all simple $f$, which implies the result by density.\n\\end{proof}\n\n\\begin{remark}~\n\\begin{enumerate}[(i)]\n\\item The weight dependence of the implicit constant in Theorem \\ref{theorem:mihlin} in the case $l=1$ is $[w]_{A_{p}(\\R^d_{\\mbs{a}})}^{\\max\\cbrace{\\frac{1}{p-1},1}}$, which is sharp. For $l\\geq 2$ the dependence our proof yields is more complicated and not sharp for all choices of $\\mbs{p} \\in (1,\\infty)^l$.\n  \\item In the proof of Theorem \\ref{theorem:mihlin} we only use the $\\mc{R}$-boundedness of the set\n  $$\n  \\cbraceb{\\abs{\\xi}_{\\mbs{a}}^{\\mbs{a} \\cdot \\theta}\\cdot  \\partial^\\theta m(\\xi): \\xi \\in \\R^d\\setminus\\cbrace{0}}\n   $$\n   for $\\theta  \\in \\cbrace{0,1}^d$. For all other $\\theta \\in \\N^d$ with $\\mbs{a} \\cdot \\theta \\leq N$ it suffices to know uniform boundedness of this set.\n\\item One could reduce the number of derivatives necessary in Theorem \\ref{theorem:mihlin}, by arguing as in \\cite{Hy04} instead of using \\cite[Lemma 4.4.6 and 4.4.7]{Li14b}. See also \\cite[Section 6]{FHL18}.\n\\item Using the sparse domination of Theorem \\ref{theorem:mihlin} one can also deduce two-weight estimates for $T_m$ as in \\cite[Section 6]{FHL18}.\n\\end{enumerate}\n\\end{remark}\n\n\\section{The Rademacher maximal function}\\label{section:maximal}\nIn this section we will apply Theorem \\ref{theorem:localsparse} to the Rademacher maximal function. The proofs will illustrate very nicely how the geometry of the Banach space plays a role in deducing the localized $\\ell^r$-estimate for this operator. In particular, we will use the type of a Banach space $X$ to deduce the localized $\\ell^r$-estimate for the Rademacher maximal function.\n\n\n The Rademacher maximal function was introduced by Hyt\\\"onen, McIntosh and Portal in \\cite{HMP08} as a vector-valued generalization of Doob's maximal function that takes into account the different ``directions'' in a Banach space. They used the Rademacher maximal function to prove a Carleson's embedding theorem for vector-valued functions in connection to Kato's square root problem in Banach spaces. The Carleson's embedding theorem for vector-valued functions has since found many other applications, like the local vector-valued $T(b)$ theorem (see \\cite{HV15}).\n\n\nLet $(S,d,\\mu)$ be a space of homogeneous type with a dyadic system $\\ms{D}$ and let $X$ be a Banach space.\nFor $f \\in L^1_{\\loc}(S;X)$ we define the \\emph{Rademacher maximal function} by\n\\begin{align*}\n  M_{\\Rad}^{\\ms{D}}f(s)&:= \\sup\\cbraces{\\nrms{\\sum_{Q \\in \\ms{D}:s \\in Q}\\varepsilon_Q\\lambda_Q \\ip{f}_{1,Q}}_{L^2(\\Omega;X)}: \\\\&\\hspace{2cm}(\\lambda_Q)_{Q \\in \\ms{D}} \\text{ finitely non-zero with } \\sum_{Q \\in \\ms{D}}\\abs{\\lambda_Q}^2\\leq 1},\n\\end{align*}\nwhere $(\\varepsilon_Q)_{Q \\in \\ms{D}}$ is a Rademacher sequence on $\\Omega$. One can interpret this maximal function as Doob's maximal function\n \\begin{equation*}\n  f^*(s): =\\sup_{Q \\in \\ms{D}:s \\in Q} \\nrmb{\\ip{f}_{1,Q}}_X, \\qquad s\\in S,\n\\end{equation*}\n with the uniform bound over the $\\ip{f}_{1,Q}$'s replaced by the $\\mc{R}$-bound. Here the $\\mc{R}$-bound of a set $U\\subseteq X$ is the $\\mc{R}$-bound of the family of operators $T_x:\\C \\to X$ given by $\\lambda \\mapsto \\lambda x$ for $x \\in U$.\n\nWe say that the Banach space $X$ has the $\\RMF$ property if $M_{\\Rad}^{\\ms{D}[0,1)}$ is a bounded operator on $L^p([0,1);X)$ for some $p \\in (1,\\infty)$, where\n$$\\ms{D}[0,1) := \\cbraceb{2^{-k} [j-1,j): k\\in \\N\\cup\\cbrace{0},\\, j=1,\\ldots,2^k}$$\nis the standard dyadic system in $[0,1)$.\nIt was shown by Hyt\\\"onen, McIntosh and Portal \\cite[Proposition 7.1]{HMP08} that this implies boundedness for all $p \\in (1,\\infty)$ and by Kemppainen \\cite[Theorem 5.1]{Ke11}  that  this implies boundedness of $M_{\\Rad}^{\\ms{D}}$  on $L^p(S;X)$ for any space of homogeneous type $(S,d,\\mu)$ with a dyadic system $\\ms{D}$.\n\nThe relation of  $\\RMF$ property to other Banach space properties is not yet fully understood. However, we do have some necessary and sufficient conditions:\n\\begin{itemize}\n  \\item The $\\mc{R}$-bound of a set $U\\subseteq X$ is equivalent to the uniform bound of that set if and only if $X$ has type $2$ (see \\cite[Proposition 8.6.1]{HNVW17}). Therefore if $X$ has type $2$ we have for any $f \\in L^1_{\\loc}([0,1);X)$ that $M_{\\Rad}^{\\ms{D}[0,1)}f \\lesssim M^{\\ms{D}[0,1)}(\\nrm{f}_X)$, so $X$ has the $\\RMF$ property.\n  \\item Any $\\UMD$ Banach lattice has the $\\RMF$ property, see also the discussion related to the Hardy--Littlewood maximal operator at the end of this section.\n  \\item Non-commutative $L^p$-spaces for $p \\in (1,\\infty)$ have the $\\RMF$ property, see \\cite[Corollary 7.6]{HMP08}.\n  \\item The $\\RMF$ property implies nontrivial type, see \\cite[Proposition 4.2]{Ke11}.\n\\end{itemize}\nIt is an open problem whether nontrivial type or even the $\\UMD$ property implies the $\\RMF$ property.\n\n\\bigskip\n\nWeighted bounds for the Rademacher maximal function in the Euclidean setting were studied by Kemppainen \\cite[Theorem 1]{Ke13}. The proof was based on a good-$\\lambda$ inequality, which does not give sharp quantitative estimates in terms of the weight characteristic. Using Theorem \\ref{theorem:localsparse} we can prove sharp quantitative weighted estimates for the Rademacher maximal function through sparse domination. We will not consider the situation in which $X$ has type $2$, as this case follows directly from\n$M_{\\Rad}^{\\ms{D}[0,1)}f \\lesssim M^{\\ms{D}[0,1)}(\\nrm{f}_X)$ and the well-known sparse domination for the Hardy--Littlewood maximal operator.\n\nWe will need a version of the Rademacher maximal function for finite collections of cubes. For a subcollection of cubes $\\mc{D} \\subseteq \\ms{D}$ we define  $M_{\\Rad}^{\\mc{D}}$ analogous to $M_{\\Rad}^{\\ms{D}}$.\n\n\\begin{theorem}\\label{theorem:RMF}\n  Let $(S,d,\\mu)$ be a space of homogeneous type with a dyadic system $\\ms{D}$ and let $X$ be a Banach space with the $\\RMF$ property. Assume that $X$ has type $r$ for $r \\in [1,2)$. For any finite collection of cubes $\\mc{D} \\subseteq \\ms{D}$ and  $f \\in L^1(S;X)$ there exists an $\\frac12$-sparse collection of cubes $\\mc{S}\\subseteq \\ms{D}$ such that\n  \\begin{equation*}\n    M_{\\Rad}^{\\mc{D}}f(s) \\lesssim_{X,S,\\ms{D},r} \\has{\\sum_{Q \\in \\mc{S}}\\ipb{\\nrm{f}_X}_{1,Q}^{(\\frac1r-\\frac12)^{-1}}\\ind_Q(s)}^{\\frac1r-\\frac12}, \\qquad s\\in S\n  \\end{equation*}\nMoreover, for all $p \\in (1,\\infty)$ and $w\\in A_p$ we have\n  \\begin{align*}\n    \\nrmb{M_{\\Rad}^{\\ms{D}}}_{L^p(S,w;X) \\to L^p(S,w;X)} &\\lesssim_{X,S,\\ms{D},p,r} [w]_{A_p}^{\\max\\cbraceb{\\frac{1}{p-1},\\frac{1}{r}-\\frac{1}{2}}}.\n  \\end{align*}\n\\end{theorem}\n\n\\begin{proof}\nFix a finite collection of cubes $\\mc{D} \\subseteq \\ms{D}$. By \\cite[Proposition 6.1]{Ke11} $M_{\\Rad}^{\\mc{D}}$  is weak $L^1$-bounded.\nWe will view $M_{\\Rad}^{\\mc{D}}$ as a bounded operator\n  \\begin{equation*}\n    M_{\\Rad}^{\\mc{D}}: L^1(S;X) \\to L^{1,\\infty}(S; \\mc{L}(\\ell^2(\\mc{D}),L^2(\\Omega;X)))\n  \\end{equation*}\n  given by\n  \\begin{equation*}\n    M_{\\Rad}^{\\mc{D}}f(s) = \\has{(\\lambda_Q)_{Q \\in \\mc{D}} \\mapsto \\sum_{Q \\in \\ms{D}:s \\in Q} \\varepsilon_Q \\lambda_Q \\,\\ip{f}_{1,Q}}, \\qquad s \\in S,\n  \\end{equation*}\n  where $(\\varepsilon_Q)_{Q \\in \\mc{D}}$ is a Rademacher sequence on $\\Omega$.\n\n\n  For $Q \\in \\ms{D}$ set\n  $$\\mc{D}(Q):=\\cbrace{P \\in\\mc{D}:P \\subseteq Q}$$\n   and define $T_Q:= M_{\\Rad}^{\\mc{D}(Q)}$. Then $\\cbrace{T_Q}_{Q \\in \\ms{D}}$ is a $1$-localization family for $M_{\\Rad}^{\\mc{D}}$.\n Furthermore we have for $f \\in L^1(S;X)$ and $s \\in Q \\in \\ms{D}$ that\n  \\begin{align*}\n    \\mc{M}^{\\#}_{M_{\\Rad}^{\\mc{D}},Q}f(s)  &= \\sup_{\\substack{Q' \\in \\ms{D}(Q):\\\\s \\in Q'}}  \\esssup_{s' ,s'' \\in Q'} \\nrmb{T_{Q\\setminus Q'}f(s')-T_{Q\\setminus Q'}f(s'')}_{\\mc{L}(\\ell^2(\\ms{D}),L^2(\\Omega;X))}\\\\&=0\n  \\end{align*}\n  where the second step follows from the fact that $T_{Q\\setminus Q'}f = M_{\\Rad}^{\\mc{D}(Q)\\setminus \\mc{D}(Q')}f$ is constant on $Q'$. So $\\mc{M}^{\\#}_{M_{\\Rad}^{\\mc{D}},Q}$ is trivially bounded from $L^1(S;X)$ to $L^{1,\\infty}(S)$.\n\nSet  $q:=(\\frac1r-\\frac12)^{-1}$. To check the localized $\\ell^q$-estimate for $M_{\\Rad}^{\\mc{D}}$ take $Q_1,\\ldots,Q_n \\in \\ms{D}$ with $Q_n\\subseteq \\ldots\\subseteq Q_1$. Let $(\\lambda_Q)_{Q \\in \\mc{D}} \\in \\ell^2(\\mc{D})$ be of norm one and let $(\\varepsilon_Q)_{Q \\in \\mc{D}}$ and $(\\varepsilon'_k)_{k=1}^n$ be Rademacher sequences on $\\Omega$ and $\\Omega'$ respectively. Define for $k=1,\\ldots,n-1$\n\\begin{equation*}\n  \\lambda_k := \\has{\\sum_{Q \\in \\mc{D}\\ha{Q_{k+1}}\\setminus \\mc{D}\\ha{Q_k}}\n\\abs{\\lambda_Q}^2}^{1\/2}, \\qquad \\lambda_{n} := \\has{\\sum_{Q \\in \\mc{D}\\ha{Q_n}} \\abs{\\lambda_Q}^2}^{1\/2}\n\\end{equation*}\n Then for $f \\in L^1(S;X)$, setting $f_Q := \\varepsilon_Q \\lambda_Q\\ip{f}_{1,Q}$,  we have\n  \\begin{align*}\n    \\nrms{\\sum_{Q \\in \\mc{D}\\ha{Q_1}} &\\varepsilon_Q \\lambda_Q\\ip{f}_{1,Q}}_{L^2(\\Omega;X)}\\\\\n    &= \\nrms{\\varepsilon_n'\\sum_{Q \\in \\mc{D}\\ha{Q_n}} f_Q+ \\sum_{k=1}^{n-1}\\varepsilon_{k}'\\sum_{Q\\in \\mc{D}\\ha{Q_{k+1}}\\setminus \\mc{D}\\ha{Q_k}} f_Q}_{L^2(\\Omega\\times \\Omega';X)}\\\\\n    &\\lesssim_{X,r} \\has{\\lambda_n^r \\nrms{\\sum_{Q \\in \\mc{D}\\ha{Q_n}}  \\lambda_n^{-1}f_Q}^r_{L^2(\\Omega;X)} \\\\&\\hspace{1cm}+ \\sum_{k=1}^{n-1}\\lambda_{k}^r \\nrms{\\sum_{Q\\in \\mc{D}\\ha{Q_{k+1}}\\setminus \\mc{D}\\ha{Q_k}} \\lambda_{k}^{-1}f_Q}_{L^2(\\Omega;X)}^r}^{1\/r}\\\\\n    &\\leq\\has{ \\nrms{\\sum_{Q \\in \\mc{D}\\ha{Q_n}}  \\varepsilon_Q \\lambda_n^{-1} \\lambda_Q\\ip{f}_{1,Q}}^{q}_{L^2(\\Omega;X)} \\\\&\\hspace{1cm}+ \\sum_{k=1}^{n-1} \\nrms{\\sum_{Q\\in \\mc{D}\\ha{Q_{k+1}}\\setminus \\mc{D}\\ha{Q_k}} \\varepsilon_Q \\lambda_{k}^{-1}\\lambda_Q\\ip{f}_{1,Q}}_{L^2(\\Omega;X)}^{q}}^{1\/q},\n  \\end{align*}\nusing randomization (see \\cite[Proposition 6.1.11]{HNVW17}) in the first step, type $r$ of $X$ in the second step, and H\\\"older's inequality and $\\sum_{k=1}^n \\lambda_k^2=1$ in the last step. Noting that for $k=1,\\ldots,n-1$\n\\[\\sum_{Q\\in \\mc{D}(Q_{k+1})\\setminus \\mc{D}(Q_k)} \\abs{\\lambda_{k}^{-1}{\\lambda_Q}}^2=1,\\qquad \\sum_{Q\\in {\\mc{D}(Q_n)} } \\abs{\\lambda_{n}^{-1}{\\lambda_Q}}^2=1,\\] this implies the  localized $\\ell^{q}$-estimate for $M_{\\Rad}^{\\mc{D}}$.\n\nHaving checked all assumptions of Theorem \\ref{theorem:localsparse} for $M_{\\Lat}^{\\mc{D}}$ it follows that for any $Q \\in \\mc{D}$ there is a $\\frac{1}{2}$-sparse collection of cubes $\\mc{S}_Q\\subseteq \\ms{D}(Q)$ such that\n\\begin{align*}\n  \\nrmb{T_Q(s)}_Y\\lesssim_{X,S,\\ms{D},r} ,\n  \\has{ \\sum_{P \\in \\mc{S}} \\ipb{\\nrm{f}_X}_{p,\\alpha P}^r \\ind_P(s)}^{1\/r},\\qquad s \\in Q.\n  \\end{align*}\nLet $\\mc{D}'$ be the maximal cubes (with respect to set inclusion) of $\\mc{D}$, which are pairwise disjoint. Then $\\mc{S}:=\\bigcup_{Q \\in \\mc{D}'} \\mc{S}_Q$ is a $\\frac{1}{2}$-sparse collection of cubes that satisfies the claimed sparse domination\nas $T_Q(s) = M_{\\Rad}^{\\mc{D}}f(s)$ for any $s\\in Q \\in \\mc{D}'$ and $M_{\\Rad}^{\\mc{D}}f$ is zero outside $\\bigcup_{Q \\in \\mc{D}'}Q$.  The weighted bounds follow from Proposition \\ref{proposition:weights} and the monotone convergence theorem.\n\\end{proof}\n\nLet us check that the weighted estimate in Theorem \\ref{theorem:RMF}, and consequently also the sparse domination in Theorem \\ref{theorem:RMF}, is sharp. We take $X=\\ell^r$ for $r \\in (1,2)$, a prototypical Banach space with type $r$.  Since $\\mc{R}$-bounds are stronger than uniform bounds, we note that for any strongly measurable $f \\colon [0,1) \\to \\ell^q$ we have\n\\begin{equation*}\n f^*(s) \\leq M_{\\Rad}^{\\ms{D}[0,1)}f(s),\\qquad s \\in [0,1).\n\\end{equation*}\nThus by the corresponding result for Doob's maximal operator (see \\cite[Proposition 3.2.4]{HNVW16}), we have for $p \\in (1,\\infty)$\n\\begin{equation}\\label{eq:radlowerbound}\n  \\nrmb{M_{\\Rad}^{\\ms{D}[0,1)}}_{L^p([0,1);\\ell^r) \\to L^p([0,1);\\ell^r) } \\geq \\frac{p}{p-1}\n\\end{equation}\nNow let $(e_n)_{n=1}^\\infty$ be the canonical basis of $\\ell^r$ and define\n\\begin{equation*}\n  f(s):= \\sum_{n=1}^\\infty \\ind_{[2^{-n},2^{-n+1})}(s) e_n, \\qquad s \\in [0,1).\n\\end{equation*}\nFor $p \\in (1,\\infty)$ we have\n\\begin{equation*}\n  \\nrm{f}_{L^p([0,1);\\ell^r)} = 1.\n\\end{equation*}\nTo compute $\\nrm{M_{\\Rad}^{\\ms{D}[0,1)}f}_{L^p([0,1);\\ell^r)}$ set $I_j:= [0,2^{-j+1}]$, take $s \\in $ and let $m \\in \\N$ be such that $2^{-m}\\leq s \\leq 2^{-m+1}$. Then we have, using $\\lambda_{I_j} = m^{-1\/2}$ for $j=1,\\ldots,m$ and the Khintchine--Maurey inequalities (see \\cite[Theorem 7.2.13]{HNVW17}), that\n\\begin{align*}\n  M_{\\Rad}^{\\ms{D}[0,1)}f(s) &\\geq \\frac{1}{m^{1\/2}}  \\nrms{\\sum_{j=1}^m \\varepsilon_j\\ip{f}_{1,I_j}}_{L^2(\\Omega;\\ell^r)}\n  \\gtrsim \\frac{1}{m^{1\/2}}\\nrms{\\has{\\sum_{j=1}^m \\ip{f}_{1,I_j}^2}^{1\/2}}_{\\ell^r}\\\\\n  &\\gtrsim \\frac{1}{m^{1\/2}}\\nrms{\\sum_{j=1}^m e_j}_{\\ell^r}\n  \\gtrsim m^{1\/r-1\/2} \\gtrsim {\\log(1\/s)}^{1\/r-1\/2}.\n\\end{align*}\nTherefore we obtain\n\\begin{align*}\n  \\nrmb{M_{\\Rad}^{\\ms{D}[0,1)}f}_{L^p([0,1);\\ell^r)}&\\gtrsim \\has{\\int_0^1 {\\log(1\/s)}^{p\/r-p\/2} \\dd s}^{1\/p}\n  \\\\&= \\has{\\int_1^\\infty x^{p\/r-p\/2} \\ee^{-x}\\dd x}^{1\/p} \\\\&\\geq \\has{\\sum_{n=2}^\\infty n^{p\/r-p\/2}\\ee^{-n}}^{1\/p} \\\\&\\gtrsim p^{1\/r-1\/2},\n\\end{align*}\nwhere we drop all terms except $n = \\ceil{p}$ in the last step. Thus combined with \\eqref{eq:radlowerbound} we find\n\\begin{equation*}\n  \\nrmb{M_{\\Rad}^{\\ms{D}[0,1)}}_{L^p([0,1);\\ell^r) \\to L^p([0,1);\\ell^r) } \\gtrsim \\max\\cbraces{\\frac{1}{p-1}, p^{1\/r-1\/2}},\n\\end{equation*}\nwhich implies that the weighted estimate in Theorem \\ref{theorem:RMF} is sharp by \\cite[Theorem 1.2]{LPR15}.\n\n\\bigskip\n\nTo finish this section we will compare the sparse domination for the Rademacher maximal operator in Theorem \\ref{theorem:RMF} with the sparse domination for the lattice Hardy--Littlewood maximal operator obtained by H\\\"anninnen and the author in \\cite[Theorem 1.3]{HL17}. Let $X$ be a Banach lattice with finite cotype and $\\ms{D}$ the standard dyadic system in $\\R^d$. For a simple function $f:\\R^d \\to X$ define  \\emph{dyadic lattice Hardy--Littlewood maximal operator} (see e.g. \\cite{GMT93}) by\n\\begin{equation}\\label{eq:maxop}\n  M_{\\Lat}^{\\ms{D}}f(s) : = \\sup_{Q \\in \\ms{D}: s \\in Q} \\ipb{\\abs{f}}_{1,Q}, \\qquad s\\in \\R^d,\n\\end{equation}\nwhere the absolute value and the supremum are taken in the lattice sense. By the Khintchine--Maurey inequalities (see e.g. \\cite[Theorem 7.2.13]{HNVW17}) we have\n      $$M_{\\Rad}^{\\ms{D}}f \\lesssim M_{\\Lat}^{\\ms{D}}f$$\n       for any simple $f\\colon \\R^d \\to X$.\nBy \\cite{Bo84,Ru86} we know  that $X$ has the $\\UMD$ property if and only if\n $M_{\\Lat}^{\\ms{D}}$ is bounded on $L^p(\\R^d;X)$ and $L^p(\\R^d;X^*)$ for some (all) $p \\in (1,\\infty)$, which  implies that any $\\UMD$ Banach lattice has the $\\RMF$ property.\n\n Comparing the sparse domination result in  Theorem \\ref{theorem:RMF} with the corresponding sparse domination result for the dyadic lattice Hardy--Littlewood maximal operator, we see that the sparse operator in Theorem \\ref{theorem:RMF} is smaller than the sparse operator in \\cite[Theorem 1.3]{HL17}. Moreover, the sparse domination for the lattice Hardy--Littlewood maximal operator  is sharp, as shown in \\cite[Theorem 1.2]{HL17}. Therefore on any $\\RMF$ Banach lattice that is not $\\infty$-convex, the operators $M_{\\Rad}^{\\ms{D}}$ and $M_{\\Lat}^{\\ms{D}}$ are incomparable,  i.e. the (dyadic) lattice Hardy--Littlewood maximal operator is strictly larger than the Rademacher maximal operator. As the only $\\infty$-convex $\\RMF$ Banach lattices are the finite dimensional ones, we have the following corollary.\n\n\\begin{corollary}\n  Let $X$ be an infinite dimensional $\\RMF$ Banach lattice. Then there is no $C>0$ such that for all simple $f\\colon \\R^d \\to X$\n  $$\n    M_{\\Lat}^{\\ms{D}}f\\leq C \\, M_{\\Rad}^{\\ms{D}}f.$$\n\\end{corollary}\n\n\n\\section{Further Applications}\\label{section:further}\n In this final section we comment on some further applications of our main theorems, for which we leave the details to the interested reader.\n\\begin{itemize}\n\\item Sparse domination and weighted bounds for variational truncations of Calder\\'on--Zygmund operators were studied in \\cite{FZ16, HLP13b, MTX15, MTX17}. The arguments presented in these references also imply the boundedness of our sharp grand maximal truncation operator and thus by Theorem \\ref{theorem:main} yield sparse domination of the variational truncations of Cal\\-der\\'on--Zygmund operators.\n\\item In \\cite{LOR17} Lerner, Ombrosi and Rivera-R\\'ios show sparse domination for commutators of a $\\BMO$ function $b$ with a Calder\\'on--Zygmund operator using sparse operators adapted to the function $b$. By a slight adaptation of the arguments presented in the proof of Theorem \\ref{theorem:localsparse}, one can prove the main result of \\cite{LOR17} in our framework and extend it to the vector-valued setting and to spaces of homogeneous type.\n\\item H\\\"ormander--Mihlin type conditions as in \\cite[Theorem IV.3.9]{GR85} imply the weak $L^{p_1}$-boundedness  of our maximal truncation operator for $p_1>n\/a$ and thus sparse domination for the associated Fourier multiplier operator by Theorem \\ref{theorem:main}. Vector-valued extensions under Fourier type assumptions can be found in \\cite{GW03,Hy04} and Theorem \\ref{theorem:main} may therefore also be used to prove weighted results in that setting.\n\\item In \\cite{Le11} Lerner used his local mean oscillation decomposition to deduce sparse domination and sharp weighted norm inequalities for various Littlewood--Paley operators. These results are also an almost immediate consequence of Theorem \\ref{theorem:localsparse} with $r=2$, using a truncation of the cone of aperture in the definition of a Littlewood--Paley operator in order to make the localized $\\ell^2$-estimate checkable. Using similar arguments one can also treat the dyadic square function with Theorem \\ref{theorem:localsparse}, which yields the sharp weighted norm inequalities as obtained by Cruz-Uribe, Martell and Perez \\cite{CMP12}.\n\n Very recently Bui and Duong \\cite{BD19} extended the results in \\cite{Le11} to square functions of a general operator $L$ which has a Gaussian heat kernel bound and a bounded holomorphic functional calculus on $L^2(S)$, where $(S,d,\\mu)$ is a space of homogeneous type. The arguments they present can also be used to estimate our sharp grand maximal truncation operator, so their result is also be treated by Theorem \\ref{theorem:localsparse}.\n\\item Fackler, Hyt\\\"onen and Lindemulder \\cite{FHL18} proved weighted vector-valued Littlewood-Paley theory on a $\\UMD$ Banach space in order to prove their weighted, anisotropic, mixed-norm Mihlin multiplier theorems. Using Theorem \\ref{theorem:main} and Proposition \\ref{proposition:weights} on the Littlewood--Paley square function with smooth cut-offs one can prove sparse domination and weighted estimates in the smooth cut-off case. This can then be transferred to sharp cut-offs by standard arguments, recovering \\cite[Theorem 3.4]{FHL18}.\n\\item  In \\cite{PSX12} Potapov, Sukochev and Xu proved extrapolation upwards of unweighted vector-valued Littlewood--Paley--Rubio de Francia inequalities. Using \\cite[Lemma 4.5]{PSX12}  one can check the weak $L^2$-boundedness of our sharp grand maximal truncation operator, which by Theorem \\ref{theorem:main} and Proposition \\ref{proposition:weights} yields sparse domination and weighted estimates for vector-valued Littlewood--Paley--Rubio de Francia estimates. In the scalar case sparse domination was shown by Garg, Roncal and Shrivastava \\cite{GRS18} using time-frequency analysis.\n \\item Theorem \\ref{theorem:fractional} can be used to show sparse domination and sharp weighted estimates for fractional integral operators as in \\cite{CB13, CB13b,Cr17,IRV18}. The boundedness of the sharp grand maximal truncation operator associated to these operators can be shown using a similar argument as we used in the proof of Theorem \\ref{theorem:A2}.\n\\item In \\cite{BFP16} Bernicot, Frey and Petermichl show that the sparse domination\nprinciple is also applicable to non-integral singular operators falling\noutside the scope of Calder\\'on--Zygmund operators. Sparse domination for square functions related to these operators was studied in \\cite{BBR20}.  The methods developed in these papers actually show the boundedness of the localized sharp grand $q$-maximal truncation operator used in Theorem \\ref{theorem:sparseform}, so these results also fit in our framework.\n\\end{itemize}\n\n\\subsection*{Acknowledgement} The author would like to thank Dorothee Frey, Bas Nieraeth and Mark Veraar for their helpful comments on the draft version of this paper. Moreover the author would like to thank Luz Roncal for bringing one of the applications in Section \\ref{section:A2} under the author's attention and Olli Tapiola for his remarks on the Lebesgue differentiation theorem in spaces of homogeneous type.\n\n\\newcommand{\\etalchar}[1]{$^{#1}$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\n\nHigh energy elastic proton-proton and proton-antiproton cross\nsections reveal very complicated dynamics which is rather\ndifficult to explain within the framework of Quantum\nChromodynamics (QCD) (see the discussion in\n\\cite{Krisch:2010hr,Fiore:2008tp,Dremin:2013xza,\nUzhinsky:2011qu,Bourrely:2012hp,Selyugin:2013cia,Martynov:2013ana,Donnachie:2013xia,Khoze:2013jsa}).\nIn a conventional approach at small transfer momentum experimental\ndata can be described quite well by the diffractive scattering\ninduced by Pomeron exchange between hadrons. At large $-t \\gg 1$\nGeV$^2$ in the popular Donnachie-Landshoff (DL) model the dominant\ncontribution comes from the exchange by Odderon which is the\n$P=C=-1$ partner of Pomeron. It was suggested that this effective\nexchange originated from the perturbative three gluon exchange in\nthe proton-proton and proton-antiproton scattering\n\\cite{Donnachie:1979yu}.\n  The experimental support for the existence  of such exchange comes\nfrom high energy ISR data on the difference in the dip structure\naround $\\mid t\\mid \\approx 1.4 $ GeV$^2$ in the proton-proton and\nproton-antiproton differential cross sections at $\\sqrt{s}=53 $\nGeV \\cite{landshoff}.\n However, there is no any signal for  Odderon at very small transfer\nmomentum. We would like to emphasize that one cannot expect the\nperturbative QCD DL approach to be valid even at the  largest\ntransfer momentum $-t\\sim 14$ GeV$^2$   accessible at ISR\nenergies. This is related to the fact that in the three-gluon\nexchange model, which is applied   to describe elastic cross\nsections in the interval $-t= 3-14 $ GeV$^2$, the average\nvirtuality of exchanged gluons $\\hat{t}\\approx t\/9 $ is quite\nsmall -$\\hat{t}=0.3-1.6 $ GeV$^2$. Therefore, in this kinematic\nregion  nonperturbative QCD effects should be taken into account.\n\nThe attempt to include some of the nonperturbative effects into\nthe DL model was made in \\cite{landshoff2}. In that paper  the\nstrength of three-gluon exchange with perturbative quark-gluon\nvertices was considered as a free parameter and its value was\nfound from the fit of the data. Therefore, a good description of\nthe large $-t$ cross sections in the paper  is not the result of\ncalculation but rather of the fine tuning to  experimental data.\n\n\nOne of the successful models of  nonperturbative effects is the\ninstanton liquid model for QCD vacuum \\cite{shuryak,diakonov}.\nInstantons describe nontrivial topological gluon field excitations\nin vacuum and their existence leads to the spontaneous chiral\nsymmetry breaking in QCD. One of the manifestations of this\nphenomenon is the appearance of dynamical quark mass and\nnonperturbative helicity-flip quark-gluon  interaction\n\\cite{Kochelev:1996pv,diakonov}. Such new interaction can be\ntreated as a nonperturbative anomalous quark chromomagnetic moment\n(AQCM). It was shown that AQCM gives a very important contribution\nthe to quark-quark scattering at large energies for both polarized\nand nonpolarized cases\n\\cite{Kochelev:1996pv,diakonov,Kochelev:2013zoa,Kochelev:2009rf,Kochelev:2006ny}.\nOne of the applications of these results is a new model for\nPomeron based on AQCM  and nonperturbative two gluon exchange\nbetween hadrons suggested in \\cite{diakonov,Kochelev:2009rf}.\n\n\nIn this paper, we extend this model to the case  of the three\ngluon colorless exchange between nucleons.  It will be shown that\na nonperturbative version of the Donnachie-Landshoff Odderon model\nbased on AQCM describes  well high energy data for the elastic\nproton-proton, proton-antiproton cross sections at large transfer\nmomentum. The spin effects in elastic scattering are also under\ndiscussion.\n\n\n\\section{Anomalous quark chromomagnetic moment and Odderon exchange}\n\n\nThe interaction vertex of a massive quark with a gluon  can be\nwritten in the following form:\n\\begin{equation}\nV_\\mu(k_1^2,k_2^2,q^2)t^a = -g_st^a[ \\gamma_\\mu\nF_1(k_1^2,k_2^2,q^2) -\n\\frac{\\sigma_{\\mu\\nu}q_\\nu}{2M_q}F_2(k_1^2,k_2^2,q^2)],\n \\label{vertex}\n \\end{equation}\nwhere the form factors $F_{1,2}$ describe nonlocality of the\ninteraction, $k_{1,2}$ is the momentum of incoming and outgoing\nquarks, respectively,  $ q=k_1-k_2$, $M_q$ is the quark mass, and\n$\\sigma_{\\mu\\nu}=(\\gamma_\\mu \\gamma_\\nu-\\gamma_\\nu \\gamma_\\mu)\/2$.\n Within the instanton\nmodel  the shape  of the form factor $F_2(k_1^2,k_2^2,q^2)$ is\n \\be\n F_2(k_1^2,k_2^2,q^2) =\\mu_a\n\\Phi_q(\\mid k_1\\mid\\rho\/2)\\Phi_q(\\mid k_2\\mid\\rho\/2)F_g(\\mid\nq\\mid\\rho) \\ , \\label{form1} \\ee where \\ba\n\\Phi_q(z)&=&-z\\frac{d}{dz}(I_0(z)K_0(z)-I_1(z)K_1(z)), \\nonumber\\\\\nF_g(z)&=&\\frac{4}{z^2}-2K_2(z) \\label{form3} \\ea are the\nFourier-transformed quark zero-mode and instanton fields,\nrespectively,  $I_{\\nu}(z)$ and $K_{\\nu}(z)$ are the modified\nBessel functions and $\\rho$ is the instanton size.\n\nAQCM is defined by formula\n\\begin{equation}\n\\mu_a=F_2(0,0,0). \\nonumber\n\\end{equation}\n\nFor our estimation below we will use the value of AQCM $\\mu_a=-1$\nwhich is within the interval $-\\mu_a\\sim 0.4 -1.6$ given by the\ninstanton model \\cite{Kochelev:2009rf}. This value  is also\nsupported by hadron spectroscopy (see \\cite{Ebert:2009ub} and\nreferences therein). Recently, a similar value of AQCM was also\nobtained within the Dyson-Schwinger equation approach with\nnonperturbative quark and gluon propagators \\cite{Cloet:2013jya}.\n In Fig.1, the  Donnachie-Landshoff perturbative QCD (pQCD)\nand nonperturbative  AQCM-induced three gluon exchange between two\nnucleons are presented.\n\n\n\\begin{figure}[htb]\n\\includegraphics[scale=1.0]{fig1.eps}\n\\caption{ The left panel is the Donnachie-Landshoff mechanism for\nthe large $-t$ proton-proton scattering. The  right panels are the\nexample of  the AQCM contribution induced by the second term in\nEq.\\ref{vertex}.}\n\\end{figure}\nWithin the DL model the differential cross-section of the\nproton-proton and proton-antiproton scattering is given by the\nformula\n\\begin{equation}\n\\frac{d\\sigma}{dt}\\approx \\frac{244P^4}{s^6t^2R^{12}}\\mid\nM_{qq}(\\theta)\\mid^6\n \\label{sigma}\n\\end{equation}\nwhere  $M_{qq}$ is the matrix element at the quark level, $\\theta$\nis the scattering angle in the center of mass,  $P$ is the\nprobability of the three quark configuration in a proton, and $R$\nis the proton radius. In the pQCD DL approach at the quark level\n\\begin{equation}\n\\mid\nM^{pQCD}_{qq}(\\theta)\\mid^2=\\frac{128\\pi^2\\alpha_s^2}{9}\\frac{\\hat{s}^2}{\\hat{t}^2},\n\\label{pQCD}\n\\end{equation}\n where  $\\hat{s}\\approx s\/9$,  at $\\hat{s}\\gg-\\hat{t}$\\ \\ $\\hat{t}\/\\hat{s} \\sim -\\sin^2\\theta\/4$,\nand  the following  values of the parameters were taken  {\\it ad hoc} :\\\\\n\\begin{equation}\nP=1\/10,\\ \\  \\alpha_s=0.3,\\ \\  R=0.3 fm.\n\\end{equation}\nWe should emphasize that  DL assumed a very small proton radius\nwhich is far away from the real  proton size $R\\approx 1$ fm.\n For more suitable values $P=1$ and $R=1$ fm, we  got\n$d\\sigma\/dt\\sim 8\\cdot 10^{-4}\/t^8$ mb\/GeV$^2$. It is about two\norders of magnitude less than high energy data $d\\sigma\/dt\\approx\n9\\cdot 10^{-2}\/t^8$ mb\/GeV$^2$  at large $-t$, Fig.2 .\n\\begin{figure}[h] \\vspace*{-0.0cm}\n\\centerline{\\epsfig{file=elasticpp.eps,width=10cm,height=10cm,angle=0}}\\\n\\caption{ \\small The contribution of pQCD exchange (dashed line)\nand AQCM   contribution (solid line)  to the elastic proton-proton\nscattering at large energy and large momentum transfer in\ncomparison with data \\cite{Nagy:1978iw}.}\n\\end{figure}\nFor the AQCM contribution at the quark level we have\n\\begin{eqnarray}\n\\mid M^{AQCM}_{qq}(\\hat{s},\\hat{t})\\mid^2&=&\\frac{16\\pi^3}{3}\n\\alpha_s (\\mid \\hat t\\mid) \\mid\\mu_a\\mid\n \\rho_c^2 F_g^2(\\sqrt{\\mid \\hat{t}\\mid}\\rho_c)\\frac{\\hat{s}^2}{\\mid\\hat{t}\\mid}\n+\\frac{\\pi^4}{2}\\mu_a^2\\rho_c^4 F_g^4(\\sqrt{\\mid\n\\hat{t}\\mid}\\rho_c)\\hat{s}^2. \\label{AQCM2}\n\\end{eqnarray}\nFor estimation, we   use  $R=1$ fm, $P=1$\\footnote{The value of\nthe strong proton radius $R\\approx 1$ fm is related to the\nconfinement scale. The probability of the three quark\nconfiguration in the proton $P=1$ is a natural assumption in our\nthree quarks on three quarks scattering model for large $-t$.},\n dynamical quark mass\n$M_q=280$ MeV, average instanton size $\\rho_c =1\/3$ fm and the\nstrong coupling constant\n\\begin{equation}\n\\alpha_s(q^2)=\\frac{4\\pi}{9\\ln((q^2+m_g^2)\/\\Lambda_{QCD}^2)},\n\\end{equation}\nwith $\\Lambda_{QCD}=0.280$ GeV  and $m_g=0.88$ GeV\n\\cite{Kochelev:2009rf}. To get Eq.\\ref{AQCM2} the approximation\n$F_1(k_1^2,k_2^2,q^2)\\approx 1$ was used and we neglected nonzero\nvirtuality of quarks in a proton.\n The final result for the AQCM\ncontribution to the proton-proton and proton-antiproton cross\nsection is presented by the solid line in Fig.2. We should mention\n that the AQCM contribution asymptotically decays as $1\/t^{11}$ due\nto the form factor Eq.\\ref{form3}. Therefore, asymptotically at\nvery large transfer momentum  perturbative $1\/t^8$ should give the\ndominating contribution. However,  in the kinematic region\naccessible at the present time in experiments  $-t \\leq 14$\nGeV$^2$, the nonperturbative AQCM contribution\n describes the available large\n$-t$ data very well, Fig.2.\n\\begin{figure}[htb]\n\\centerline{\\epsfig{file=fig3.eps,width=14cm,height=4cm,angle=0}}\\\n\\caption{ The interference between a) DL-type AQCM  diagram and b)\nPomeron spin-flip induced by AQCM.}\n\\end{figure}\nFinally,  some part of the difference between the structure of the\ndip around $-t\\approx 1-2 $ GeV$^2$ in the proton-proton and\nproton-antiproton elastic scatterating at ISR energies might be\nrelated to the difference in the sign of the interference between\nthe AQCM Odderon and Pomeron spin-flip amplitudes, Fig.3.\n\n\nIn our approach the  spin-flip component, which is proportional to\n$t$, gives the dominating contribution to the negative charge\nparity Odderon amplitude. In the region of small transfer momentum\nthis contribution to the amplitude of the  $PP$ and $P\\bar P$\nscattering has the dependence\n\\begin{equation}\n M\\sim \\frac{\\sqrt{-t}}{(m_g^2-t)^3}, \\label{small}\n\\end{equation}\ndue to  quark spin-flip induced by AQCM. In Eq.\\ref{small},\n$m_g\\approx 0.4$ GeV is the dynamical gluon mass\n\\cite{Aguilar:2013hoa}.\n Therefore, the difference in the $PP$\nand $P\\bar P$ differential cross sections at small $-t$ and the\ndifference in the total  $PP$ and $P\\bar P$ cross sections should\nbe very small at high energies. This is in  agreement with\nexperimental data.\n\nOf course, one can describe  $PP$ and $\\bar PP$ large $-t> 3.5 $\nGeV$^2$ data  by using the assumption about a specific $t$\ndependence of the Pomeron trajectory (see, for example\n\\cite{Bugrii:1979zh}). However, in anyway, it is necessary   to\nintroduce the additional $C=-1$ exchange with high intercept to\ndescribe the difference in the $PP$ and $\\bar PP$ elastic cross\nsections at $\\sqrt{s}=53$ GeV. A natural candidate for such\nexchange is the nonperturbative three gluon DL-type  exchange. We\nwould like to mention that  the sizable contribution from the\nconventional Pomeron exchange at large $-t>3.5$ GeV$^2$ is not\nexpected due to the huge suppression factor at large energies,\n$(s\/s_0)^{2\\alpha^\\prime_Pt}$,  which comes from the nonzero slope\nof the Pomeron trajectory $\\alpha^\\prime_P\\approx 0.25$\nGeV$^{-2}$.\n\nIn the estimation above we assume, as in the DL model, that\nmomenta of exchanged gluons  are approximately equal. The\njustification of this assumption   is  quite clear. To keep a\nproton as a bound state of three quarks at large transfer\nmomentum, all quarks in the proton should scatter approximately to\nthe same angle. In fact, one  can also consider more complicated\nmultigluon contributions to elastic scattering, but we believe\nthat such contribution will be suppressed  by either additional\nfactors $\\alpha_s$ or by extra factors $1\/t^n$ coming from gluon\npropagators and\/or from form factors in the quark-gluon vertices.\n\n\n\n\\section{\n Single-spin asymmetry $A_N$ in $PP$ and $P\\bar P$ elastic\n scattering}\n\nOne of the long-standing problems of QCD is the  understanding of\nthe large spin effects observed in the different high energy\nreactions \\cite{Krisch:2010hr}, \\cite{Leader:2001gr}. Recently, we\nhave shown that the AQCM contribution leads to  very large\nsingle-spin asymmetry (SSA) in the quark-quark scattering\n\\cite{Kochelev:2013zoa} and, therefore, it can be considered as a\nfundamental mechanism for explanation of anomalously large SSA\nobserved in different inclusive and exclusive reactions at high\nenergy. In elastic scattering, large SSA  was found in the\nproton-proton scattering at AGS energies at large transfer\nmomentum, Fig.4.\n\\begin{figure}[htb] \\vspace*{-0.0cm}\n\\centerline{\\epsfig{file=ANpp.eps,width=8cm,height=8cm,angle=0}}\\\n\\caption{ \\small Single-spin asymmetry in the elastic\n$PP\\rightarrow PP$ scattering at large momentum transfer at AGS\n\\cite{Crabb:1990as}.}\n\\end{figure}\nIn the bases of the c.m. helicity amplitudes SSA is given by the\nformula\n\\begin{equation}\nA_N=-\\frac{2Im[\\Phi_5^*(\\Phi_1+\\Phi_2+\\Phi_3-\\Phi_4)]}{\\mid\\Phi_1\\mid^2+\\mid\\Phi_2\\mid^2+\\mid\\Phi_3\\mid^2+\n\\mid\\Phi_4\\mid^2+4\\mid\\Phi_5\\mid^2},\\label{SSA}\n\\end{equation}\n where the helicity amplitudes $\\Phi_1=<++\\mid++>$,\n$\\Phi_2=<++\\mid-->$, $\\Phi_3=<+-\\mid+->$, $\\Phi_4=<++\\mid-->$ and\n$\\Phi_5=<++\\mid-+>$. It is evident that due to the negative charge\nparity Odderon contribution, the helicity-flip amplitude $\\Phi_5$\nshould have a different sign for the proton-proton and\nproton-antiproton scattering. Therefore, SSA in the case of the\nelastic proton-antiproton scattering flips the sign in comparison\nwith the proton-proton scattering. This prediction can be tested\nby the PAX Collaboration at HESR \\cite{Barone:2005pu}. Due to the\ndominance of spin one $t$-channel gluon exchanges in the structure\nof Pomeron and Odderon, we can also expect that single-spin\nasymmetry at large $-t$ should have a weak energy dependence. This\nprediction can be checked in the polarized  proton-proton elastic\nscattering in the pp2pp experiment  at RHIC in  case of extending\ntheir kinematics to the large transfer momentum region\n\\cite{pp2pp} \\footnote{We would like to thank Jacek Turnau for the\ndiscussion of  this problem.}. However, the calculation of\nabsolute value of SSA in the elastic $PP$\n and $P\\bar P$ scattering at large transfer momenta is a very difficult task, because one\n needs\n to know spin-flip and non-spin flip components of both Odderon\n and Pomeron exchanges. Furthermore, in the region of small\n transfer momenta and low energies it is needed to include the effects\n of secondary Reggion exchanges as well.\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\nIn summary, it is shown that the anomalous quark-gluon\nnonperturbative vertex gives a large contribution to the elastic\nproton-proton and proton-antiproton  scattering at large momentum\ntransfer. One can treat three-gluon exchange induced by this\nvertex as effective Odderon exchange with the spin-flip dominance\nin its amplitude.\n We should mention that the anomalous quark\nchromomagnetic moment is proportional to $1\/\\alpha_s$\n\\cite{Kochelev:1996pv}. Therefore, non-spin flip component in\nOdderon due to perturbative vertex should be suppressed by\n$\\alpha_s$ factor.\n We argue that a strong spin\ndependence of the Odderon amplitude might lead to the large spin\neffects in the proton-proton and proton-antiproton scattering at\nlarge momentum transfer.\n\n Our approach is based on the existence of two\nquite different scales in hadron physics. One of them  is related\nto the confinement radius $ R\\approx 1$ fm and it is consistent,\nas well,  with an average distance between instanton and\nantiinstanton within the  instanton liquid model, $R_{I\\bar\nI}\\approx 1$ fm \\cite{shuryak,diakonov} . This scale is\nresponsible for the diffractive type scattering at small momentum\ntransfer. Another one  is fixed by the scale of spontaneous\nsymmetry breaking. Within the instanton model it is given by an\naverage instanton size in QCD vacuum $\\rho_c\\approx 1\/3$ fm. This\nscale leads to the appearance of a large dynamical quark mass and\nlarge anomalous quark chromomagnetic moment and is responsible for\nthe dynamics of the hadron-hadron elastic scattering at large\nmomentum transfer. We would like to mention that the two scale\nmodel for the hadron structure was discussed in different aspects\nin the papers \\cite{Dorokhov:1993fc,Schweitzer:2012hh}.\n\n\\section{Acknowledgments}\nThe authors are very grateful to  A.E. Dorokhov, A.V. Efremov,\nS.~V.~Goloskokov, E.A.~Kuraev, N.N.~Nikolaev, L.N. Lipatov, O.V.\nSelyugin and V.V. Uzhinsky  for useful discussions.\n The work of N.K. was supported in part by  a visiting\nscientist grant from the University of Valencia and  by the MEST\nof the Korean Government (Brain Pool Program No. 121S-1-3-0318).\nWe also acknowledge that this work was initiated through the\nseries of APCTP-BLTP JINR Joint Workshops.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ Abelian Lie algebras}\nIn this section, we show by an elementary argument that any abelian Lie algebra ( i.e. any vector space over $\\mathbb{Q}$), is potentially one dimensional, i.e. for any vector space $L$ over $\\mathbb{Q}$, there exists a field $E$ of characteristic zero, such that $\\dim_E L=1$. Then we show that this can not be generalized for nilpotent Lie algebras of class $2$.\n\n\\begin{proposition}\nLet $L$ be a vector space over $\\mathbb{Q}$. Then there exists a field $E$ of characteristic zero, such that $L$ is a vector space over $E$ and $\\dim_E L=1$.\n\\end{proposition}\n\n\\begin{proof}\nFirst, let $\\dim_{\\mathbb{Q}}L=n$ and $x_1, \\ldots, x_n$ be a basis of $L$. Suppose\n$$\nq(t)=a_0+a_1t+\\cdots+a_{n-1}t^{n-1}+t^n\\in \\mathbb{Q}[t]\n$$\nbe an irreducible polynomial. Let $\\lambda\\in \\mathbb{C}$ be any root of $q(t)$ and $E=\\mathbb{Q}(\\lambda)$. Define an action of $E$ on $L$ by the $\\mathbb{Q}$-bilinear extension of following rules\n\\begin{eqnarray*}\n\\lambda.x_1&=&x_2\\\\\n\\lambda.x_2&=&x_3\\\\\n           &\\vdots&\\\\\n\\lambda.x_n&=&-a_0x_1-a_1x_2-\\cdots-a_{n-1}x_n.\n\\end{eqnarray*}\nIt is easy to check that $L$ is a vector space over $E$ of dimension $1$. Now, suppose $\\dim_\\mathbb{Q}L=\\alpha$ is infinite. Let $X$ be any set with the cardinality $\\alpha$ and suppose $\\mathbb{Q}(X)$ is the field rational functions with variables in the set $X$. Note that\n$$\n\\mathbb{Q}(X)=\\bigcup_{Y\\subset X}\\mathbb{Q}(Y),\n$$\nwhere $Y$ varies in the set of all finite subsets of $X$. Since every $\\mathbb{Q}(Y)$ is countable so we have $|\\mathbb{Q}(X)|=|X|$. Hence as $\\mathbb{Q}$-spaces, we have $L\\cong\\mathbb{Q}(X)$. Let $f:L\\to \\mathbb{Q}(X)$ be a $\\mathbb{Q}$-isomorphism. Now for $q\\in \\mathbb{Q}(X)$ and $x\\in L$, define\n$$\nq.x=f^{-1}(qf(x)).\n$$\nSo, by assuming $E=\\mathbb{Q}(X)$, $L$ is an $E$-space with $\\dim_E L=1$.\n\\end{proof}\n\nOne may wants to generalize this proposition for the case of nilpotent Lie algebras of higher nilpotency classes. But this is not true in general, since the three dimensional Heisenberg algebra $H=\\langle x, y, z\\rangle$ with $[x,y]=z$ and $z\\in Z(H)$ is a nilpotent rational Lie algebra of class two but there is no field $E$ with $\\dim_E H=2$. In general, the extension degree $[E:\\mathbb{Q}]$ is restricted by the numbers $\\dim_{\\mathbb{Q}}C_L(x)$, $x\\in L$. To see this, let $L$ be any Lie algebra over $E$. Then clearly for any $x\\in L$, we have $Span_E(x)\\subseteq C_L(x)$, so\n$$\n[E:\\mathbb{Q}]=\\dim_{\\mathbb{Q}}Span_E(x)\\leq \\dim_{\\mathbb{Q}}C_L(x).\n$$\nTherefore, we have\n\n\\begin{lemma}\nLet $L$ be a rational Lie algebra and $E$ be any field such that $(L, E)$ is a completion for $L$. Then\n$$\n[E:\\mathbb{Q}]\\leq \\min_{x\\in L}\\ \\dim_{\\mathbb{Q}}C_L(x).\n$$\n\\end{lemma}\n\nBy a similar argument, we can prove the following proposition.\n\n\\begin{proposition}\nLet $L$ be an infinite dimensional rational Lie algebra which has a finite dimensional completion of the form $(L, E)$. Then\n\n\\ i- either $L$ is solvable or for any ordinal $\\alpha$ we have $\\dim_{\\mathbb{Q}}L^{(\\alpha)}=\\infty$.\n\nii- either $L$ is nilpotent or for any ordinal $\\alpha$ we have $\\dim_{\\mathbb{Q}}\\gamma_{\\alpha}(L)=\\infty$.\n\\end{proposition}\n\n\\section{The class of completions}\nIn this section, we show that the class of completions of any finite dimensional rational Lie algebra is elementary ( axiomatizable in some first order language). We also show that this is not true for the class of all fields $E$ in which $L$ is a Lie algebra over $E$. Remember that a pair $(K, E)$ is a completion of $L$, if\n\n\\ \\ i- $E$ is a field of characteristic zero,\n\n\\ ii- $K$ is a Lie algebra over $E$,\n\niii- $L\\leq_{\\mathbb{Q}}K$ ( a $\\mathbb{Q}$-Lie subalgebra),\n\niv- $K=Span_E(L)$.\n\nLet $\\mathfrak{X}_L$ be the class of all completions of $L$.\n\n\\begin{theorem}\nIf $\\dim_{\\mathbb{Q}}L$ is finite, then the class $\\mathfrak{X}_L$ is elementary.\n\\end{theorem}\n\n\\begin{proof}\nLet $v_1, \\ldots, v_n$ be a $\\mathbb{Q}$-basis of $L$. We introduce a first order language\n$$\n\\mathcal{L}=(0, 1, +, \\times, q, (a^{\\ast}, a\\in L))\n$$\nwhich contains constant symbols $0$, $1$ and $a^{\\ast}$ for all $a\\in L$. It also has two binary functional symbols $+$ and $\\times$, and $q$ is a unary predicate symbol which will be interpreted as \"{\\em to be scalar}\". Let $\\Sigma$ be the following set of axioms:\\\\\n\n\\ 1- $\\forall x\\forall y\\ (q(x)\\wedge q(y))\\Rightarrow (q(x+y)\\wedge q(xy))$.\n\n\\ 2- $\\forall x\\forall y\\ (q(x)\\wedge q(y))\\Rightarrow (x+y=y+x \\wedge x\\times y=y\\times x)$.\n\n\\ 3- $\\forall x\\forall y\\forall z\\ (q(x)\\wedge q(y)\\wedge q(z))\\Rightarrow ((x+y)+z=x+(y+z)\\wedge (x\\times y)\\times z=x\\times (y\\times z))$.\n\n\\ 4- $\\forall x\\ q(x)\\Rightarrow x+0=x$.\n\n\\ 5- $\\forall x\\ 1\\times x=x$.\n\n\\ 6- $\\forall x\\ q(x)\\Rightarrow (\\exists y\\ q(y)\\wedge x+y=0)$.\n\n\\ 7- $\\forall x\\forall y\\forall z \\ (q(x)\\wedge q(y)\\wedge q(z))\\Rightarrow (x\\times (y+z)=x\\times y+x\\times z)$.\n\n\\ 8- $\\forall x\\ (q(x)\\wedge x\\neq 0)\\Rightarrow (\\exists y\\ q(y)\\wedge x\\times y=1)$.\n\n\\ 9- $\\forall x\\forall y\\ (\\neg q(x)\\wedge \\neg q(y))\\Rightarrow(\\neg q(x+y)\\wedge q(x\\times y))$.\n\n10- $\\forall x\\forall y\\ (\\neg q(x)\\wedge \\neg q(y))\\Rightarrow (x+y)=y+x$.\n\n11- $\\forall x\\ \\neg q(x)\\Rightarrow x\\times x=0^{\\ast}$.\n\n12- $\\forall x\\forall y\\forall z\\ (\\neg q(x)\\wedge \\neg q(y)\\wedge \\neg q(z))\\Rightarrow (x+y)+z=x+(y+z)$.\n\n13- $\\forall x\\ \\neg q(x)\\Rightarrow x+0^{\\ast}=x$.\n\n14- $\\forall x\\ \\neg q(x)\\Rightarrow(\\exists y\\  \\neg q(y)\\wedge x+y=0^{\\ast})$.\n\n15- $\\forall x\\forall y\\forall z( \\neg q(x)\\wedge \\neg q(y)\\wedge \\neg q(z))\\Rightarrow (x\\times y)\\times z+(y\\times z)\\times x+(z\\times x)\\times y=0^{\\ast}$.\n\n16- $\\forall x\\forall y\\forall z(\\neg q(y)\\wedge \\neg q(z))\\Rightarrow x\\times(y+z)=x\\times y+x\\times z$.\n\n\n17- $\\forall x\\forall y(q(x)\\wedge \\neg q(y))\\Rightarrow \\neg q(x\\times y)$.\n\n18- $\\forall x\\forall y\\forall z( q(x)\\wedge \\neg q(y)\\wedge \\neg q(z))\\Rightarrow x\\times (y\\times z)=(x\\times y)\\times z= y\\times (x\\times z)$.\n\n19- $\\forall x\\forall y\\forall z( q(x)\\wedge q(y)\\wedge \\neg q(z))\\Rightarrow (x\\times y)\\times z= x\\times (y\\times z)$.\n\n20- $\\forall x\\forall y\\forall z( q(x)\\wedge q(y)\\wedge \\neg q(z))\\Rightarrow (x+ y)\\times z= x\\times z+ y\\times z$.\n\n21- for all $a\\in L$ the axiom: $\\neg q(a^{\\ast})$.\n\n22- for all $a, b\\in L$ the axiom: $a^{\\ast}\\neq b^{\\ast}$.\n\n23- for all $a, b\\in L$ the axioms: $(a+b)^{\\ast}=a^{\\ast}+b^{\\ast}$ and $[a^{\\ast}, b^{\\ast}]=a^{\\ast}\\times b^{\\ast}$.\n\n24- for all natural number $m$ the axiom:  $\\forall x\\ (q(x)\\wedge mx=0)\\Rightarrow x=0 $.\n\n25- $\\forall x\\ \\neg q(x)\\Rightarrow (\\exists x_1\\exists x_2 \\ldots \\exists x_n\\ (\\bigwedge_{i=1}^n q(x_i))\\wedge (x=\\sum_{i=1}^nx_i\\times v_i))$.\\\\\n\nNow, let $M\\in Mod(\\Sigma)$ be a model of $\\Sigma$. Define\n$$\nK=\\{ x\\in M: \\neg q(x)\\}\n$$\nand\n$$\nE=\\{x\\in M: q(x)\\}.\n$$\nThen it is easy to verify that $(K, E)$ is a completion of $L$. Conversely if $(K, E)$ is a completion of $L$ then we let $M=K\\cup E$. For all $x, y\\in M$, define $x+y$ to be their sum in $E$, if $x, y\\in E$, their sum in $K$ if, $x, y\\in K$, and $0^{\\ast}$ otherwise. Similarly, define $x\\times y$ to be $xy$, if $x, y\\in E$, $[x,y]$, if $x, y\\in K$, and the scalar product otherwise. It can be verified that $M$ is a model for $\\Sigma$ and so $\\Sigma$ axiomatizes the class $\\mathfrak{X}_L$.\n\\end{proof}\n\nIt is clear that the class of all Lie algebras over a fixed field $E$ is elementary. We show that the class of all fileds for a fixed rational Lie algebra is not elementary. Let $L$ be such a Lie algebra. By $\\mathfrak{F}_L$ we denote the class of all fields $E$ such that $(L, E)$ is a completion of $L$. Note that every element of $\\mathfrak{F}_L$ is a sub-ring of the centroid $\\Gamma(L)$.\n\n\\begin{proposition}\nLet $L$ be a finite dimensional Lie algebra over $\\mathbb{Q}$. Then $\\mathfrak{F}_L$ is not elementary.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\mathcal{L}$ be a first order language and $\\Sigma$ be a set of axioms of $\\mathfrak{F}_L$. Since $L$ is countable, we may assume that $\\mathcal{L}$ is also countable. Now $\\mathbb{Q}$ is a model for $\\Sigma$ and $|\\mathbb{Q}|\\geq |\\mathcal{L}|$. So by the Lowenheim-Skolm theorem, for any infinite cardinal $\\alpha$, the set $\\Sigma$ has a model $E$ of size $\\alpha$. Now\n$$\n\\dim_{\\mathbb{Q}}L=[E:\\mathbb{Q}]\\dim_E L,\n$$\nand the left hand side is finite. So $[E:\\mathbb{Q}]$ is finite, contradicting $|E|=\\alpha$.\n\\end{proof}\n\nSimilarly, one can prove that the class $\\mathfrak{F}_L$ is not elementary for any abelian Lie algebra $L$.\n\n\\section{Entangled ideals}\nIn this section, we introduce the concept of {\\em entangled} ideals in a tensor product $E\\otimes_{\\mathbb{Q}}L$ and we show that they have close connections with the completions of $L$. For a rational Lie algebra $L$ and any field $E$ of characteristic zero, we say that a Lie algebra $K$ is an $E$-completion of $L$, if $(K,E)$ is a completion for $L$.\n\n\\begin{definition}\nAn ideal $N$ of the $E$-algebra $E\\otimes_{\\mathbb{Q}}L$ is called entangled, if $N\\cap (1\\otimes L)=0$.\n\\end{definition}\n\n\\begin{proposition}\nLet $K$ be an $E$-completion of $L$. Then there exists an entangled ideal $N\\unlhd E\\otimes_{\\mathbb{Q}}L$ such that\n$$\nK=\\frac{E\\otimes_{\\mathbb{Q}}L}{N}.\n$$\nConversely, for any entangled ideal $N$, the Lie algebra $K$ defined by the above equality is an $E$-completion.\n\\end{proposition}\n\n\\begin{proof}\nDefine a map $\\varphi:E\\times L\\to K$ by $\\varphi(x, a)=xa$. This map is  $\\mathbb{Q}$-bilinear and so there is a $\\mathbb{Q}$-linear map $\\varphi^{\\ast}:E\\otimes_{\\mathbb{Q}}L\\to K$ with $\\varphi^{\\ast}(x\\otimes a)=xa$. Since $K=Span_E(L)$, so $\\varphi^{\\ast}$ is surjective. Also for any $\\lambda\\in E$, we have\n$$\n\\varphi^{\\ast}(\\lambda(x\\otimes a))=\\varphi^{\\ast}((\\lambda x)\\otimes a)= \\lambda \\varphi^{\\ast}(x\\otimes a),\n$$\nhence $\\varphi^{\\ast}$ is $E$-linear. Further\n\\begin{eqnarray*}\n\\varphi^{\\ast}([x\\otimes a, y\\otimes b])&=& \\varphi^{\\ast}((xy)\\otimes [a,b])\\\\\n                                        &=&xy[a,b]\\\\\n                                        &=&[\\varphi^{\\ast}(x\\otimes a), \\varphi^{\\ast}(y\\otimes b)].\n\\end{eqnarray*}\nSo, $\\varphi^{\\ast}$ is an epimorphism of $E$-Lie algebras. Let $N=\\ker \\varphi^{\\ast}$. Then\n$$\nK\\cong \\frac{E\\otimes_{\\mathbb{Q}}L}{N}.\n$$\nNote that\n$$\nN=\\{ \\sum_i x_i\\otimes a_i\\in E\\otimes_{\\mathbb{Q}}L:\\ \\sum_ix_ia_i=0\\},\n$$\nso if $1\\otimes a\\in N$, then $a=0$ and therefore $N$ is an entangled ideal.\n\nConversely, let $N\\unlhd E\\otimes_{\\mathbb{Q}}L$ be an entangled ideal. Let $K$ be defined as above. Then the map $a\\mapsto 1\\otimes a+N$ is a $\\mathbb{Q}$-embedding of $L$ in $K$. So, we may assume that $L\\leq_\\mathbb{Q}K$. Now clearly $K=Span_E(L)$ and so $K$ is an $E$-completion of $L$.\n\\end{proof}\n\n\\begin{example}\nLet $L$ be a Lie algebra over $E$. Then in the same time $L$ is a Lie algebra over $\\mathbb{Q}$. Let $I\\unlhd_{\\mathbb{Q}}L$, $f\\in Aut_{\\mathbb{Q}}(L)$, and $\\sigma\\in Aut(E)$. Define\n$$\nN(I,f,\\sigma)=\\{ \\sum_i x_i\\otimes a_i\\in E\\otimes_{\\mathbb{Q}}I:\\ \\sum_i\\sigma(x_i)f(a_i)=0\\}.\n$$\nNote that the map $\\varphi:E\\times I\\to L$ defined by $\\varphi(x,a)=\\sigma(x)f(a)$ is $\\mathbb{Q}$-bilinear and so $N(I,f,\\sigma)$ is well-defined. It can be easily seen that $N(I,f,\\sigma)$ is an entangled ideal. So we have a completion\n$$\nK(I,f,\\sigma)=\\frac{E\\otimes_{\\mathbb{Q}}L}{N(I,f,\\sigma)}.\n$$\n\\end{example}\n\n\n\\begin{proposition}\nLet $E$ be a finite extension of $\\mathbb{Q}$ and $L$ be a finite dimensional Lie algebra over $E$. With the above notations, we have\n$$\n\\dim_E K(I,f,\\sigma)=\\dim_E K(f(I), 1,1),\n$$\nand in the special case,\n$$\ndim_E K(L, f, \\sigma)=\\dim_E L.\n$$\n\\end{proposition}\n\n\\begin{proof}\nDefine a $\\mathbb{Q}$-linear map $\\sigma\\otimes f:E\\otimes_{\\mathbb{Q}}L\\to E\\otimes_{\\mathbb{Q}}L$ by $(\\sigma\\otimes f)(x\\otimes a)=\\sigma(x)f(a)$. This is an invertible map and we have\n$$\n(\\sigma\\otimes f)(N(I,f,\\sigma))=N(f(I),1,1).\n$$\nTherefore $\\dim_{\\mathbb{Q}}N(I,f,\\sigma)=\\dim_{\\mathbb{Q}}N(f(I), 1,1)$. Now, since $[E:\\mathbb{Q}]$ is finite, so\n$$\n\\dim_EK(I,f,\\sigma)=\\dim_EK(f(I),1,1).\n$$\nAs an special case, when $I=L$, we have $\\dim_E K(L,f,\\sigma)=\\dim_E K(L,1,1)$. But by an easy argument, one can see that\n$$\n\\dim_E N(L, 1,1)=([E:\\mathbb{Q}]-1)\\dim_E L,\n$$\nso we have\n$$\n\\dim_E K(L,f,\\sigma)=\\dim_E L.\n$$\n\\end{proof}\n\nWe can say more about the structure of $K(L,f,\\sigma)$. Let $a\\to \\bar{a}$ be the embedding of $L$ in $K(L,f,\\sigma)$. We have,\n\n\\begin{proposition}\nBy the above notations,\n$$\nK(L,f,\\sigma)=\\{ \\bar{a}:\\ a\\in L\\}\n$$\nand for all $x\\in E$, $x.\\bar{a}=\\overline{f^{-1}(\\sigma(x)f(a))}$.\n\\end{proposition}\n\n\\begin{proof}\nLet $N=N(L,f,\\sigma)$ and $X=\\sum x_i\\otimes a_i+N\\in K(L,f,\\sigma)$. We know that $\\sum\\sigma(x_i)f(a_i)\\in L$, so for some $a\\in L$, we have\n$f(a)=\\sum \\sigma(x_i)f(a_i)$. Therefore $1\\otimes a-\\sum x_i\\otimes a_i\\in N$ and hence $X=\\bar{a}$. Now, if $x\\in E$, then\n$$\nx.\\bar{a}=x\\otimes a+N=1\\otimes b+N,\n$$\nfor some $b$. Clearly, $f(b)=\\sigma(x)f(a)$, so $b=f^{-1}(\\sigma(x)f(a))$. Hence\n$$\nx.\\bar{a}=\\overline{f^{-1}(\\sigma(x)f(a))}.\n$$\n\\end{proof}\n\nNow, we show that nilpotency (solvablity) of $L$ implies nilpotency (solvablity) of its completions.\n\n\\begin{proposition}\nLet $L$ be a rational Lie algebra and $K$ be a completion for $L$ over some field $E$. If $L$ is nilpotent, then $K$ is nilpotent of the same class. If $L$ is solvable, then $K$ is also solvable of the same derived length.\n\\end{proposition}\n\n\\begin{proof}\nWe know that there is an entangled ideal $N$ such that\n$$\nK=\\frac{E\\otimes_{\\mathbb{Q}}L}{N}.\n$$\nSo, for any $n$ we have\n$$\n\\gamma_n(K)=\\frac{E\\otimes_{\\mathbb{Q}}\\gamma_n(L)+N}{N}, \\ \\ K^{(n)}=\\frac{E\\otimes_{\\mathbb{Q}}L^{(n)}+N}{N}.\n$$\nThis shows that, if $\\gamma_n(L)=0$ ($L^{(n)}=0$), then $\\gamma_n(K)=0$ ($K^{(n)}=0$). On the other hand, if $\\gamma_d(K)=0$ ($K^{(d)}=0$), then since $L\\subseteq K$, we must have $\\gamma_d(L)=0$ ($L^{(d)}=0$).\n\\end{proof}\n\nA Lie algebra $L$ over $\\mathbb{Q}$ is potentially finite dimensional, if it has a finite dimensional completion. Using a theorem of Ado (see \\cite{Jac}), it is easy to prove that $L$ is potentially finite dimensional, if and only if $L\\leq_{\\mathbb{Q}}\\mathfrak{g}\\mathfrak{l}_n(E)$ for some field $E$ of characteristic zero and a natural number $n$. By the above proposition, we have\n\n\\begin{corollary}\nLet $L$ be a potentially finite dimensional nilpotent rational Lie algebra. Then $L\\leq_{\\mathbb{Q}}\\mathfrak{n}^+_n(E)$, for some field $E$ and $n\\geq 1$, where $\\mathfrak{n}^+_n(E)$ is the Lie algebra of strictly upper triangular $n\\times n$ matrices over $E$.\n\\end{corollary}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nA full century after the conception of general relativity (GR) the direct observation of gravitational waves (GWs) from merging black holes\nby the advanced LIGO-Virgo network of detectors~\\cite{Abbott:2016blz, TheLIGOScientific:2016pea, Abbott:2016nmj, Abbott:2017vtc, GW170814}\nhas finally opened the door to tests of relativistic gravity in the truly nonlinear strong field regime. Among the  prime objectives of present and\nnear-future GW astronomy is the materialisation of `black hole spectroscopy', that is, the observation of the so-called ringdown signal at the very\nend of the merger and the extraction\/identification of the final black hole's quasinormal modes (QNMs). The power of this method, much like its\natomic physics kin, lies in the fact that GR predicts a unique spectrum of complex QNM frequencies for a given black hole mass and spin and therefore\nthe simultaneous observation of more than two QNMs should, in principle, allow the Kerr hypothesis to be tested \\cite{Detweiler:1980gk,Dreyer:2003bv,Berti:2005ys}.\n\nAny programme aiming at probing the true nature of black holes should allow for deviations from GR's Kerr spacetime as well as for\ntheoretical input from alternative to GR theories of gravity. Perhaps the simplest `beyond-Kerr' strategy  is to use parametrised schemes (with the parameters controlling\nthe deformation away from GR) both for the black hole's spacetime metric~\\cite{Johannsen:2011dh, Johannsen2013PhRvD,Konoplya:2016jvv,Cardoso:2014rha}\nand the associated QNMs \\cite{Glampedakis:2017, Cardoso_etal2019,McManus_etal2019} without the need to commit to any particular theory\nof gravity.  The main drawback of this approach is that it may constitute nothing more than a null test of the Kerr metric, in the sense that the\ndeformations may not actually map onto any specific gravity theory. The alternative, more rigorous (and far more laborious) strategy is the\ntheoretical calculation of black hole spacetimes and their GW signature on a case-to-case basis within the zoo of modified gravity theories.\nNot surprisingly, this second approach is much more difficult to implement and as a result QNMs of non-GR black holes have been computed\nonly for a handful of cases, usually under the assumption of spherical\nsymmetry, e.g.~\\cite{molina2010, Kobayashi:2012kh, Kobayashi:2014wsa,Blazquez-Salcedo:2016enn, Blazquez-Salcedo:2016enn,Blazquez-Salcedo:2017txk, Brito:2018hjh, Tattersall_2018a, Tattersall_etal2018}\n(for a comprehensive review and further references see~\\cite{Bertietal2015}).\n\nIn this paper we study QNMs of black holes beyond GR by combining elements of the two aforementioned approaches.\nWe assume \\emph{spherically symmetric} black holes in which case, after separating out the angular dependence,  the wave dynamics\nand QNMs of the system are described by a set of radial wave equations for the perturbed spacetime metric (the tensorial field)\nand the other fields that are generically present in modified theories of gravity. In some instances one of the two symmetry sectors of the tensorial field\n(axial\/odd of polar\/even modes) couples to the other fields while the remaining one is described by the same Regge-Wheeler or Zerilli equation\nas in GR. Focusing, for obvious reasons, on the \\emph{coupled} case, we adopt a largely theory-agnostic approach and postulate\na pair of wave equations for the tensorial and the extra `scalar' field with parametrised potentials.\nAlthough far from representing the most general situation~\\cite{Tattersall_etal2018}, our parametrised model provides a useful benchmark\nfor describing perturbed non-GR black holes; in addition it has the merit of including as a special case at least a pair of modified theories of gravity,\nnamely, dynamical Chern-Simons gravity~\\cite{molina2010} and the sixth-order Proca theory~\\cite{Heisenberg:2017hwb,Tattersall_etal2018}.\n\n\nOur main main tool -- and second main simplification -- is the use of the \\emph{eikonal} (or geometric optics) approximation\nfor obtaining QNM solutions from the wave equations. The eikonal approximation  has a long and successful history in the study of\nSchwarzschild and Kerr black holes, dating back to the early 1970s~\\cite{Press:1971wr, Goebel:1972}\n(see~\\cite{Kokkotas:1999bd, Berti:2009kk} for reviews on the subject). According to the established eikonal picture, the fundamental QNM\ncan be visualised as a wavepacket localised in the radial direction at the peak of the wave potential;  in this approximation the peak itself coincides\nwith the location of the photon ring of null geodesics. The real part of the QNM frequency is found to be an integer multiple of the orbital angular frequency\nat the photon ring. Similarly, the decay rate (imaginary part) of the QNM is related to the Lyapunov exponent of the unstable null orbits at the photon ring\nradius~\\cite{Ferrari:1984zz, Mashhoon:1985cya,Cardoso:2008bp}. The same intuitive eikonal model has been used to establish a\nconnection between the $\\ell > |m|$ QNMs of Kerr black holes (where $\\ell, m$ are the usual spherical harmonic integers) and the nonequatorial\nspherical photon orbits~\\cite{Dolan:2010wr, Yang:2012he}. More recently, and inspired by the QNM-photon ring relation,\nan eikonal post-Kerr parametrised scheme was developed as a model for the fundamental $\\ell=|m|$ mode of non-GR black holes~\\cite{Glampedakis:2017}.\nAlthough this model can describe the astrophysically more interesting case of rotating black holes it also has the drawback that it does not\naccount for any extra field degrees of freedom. On the same topic of non-GR black holes it should be noted that some recent work has criticised\nthe validity of the connection between QNMs and photon geodesics~\\cite{Khanna:2016yow,Konoplya:2017wot}, although the models and arguments\nused in these papers are far from being conclusive.\n\nThe purpose of the eikonal calculation presented in this paper is therefore twofold. First, by including a coupling to an extra field,\nwe develop a parametrised eikonal QNM model that surpasses in rigorousness that of Ref.~\\cite{Glampedakis:2017}\n(although in doing so we restrict ourselves to the less physically interesting case of  spherical symmetry). Second, we examine to what\nextent these novel eikonal formulae preserve some connection between the fundamental QNM and photon geodesics.\nOur implementation of the eikonal approximation differs in one key aspect with respect to what is\nusually done in the context of GR black holes. In this latter case, the eikonal approximation consists in\ntaking the angular limit $\\ell \\gg 1$  in the Bohr-Sommerfeld formula originating from the WKB-approximated radial\nwave equations (for example, see \\cite{Berti:2014bla}). As no WKB approximation appears to exist (to the best of our knowledge)\nfor a system of coupled wave equations, we are forced to take the eikonal limit of the equations themselves both in the radial and angular\ndirections. In essence, our approach is very similar to that of Ref.~\\cite{Dolan:2010wr} for Kerr black holes, and this will become\napparent in the discussion of the following section.\n\nThe remainder of this paper is organised as follows. In Section~\\ref{sec:wave_eik}, and before embarking on\nour analysis of coupled wave equations, we study the eikonal limit of a single wave equation, albeit a generalised one that allows\nfor deviations from GR. Section~\\ref{sec:nonGR} contains the main calculation of this paper, namely, the derivation of eikonal formulae\nfor non-GR QNMs described by a pair of coupled wave equations in the context of modified theories of gravity with an extra scalar field degree of\nfreedom besides the standard tensorial one. A summary of these results can be found in Section~\\ref{sec:summary}.\nSection~\\ref{sec:CS} provides an application of our eikonal formulae for the case of QNMs\nof Schwarzschild black holes in dynamical Chern-Simons gravity. Our concluding remarks can be found in Section~\\ref{sec:conclusions}\nwhile the two appendices contain some additional material on photon geodesics and their correspondence to the eikonal limit.\n\nThroughout this paper we adopt geometric units $G=c=1$ and assume an $ e^{-i\\omega t}$ time dependence for the perturbed fields.\nWe use a prime to denote a $d\/dr$ radial derivative. For any function $f(r)$ we use abbreviations  $f_{\\rm ph} = f(r_{\\rm ph}), f_{\\rm max} = f(r_{\\rm max})$, etc.\n\n\n\\section{Generalised wave equation in a spherical black hole spacetime}\n\\label{sec:wave_eik}\n\n\\subsection{Eikonal approximation: leading order}\n\nIn order to set the stage for calculating QNMs of non-GR black holes in the eikonal approximation we first consider the case\nof a single (radial) wave equation describing perturbations of a field $\\psi$ in a background spherical spacetime.\nWe assume an equation,\n\\begin{equation}\n\\frac{d^2 \\psi}{dx^2} + \\left ( \\omega^2 - U  \\right ) \\psi = 0,\n\\label{gen_wave}\n\\end{equation}\nwhere $x(r)$ is a suitably defined tortoise coordinate which maps the black hole horizon (located at $r=r_{\\rm H}$) and infinity ($r=\\infty$)\nonto $x= -\\infty$ and $x=+\\infty$ respectively. The wave potential is assumed to be of the form\\footnote{Single-field wave equations of the\nform (\\ref{gen_wave})-(\\ref{UsinglePot}) can appear in extensions of GR like the higher-dimensional Einstein-Gauss-Bonnet\ntheory~\\cite{Konoplya:2017wot}.},\n\\begin{equation}\nU = f(r) \\left [\\, \\frac{\\ell(\\ell+1)}{r^2} \\alpha(r) - \\frac{6M}{r^3} \\zeta (r) \\, \\right ],\n\\label{UsinglePot}\n\\end{equation}\nwhere $\\ell$ is the familiar angular integer multipole and $f(r)$ a function with asymptotic behaviours $f(r\\to r_{\\rm H}) \\to 0$ and $f(r \\to \\infty)\\to 1$.\nThe functions $\\alpha (r), \\zeta (r)$ are assumed to carry no $\\ell$-dependence but are otherwise unspecified. Crucially, we require\n$U$ to be `black hole-like', that is, with a single peak and $U(x\\to \\pm \\infty) \\to 0 $.\n\nAccording to the eikonal\/geometric optics prescription  we look for wave solutions of (\\ref{gen_wave}) of the form,\n\\begin{equation}\n\\psi (x) = A(x) e^{iS(x)\/\\epsilon},\n\\end{equation}\nwhere $\\epsilon$ is the customary bookkeeping parameter. Then,\n\\begin{eqnarray}\n\\frac{d^2 \\psi}{dx} = e^{iS\/\\epsilon} \\left [\\, A_{,xx} + \\frac{i}{\\epsilon} \\left (\\,  2S_{,x} A_{,x} + S_{,xx} A \\,\\right ) - \\frac{(S_{,x})^2}{\\epsilon^2} A   \\, \\right ],\n\\nonumber \\\\\n\\end{eqnarray}\nand the wave equation becomes,\n\\begin{align}\n    A_{,xx} &+ \\frac{i}{\\epsilon} \\left (\\,  2S_{,x} A_{,x} + S_{,xx} A \\,\\right )\n + \\Big [ \\omega^2  - \\frac{(S_{,x})^2}{\\epsilon^2}\n \\nonumber \\\\\n & - f\\left (\\, \\frac{\\ell(\\ell+1)}{r^2} \\alpha - \\frac{6M}{r^3} \\zeta \\, \\right ) \\Big ] A = 0.\n \\label{full_eik}\n\\end{align}\nThe double limit $\\epsilon \\ll 1$ and $\\ell \\gg 1$ enforces the eikonal limit in the radial and angular directions\nand leads to the following leading-order equation,\n\\begin{equation}\n - \\frac{(S_{,x})^2}{\\epsilon^2} + \\omega^2 - \\ell^2 \\frac{f \\alpha}{r^2}  = 0 + {\\cal O}(\\ell, \\epsilon^{-1}).\n \\label{lead_eik}\n\\end{equation}\nThis expression, reminiscent of a radial Hamilton-Jacobi equation, only makes sense if the two expansion parameters balance,\n$\\epsilon \\ell = {\\cal O}(1)$. The physical picture behind this balance is that of wave packets equally localised in the radial and angular directions.\nThe same reasoning dictates a scaling $\\omega = {\\cal O} (\\ell)$.\n\nFor a QNM wave solution we need to impose the boundary conditions,\n\\begin{equation}\nS (x\\to  \\pm \\infty) = \\pm \\omega x.\n\\end{equation}\nFollowing a reasoning similar to the eikonal analysis of Ref. \\cite{Dolan:2010wr},\nwe expect the phase function $S$ to switch from outgoing ($x >0$) to ingoing ($x<0$) at the location of the potential peak (located at $x\\approx 0$);\nin other words $S$ should have a minimum at that radius.\n\n\nIf we define $\\tilde{U}$ as\n\\begin{equation}\nU(r) \\approx  \\ell^2   \\frac{f \\alpha}{r^2} \\equiv \\ell^2 \\tilde{U} (r), \\qquad (\\ell \\gg 1),\n\\end{equation}\nthe potential peak in the eikonal limit, $r=r_0$, should be\n\\begin{equation}\n\\tilde{U}^\\prime_0 = \\left (  \\frac{f \\alpha}{r^2} \\right )^\\prime_0 = 0,\n\\label{dU0eq}\n\\end{equation}\nwhere a prime stands for a derivative with respect to $r$.\nIn fact, the association of $S_{,x} = 0 $ with the potential peak can be enforced rather than being merely assumed\nby taking the $x$-derivative of (\\ref{lead_eik}):\n\\begin{equation}\n \\frac{2}{\\epsilon^2} S_{,x} S_{,xx} = -  \\ell^2 \\frac{dr}{dx} \\tilde{U}^\\prime ~\\Rightarrow ~  S_{,x} (r_0) = 0.\n\\end{equation}\nThe evaluation of (\\ref{lead_eik}) at the potential peak $r=r_0$ singles out the black hole's fundamental QNM frequency (at leading\neikonal order). We obtain\n\\begin{equation}\n\\omega^2 =  \\ell^2 \\tilde{U}_0 = \\ell^2 \\frac{f_0\\, \\alpha_0}{r^2_0}.\n\\label{omRgen}\n\\end{equation}\nNote that we could have arrived at the same result via the $\\ell \\gg 1$ limit of the standard WKB formula~\\cite{SchutzWill}\nwhich is still applicable for our wave equation with the assumed single-peak potential. In the GR limit of a Schwarzschild spacetime\nwe have $f = 1 -2M\/r, ~ \\alpha =1$, and our result reduces to the correct expression (see e.g.~\\cite{Berti:2009kk}).\n\nWriting the complex QNM frequency as\n\\begin{equation}\n\\omega = \\omega_R + i \\omega_I,\n\\end{equation}\nwe would normally expect $ \\omega_R \\gg |\\omega_I|$ in the eikonal limit as in, e.g. Schwarzschild black holes. In that case\n\\begin{equation}\n\\omega_R  \\approx  \\ell \\, \\tilde{U}_0^{1\/2}.\n\\label{omReik1}\n\\end{equation}\nIn principle, though, the leading-order result (\\ref{omRgen}) should apply equally well for a QNM with the opposite arrangement\n$|\\omega_I| \\gg \\omega_R$ of real and imaginary parts.\n\n\n\\subsection{Eikonal approximation: subleading order}\n\nMoving beyond the leading-order eikonal calculation, we expect  $\\omega_I$ to be determined by the ${\\cal O} (\\epsilon^{-1})$ terms\nof Eq.~(\\ref{full_eik}) evaluated at $r=r_0$.  At this order we should also account for the subleading piece of $\\omega_R$.  That is,\n\\begin{equation}\n\\omega_R = \\omega_R^{(0)} + \\omega_R^{(1)} + {\\cal O} (\\ell^{-1}),\n\\quad  \\omega_R^{(0)} = \\ell \\, \\tilde{U}_0^{1\/2},\n\\end{equation}\nwhere both $\\omega_I,~ \\omega_R^{(1)}$ are ${\\cal O} (1)$ quantities.\nThen,\n\\begin{equation}\n  \\frac{2 i}{\\epsilon} S_{,x} A_{,x} + \\left ( \\frac{i}{\\epsilon} S_{,xx} +2 i \\omega_R^{(0)} \\omega_I  + 2\\omega_R^{(0)} \\omega_R^{(1)}\n - \\ell \\tilde{U} \\right ) A = 0.\n \\end{equation}\n Setting $r=r_0$,\n \\begin{equation}\n\\frac{i}{\\epsilon} S_{,xx} (r_0) +2 i \\omega_R^{(0)} \\omega_I + 2\\omega_R^{(0)} \\omega_R^{(1)} - \\ell \\, \\tilde{U}_0 = 0.\n \\label{sublead_eik}\n\\end{equation}\nTo proceed, we must obtain $S_{,xx}$. By Taylor-expanding Eq.~(\\ref{lead_eik}) around $r_0$ and making use of Eqs.~(\\ref{dU0eq}) and (\\ref{omReik1})\nwe find\n\\begin{equation}\n \\frac{(S_{,x})^2}{\\epsilon^2}  = -\\frac{\\ell^2}{2}   \\tilde{U}^{\\prime\\prime}_0 (r-r_0)^2.\n\\end{equation}\nTaking the positive square root and then differentiating, we get\n \\begin{equation}\n \\frac{S_{,xx}}{\\epsilon}  =  \\frac{\\ell}{\\sqrt{2}}   \\frac{dr}{dx} | \\tilde{U}^{\\prime\\prime}_0 |^{1\/2}.\n \\label{SxxEq1}\n \\end{equation}\nThe imaginary part of (\\ref{sublead_eik}) gives\n\\begin{equation}\n\\omega_I =  -\\frac{1}{2} \\left (\\frac{dr}{dx}\\right )_0  \\sqrt{\\frac{| \\tilde{U}^{\\prime\\prime}_0 |}{2\\tilde{U}_0}}.\n\\label{omIeik1}\n\\end{equation}\nAs in the case of $\\omega_R^{(0)}$, this $\\omega_I$ result could have been derived by taking the $\\ell \\gg 1$\nlimit of a WKB formula.\n\nMoving on, we can see that the real part of (\\ref{sublead_eik}) furnishes the $\\omega_R^{(1)}$ correction,\n\\begin{equation}\n\\omega_R^{(1)} = \\frac{1}{2}  \\tilde{U}_0^{1\/2}.\n\\end{equation}\nThis means that we can write an expression for $\\omega_R$ featuring the famous `$\\ell +1\/2$' Langer factor,\n\\begin{equation}\n\\omega_R = \\left (\\ell + \\frac{1}{2} \\right )  \\tilde{U}_0^{1\/2} + {\\cal O} (\\ell^{-1}).\n\\label{omReik2}\n\\end{equation}\nTogether with Eq.~(\\ref{omIeik1}) this result completes our eikonal approximation analysis of the wave equation (\\ref{gen_wave}).\n\n\n\\subsection{An eikonal QNM-photon geodesics connection?}\n\\label{sec:eik_geod}\n\nAs already discussed in some detail, the eikonal analysis of Schwarzschild (or Kerr) QNMs establishes a direct\nconnection between the fundamental QNM associated with the peak of the wave potential and the geodesic photon\nring~\\cite{Ferrari:1984zz,Mashhoon:1985cya,Cardoso:2008bp, Dolan:2010wr, Berti:2009kk}. We can explore the validity\nof this connection within the general setup of the previous section by assuming a Schwarzschild background spacetime\nwith $f=1-2M\/r$.\n\nThe geodesic photon ring at $r_{\\rm ph}=3M$ is the solution of $ 2f= r f^\\prime$ (see Appendix~\\ref{sec:geodesics}\nfor a discussion of photon rings in a general spherical metric).\nOn the other hand, the peak $r_0$ of the eikonal wave potential solves the slightly different equation [see Eq.~(\\ref{dU0eq})]\n\\begin{equation}\nr ( f\\alpha)^\\prime = 2 f \\alpha,\n\\label{dU0eqx}\n\\end{equation}\nand therefore  $r_0 \\neq r_{\\rm ph}$, \\emph{unless} $\\alpha=\\mbox{const.}$ In this case\nwe inevitably face a breakdown of the eikonal QNM-photon ring correspondence [here we note that Ref.~\\cite{Konoplya:2017wot}\nconsiders higher-dimensional wave equations of the type (\\ref{gen_wave}) and arrives at a similar conclusion].\n\nAs it turns out, however, the eikonal $\\omega_R$ does `see' an \\emph{effective} spherical metric with,\n\\begin{equation}\n-g^{\\rm eff}_{tt} = \\frac{1}{ g_{rr}^{\\rm eff}} = f (r) \\alpha (r).\n\\label{geffsingle}\n\\end{equation}\nNull geodesics in this metric obey the radial equation (see Appendix~\\ref{sec:geodesics})\n\\begin{equation}\n(u^r)^2 = 1 - b^2 \\tilde{U} (r)  = V_r (r,b),\n\\end{equation}\nwhere $b$ is the impact parameter. Circular photon orbits are singled out by the two conditions  $V_r =  V^\\prime_r = 0$; these\nlead to the following equation for the `photon ring' radius [see Eq.~(\\ref{rpheqsph2})],\n\\begin{equation}\n r (g_{tt}^{\\rm eff} )^\\prime = 2 g_{tt}^{\\rm eff},\n\\label{rpheqsph2x}\n\\end{equation}\nwhich is identical to (\\ref{dU0eqx}); therefore, the photon ring coincides with the peak $r_0$.\nAt the same time, the photon ring's `angular frequency' $\\Omega_0$ is given by [this is Eq.~(\\ref{Omph}) evaluated at\n$r_{\\rm ph} = r_0$]\n\\begin{equation}\n\\Omega_0 = \\frac{\\sqrt{-g_{tt}^{\\rm eff}(r_0)}}{r_0} = \\frac{(f\\alpha)^{1\/2}_0}{r_0},\n\\label{Omphx}\n\\end{equation}\nand our earlier result (\\ref{omReik1}) becomes $\\omega_R \\approx \\ell \\, \\Omega_0$, i.e. the familiar relation between the\neikonal $\\omega_R$ and the photon ring's angular frequency~\\cite{Ferrari:1984zz,Mashhoon:1985cya,Cardoso:2008bp, Dolan:2010wr, Berti:2009kk}.\n\nEncouraged by this result, the next objective is to examine whether the QNM's correspondence to an effective geodesic photon ring\nextends to the imaginary part $\\omega_I$ via a relation to the photon ring's  Lyapunov exponent.\nThe latter parameter is given by [see Ref.~\\cite{Cardoso:2008bp} and Eq.~(\\ref{Lyap1})]\n\\begin{equation}\n\\gamma^2_0 = -\\frac{1}{2} \\, r^2_0 \\, g_{tt}^{\\rm eff} (r_0)  \\left ( \\frac{g_{tt}^{\\rm eff}}{r^2} \\right )^{\\prime\\prime}_0.\n\\label{Lyap1x}\n\\end{equation}\nWe can then see that (\\ref{omIeik1}) becomes, after using the identification~\\eqref{geffsingle},\n\\begin{equation}\n\\omega_I = - \\frac{1}{2} \\left (\\frac{dr\/dx}{f \\alpha}  \\right )_0 |\\gamma_0 |.\n\\end{equation}\nThis reduces to the standard expression $\\omega_I = - |\\gamma_0 |\/2 $ provided\n\\begin{equation}\n\\frac{dr}{dx} = f \\alpha = -g_{tt}^{\\rm eff}.\n\\end{equation}\nIndeed, this is the same $x$ coordinate required for writing the scalar wave equation $ g_{\\mu\\nu}^{\\rm eff} \\nabla^\\mu \\nabla^\\nu \\Phi =0$\nin the form (\\ref{gen_wave}), see Appendix~\\ref{sec:qnmgeod} for details.\n\nWe can thus conclude that the eikonal QNM of the generalised wave equation (\\ref{gen_wave}), and in terms of the effective metric (\\ref{geffsingle}),\nretains the physical interpretation it enjoys in GR; i.e. it describes unstable null orbits in the photon ring of that metric.\n\nIn a slightly different scenario than the one considered here (see Appendix~\\ref{sec:qnmgeod}), one can show that the eikonal QNM\nof the usual scalar wave equation in a general spherical metric (and assuming that the present\nsection's model applies) is related to the properties of the metric's \\emph{true} photon ring via the same leading-order relations\n$\\omega_R = \\ell \\, \\Omega_{\\rm ph},~\\omega_I =  -|\\gamma_{\\rm ph} |\/2$. In the light of this section's calculation this\nresult should not come as a total surprise given that the eikonal potential of the scalar wave equation (\\ref{radscalar})\nis $-\\ell^2 \\, g_{tt}\/r^2$ in perfect analogy with this section's potential $U = -\\ell^2 \\, g_{tt}^{\\rm eff}\/r^2 $.\n\n\n\n\\section{Eikonal QNM of non-GR black holes}\n\\label{sec:nonGR}\n\n\\subsection{Coupled wave equations in a spherical black hole spacetime}\n\\label{sec:coupled}\n\nThe perturbation theory underpinning the calculation of QNMs of spherically symmetric black holes in modified theories of gravity\ntypically boils down to a coupled system of wave equations for the axial or polar components of the metric-tensor field ($\\psi$) and the\nadditional field degrees of freedom; see for example~\\cite{molina2010, Kobayashi:2012kh, Kobayashi:2014wsa, Blazquez-Salcedo:2016enn,Blazquez-Salcedo:2017txk,\nBertietal2015}. Our own discussion here is meant to be theory independent, but in order to keep this first analysis\nsimple, we assume a system comprising two perturbed fields $\\psi$ and $\\Theta$ (the scalar field, governed by a pair of coupled wave\nequations,\n\\begin{subequations}\n\\begin{align}\n&\\frac{d^2 \\psi}{dx^2} + \\left [\\, \\omega^2 - V_\\psi(r) \\, \\right ] \\psi = \\beta_\\psi (r) \\Theta,\n\\label{waveT}\n \\\\\n& \\frac{d^2 \\Theta}{dx^2} + g(r)  \\frac{d\\Theta}{dx} + \\left [\\, \\omega^2 - V_\\Theta (r) \\, \\right ] \\Theta = \\beta_\\Theta (r) \\psi,\n\\label{waveS}\n\\end{align}\n\\end{subequations}\nwhere $x$ is a common tortoise coordinate. The potential $V_\\psi$ is assumed to be identical to the Schwarzschild's spacetime\nRegge-Wheeler or Zerilli potential~\\cite{ChandraBook} while the scalar potential $V_\\Theta$ is allowed to deviate from GR,\n\\begin{equation}\nV_\\psi = \\{ V_{\\rm RW}, V_{\\rm Z} \\},\n\\quad V_\\Theta = f \\left [ \\frac{\\ell(\\ell+1)}{r^2}\\alpha + \\frac{2M}{r^3} \\zeta  \\right ],\n\\end{equation}\nwith $f=1-2M\/r$. The coupling functions $\\beta_\\psi(r)$,  $\\beta_{\\Theta} (r)$ and the potential functions\n$\\{g(r), \\alpha(r), \\zeta(r)\\}$ are left undetermined, but we expect them to scale with negative powers of $r$ so that\nasymptotic flatness is preserved\\footnote{This requirement implies a massless scalar field.}.\nWe also assume that the $\\ell (\\ell +1)$ factor represents the entire $\\ell$-dependence of the left-hand-side of (\\ref{waveS}).\nThis `reduced' system is modeled after the perturbation equations for the axial tensor-scalar\nperturbations of Schwarzschild black holes in Chern-Simons gravity~\\cite{molina2010}, and it includes them as\na special case.\n\nOur coupled system admits a QNM solution provided the appropriate boundary conditions are satisfied:\n\\begin{equation}\n\\psi (x\\to  \\pm \\infty) \\sim e^{\\pm i\\omega x}, \\quad \\Theta (x\\to  \\pm \\infty) \\sim e^{\\pm i\\omega x}.\n\\end{equation}\nFor this to be possible $\\{ V_\\psi$, $V_\\Theta\\}$ and $\\{\\beta_\\psi, \\beta_{\\Theta}, g\\}$ should vanish at $x = \\pm \\infty$.\nThe first set does indeed comply with this requirement provided $\\alpha$ and $\\zeta$ do not grow faster than $r^2$ and\n$r^3$, respectively. The second set of parameters follows suit if these parameters scale with $f$ and negative powers of $r$.\n\nWith regard to approximation methods for a system like  (\\ref{waveT})-(\\ref{waveS}), and as far as we are aware,\nno WKB formulae appear to exist in the literature. Fortunately, though, the eikonal\/geometric optics approximation is flexible\nenough to be applicable to a coupled system. To this end we assume the eikonal solution ansatz,\n\\begin{equation}\n\\psi (x) = A(x) e^{iS(x)\/\\epsilon}, \\quad \\Theta (x) = B(x) e^{i H(x)\/\\epsilon},\n\\end{equation}\nwhere we have allowed for different phase functions for the two fields.\n\n\nThe best strategy for dealing with the coupled system at hand is to first combine the two equations and subsequently\ntake the eikonal limit of the resulting expression. As in the previous case of the single wave equation, it quickly becomes\nevident that the two expansion parameters should balance, i.e.  $\\epsilon \\ell = {\\cal O}(1)$, and that $\\omega = {\\cal O} (\\ell)$.\nFollowing this recipe, we obtain the following expression after having eliminated the $B$ amplitude between the two wave equations:\n\\begin{align}\n& \\omega^4 -\\omega^2 \\left [\\,   \\ell^2  \\tilde{V}  (1+\\alpha)  + \\frac{1}{\\epsilon^2} \\left \\{ (S_{,x})^2 + (H_{,x})^2 \\right \\} \\, \\right ] -\\beta_{\\psi\\Theta}\n\\nonumber \\\\\n&+  \\ell^2  \\tilde{V}   \\left [  \\alpha \\left \\{ \\ell^2 \\tilde{V}  + \\frac{(S_{,x})^2}{\\epsilon^2} \\right \\}  + \\frac{(H_{,x})^2}{\\epsilon^2}   \\right ]\n+ \\frac{(S_{,x} H_{,x})^2}{\\epsilon^4}\n\\nonumber \\\\\n&   + \\frac{\\beta_\\psi}{A} e^{i(H-S)\/\\epsilon} \\left [ \\left (g + \\frac{2i}{\\epsilon} H_{,x} \\right ) B_{,x} + B_{,xx} \\right ] + 2 \\ell^3 \\tilde{V}^2  \\alpha\n\\nonumber \\\\\n& - \\frac{i}{\\epsilon^3}  \\left [\\, \\left ( g H_{,x}  +  H_{,xx} \\right )  (S_{,x} )^2  + \\left ( \\frac{2 A_{,x}}{A} S_{,x}  + S_{,xx} \\right ) (H_{,x})^2\\, \\right ]\n\\nonumber \\\\\n&- \\frac{i}{\\epsilon} \\ell^2 \\tilde{V} \\Big [\\, \\left ( \\frac{2A_{,x}}{A} S_{,x} +  S_{,xx} \\right ) \\alpha + g H_{,x}  + H_{,xx}   \\, \\Big ]\n\\nonumber \\\\\n&+  \\frac{\\ell \\tilde{V}}{\\epsilon^2} \\left [ \\alpha (S_{,x})^2 + (H_{,x})^2 \\right ]\n- \\omega^2 \\Big [\\, \\ell \\tilde{V} (1+\\alpha)  - \\frac{i}{\\epsilon} \\Big ( g H_{,x}\n\\nonumber \\\\\n& +  H_{,xx}  +  \\frac{2 A_{,x}}{A} S_{,x}  + S_{,xx} \\Big ) \\,\\Big ]  = 0  +  {\\cal O} \\left (\\epsilon^{-2} \\right),\n\\label{waveTSeik}\n\\end{align}\nwhere we have defined $\\beta_{\\psi \\Theta} \\equiv \\beta_\\psi \\beta_\\Theta$ and $\\tilde{V} (r) \\equiv f(r)\/r^2$. This expression includes all\nterms up to ${\\cal O}(\\epsilon^{-3})$ order; all terms beyond that order have been omitted as they will not play any role in the subsequent analysis.\nAt the same time we have retained all $\\beta$-coupling terms as a result of their unspecified eikonal order. From the above equation\nit is clear that $\\beta_{\\psi\\Theta} \\leq {\\cal O} (\\ell^4)$, which is the highest allowed eikonal order consistent with the rest of the terms.\nThe presence of the exponential term would in principle be a cause for concern; assuming a $\\beta_\\psi \\leq {\\cal O} (\\ell^2)$ pushes this\nterm into the group of ${\\cal O} (\\epsilon^{-3})$ subleading order terms where, as we shall see below, it makes no impact to the final results.\nThe same is true for the remaining amplitude-dependent terms.\n\nTaking the eikonal limit,  $\\epsilon \\ll 1$ and $\\ell \\gg 1$, we have at leading order [i.e. the $ {\\cal O} ( \\epsilon^{-4})$ terms of (\\ref{waveTSeik})]\n\\begin{align}\n& \\omega^4 -\\omega^2 \\left [\\, \\ell^2 \\tilde{V}  (1+\\alpha)  + \\frac{1}{\\epsilon^2} \\left \\{ (S_{,x})^2 + (H_{,x})^2 \\right \\} \\, \\right ]\n\\nonumber \\\\\n&+ \\ell^2  \\tilde{V}   \\left [  \\alpha \\left \\{ \\ell^2 \\tilde{V}  + \\frac{(S_{,x})^2}{\\epsilon^2} \\right \\}  + \\frac{(H_{,x})^2}{\\epsilon^2}   \\right ]\n+ \\frac{(S_{,x} H_{,x})^2}{\\epsilon^4} = \\beta_{\\psi\\Theta}.\n\\label{leadEq1}\n\\end{align}\n\n\nThe simplest situation is the one in which  $\\beta_{\\psi \\Theta} \\leq {\\cal O} (\\ell^3)$; in that\ncase the coupling term in (\\ref{leadEq1}) is subdominant with respect to the other terms\nand the system of the two equations effectively decouples. We discuss this case in more detail in the following section.\n\n\n\n\\subsection{The case of decoupled wave equations}\n\nHere we focus on the scenario of  `weak $\\ell$-coupling' where the term $\\beta_{\\psi\\Theta} $ does not appear\nin the leading-order eikonal equation (\\ref{leadEq1}). This obviously includes the trivial case of \\emph{no} coupling\n$\\beta_\\psi = \\beta_\\Theta = 0$. Under these circumstances (\\ref{leadEq1}) becomes the product of two factors,\n\\begin{equation}\n\\frac{(S_{,x})^2}{\\epsilon^2} = \\omega^2 - \\ell^2 \\tilde{V}, \\quad\n\\frac{(H_{,x})^2}{\\epsilon^2} = \\omega^2 -   \\ell^2 \\tilde{V}  \\alpha.\n\\end{equation}\nEach of these equations is a special case of the general wave equation of Sec.~\\ref{sec:wave_eik} and as a\nconsequence we can use the results obtained there.  For the $\\psi$-field we recover the standard Schwarzschild result,\n$\\omega_R \\approx \\ell \\, \\tilde{V}_0^{1\/2} =  \\ell \\, \\Omega_{\\rm ph}$ with $r_0 =r_{\\rm ph}=3M$.\nFor the $\\Theta$-field  we similarly obtain $\\omega_R\\approx \\ell  \\, ( \\tilde{V} \\alpha )_0^{1\/2}$ with a different peak\n$\\tilde{r}_0 \\neq 3M$ associated with the eikonal potential  $\\ell^2 \\tilde{V} \\alpha$ where $H_{,x} =0$.\n\nClearly, due to the presence of the $\\alpha(r)$ function the two eikonal frequencies do not coincide. This implies\nthat the two fields propagate independently, each one with its own QNM frequency.\n\n\n\n\\subsection{Leading-order analysis of the coupled system.}\n\\label{sec:fullcoupled}\n\nThe lesson from the preceding calculation is clear: the existence of a mixed scalar-tensor QNM wave\nwith a single frequency $\\omega$ in the eikonal limit \\emph{requires} the presence of a term\n$\\beta_{\\psi \\Theta} = {\\cal O}(\\ell^4)$ in the leading-order equation~(\\ref{leadEq1}).\n\nThe next step is to assume that both phase functions are simultaneously minimised, $S_{,x}  = H_{,x} = 0$,\nat the \\emph{same} radius $r_{\\rm m}$.  Provided $\\beta_{\\psi\\Theta}$ is $\\omega$-independent, the resulting expression\nis a  biquadratic equation  in $\\omega$:\n\\begin{equation}\n\\omega^4 - \\omega^2 \\ell^2 \\tilde{V}_{\\rm m} ( 1+ \\alpha_{\\rm m})\n+ \\ell^4  \\tilde{V}^2_{\\rm m} \\alpha_{\\rm m}   -  (\\beta_{\\psi \\Theta})_{\\rm m} = 0,\n\\label{coupled5}\n\\end{equation}\nadmitting the pair of roots\n\\begin{align}\n\\omega^2_{\\pm} = \\frac{\\ell^2}{2}  \\left [\\, \\tilde{V}_{\\rm m} (1+\\alpha_{\\rm m})\n\\pm \\sqrt{  \\tilde{V}^2_{\\rm m} (1-\\alpha_{\\rm m})^2 +  4 ( \\tilde{\\beta}_{\\psi \\Theta})_{\\rm m}} \\,  \\right ],\n\\nonumber \\\\\n\\label{coupled6}\n\\end{align}\nwhere $\\tilde{\\beta}_{\\psi \\Theta} = \\beta_{\\psi\\Theta}\/\\ell^4$ is $\\ell$-independent. Apart from its\nself-consistent scaling, $\\omega_{\\pm} = {\\cal O} (\\ell)$, this result also has the correct GR limit\n ($\\alpha \\to 1,~ \\beta_{\\psi \\Theta} \\to 0 $) provided $r_{\\rm m} \\to 3M$.\n\nAs pointed out earlier, the eikonal $\\omega_{\\pm}$ may not necessarily coincide with the QNM with frequency\n$\\omega_R \\gg |\\omega_I |$. In the spirit of our theory-agnostic framework, we should not rule out modes with\nthe opposite arrangement $| \\omega_I| \\gg \\omega_R $ or indeed ones with $| \\omega_I | \\sim \\omega_R$ in the eikonal limit\n[this last scenario requires at least one of the functions $\\alpha, \\beta_\\psi, \\beta_\\Theta$ to be complex-valued and\/or\n$( \\tilde{\\beta}_{\\psi \\Theta})_{\\rm m} < 0 $].\n\n\nThe radius $r_{\\rm m}$ can be calculated by taking the derivative of (\\ref{leadEq1}) and then evaluating it at\n$(r,\\omega)=(r_{\\rm m}, \\omega_\\pm)$,\n\\begin{equation}\n\\frac{\\omega^2_\\pm}{\\ell^2} \\left [  \\tilde{V} ( 1+ \\alpha)  \\right ]^\\prime_{\\rm m} - ( \\alpha  \\tilde{V}^2  )^\\prime_{\\rm m}\n+  (\\tilde{\\beta}_{\\psi \\Theta})^\\prime_{\\rm m} = 0.\n\\label{coupled8}\n\\end{equation}\nThis equation determines $r_{\\rm m}$; in the GR limit  it reduces to $ \\tilde{V}^\\prime =0$ and $ r_{\\rm m} = r_{\\rm ph}$.\n\n\nRemarkably, $r_{\\rm m} $ turns out to be a `peak' (in reality a local minimum or maximum)\nof an \\emph{effective} potential, albeit a frequency-dependent one. This can be easily verified by first defining\n\\begin{equation}\nV_{\\rm eff}(r,\\omega) \\equiv \\ell^2  \\tilde{V} \\left [   \\omega^2 ( 1+ \\alpha)  - \\ell^2   \\tilde{V} \\alpha \\right ] + \\ell^4  \\tilde{\\beta}_{\\psi \\Theta},\n\\label{Veff}\n\\end{equation}\nand then noticing that (\\ref{coupled8}) is equivalent to\n\\begin{equation}\nV_{\\rm eff}^\\prime (r,\\omega)=0, \\quad \\mbox{at}~ (r,\\omega)=(r_{\\rm m},\\omega_\\pm).\n\\label{dVeff}\n\\end{equation}\nWith the effective potential $V_{\\rm eff} (r, \\omega)$ defined in this way, Eq. (\\ref{coupled5}) becomes\n\\begin{equation}\n\\omega^4_\\pm = V_{\\rm eff} (r_{\\rm m}, \\omega_\\pm).\n\\label{omVeff}\n\\end{equation}\nInterestingly, one can define an alternative effective potential which is equivalent to (\\ref{Veff}) in the sense\nthat both potentials satisfy the condition (\\ref{dVeff}) and therefore have the same `peak' $r_{\\rm m}$.\nThis second potential can be produced from (\\ref{Veff}) after changing $\\omega \\to \\omega_\\pm(r)$, where\n$\\omega_\\pm (r)$ is the function of Eq.~(\\ref{coupled6}) with $r_{\\rm m} \\to r$.\nFor the same reason (\\ref{omVeff}) becomes $\\omega^4_\\pm (r) = V_{\\rm eff} (r,\\omega_\\pm (r))$. Then,\n\\begin{equation}\n\\frac{d}{dr}  V_{\\rm eff} (r,\\omega_\\pm (r)) = V_{\\rm eff}^\\prime + V_{\\rm eff,\\omega_\\pm}\\, \\omega^\\prime_\\pm,\n\\end{equation}\nand at the same time,\n\\begin{equation}\n\\\\\n4 \\omega^3_\\pm \\omega^\\prime_\\pm = \\frac{d}{dr}  V_{\\rm eff} (r,\\omega_\\pm (r)).\n\\end{equation}\nThe combination of these two equations implies $V_{\\rm eff}^\\prime =0 \\Leftrightarrow \\omega^\\prime_\\pm =0$,\nthus ensuring the equivalence of the two potentials.\n\n\n\\subsection{The coupled system at subleading eikonal order}\n\\label{sec:coupled_sub}\n\nGoing beyond the leading eikonal order, we assume a `canonical' QNM with $\\omega_R \\gg |\\omega_I | $\nand\n\\begin{equation}\n\\omega = \\omega_R^{(0)} + \\omega_R^{(1)} + i \\omega_I +  {\\cal O} (\\ell^{-1}), \\quad \\omega_R^{(0)} = \\omega_\\pm.\n\\end{equation}\nWe also expand the coupling parameter as\n\\begin{equation}\n\\beta_{\\psi\\Theta} =  \\beta_{\\psi\\Theta}^{(0)} +  \\beta_{\\psi\\Theta}^{(1)}+ {\\cal O}(\\ell^2), \\quad\n \\beta_{\\psi\\Theta}^{(0)} = \\ell^4 \\tilde{\\beta}_{\\psi\\Theta}.\n\\end{equation}\nThe subleading order eikonal equation consists of the ${\\cal O} (\\epsilon^{-3})$ terms of the general expression (\\ref{waveTSeik}).\nAfter setting $r=r_{\\rm m}$, all terms with $S_{,x}$ and $H_{,x}$ factors vanish (including terms with the `frictional' parameter $g$\nand exponentials) and we obtain\n\\begin{align}\n&   2 \\omega_\\pm \\omega_R^{(1)} \\left [\\,  2 \\omega^2_\\pm -  \\ell^2 \\tilde{V}_{\\rm m} (1+ \\alpha_{\\rm m}) \\, \\right ]\n-\\ell \\tilde{V}_{\\rm m}\\Big [\\, \\omega_\\pm^2   (1+ \\alpha_{\\rm m} )\n\\nonumber \\\\\n&- 2 \\ell^2 \\tilde{V}_{\\rm m}  \\alpha_{\\rm m}  \\Big ] - (\\beta_{\\psi\\Theta}^{(1)})_{\\rm m}\n+ i \\Big [ \\,  (H_{,xx})_{\\rm m} \\left ( \\omega^2_\\pm -\\ell^2 \\tilde{V}_{\\rm m} \\right )\n \\nonumber \\\\\n&  + (S_{,xx})_{\\rm m} \\left ( \\omega^2_\\pm -  \\ell^2 \\tilde{V}_{\\rm m}  \\alpha_{\\rm m} \\right )\n+ 2 \\omega_\\pm \\omega_I \\Big \\{\\,  2 \\omega^2_\\pm\n\\nonumber \\\\\n& -  \\ell^2 \\tilde{V}_{\\rm m} (1+ \\alpha_{\\rm m}) \\, \\Big \\}  \\, \\Big ]  = 0.\n\\end{align}\nWe decompose this into real and imaginary parts, assuming for simplicity an entirely real $\\beta_{\\psi\\Theta}$,\n\\begin{subequations}\n\\begin{align}\n&   2 \\omega_\\pm \\omega_R^{(1)} \\left [\\,  2 \\omega^2_\\pm -  \\ell^2 \\tilde{V}_{\\rm m} (1+ \\alpha_{\\rm m}) \\, \\right ]\n-\\ell \\tilde{V}_{\\rm m}\\Big [\\, \\omega_\\pm^2   (1+ \\alpha_{\\rm m} )\n\\nonumber \\\\\n&- 2 \\ell^2 \\tilde{V}_{\\rm m}  \\alpha_{\\rm m}  \\Big ] - (\\beta_{\\psi\\Theta}^{(1)})_{\\rm m} = 0,\n\\label{subleadEqR}\n\\\\\n\\nonumber \\\\\n& 2 \\omega_\\pm \\omega_I \\left [\\,  2 \\omega^2_\\pm  -  \\ell^2 \\tilde{V}_{\\rm m} (1+ \\alpha_{\\rm m}) \\, \\right ]\n+ (H_{,xx})_{\\rm m} \\left ( \\omega^2_\\pm -\\ell^2 \\tilde{V}_{\\rm m} \\right )\n\\nonumber \\\\\n&+ (S_{,xx})_{\\rm m} \\left ( \\omega^2_\\pm -  \\ell^2 \\tilde{V}_{\\rm m}  \\alpha_{\\rm m} \\right ) = 0.\n\\label{subleadEqIm}\n\\end{align}\n\\end{subequations}\nThis is a decoupled system of equations for the subleading frequency corrections $\\{\\omega_R^{(1)},\\omega_I \\}$\nwith roots\n\\begin{align}\n& \\omega_I =  \\frac{(H_{,xx})_{\\rm m} \\left ( \\omega^2_\\pm  -\\ell^2 \\tilde{V}_{\\rm m} \\right )\n+ (S_{,xx})_{\\rm m} \\left [ \\omega^2_\\pm - \\ell^2 \\tilde{V}_{\\rm m}  \\alpha_m  \\right ]}\n{2\\omega_\\pm \\left [\\,  \\ell^2 \\tilde{V}_{\\rm m} (1+ \\alpha_{\\rm m} ) -2 \\omega^2_\\pm \\,\\right ]},\n\\label{omIroot1}\n\\\\\n\\nonumber \\\\\n& \\omega_R^{(1)} = \\frac{ \\ell \\tilde{V}_{\\rm m} \\left [\\, \\omega^2_\\pm (1+\\alpha_{\\rm m})\n- 2\\ell^2 \\tilde{V}_{\\rm m}^2 \\alpha_{\\rm m}  \\, \\right ] + (\\beta^{(1)}_{\\psi\\Theta})_{\\rm m} }{2\\omega_\\pm \\left [\\, 2 \\omega^2_\\pm\n-  \\ell^2\\tilde{V}_{\\rm m} (1+ \\alpha_{\\rm m} ) \\,\\right ]}.\n\\label{omR1root1}\n\\end{align}\nIn order to calculate the second derivatives $\\{ S_{,xx}, H_{,xx}\\}$ we rewrite (\\ref{leadEq1}) in the equivalent form\n\\begin{align}\n& \\omega^4 -V_{\\rm eff} (r,\\omega)  - \\frac{(S_{,x})^2}{\\epsilon^2}  \\left ( \\omega^2 - \\ell^2 \\tilde{V} \\alpha \\right )\n \\nonumber \\\\\n& -\\frac{(H_{,x})^2}{\\epsilon^2} \\left (  \\omega^2 - \\ell^2 \\tilde{V} \\right ) + \\frac{(S_{,x} H_{,x})^2}{\\epsilon^4} = 0,\n\\end{align}\nand then expand around $r=r_{\\rm m}$. With the help of\n\\begin{align}\nS_{,x} (r) &\\approx \\left ( \\frac{dx}{dr} \\right )_{\\rm m} (S_{,xx})_{\\rm m} (r-r_{\\rm m}),\n\\\\\nH_{,x} (r) &\\approx  \\left ( \\frac{dx}{dr} \\right )_{\\rm m} (H_{,xx})_{\\rm m} (r-r_{\\rm m}),\n\\\\\nV_{\\rm eff} (r,\\omega_\\pm) & \\approx \\omega^4_\\pm  + \\frac{1}{2} V_{\\rm eff}^{\\prime\\prime}  (r_{\\rm m}, \\omega_\\pm) (r-r_{\\rm m} )^2,\n\\end{align}\nwe obtain at leading order in $r-r_{\\rm m}$,\n\\begin{align}\n& \\frac{(S_{,xx})^2_{\\rm m}}{\\epsilon^2} \\left ( \\omega^2_\\pm -\\ell^2 \\tilde{V}_{\\rm m}  \\alpha_{\\rm m} \\right )\n + \\frac{(H_{,xx})^2_{\\rm m}}{\\epsilon^2} \\left ( \\omega^2_\\pm - \\ell^2 \\tilde{V}_{\\rm m} \\right )\n\\nonumber \\\\\n& =  - \\frac{1}{2}   \\left ( \\frac{dr}{dx} \\right )_{\\rm m}^2 V_{\\rm eff}^{\\prime\\prime}  (r_{\\rm m}, \\omega_\\pm).\n\\label{coupled9}\n\\end{align}\nAs it stands, this expression cannot be manipulated further unless we assume a relation between the\nderivatives of the phase functions. The simplest choice is to set $ (S_{,xx})^2_{\\rm m} = (H_{,xx})^2_{\\rm m}$;\nthis would clearly be the case if the phase functions were equal (up to a constant)\\footnote{The equality between the phase functions\nwould imply a common wave propagation speed for the two fields.}, i.e. $S_{,x} =H_{,x}$. With this assumption (\\ref{coupled9}) becomes\n\\begin{align}\n& \\left [ 2 \\omega^2_\\pm  -\\ell^2 \\tilde{V}_{\\rm m} (1+  \\alpha_{\\rm m} )  \\right ] \\frac{(S_{,xx})^2_{\\rm m} }{\\epsilon^2}\n=  \\pm \\ell^2 \\Big [  \\tilde{V}_{\\rm m}^2 (1-  \\alpha_{\\rm m} )^2\n\\nonumber \\\\\n& +  4 ( \\tilde{\\beta}_{\\psi \\Theta} )_{\\rm m}   \\Big ]^{1\/2} \\frac{(S_{,xx})^2_{\\rm m} }{\\epsilon^2}\n= -\\frac{1}{2}  \\left ( \\frac{dr}{dx} \\right )_{\\rm m}^2 V_{\\rm eff}^{\\prime\\prime} (r_{\\rm m}, \\omega_\\pm ).\n\\label{coupled10}\n\\end{align}\nAssuming a potential peak, $V_{\\rm eff}^{\\prime\\prime} (r_{\\rm m}, \\omega_\\pm ) < 0 $, it is always possible to choose the sign\nof the square root so that $(S_{,xx})_{\\rm m} >0$. We then have\n\\begin{align}\n\\frac{(S_{,xx})_{\\rm m}}{\\epsilon} &=   \\frac{1}{\\sqrt{2} \\ell} \\left (\\frac{dr}{dx} \\right )_{\\rm m}\n | V_{\\rm eff}^{\\prime\\prime} (r_{\\rm m}, \\omega_\\pm ) | ^{1\/2}\n \\nonumber \\\\\n&\\quad \\times \\left [ \\tilde{V}_{\\rm m}^2 (1-\\alpha_{\\rm m})^2 + 4 ( \\tilde{\\beta}_{\\psi\\Theta} )_{\\rm m}   \\right ]^{-1\/4},\n\\end{align}\nand solution (\\ref{omIroot1}) for $\\omega_I$ becomes\n\\begin{align}\n\\omega_I & = - \\frac{(S_{,xx})_{\\rm m}}{2 \\omega_\\pm}\n\\nonumber \\\\\n&= -\\frac{(dr\/dx)_{\\rm m}}{2 \\sqrt{2} \\omega_\\pm \\ell}  \\frac{  | V_{\\rm eff}^{\\prime\\prime} (r_{\\rm m}, \\omega_\\pm ) |^{1\/2}}\n{ \\left [\\,  \\tilde{V}_{\\rm m}^2 (1-\\alpha_{\\rm m})^2 + 4 ( \\tilde{\\beta}_{\\psi \\Theta} )_{\\rm m}  \\, \\right ]^{1\/4} }.\n\\label{omIroot2}\n\\end{align}\n\n\n\n\n\n\\subsection{Summary of eikonal formulae}\n\\label{sec:summary}\n\nHere we recap the eikonal results of this section for the complex QNM frequency $\\omega = \\omega_R + i \\omega_I$\nof the coupled system (\\ref{waveT}) and (\\ref{waveS}).\nWe have\n\\begin{equation}\n\\omega = \\omega_R^{(0)} + \\omega_R^{(1)} + i \\omega_I +  {\\cal O} (\\ell^{-1}),\n\\end{equation}\nwith\n\\begin{align}\n \\omega_R^{(0)}  &= \\omega_{\\pm} = \\frac{\\ell}{\\sqrt{2}}  \\Big [\\, \\tilde{V}_{\\rm m} (1+\\alpha_{\\rm m})\n \\nonumber \\\\\n&\\quad\\quad\\quad\\quad \\pm \\sqrt{  \\tilde{V}^2_{\\rm m} (1-\\alpha_{\\rm m})^2 +  4 ( \\tilde{\\beta}_{\\psi \\Theta})_{\\rm m}} \\,  \\Big ]^{1\/2},\n\\label{omR0_final}\n\\\\\n\\nonumber \\\\\n\\omega_R^{(1)} &= \\frac{ \\ell \\tilde{V}_{\\rm m} \\left [\\, \\omega^2_\\pm (1+\\alpha_{\\rm m})\n- 2\\ell^2 \\tilde{V}_{\\rm m}^2 \\alpha_{\\rm m}  \\, \\right ] + (\\beta^{(1)}_{\\psi\\Theta})_{\\rm m} }{2\\omega_\\pm \\left [\\, 2 \\omega^2_\\pm\n-  \\ell^2\\tilde{V}_{\\rm m} (1+ \\alpha_{\\rm m} ) \\,\\right ]},\n\\label{omR1_final2}\n\\end{align}\nfor the real part and\n\\begin{equation}\n\\omega_I = -\\frac{(dr\/dx)_{\\rm m}}{2 \\sqrt{2} \\omega_\\pm \\ell}  \\frac{  | V_{\\rm eff}^{\\prime\\prime} (r_{\\rm m}, \\omega_\\pm ) |^{1\/2} }\n{ \\left [\\,  \\tilde{V}_{\\rm m}^2 (1-\\alpha_{\\rm m})^2 + 4 ( \\tilde{\\beta}_{\\psi\\Theta} )_{\\rm m}  \\, \\right ]^{1\/4} },\n\\label{omI_final}\n\\end{equation}\nfor the imaginary part. For the coupling functions we have assumed $\\beta_\\psi \\leq {\\cal O} (\\ell^2)$\nand that $\\beta_{\\psi\\Theta} = \\beta_\\psi \\beta_\\Theta $ is real-valued, $\\omega$-independent and with an $\\ell \\gg 1$\nexpansion $ \\beta_{\\psi\\Theta} = \\ell^4   \\tilde{\\beta}_{\\psi\\Theta} + \\beta_{\\psi\\Theta}^{(1)} + {\\cal O}(\\ell^2) $.\nThe potentials appearing in these expressions are given by\n\\begin{align}\n    V_{\\rm eff} (r,\\omega) & =  \\ell^2  \\tilde{V} \\left[   \\omega^2 ( 1+ \\alpha)  - \\ell^2   \\tilde{V} \\alpha \\right] + \\ell^4  \\tilde{\\beta}_{\\psi \\Theta},\n\\label{Veff_final}\n\\\\\n\\tilde{V} (r) &= \\frac{f}{r^2},\n\\end{align}\nand the potential peak radius $r_{\\rm m}$ solves $V_{\\rm eff}^\\prime (r_{\\rm m},\\omega_{\\pm})=0$.\nIt is straightforward to verify that in the GR limit,\n\\begin{equation}\n V_{\\rm eff}^{\\prime\\prime} (r_{\\rm m}, \\omega_\\pm ) \\to 2 \\ell^2 \\left [\\, \\tilde{V}_{\\rm m}^{\\prime\\prime} (\\omega^2_\\pm -\\ell^2 \\tilde{V}_{\\rm m})\n -\\ell^2 (\\tilde{V}^\\prime_{\\rm m})^2 \\, \\right ],\n\\end{equation}\nand $ \\tilde{V}^\\prime_{\\rm m} \\to 0$. Then, the above results reduce to the familiar expressions\n\\begin{align}\n& \\omega_R^{(0)} \\to \\ell \\tilde{V}_{\\rm m}^{1\/2}, \\quad\n\\omega_R^{(1)}  \\to \\frac{1}{2} \\tilde{V}_{\\rm m}^{1\/2},\n\\\\\n&\\omega_I  \\to  -\\frac{1}{2} \\left (\\frac{dr}{dx} \\right )_{\\rm m} \\sqrt{ \\frac{|\\tilde{V}^{\\prime\\prime}_{\\rm m}|}{ 2\\tilde{V}_{\\rm m}} },\n\\end{align}\nwith $r_{\\rm m} \\to 3 M$.\n\n\n\n\\subsection{A `photon ring' for the coupled system}\n\\label{coupled_geod}\n\nThe emergence of $V_{\\rm eff}$ in the eikonal calculation of the preceding sections prompts us to explore the possibility of\na `geodesic' connection to an effective photon ring. The form of $V_{\\rm eff}$  and the correspondence\n$\\ell\/\\omega \\to b$ suggests an effective radial geodesic potential,\n\\begin{equation}\nV_r^{\\rm eff} (r,b) = 1 -  b^2  \\left [\\, \\tilde{V}(1 + \\alpha) - b^2 \\tilde{V}^2 \\alpha -  b^2 \\tilde{\\beta}_{\\psi\\Theta} \\,\\right ].\n\\label{Vreff}\n\\end{equation}\nThe clear dissimilarity to the radial potential (\\ref{pheom}) of null geodesics in a spherical metric suggests that we should\nnot be too optimistic about attaching a geodesic analogy to the eikonal QNM of the coupled system.\n\nA `photon ring' in the potential (\\ref{Vreff}) is defined by\n\\begin{equation}\nV_r^{\\rm eff} = 0, \\quad (V_{r}^{\\rm eff} )^\\prime = 0.\n\\end{equation}\nThe first condition leads backs to $b_\\pm^2 = \\ell^2\/\\omega_{\\rm \\pm}^2 $ and then the second one can be identified with Eq.~(\\ref{coupled8})\nfor $r_{\\rm m}$. In other words, $r_{\\rm m }$ is the photon ring radius of the `geodesic' potential $V^{\\rm eff}_r$.\n\nWe can rewrite the impact parameter result as,\n\\begin{equation}\n\\omega_\\pm  = \\ell \\, b_\\pm^{-1} \\equiv \\ell\\, \\Omega_\\pm,\n\\end{equation}\nin accordance with the definition (\\ref{Omph}) of angular frequency in spherical symmetry.\nThus defined, $\\Omega_\\pm$ represents the effective photon ring's angular frequency.\n\nThis is as far as the geodesic analogy can be pushed.  Although a Lyapunov exponent can be defined for the potential\n(\\ref{Vreff}), we find that it is not related to the $\\omega_I$ given by (\\ref{omI_final}) in the same way as in the\nsingle wave equation case.\nWith our eikonal analysis of the coupled system brought to an end we are now ready to study a specific example of\na modified theory of gravity.\n\n\n\\section{Application: Schwarzschild black holes in Chern-Simons gravity}\n\\label{sec:CS}\n\nChern-Simons gravity (dCS), in its dynamical version, represents an extension of GR achieved by adding a parity-violating term to\nthe standard Einstein-Hilbert action; see \\cite{jackiw2003,alexander2009, yunes2009} for further details. As a result of this modification\nthe gravitational field in this theory acquires a dynamical scalar field degree of freedom in addition to the standard tensorial one.\nAs far as black holes are concerned, the theory admits the Schwarzschild metric as a spherically symmetric vacuum solution with a\nvanishing background scalar field~\\cite{Yunes2008,Cardoso:2009pk}. The polar perturbations of these black holes are described by the\nfamiliar general relativistic Zerilli equation~\\cite{ChandraBook} while the axial sector ($\\psi$) of the perturbations couples to the perturbed\nscalar field ($\\Theta$). This coupling signals the breakdown of isospectrality between the polar and axial QNM sectors; the former modes\nremain the same as in GR while the latter are governed by a system of coupled wave equations of the form (\\ref{waveT}) and (\\ref{waveS})\nwith~\\cite{Cardoso:2009pk, molina2010}\n\\begin{subequations}\n\\begin{align}\n& V_\\psi = V_{\\rm RW} = f \\left [ \\frac{\\ell (\\ell+1)}{r^2} -\\frac{6M}{r^3}\\right ], \\quad g = 0,\n\\\\\n&  \\frac{dr}{dx} = f = 1 - \\frac{2M}{r}, \\quad \\alpha =1 + \\frac{576 \\pi M^2}{\\beta r^6}, \\quad \\zeta =1,\n\\\\\n& \\beta_\\psi = 96  \\pi M \\frac{f }{r^5}, \\quad \\beta_\\Theta =  \\frac{6 M}{ \\beta} \\frac{(\\ell+2)!}{(\\ell-2)!} \\frac{f}{r^5},\n \\end{align}\n\\end{subequations}\nwhere $\\beta$ is the theory's coupling constant (GR is formally recovered in the limit $ M^4 \\beta \\to \\infty$).\nWe can observe that the dependence of these functions on $\\{r,\\ell, f\\}$ is consistent with the constraints discussed\nin Secs.~\\ref{sec:coupled} and \\ref{sec:fullcoupled} in relation to the necessary QNM boundary conditions and\nthe $\\ell$-scaling of $\\beta_{\\psi\\Theta}$.\n\nThe aim of this section is to compare the numerically computed axial tensor-scalar QNMs of dCS black holes\nreported in Ref.~\\cite{molina2010} to the leading-order eikonal formulae (\\ref{omR0_final}) and (\\ref{omI_final}).\nWe limit our discussion to the more physically relevant case\\footnote{A $\\beta < 0$ leads to a negative kinetic energy term\nin the action and as a result the theory is infested with ghostlike instabilities.} $\\beta > 0$ \\cite{molina2010}.\n\nThe eikonal limit of the coupling functions,\n\\begin{equation}\n\\beta_{\\psi\\Theta} = \\frac{576 \\pi M^2}{\\beta} \\frac{f^2}{r^{10}} \\left [ \\ell^4 + 2 \\ell^3 -\\ell^2 + {\\cal O}(\\ell) \\right ],\n\\end{equation}\nallows for easy identification of $\\tilde{\\beta}_{\\psi \\Theta}$ and $\\beta_{\\psi\\Theta}^{(1)}$.\nFor the real part of the eikonal QNM, we find\n\\begin{align}\n\\omega^2_\\pm =  \\frac{\\ell^2 f_{\\rm m}}{r^2_{\\rm m}} \\left[  1 +   \\frac{288 \\pi M^2}{\\beta r_{\\rm m}^6}\n\\left (\\, 1 \\pm  \\sqrt{  1+ \\frac{\\beta r^6_{\\rm m} }{ 144 \\pi M^2} } \\, \\right ) \\right].\n\\nonumber \\\\\n\\label{omCS}\n\\end{align}\nThe expression for $\\omega_I$ is somewhat cumbersome and, as a consequence, is not shown here.\nEquation~(\\ref{coupled8}) for the potential peak radius $r_{\\rm m}$ takes the form\n\\begin{align}\n&  \\Big [ 288 \\pi M^2 \\left ( 4 r_{\\rm m}  -9 M  \\right ) +  \\beta r^6_{\\rm m}   (r_{\\rm m}  - 3 M )  \\Big ]\n\\nonumber \\\\\n& \\times \\Big [ 288 \\pi M^2   + \\beta r^6_{\\rm m}  \\pm 24M \\sqrt{\\pi(144 \\pi M^2  + \\beta r^6_{\\rm m}  )}  \\Big ]\n \\nonumber \\\\\n& - r^{12}_{\\rm m}   (r_{\\rm m} -3M) \\beta^2 = 0.\n\\label{rmCS}\n\\end{align}\nThe fact that the right-hand side of (\\ref{omCS}) is real  implies $\\omega_R \\gg | \\omega_I|$ or\n$| \\omega_I | \\gg \\omega_R$ for the eikonal modes. Considering the strong `anti-GR' coupling limit $ M^4 \\beta \\ll 1$,\nwe find the asymptotic expressions,\n\\begin{equation}\n\\omega_{+} \\approx \\frac{ 2048\\ell }{6561} \\sqrt{\\frac{\\pi}{\\beta}},\n\\quad r_{\\rm m} \\approx \\frac{9}{4} M \\left ( 1 + \\frac{19683}{524288} \\frac{\\beta}{\\pi} \\right ),\n\\end{equation}\nfor the `+ mode' and\n\\begin{equation}\n\\omega_{-} \\approx \\frac{\\ell f_{\\rm m}^{1\/2}}{r_{\\rm m}} \\left [ 1 -12 \\left (\\frac{\\pi M^2}{\\beta r_{\\rm m}^6} \\right )^{1\/2} \\right ],\n\\quad \\frac{\\beta r_{\\rm m}^6}{M^2} \\gg 1,\n\\end{equation}\nfor the `- mode'.  The divergence of $\\omega_+$ as $\\beta \\to 0$ suggests that this mode may not be physically relevant in the strong\ncoupling regime.\n\n\nTable~\\ref{tab:CSqnms} collects numerical QNM data from Ref.~\\cite{molina2010} and our eikonal frequencies\nfor the same values of $\\beta$ and $\\ell$. In all cases Eq.~(\\ref{rmCS}) leads to a unique real solution with $r_{\\rm m} > 2M$.\nStarting from the intermediate coupling regime, $M^4 \\beta \\sim 1$, we find good agreement between the two sets of results\nwith an overall precision ($\\sim 10\\%-20\\%$) which is typical of the eikonal approximation in the low-$\\ell$ regime.\nWe can observe that among the two eikonal modes, $\\omega_{-}$ is the least damped and the one that lies closer to a numerical QNM.\nThe precision of the $\\omega_I$ results can be seen to be far better than that of $\\omega_R$; in the examples\nshown here the agreement can extend to two or three significant digits!\n\n\nMoving towards the GR limit,  $M^4 \\beta \\gg 1$, the two eikonal modes approach each other, while the numerical modes converge to the usual\nQNM frequencies of decoupled gravitational and scalar perturbations in the Schwarzschild spacetime (while, at the same time, $r_{\\rm m} \\to 3M$).\nInterestingly, the eikonal $\\omega_R$ ($\\omega_I$) is seen to converge towards the corresponding component of the gravitational (scalar) QNM frequency.\n\n\\begin{figure*}[htb!]\n\\includegraphics[width=0.49\\textwidth]{PLOTS\/effective_potential_beta_5em1_minus.eps}\n\\includegraphics[width=0.49\\textwidth]{PLOTS\/effective_potential_beta_5em1_plus.eps}\n\\caption{\\emph{The effective potentials for axial tensor-scalar QNMs of Schwarzschild black holes in dCS gravity}.\nWe show the $M^4\\beta=0.5,~\\ell=2$ potentials $V_{\\rm eff} (r, \\omega_\\pm)$ and $V_{\\rm eff}(r, \\omega_\\pm (r))$ defined in Sec.~\\ref{sec:fullcoupled}.\nThe left (right) panel corresponds to the $-$ ($+$) mode solution. The corresponding $r_{\\rm m}$ solutions (cf. Table~\\ref{tab:CSqnms}) mark the common\nextrema of the two potentials. In the right panel, these extrema are both maxima, while in the left panel (see inset), it is a maximum for $V_{\\rm eff} (r, \\omega_{-}(r))$\nand a minimum for $V_{\\rm eff} (r, \\omega_{-})$.\n}\n\\label{fig:VeffCS}\n\\end{figure*}\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{c c c c c c c}\n\\hline\\hline\n$ M^4 \\beta$      & $\\ell$  & $+\/-$  & $ r_{\\rm m} \/M  $  & $ M \\omega_{\\rm eik}  $  & $ M \\omega_{\\rm num} $ & $\\Delta \\omega$ [\\%]  \\\\\n\\hline\n0.005             &  2     & $-$   &  9.96       &  0.133-0.0642$i$    & 0.186-0.0606$i$   & 39.8+5.6$i$ \\\\\n                  &  2     & $+$   &  2.25       &  15.7-0.148$i$     &                    &  \\\\\n                  & 10     & $-$   &  9.96       &  0.665-0.0642$i$    & 0.696-0.0636$i$   & 4.7+0.9$i$\\\\\n                  & 10     & $+$   &  2.25       &  78.3-0.14$i$      &                    & \\\\\n\\hline\n0.04              & 2      & $-$   &  7.24       &  0.178-0.0793$i$    & 0.220-0.0760$i$  & 23.6+4.1$i$  \\\\\n                  & 2      & $+$   &  2.25       &  5.55-0.148$i$     &                     \\\\\n                  & 7      & $-$   &  7.24       &  0.624-0.0793$i$    & 0.662-0.0687$i$ & 6.1+13.4$i$   \\\\\n                  & 7      & $+$   &  2.25       &  19.4-0.148$i$     &                     \\\\\n\\hline\n0.5               & 2      & $-$   &  5.07       &  0.244-0.0936$i$    & 0.276-0.0936$i$  & 13.1+0$i$ \\\\\n                  &        & $+$   &  2.26       &  1.620-0.144$i$     & 1.97-0.144$i$   & 21.6+0$i$ \\\\\n\\hline\n1                 & 2      & $-$   &  4.65       &  0.263-0.0959$i$    & 0.292-0.0971$i$  & 11.0+1.3$i$  \\\\\n                  &        & $+$   &  2.28       &  1.183-0.141$i$     & 1.43-0.142$i$   & 20.9+0.7$i$   \\\\\n\\hline\n100               & 2      & $-$   &  3.23       &  0.359-0.0968$i$   & 0.367-0.092$i$    & 2.2+4.9$i$ \\\\\n                  &        & $+$   &  2.77       &  0.421-0.0976$i$    & 0.501-0.0954$i$  & 19.0+2.3$i$   \\\\\n\\hline\n$10^4$            & 2      & $-$   &  3.02       &  0.382-0.0962$i$    & 0.374-0.0889$i$  & 2.1+7.6$i$  \\\\\n                  &        & $+$   &  2.98       &  0.388-0.0962$i$    & 0.484-0.0968$i$  & 24.7+0.6$i$  \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\emph{Axial tensor-scalar QNMs of a Schwarzschild black hole in dCS gravity}.  We show\neikonal results ($M \\omega_{\\rm eik}$) corresponding to the modes $\\omega_{\\pm}$\ncomputed with the help of Eqs.~(\\ref{omI_final}), (\\ref{omCS}) and (\\ref{rmCS}) against\nnumerical data ($M \\omega_{\\rm num}$)~\\cite{molina2010,molina_private} for a variety of coupling strengths $M^4 \\beta$ and multipoles $\\ell$.\nThe relative error $\\Delta \\omega = | [ (\\omega_{\\rm eik})_{R,I} -(\\omega_{\\rm num})_{R,I}]\/(\\omega_{\\rm eik})_{R,I}| $ is shown as a complex\nnumber in the last column. We also tabulate the radial location $r_{\\rm m}$ of the `peak' of the effective potential $V_{\\rm eff}$, as obtained\nfrom (\\ref{rmCS}).}\n\\label{tab:CSqnms}\n\\end{table}\nIn the opposite limit of strong coupling, $M^4 \\beta \\ll 1$, Ref.~\\cite{molina2010} finds that the black hole's ringdown is dominated\nby nonoscillatory QNMs. At first glance, this seems to be at odds with the eikonal formulae's prediction of oscillatory modes\n(no matter which of $\\omega_R, \\omega_I$ is the dominant component). As it turns out, however, these nonoscillatory modes do not\nrepresent the black hole's fundamental mode. Indeed, nonoscillatory modes also appear in the late part of the intermediate regime\n($M^4 \\beta \\sim 1$) time evolutions, immediately after the ringdown of the fundamental mode, see Fig. 2 in~\\cite{molina2010}.\nAs $\\beta$ is reduced,  the nonoscillatory mode emerges at an earlier stage thus dominating most of the time domain signal.\nA recent unpublished (and preliminary) QNM calculation~\\cite{molina_private} reveals the presence of oscillatory modes\nin the time domain signal of perturbed, strong coupling regime black holes; this are the data displayed in the top two rows of\nTable~\\ref{tab:CSqnms} for $M^4 \\beta = \\{0.005, 0.04\\}$.  The $\\ell=2$ eikonal frequency exhibits a deteriorated precision\n($\\sim 20\\% - 40 \\%$) with respect to the precision in the $M^4 \\beta \\gtrsim 1$ regime. On the other hand, the accuracy of the\neikonal damping rate is much better ($\\sim 5 \\%$). As expected, the accuracy of the eikonal results gets better for the\nhigher $\\ell$ multipoles. It is also likely that the overall precision of the eikonal approximation in the $M^4 \\beta \\ll 1$ regime\nmay get better when more accurate numerical QNM results become available.\n\n\nBesides the frequencies themselves, it is interesting to study the form of the effective potential $V_{\\rm eff}$ as defined in two\nequivalent ways at the end of Sec.~\\ref{sec:fullcoupled}. Figure~\\ref{fig:VeffCS} displays a typical example of this potential for the\n$M^4 \\beta=0.5$, $\\ell=2$ QNMs appearing in Table~\\ref{tab:CSqnms}. The shape of the potential depends on which definition we adopt, namely,\n$V_{\\rm eff} (r,\\omega_\\pm)$ or $V_{\\rm eff} (r,\\omega_\\pm (r))$. In this particular example, three out of the four potentials are found to be\nblack hole-like, with a single hump located at the same $r_{\\rm m}$ for a given mode. In contrast, the remaining fourth potential has a local\nminimum at $r_{\\rm m}$ and two maxima at different radii (see inset in Fig.~\\ref{fig:VeffCS}). In all cases $V_{\\rm eff} \\to 0$ as $r\\to \\{2M, \\infty\\}$\nin agreement with the QNM boundary conditions. Moreover, by taking the $M^4 \\beta \\to \\infty$ limit in (\\ref{Veff_final}),\nthe effective potential converges to $\\omega^2 \\ell^2 \\tilde{V}$.\n\n\n\n\n\\section{Concluding remarks}\n\\label{sec:conclusions}\n\nThe main results of this paper, Eqs.~(\\ref{omR0_final})--(\\ref{omI_final}), represent the leading-order\neikonal QNM complex frequency for black hole perturbations described by the general coupled system of\nwave equations (\\ref{waveT}) and (\\ref{waveS}). Furthermore, this eikonal mode (which is an approximation\nto the black hole's fundamental mode) can be associated with the peak of a single effective potential, Eq.~(\\ref{Veff_final}),\nin much the same way as it happens for the eikonal modes of Schwarzschild or Kerr black holes in GR~\\cite{Berti:2009kk}.\nAs a performance benchmark of our results, we have computed eikonal modes of Schwarzschild black holes in dCS gravity and found\nthem to be in good agreement with numerically computed data~\\cite{molina2010, molina_private}.\n\n\n\nThe strength of the eikonal method lies in the analytic form of its results and the present study is no exception to the rule.\nAlthough the QNM spectrum of spherical black holes in a given modified theory of gravity could, in principle, be obtained by means\nof direct numerical integration, our eikonal formulae have the key merit of describing the fundamental mode of black holes in a\nparametrised `post-GR' form and explicitly displaying the mode's dependence on the coupling parameters that generically\nappear in modified theories of gravity.\nIn addition, our framework is sufficiently general to cover background black hole solutions which are either\ngiven analytically or numerically.\n\nOur exploration of the connection between the eikonal QNMs and photon geodesics has been partially successful.\nWe have been able to identify an effective metric and photon ring that can be mapped on the QNM of a general class\nof single-field wave equations. The more complicated (and physically relevant) coupled system of equations has only allowed us to\nassociate an effective potential and its peak to the eikonal QNM. However, the connection of this potential to the true photon\ngeodesics of a given theory is an open question.\n\nSeveral gravity theories lead to more complex perturbation systems than the one considered here, featuring\nmore than two wave equations\/perturbed fields (including massive scalar fields) and\/or higher-order derivative terms (see for\nexample~\\cite{Blazquez-Salcedo:2016enn, Blazquez-Salcedo:2017txk, Brito:2018hjh, Tattersall_2018a, Tattersall_etal2018}).\nIn principle, the eikonal scheme of this paper should be adaptable to these more general scenarios at the cost of an increased\nalgebraic complexity. An equally important extension of this work -- and one that could lift our capability of testing the Kerr hypothesis\nwith GW observations beyond the level of null tests -- would be towards the study of rotating black holes, perhaps initially\nwithin a slow-rotation approximation. These are all important issues that need to be addressed in future work.\n\n\n\\acknowledgements\n\nWe are grateful to Carlos Molina for providing us with black hole QNM data in dCS gravity.\nWe also wish to thank Caio F. B. Macedo for helpful feedback on our paper.\nK.G. acknowledges networking support by the COST Actions GWverse CA16104 and PHAROS  CA16214.\nH.O.S. acknowledges financial support through NASA Grants No.~NNX16AB98G and No.~80NSSC17M0041.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this article, we describe a new mechanism for obtaining a scalar that is parametrically lighter than the UV cutoff of the \ntheory.  This mechanism involves a discrete $\\mathbb{Z}_N$ symmetry that is non-linearly realized on a scalar as a shift symmetry\nand manifests as an exchange symmetry on the $N$ copies of particles with which it interacts~\\footnote{Due to the presence of additional discrete symmetries, the $\\mathbb{Z}_N$ symmetry may be enhanced to a $\\mathbb{D}_N$ symmetry.}.  This approach allows one to obtain\na hierarchy between the mass of the scalar and the UV cutoff that is exponential in $N$, meaning that $N=3$ already solves most problems.\n\nWe first consider an explicit example of a Yukawa coupling between a scalar and a fermion.\nThe starting point is a periodic scalar $\\phi$ with period $2 \\pi f$.  $\\phi$ \nhas a spurion $\\epsilon$ which breaks the arbitrary shift symmetry $\\phi \\rightarrow \\phi + \\alpha$ down to \n$\\phi \\rightarrow \\phi + 2 \\pi f$.  Thus $\\phi$ only appears in the Lagrangian as\n\\begin{eqnarray}\n\\epsilon \\sin \\left ( \\frac{\\phi}{f} + \\theta \\right ).\n\\end{eqnarray}\nThis structure can result from a perturbative UV completion where $\\phi$ is the pseudo-goldstone\nof a complex scalar $\\Phi \\sim e^{i \\phi\/f}$.\n\nThe theory also has a discrete $\\mathbb{Z}_N$ symmetry under which $\\phi$ transforms as \n\\begin{eqnarray}\n\\phi \\rightarrow \\phi + \\frac{2 \\pi f}{N}.\n\\end{eqnarray}\nIt is crucial to the mechanism that the spurion of shift symmetry breaking is not invariant under this symmetry.\nNext, we introduce Weyl fermions $\\psi_k$ and $\\psi_k^c$ for $\\phi$ to couple to.  Under the $\\mathbb{Z}_N$ symmetry, the fermions transform as\n\\begin{eqnarray}\n\\psi_k , \\psi_k^c \\rightarrow \\psi_{k+1} ,  \\psi_{k+1}^c,\n\\end{eqnarray}  \nwhere we have identified $\\psi_0^{(c)}$ and $\\psi_N^{(c)}$.\n$\\phi$ and $\\psi$ are coupled in a $\\mathbb{Z}_N$-symmetric manner via the interaction\n\\begin{eqnarray}\n\\label{Eq: interact}\n\\mathcal{L} = \\sum_{k=1}^N \\left ( m_\\psi + \\epsilon \\sin \\left ( \\frac{\\phi}{f} + \\frac{2 \\pi k }{N} \\right ) \\right )  \\psi_k \\psi^c_k.\n\\end{eqnarray}\n\nThis Lagrangian describes a scalar $\\phi$ coupled to fermions with a Yukawa \ncoupling $\\sim \\epsilon\/f$.  Thus the natural \nexpectation for the mass of $\\phi$ is \n\\begin{eqnarray}\nm_\\phi \\gtrsim \\frac{\\epsilon \\Lambda}{f}.\n\\end{eqnarray}\nwhere the UV cutoff is $\\Lambda$.\nThis intuition turns out to be incorrect as the scalar described in Eq.~\\ref{Eq: interact} has its potential cancelled to\na very high degree of accuracy.  $N$ insertions of $\\epsilon$ will be needed until a non-trivial potential for $\\phi$ can be\nobtained.\n\nTo see the cancelations in action, we take $N > 2$ and consider the potential induced for $\\phi$ by closing the fermion loop with two insertions of $\\epsilon$:\n\\begin{eqnarray}\nV_{1-loop} \\supset \\sum_{k=1}^N \\epsilon^2 \\sin^2 \\left ( \\frac{\\phi}{f} + \\frac{2 \\pi k }{N} \\right ) \\Lambda^2 = \\frac{N}{2} \\epsilon^2 \\Lambda^2.\n\\end{eqnarray}\nThe quadratic divergence to the $\\phi$ mass coming from a fermion $\\psi_k$ has been cancelled by its $N-1$ partners.\nThere is a stronger version of the above cancelation, which is that for all integer $m$ with  $N > m \\geq 0$,\n\\begin{eqnarray}\n\\sum_{k=1}^N \\sin^m \\left ( \\frac{\\phi}{f} + \\frac{2 \\pi k }{N} \\right ) &=& \\begin{cases}\n  0 \\qquad \\qquad \\quad \\, \\, \\, m = \\text{odd} \\\\\n  \\frac{N}{2^m} \\frac{m!}{(m\/2)!^2} \\qquad m = \\text{even}\n\\end{cases}\n\\end{eqnarray}\nThus the leading-order UV contribution to the potential of $\\phi$ comes from the higher-dimensional operator with $N$ insertions of the spurion $\\epsilon$.\n\\begin{eqnarray}\nV(\\phi) &\\propto& \\frac{\\epsilon^N}{\\Lambda^{N-4}} \\sum_{k=1}^N \\sin^N \\left ( \\frac{\\phi}{f} + \\frac{2 \\pi k }{N} \\right )  \\\\\n&=& \n  (-1)^{ \\lfloor N\/2 \\rfloor} \\frac{N \\epsilon^N}{2^{N-1} \\Lambda^{N-4}} \\cos \\left ( \\frac{N \\phi}{f} -\\frac{\\pi}{2} (N \\% 2) \\right ) .\\nonumber\n\\end{eqnarray}\nThe UV contribution to the mass of $\\phi$ scales as\n\\begin{eqnarray}\nm_\\phi^2 \\sim \\frac{N^3 \\epsilon^N}{f^2 \\Lambda^{N-4}}\n\\end{eqnarray}\nand is exponentially suppressed in $N$.\n\nTo get an intuitive understanding of why the potential is so suppressed, we analyze the potential for $\\phi$ from a symmetry perspective.  Due to the $\\mathbb{Z}_N$ symmetry, the potential for $\\phi$ must be\n$2 \\pi f\/N$ periodic so that we can express it as\n\\begin{eqnarray}\nV(\\phi) = \\sum_k c_k \\sin \\left (\\frac{N k \\phi}{f} + \\theta_k \\right ) .\n\\end{eqnarray}\nWe will see below that under a broad set of assumptions,\n\\begin{eqnarray}\n\\label{Eq: ci}\nc_k \\sim \\epsilon^{N k}.\n\\end{eqnarray}\nso that by taking $\\epsilon$ small and $N$ large, we can parametrically separate the mass of the scalar from the UV cutoff.\n\nThe quick and dirty way to derive that $c_k \\sim \\epsilon^{N k}$ is to note that $e^{i N \\phi\/f} = ( e^{i \\phi\/f} )^N$ so that $N$ insertions of the symmetry-breaking spurion $\\epsilon e^{i \\phi\/f}$ are needed to generate a potential for $\\phi$.  Alternatively, the relation can be\nobtained by considering the potential in frequency space.\nThe interaction frequency of $\\phi$ is $\\omega = 1\/f$ ($ V \\sim \\cos(\\omega \\phi) \\psi \\psi^c$). \nThe frequency associated with the mass term and every other term in the potential for $\\phi$ is $N \\omega = N\/f$ due to the $\\mathbb{Z}_N$ symmetry.  \nIn order\nto construct the high-frequency mass term, $N$ contributions of the lower frequency $\\omega$ are needed.\nEach of these comes with its own factor of $\\epsilon$ giving the scaling shown in Eq.~\\ref{Eq: ci}.\n\nA critical assumption that was made implicitly in the previous discussion is that there are no phase transitions or massless particles as $\\phi$ varies in field space.  \nPhase transitions introduce discontinuities.  By cutting up a $2 \\pi f$ symmetric potential, a $2 \\pi f\/N$ symmetric potential can\nbe easily generated.  This point will be exploited later in the paper to obtain a light Higgs boson.\n\nSince $\\epsilon$ is a dimension-one number, we need to specify what the dimensionless expansion parameter is.\nWhen dealing with UV contributions to the potential, it is clear that the expansion parameter is\n$\\epsilon\/\\Lambda$.  In this case,\n\\begin{eqnarray}\nc_k \\sim \\Lambda^4 \\left ( \\frac{\\epsilon}{\\Lambda} \\right )^{N k} .\n\\end{eqnarray}\n\nIn addition to UV contributions to the potential, there will also be IR contributions.\nAs an example of how the IR contributions behave, consider the coupling to the fermions discussed before in Eq.~\\ref{Eq: interact}.  The effective potential \n will depend on the fermion mass $m_\\psi$ in some manner, e.g. $m_\\psi^4 \\log m_\\psi$.  From this,\n one sees that the expansion parameter for IR contributions to the potential is $\\epsilon\/m_\\psi$ and that\n\\begin{eqnarray}\nc_k \\sim \\Lambda_{IR}^4 \\left ( \\frac{\\epsilon}{m_\\psi} \\right )^{N k} .\n\\end{eqnarray}\nDepending on other IR parameters in the Lagrangian, the IR potential for $\\phi$ can be relatively unsuppressed.  Thus scalars of this type are sensitive to the IR but insensitive to the UV.\n\n\\section{Analytic bounds}\n\nIn the example given in the Introduction, we saw that a scalar could couple strongly to \nmatter yet remain light.  \nIn this section, we extend the previous result to more general situations. We describe\nunder what circumstances the mass of a scalar $\\phi$ is exponentially suppressed in $N$\nand under what circumstances it is only power-law suppressed in $N$.\n\nDue to the $\\mathbb{Z}_N$ symmetry, \nthe potential for $\\phi$ is of the form\n\\begin{eqnarray}\n\\label{Eq: assumption}\nV(\\phi) \\propto \\sum_{k=0}^{N-1} F(\\frac{\\phi}{f} + \\frac{2 \\pi k}{N} ).\n\\end{eqnarray}\nUnlike before, we do not assume the existence of a small coupling $\\epsilon$.  The results \npresented in this section are valid for any choice of the function $F$ and for any value of $N$ but are most useful in the case where $F$ does not depend explicitly on $N$.\n\nThe large-$N$ limit of Eq.~\\ref{Eq: assumption} is easy to understand as the sum is simply a Riemann sum that converges to an integral :  \n\\begin{eqnarray}\nV(\\phi) &\\propto& \\sum_{k=0}^{N-1} F(\\frac{\\phi}{f} + \\frac{2 \\pi k}{N} ) \\nonumber \\\\\n\\label{Eq: Riemann}\n&=& \\frac{N}{2 \\pi} \\int_0^{2\\pi} F(\\theta) d\\theta + \\mathcal{O}(N^0) .\n\\end{eqnarray}\nThe leading-order piece is completely $\\phi$ independent so that the mass of $\\phi$\nis subleading in the large-$N$ limit.  Let us denote the subleading piece that generates the\nmass of $\\phi$ as \n\\begin{eqnarray}\nE_N(F) = \\int_0^{2\\pi} F(\\theta) d\\theta - \\frac{2 \\pi}{N} \\sum_{k=0}^{N-1} F(\\frac{\\phi}{f} + \\frac{2 \\pi k}{N} ).\n\\end{eqnarray}\nThe scaling of the mass of $\\phi$ with $N$ is an issue of estimating the convergence rate of \nthe Riemann sum of $F''$.  Riemann sums of periodic functions are known to converge extremely \nquickly, and we present two useful theorems (the proofs can be found in Section 9.4 of Ref.~\\cite{book}).\n\nThe first theorem is a special case of the Euler-Maclaurin theorem.  If the function $F$ is $2 \\pi$\nperiodic and $2m + 1$ times differentiable, then\n\\begin{eqnarray}\n| E_N(F) | \\leq \\frac{2}{N^{2m+1}} \\left ( \\sum_{j=1}^\\infty \\frac{1}{j^{2m+1}} \\right ) \\int_{0}^{2 \\pi} | F^{(2 m + 1)} (\\theta) | d\\theta . \\nonumber\n\\end{eqnarray}\nPotentials where there are massless particles as $\\phi$ varies will have discontinuities and are not infinitely \ndifferentiable.  In these cases, the mass of $\\phi$ is only power-law suppressed by $N$.\n\n\nThe second theorem can be shown via the residue theorem.  Let $F$ be a $2 \\pi$-periodic function \nthat is also analytic.  Then there exists an open strip, which includes the real axis and the \ncomplex axis from $-i a$ to $i a$ with $a > 0$, upon which $F$ can be extended into a holomorphic, $2 \\pi$-periodic, bounded function with bound $M$.  \nIn this case,\n\\begin{eqnarray}\n| E_N(F) | \\leq \\frac{4 \\pi M}{e^{N a} - 1} .\n\\end{eqnarray}\nPotentials that are infinitely differentiable give\nan exponentially suppressed mass for $\\phi$ even if there is no small parameter in the problem.\n\nThese two theorems demonstrate how much the mass of $\\phi$ can be suppressed for any given\npotential.  In the case where there are massless particles as $\\phi$ varies, the mass of $\\phi$ is suppressed by\nhow differentiable the potential is.  In the case with no \ndiscontinuities, the mass is exponentially suppressed even if there is no small number\nin the problem.\n\n\n\\section{Examples}\n\nIn the example given in the Introduction, we showed how a light scalar (e.g. from a fifth force or dark matter) Yukawa coupled to the SM that naively looks tuned can actually arise naturally.  In this \nsection, we provide two more examples of how this new solution to the Hierarchy Problem can be applied to theories of interest.  The first example is that of a light axion.  \nThis example serves to highlight how different this solution is from other solutions such as supersymmetry, \nwhich cannot make the axion lighter than its QCD contributions.\nThe second example is a solution to the Little Hierarchy Problem.\n\n\\subsection{A light axion}\n\nUnlike the example in the Introduction, the axion~\\cite{Peccei:1977hh,Peccei:1977ur,Weinberg:1977ma,Wilczek:1977pj} does not have a small parameter $\\epsilon$ characterizing its couplings.  \nHowever, the axion potential is analytic so that by the\ntheorems presented before, its mass must be exponentially suppressed in the large-$N$ limit.\nAs before, we have a $\\mathbb{Z}_N$ symmetry with $N$ copies of the SM that are interchanged under the \nsymmetry, and an axion that non-linearly realizes the discrete symmetry.\nThe axion couples to the $N$ different sectors via the coupling\n\\begin{eqnarray}\n\\mathcal{L} = \\sum_k \\left ( \\frac{a}{f} + \\frac{2 \\pi k}{N} + \\theta \\right ) G_k \\tilde G_k .\n\\end{eqnarray}\n\n\nDue to confinement of the $N$ different QCD sectors, there will be a potential for the axion whose leading order contribution is\n\\begin{eqnarray}\nV(a) = &-& m_\\pi^2 f_\\pi^2 \\sum_k \\sqrt{1 - 4 \\frac{m_u m_d}{(m_u + m_d)^2} \\sin^2 \\left ( \\frac{a}{2f} + \\frac{\\pi k}{N} \\right ) } \\nonumber \\\\\n\\label{Eq: axion}\n&+& \\mathcal{O}(m_\\pi^4).\n\\end{eqnarray}\nAs the theta angles are all identical, we have shifted them away.\nWith a bit of algebra, one can show that $\\theta = 2 \\pi k\/N$ for integer $k$ is a minimum (maximum) for $N$ odd (even).\n\nTo use the results of the convergence theorems presented before, we define\n\\begin{eqnarray}\nF(z) = \\sqrt{1 - 4 \\frac{m_u m_d}{(m_u + m_d)^2} \\sin^2 \\left ( \\frac{z}{2} \\right ) } ,\n\\end{eqnarray}\nwhich is holomorphic until the square root of a negative number is taken.  We are thus considering\nthe open strip $(-i a , i a)$ where \n\\begin{eqnarray}\na = \\log ( c + \\sqrt{c^2 - 1} ), \\qquad c = \\frac{(m_u + m_d)^2}{2 m_u m_d} - 1 .\n\\end{eqnarray}\nInserting the measured values of the quark masses, we find that the mass of the axion is bounded to be $\\approx 1\/2^{N\/2}$ smaller than its natural value where we have neglected the subleading terms proportional to powers of $N$.\n\nEq.~\\ref{Eq: axion} contains subleading terms suppressed by more powers of the quark masses.  \nThese terms are also analytic and thus also give exponentially suppressed contributions to the axion mass.  \nThese contributions are also suppressed by at least $2^{N\/2}$.  To understand this scaling behavior,\nwe note that if $m_u = m_d$, there is a first-order phase transition at $\\theta = \\pi$~\\cite{Witten:1980sp}.  If the quark masses are equal, the axion\npotential is non-analytic and thus not exponentially suppressed.  Any non-zero mass difference\nresults in an analytic potential, and the parameter governing the exponential suppression is necessarily $m_u\/m_d \\sim 1\/2$.\nAs the exponential suppression of the mass is due to how far away the potential is from discontinuities, a.k.a. phase transitions, the axion mass\nis necessarily suppressed by at least $1\/2^{N\/2}$.\n\nAside from analytically bounding the mass of the axion in the large-$N$ limit, we also numerically fit the exponential dependence of the axion mass on $N$ and find that\n\\begin{eqnarray}\n\\frac{m_a(N)}{m_a(N=1)} \\sim \\frac{4}{2^{N\/2}}\n\\end{eqnarray}\nThere is good agreement between the analytically-derived limit on the mass and actual mass.\n\nThe price of obtaining an exponentially lighter axion is the linear problem associated with the fact that only one of the $N$ copies of the SM has $\\theta = 0$.  The rest have $\\theta = 2 \\pi k\/N$.  Thus one has traded an exponential fine tuning in the mass for a linear tuning of why we are in the sector with $\\theta = 0$.  This secondary problem may be solvable via other mechanisms.\n\n\n\\subsection{A naturally light Higgs boson} \\label{Sec: Higgs}\n\nSolving the Hierarchy Problem is fundamentally about finding a reason that a Higgs mass of zero is special.  \nThe scalars discussed in this article are sensitive to phase transitions.  As a phase transition occurs when the Higgs mass crosses zero,\nthese scalars should be able to favor a small Higgs mass.\n\nFollowing this train of thought, in this subsection we develop a theory for the modulus of the Higgs mass.  \nA modulus coupling to $N=3$ or $4$ copies of the Higgs bosons can result in one of them\nbeing lighter than what naturalness would otherwise imply by a factor of 10, thereby solving the Little Hierarchy Problem.\nIf the other $N-1$ copies of the Higgs boson had positive masses, then there would only need to be N copies of the Higgs boson with the rest of the SM transforming trivially under the $\\mathbb{Z}_N$.\nIn the model presented below, the other Higgs bosons obtain a negative mass squared so that the entire SM needs to be copied.\n\n\\subsubsection{$N=3$\/$N=4$ case}\n\nConsider a $\\mathbb{Z}_N$ symmetry under which there are $N$ copies of the SM\nand a scalar $\\phi$, which is the modulus of the Higgs mass~\\footnote{In principle, $SU(3)_c \\times U(1)_Y$ could transform trivially under this exchange symmetry,\nbut the resulting light colored and charged particles have been excluded by experiment.}.  For the rest of this section,\n$N$ will be 3 or 4.  \nWe couple $\\phi$ to the Higgs with a shift symmetry-breaking parameter $\\epsilon^2$:\n\\begin{eqnarray}\n\\label{Eq: Higgs}\nV &=& \\sum_k m_{H,k}^2(\\phi) H_k H^\\dagger_k + \\lambda (H_k H^\\dagger_k)^2 ,\\\\\n\\nonumber m_{H,k}^2(\\phi) &=& -m_H^2 + \\epsilon^2 \\cos \\left ( \\frac{\\phi}{f} + \\frac{2 \\pi k}{N} \\right ) .\n\\end{eqnarray}\nFor simplicity, we will take all cross-quartic couplings between the Higgses to be zero, but our results will not depend\non this assumption.  We will take $m_H^2 > 0$ and \n$\\Lambda^2 > \\epsilon^2 \\gtrsim m_H^2 = 3 y_t^2 \\Lambda^2\/8 \\pi^2$.  As discussed before, the UV contribution to the $\\phi$ potential\nwill be suppressed by $\\epsilon^{2N}\/\\Lambda^{2N-4}$.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{3negative.pdf}\n  \\caption{ The $N=3$ tree-level potential when $\\epsilon^2 = 1.3 \\, m_H^2$.  The solid line is the potential with no UV contribution, while the dotted (dashed) line includes the UV generated potential with $\\theta = 0$ ($\\theta = \\pi$) in Eq.~\\ref{Eq: potential}.  If the UV-generated potential has a minimum where the Higgs masses are all negative, then all Higgs masses are at their large natural value.  If the UV-generated potential has a minimum where one of the Higgs masses is positive, then there is a light Higgs.\n    \\label{fig}}\n\\end{figure}\n\nThe Higgs naturally has a phase transition when its mass changes sign.  Thus the potential for $\\phi$ will be sensitive to changes in\nthe sign of the Higgs mass.  To see this effect explicitly, we\nintegrate out the Higgs classically.  Only if the total Higgs mass is negative will the Higgs induce a non-zero tree-level potential for \n$\\phi$.  The tree-level potential for $\\phi$ is\n\\begin{eqnarray}\nV = &-& \\frac{\\epsilon^{2N}}{\\Lambda^{2N-4}} \\cos \\left( N \\frac{\\phi}{f} + \\theta \\right )  \\nonumber \\\\\n\\label{Eq: potential}\n&-& \\sum_k \\frac{m_{H,k}^4(\\phi)}{4 \\lambda} \\quad \\Theta \\left ( -m_{H,k}^2(\\phi) \\right ) .\n\\end{eqnarray}\nBy previous arguments, if all three Higgs masses are negative~\\footnote{Requiring that all three Higgs masses are negative for certain values of $\\phi$ and that one of them becomes positive for other values of $\\phi$ corresponds to the choice that\n$2 m_H^2 > \\epsilon^2 > m_H^2$.},\n then the contribution to the potential from the Higgs is $\\phi$ independent.  However, as soon as some of the Higgs masses become positive, a phase transition occurs and there is a potential for $\\phi$.  \n\n\n\nAn example $N=3$ potential is shown in Fig.~\\ref{fig} for some specific choices of parameters.\nThe preference for small Higgs masses can be seen by considering the Higgs's contribution to the\npotential of $\\phi$.  \nOver some of parameter space, all three Higgs vevs are negative and $\\phi$ does not acquire a potential \nfrom the Higgses.  However, whenever one of the Higgs masses becomes positive, there is no longer \na cancelation and the potential quickly increases.  Thus this contribution to the potential has a \nminimum whenever all of the Higgs masses are negative.\nThis preference for negative Higgs masses is balanced against the $\\epsilon$-suppressed UV contribution to the potential.  Choosing the phase of the UV contribution to favor positive Higgs masses gives a theory where at the minimum of the potential, one of the sectors has a Higgs with a small positive mass.\n\nWhen $N \\gtrsim 3$, the UV contribution becomes subdominant to the 1-loop potential for $\\phi$.\nThe 1-loop Coleman-Weinberg potential gives a potential for $\\phi$ that is of the form\n\\begin{eqnarray}\nV_\\text{1-loop} = \\frac{\\beta}{16 \\pi^2} \\sum_k \\left ( H_k H_k^{\\dagger} \\right )^2 \\log H_k H_k^{\\dagger} \/m_H^2.\n\\end{eqnarray}\nThe sign and value of $\\beta$ is determined by the beta functions at the natural scale of the Higgs masses.\nThe $N=4$ potential including the 1-loop potential ($\\beta = 0.2$) is shown in Fig.~\\ref{fig2}.  \n\n\nThe previous two examples of $N=3$ and $N=4$ gave a small positive Higgs mass, as opposed to the observed small negative Higgs mass.  There are two simple ways of obtaining a small negative Higgs mass.  The first is to introduce a small amplitude but high-frequency sine wave\npotential for $\\phi$,\n\\begin{eqnarray}\nV = \\alpha \\cos \\left ( \\frac{M \\phi}{f} \\right ) .\n\\end{eqnarray}\nThis will introduce additional minima, but can result in a small negative Higgs\nmass  in the {\\it absolute} minimum.  For example, including the 1-loop potential for $N=4$, the\nvalues $\\alpha = -0.01 \\, m_H^4$, $M=36$ and $\\beta = 0.1$ give a negative Higgs mass at the absolute minimum.\nFor this parameter set, the UV cutoff is 10 TeV.\n\nAnother way of obtaining a small negative Higgs mass is to introduce an additional $\\mathbb{Z}_2$ symmetry under which the $N$ copies of the SM are taken to another $N$ copies.  In the limit\nof an exact $\\mathbb{Z}_2$ symmetry, there are two light Higgses with identical small positive masses.  \nThe $\\mathbb{Z}_2$ symmetry is softly broken by giving the two sectors slightly different Higgs masses, $m_H^2$ and $m_H^2 + \\delta^2$.  A small negative Higgs mass can result if $\\delta \\gtrsim 125$ GeV.\nFor example, in the $N=4$ case above, a small $\\delta^2 \\sim 0.02 m_H^2$ and $\\beta = 0.05$ results in two light Higgses each a factor of 10 lighter than the bare mass.  One of the two has a positive mass squared while the other has a negative mass squared.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{4negative.pdf}\n  \\caption{ The $N=4$ potential including 1-loop effects for $\\epsilon^2 = 1.3 \\, m_H^2$ and $\\beta = 0.2$.  At the minimum, one of the four Higgs masses is $\\sim 0.1 \\, m_H$ and is positive.  As the Higgs is a factor of $\\sim 10$ lighter than its natural value, it solves the Little Hierarchy Problem modulo getting the sign of the Higgs mass wrong.\n    \\label{fig2}}\n\\end{figure}\n\nTaking the large-$N$ limit of the above solution to the Little Hierarchy Problem does not produce any parametric\nenhancements, though there are small numerical benefits going beyond $N=3$ or $4$.  The main issue is that requiring that there is only one Higgs mass that is positive\nat a time means that $m_H^2\/\\cos \\left ( \\pi\/N \\right ) > \\epsilon^2 > m_H^2$.  In the large-$N$ limit, this amounts to a $1\/N^2$ tuning.\nEven accepting this tuning, the other major issue is that the 1-loop Coleman-Weinberg potential is not parametrically suppressed.  By explicit calculation, the Coleman-Weinberg potential is only $1\/N^2$ suppressed, which does not result in a small enough Higgs mass.\n\nMany intriguing possibilities arise if the Coleman-Weinberg potential were highly suppressed due to very small beta functions.  Taking $N \\sim 10$ would suppress the UV contributions to a point where this would be a solution to the full Hierarchy problem.  Additionally, the minimum with a negative Higgs mass near 100 GeV could be generated by features in the beta function, e.g. a small increase and decrease in the quartic around 100 GeV.  This offers a unique twist on how conformal dynamics might result in a light Higgs mass.\n\n\\subsubsection{Phenomenology}\n\nThe phenomenology of this solution to the Little Hierarchy Problem will be discussed in detail in future \nwork.  Here we summarize the salient features.\n\nPreferential reheating of the sector with a light negative Higgs mass is built into the model as $\\phi$ couples to the SM \nHiggses through scalar mixing and is exactly the scalar reheating model described in \nNNaturalness~\\cite{Arkani-Hamed:2016rle}.  Since $\\phi$ is naturally lighter than the lightest Higgs, its decays\npreferentially reheat the sector with the lightest Higgs, evading all current cosmological \nconstraints.  Thus if $\\phi$ mediates reheating to the SM, then all cosmological problems are \nnaturally avoided.\n\nAnother feature of this solution is that the only particle required to interact with the Higgs is $\\phi$.  Much like the axion solution to the \nstrong CP problem, this mechanism works for any value of $f$ and a large value of $f$ results in $\\phi$ being very difficult to detect.  \nIn this limit, $\\phi$ shares all of the same\nbenefits and problems as the axion, e.g. care is needed so as not to overclose the universe, but on the flip side, $\\phi$ can provide a dark matter candidate.\n\nThe details will depend on the particular models, but parametrically $m_\\phi \\sim \\text{TeV}^2\/f$ and the mixing with the Higgs scales as $\\sim 10$ TeV\/$f$.\n$\\phi$ with masses down to $\\sim$ 0.1 GeV ($f \\lesssim 10^7$ GeV) are excluded by meson decays and beam dumps (see Ref.~\\cite{Alekhin:2015byh,Flacke:2016szy} for a compendium of constraints).\nHorizontal branch star cooling constrains scalar couplings to electrons and excludes $\\phi$ in the range $10^{13}$ GeV $\\gtrsim f \\gtrsim 10^{10}$ GeV, while fifth-force experiments exclude $f \\gtrsim 10^{17}$ GeV (see Ref.~\\cite{Raffelt:2012sp,Hardy:2016kme} for a compendium of constraints).  These estimates are very rough and detailed constraints will be model dependent.\n\nFinally, if there is a non-zero cross-quartic coupling between the Higgses, then the Higgs can mix with\nthe other Higgses.  Due to the requirement of vacuum stability and the relatively small value of the SM quartic coupling,\nnegative cross-quartics larger than a few percent are excluded.  Positive cross-quartics larger than a few percent are\nalso excluded, as large cross-quartics generally push the theory out of the parameter space where all three Higgses\ncan obtain vevs.  As a result, the mixing between the multiple Higgses is suppressed by $\\approx$ few $\\times \\, 10^{-3}$.\nThe resulting exotic collider signatures are very similar to Twin Higgs models (see e.g. Refs.~\\cite{Burdman:2014zta,Craig:2015pha}) only with much smaller production rates.\nAnother difference is the absence of the $v\/f$ tuning needed to make electroweak symmetry breaking\nwork in Twin Higgs models.\n\n\n\\section{Conclusion}\n\nTo conclude, we briefly compare our new solution to similar solutions to the Hierarchy Problem,\nTwin Higgs~\\cite{Chacko:2005pe} and Little Higgs~\\cite{ArkaniHamed:2001nc}.  In unitary gauge, both Little Higgs and Twin\nHiggs solve the gauge and Yukawa divergences by coupling the Higgs as $\\sin v\/f$ to our sector and as $\\cos v\/f$\nto our partners.  The cancelations are then just a $\\mathbb{Z}_4$ version of the previous arguments, where two copies\nhave been removed because gauge invariance cancels the odd powers of $v$ so that the extra two copies are not needed\nfor the cancelation.  This discrete symmetry solution utilizes a generalization of these sin\/cos identities though in a completely different manner.\n\nThe approach most similar to the one presented in this paper are dimensional deconstructions of an extra dimension where the scalars \nonly pick up a mass non-locally~\\cite{ArkaniHamed:2001ca,ArkaniHamed:2001nc}.  The dimensional deconstruction based theories result in effective field theories that are a \nsubset of our more general approach.  In that language, the scalar is light due to 5D gauge invariance and locality.\nIn contrast, as can be seen from our discrete symmetry based approach, the $N$ sectors do not need to be Higgsed down to the \ndiagonal subgroup (no fifth dimension is required) and no sense of locality is needed in\ntheory space.  Any interaction is allowed as long as the interactions satisfy the $\\mathbb{Z}_N$ symmetry.\n\n\nThe most obvious extension for this approach is to use it on the Higgs boson directly.  There are two challenges for this approach.  The first is that our solution to the Hierarchy Problem suppresses the Higgs quartic.\nThe second is that while gauge charged scalars can be made compact, they cannot be made \nperiodic.  If there were a non-abelian equivalent  of frequency, then there might be a way to solve the quadratic divergences coming from the Yukawa \ncouplings in this manner.\n\nThere is still much to be explored with this new solution to the Hierarchy Problem.  We briefly list a few below:\n\\begin{itemize}\n\\item The $\\mathbb{Z}_N$ solutions to the Little Hierarchy Problem need to be \nexplored both theoretically and phenomenologically.  Ideally there exists a model where the only particles transforming under the $\\mathbb{Z}_N$ symmetry are the Higgs boson and the modulus.  In this case, the top quark would be its own partner.\n\\item UV completions of these theories would also be interesting, since moduli frequently appear in String Theory and very often have discrete symmetries associated with them.\n\\item Many theories of flavor involve discrete symmetries, hence a scalar that realizes the discrete symmetry non-linearly \nmay allow for interesting flavor physics and possibly even explain why the universe has three generations.\n\\item This new solutions allows for fifth forces and scalar dark matter that would otherwise appear tuned.\n\\item The moduli can be dark matter, leading to new and interesting signatures in the early universe and at experiments.\n\\item This mechanism allows for the inflaton to have large couplings and not ruin its flat potential.\n\\end{itemize}\n\nVery optimistically, the sensitivity of this scalar to phase transitions\nleads one to hope that this approach could help solve the cosmological constant problem as well.\nAt the very least, the recent spate of solutions to the Hierarchy Problem~\\cite{Graham:2015cka,Arkani-Hamed:2016rle} demonstrates that there is still much to be learned about naturalness.\n\n\n\n\\section*{Acknowledgements}\n\nA.H. is supported by NSF Grant PHY-1620074, DOE Grant DE- SC0012012 and by the Maryland Center for Fundamental Physics (MCFP).  A.H. thanks Savas Dimopolous, Junwu Huang, and Raman Sundrum for useful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ Introduction } \n\nKibble-Zurek mechanism (KZM) evolved from the scenario for defect creation in cosmological symmetry-breaking phase transitions \\cite{K-a, *K-b, *K-c}. As the post-Big-Bang Universe cools, causally disconnected regions must choose broken symmetry vacuum independently. Such random choices lead to topologically nontrivial configurations that survive phase ordering as topological defects. In the cosmological setting average size of the causally connected regions (hence, the average density of defects) is set by the Hubble radius at the time of the transition. This early Universe scenario relies on the speed of light and does not apply to the laboratory phase transitions. However, it was the point of departure for the dynamical theory \\cite{Z-a, *Z-b, *Z-c, Z-d} that employs critical exponents of the transition and the quench time to predict the scaling of the resulting density of defects. KZM was successfully tested using numerical simulations \\cite{KZnum-a,KZnum-b,KZnum-c,KZnum-d,KZnum-e,KZnum-f,KZnum-g,KZnum-h,KZnum-i,KZnum-j,KZnum-k,KZnum-l,KZnum-m} and laboratory experiments in condensed matter systems \\cite{KZexp-a,KZexp-b,KZexp-c,KZexp-d,KZexp-e,KZexp-f,KZexp-g,KZexp-gg,KZexp-h,KZexp-i,KZexp-j,KZexp-k,KZexp-l,KZexp-m,KZexp-n,KZexp-o,KZexp-p,KZexp-q,KZexp-r,KZexp-s,KZexp-t,KZexp-u,KZexp-v,KZexp-w,KZexp-x}. More recently, KZM was adapted to quantum phase transitions \\cite{Polkovnikov2005,QKZ1,QKZ2,d2005,d2010-a, d2010-b}. Theoretical developments \\cite{QKZteor-a,QKZteor-b,QKZteor-c,QKZteor-d,QKZteor-e,QKZteor-f,QKZteor-g,QKZteor-h,QKZteor-i,QKZteor-j,QKZteor-k,QKZteor-l,QKZteor-m,QKZteor-n,QKZteor-o,QKZteor-p,QKZteor-q,QKZteor-r,QKZteor-s,QKZteor-t} and experimental tests \\cite{QKZexp-a, QKZexp-b, QKZexp-c, QKZexp-d, QKZexp-e, QKZexp-f, QKZexp-g,deMarco2,Lukin18} of quantum KZM (QKZM) followed. The recent experiment \\cite{Lukin18}, where a quantum Ising chain in the transverse field is emulated using Rydberg atoms, is fully consistent with the predicted scaling \\cite{QKZ2,d2005}. \n\nThe KZM is often presented in its cartoon version where -- due to the critical slowing down \/ closing of the energy gap -- the dynamics of the system literally freezes-out in the neighborhood of the critical point. Today, as the experiments are able not only to count the final number of defects but can also monitor and probe the state of the system during the transition, it is timely to re-investigate the causally limited spreading of correlations during the putative ``freeze-out'' stage of the evolution. \n\n\\begin{figure}[t]\n\\vspace{-0.5cm}\n\\includegraphics[width=0.9\\columnwidth,clip=true]{Fig1.pdf}\n\\vspace{-0.1cm}\n\\caption{  \n{\\bf Adiabatic-impulse-adiabatic view of KZM.} \nLinear ramp crosses the critical point at time $t=0$. \nThe instantaneous transition rate, $\\left|\\dot{\\epsilon}\/\\epsilon\\right| = 1\/|t|$, diverges at the critical point and the relevant energy gap closes like $|\\epsilon|^{z\\nu}$. Consequently, while before $-\\hat t$ the state follows the adiabatic ground state, near the critical point (between $-\\hat t$ and $\\hat t$) its evolution is non-adiabatic.\nThe freeze-out assumes that the state is ``frozen'' at $-\\hat t$ \n-- size of the domains of the nascent phase does not change until $+\\hat t$, where the state starts to ``catch up'' with the Hamiltonian. \nThis version of KZM ignores propagation of the new phase front in the time interval $(- \\hat t, +\\hat t)$. It yields correct scalings, but it does not capture what happens -- for example -- in the paramagnetic-ferromagnetic quantum phase transition in the quantum Ising chain in transverse field \\cite{KZscaling1,Francuzetal}. Nevertheless, it may well be relevant in phase transitions where the conserved order parameter or other causes (localization) impede propagation of phase fronts of the broken symmetry phase.\n}\n\\label{fig:KZcartoon}\n\\end{figure}\n\nIn QKZM a system initially prepared in its ground state is smoothly ramped across a critical point to the other side of the quantum phase transition. A distance from the critical point, measured by a dimensionless parameter $\\epsilon$ controlling a Hamiltonian, can be linearized close to the critical point as\n\\begin{equation}\n\\epsilon(t)=\\frac{t}{\\tau_Q}.  \n\\label{epsilont}\n\\end{equation}\nHere $\\tau_Q$ is a quench time. Initially, far from the critical point, the evolution is adiabatic, and the system follows its adiabatic ground state, see Fig.~\\ref{fig:KZcartoon}. The adiabaticity fails at $-\\hat t$ when the reaction time of the system given by the inverse of the gap becomes slower than the timescale $|\\epsilon\/\\dot \\epsilon| = |t|$ on which the transition is being imposed.\nThe gap closes like $\\Delta\\simeq|\\epsilon|^{z\\nu}$, where $z$ and $\\nu$ are the dynamical and correlation length exponents, respectively. From the equation $|t|\\simeq |t\/\\tau_Q|^{-z\\nu}$ we obtain $\\hat t\\simeq \\tau_Q^{z\\nu\/(1+z\\nu)}$ and the corresponding $\\hat\\epsilon=\\hat t\/\\tau_Q\\simeq\\tau_Q^{1\/(1+z\\nu)}$. In the naive ``freeze-out'' version of the impulse approximation the ground state at $-\\hat\\epsilon$, with a corresponding correlation length\n\\begin{equation}\n\\hat\\xi \\simeq \\tau_Q^{\\nu\/(1+z\\nu)},  \n\\label{hatxi}\n\\end{equation}\nis expected to characterize the state of the system until $+\\hat t$, when the evolution can restart. In this way, $\\hat\\xi$ becomes imprinted on the initial state for the final adiabatic stage of the evolution after $+\\hat t$. Simplistic as it is, the adiabatic-impulse-adiabatic approximation correctly predicts the scaling of the characteristic lengthscale $\\hat\\xi$ and the timescale \n\\begin{equation} \n\\hat t\\simeq \\hat\\xi^z,\n\\label{hatt}\n\\end{equation} \nwith the critical exponents and $\\tau_Q$.\nThey both diverge in the adiabatic limit, $\\tau_Q\\to\\infty$, where they become the unique relevant scales in the KZ scaling ansatz~\\cite{KZscaling1,KZscaling2,Francuzetal}. For instance, a two-point correlation function $C_R(t)$, between two sites separated by a distance $R$, should satisfy\n\\begin{equation} \n\\hat\\xi^{\\Delta} C_R(t) = F\\left(t\/\\hat\\xi^z,R\/\\hat\\xi\\right).\n\\label{CRscaling}\n\\end{equation}\nHere $\\Delta$ is a scaling dimension and $F$ a non-universal scaling function. Eq.~\\eqref{CRscaling} is expected to be accurate in the long-wavelength and low-frequency limit.\nIt is worth to observe here, that the crude adiabatic-impulse-adiabatic approximation is consistent with the scaling hypothesis \\eqref{CRscaling}. However, it implies a particular  (time independent) form of the scaling function $F$.\n\n\\begin{figure}[t]\n\\vspace{-0.5cm}\n\\includegraphics[width=0.9\\columnwidth,clip=true]{Fig2.pdf}\n\\vspace{-0.1cm}\n\\caption{  \n{\\bf Sonic horizon view of KZM.} Initially, the correlation length $\\xi$ follows adiabatically the equilibrium healing length that -- in the adiabatic ground state (black) -- diverges at the critical point. Critical slowing down means that the size of the correlation length will begin to lag behind the values dictated by the ground state of the Hamiltonian at about $- \\hat t$. Pre-transition fluctuations reach size $\\hat \\xi$ at that instant and seed subsequent evolution of the system. The new broken symmetry phase is therefore selected by fluctuations in domains if size $\\hat \\xi$ at $-\\hat t$. Broken symmetry spreads within the impulse time interval of $2 \\hat t$ with the velocity $2\\hat v$ in every direction, enlarging the resulting ``sound cone'' to roughly $ 5 \\hat \\xi$ by $\\hat t$. In the freeze-out approximation (blue), after $-\\hat t$ the correlation length freezes, and remains close to the adiabatic correlation length at $-\\hat t$. Both the freeze-out and the sonic horizon views lead to the same scalings, but they result in different estimates of the pre-factors for domain sizes and defect densities.\n}\n\\label{fig:KZreal}\n\\end{figure}\n\n\nAs emphasized already in the early papers, see Ref.~\\onlinecite{Z-a, Z-b, Z-c, Z-d}, the freeze-out is not the complete story, and often not even a good approximation. A simple ``sonic horizon'' argument appealing to causality that goes beyond the impulse approximation is often more accurate. It is illustrated schematically in Fig.~\\ref{fig:KZreal}. As long as the evolution is adiabatic, the rate of growth of the diverging adiabatic correlation length, $\\xi\\simeq|\\epsilon|^{-\\nu}$, is  \n\\begin{equation} \n\\frac{d\\xi}{dt}=\n\\frac{d\\epsilon}{dt}\\frac{d\\xi}{d\\epsilon}=\n\\frac{1}{\\tau_Q}\n\\frac{\\nu}{|\\epsilon|^{\\nu+1}}. \n\\end{equation} \nThis rate diverges at the critical point. Hence there must be time $-\\hat t$ when it exceeds the speed limit set by twice\n\\begin{equation} \n\\hat v\\simeq\\frac{\\hat\\xi}{\\hat t}\\simeq\\tau_Q^{-\\nu(z-1)\/(1+z\\nu)}.\n\\label{hatv}\n\\end{equation} \nThe scaling of $-\\hat t$ obtained in this way is the same as in Eq.~\\eqref{hatt}. \n\nCausality and the KZ velocity $\\hat v$ are also central for the short-cuts to adiabaticity via inhomogeneous KZM. Therein, the external driving field has a smooth position dependence, gradually taking the system across the critical point---one part after another. Velocity of the driven critical front below $\\hat v$ (which in general depends on the shape of the above position dependence) is expected to pave the way to adiabatic dynamics, both for classical\\cite{ inhomo_classical-a, KZnum-c, inhomo_classical-c, inhomo_classical-d, inhomo_classical-e, inhomo_classical-f} and for quantum\\cite{inhomo_quantum-a, *inhomo_quantum-aa, inhomo_quantum-b, *inhomo_quantum-c, inhomo_quantum-d, inhomo_quantum-e, inhomo_quantum-f, inhomo_quantum-g} systems.\n\nIn the QKZM the speed limit is central to the causal argument. It originates from the dispersion of quasiparticles at the critical point: $\\omega\\simeq k^z$. Their speed for a quasimomentum $k$ is $v=d\\omega\/dk\\simeq k^{z-1}$. Between $-\\hat t$ and $\\hat t$ the quench excites quasiparticles with the magnitude of $k$ up to $\\hat k\\simeq \\hat\\xi^{-1}$. The speed of quasiparticles with the largest excited $k$ is therefore $\\hat v\\simeq\\hat k^{z-1}\\simeq\\hat\\xi^{1-z}\\simeq\\hat\\xi\/\\hat t$. When $z\\geq1$, $\\hat v$ is an upper bound on the velocity of quasiparticles. A quench in a translationally invariant system excites entangled pairs of quasiparticles with opposite quasimomenta: $k$ and $-k$. When moving apart, they are spreading correlations across the system. For $z\\geq1$ the rate of correlation spreading is limited by twice the speed $\\hat v$ of the fastest quasiparticles. \n\nIn the crudest version, neglecting in particular dependence of $\\hat v$ on the distance from the critical point, the argument implies that after $-\\hat t$ the correlation range (i.e., the ``sonic cone'') continues to grow at the rate $2\\hat v$ until $+\\hat t$. By this time, the range increases from the initial $\\hat\\xi$ at $-\\hat t$ to a final $\\hat\\xi+2\\hat v \\times 2\\hat t\\approx 5\\hat\\xi$. The growth is roughly five-fold. Even if the $5$ is just a very rough estimate, it shows how much the actual evolution can differ from the impulse approximation. Nevertheless, this argument confirms the role of $\\hat\\xi$ as the key relevant scale of length. With or without the prefactor of $5$, the final correlation length is proportional to $\\hat\\xi$, i.e., it scales with $\\tau_Q$ in the same way as given by Eq.~\\eqref{hatxi}. \n\nFrom another perspective, in the impulse approximation, the adiabatic ground state $|\\psi\\rangle$ corresponding to $-\\hat t$ freezes out as the state of the system. After $-\\hat t$ the adiabatic ground state departs from the frozen state. The frozen state becomes a superposition over adiabatic eigenbasis $|n\\rangle$: $\\sum_n c_n(t) |n\\rangle$, where $c_n(t)=\\langle n|\\psi\\rangle$. As a first step beyond the impulse approximation, we can include approximate dynamical phases: $\\sum_n c_n(t) e^{-i\\omega_n t}|n\\rangle$, where $\\omega_n$ is the adiabatic eigenfrequency at the critical point. In a (non-interacting) translationally invariant system, the eigenstates consist of pairs of excited quasiparticles, $|k,-k\\rangle$, and the eigenfrequencies are sums of $2\\omega_k$. The dynamical phase factors become scrambled -- and the phases begin to appear random -- when the largest of them, $2\\omega_{\\hat k}t\\propto\\hat k^z t$, becomes comparable to $1$ near $\\hat t$. The dephasing begins when the evolution crosses over from the non-adiabatic KZ stage to the post-KZ adiabatic stage. That is when the phases definitely can no longer be ignored, but even before the cross-over the phase factors $e^{-2i\\omega_k t}$ make the quasiparticle phase fronts propagate and let the quasiparticles spread the correlations across the system.\n\nFor $z=1$, when the dispersion is linear in $k$ and the quasiparticles have a definite speed of sound. This effect was termed the quasiparticle event horizon \\cite{EventHorizon}. In the QKZM context, it was considered in Refs. \\onlinecite{KZscaling1,Francuzetal}---see Fig.~\\ref{fig:Ising} for an example of the\nprototypical 1D quantum Ising model.\n\n\n\n\\begin{figure}[t]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\columnwidth]{fig2a.pdf}\n\\vspace{-0.7cm}\n\\caption{ {\\bf Quantum Ising chain.} \nThe correlation length during the quench in the 1D quantum Ising model, $H=-\\sum_n(1-\\epsilon) \\sigma_n^z + \\sigma^x_n \\sigma^x_{n+1} $, where the dynamical critical exponent $z=1$ and the excited quasiparticles posses definite speed of sound. Compare with Fig.~\\ref{fig:KZreal}.\nData from Ref.~\\onlinecite{Francuzetal}.\n}\n\\label{fig:Ising}\n\\end{figure}\n\n\nIn this paper we go beyond $z=1$ and present two examples with $z>1$: the classical Ising model with Glauber dynamics in Section \\ref{sec:Onsager} and the generalized quantum XY chain in Section \\ref{sec:XY}. They both exhibit an effective event horizon with a speed limit that depends on the quench time $\\tau_Q$.\nThe generic scenario is delimited by two examples where the sonic horizon effect cease to manifest because one of its underlying assumptions is not satisfied. The first one is the random Ising model in Section \\ref{sec:Random}, where localization of excited quasiparticles prevents the spreading of correlations, thus yielding in effect a ``freeze-out''. The other is the extended Ising model with long-range interactions in Section \\ref{sec:LR}, where the dynamical exponent $z$ is less than $1$. The excited quasiparticles with $k\\to0$ have infinite velocity, and the speed $\\hat v$ at the maximal excited $\\hat k$ is not an upper but a lower speed limit. Consequently, there is no sonic horizon effect, and the correlations have a long-range power-law tail that can evolve in time. After $-\\hat t$ the tail begins to lag behind its adiabatic evolution. However, instead of completely freezing out, it continues to grow at a finite rate.\n\n\\begin{figure}[b]\n\\vspace{-0.5cm}\n\\includegraphics[width=\\columnwidth]{fig3.pdf}\n\\vspace{-0.7cm}\n\\caption{ {\\bf Classical Ising in 2D on a square lattice.} \nIn (a),\nenergy per site, $E$, as a function of $\\beta(t)$ for different quench times $\\tau_Q$.\nFor reference, \nthe black line is the equilibrium internal energy $U(\\beta)$. \nWith increasing quench time, the quench curves converge to the equilibrium one.\nIn (b),\nexcitation energy per site, $\\Delta E=E-U$, as a function of scaled time for different quench times $\\tau_Q$.\nHere, both $\\Delta E$ and $t$ are rescaled according to the KZM predictions.\nFor large $\\tau_Q\\gg1$ the scaling makes them collapse in the KZ regime between \n$t\\approx-\\hat t$ and $t\\approx\\hat t$, corresponding to the rescaled times $-1$ and $+1$, respectively.\nAt later times the phase ordering kinetics steps in, which goes beyond the KZ physics.\n}\n\\label{fig:onsagerenergy}\n\\end{figure}\n\\section{ Classical Ising model: ${\\mathbf z=2}$ } \n\\label{sec:Onsager}\n\nWe begin with the classic example of the classical Ising model on a periodic square lattice of size $L\\times L$:\n\\begin{equation}\nH~=~-\\sum_{\\langle j,j'\\rangle} \\sigma^z_j \\sigma^z_{j'} ~.\n\\label{Honsager}\n\\end{equation}\nOn an infinite lattice, the critical inverse temperature would be $\\beta_c=\\ln(1+\\sqrt2)\/2\\simeq 0.4407$.\nThe relevant equilibrium exponents are $\\nu=1$ and $\\eta=1\/4$ \\cite{Schultz1964}.\nWe model relaxation to an external heat bath with the Glauber dynamics: \nMonte Carlo update thermalizes one spin (chosen at random) at a time.\nThe time needed for $L^2$ such one-spin updates sets unit of time.   \nFor such simple relaxation, the dynamical exponent is $z=2$, belonging to the universality class of model-A dynamics~\\cite{hohenberg1977theory}. \nWe performed all our numerical simulations on a $4096\\times 4096$ lattice. This lattice size is $100$ times longer than the longest correlation range encountered in the simulations, hence any finite size effects are eliminated with a wide safety margin. All results were averaged over $50$ repetitions of the quench, each of them starting from a different initial random spin configuration at infinite temperature.\n\n\\begin{figure*}[t]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\textwidth]{fig4.pdf}\n\\vspace{-0.5cm}\n\\caption{  \n{\\bf Classical Ising model in 2D.} The scaled correlator $\\hat\\xi^{\\eta}C_R$ as a function of the scaled distance $R\/\\hat\\xi$\nfor scaled times $t\/\\hat t=-1,0,1$ (left to right) and different quench times $\\tau_Q$.\nFor each scaled time,\nwhen $\\tau_Q\\gg1$ the plots with different $\\tau_Q$ collapse to a single scaling function $F(t\/\\hat t,R\/\\hat\\xi)$\ndemonstrating the KZ scaling (\\ref{CRscaling}) hypothesis for slow enough quenches.\n}\n\\label{fig:corrcollapse}\n\\end{figure*}\n\n\nThe inverse temperature of the heat bath is ramped linearly in time,\n\\begin{equation} \n\\beta(t)=\\beta_c\\left(1+\\frac{t}{\\tau_Q}\\right),\n\\end{equation}\nstarting with random spin configuration at $\\beta(-\\tau_Q)=0$. Figure~\\ref{fig:onsagerenergy}(a) shows the energy $E$ during the ramp as a function of $\\beta$ for different quench times $\\tau_Q$. Generally, the system is more ordered for slower quenches. For slow enough quenches, the KZ picture emerges. The system evolves adiabatically until it begins to go out of equilibrium around $-\\hat t$, where $\\hat t\\simeq\\tau_Q^{2\/3}$ is the KZ timescale.\n\nIn order to see how the excitation energy should depend on the quench time, let us consider the equilibrium internal energy. Near the critical point it is\n\\begin{equation} \nU(\\beta) = - \\left[ 1 + A (\\beta-\\beta_c) \\ln\\left|\\beta-\\beta_c\\right| \\right]\/\\tanh\\beta_c,\n\\end{equation}\nwhere $A\\simeq 1$ is a constant. In the adiabatic-impulse approximation the state becomes effectively frozen near $-\\hat t$ when $\\beta_c-\\hat\\beta\\simeq\\tau_Q^{-1\/3}$. At $\\beta_c$ the energies of the frozen state (i.e. the instantaneous state at $-\\hat t$) and the equilibrium one differ by\n\\bea\n\\Delta E(\\beta_c) &=&\nU(\\hat\\beta)-U(\\beta_c) \\nonumber\\\\\n&=&\nA (\\beta_c-\\hat\\beta) \\ln\\left|\\hat\\beta-\\beta_c\\right|\/\\tanh\\beta_c \\nonumber\\\\\n&\\simeq &\n\\tau_Q^{-1\/3}\\ln\\left(\\tau_Q\/\\tau_0\\right).\n\\end{eqnarray} \nHere $\\tau_0\\simeq1$ is a constant. We can see that, up to a subleading logarithmic correction, the KZ scale of energy is $\\simeq\\tau_Q^{-1\/3}$. Accordingly, in Figure~\\ref{fig:onsagerenergy}(b) we show scaled excitation energy $\\tau_Q^{1\/3}(E-U)$ in function of scaled time $t\/\\hat t$. For $\\tau_Q\\gg1$ the plots for different $\\tau_Q$ collapse in the KZ regime: $-1<t\/\\hat t<1$. There is no collapse at later times when phase ordering kinetics \\cite{POK} brings in new physics beyond the KZ mechanism.\nHaving seen how the excitations build up, we can have a closer look at the ferromagnetic correlator\n\\begin{equation} \nC_R(t) =\n\\langle \\sigma^z_i \\sigma^z_{j} \\rangle -\n\\langle \\sigma^z_i \\rangle \\langle \\sigma^z_{j} \\rangle.\n\\end{equation}\n\n\\begin{figure}[b]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\columnwidth]{fig5.pdf}\n\\vspace{-0.3cm}\n\\caption{  {\\bf Classical Ising model in 2D.}\nIn (a),\nthe scaled correlator $\\hat\\xi^{\\eta}C_R$ as a function of the scaled distance $R\/\\hat\\xi$ for $\\tau_Q=2^{15}$ at different scaled times $t\/\\hat t\\in[-1,1]$. The dashed line marks a cut-off $c$ (here $c=0.1$) whose crossing point is a working definition of a scaled correlation range $\\xi_c\/\\hat\\xi$.\nIn (b),\nthe scaled correlation range $\\xi_c\/\\hat\\xi$ as a function of the scaled time $t\/\\hat t$ for the cut-off $c=0.1$.\nThe best linear fit yields a slope $0.72$ as an estimate of the scaled velocity.\nIn (c),\nthe scaled velocity (slope) as a function of the cut-off. A linear extrapolation to zero cut-off yields $0.93$. This number is an estimate for (scaled) velocity at which the furthest correlation tail is spreading.\n}\n\\label{fig:spread}\n\\end{figure}\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\vspace{-0.1cm}\n\\includegraphics[width=\\textwidth]{fig6.pdf}\n\\vspace{-0.7cm}\n\\end{center}\n\\caption{{\\bf Extended quantum XY chain.} Scaled ferromagnetic correlations $C^{xx}_R$ as a function of scaled distance $R\/\\hat{\\xi}$ for the extended XY model in (\\ref{XYHsigma}) plotted for different quench times $\\tau_Q$ and at different scaled times $t\/\\hat{t}=-1,0,1$. For large enough $\\tau_Q$ plots collapse towards a universal scaling function $F(t\/\\hat{t},R\/\\hat{\\xi})$ demonstrating the KZ scaling hypothesis (\\ref{CRscaling}).\n}\n\\label{fig:XYcollapse}\n\\end{figure*}\n\n\n\nHere, the sites $i$ and $j$ are separated by a distance $R$ along one of the axes and the average is taken in the state at time $t$. Figure~\\ref{fig:corrcollapse} shows the scaled correlator $\\hat\\xi^\\eta C_R$ in function of the scaled distance $R\/\\hat\\xi$ for three scaled times $t\/\\hat t=-1,0,1$. Here $\\hat\\xi=\\tau_Q^{1\/3}$ is the scale of length in Eq.~(\\ref{hatxi}) and $\\hat t=\\hat\\xi^z=\\tau_Q^{1\/3}$ is the scale of time in Eq.~(\\ref{hatt}). In accordance with the scaling hypothesis (\\ref{CRscaling}), for each scaled time the plots with different $\\tau_Q$ collapse to a common scaling function when $\\tau_Q$ is large enough. The plot for the longest $\\tau_Q=2^{15}$ is practically equal to the scaling function $F(t\/\\hat t,R\/\\hat\\xi)$.\n\n\nComparing the three panels in Figure~\\ref{fig:corrcollapse} we can see that (at odds with the impulse approximation) the range of correlations increases several times between $-\\hat t$ and $\\hat t$. This observation is further corroborated by Fig.~\\ref{fig:spread}(a) where we collect scaled plots for $\\tau_Q=2^{15}$ at different scaled times. Using $\\hat\\xi$ as a natural unit of length, this figure shows how the correlation spreads in the KZ regime between $-\\hat t$ and $+\\hat t$. \n\nFigure~\\ref{fig:spread}(b) shows how a scaled range of the correlator in Fig.~\\ref{fig:spread}(a) depends on the scaled time $s=t\/\\hat t$. Here, the range is defined as the distance at which the scaled correlator falls below a threshold---set here at $0.1$. The nearly linear time dependence is fitted with a line whose slope estimates the scaled velocity of correlation spreading. We repeated the same procedure for thresholds down to $0.02$ (below $0.02$ the tail of the correlator becomes too noisy for an unambiguous estimation of the correlation range). The estimated slopes\/velocities are collected in Fig.~\\ref{fig:spread}(c). The velocity increases rather slowly with a decreasing threshold. In the limit of zero threshold -- probing the longest correlation tail -- it extrapolates to $0.93$. We can conclude that the speed limit in this model is $\\hat v$, which allows the size of the sonic horizon to expand with\n\\begin{equation} \n2\\hat v = \n0.93~ \\frac{\\hat\\xi}{\\hat t} = \n0.93~ \\tau_Q^{-1\/2}.\n\\end{equation}\nIn natural units, in accordance with KZ prediction in Eq.~\\eqref{hatv}, the speed limit becomes slower for slower quenches.\n\\section{ Extended quantum XY chain: ${\\mathbf z=3}$ } \n\\label{sec:XY}\n\n\\begin{figure*}[t]\n\\includegraphics[width=\\textwidth]{fig7.pdf}\n\\caption{{\\bf Extended quantum XY chain.}\nIn (a), \nfunction (\\ref{fitting}) is fitted (dashed lines) to the scaled ferromagnetic correlators $\\hat{\\xi}^{1\/4}C^{xx}_R$ for different scaled times during the non-adiabatic stage of the evolution: $t\/\\hat{t}\\in\\left[-2,2\\right]$.\nHere $\\tau_Q=2^{20}$.\nIn (b),\nthe fits provide the scaled correlation length $\\xi\/\\hat{\\xi}$ plotted against the scaled time $t\/\\hat{t}$. The slope of this plot is an estimate of the scaled velocity $v\/\\hat{v}=0.84$ at which the correlations are spreading. \nIn (c),\na more accurate approximation of the velocity is obtained by taking a derivative of the plot (b) with respect to the scaled time, which yields the scaled velocity $v\/\\hat{v}$ as a function of the scaled time.\nMaximal velocity is $v_{\\rm max}\/\\hat v=0.84$ for $\\tau_Q=2^{20}$. \nIn (d),\nthe maximal velocity is shown as a function of $\\tau_Q$.\nThe red line is a fit, which, for large $\\tau_Q$, saturates at $v_{max}\/\\hat v= 0.868$. \n}\n\\label{fig:XYspeed}\n\\end{figure*}\n\n\nWe step into the quantum realm \nwith the extended quantum XY periodic spin chain,\n\\bea\nH&=&-\\sum_{n=1}^N\n\\left[\n(1-\\epsilon) \\sigma^z_n +\nJ^{xx}\\sigma^x_n\\sigma^x_{n+1}+ \nJ^{yy}\\sigma^y_n\\sigma^y_{n+1}+\n\\right. \\nonumber \\\\\n&&\n\\left.\nJ^{xzx}\\sigma^x_n \\sigma^z_{n+1} \\sigma^x_{n+2}+ \nJ^{yzy}\\sigma^y_n \\sigma^z_{n+1} \\sigma^y_{n+2}\n\\right],\n\\label{XYHsigma}\n\\end{eqnarray} \na generalization of the quantum XY chain. It combines the desired $z>1$ with exact solvability.\nHere we consider an anisotropic model:\n\\bea\n&&\\left(J^{xx},J^{yy},J^{xzx},J^{yzy}\\right)=\\nonumber\\\\\n&&\n\\left(a\\frac{1+\\gamma}{2},a\\frac{1-\\gamma}{2},\n      b\\frac{1+\\delta}{2},b\\frac{1-\\delta}{2}\\right)\n\\end{eqnarray}\nwith the external magnetic field parametrized by $\\epsilon\\in[-1,1]$. The parameter is driven linearly (\\ref{epsilont}) from an initial $\\epsilon=-1$ in the paramagnetic phase, across the critical point at $\\epsilon=0$, to a final $\\epsilon=1$ in the ferromagnetic phase. After the Jordan-Wigner and Fourier transformations (\\ref{JordanWigner},\\ref{Fourier}), the Hamiltonian becomes:\n\\bea\nH &=\\sum_{k>0} &\nA_k(\\epsilon) \\left(c_k^\\dag c_k+c_{-k}^\\dag c_{-k}\\right) \\nonumber \\\\ \n&&\n+B_k \\left(c_kc_{-k}+c_{-k}^\\dag c_k^\\dag\\right),\n\\label{XYHk}\n\\end{eqnarray}\nwhere,\n\\bea \nA_k(\\epsilon) &=& 1-\\epsilon-a\\cos k-b\\cos 2k,  \\nonumber\\\\  \nB_k           &=& a\\gamma\\sin k+b\\delta\\sin 2k.\n\\end{eqnarray}\nIt is diagonalized by eigenmodes of the stationary Bogoliubov-de Gennes equations:\n\\bea\n\\omega_k\n\\left(\n\\begin{array}{c}\nU_k \\\\\nV_k\n\\end{array}\n\\right)&=&\n2\\left[\\sigma^z A_k(\\epsilon)+\\sigma^x B_k \\right] \n\\left(\n\\begin{array}{c}\nU_k \\\\\nV_k\n\\end{array}\n\\right),\n\\label{XYBdGk}\n\\end{eqnarray}\nwith eigenfrequencies\n$\n\\omega_k=2\\sqrt{A_k^2(\\epsilon)+B_k^2}.\n$\nHere, we set $a = 4\/3, b=-1\/3, \\gamma =1\/2, \\delta=1$. For the critical $\\epsilon=0$, we obtain a cubic dispersion relation\n\\begin{equation} \n\\omega_k\\simeq |k|^3,\n\\label{XYomegakz}\n\\end{equation} \nexpanding $\\omega_k$ around $k=0$, and the dynamical exponent $z=3$. On the other hand, setting $k=0$ and expanding in small $\\epsilon$, we obtain the gap opening as $\\omega_0\\simeq|\\epsilon|^1$. Hence $z\\nu=1$ and $\\nu=1\/3$. The dominant static ferromagnetic correlations have oscillatory tails:\n\\bea\nC_R     &=&\\langle\\sigma_i^x\\sigma_{i+R}^x\\rangle-\n           \\langle\\sigma_i^x\\rangle\\langle\\sigma_{i+R}^x\\rangle \\nonumber \\\\\n        &\\sim & R^{-\\eta} e^{-R\/\\xi}\\cos(b~R\/\\xi+c).\n\\end{eqnarray} \nHere $\\xi\\sim|\\epsilon|^{-\\nu}$ is a static correlation length and $\\eta=1\/4$. Given the exponents $z$ and $\\nu$, we can define the dynamical length (\\ref{hatxi}) and time (\\ref{hatt}) scales: $\\hat\\xi=\\tau_Q^{1\/6}$ and $\\hat t=\\tau_Q^{1\/2}$, respectively. The fastest excited quasiparticles have velocity $\\hat v\\simeq\\tau_Q^{-1\/3}$, see Eq.~\\eqref{hatv}. \n\nThe time-dependent quench is solved in appendix \\ref{XYtime} in a standard way \\cite{d2005} by mapping to the Landau-Zener problem, see appendix \\ref{LZ}. The KZ scaling hypothesis (\\ref{CRscaling}) is demonstrated in Fig.~\\ref{fig:XYcollapse} by collapse of the plots for different $\\tau_Q$. A perfect collapse requires very large $\\tau_Q\\simeq2^{20}$, as explained by Eq.~\\eqref{rescuv}. The oscillatory behaviour of $C_R$ in the original paramagnetic phase survives through the transition. In order to estimate the range of the scaled correlators, in Fig.~\\ref{fig:XYcollapse} we fit their tails with oscillatory functions of the form:\n\\begin{equation} \n\\hat{\\xi}^{1\/4} C_R(t)=\na~(R\/\\hat{\\xi})^{-1\/4}~\\mathrm{e}^{-\\frac{R\/\\hat{\\xi}}{\\xi\/\\hat{\\xi}}}\\cos(b~R\/\\hat{\\xi}+c),\n\\label{fitting}\n\\end{equation}\nwhere $a,b,c$ and $\\xi\/\\hat\\xi$ are the fitting parameters. We are interested how fast the scaled correlation length $\\xi\/\\hat\\xi$ grows with the scaled time $t\/\\hat t$. In order to reveal the universal behavior undisturbed by any short-range effects, we perform the fit in the range of scaled distances $R\/\\hat\\xi>2.5$. Ferromagnetic correlations for different scaled times $t\/\\hat{t}$, together with the fits, are shown in Fig.~\\ref{fig:XYspeed}(a) (for $\\tau_Q=2^{20}$). The correlation length $\\xi\/\\hat{\\xi}$ as a function of scaled time is shown in Fig.~\\ref{fig:XYspeed}(b). Its slope, equal to $0.838$, provides an estimate of the scaled velocity at which the correlations are spreading. For completeness, in Fig.~\\ref{fig:XYspeed}(c) we plot the derivative of the plot in Fig. \\ref{fig:XYspeed}(b) with respect to the scaled time. Its maximal value is shown in Fig.~\\ref{fig:XYspeed}(d) as a function of $\\tau_Q\\in [2^{10},2^{20}]$. For large $\\tau_Q$ it extrapolates to\n\\begin{equation} \n2 \\hat v = 0.868~ \\frac{\\hat{\\xi}}{\\hat{t}} = 0.868~ \\tau_Q^{-1\/3}.\n\\end{equation}\nAs in the classical 2D Ising model, where also $z>1$, the correlation spreading becomes slower for slower quenches.\n\n\\section{ Random Ising model } \n\\label{sec:Random}\n\nIn order to break the translational invariance and impede the propagation of excited quasiparticles that could spread correlations, next we consider the random Ising model (RIM) in one dimension defined by the Hamiltonian\n\\bea\nH=\n-\\sum_{n=1}^N\n\\left[ (h_{n} - \\epsilon)\\sigma^z_n \\ +\n J_{n}\\sigma^x_n\\sigma^x_{n+1}\n\\right].\n\\label{Hamil_RIM}\n\\end{eqnarray} \nHere both transverse fields $h_n$'s and nearest-neighbor couplings $J_n$'s are randomly selected from a uniform distribution between $0$ and $1$. The quantum critical point at $\\epsilon=0$ is separating the paramagnetic phase ($\\epsilon<0$) from the ferromagnetic one ($\\epsilon>0$). The critical point in this model is surrounded by Griffiths region \\cite{Griffiths-a, Griffiths-b, Griffiths-c}, where the presence of the so-called rare regions primarily manifests in two features: the activated dynamical scaling (the dynamical exponent $z\\rightarrow\\infty$) at the quantum critical point and the existence of singular regions where the linear susceptibility diverges even away from the critical point. The locally ordered rare regions act as giant spins that flip as a whole and are responsible for the exponentially slow dynamics near the critical point. These features are encapsulated in a dynamical exponent that diverges at the critical point as $z \\sim|\\epsilon|^{-1}$, see Ref. \\onlinecite{Fisher1-a, Fisher1-b} which also shows that the correlation length exponent is $\\nu = 2$. The average correlation function at criticality is a power law\\cite{McCoyWu}:\n\\begin{equation} \nC^{xx}_R=\\langle \\sigma_{i}^{x}\\sigma_{i+R}^{x}\\rangle \\sim R^{-\\eta},\n\\end{equation}\nwhere $\\eta = \\frac{3-\\sqrt{5}}{2}\\approx 0.38$. \n\nWe consider ramping the parameter $\\epsilon$ linearly as a function of time (\\ref{epsilont}) driving the Hamiltonian from the initial paramagnetic ground state, across the critical point at $t=0$, into the ordered phase. Ref. \\onlinecite{dziarmagaRIM-a, dziarmagaRIM-b, dziarmagaRIM-c} showed that the KZ correlation length $\\hat \\xi$ of the model varies logarithmically with the quench rate $\\tau_Q$:\n\\bea\n\\centering\n\\hat{\\xi} = \\ln^2{(\\tau_Q\/a)},\n\\label{hatxiRIM}\n\\end{eqnarray}\nwhen $\\ln{(\\tau_Q\/a)}\\gg 1$. Here $a\\simeq1$ is a non-universal constant. We can see, that the dependence on $\\tau_Q$ is very weak compared to any of the usual KZ power-law scalings (\\ref{hatxi}). Taking into account that the correlation-length exponent near the critical point is $\\nu = 2$, we get an estimate of the characteristic timescale,\n\\bea\n\\centering\n\\hat{t} = \\frac{\\tau_{Q}}{\\ln{(\\tau_{Q}\/a)}},\n\\label{hattRIM}\n\\end{eqnarray}\n for $\\ln{(\\tau_Q\/a)} \\gg 1$. \n\n\n\\begin{figure}[t]\n\\includegraphics[width=\\columnwidth]{fig8.pdf}\n\\caption{{\\bf Random Ising model.} Average scaled density of excitations $\\hat\\xi d_{\\rm ex}$ during the quench as a function of scaled time $t\/\\hat t$ for different quench times $\\tau_Q$. Here, we use Eqs.~(\\ref{hatxiRIM},\\ref{hattRIM}) with $a = 0.118$. This $a$ was tuned to obtain the best possible collapse at $t\/\\hat t=0$ but the plots collapse well during the whole quench. The KZ diabatic stage (shaded in green) extends roughly from $-0.25t\/\\hat t$ to $0.25t\/\\hat t$. This is where the excitation grows before it saturates in the last adiabatic stage. Averaging was done over 30 random realizations for a lattice of size $N = 128$.\n}\n\\label{FigDoERIM}\n\\end{figure}\n\\begin{figure*}[t]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\textwidth,clip=true]{fig9.pdf}\n\\vspace{-0.1cm}\n\\caption{ {\\bf Random Ising model.} Average scaled ferromagnetic correlations $\\hat\\xi^\\eta C^{xx}_R$ as a function of scaled distance $R\/\\hat{\\xi}$ for the random Ising model (\\ref{Hamil_RIM}) plotted for different quench times $\\tau_Q$ and at different scaled times $t\/\\hat{t}=-0.25,0,0.25$. For large enough $\\tau_Q$ plots collapse towards a universal scaling function $F(t\/\\hat{t},R\/\\hat{\\xi})$ demonstrating the KZ scaling hypothesis\n(\\ref{CRscaling}).\n}\n\\label{fig:RIMcollapse}\n\\end{figure*}\n\nIn order to solve the dynamics, we map the Hamiltonian by the Jordan-Wigner transformation (\\ref{JordanWigner}) to a quadratic spinless free-fermionic model\n\\bea\nH = \\sum_{n}(h_{n}-\\epsilon)c_{n}^\\dag c_{n} -J_{n}c_{n}^\\dag c_{n+1} - J_{n}c_{n+1} c_{n} + {\\rm h.c}.\n\\label{JW_RIM}\n\\end{eqnarray}\nFollowing the convention of this article, we confine ourselves to the subspace of even parity of $c$-quasiparticles (anti-periodic boundary conditions, i.e., $c_{N+1} = -c_1$) and in this subspace, we diagonalize the Hamiltonian by a Bogoliubov transformation:\n\\bea\n\\centering\nc_{n} = \\sum\\limits_{m=1}^{N}(U_{nm}\\gamma_{m} + V_{nm}^* \\gamma_{m}^\\dag)\n\\label{bogoliubov_RIM}\n\\end{eqnarray}\nThe index $m$ labels the (Bogoliubov) eigenmodes of the stationary Bogoliubov-de Gennes equation\n\\bea\n\\centering\n\\omega_{m}U_{n,m}^{\\pm} = 2h_{n}U_{n,m}^{\\mp} - 2j_{n-1}U_{n-1,m}^{\\mp},\n\\label{stationary_B-DeG}\n\\end{eqnarray}\nwhere $\\omega_{m}>0$, $U_{nm}^{\\pm} = U_{nm}\\pm V_{nm}$ and anti-periodic boundary conditions $(U_{N+1,m}^{\\pm} = -U_{1,m}^{\\pm},U_{0,m}^{\\pm}=-U_{N,m}^{\\pm})$ is implemented. The eigenstates ($U_{nm},V_{nm}$), with positive energy, $\\omega_{m}>0$ normalized so that $\\sum_{n}(|U_{nm}|^2 + |V_{nm}|^2) = 1$, define quasiparticle operators $\\gamma_{m}=U_{nm}^{*}c_{n}+V_{nm}c_{n}^{\\dag}$. We have corresponding negative energy components of the eigenstates labelled ($U_{nm}^{neg},V_{nm}^{neg}$) with energy $-\\omega_{m}$, which defines a quasiparticle operator $\\gamma_{m}^{neg}=(U_{nm}^{neg})^{*}c_{n}+V_{nm}^{neg}c_{n}^{\\dag}$. The Bogoliubov transformation renders the Hamiltonian to be $H = \\sum_{m = 1}^{N}\\omega_{m}(\\gamma_{m}^{\\dag}\\gamma_{m} - \\frac{1}{2})$. In the even parity subspace only states with even number of quasi-particles belong to the spectrum of $H$.\n\nIn order to find whether the RIM fits into our narrative of the uniform scaled velocity of the correlation spreading by quasiparticles,\nwe proceed with the numerical simulation of the quench. \nWe prepare the initial state of the system deep in the paramagnetic phase in the ground state, i.e., in the Bogoliubov vacuum state for quasiparticles at an initial time $t_{0}$ where $\\epsilon(t_{0}) = 5$. As we tune the parameter $\\epsilon(t)$ towards $0$, see Eq.~\\eqref{epsilont}, the state of the systems departs from its adiabatic ground state and gets excited due to closing of the energy gap near the critical point. We work in the Heisenberg picture where we assume that the excited state is a Bogoliubov vacuum while the time-dependence is ascribed to a set of time-dependent quasi-particle operators\n\\bea\n\\centering\n\\gamma_{m}(t)=u_{nm}^{*}(t)c_{n}+v_{nm}^{*}(t)c_{n}^{\\dag}\n\\end{eqnarray}\nThe Bogoliubov modes $u_{nm}$ and $v_{nm}$ solve the time-dependent Bogoliubov-de Gennes  equations:\n\\bea\n\\centering\ni\\frac{du_{nm}^{\\pm}}{dt} = 2(h_{n}-\\epsilon(t))u_{nm}^{\\mp}-2J_{n}u_{n\\mp1,m}^{\\mp}\n\\label{t_B-deG}\n\\end{eqnarray}\nWe integrate equation (\\ref{t_B-deG}) numerically using the 2nd order Suzuki-Trotter method, see Appendix \\ref{tB-deG_RIM}. The density of excited Bogoliubov quasiparticles $d_{ex}(t)$ can be calculated at each time step by projecting the time-dependent Bogoliubov modes ($u_{nm}(t),v_{nm}(t)$) onto the corresponding instantaneous static negative Bogoliubov modes ($U_{nm}^{\\text{neg}},V_{nm}^{\\text{neg}}$):\n\\bea\n\\centering\nd_{ex}(t) = \\frac{1}{N} \\sum_{s}^N\\sum_{p}^N \\left|\\langle U_{p}^{\\text{neg}},V_{p}^{\\text{neg}}|u_{s}(t),v_{s}(t)\\rangle\\right|^2.\n\\label{doe_RIM}\n\\end{eqnarray}\n\nWhile in the initial adiabatic stage, the system remains in the instantaneous ground state and hence the density of quasiparticle excitations is $0$. On tuning the field $\\epsilon(t)$ towards its critical value at $t\/\\hat{t} = 0$, the system gets excited and consequently $d_{ex}$ starts to grow. As we keep increasing the field $\\epsilon(t)$, $d_{ex}$ saturates as the system ends its non-adiabatic journey across the critical point. In Fig.~\\ref{FigDoERIM} we see that the scaled plots (averaged over disorder) for different $\\tau_Q$ collapse in accordance with the dynamical scaling hypothesis (\\ref{CRscaling}). \n\n\\begin{figure}[b]\n\\includegraphics[width=\\columnwidth]{fig10.pdf}\n\\caption{   {\\bf Random Ising model.}\nIn (a),\nthe scaled correlator $\\hat\\xi^{\\eta}C_R$ as a function of the scaled distance $R\/\\hat\\xi$ for $\\tau_Q=2^{11}$\nat different scaled times $t\/\\hat t$ in $[-0.25,0.25]$.\nThe dashed line marks a cut-off (here $1$) whose crossing point \nis a working definition of a scaled correlation range $\\xi_c\/\\hat\\xi$.\nIn (b),\nthe scaled correlation range $\\xi_c\/\\hat\\xi$ as a function of the scaled time $t\/\\hat{t}$ for the cut-off $c=0.5$. The best linear fit yields a slope of $0.113$ as an estimate of the scaled velocity.\nIn (c),\nthe scaled velocity (slope) as a function of the cut-off, with the error bars indicating fitting errors.\n}\n\\label{fig:RIMspread}\n\\end{figure}\n\nThe hypothesis for correlations in Eq.~\\eqref{CRscaling} is verified by a similar collapse of the plots in Fig.~\\ref{fig:RIMcollapse}. We consider large $\\tau_Q$ as the logarithmic KZ scaling laws are only valid in that regime. The correlation function is averaged over $100$ instances of disorder and lattice translations on a lattice of $N=256$. This lattice is a few times longer than the longest correlation range. The details of calculating the correlation functions are given in Appendix \\ref{appcorrRIM}.\n\nA close look at Fig.~\\ref{fig:RIMcollapse} reveals that the scaled dynamical correlation function grows with scaled time. In order to appreciate the growth, we collect the scaled correlators for different scaled times in Fig.~\\ref{fig:RIMspread}(a). In order to estimate the scaled correlation range $\\xi_c\/\\hat{\\xi}$ as a function of scaled time, we choose a cut-off $c$ for the scaled correlation and calculate scaled distances corresponding to the cut-off by spline interpolation. Fig.~\\ref{fig:RIMspread}(b) shows the scaled range in function of the scaled time for the cut-off (selected equal to $0.5$) in Fig.~\\ref{fig:RIMspread}(a).\nNear the critical point, we approximate it with linear dependence where the slope is $0.113\\hat\\xi\/\\hat t$. We repeat the same procedure for other cut-offs. We present the enumerated slopes for various cut-offs in Fig.~\\ref{fig:RIMspread}(c), with the error bars indicating the fitting error. Finally, we estimate the sonic horizon expansion speed in this model with the maximal observed value\n\\bea\n\\centering\n2\\hat{v} = 0.12 \\frac{\\hat{\\xi}}{\\hat{t}} = 0.12 \\frac{\\ln^3(\\tau_Q\/a)}{\\tau_Q}\n\\end{eqnarray}\nCompared to the previous models, with the prefactor $0.12$, the speed limit is significantly below expectations in this case. The entangled Bogoliubov quasiparticles get excited but, due to their localization by disorder, they do not propagate to spread correlations appreciably. In the spin language, the locally ordered regions are excited, but they essentially stay where they are. One may even argue that in this case, the adiabatic-impulse-adiabatic freeze-out approximation captures the essential physics.\n\n\n\\section{Long-range extended Ising model: $0<z<1$ } \n\\label{sec:LR}\n\n\nThe long-range extended Ising model is a further generalization of the extended XY model that we have discussed in Sec. \\ref{sec:XY}. Now the Hamiltonian of the system includes not only the nearest-neighbor and next-nearest-neighbor interaction, but all possible long-range cluster terms: \n\\bea\nH=\n-\\sum_{n=1}^N\n\\left[\n (1-\\epsilon) \\sigma^z_n +\n\\sum_{r=1}^{N-1} \nJ_r\\sigma^x_n \\sigma^x_{n+r} \\prod_{i=n+1}^{n+r-1} \\sigma^z_i\n\\right].\n\\label{Hamil_LR}\n\\end{eqnarray} \nHere \n\\begin{equation}\nJ_r ~=~ \\frac{1}{\\zeta(\\alpha)} ~ \\frac{1}{r^\\alpha},\n\\end{equation}\nand the normalization by the Riemann zeta function, possible for $\\alpha>1$, is such that $\\sum_r J_r=1$. The case of $\\alpha>2$ is not the most interesting here because the model behaves effectively like the short-range one \\cite{LRKitaev1,LRKitaev2,LR1,LR2,LRBosonic}. Similarly, when $0\\le\\alpha\\le1$ we would need to restrict ourselves to a finite system because the thermodynamic limit does not exist in this case, and the model behaves effectively like the Lipkin-Meshkov-Glick model \\cite{LMG} with infinite-range interactions. Therefore, we focus here on the intermediate $1<\\alpha<2$, where a cross-over between the short and infinite-ranges happens. In this regime, we consider the linear quench (\\ref{epsilont}) driving the system from the initial paramagnetic phase at $t=-\\infty$, across the critical point at $t=0$, to the final ferromagnetic phase. \n\n\\begin{figure}[b]\n\\includegraphics[width=\\columnwidth]{fig11.pdf}\n\\caption{  {\\bf Long-range extended Ising model.}\nIn (a),\nthe static correlation function $C_R$ at the critical $\\epsilon=0$ for $\\alpha=3\/2$. \nThe solid line is a linear fit with a slope $-1.002\\approx -1 $.\nThe same panel shows the same function far from the critical point at $\\epsilon=-2$. \nThe solid line is a linear fit with a slope $-1.45 \\approx -(3 -\\alpha) $.  \nIn (b),\nthe scaled static correlation function $\\xi C_R$ as a function of scaled distance $R\/\\xi$.\nIn accordance with the static scaling hypothesis (\\ref{staticSH}),\nthe plots for different $\\epsilon$ collapse to a unique scaling function.\nWe can see a crossover from the critical $(R\/\\xi)^{-1}$ to the off-critical $(R\/\\xi)^{-3\/2}$\nnear $R\/\\xi=1$. The straight lines are linear fits with slopes $-1.085\\approx-1$ and $-1.42\\approx -(3-\\alpha)$ for smaller and larger $R\/\\xi$, respectively. \n}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\includegraphics[width=\\textwidth]{fig12.pdf}\n\\caption{ {\\bf Long-range extended Ising model.}\nThe scaled dynamical correlation function $\\hat\\xi C_R$ for different quench times $\\tau_Q$ at different scaled times, $s={t}\/{\\hat{t}}$, during the quench.\nHere $\\alpha=3\/2$.\nFrom left to right: $s=-1,0,0.35,0.4$. \nIn (a) at $s=-1$,\nthe evolution is still adiabatic and the plots for different $\\tau_Q$ collapse according to the static scaling hypothesis which is equivalent to the KZ hypothesis in this early adiabatic regime.\nIn (b) at $s=0$,\nwhen crossing the critical point the evolution is diabatic. Different $\\tau_Q$ collapse to a common scaling function $F(s,R\/\\hat\\xi)$ in accordance with the KZ scaling hypothesis (\\ref{CRscaling}).\nIn (c) at $s=0.35$,\nthe correlation function begins to bend down at distances between $R\/\\hat\\xi=10$ and $100$.\nDifferent $\\tau_Q$ collapse, as the evolution is still in the KZ regime.\nFinally in (d) at $s=0.4$,\nthe collapse fails due to the phase ordering after the system left the KZ regime and entered the adiabatic one. At the distances between $R\/\\hat\\xi=10$ and $100$ the correlations become negative for longer $\\tau_Q$.\n}\n\\label{fig:corrp0m}\n\\end{figure*}\n\n\nThanks to the string operator in the long-range interaction terms, the model can be mapped to a quadratic free fermion model and solved analytically. After the Jordan-Wigner transformation (\\ref{JordanWigner}) the Hamiltonian (\\ref{Hamil_LR}) becomes:\n\\bea\nH &=& -\\sum_n \\left(1-\\epsilon\\right)\\left(c_nc_n^\\dag-c_n^\\dag c_n\\right) \\nonumber \\\\\n&&- \\sum_{n,r} J_r\\left( c_n^\\dag c_{n+r} + c_n^\\dag c_{n+r}^\\dag + {\\rm h.c.}  \\right),\n\\label{HcLR}\n\\end{eqnarray}\nfor the anti-periodic boundary conditions. This representation is also known as the long-range Kitaev model \\cite{LRKitaev1, LRKitaev2}, where the hopping and pairing terms are of equal strength. After Fourier transformation (\\ref{Fourier}),\n\\bea\nH &=&\n-2\\sum_{k>0} \n(1-\\epsilon -  \\Re(\\tilde{J_k})) \\left(c_k^\\dag c_k+c_{-k}^\\dag c_{-k}\\right) \\nonumber \\\\\n&&+  ~\\Im(\\tilde{J_k})\\left(c_{k}^\\dag c_{-k}^\\dag + c_{-k}c_{k}\\right),\n\\label{HkLR}\n\\end{eqnarray}\nwhere  $\\Re(\\tilde{J_k})$ and  $\\Im(\\tilde{J_k})$ are, respectively, real and imaginary parts of the Fourier transform $\\tilde{J_k}=\\sum_r J_r e^{ikr}$. We have $\\tilde{J_k}=\\frac{{\\rm Li}_{\\alpha}(e^{ik})}{\\zeta(\\alpha)}$, where ${\\rm Li}$ is the polylogarithm function: ${\\rm Li}_\\alpha(x)=\\sum_{n=1}^\\infty\\frac{x^n}{n^\\alpha}$. \n\n\n\nWe can now find the stationary Bogoliubov-de Gennes equations (\\ref{XYBdGk}) with\n$\nA_k(\\epsilon)=1-\\epsilon-\\Re(\\tilde{J_k}),~~ B_k=\\Im(\\tilde{J_k})\n$\nand eigenfrequencies\n$\n\\omega_k=2\\sqrt{A_k^2(\\epsilon)+B_k^2}.\n$\nWe have a critical point at $\\epsilon=0$ where the gap closes for $k=0$. Another critical point, \nnot to be considered here, is at $\\epsilon=2(1-2^{-\\alpha})$ and $k=\\pi$. The dispersion relation at the critical $\\epsilon=0$ is\n\\bea\n\\omega_k \\simeq |k|^{\\alpha-1},\n\\end{eqnarray}\nhence the dynamical exponent is $z=\\alpha-1\\in(0,1)$. On the other hand, for small $\\epsilon$, the gap at $k=0$ closes as\n$ \n\\omega_0 = 2|\\epsilon|,\n$\nhence $z\\nu=1$. \n\n\n\nIn a short range model for large $R$ the correlation function decays exponentially with $R$ when the system is away from the critical point but this does not need to hold for long-range interactions. Indeed, our system has a power law scaling even far away from the critical point \\cite{LRKitaev1,LRKitaev2}:\n\\bea \nC_R &=&\n\\langle \\sigma_i^x \\sigma_{i+1}^z \\ldots \\sigma_{i+R-1}^z \\sigma_{i+R}^x\\rangle\n\\sim 1\/R^{3-\\alpha},\n\\label{Corr}\n\\end{eqnarray}\ncompare Fig.~\\ref{fig2}(a). On the other hand, as discussed in more detail in Ref.~\\onlinecite{natun}, at the critical point we expect a critical power law $C_R \\sim 1\/r^{\\eta}$ with $\\eta<\\alpha$. Indeed, in Fig.~\\ref{fig2}(a) we find that $\\eta=1$ for $\\alpha=3\/2$. Similarly as for short-range interactions, for a small $\\epsilon$ we expect a cross-over between the two power laws when $R$ is close to $\\xi\\sim\\epsilon^{-\\nu}$. Note that here $\\xi$ is not the usual exponential correlation length, even though it scales with $\\epsilon$ in the characteristic way. The crossover can be verified with a static scaling hypothesis:\n\\bea \n\\xi^\\eta ~C_R = F_{\\rm st}\\left( R\/\\xi \\right).\n\\label{staticSH}\n\\end{eqnarray} \nWith $\\xi=\\epsilon^{-\\nu}$ the plots of the scaled correlator $\\xi^\\eta C_R$ as a function of the scaled distance $R\/\\xi$ for different $\\epsilon$ should collapse to a common static scaling function $F_{\\rm st}(x)$. We expect $F_{\\rm st}(x)$ to cross-over around $x=1$ between the critical tail $x^{-1}$ for small $x$ and $x^{3-\\alpha}$ for large $x$. Indeed, this is what we see in Fig.~\\ref{fig2}(b). The unscaled leading static tail is, therefore, \n\\begin{equation} \nC_R\\simeq \\xi^{1\/2} R^{-3\/2}\n\\label{lrtailstatic}\n\\end{equation} \nfor $R\\gg\\xi$.\n\n\n\\begin{figure}[b]\n\\vspace{-0.5cm}\n\\includegraphics[width=\\columnwidth]{fig13.pdf}\n\\vspace{-0.5cm}\n\\caption{ {\\bf Long-range extended Ising model.}\nIn (a), \nscaled correlator $\\hat\\xi C_R$ for $\\alpha=3\/2$ as a function of scaled distance for different scaled times in the log-log scale.\nIn (b),\nin the inset the coefficients $A,B,C$ in Eq.~\\eqref{ABC(s)fit} fitted to the tail of the scaled correlator are shown as a function of scaled time $s=t\/\\hat t$. The main picture shows $A(t\/\\hat t)$ together with the adiabatic $A_{\\rm ad}(t\/\\hat t)$.\nIn the initial adiabatic stage $A$ follows $A_{\\rm ad}$. \nIn the KZ stage $A_{\\rm ad}$ diverges to infinity at the critical point forcing $A$ to lag behind.\n}\n\\label{fig:fig2a}\n\\end{figure}\n\n\nHaving verified the static hypothesis, we are encouraged to propose its dynamical version in the same form (\\ref{CRscaling}) as for local interactions. The dynamics is described by the time-dependent Bogoliubov-de Gennes equations (\\ref{tBdGk}) whose solution is presented in Appendix \\ref{AppLR}. The KZ mechanism has recently been verified in a similar model~\\cite{amitdutta}, where defect density was found to scale as $\\tau_Q^{-1\/2(\\alpha-1)}$ in agreement with the KZ prediction $\\tau_Q^{-\\nu\/(1+z\\nu)}$, see also Ref.~\\cite{KZLR2}. Further developments, like scaling of excitation energy (consistent with KZ) or possible experimental implementation, were considered in Refs.~\\onlinecite{MorigiLR,LRPlenio}.\nHere, we are interested in the build-up of correlations during the quench. In Fig.~\\ref{fig:corrp0m}(a) we show scaled correlation functions at different scaled times. For definiteness, in this figure we set $\\hat t=\\tau_Q^{1\/2}$ and $\\hat\\xi=\\tau_Q^{1\/2(\\alpha-1)}$ with both numerical prefactors equal to $1$. We can see that up to $t=0.35\\hat t$ the scaled correlators for different $\\tau_Q$ collapse demonstrating the KZ scaling hypothesis in Eq.~\\eqref{CRscaling}. At $t\/\\hat t\\approx 0.4$ they already fail to collapse, signalling the end of the KZ regime and the beginning of phase ordering in the second adiabatic stage. \n\nIn Fig.~\\ref{fig:fig2a}(a), we collect scaled correlators for different scaled times, $s=t\/\\hat t$, in the KZ regime. In order to gain better insight, in Fig.~\\ref{fig:fig2a}(b) we fit their tails with a function\n\\begin{equation} \n\\hat \\xi C_R(s)=A(s)(R\/\\hat\\xi)^{-3\/2}+B(s)(R\/\\hat\\xi)^{-2}+C(s)(R\/\\hat\\xi)^{-5\/2}.\n\\label{ABC(s)fit}\n\\end{equation} \nIts form is motivated by correlation tails after a sudden quench considered in appendices \\ref{SuddenAppInfinite} and \\ref{SuddenAppFinite}. Here, the coefficients $A,B,C$ are functions of the scaled time shown in Fig.~\\ref{fig:fig2a}(c). On top of the leading $A(s)$ tail, sub-leading dynamical correlations build up in time, with $B(s)$ growing negative and $C(s)$ positive. Their behavior is similar to what is observed after a sudden quench to the critical $\\epsilon=0$ either from $\\epsilon=-\\infty$ or $\\epsilon=-1$, see appendices \\ref{SuddenAppInfinite} and \\ref{SuddenAppFinite}, respectively. The main Fig.~\\ref{fig:fig2a}(b) shows $A(s)$, i.e., the coefficient of the leading long-range tail $\\propto R^{-3\/2}$. \n\n\nThe same figure shows $A_{\\rm ad}(s)$, i.e., the same coefficient but in case the evolution were adiabatic. $A_{\\rm ad}(s)$ was obtained by a fit to the far tail of the static correlation function at $\\epsilon(s)=s\/(\\tau_Q\/\\hat t)$. The same $A_{\\rm ad}(s)$ can be obtained by equating the static tail (\\ref{lrtailstatic}) with the adiabatic tail $C_R=A_{\\rm ad}(s)\\hat\\xi^{1\/2}\/R^{3\/2}$, compare (\\ref{ABC(s)fit}):\n\\begin{equation} \nA_{\\rm ad}(s) \\simeq\n\\left[\\frac{\\xi(s)}{\\hat\\xi}\\right]^{1\/2} \\simeq\n\\left[\\frac{\\epsilon(s)^{-\\nu}}{\\hat\\xi}\\right]^{1\/2} \\sim\ns^{-\\nu\/2}.\n\\end{equation} \n$A_{\\rm ad}(s)\\sim s^{-1}$ for $\\alpha=3\/2$ is consistent with the divergence we observe in Fig.~\\ref{fig:fig2a}(b). We can also see that, as expected, $A(s)$ follows $A_{\\rm ad}(s)$ in the initial adiabatic stage up to $s\\simeq -1$. After that, in the diabatic KZ stage, it lags behind as it cannot catch up with the diverging $A_{\\rm ad}(s)$. If it did not, and the evolution were adiabatic, then the correlation range -- defined by $R$ where $C_R$ falls below a fixed small cutoff -- would diverge near the critical point as $|t|^{-2\/3}$. \n\n\n\nAfter $A(s)$ begins to lag behind the adiabatic evolution, it continues to grow at a finite rate until the critical point. This is the stage where the long-range model is the most reminescent of the standard causal KZ picture. After the critical point $A(s)$ begins to dip down around $s\\approx 0.2$ but this is where the KZ scaling hypothesis (\\ref{CRscaling}) begins to be violated, see Fig.~\\ref{fig:corrp0m}(d).\n\n\\section{ Conclusion } \n\\label{SectionConclusion}\n\nWe recall the causal\/sonic horizon version of the Kibble-Zurek mechanism (KZM).\nIn the initial adiabatic stage of the evolution, \nthe correlation range follows its adiabatic counterpart.\nAt $-\\hat t$ it begins to lag behind the diverging adiabatic range and continues to grow at a finite rate set by the speed limit $\\simeq \\hat\\xi\/\\hat t$.\nThis way, between $-\\hat t$ and $+\\hat t$, the correlation range can increase several times at odds with the impulse approximation where it remains frozen. \n\nThere are notable exceptions, like the random quantum Ising chain, where the increase is small due to localization of excited quasiparticles that prevents them from spreading entanglement. There are interesting generalizations, like the long-range extended Ising model, where it is the power law correlation tail, rather than the exponential correlation length, that lags behind its diverging adiabatic evolution.\n\n\n\\acknowledgements\nAS would like to thank Titas Chanda for useful discussions.\nWe acknowledge funding by National Science Centre (NCN), \nPoland under projects No.~2016\/23\/B\/ST3\/00830 (JS,AF) \nand No. 2016\/23\/D\/ST3\/00384 (MMR), \nNCN together with European Union through QuantERA ERA NET \nprogram No.~2017\/25\/Z\/ST2\/03028 (AS,DS,JD),\nand Department of Energy under the Los Alamos National Laboratory \nLDRD Program (WHZ).\nWHZ was also supported by the U.S. Department of Energy, Office of Science, Basic Energy\nSciences, Materials Sciences and Engineering Division, Condensed Matter Theory Program.\nAF acknowledges financial support by Polish Ministry of Science and Education, project No. DI2015 021345, from the budget funds for science in 2016-2020 under the \"Diamond Grant\" program.\nThis research was carried out with the equipment purchased thanks to the financial support of the European Regional Development Fund in the framework of the Polish Innovation Economy Operational Program (contract No. POIG.02.01.00-12-023\/08). \n\n\n\n\\section{ Introduction } \n\nKibble-Zurek mechanism (KZM) evolved from the scenario for defect creation in cosmological symmetry-breaking phase transitions \\cite{K-a, *K-b, *K-c}. As the post-Big-Bang Universe cools, causally disconnected regions must choose broken symmetry vacuum independently. Such random choices lead to topologically nontrivial configurations that survive phase ordering as topological defects. In the cosmological setting average size of the causally connected regions (hence, the average density of defects) is set by the Hubble radius at the time of the transition. This early Universe scenario relies on the speed of light and does not apply to the laboratory phase transitions. However, it was the point of departure for the dynamical theory \\cite{Z-a, *Z-b, *Z-c, Z-d} that employs critical exponents of the transition and the quench time to predict the scaling of the resulting density of defects. KZM was successfully tested using numerical simulations \\cite{KZnum-a,KZnum-b,KZnum-c,KZnum-d,KZnum-e,KZnum-f,KZnum-g,KZnum-h,KZnum-i,KZnum-j,KZnum-k,KZnum-l,KZnum-m} and laboratory experiments in condensed matter systems \\cite{KZexp-a,KZexp-b,KZexp-c,KZexp-d,KZexp-e,KZexp-f,KZexp-g,KZexp-gg,KZexp-h,KZexp-i,KZexp-j,KZexp-k,KZexp-l,KZexp-m,KZexp-n,KZexp-o,KZexp-p,KZexp-q,KZexp-r,KZexp-s,KZexp-t,KZexp-u,KZexp-v,KZexp-w,KZexp-x}. More recently, KZM was adapted to quantum phase transitions \\cite{Polkovnikov2005,QKZ1,QKZ2,d2005,d2010-a, d2010-b}. Theoretical developments \\cite{QKZteor-a,QKZteor-b,QKZteor-c,QKZteor-d,QKZteor-e,QKZteor-f,QKZteor-g,QKZteor-h,QKZteor-i,QKZteor-j,QKZteor-k,QKZteor-l,QKZteor-m,QKZteor-n,QKZteor-o,QKZteor-p,QKZteor-q,QKZteor-r,QKZteor-s,QKZteor-t} and experimental tests \\cite{QKZexp-a, QKZexp-b, QKZexp-c, QKZexp-d, QKZexp-e, QKZexp-f, QKZexp-g,deMarco2,Lukin18} of quantum KZM (QKZM) followed. The recent experiment \\cite{Lukin18}, where a quantum Ising chain in the transverse field is emulated using Rydberg atoms, is fully consistent with the predicted scaling \\cite{QKZ2,d2005}. \n\nThe KZM is often presented in its cartoon version where -- due to the critical slowing down \/ closing of the energy gap -- the dynamics of the system literally freezes-out in the neighborhood of the critical point. Today, as the experiments are able not only to count the final number of defects but can also monitor and probe the state of the system during the transition, it is timely to re-investigate the causally limited spreading of correlations during the putative ``freeze-out'' stage of the evolution. \n\n\\begin{figure}[t]\n\\vspace{-0.5cm}\n\\includegraphics[width=0.9\\columnwidth,clip=true]{Fig1.pdf}\n\\vspace{-0.1cm}\n\\caption{  \n{\\bf Adiabatic-impulse-adiabatic view of KZM.} \nLinear ramp crosses the critical point at time $t=0$. \nThe instantaneous transition rate, $\\left|\\dot{\\epsilon}\/\\epsilon\\right| = 1\/|t|$, diverges at the critical point and the relevant energy gap closes like $|\\epsilon|^{z\\nu}$. Consequently, while before $-\\hat t$ the state follows the adiabatic ground state, near the critical point (between $-\\hat t$ and $\\hat t$) its evolution is non-adiabatic.\nThe freeze-out assumes that the state is ``frozen'' at $-\\hat t$ \n-- size of the domains of the nascent phase does not change until $+\\hat t$, where the state starts to ``catch up'' with the Hamiltonian. \nThis version of KZM ignores propagation of the new phase front in the time interval $(- \\hat t, +\\hat t)$. It yields correct scalings, but it does not capture what happens -- for example -- in the paramagnetic-ferromagnetic quantum phase transition in the quantum Ising chain in transverse field \\cite{KZscaling1,Francuzetal}. Nevertheless, it may well be relevant in phase transitions where the conserved order parameter or other causes (localization) impede propagation of phase fronts of the broken symmetry phase.\n}\n\\label{fig:KZcartoon}\n\\end{figure}\n\nIn QKZM a system initially prepared in its ground state is smoothly ramped across a critical point to the other side of the quantum phase transition. A distance from the critical point, measured by a dimensionless parameter $\\epsilon$ controlling a Hamiltonian, can be linearized close to the critical point as\n\\begin{equation}\n\\epsilon(t)=\\frac{t}{\\tau_Q}.  \n\\label{epsilont}\n\\end{equation}\nHere $\\tau_Q$ is a quench time. Initially, far from the critical point, the evolution is adiabatic, and the system follows its adiabatic ground state, see Fig.~\\ref{fig:KZcartoon}. The adiabaticity fails at $-\\hat t$ when the reaction time of the system given by the inverse of the gap becomes slower than the timescale $|\\epsilon\/\\dot \\epsilon| = |t|$ on which the transition is being imposed.\nThe gap closes like $\\Delta\\simeq|\\epsilon|^{z\\nu}$, where $z$ and $\\nu$ are the dynamical and correlation length exponents, respectively. From the equation $|t|\\simeq |t\/\\tau_Q|^{-z\\nu}$ we obtain $\\hat t\\simeq \\tau_Q^{z\\nu\/(1+z\\nu)}$ and the corresponding $\\hat\\epsilon=\\hat t\/\\tau_Q\\simeq\\tau_Q^{1\/(1+z\\nu)}$. In the naive ``freeze-out'' version of the impulse approximation the ground state at $-\\hat\\epsilon$, with a corresponding correlation length\n\\begin{equation}\n\\hat\\xi \\simeq \\tau_Q^{\\nu\/(1+z\\nu)},  \n\\label{hatxi}\n\\end{equation}\nis expected to characterize the state of the system until $+\\hat t$, when the evolution can restart. In this way, $\\hat\\xi$ becomes imprinted on the initial state for the final adiabatic stage of the evolution after $+\\hat t$. Simplistic as it is, the adiabatic-impulse-adiabatic approximation correctly predicts the scaling of the characteristic lengthscale $\\hat\\xi$ and the timescale \n\\begin{equation} \n\\hat t\\simeq \\hat\\xi^z,\n\\label{hatt}\n\\end{equation} \nwith the critical exponents and $\\tau_Q$.\nThey both diverge in the adiabatic limit, $\\tau_Q\\to\\infty$, where they become the unique relevant scales in the KZ scaling ansatz~\\cite{KZscaling1,KZscaling2,Francuzetal}. For instance, a two-point correlation function $C_R(t)$, between two sites separated by a distance $R$, should satisfy\n\\begin{equation} \n\\hat\\xi^{\\Delta} C_R(t) = F\\left(t\/\\hat\\xi^z,R\/\\hat\\xi\\right).\n\\label{CRscaling}\n\\end{equation}\nHere $\\Delta$ is a scaling dimension and $F$ a non-universal scaling function. Eq.~\\eqref{CRscaling} is expected to be accurate in the long-wavelength and low-frequency limit.\nIt is worth to observe here, that the crude adiabatic-impulse-adiabatic approximation is consistent with the scaling hypothesis \\eqref{CRscaling}. However, it implies a particular  (time independent) form of the scaling function $F$.\n\n\\begin{figure}[t]\n\\vspace{-0.5cm}\n\\includegraphics[width=0.9\\columnwidth,clip=true]{Fig2.pdf}\n\\vspace{-0.1cm}\n\\caption{  \n{\\bf Sonic horizon view of KZM.} Initially, the correlation length $\\xi$ follows adiabatically the equilibrium healing length that -- in the adiabatic ground state (black) -- diverges at the critical point. Critical slowing down means that the size of the correlation length will begin to lag behind the values dictated by the ground state of the Hamiltonian at about $- \\hat t$. Pre-transition fluctuations reach size $\\hat \\xi$ at that instant and seed subsequent evolution of the system. The new broken symmetry phase is therefore selected by fluctuations in domains if size $\\hat \\xi$ at $-\\hat t$. Broken symmetry spreads within the impulse time interval of $2 \\hat t$ with the velocity $2\\hat v$ in every direction, enlarging the resulting ``sound cone'' to roughly $ 5 \\hat \\xi$ by $\\hat t$. In the freeze-out approximation (blue), after $-\\hat t$ the correlation length freezes, and remains close to the adiabatic correlation length at $-\\hat t$. Both the freeze-out and the sonic horizon views lead to the same scalings, but they result in different estimates of the pre-factors for domain sizes and defect densities.\n}\n\\label{fig:KZreal}\n\\end{figure}\n\n\nAs emphasized already in the early papers, see Ref.~\\onlinecite{Z-a, Z-b, Z-c, Z-d}, the freeze-out is not the complete story, and often not even a good approximation. A simple ``sonic horizon'' argument appealing to causality that goes beyond the impulse approximation is often more accurate. It is illustrated schematically in Fig.~\\ref{fig:KZreal}. As long as the evolution is adiabatic, the rate of growth of the diverging adiabatic correlation length, $\\xi\\simeq|\\epsilon|^{-\\nu}$, is  \n\\begin{equation} \n\\frac{d\\xi}{dt}=\n\\frac{d\\epsilon}{dt}\\frac{d\\xi}{d\\epsilon}=\n\\frac{1}{\\tau_Q}\n\\frac{\\nu}{|\\epsilon|^{\\nu+1}}. \n\\end{equation} \nThis rate diverges at the critical point. Hence there must be time $-\\hat t$ when it exceeds the speed limit set by twice\n\\begin{equation} \n\\hat v\\simeq\\frac{\\hat\\xi}{\\hat t}\\simeq\\tau_Q^{-\\nu(z-1)\/(1+z\\nu)}.\n\\label{hatv}\n\\end{equation} \nThe scaling of $-\\hat t$ obtained in this way is the same as in Eq.~\\eqref{hatt}. \n\nCausality and the KZ velocity $\\hat v$ are also central for the short-cuts to adiabaticity via inhomogeneous KZM. Therein, the external driving field has a smooth position dependence, gradually taking the system across the critical point---one part after another. Velocity of the driven critical front below $\\hat v$ (which in general depends on the shape of the above position dependence) is expected to pave the way to adiabatic dynamics, both for classical\\cite{ inhomo_classical-a, KZnum-c, inhomo_classical-c, inhomo_classical-d, inhomo_classical-e, inhomo_classical-f} and for quantum\\cite{inhomo_quantum-a, *inhomo_quantum-aa, inhomo_quantum-b, *inhomo_quantum-c, inhomo_quantum-d, inhomo_quantum-e, inhomo_quantum-f, inhomo_quantum-g} systems.\n\nIn the QKZM the speed limit is central to the causal argument. It originates from the dispersion of quasiparticles at the critical point: $\\omega\\simeq k^z$. Their speed for a quasimomentum $k$ is $v=d\\omega\/dk\\simeq k^{z-1}$. Between $-\\hat t$ and $\\hat t$ the quench excites quasiparticles with the magnitude of $k$ up to $\\hat k\\simeq \\hat\\xi^{-1}$. The speed of quasiparticles with the largest excited $k$ is therefore $\\hat v\\simeq\\hat k^{z-1}\\simeq\\hat\\xi^{1-z}\\simeq\\hat\\xi\/\\hat t$. When $z\\geq1$, $\\hat v$ is an upper bound on the velocity of quasiparticles. A quench in a translationally invariant system excites entangled pairs of quasiparticles with opposite quasimomenta: $k$ and $-k$. When moving apart, they are spreading correlations across the system. For $z\\geq1$ the rate of correlation spreading is limited by twice the speed $\\hat v$ of the fastest quasiparticles. \n\nIn the crudest version, neglecting in particular dependence of $\\hat v$ on the distance from the critical point, the argument implies that after $-\\hat t$ the correlation range (i.e., the ``sonic cone'') continues to grow at the rate $2\\hat v$ until $+\\hat t$. By this time, the range increases from the initial $\\hat\\xi$ at $-\\hat t$ to a final $\\hat\\xi+2\\hat v \\times 2\\hat t\\approx 5\\hat\\xi$. The growth is roughly five-fold. Even if the $5$ is just a very rough estimate, it shows how much the actual evolution can differ from the impulse approximation. Nevertheless, this argument confirms the role of $\\hat\\xi$ as the key relevant scale of length. With or without the prefactor of $5$, the final correlation length is proportional to $\\hat\\xi$, i.e., it scales with $\\tau_Q$ in the same way as given by Eq.~\\eqref{hatxi}. \n\nFrom another perspective, in the impulse approximation, the adiabatic ground state $|\\psi\\rangle$ corresponding to $-\\hat t$ freezes out as the state of the system. After $-\\hat t$ the adiabatic ground state departs from the frozen state. The frozen state becomes a superposition over adiabatic eigenbasis $|n\\rangle$: $\\sum_n c_n(t) |n\\rangle$, where $c_n(t)=\\langle n|\\psi\\rangle$. As a first step beyond the impulse approximation, we can include approximate dynamical phases: $\\sum_n c_n(t) e^{-i\\omega_n t}|n\\rangle$, where $\\omega_n$ is the adiabatic eigenfrequency at the critical point. In a (non-interacting) translationally invariant system, the eigenstates consist of pairs of excited quasiparticles, $|k,-k\\rangle$, and the eigenfrequencies are sums of $2\\omega_k$. The dynamical phase factors become scrambled -- and the phases begin to appear random -- when the largest of them, $2\\omega_{\\hat k}t\\propto\\hat k^z t$, becomes comparable to $1$ near $\\hat t$. The dephasing begins when the evolution crosses over from the non-adiabatic KZ stage to the post-KZ adiabatic stage. That is when the phases definitely can no longer be ignored, but even before the cross-over the phase factors $e^{-2i\\omega_k t}$ make the quasiparticle phase fronts propagate and let the quasiparticles spread the correlations across the system.\n\nFor $z=1$, when the dispersion is linear in $k$ and the quasiparticles have a definite speed of sound. This effect was termed the quasiparticle event horizon \\cite{EventHorizon}. In the QKZM context, it was considered in Refs. \\onlinecite{KZscaling1,Francuzetal}---see Fig.~\\ref{fig:Ising} for an example of the\nprototypical 1D quantum Ising model.\n\n\n\n\\begin{figure}[t]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\columnwidth]{fig2a.pdf}\n\\vspace{-0.7cm}\n\\caption{ {\\bf Quantum Ising chain.} \nThe correlation length during the quench in the 1D quantum Ising model, $H=-\\sum_n(1-\\epsilon) \\sigma_n^z + \\sigma^x_n \\sigma^x_{n+1} $, where the dynamical critical exponent $z=1$ and the excited quasiparticles posses definite speed of sound. Compare with Fig.~\\ref{fig:KZreal}.\nData from Ref.~\\onlinecite{Francuzetal}.\n}\n\\label{fig:Ising}\n\\end{figure}\n\n\nIn this paper we go beyond $z=1$ and present two examples with $z>1$: the classical Ising model with Glauber dynamics in Section \\ref{sec:Onsager} and the generalized quantum XY chain in Section \\ref{sec:XY}. They both exhibit an effective event horizon with a speed limit that depends on the quench time $\\tau_Q$.\nThe generic scenario is delimited by two examples where the sonic horizon effect cease to manifest because one of its underlying assumptions is not satisfied. The first one is the random Ising model in Section \\ref{sec:Random}, where localization of excited quasiparticles prevents the spreading of correlations, thus yielding in effect a ``freeze-out''. The other is the extended Ising model with long-range interactions in Section \\ref{sec:LR}, where the dynamical exponent $z$ is less than $1$. The excited quasiparticles with $k\\to0$ have infinite velocity, and the speed $\\hat v$ at the maximal excited $\\hat k$ is not an upper but a lower speed limit. Consequently, there is no sonic horizon effect, and the correlations have a long-range power-law tail that can evolve in time. After $-\\hat t$ the tail begins to lag behind its adiabatic evolution. However, instead of completely freezing out, it continues to grow at a finite rate.\n\n\\begin{figure}[b]\n\\vspace{-0.5cm}\n\\includegraphics[width=\\columnwidth]{fig3.pdf}\n\\vspace{-0.7cm}\n\\caption{ {\\bf Classical Ising in 2D on a square lattice.} \nIn (a),\nenergy per site, $E$, as a function of $\\beta(t)$ for different quench times $\\tau_Q$.\nFor reference, \nthe black line is the equilibrium internal energy $U(\\beta)$. \nWith increasing quench time, the quench curves converge to the equilibrium one.\nIn (b),\nexcitation energy per site, $\\Delta E=E-U$, as a function of scaled time for different quench times $\\tau_Q$.\nHere, both $\\Delta E$ and $t$ are rescaled according to the KZM predictions.\nFor large $\\tau_Q\\gg1$ the scaling makes them collapse in the KZ regime between \n$t\\approx-\\hat t$ and $t\\approx\\hat t$, corresponding to the rescaled times $-1$ and $+1$, respectively.\nAt later times the phase ordering kinetics steps in, which goes beyond the KZ physics.\n}\n\\label{fig:onsagerenergy}\n\\end{figure}\n\\section{ Classical Ising model: ${\\mathbf z=2}$ } \n\\label{sec:Onsager}\n\nWe begin with the classic example of the classical Ising model on a periodic square lattice of size $L\\times L$:\n\\begin{equation}\nH~=~-\\sum_{\\langle j,j'\\rangle} \\sigma^z_j \\sigma^z_{j'} ~.\n\\label{Honsager}\n\\end{equation}\nOn an infinite lattice, the critical inverse temperature would be $\\beta_c=\\ln(1+\\sqrt2)\/2\\simeq 0.4407$.\nThe relevant equilibrium exponents are $\\nu=1$ and $\\eta=1\/4$ \\cite{Schultz1964}.\nWe model relaxation to an external heat bath with the Glauber dynamics: \nMonte Carlo update thermalizes one spin (chosen at random) at a time.\nThe time needed for $L^2$ such one-spin updates sets unit of time.   \nFor such simple relaxation, the dynamical exponent is $z=2$, belonging to the universality class of model-A dynamics~\\cite{hohenberg1977theory}. \nWe performed all our numerical simulations on a $4096\\times 4096$ lattice. This lattice size is $100$ times longer than the longest correlation range encountered in the simulations, hence any finite size effects are eliminated with a wide safety margin. All results were averaged over $50$ repetitions of the quench, each of them starting from a different initial random spin configuration at infinite temperature.\n\n\\begin{figure*}[t]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\textwidth]{fig4.pdf}\n\\vspace{-0.5cm}\n\\caption{  \n{\\bf Classical Ising model in 2D.} The scaled correlator $\\hat\\xi^{\\eta}C_R$ as a function of the scaled distance $R\/\\hat\\xi$\nfor scaled times $t\/\\hat t=-1,0,1$ (left to right) and different quench times $\\tau_Q$.\nFor each scaled time,\nwhen $\\tau_Q\\gg1$ the plots with different $\\tau_Q$ collapse to a single scaling function $F(t\/\\hat t,R\/\\hat\\xi)$\ndemonstrating the KZ scaling (\\ref{CRscaling}) hypothesis for slow enough quenches.\n}\n\\label{fig:corrcollapse}\n\\end{figure*}\n\n\nThe inverse temperature of the heat bath is ramped linearly in time,\n\\begin{equation} \n\\beta(t)=\\beta_c\\left(1+\\frac{t}{\\tau_Q}\\right),\n\\end{equation}\nstarting with random spin configuration at $\\beta(-\\tau_Q)=0$. Figure~\\ref{fig:onsagerenergy}(a) shows the energy $E$ during the ramp as a function of $\\beta$ for different quench times $\\tau_Q$. Generally, the system is more ordered for slower quenches. For slow enough quenches, the KZ picture emerges. The system evolves adiabatically until it begins to go out of equilibrium around $-\\hat t$, where $\\hat t\\simeq\\tau_Q^{2\/3}$ is the KZ timescale.\n\nIn order to see how the excitation energy should depend on the quench time, let us consider the equilibrium internal energy. Near the critical point it is\n\\begin{equation} \nU(\\beta) = - \\left[ 1 + A (\\beta-\\beta_c) \\ln\\left|\\beta-\\beta_c\\right| \\right]\/\\tanh\\beta_c,\n\\end{equation}\nwhere $A\\simeq 1$ is a constant. In the adiabatic-impulse approximation the state becomes effectively frozen near $-\\hat t$ when $\\beta_c-\\hat\\beta\\simeq\\tau_Q^{-1\/3}$. At $\\beta_c$ the energies of the frozen state (i.e. the instantaneous state at $-\\hat t$) and the equilibrium one differ by\n\\bea\n\\Delta E(\\beta_c) &=&\nU(\\hat\\beta)-U(\\beta_c) \\nonumber\\\\\n&=&\nA (\\beta_c-\\hat\\beta) \\ln\\left|\\hat\\beta-\\beta_c\\right|\/\\tanh\\beta_c \\nonumber\\\\\n&\\simeq &\n\\tau_Q^{-1\/3}\\ln\\left(\\tau_Q\/\\tau_0\\right).\n\\end{eqnarray} \nHere $\\tau_0\\simeq1$ is a constant. We can see that, up to a subleading logarithmic correction, the KZ scale of energy is $\\simeq\\tau_Q^{-1\/3}$. Accordingly, in Figure~\\ref{fig:onsagerenergy}(b) we show scaled excitation energy $\\tau_Q^{1\/3}(E-U)$ in function of scaled time $t\/\\hat t$. For $\\tau_Q\\gg1$ the plots for different $\\tau_Q$ collapse in the KZ regime: $-1<t\/\\hat t<1$. There is no collapse at later times when phase ordering kinetics \\cite{POK} brings in new physics beyond the KZ mechanism.\nHaving seen how the excitations build up, we can have a closer look at the ferromagnetic correlator\n\\begin{equation} \nC_R(t) =\n\\langle \\sigma^z_i \\sigma^z_{j} \\rangle -\n\\langle \\sigma^z_i \\rangle \\langle \\sigma^z_{j} \\rangle.\n\\end{equation}\n\n\\begin{figure}[b]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\columnwidth]{fig5.pdf}\n\\vspace{-0.3cm}\n\\caption{  {\\bf Classical Ising model in 2D.}\nIn (a),\nthe scaled correlator $\\hat\\xi^{\\eta}C_R$ as a function of the scaled distance $R\/\\hat\\xi$ for $\\tau_Q=2^{15}$ at different scaled times $t\/\\hat t\\in[-1,1]$. The dashed line marks a cut-off $c$ (here $c=0.1$) whose crossing point is a working definition of a scaled correlation range $\\xi_c\/\\hat\\xi$.\nIn (b),\nthe scaled correlation range $\\xi_c\/\\hat\\xi$ as a function of the scaled time $t\/\\hat t$ for the cut-off $c=0.1$.\nThe best linear fit yields a slope $0.72$ as an estimate of the scaled velocity.\nIn (c),\nthe scaled velocity (slope) as a function of the cut-off. A linear extrapolation to zero cut-off yields $0.93$. This number is an estimate for (scaled) velocity at which the furthest correlation tail is spreading.\n}\n\\label{fig:spread}\n\\end{figure}\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\vspace{-0.1cm}\n\\includegraphics[width=\\textwidth]{fig6.pdf}\n\\vspace{-0.7cm}\n\\end{center}\n\\caption{{\\bf Extended quantum XY chain.} Scaled ferromagnetic correlations $C^{xx}_R$ as a function of scaled distance $R\/\\hat{\\xi}$ for the extended XY model in (\\ref{XYHsigma}) plotted for different quench times $\\tau_Q$ and at different scaled times $t\/\\hat{t}=-1,0,1$. For large enough $\\tau_Q$ plots collapse towards a universal scaling function $F(t\/\\hat{t},R\/\\hat{\\xi})$ demonstrating the KZ scaling hypothesis (\\ref{CRscaling}).\n}\n\\label{fig:XYcollapse}\n\\end{figure*}\n\n\n\nHere, the sites $i$ and $j$ are separated by a distance $R$ along one of the axes and the average is taken in the state at time $t$. Figure~\\ref{fig:corrcollapse} shows the scaled correlator $\\hat\\xi^\\eta C_R$ in function of the scaled distance $R\/\\hat\\xi$ for three scaled times $t\/\\hat t=-1,0,1$. Here $\\hat\\xi=\\tau_Q^{1\/3}$ is the scale of length in Eq.~(\\ref{hatxi}) and $\\hat t=\\hat\\xi^z=\\tau_Q^{1\/3}$ is the scale of time in Eq.~(\\ref{hatt}). In accordance with the scaling hypothesis (\\ref{CRscaling}), for each scaled time the plots with different $\\tau_Q$ collapse to a common scaling function when $\\tau_Q$ is large enough. The plot for the longest $\\tau_Q=2^{15}$ is practically equal to the scaling function $F(t\/\\hat t,R\/\\hat\\xi)$.\n\n\nComparing the three panels in Figure~\\ref{fig:corrcollapse} we can see that (at odds with the impulse approximation) the range of correlations increases several times between $-\\hat t$ and $\\hat t$. This observation is further corroborated by Fig.~\\ref{fig:spread}(a) where we collect scaled plots for $\\tau_Q=2^{15}$ at different scaled times. Using $\\hat\\xi$ as a natural unit of length, this figure shows how the correlation spreads in the KZ regime between $-\\hat t$ and $+\\hat t$. \n\nFigure~\\ref{fig:spread}(b) shows how a scaled range of the correlator in Fig.~\\ref{fig:spread}(a) depends on the scaled time $s=t\/\\hat t$. Here, the range is defined as the distance at which the scaled correlator falls below a threshold---set here at $0.1$. The nearly linear time dependence is fitted with a line whose slope estimates the scaled velocity of correlation spreading. We repeated the same procedure for thresholds down to $0.02$ (below $0.02$ the tail of the correlator becomes too noisy for an unambiguous estimation of the correlation range). The estimated slopes\/velocities are collected in Fig.~\\ref{fig:spread}(c). The velocity increases rather slowly with a decreasing threshold. In the limit of zero threshold -- probing the longest correlation tail -- it extrapolates to $0.93$. We can conclude that the speed limit in this model is $\\hat v$, which allows the size of the sonic horizon to expand with\n\\begin{equation} \n2\\hat v = \n0.93~ \\frac{\\hat\\xi}{\\hat t} = \n0.93~ \\tau_Q^{-1\/2}.\n\\end{equation}\nIn natural units, in accordance with KZ prediction in Eq.~\\eqref{hatv}, the speed limit becomes slower for slower quenches.\n\\section{ Extended quantum XY chain: ${\\mathbf z=3}$ } \n\\label{sec:XY}\n\n\\begin{figure*}[t]\n\\includegraphics[width=\\textwidth]{fig7.pdf}\n\\caption{{\\bf Extended quantum XY chain.}\nIn (a), \nfunction (\\ref{fitting}) is fitted (dashed lines) to the scaled ferromagnetic correlators $\\hat{\\xi}^{1\/4}C^{xx}_R$ for different scaled times during the non-adiabatic stage of the evolution: $t\/\\hat{t}\\in\\left[-2,2\\right]$.\nHere $\\tau_Q=2^{20}$.\nIn (b),\nthe fits provide the scaled correlation length $\\xi\/\\hat{\\xi}$ plotted against the scaled time $t\/\\hat{t}$. The slope of this plot is an estimate of the scaled velocity $v\/\\hat{v}=0.84$ at which the correlations are spreading. \nIn (c),\na more accurate approximation of the velocity is obtained by taking a derivative of the plot (b) with respect to the scaled time, which yields the scaled velocity $v\/\\hat{v}$ as a function of the scaled time.\nMaximal velocity is $v_{\\rm max}\/\\hat v=0.84$ for $\\tau_Q=2^{20}$. \nIn (d),\nthe maximal velocity is shown as a function of $\\tau_Q$.\nThe red line is a fit, which, for large $\\tau_Q$, saturates at $v_{max}\/\\hat v= 0.868$. \n}\n\\label{fig:XYspeed}\n\\end{figure*}\n\n\nWe step into the quantum realm \nwith the extended quantum XY periodic spin chain,\n\\bea\nH&=&-\\sum_{n=1}^N\n\\left[\n(1-\\epsilon) \\sigma^z_n +\nJ^{xx}\\sigma^x_n\\sigma^x_{n+1}+ \nJ^{yy}\\sigma^y_n\\sigma^y_{n+1}+\n\\right. \\nonumber \\\\\n&&\n\\left.\nJ^{xzx}\\sigma^x_n \\sigma^z_{n+1} \\sigma^x_{n+2}+ \nJ^{yzy}\\sigma^y_n \\sigma^z_{n+1} \\sigma^y_{n+2}\n\\right],\n\\label{XYHsigma}\n\\end{eqnarray} \na generalization of the quantum XY chain. It combines the desired $z>1$ with exact solvability.\nHere we consider an anisotropic model:\n\\bea\n&&\\left(J^{xx},J^{yy},J^{xzx},J^{yzy}\\right)=\\nonumber\\\\\n&&\n\\left(a\\frac{1+\\gamma}{2},a\\frac{1-\\gamma}{2},\n      b\\frac{1+\\delta}{2},b\\frac{1-\\delta}{2}\\right)\n\\end{eqnarray}\nwith the external magnetic field parametrized by $\\epsilon\\in[-1,1]$. The parameter is driven linearly (\\ref{epsilont}) from an initial $\\epsilon=-1$ in the paramagnetic phase, across the critical point at $\\epsilon=0$, to a final $\\epsilon=1$ in the ferromagnetic phase. After the Jordan-Wigner and Fourier transformations (\\ref{JordanWigner},\\ref{Fourier}), the Hamiltonian becomes:\n\\bea\nH &=\\sum_{k>0} &\nA_k(\\epsilon) \\left(c_k^\\dag c_k+c_{-k}^\\dag c_{-k}\\right) \\nonumber \\\\ \n&&\n+B_k \\left(c_kc_{-k}+c_{-k}^\\dag c_k^\\dag\\right),\n\\label{XYHk}\n\\end{eqnarray}\nwhere,\n\\bea \nA_k(\\epsilon) &=& 1-\\epsilon-a\\cos k-b\\cos 2k,  \\nonumber\\\\  \nB_k           &=& a\\gamma\\sin k+b\\delta\\sin 2k.\n\\end{eqnarray}\nIt is diagonalized by eigenmodes of the stationary Bogoliubov-de Gennes equations:\n\\bea\n\\omega_k\n\\left(\n\\begin{array}{c}\nU_k \\\\\nV_k\n\\end{array}\n\\right)&=&\n2\\left[\\sigma^z A_k(\\epsilon)+\\sigma^x B_k \\right] \n\\left(\n\\begin{array}{c}\nU_k \\\\\nV_k\n\\end{array}\n\\right),\n\\label{XYBdGk}\n\\end{eqnarray}\nwith eigenfrequencies\n$\n\\omega_k=2\\sqrt{A_k^2(\\epsilon)+B_k^2}.\n$\nHere, we set $a = 4\/3, b=-1\/3, \\gamma =1\/2, \\delta=1$. For the critical $\\epsilon=0$, we obtain a cubic dispersion relation\n\\begin{equation} \n\\omega_k\\simeq |k|^3,\n\\label{XYomegakz}\n\\end{equation} \nexpanding $\\omega_k$ around $k=0$, and the dynamical exponent $z=3$. On the other hand, setting $k=0$ and expanding in small $\\epsilon$, we obtain the gap opening as $\\omega_0\\simeq|\\epsilon|^1$. Hence $z\\nu=1$ and $\\nu=1\/3$. The dominant static ferromagnetic correlations have oscillatory tails:\n\\bea\nC_R     &=&\\langle\\sigma_i^x\\sigma_{i+R}^x\\rangle-\n           \\langle\\sigma_i^x\\rangle\\langle\\sigma_{i+R}^x\\rangle \\nonumber \\\\\n        &\\sim & R^{-\\eta} e^{-R\/\\xi}\\cos(b~R\/\\xi+c).\n\\end{eqnarray} \nHere $\\xi\\sim|\\epsilon|^{-\\nu}$ is a static correlation length and $\\eta=1\/4$. Given the exponents $z$ and $\\nu$, we can define the dynamical length (\\ref{hatxi}) and time (\\ref{hatt}) scales: $\\hat\\xi=\\tau_Q^{1\/6}$ and $\\hat t=\\tau_Q^{1\/2}$, respectively. The fastest excited quasiparticles have velocity $\\hat v\\simeq\\tau_Q^{-1\/3}$, see Eq.~\\eqref{hatv}. \n\nThe time-dependent quench is solved in appendix \\ref{XYtime} in a standard way \\cite{d2005} by mapping to the Landau-Zener problem, see appendix \\ref{LZ}. The KZ scaling hypothesis (\\ref{CRscaling}) is demonstrated in Fig.~\\ref{fig:XYcollapse} by collapse of the plots for different $\\tau_Q$. A perfect collapse requires very large $\\tau_Q\\simeq2^{20}$, as explained by Eq.~\\eqref{rescuv}. The oscillatory behaviour of $C_R$ in the original paramagnetic phase survives through the transition. In order to estimate the range of the scaled correlators, in Fig.~\\ref{fig:XYcollapse} we fit their tails with oscillatory functions of the form:\n\\begin{equation} \n\\hat{\\xi}^{1\/4} C_R(t)=\na~(R\/\\hat{\\xi})^{-1\/4}~\\mathrm{e}^{-\\frac{R\/\\hat{\\xi}}{\\xi\/\\hat{\\xi}}}\\cos(b~R\/\\hat{\\xi}+c),\n\\label{fitting}\n\\end{equation}\nwhere $a,b,c$ and $\\xi\/\\hat\\xi$ are the fitting parameters. We are interested how fast the scaled correlation length $\\xi\/\\hat\\xi$ grows with the scaled time $t\/\\hat t$. In order to reveal the universal behavior undisturbed by any short-range effects, we perform the fit in the range of scaled distances $R\/\\hat\\xi>2.5$. Ferromagnetic correlations for different scaled times $t\/\\hat{t}$, together with the fits, are shown in Fig.~\\ref{fig:XYspeed}(a) (for $\\tau_Q=2^{20}$). The correlation length $\\xi\/\\hat{\\xi}$ as a function of scaled time is shown in Fig.~\\ref{fig:XYspeed}(b). Its slope, equal to $0.838$, provides an estimate of the scaled velocity at which the correlations are spreading. For completeness, in Fig.~\\ref{fig:XYspeed}(c) we plot the derivative of the plot in Fig. \\ref{fig:XYspeed}(b) with respect to the scaled time. Its maximal value is shown in Fig.~\\ref{fig:XYspeed}(d) as a function of $\\tau_Q\\in [2^{10},2^{20}]$. For large $\\tau_Q$ it extrapolates to\n\\begin{equation} \n2 \\hat v = 0.868~ \\frac{\\hat{\\xi}}{\\hat{t}} = 0.868~ \\tau_Q^{-1\/3}.\n\\end{equation}\nAs in the classical 2D Ising model, where also $z>1$, the correlation spreading becomes slower for slower quenches.\n\n\\section{ Random Ising model } \n\\label{sec:Random}\n\nIn order to break the translational invariance and impede the propagation of excited quasiparticles that could spread correlations, next we consider the random Ising model (RIM) in one dimension defined by the Hamiltonian\n\\bea\nH=\n-\\sum_{n=1}^N\n\\left[ (h_{n} - \\epsilon)\\sigma^z_n \\ +\n J_{n}\\sigma^x_n\\sigma^x_{n+1}\n\\right].\n\\label{Hamil_RIM}\n\\end{eqnarray} \nHere both transverse fields $h_n$'s and nearest-neighbor couplings $J_n$'s are randomly selected from a uniform distribution between $0$ and $1$. The quantum critical point at $\\epsilon=0$ is separating the paramagnetic phase ($\\epsilon<0$) from the ferromagnetic one ($\\epsilon>0$). The critical point in this model is surrounded by Griffiths region \\cite{Griffiths-a, Griffiths-b, Griffiths-c}, where the presence of the so-called rare regions primarily manifests in two features: the activated dynamical scaling (the dynamical exponent $z\\rightarrow\\infty$) at the quantum critical point and the existence of singular regions where the linear susceptibility diverges even away from the critical point. The locally ordered rare regions act as giant spins that flip as a whole and are responsible for the exponentially slow dynamics near the critical point. These features are encapsulated in a dynamical exponent that diverges at the critical point as $z \\sim|\\epsilon|^{-1}$, see Ref. \\onlinecite{Fisher1-a, Fisher1-b} which also shows that the correlation length exponent is $\\nu = 2$. The average correlation function at criticality is a power law\\cite{McCoyWu}:\n\\begin{equation} \nC^{xx}_R=\\langle \\sigma_{i}^{x}\\sigma_{i+R}^{x}\\rangle \\sim R^{-\\eta},\n\\end{equation}\nwhere $\\eta = \\frac{3-\\sqrt{5}}{2}\\approx 0.38$. \n\nWe consider ramping the parameter $\\epsilon$ linearly as a function of time (\\ref{epsilont}) driving the Hamiltonian from the initial paramagnetic ground state, across the critical point at $t=0$, into the ordered phase. Ref. \\onlinecite{dziarmagaRIM-a, dziarmagaRIM-b, dziarmagaRIM-c} showed that the KZ correlation length $\\hat \\xi$ of the model varies logarithmically with the quench rate $\\tau_Q$:\n\\bea\n\\centering\n\\hat{\\xi} = \\ln^2{(\\tau_Q\/a)},\n\\label{hatxiRIM}\n\\end{eqnarray}\nwhen $\\ln{(\\tau_Q\/a)}\\gg 1$. Here $a\\simeq1$ is a non-universal constant. We can see, that the dependence on $\\tau_Q$ is very weak compared to any of the usual KZ power-law scalings (\\ref{hatxi}). Taking into account that the correlation-length exponent near the critical point is $\\nu = 2$, we get an estimate of the characteristic timescale,\n\\bea\n\\centering\n\\hat{t} = \\frac{\\tau_{Q}}{\\ln{(\\tau_{Q}\/a)}},\n\\label{hattRIM}\n\\end{eqnarray}\n for $\\ln{(\\tau_Q\/a)} \\gg 1$. \n\n\n\\begin{figure}[t]\n\\includegraphics[width=\\columnwidth]{fig8.pdf}\n\\caption{{\\bf Random Ising model.} Average scaled density of excitations $\\hat\\xi d_{\\rm ex}$ during the quench as a function of scaled time $t\/\\hat t$ for different quench times $\\tau_Q$. Here, we use Eqs.~(\\ref{hatxiRIM},\\ref{hattRIM}) with $a = 0.118$. This $a$ was tuned to obtain the best possible collapse at $t\/\\hat t=0$ but the plots collapse well during the whole quench. The KZ diabatic stage (shaded in green) extends roughly from $-0.25t\/\\hat t$ to $0.25t\/\\hat t$. This is where the excitation grows before it saturates in the last adiabatic stage. Averaging was done over 30 random realizations for a lattice of size $N = 128$.\n}\n\\label{FigDoERIM}\n\\end{figure}\n\\begin{figure*}[t]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\textwidth,clip=true]{fig9.pdf}\n\\vspace{-0.1cm}\n\\caption{ {\\bf Random Ising model.} Average scaled ferromagnetic correlations $\\hat\\xi^\\eta C^{xx}_R$ as a function of scaled distance $R\/\\hat{\\xi}$ for the random Ising model (\\ref{Hamil_RIM}) plotted for different quench times $\\tau_Q$ and at different scaled times $t\/\\hat{t}=-0.25,0,0.25$. For large enough $\\tau_Q$ plots collapse towards a universal scaling function $F(t\/\\hat{t},R\/\\hat{\\xi})$ demonstrating the KZ scaling hypothesis\n(\\ref{CRscaling}).\n}\n\\label{fig:RIMcollapse}\n\\end{figure*}\n\nIn order to solve the dynamics, we map the Hamiltonian by the Jordan-Wigner transformation (\\ref{JordanWigner}) to a quadratic spinless free-fermionic model\n\\bea\nH = \\sum_{n}(h_{n}-\\epsilon)c_{n}^\\dag c_{n} -J_{n}c_{n}^\\dag c_{n+1} - J_{n}c_{n+1} c_{n} + {\\rm h.c}.\n\\label{JW_RIM}\n\\end{eqnarray}\nFollowing the convention of this article, we confine ourselves to the subspace of even parity of $c$-quasiparticles (anti-periodic boundary conditions, i.e., $c_{N+1} = -c_1$) and in this subspace, we diagonalize the Hamiltonian by a Bogoliubov transformation:\n\\bea\n\\centering\nc_{n} = \\sum\\limits_{m=1}^{N}(U_{nm}\\gamma_{m} + V_{nm}^* \\gamma_{m}^\\dag)\n\\label{bogoliubov_RIM}\n\\end{eqnarray}\nThe index $m$ labels the (Bogoliubov) eigenmodes of the stationary Bogoliubov-de Gennes equation\n\\bea\n\\centering\n\\omega_{m}U_{n,m}^{\\pm} = 2h_{n}U_{n,m}^{\\mp} - 2j_{n-1}U_{n-1,m}^{\\mp},\n\\label{stationary_B-DeG}\n\\end{eqnarray}\nwhere $\\omega_{m}>0$, $U_{nm}^{\\pm} = U_{nm}\\pm V_{nm}$ and anti-periodic boundary conditions $(U_{N+1,m}^{\\pm} = -U_{1,m}^{\\pm},U_{0,m}^{\\pm}=-U_{N,m}^{\\pm})$ is implemented. The eigenstates ($U_{nm},V_{nm}$), with positive energy, $\\omega_{m}>0$ normalized so that $\\sum_{n}(|U_{nm}|^2 + |V_{nm}|^2) = 1$, define quasiparticle operators $\\gamma_{m}=U_{nm}^{*}c_{n}+V_{nm}c_{n}^{\\dag}$. We have corresponding negative energy components of the eigenstates labelled ($U_{nm}^{neg},V_{nm}^{neg}$) with energy $-\\omega_{m}$, which defines a quasiparticle operator $\\gamma_{m}^{neg}=(U_{nm}^{neg})^{*}c_{n}+V_{nm}^{neg}c_{n}^{\\dag}$. The Bogoliubov transformation renders the Hamiltonian to be $H = \\sum_{m = 1}^{N}\\omega_{m}(\\gamma_{m}^{\\dag}\\gamma_{m} - \\frac{1}{2})$. In the even parity subspace only states with even number of quasi-particles belong to the spectrum of $H$.\n\nIn order to find whether the RIM fits into our narrative of the uniform scaled velocity of the correlation spreading by quasiparticles,\nwe proceed with the numerical simulation of the quench. \nWe prepare the initial state of the system deep in the paramagnetic phase in the ground state, i.e., in the Bogoliubov vacuum state for quasiparticles at an initial time $t_{0}$ where $\\epsilon(t_{0}) = 5$. As we tune the parameter $\\epsilon(t)$ towards $0$, see Eq.~\\eqref{epsilont}, the state of the systems departs from its adiabatic ground state and gets excited due to closing of the energy gap near the critical point. We work in the Heisenberg picture where we assume that the excited state is a Bogoliubov vacuum while the time-dependence is ascribed to a set of time-dependent quasi-particle operators\n\\bea\n\\centering\n\\gamma_{m}(t)=u_{nm}^{*}(t)c_{n}+v_{nm}^{*}(t)c_{n}^{\\dag}\n\\end{eqnarray}\nThe Bogoliubov modes $u_{nm}$ and $v_{nm}$ solve the time-dependent Bogoliubov-de Gennes  equations:\n\\bea\n\\centering\ni\\frac{du_{nm}^{\\pm}}{dt} = 2(h_{n}-\\epsilon(t))u_{nm}^{\\mp}-2J_{n}u_{n\\mp1,m}^{\\mp}\n\\label{t_B-deG}\n\\end{eqnarray}\nWe integrate equation (\\ref{t_B-deG}) numerically using the 2nd order Suzuki-Trotter method, see Appendix \\ref{tB-deG_RIM}. The density of excited Bogoliubov quasiparticles $d_{ex}(t)$ can be calculated at each time step by projecting the time-dependent Bogoliubov modes ($u_{nm}(t),v_{nm}(t)$) onto the corresponding instantaneous static negative Bogoliubov modes ($U_{nm}^{\\text{neg}},V_{nm}^{\\text{neg}}$):\n\\bea\n\\centering\nd_{ex}(t) = \\frac{1}{N} \\sum_{s}^N\\sum_{p}^N \\left|\\langle U_{p}^{\\text{neg}},V_{p}^{\\text{neg}}|u_{s}(t),v_{s}(t)\\rangle\\right|^2.\n\\label{doe_RIM}\n\\end{eqnarray}\n\nWhile in the initial adiabatic stage, the system remains in the instantaneous ground state and hence the density of quasiparticle excitations is $0$. On tuning the field $\\epsilon(t)$ towards its critical value at $t\/\\hat{t} = 0$, the system gets excited and consequently $d_{ex}$ starts to grow. As we keep increasing the field $\\epsilon(t)$, $d_{ex}$ saturates as the system ends its non-adiabatic journey across the critical point. In Fig.~\\ref{FigDoERIM} we see that the scaled plots (averaged over disorder) for different $\\tau_Q$ collapse in accordance with the dynamical scaling hypothesis (\\ref{CRscaling}). \n\n\\begin{figure}[b]\n\\includegraphics[width=\\columnwidth]{fig10.pdf}\n\\caption{   {\\bf Random Ising model.}\nIn (a),\nthe scaled correlator $\\hat\\xi^{\\eta}C_R$ as a function of the scaled distance $R\/\\hat\\xi$ for $\\tau_Q=2^{11}$\nat different scaled times $t\/\\hat t$ in $[-0.25,0.25]$.\nThe dashed line marks a cut-off (here $1$) whose crossing point \nis a working definition of a scaled correlation range $\\xi_c\/\\hat\\xi$.\nIn (b),\nthe scaled correlation range $\\xi_c\/\\hat\\xi$ as a function of the scaled time $t\/\\hat{t}$ for the cut-off $c=0.5$. The best linear fit yields a slope of $0.113$ as an estimate of the scaled velocity.\nIn (c),\nthe scaled velocity (slope) as a function of the cut-off, with the error bars indicating fitting errors.\n}\n\\label{fig:RIMspread}\n\\end{figure}\n\nThe hypothesis for correlations in Eq.~\\eqref{CRscaling} is verified by a similar collapse of the plots in Fig.~\\ref{fig:RIMcollapse}. We consider large $\\tau_Q$ as the logarithmic KZ scaling laws are only valid in that regime. The correlation function is averaged over $100$ instances of disorder and lattice translations on a lattice of $N=256$. This lattice is a few times longer than the longest correlation range. The details of calculating the correlation functions are given in Appendix \\ref{appcorrRIM}.\n\nA close look at Fig.~\\ref{fig:RIMcollapse} reveals that the scaled dynamical correlation function grows with scaled time. In order to appreciate the growth, we collect the scaled correlators for different scaled times in Fig.~\\ref{fig:RIMspread}(a). In order to estimate the scaled correlation range $\\xi_c\/\\hat{\\xi}$ as a function of scaled time, we choose a cut-off $c$ for the scaled correlation and calculate scaled distances corresponding to the cut-off by spline interpolation. Fig.~\\ref{fig:RIMspread}(b) shows the scaled range in function of the scaled time for the cut-off (selected equal to $0.5$) in Fig.~\\ref{fig:RIMspread}(a).\nNear the critical point, we approximate it with linear dependence where the slope is $0.113\\hat\\xi\/\\hat t$. We repeat the same procedure for other cut-offs. We present the enumerated slopes for various cut-offs in Fig.~\\ref{fig:RIMspread}(c), with the error bars indicating the fitting error. Finally, we estimate the sonic horizon expansion speed in this model with the maximal observed value\n\\bea\n\\centering\n2\\hat{v} = 0.12 \\frac{\\hat{\\xi}}{\\hat{t}} = 0.12 \\frac{\\ln^3(\\tau_Q\/a)}{\\tau_Q}\n\\end{eqnarray}\nCompared to the previous models, with the prefactor $0.12$, the speed limit is significantly below expectations in this case. The entangled Bogoliubov quasiparticles get excited but, due to their localization by disorder, they do not propagate to spread correlations appreciably. In the spin language, the locally ordered regions are excited, but they essentially stay where they are. One may even argue that in this case, the adiabatic-impulse-adiabatic freeze-out approximation captures the essential physics.\n\n\n\\section{Long-range extended Ising model: $0<z<1$ } \n\\label{sec:LR}\n\n\nThe long-range extended Ising model is a further generalization of the extended XY model that we have discussed in Sec. \\ref{sec:XY}. Now the Hamiltonian of the system includes not only the nearest-neighbor and next-nearest-neighbor interaction, but all possible long-range cluster terms: \n\\bea\nH=\n-\\sum_{n=1}^N\n\\left[\n (1-\\epsilon) \\sigma^z_n +\n\\sum_{r=1}^{N-1} \nJ_r\\sigma^x_n \\sigma^x_{n+r} \\prod_{i=n+1}^{n+r-1} \\sigma^z_i\n\\right].\n\\label{Hamil_LR}\n\\end{eqnarray} \nHere \n\\begin{equation}\nJ_r ~=~ \\frac{1}{\\zeta(\\alpha)} ~ \\frac{1}{r^\\alpha},\n\\end{equation}\nand the normalization by the Riemann zeta function, possible for $\\alpha>1$, is such that $\\sum_r J_r=1$. The case of $\\alpha>2$ is not the most interesting here because the model behaves effectively like the short-range one \\cite{LRKitaev1,LRKitaev2,LR1,LR2,LRBosonic}. Similarly, when $0\\le\\alpha\\le1$ we would need to restrict ourselves to a finite system because the thermodynamic limit does not exist in this case, and the model behaves effectively like the Lipkin-Meshkov-Glick model \\cite{LMG} with infinite-range interactions. Therefore, we focus here on the intermediate $1<\\alpha<2$, where a cross-over between the short and infinite-ranges happens. In this regime, we consider the linear quench (\\ref{epsilont}) driving the system from the initial paramagnetic phase at $t=-\\infty$, across the critical point at $t=0$, to the final ferromagnetic phase. \n\n\\begin{figure}[b]\n\\includegraphics[width=\\columnwidth]{fig11.pdf}\n\\caption{  {\\bf Long-range extended Ising model.}\nIn (a),\nthe static correlation function $C_R$ at the critical $\\epsilon=0$ for $\\alpha=3\/2$. \nThe solid line is a linear fit with a slope $-1.002\\approx -1 $.\nThe same panel shows the same function far from the critical point at $\\epsilon=-2$. \nThe solid line is a linear fit with a slope $-1.45 \\approx -(3 -\\alpha) $.  \nIn (b),\nthe scaled static correlation function $\\xi C_R$ as a function of scaled distance $R\/\\xi$.\nIn accordance with the static scaling hypothesis (\\ref{staticSH}),\nthe plots for different $\\epsilon$ collapse to a unique scaling function.\nWe can see a crossover from the critical $(R\/\\xi)^{-1}$ to the off-critical $(R\/\\xi)^{-3\/2}$\nnear $R\/\\xi=1$. The straight lines are linear fits with slopes $-1.085\\approx-1$ and $-1.42\\approx -(3-\\alpha)$ for smaller and larger $R\/\\xi$, respectively. \n}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\includegraphics[width=\\textwidth]{fig12.pdf}\n\\caption{ {\\bf Long-range extended Ising model.}\nThe scaled dynamical correlation function $\\hat\\xi C_R$ for different quench times $\\tau_Q$ at different scaled times, $s={t}\/{\\hat{t}}$, during the quench.\nHere $\\alpha=3\/2$.\nFrom left to right: $s=-1,0,0.35,0.4$. \nIn (a) at $s=-1$,\nthe evolution is still adiabatic and the plots for different $\\tau_Q$ collapse according to the static scaling hypothesis which is equivalent to the KZ hypothesis in this early adiabatic regime.\nIn (b) at $s=0$,\nwhen crossing the critical point the evolution is diabatic. Different $\\tau_Q$ collapse to a common scaling function $F(s,R\/\\hat\\xi)$ in accordance with the KZ scaling hypothesis (\\ref{CRscaling}).\nIn (c) at $s=0.35$,\nthe correlation function begins to bend down at distances between $R\/\\hat\\xi=10$ and $100$.\nDifferent $\\tau_Q$ collapse, as the evolution is still in the KZ regime.\nFinally in (d) at $s=0.4$,\nthe collapse fails due to the phase ordering after the system left the KZ regime and entered the adiabatic one. At the distances between $R\/\\hat\\xi=10$ and $100$ the correlations become negative for longer $\\tau_Q$.\n}\n\\label{fig:corrp0m}\n\\end{figure*}\n\n\nThanks to the string operator in the long-range interaction terms, the model can be mapped to a quadratic free fermion model and solved analytically. After the Jordan-Wigner transformation (\\ref{JordanWigner}) the Hamiltonian (\\ref{Hamil_LR}) becomes:\n\\bea\nH &=& -\\sum_n \\left(1-\\epsilon\\right)\\left(c_nc_n^\\dag-c_n^\\dag c_n\\right) \\nonumber \\\\\n&&- \\sum_{n,r} J_r\\left( c_n^\\dag c_{n+r} + c_n^\\dag c_{n+r}^\\dag + {\\rm h.c.}  \\right),\n\\label{HcLR}\n\\end{eqnarray}\nfor the anti-periodic boundary conditions. This representation is also known as the long-range Kitaev model \\cite{LRKitaev1, LRKitaev2}, where the hopping and pairing terms are of equal strength. After Fourier transformation (\\ref{Fourier}),\n\\bea\nH &=&\n-2\\sum_{k>0} \n(1-\\epsilon -  \\Re(\\tilde{J_k})) \\left(c_k^\\dag c_k+c_{-k}^\\dag c_{-k}\\right) \\nonumber \\\\\n&&+  ~\\Im(\\tilde{J_k})\\left(c_{k}^\\dag c_{-k}^\\dag + c_{-k}c_{k}\\right),\n\\label{HkLR}\n\\end{eqnarray}\nwhere  $\\Re(\\tilde{J_k})$ and  $\\Im(\\tilde{J_k})$ are, respectively, real and imaginary parts of the Fourier transform $\\tilde{J_k}=\\sum_r J_r e^{ikr}$. We have $\\tilde{J_k}=\\frac{{\\rm Li}_{\\alpha}(e^{ik})}{\\zeta(\\alpha)}$, where ${\\rm Li}$ is the polylogarithm function: ${\\rm Li}_\\alpha(x)=\\sum_{n=1}^\\infty\\frac{x^n}{n^\\alpha}$. \n\n\n\nWe can now find the stationary Bogoliubov-de Gennes equations (\\ref{XYBdGk}) with\n$\nA_k(\\epsilon)=1-\\epsilon-\\Re(\\tilde{J_k}),~~ B_k=\\Im(\\tilde{J_k})\n$\nand eigenfrequencies\n$\n\\omega_k=2\\sqrt{A_k^2(\\epsilon)+B_k^2}.\n$\nWe have a critical point at $\\epsilon=0$ where the gap closes for $k=0$. Another critical point, \nnot to be considered here, is at $\\epsilon=2(1-2^{-\\alpha})$ and $k=\\pi$. The dispersion relation at the critical $\\epsilon=0$ is\n\\bea\n\\omega_k \\simeq |k|^{\\alpha-1},\n\\end{eqnarray}\nhence the dynamical exponent is $z=\\alpha-1\\in(0,1)$. On the other hand, for small $\\epsilon$, the gap at $k=0$ closes as\n$ \n\\omega_0 = 2|\\epsilon|,\n$\nhence $z\\nu=1$. \n\n\n\nIn a short range model for large $R$ the correlation function decays exponentially with $R$ when the system is away from the critical point but this does not need to hold for long-range interactions. Indeed, our system has a power law scaling even far away from the critical point \\cite{LRKitaev1,LRKitaev2}:\n\\bea \nC_R &=&\n\\langle \\sigma_i^x \\sigma_{i+1}^z \\ldots \\sigma_{i+R-1}^z \\sigma_{i+R}^x\\rangle\n\\sim 1\/R^{3-\\alpha},\n\\label{Corr}\n\\end{eqnarray}\ncompare Fig.~\\ref{fig2}(a). On the other hand, as discussed in more detail in Ref.~\\onlinecite{natun}, at the critical point we expect a critical power law $C_R \\sim 1\/r^{\\eta}$ with $\\eta<\\alpha$. Indeed, in Fig.~\\ref{fig2}(a) we find that $\\eta=1$ for $\\alpha=3\/2$. Similarly as for short-range interactions, for a small $\\epsilon$ we expect a cross-over between the two power laws when $R$ is close to $\\xi\\sim\\epsilon^{-\\nu}$. Note that here $\\xi$ is not the usual exponential correlation length, even though it scales with $\\epsilon$ in the characteristic way. The crossover can be verified with a static scaling hypothesis:\n\\bea \n\\xi^\\eta ~C_R = F_{\\rm st}\\left( R\/\\xi \\right).\n\\label{staticSH}\n\\end{eqnarray} \nWith $\\xi=\\epsilon^{-\\nu}$ the plots of the scaled correlator $\\xi^\\eta C_R$ as a function of the scaled distance $R\/\\xi$ for different $\\epsilon$ should collapse to a common static scaling function $F_{\\rm st}(x)$. We expect $F_{\\rm st}(x)$ to cross-over around $x=1$ between the critical tail $x^{-1}$ for small $x$ and $x^{3-\\alpha}$ for large $x$. Indeed, this is what we see in Fig.~\\ref{fig2}(b). The unscaled leading static tail is, therefore, \n\\begin{equation} \nC_R\\simeq \\xi^{1\/2} R^{-3\/2}\n\\label{lrtailstatic}\n\\end{equation} \nfor $R\\gg\\xi$.\n\n\n\\begin{figure}[b]\n\\vspace{-0.5cm}\n\\includegraphics[width=\\columnwidth]{fig13.pdf}\n\\vspace{-0.5cm}\n\\caption{ {\\bf Long-range extended Ising model.}\nIn (a), \nscaled correlator $\\hat\\xi C_R$ for $\\alpha=3\/2$ as a function of scaled distance for different scaled times in the log-log scale.\nIn (b),\nin the inset the coefficients $A,B,C$ in Eq.~\\eqref{ABC(s)fit} fitted to the tail of the scaled correlator are shown as a function of scaled time $s=t\/\\hat t$. The main picture shows $A(t\/\\hat t)$ together with the adiabatic $A_{\\rm ad}(t\/\\hat t)$.\nIn the initial adiabatic stage $A$ follows $A_{\\rm ad}$. \nIn the KZ stage $A_{\\rm ad}$ diverges to infinity at the critical point forcing $A$ to lag behind.\n}\n\\label{fig:fig2a}\n\\end{figure}\n\n\nHaving verified the static hypothesis, we are encouraged to propose its dynamical version in the same form (\\ref{CRscaling}) as for local interactions. The dynamics is described by the time-dependent Bogoliubov-de Gennes equations (\\ref{tBdGk}) whose solution is presented in Appendix \\ref{AppLR}. The KZ mechanism has recently been verified in a similar model~\\cite{amitdutta}, where defect density was found to scale as $\\tau_Q^{-1\/2(\\alpha-1)}$ in agreement with the KZ prediction $\\tau_Q^{-\\nu\/(1+z\\nu)}$, see also Ref.~\\cite{KZLR2}. Further developments, like scaling of excitation energy (consistent with KZ) or possible experimental implementation, were considered in Refs.~\\onlinecite{MorigiLR,LRPlenio}.\nHere, we are interested in the build-up of correlations during the quench. In Fig.~\\ref{fig:corrp0m}(a) we show scaled correlation functions at different scaled times. For definiteness, in this figure we set $\\hat t=\\tau_Q^{1\/2}$ and $\\hat\\xi=\\tau_Q^{1\/2(\\alpha-1)}$ with both numerical prefactors equal to $1$. We can see that up to $t=0.35\\hat t$ the scaled correlators for different $\\tau_Q$ collapse demonstrating the KZ scaling hypothesis in Eq.~\\eqref{CRscaling}. At $t\/\\hat t\\approx 0.4$ they already fail to collapse, signalling the end of the KZ regime and the beginning of phase ordering in the second adiabatic stage. \n\nIn Fig.~\\ref{fig:fig2a}(a), we collect scaled correlators for different scaled times, $s=t\/\\hat t$, in the KZ regime. In order to gain better insight, in Fig.~\\ref{fig:fig2a}(b) we fit their tails with a function\n\\begin{equation} \n\\hat \\xi C_R(s)=A(s)(R\/\\hat\\xi)^{-3\/2}+B(s)(R\/\\hat\\xi)^{-2}+C(s)(R\/\\hat\\xi)^{-5\/2}.\n\\label{ABC(s)fit}\n\\end{equation} \nIts form is motivated by correlation tails after a sudden quench considered in appendices \\ref{SuddenAppInfinite} and \\ref{SuddenAppFinite}. Here, the coefficients $A,B,C$ are functions of the scaled time shown in Fig.~\\ref{fig:fig2a}(c). On top of the leading $A(s)$ tail, sub-leading dynamical correlations build up in time, with $B(s)$ growing negative and $C(s)$ positive. Their behavior is similar to what is observed after a sudden quench to the critical $\\epsilon=0$ either from $\\epsilon=-\\infty$ or $\\epsilon=-1$, see appendices \\ref{SuddenAppInfinite} and \\ref{SuddenAppFinite}, respectively. The main Fig.~\\ref{fig:fig2a}(b) shows $A(s)$, i.e., the coefficient of the leading long-range tail $\\propto R^{-3\/2}$. \n\n\nThe same figure shows $A_{\\rm ad}(s)$, i.e., the same coefficient but in case the evolution were adiabatic. $A_{\\rm ad}(s)$ was obtained by a fit to the far tail of the static correlation function at $\\epsilon(s)=s\/(\\tau_Q\/\\hat t)$. The same $A_{\\rm ad}(s)$ can be obtained by equating the static tail (\\ref{lrtailstatic}) with the adiabatic tail $C_R=A_{\\rm ad}(s)\\hat\\xi^{1\/2}\/R^{3\/2}$, compare (\\ref{ABC(s)fit}):\n\\begin{equation} \nA_{\\rm ad}(s) \\simeq\n\\left[\\frac{\\xi(s)}{\\hat\\xi}\\right]^{1\/2} \\simeq\n\\left[\\frac{\\epsilon(s)^{-\\nu}}{\\hat\\xi}\\right]^{1\/2} \\sim\ns^{-\\nu\/2}.\n\\end{equation} \n$A_{\\rm ad}(s)\\sim s^{-1}$ for $\\alpha=3\/2$ is consistent with the divergence we observe in Fig.~\\ref{fig:fig2a}(b). We can also see that, as expected, $A(s)$ follows $A_{\\rm ad}(s)$ in the initial adiabatic stage up to $s\\simeq -1$. After that, in the diabatic KZ stage, it lags behind as it cannot catch up with the diverging $A_{\\rm ad}(s)$. If it did not, and the evolution were adiabatic, then the correlation range -- defined by $R$ where $C_R$ falls below a fixed small cutoff -- would diverge near the critical point as $|t|^{-2\/3}$. \n\n\n\nAfter $A(s)$ begins to lag behind the adiabatic evolution, it continues to grow at a finite rate until the critical point. This is the stage where the long-range model is the most reminescent of the standard causal KZ picture. After the critical point $A(s)$ begins to dip down around $s\\approx 0.2$ but this is where the KZ scaling hypothesis (\\ref{CRscaling}) begins to be violated, see Fig.~\\ref{fig:corrp0m}(d).\n\n\\section{ Conclusion } \n\\label{SectionConclusion}\n\nWe recall the causal\/sonic horizon version of the Kibble-Zurek mechanism (KZM).\nIn the initial adiabatic stage of the evolution, \nthe correlation range follows its adiabatic counterpart.\nAt $-\\hat t$ it begins to lag behind the diverging adiabatic range and continues to grow at a finite rate set by the speed limit $\\simeq \\hat\\xi\/\\hat t$.\nThis way, between $-\\hat t$ and $+\\hat t$, the correlation range can increase several times at odds with the impulse approximation where it remains frozen. \n\nThere are notable exceptions, like the random quantum Ising chain, where the increase is small due to localization of excited quasiparticles that prevents them from spreading entanglement. There are interesting generalizations, like the long-range extended Ising model, where it is the power law correlation tail, rather than the exponential correlation length, that lags behind its diverging adiabatic evolution.\n\n\n\\acknowledgements\nAS would like to thank Titas Chanda for useful discussions.\nWe acknowledge funding by National Science Centre (NCN), \nPoland under projects No.~2016\/23\/B\/ST3\/00830 (JS,AF) \nand No. 2016\/23\/D\/ST3\/00384 (MMR), \nNCN together with European Union through QuantERA ERA NET \nprogram No.~2017\/25\/Z\/ST2\/03028 (AS,DS,JD),\nand Department of Energy under the Los Alamos National Laboratory \nLDRD Program (WHZ).\nWHZ was also supported by the U.S. Department of Energy, Office of Science, Basic Energy\nSciences, Materials Sciences and Engineering Division, Condensed Matter Theory Program.\nAF acknowledges financial support by Polish Ministry of Science and Education, project No. DI2015 021345, from the budget funds for science in 2016-2020 under the \"Diamond Grant\" program.\nThis research was carried out with the equipment purchased thanks to the financial support of the European Regional Development Fund in the framework of the Polish Innovation Economy Operational Program (contract No. POIG.02.01.00-12-023\/08). \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nThe theoretical study of anomalously slow relaxation processes in time-dependent force fields \nconstitutes a challenge of current research interest which is not free of ambiguity. \nIt is known that there is no unique physical mechanism responsible for the \noccurrence of subdiffusion in condensed media \\cite{Bouchaud1990}. \nOne possible mechanism, which will be addressed in this work, corresponds to \ndisordered glassy-like media consisting of trapping domains where the traveling particle can dwell\nfor a random time with divergent mean value \\cite{Scher, ScherMontroll, shlesinger1974}. \nThe successive residence times in traps are assumed to be mutually uncorrelated. \nDiffusion is nevertheless a non-Markovian (semi-Markovian) process exhibiting long\n(quasi-infinite) time correlations in the particle positions with a weak ergodicity breaking \\cite{BelBarkai}. \nMathematically this physical picture can be described by a continuous time random walk (CTRW) \nmodel \\cite{ScherMontroll, shlesinger1974} which in the continuous space limit leads\nto the fractional Fokker-Planck equation (FFPE) \\cite{metzler2000R, letter, heinsalu2006b}. \nThis latter formulation is incomplete, as no non-Markovian master equation can define \nthe underlying non-Markovian  stochastic process \\cite{HT}. \nHowever, the FFPE is very useful and has a tightly associated, complete description with (ordinary)  \nLangevin equation in subordinated, random operational time \\cite{Fogedby1994, Stanislavsky, Magdziartz}.\n\nThe generalization of the CTRW and FFPE to time-dependent\nforces is a highly non-trivial matter since the force changes in\nthe real and not in the operational time \\cite{heinsalu2007b, goychuk07}. \nAlso, how a field varying in time affects the distribution of the residence \ntimes in the traps is not clear without\nspecifying a concrete mechanism or some plausible model,\nespecially when the mean residence time does not exist \\cite{PRE04}. \nThe FFPE describing the dynamics in time-dependent force fields $F(x,t)$  \nbecomes ambiguous with a frequently (ab)used, \\textit{ad hoc} version \\cite{sokolov2001, Sample} \nwhich lacks clear theoretical grounds \\cite{heinsalu2007b}. \nThe correct version of the FFPE for time-dependent fields was first given in \nRefs.~\\cite{SokolovKlafter06, heinsalu2007b}: differently from the FFPE for a \ntime-independent force, in the case of a time-dependent field the fractional derivative \ndoes not stand in front of the Fokker-Planck operator but after it. \nAs we explain with this work in more detail, such a FFPE can be justified beyond \nthe linear response approximation within a CTRW approach only for a special \nclass of dichotomously fluctuating fields.\nThe derivation of the FFPE for subdiffusion in such time-dependent fields \nis presented in Sec.~\\ref{derivation}. \n\nIn Sec.~\\ref{results} we apply the derived FFPE to study the influence of \ntime-periodic rectangular fields on subdiffusive motion. \nAnalytical solutions of the FFPE are confirmed by stochastic \nMonte Carlo simulations of the underlying CTRW.  \nIn particular, we show with this work that the universal scaling relation between \nthe biased anomalous diffusion and sub-current \n\\cite{ScherMontroll, shlesinger1974, letter, heinsalu2006b} is not affected by the periodic driving.  \nNeither current nor diffusion are influenced asymptotically by the time-periodic field. \nThis is in spite of the fact that unbiased subdiffusion of the studied kind can be  strongly\nenhanced in the time-periodic field \\cite{SokolovKlafter06, heinsalu2007b}.  \n\n\n\n\n\\section{Derivation of the FFPE for time-dependent fields from the underlying CTRW} \\label{derivation}\n\n\n\nSince the FFPE does not define the underlying stochastic non-Markovian process, \nits generalization to include the influence of a time-dependent field should \nstart from the underlying CTRW \\cite{metzler2000R, letter}.\nFollowing the general picture of the CTRW we introduce a\none-dimensional lattice $\\{ x_i = i \\Delta x\\}$ with a lattice period\n$\\Delta x $ and $i = 0, \\pm 1, \\pm 2, \\ldots$\nLet us first assume that there is no time-dependent field.  \nAfter a random trapping time $\\tau$ a particle at site $i$ hops with  probability\n$q _i ^\\pm $ to one of the nearest neighbor sites $i \\pm 1$; $q_i^+ + q_i^-=1$.  \nThe random time $\\tau$ is extracted from a site-dependent \nresidence time distribution (RTD) $\\psi_i (\\tau)$. \nThe corresponding generalized master equation for\n populations $P_i(t)$ reads \\cite{Kenkre1973,Hughes,Weiss}\n\\begin{eqnarray} \\label{GME}\n\\dot P_i(t) &=& \\int_0^t \\{K_{i-1}^{+}(t-t') P_{i-1}(t') +\nK_{i+1}^{-}(t-t') P_{i+1}(t') \\nonumber \\\\\n&-& [K_i^{+}(t-t') + K_i^{-}(t-t')] P_i(t')\\} \\, \\mathrm{d} t' \\, .\n\\end{eqnarray} \nThe Laplace-transform of the kernel $K_i^{\\pm}(t)$ is related to the Laplace-transform \nof the RTD $\\psi_i (\\tau)$ via $\\tilde K_i^{\\pm}(s) = q_i^{\\pm} s\\tilde \\psi_i(s) \/ [1 - \\tilde \\psi_i(s)]$. \nIn the presence of a time-dependent field the kernels become generally functions \nof both instants of time and not only of their difference, i.e. $K_i^{\\pm}(t-t')\\to K_i^{\\pm}(t,t')$. \nOne can relate $K_i^{\\pm}(t,t')$ with the corresponding time-inhomogeneous RTDs\n $\\psi_i^{\\pm}(t+\\tau,t)\\equiv \\psi_i^{\\pm}(\\tau|t)$, which are conditioned on the entrance time $t$. \nHowever, one always needs a concrete and physically meaningful model to proceed further \\cite{PRE04}. \nA simple example is a Markovian process with time-dependent rates $g_i^{\\pm}(t)$, where \n$\\psi_i^{\\pm}(\\tau|t) = g_i^{\\pm}(t+\\tau) \\exp\\left \\{ -\\int_{t}^{t+\\tau}[g_i^{+}(t') + g_i^{-}(t')]dt'\\right \\}$ \nand $K_i^{\\pm}(t,t') = 2g^{\\pm}_i(t) \\, \\delta(t-t')$. \nThis yields in Eq.~(\\ref{GME}) the standard master equation for a \ntime-inhomogeneous Markovian process. \nHow the non-exponential RTDs will be modified for a time-inhomogeneous \nprocess is generally not clear \\cite{PRE04}.\nIn the present case, one can assume that the trapping occurs due to the \nexistence of  direction(s) orthogonal to the $x$-coordinate. \nAccording to the modeling in Refs.~\\cite{SokolovKlafter06, Magdziartz}, \nan external field directed along $x$ \nwould not affect the motion in the orthogonal direction(s).\nHowever, it is not correct to think that the RTD in the trap will not be \ninfluenced by the field acting in the direction of $x$, as it will change the rates \n(let us assume this simplest, tractable model) \nfor moving left or right when escaping from the trap. \nTherefore, the RTD will generally be affected, see e.g. Ref.~\\cite{PRE05}.\nObviously, the only situation when the RTD in the trap will not be changed is \nwhen the sum of the rates to escape from the trap, either left or right, is constant. \nIn that case, only the probabilities $q_i^{\\pm}(t)$ acquire additional \ntime-dependence and not the RTDs $\\psi_i(\\tau)$. \nThis corresponds to the special class of dichotomously fluctuating force fields \n$F (x_i, t) = F (x_i) \\xi (t)$, where $\\xi (t) = \\pm 1$. \nBeyond this class, at most the linear response approximation can work \\cite{SokolovKlafter06}. \nTherefore, we restrict our treatment to the above class of fluctuating potentials.\nIn this case, we can write $K_i^{\\pm}(t,t') = q^{\\pm}_i(t)K_i(t-t')$,\nwhere $\\tilde K_i(s) = s\\tilde \\psi_i(s) \/ [1 - \\tilde \\psi_i(s)]$.\nFurthermore, we use the Mittag-Leffler distribution for the residence times \\cite{metzler2000R},\n\\begin{equation} \\label{ML}\n\\psi_i(\\tau) = -\\frac{\\mathrm{d}}{\\mathrm{d} \\tau} E_{\\alpha}[\n-(\\nu_i \\tau)^{\\alpha}] \\, .\n\\end{equation}\nHere $E_{\\alpha}(z) = \\sum_{n = 0}^{\\infty} z^n \/ \\Gamma(n \\alpha + 1)$ denotes \nthe Mittag-Leffler function, $\\alpha \\in (0,1)$ is the index of subdiffusion, and \n$\\nu_i = [g_i^+(t) + g_i^-(t)]^{1\/\\alpha}$ the time scaling parameter; \n$g_{i}^\\pm (t) = q^{\\pm}_i(t) \\nu_i^{\\alpha}$.\nThen $\\tilde K_i(s) = \\nu_i^{\\alpha} s^{1-\\alpha}$ and we get\n\\begin{eqnarray} \\label{FME}\n\\dot P_i(t) &=& g_{i-1}^+(t)\\sideset{_0}{_t} {\\mathop{\\hat\nD}^{1-\\alpha}} P_{i-1}(t) + g_{i+1}^-(t) \\sideset{_0}{_t} {\\mathop{\\hat\nD}^{1-\\alpha}} P_{i + 1}(t)  \\nonumber \\\\\n&-& [g_i^+(t) + g_i^-(t)]\\sideset{_0}{_t} {\\mathop{\\hat\nD}^{1-\\alpha}} P_i(t) \\, ,\n\\end{eqnarray}\nwhere the symbol $\\sideset{_0}{_t}{\\mathop{\\hat D}^{1-\\alpha}}$ stands for the \nintegro-differential operator of the Riemann-Liouville fractional derivative \nacting on a generic function of time $\\chi (t)$ as \n\\begin{equation} \\label{RL}\n\\sideset{_0}{_t}{\\mathop{\\hat D}^{1 - \\alpha}} \\chi (t) \n= \\frac{1}{\\Gamma (\\alpha)} \\frac{\\partial}{\\partial t} \\int_{0}^{t} \\mathrm{d} t' \\, \\frac{\\chi (t')}{(t - t')^{1 - \\alpha}} \\, ;\n\\end{equation}\n$\\Gamma (\\alpha)$ is the gamma-function.\nIn a time-dependent potential $U(x,t)$, one can set\n\\begin{eqnarray} \\label{f-rate}\ng_i^\\pm(t) &=& (\\kappa_{\\alpha} \/ \\Delta x ^2)\n\\exp \\{ -\\beta[U_{i \\pm 1\/2}(t) - U_i(t)] \\} \\nonumber \\\\\n&\\approx & (\\kappa_{\\alpha} \/ \\Delta x ^2)\n\\exp [\\pm \\beta F(x_i,t)\\Delta x\/2] \\, ,\n\\end{eqnarray}\nso that the Boltzmann relation $g_{i-1}^+(t)\/g_i^-(t) = \\exp \\{ \\beta[U_{i-1}(t)-U_i(t)]\\}$\nis satisfied exactly and the time-independence of $g_i^+(t) + g_i^-(t) = \\nu_i^{\\alpha} = \\mathrm{const}$\nis also maintained for small $\\Delta x$ and a sufficiently smooth potential.\nWe have used here the notation $U_i(t) \\equiv U(i \\Delta x,t)$ and \n$U_{i \\pm 1\/2}(t) \\equiv U(i \\Delta x \\pm \\Delta x \/2,t)$; \n$\\beta = k_B T$ is the inverse of temperature and $\\kappa_{\\alpha}$ free fractional \ndiffusion coefficient with dimension $\\mathrm{cm}^2 \\mathrm{s}^{-\\alpha}$.\nBy passing to the continuous space limit $\\Delta x\\to 0$ \nas in Ref.~\\cite{letter}, one finally obtains,\n\\begin{eqnarray} \\label{FFPEmod}\n\\frac{ \\partial }{\\partial t} P (x, t) = \n\\left [ - \\frac{\\partial}{\\partial x} \\frac{F (x, t)}{\\eta _\\alpha} \n+ \\kappa _\\alpha \\frac{\\partial ^2}{\\partial x^2} \\right ] \n\\sideset{_0}{_t}{\\mathop{\\hat D}^{1 - \\alpha}} P (x, t) \\, .\n\\end{eqnarray}\nIn the latter equation $\\eta _\\alpha = ( \\beta \\kappa_{\\alpha})^{-1}$ \nis the fractional friction coefficient.\n\nIn the following we use Eq.~(\\ref{FFPEmod}) to study analytically \nthe subdiffusion in time-periodic rectangular fields. \nOur study is complemented by stochastic simulations of the\nunderlying CTRW using the algorithm detailed in Ref.~\\cite{heinsalu2006b}.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=7.5cm]{figure1.eps}\n\\caption{(Color online) \nAverage particle position $\\langle x (t) \\rangle$ for various values of \nthe parameter $r$ (average force $\\bar{F}$) and anomalous exponent $\\alpha $.\nSymbols represent the results from the numerical simulations of the CTRW obtained by \naveraging over $10 ^5$ trajectories (for $r = 0.5$ over $10 ^6$ trajectories). \nContinuous lines represent the analytical solution (\\ref{xx}) of the FFPE (\\ref{FFPEmod}). \nThe time-period of the rectangular force (\\ref{Fper}) is \n$\\tau _0 = 1$ and the fractional exponent $\\alpha = 0.5$;  \nin numerical simulations $F_0 \/ (\\eta _{\\alpha} \\sqrt{\\kappa _{\\alpha}}) = 1$ is used. \nThe value $r = 0.9$ corresponds to $\\bar{F} = 0.8 F_0$ and $\\sigma = 0.6 F_0$;\n$r = 0.8$ corresponds to $\\bar{F} = 0.6 F_0$ and $\\sigma = 0.8 F_0$; \n$r = 0.7$ corresponds to $\\bar{F} = 0.4 F_0$ and $\\sigma\\approx 0.92 F_0$; \n$r = 0.6$ corresponds to $\\bar{F} = 0.2 F_0$, $\\sigma\\approx 0.98 F_0$; \nand $r = 0.5$ to $\\bar{F} = 0$, $\\sigma = F_0$.} \n\\label{Fig1}\n\\end{figure}\n\n\n\n\n\\section{Driven subdiffusion} \\label{results}\n\n\n\nWe consider a dichotomous modulation  of a\nbiased subdiffusion where the absolute value of the bias is\nfixed but its direction flips periodically in time, i.e.\n\\begin{equation} \\label{Fper}\nF (t) = F_0 \\xi (t)\n\\end{equation}\nwith\n\\begin{eqnarray} \\label{FperPM}\n\\xi (t) = \\left\\{ \\begin{array}{ll} + 1 & \\textrm{~for}~~ n \\tau_0 < t < (n+r) \\tau_0 \\\\\n- 1 & \\textrm{~for}~~ (n+r) \\tau_0 < t < (n+1) \\tau _0\n\\end{array} \\right. .\n\\end{eqnarray}\nHere $\\tau _0$ is the period of the time-dependent force and $n = 0, 1, 2 \\ldots $ \nThe quantity $r \\in (0,1)$ determines the value of the average force: \n\\begin{equation}\n\\bar{F} = \\langle F (t) \\rangle _{\\tau _0} = F_0 (2r - 1) \\, .\n\\end{equation}\nFor $r = 0.5$ the average bias is zero and we recover the model investigated\nin Ref.~\\cite{heinsalu2007b}. \nNotice that the force $F(t)$ can be decomposed in the following way: $F(t) = \\bar{F} + \\tilde F(t)$. \nThe asymmetric driving, \n\\begin{eqnarray} \\label{Fper2}\n\\tilde F (t) \\!= \\!\n\\left\\{\\!\\begin{array}{ll} 2 F_0 (1-r) & \\textrm{~for}~~ n \\tau_0 < t < (n+r) \\tau_0 \\\\\n- 2 F_0 r & \\textrm{~for}~~ (n+r) \\tau_0 < t < (n+1) \\tau _0\n\\end{array} \\right. \\!,\n\\end{eqnarray}\nhas a zero mean value, $\\langle \\tilde F (t) \\rangle _{\\tau _0} = 0$, \nand the driving root-mean-squared (rms) amplitude is \n$\\sigma = \\langle \\tilde F^2 (t) \\rangle^{1\/2} _{\\tau _0} = 2F_0 \\sqrt{r(1-r)}$.\nFor a fixed average bias $\\bar{F}$, this yields\n\\begin{eqnarray}\n\\sigma = 2 \\bar{F} \\frac{\\sqrt{r(1-r)}}{2r-1}\n\\end{eqnarray}\nand therefore one can vary the ratio $\\sigma\/ \\bar{F}$ between $0$ for\n$r=1$ and $\\infty$ for $r=1\/2+\\epsilon$, $\\epsilon\\to 0$.\nThis offers the way to study the influence of an asymmetric, zero-mean\ndriving $\\tilde{F}(t)$ with period $\\tau_0$ and rms amplitude $\\sigma$ \non the subdiffusion under constant bias $\\bar{F}$.\n\nLet us begin by finding the recurrence relation  for the moments $\\langle x ^n (t) \\rangle$.  \nAssuming in Eq.~(\\ref{FFPEmod}) the force of the form \n(\\ref{Fper}) with (\\ref{FperPM}), multiplying both sides of Eq.~(\\ref{FFPEmod}) by \n$x ^n$, and integrating over the $x$-coordinate one obtains,\n\\begin{eqnarray} \\label{a2}\n\\frac{d \\langle x ^n (t) \\rangle}{d t} \n&=& n v _\\alpha \\xi (t) \\sideset{_0}{_t}{\\mathop{\\hat D}^{1 - \\alpha}} \\langle x^{n-1} (t) \\rangle \\nonumber \\\\\n&+& n (n - 1) \\kappa _\\alpha \\sideset{_0}{_t}{\\mathop{\\hat D}^{1 - \\alpha}} \\langle x ^{n - 2} (t) \\rangle \\, ,\n\\end{eqnarray}\nwith subvelocity $v _{\\alpha} = F_0 \/ \\eta _{\\alpha}$ ($n > 1$). \nFor $n = 1$ the last term on the right hand side of Eq.~(\\ref{a2}) is absent,\n\\begin{eqnarray} \\label{xav}\n\\frac{d \\langle x(t) \\rangle}{d t} = \\frac{v _\\alpha}{\\Gamma (\\alpha)} \n\\, \\xi (t) \\, t ^{\\alpha - 1} \\, .\n\\end{eqnarray}\nEquations (\\ref{a2}) and (\\ref{xav}) will be used to calculate the \naverage particle position and the mean square displacement.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=7.5cm]{figure2.eps}\n\\caption{(Color online) \nThe analytical solution (\\ref{xx}) for average particle position \n$\\langle x (t) \\rangle$ obtained from the FFPE (\\ref{FFPEmod}) is presented \nfor various values of the parameter $r$ (average bias $\\bar{F}$) and anomalous exponent $\\alpha $. \nThe time-period of the force is $\\tau _0 = 1$, however, in the long time limit \nthe same asymptotic value is obtained for any value of $\\tau _0$. \nThe relation between $r$ and $\\bar{F}$ and $\\sigma$  is the same as in Fig.~\\ref{Fig1}.} \n\\label{Fig2}\n\\end{figure}\n\n\n\n\n\\subsection{Average particle position}\n\n\n\nUpon integrating Eq.~(\\ref{xav}) in time with $\\xi (t)$ given by (\\ref{FperPM}), \nthe solution for the average particle position reads:\n\\begin{eqnarray} \\label{xx}\n&& \\langle x (t) \\rangle \\nonumber \\\\\n&& =\\left\\{ \n\\begin{array}{l@{\\quad \\quad}}\nx _N + \\frac{v _{\\alpha} t ^\\alpha}{\\Gamma (\\alpha + 1)} \\, , \\quad N \\tau _0 \\le t < (N + r) \\tau _0 \\\\\nx _N' - \\frac{v _{\\alpha} t ^\\alpha}{\\Gamma (\\alpha + 1)} \\, , \\quad (N + r) \\tau _0 \\le t < (N + 1) \\tau _0 \n\\end{array} \n\\right.\n\\end{eqnarray}\nwith\n\\begin{eqnarray} \\label{xn1}\nx _N &=& \\langle x (0) \\rangle - \\frac{v _{\\alpha} (N \\tau _0) ^\\alpha}{\\Gamma (\\alpha + 1)} \n+ \\frac{v _{\\alpha} \\tau _0 ^\\alpha}{\\Gamma (\\alpha + 1)} \\nonumber \\\\\n&\\times& \\sum_{n = 0} ^{N - 1} \\left [2(n + r) ^\\alpha -n ^\\alpha - (n + 1) ^\\alpha \\right ] \\, , \\\\\nx _N' &=& x _N + \\frac{2 v _{\\alpha} \\tau _0 ^\\alpha}{\\Gamma (\\alpha + 1)} (N + r) ^\\alpha \\, ;\n\\end{eqnarray}\n$N$ counts the number of time periods passed.\n\nWhen the average bias is zero, i.e., $r = 0.5$, in the long time limit\nthe mean particle position approaches the constant value\n\\begin{eqnarray} \\label{asympt}\n\\langle x (\\infty) \\rangle = v _{\\alpha} \\tau _0 ^{\\alpha} b (\\alpha ) \/ \\Gamma(\\alpha + 1),\n\\end{eqnarray}\nwith $b (\\alpha ) = \\sum _{n = 0} ^{\\infty}[2 (n + 1\/2) ^{\\alpha} - n ^{\\alpha} - (n + 1) ^{\\alpha}]$. \nThe function $b (\\alpha )$ changes monotonously from $b (0) = 1$ to $b (1) = 0$. \nIt describes the initial field phase effect which the system remembers forever when \n$\\alpha \\in (0, 1)$ (see also Ref.~\\cite{heinsalu2007b}). \nThis is one of the main differences between the anomalous motion in the absence \nof a force and in the presence of a time-dependent field with zero average value. \n\nIn Fig.~\\ref{Fig1} the analytical solution (\\ref{xx}) for the mean\nparticle position $\\langle x (t) \\rangle$, obtained from the\nFFPE (\\ref{FFPEmod}), is compared with the numerical solution of the CTRW\nfor different values of $r$, i.e. for different values of the average bias $\\bar{F}$.  \nIn Fig.~\\ref{Fig2} the solution (\\ref{xx}) is presented in the \nlong time limit for various values of $r$ and $\\alpha $. \nFigures \\ref{Fig1} and \\ref{Fig2} demonstrate that in the presence of an\naverage bias the mean particle position grows as $t ^\\alpha$.\nFor all values of $r$, the asymptotic value of subvelocity corresponds\nto the averaged bias $\\bar{F} = F_0 (2r-1)$, indicating that the periodic\nunbiased field does not affect the subdiffusion current for different values\n$\\bar{F}$ and the field rms-amplitude $\\sigma$.\n\nFurthermore, the results depicted in Figs.~\\ref{Fig1} and \\ref{Fig2} clearly show the \nphenomenon of the ``death of linear response'' of the fractional kinetics to time-dependent \nfields in the limit $t \\to \\infty$: the amplitude of the oscillations decays to\nzero as $1 \/ t ^{1 - \\alpha}$, Eq.~(\\ref{xav}) (see also Refs.~\\cite{Barbi, SokolovKlafter06}).  \nThe amplitude of the oscillations is larger for larger values of \n$\\alpha $ and of $\\tau _0$.\nHowever, $\\Gamma (1+\\alpha) \\langle x(t) \\rangle \/ t^\\alpha$ reaches asymptotically \nthe same value for any $\\alpha$ and $\\tau _0$.\n\n\n\n\n\\subsection{Mean square displacement}\n\n\n\nLet us now study the mean square displacement, defined as\n\\begin{equation} \\label{MSD}\n\\langle \\delta x ^2 (t) \\rangle = \\langle x ^2 (t) \\rangle - \\langle x (t) \\rangle ^2 \\,.\n\\end{equation}\nFor $n = 2$ one obtains from Eq.~(\\ref{a2}),\n\\begin{equation} \\label{AP1}\n\\frac{d \\langle x ^2 (t) \\rangle}{d t} \n= 2 v _\\alpha \\xi (t) \\sideset{_0}{_t}{\\mathop{\\hat D}^{1 - \\alpha}} \\langle x (t) \\rangle \n+ \\frac{ 2 \\kappa _\\alpha }{\\Gamma (\\alpha)} \\, t ^{\\alpha - 1} \\, .\n\\end{equation}\nIn order to find the analytical solution for the mean square displacement, \nwe use the Laplace-transform method and the Fourier series expansion for \n$\\xi (t) = \\xi (t + \\tau _0)$ given wuth Eq.~(\\ref{FperPM}),\n\\begin{equation} \\label{AP3}\n\\xi (t) = \\sum_{n = - \\infty}^{\\infty} f _n \\exp (i n \\omega _0 t) \\, ,\n\\end{equation}\nwith\n\\begin{eqnarray} \\label{AP4}\nf _n &=& \\frac{1}{\\tau _0} \\int _{0} ^{\\tau _0} \\xi (t) \\exp ( - in \\omega _0 t) dt \\nonumber \\\\\n&=& [1 - \\exp ( - inr 2 \\pi )] \/ ( in \\pi )\n\\end{eqnarray}\nand $\\omega _0 = 2 \\pi \/ \\tau _0$. \nApplying them to Eq.~(\\ref{AP1}) and assuming $\\langle x (0) \\rangle = 0$ and \n$\\langle x ^2 (0) \\rangle = 0$ we obtain that in the long time limit \n(see Appendix~\\ref{APPENDIX-1}),\n\\begin{eqnarray} \\label{MSD-1}\n\\langle x ^2 (t) \\rangle &=& \\frac{2v _\\alpha ^2 (2r - 1) ^2 }{\\Gamma (2 \\alpha + 1)} t ^{2 \\alpha } \n+ \\frac{2 \\kappa _\\alpha}{\\Gamma (\\alpha + 1)} t ^\\alpha \\nonumber \\\\\n&+& \\frac{2 v _\\alpha ^2 (2r - 1) S_1}{\\omega _0 ^\\alpha \\Gamma (\\alpha + 1)} t ^\\alpha\n + \\frac{8v _\\alpha ^2 \\cos (\\alpha \\pi \/ 2)}{\\pi ^2 \\omega _0 ^\\alpha \\Gamma (\\alpha + 1)}  \\nonumber \\\\\n&\\times & \\left [\\zeta (2 + \\alpha ) - \\sum _{n = 1} ^{\\infty } \\frac{\\cos (nr 2 \\pi )}{n ^{2 + \\alpha }} \\right ] t ^\\alpha \\, ;\n\\end{eqnarray}\nhere $\\zeta (x)$ is the Riemann's zeta-function and $S_1$ is a function of \n$\\alpha$ and $r$ as given by Eq.~(\\ref{AP13}) in Appendix~\\ref{APPENDIX-1}.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=7.5cm]{figure3.eps}\n\\caption{(Color online) Scaled effective fractional diffusion coefficient $\\kappa _\\alpha ^{\\mathrm{(eff)}}$ \nversus fractional exponent $\\alpha$ for different driving periods $\\tau _0$. \nThe average bias is zero ($r = 0.5$). \nThe analytical prediction (\\ref{basic}) (continuous lines) is compared with the results (symbols) \nobtained from the numerical simulation of the CTRW by averaging over $10 ^5$ trajectories. \nFor $\\tau _0 > \\tau _0^* \\approx 8.818$ \\cite{heinsalu2007b} the effective fractional diffusion coefficient \n$\\kappa _\\alpha ^{\\mathrm{(eff)}}(\\alpha )$ exhibits a maximum.}\n\\label{Fig3}\n\\end{figure}\n\n\nFor $r = 0.5$ (average zero bias) the first and third term in Eq.~(\\ref{MSD-1}) are equal to zero. \nFurthermore, in the long time limit the average particle position $\\langle x (\\infty ) \\rangle $ is a finite constant. \nThe asymptotic behavior of the mean square displacement is thus proportional to $t ^{\\alpha}$ as in the force\nfree case, however, characterized by an effective fractional diffusion coefficient \n$\\kappa _{\\alpha} ^\\mathrm{(eff)}$ instead of the free fractional diffusion coefficient $\\kappa _\\alpha$, i.e.\n$\\langle \\delta x ^2 (t) \\rangle = 2\\kappa _{\\alpha} ^\\mathrm{(eff)} t^ {\\alpha} \/ \\Gamma (1 + \\alpha )$ for $t \\to \\infty$. \nThe effective diffusion coefficient is \\cite{heinsalu2007b},\n\\begin{eqnarray} \\label{basic}\n\\kappa _\\alpha ^{\\mathrm{(eff)}} &=& \\kappa _\\alpha \n+ \\frac{8 F_0^2}{\\pi ^2 \\eta _\\alpha ^2 \\omega _0 ^\\alpha } \\, \\zeta (\\alpha + 2) \\nonumber \\\\\n&\\times& \\left(1- \\frac{1}{2 ^{\\alpha + 2}} \\right ) \\cos (\\alpha \\pi \/ 2) \\, .\n\\end{eqnarray}\nThe driving-induced part of the effective subdiffusion coefficient is directly proportional \nto the square of driving amplitude $F_0$ and inversely proportional to $\\omega _0 ^\\alpha $. \nFor slowly oscillating force fields this leads to a profound \nacceleration of subdiffusion compared with the force free case: \nan optimal value of the fractional exponent $\\alpha$ exists, at which the driving-induced part \nof the effective fractional diffusion coefficient possesses a maximum (see Fig.~\\ref{Fig3}).\n\nWhen $r \\neq 0.5$ (finite average force) we obtain in the long time limit\nfor $\\langle x (t) \\rangle ^2$ (see Appendix~\\ref{APPENDIX-2}),\n\\begin{equation} \\label{MSD-2}\n\\langle x (t) \\rangle ^2 = \\frac{v _\\alpha ^2 (2r - 1) ^2}{\\Gamma ^2 (\\alpha + 1)} t ^{2 \\alpha } + \n\\frac{2v _\\alpha ^2 (2r - 1) S_1}{\\omega _0 ^\\alpha \\Gamma (\\alpha + 1)} t ^\\alpha\n\\end{equation}\n[$S_1$ is given by Eq.~(\\ref{AP13}) in Appendix~\\ref{APPENDIX-1}].\nClearly, the leading term in Eq.~(\\ref{MSD-2}) corresponds to subvelocity in constant \nfield $\\bar{F}$ (averaged bias), i.e. the influence of periodic, unbiased driving \n$\\tilde F(t)$ dies asymptotically out, as illustrated in Fig.~\\ref{Fig2}.\n\nThe results (\\ref{MSD-1}) and (\\ref{MSD-2}) indicate that in the \npresence of a rectangular time-periodic force with a finite average value the\ngeneral behavior of the mean square displacement is similar to the\ncase of a constant force, i.e. the mean square displacement \n$\\langle \\delta x ^2 (t)\\rangle $ consists of terms proportional to \n$t ^\\alpha $ and $t ^{2 \\alpha }$.\nIn fact, for the leading term proportional to $t ^{2 \\alpha }$ in the \nmean square displacement one obtains the coefficient\n\\begin{equation*} \\label{AP22}\n\\frac{{\\bar{F}}^2}{\\eta _\\alpha ^2} \\left [ \\frac{2}{\\Gamma (2 \\alpha + 1)} \n- \\frac{1}{\\Gamma ^2 (\\alpha + 1)} \\right ] \\, .\n\\end{equation*}\nThis coefficient is the same as in the case of the subdiffusive motion under the influence \nof a constant force if the value of the constant force would be $\\bar{F}$.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=7.5cm]{figure4.eps}\n\\caption{(Color online) \nThe asymptotic scaling relation (\\ref{anomal}). \nSymbols correspond to the numerical results obtained from the CTRW \nfor different values of $r$ ($r \\neq 0.5$) and $\\tau_0$.\nSolid curve corresponds to the analytical result (\\ref{anomal}). \nDifferent values of the parameter $r$ correspond to different\nvalues of the bias $\\bar{F}$ and the periodic field rms $\\sigma$, as described in\nFig.~\\ref{Fig1}. \nThe field parameters do not influence the results within the statistical errors.} \n\\label{Fig4}\n\\end{figure}\n\n\nFurthermore, similarly to the case of a constant bias, the asymptotic scaling relation holds \n($r \\neq 0.5 $, $\\bar{F} \\neq 0 $) between the mean square displacement and\naverage particle position.\nIn the limit $t \\to \\infty$ the mean square displacement grows as \n$\\langle \\delta x ^2 (t) \\rangle \\propto t ^{2 \\alpha }$  and\n\\begin{eqnarray} \\label{anomal}\n\\lim_{t \\to \\infty} \\frac{\\langle \\delta x^2 (t) \\rangle}{ \\langle x (t) \\rangle ^2} \n= \\frac{2 \\Gamma ^2 (\\alpha + 1)}{\\Gamma (2 \\alpha + 1)} - 1 \\, , \\quad \\bar{F} \\neq 0 \\, .\n\\end{eqnarray}\nIt is illustrated by Fig.~\\ref{Fig4},  where the analytical curve\n[Eq.~(\\ref{anomal})] is compared with the numerical results. \nThe universality of relation (\\ref{anomal}) under the unbiased driving $\\tilde F(t)$\nmeans that the biased diffusion is not affected by the driving. \nThis is in a sharp contrast with the unbiased diffusion in Fig.~\\ref{Fig3}.\n\n\n\n\n\\section{Conclusion}\n\n\n\nWith this work we presented the derivation of the FFPE (\\ref{FFPEmod}) for a \nspecial class of space- and time-dependent force fields from the underlying CTRW picture. \nOur derivation shows along with the corresponding discussion\nthat it is difficult to justify this equation for time-dependent forces \ndifferent from $F(x, t) = F(x) \\xi (t)$ with $\\xi (t) = \\pm 1$\nbeyond the linear response approximation. \nUsing the FFPE (\\ref{FFPEmod}) we demonstrated that the universal \nscaling relation (\\ref{anomal}) for a biased subdiffusion is not affected \nby the additional action of a time-periodic zero-mean rectangular driving;\nneither is the asymptotic anomalous current nor the anomalous biased diffusion.\nWe argue that this result is general and it is valid for other driving forms\nas well.\nThis driving-immunity is due to the fact that the CTRW subdiffusion occurs \nin a random operational time which lacks mean value, whereas any physical field\nchanges in the real, physical time. \nThe CTRW-based subdiffusion fails to respond \nasymptotically to such time-dependent fields while on its intrinsic random \noperational time scale any real, alternating field is acting infinitely fast\nand it makes effectively no influence in a long run \\cite{heinsalu2007b,goychuk07},\nunless the rate of its change is precisely zero. \nThis is the main reason for the observed anomalies. \nThe remarkable enhancement of the unbiased subdiffusion within the CTRW framework \nby time-periodic rectangular fields is rather an exception than the rule. \n\n\n\n\n\\begin{acknowledgements}\n\n\n\nThis work has been supported by the targeted financing project SF0690030s09, \nEstonian Science Foundation via grant no. 7466 (M.P., E.H.), \nSpanish MICINN and FEDER through project FISICOS (FIS2007-60327) (E.H.), \nthe EU NoE BioSim, LSHB-CT-2004-005137 (M.P.),\nthe DFG-SFB-486, and by the Volks\\-wagen Foundation, no. I\/80424 (P.H.).\n\\end{acknowledgements}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and preliminaries}\n\n\n\n\n\\begin{abstract}\n\n\nThe paper studies the containment companion of a logic $\\vdash$. This consists of the consequence relation $\\vdash^{r}$ which satisfies all the inferences of $\\vdash$, where the variables of the conclusion are \\emph{contained} into those of the (set of) premises. In accordance with the work started in \\cite{BonzioMorascoPrabaldi}, we show that a different generalization of the  P\\l onka sum construction, adapted from algebras to logical matrices, allows us to provide a matrix-based semantics for containment logics. In particular, we provide an appropriate completeness theorem for a wide family of containment logics, and we show how to produce a complete Hilbert style axiomatization.\n\\end{abstract}\n\n\n\\section{introduction}\n\nIt is a recent discovery (see \\cite{Bonzio16}) that the algebraic counterparts of weak Kleene logics are formed by a regularized variety, whose members coincide with the \\emph{P\\l onka sum} of Boolean algebras, the algebraic semantics of propositional classical logic. Subsequently, the abstract construction of the P\\l onka sum of algebras has been generalized to logical matrices \\cite{BonzioMorascoPrabaldi}. The main outcome is that the suggested notion provides an algebra-based semantics for a class of propositional logics, called \\emph{logics of left variable inclusion}, of which paraconsistent weak Kleene represents the most prominent example. \n\nThe logics in the weak Kleene ``family'' -- essentially, Bochvar \\cite{BochvarBergmann} and paraconsistent weak Kleene \\cite{Hallden} --  are syntactically characterized by imposing certain limitations on the inclusions of variables to classical propositional logic \\cite{Urquhart2001,CiuniCarrara}.   \n\nThe extension of the construction of P\\l onka sums to logical matrices, introduced in \\cite{BonzioMorascoPrabaldi}, allows for an insightful investigation into the algebraic features of those logics, where the inclusion of variables runs from premises to conclusions. However, this is just one side of the coin of the \\emph{logics of variable inclusion}; the other side consisting of those logics verifying inferences in which the variables occurring in the conclusion are contained into the ones occurring in the premises. Consequence relations satisfying this feature are usually know as \\emph{containment logics}; the syntactic requirement which they share is a strengthen form of what Ferguson \\cite{ferguson2017meaning} understands as \\emph{Proscriptive Principle}, which also resembles the one defining logics of \\emph{analytic containment} \\cite{ledda2019algebraic, epstein1990semantic}. \n\n\nThe most famous example of containment logic is \\emph{Bochvar logic} $\\mathsf{B_{3}}$ \\cite{BochvarBergmann}, defined by a single matrix which features the presence of an infectious truth-value (a peculiarity shared by the twin-sister paraconsistent weak Kleene). $\\mathsf{B_{3}}$ has been successfully applied in different contexts: avoiding paradoxes in set-theory \\cite{BochvarBergmann}, modeling computer programs affected by errors \\cite{Ferguson} and non-sensical information databases \\cite{Ciuni1}, capturing the notion of truth in relations with on\/off topic arguments \\cite{beall2016off}. \n\n  \nThe main condition defining containment logics mirrors the syntactic requirement defining logics of left variable inclusion. For this reason, the present paper may be understood as an ideal continuation of the path started in \\cite{BonzioMorascoPrabaldi}.\nThis amounts to answer the very natural question on whether it is possible to build a new generalization of the P\\l onka sum construction, suitable for obtaining a matrix semantics for containment logics. Due to the intrinsic difference between the  variable inclusion constraints at stake, the solution is not self-evident.\n\n\nThe paper is structured as follows.\n\n\nIn Section \\ref{sec: preliminari}, we recall all the preliminary notions needed to go trough the reading of the whole paper. They basically consist of the basic notions of abstract algebraic logic and of the theory of P\\l onka sums.\n \nIn Section \\ref{sec:compl}, we formally introduce containment logics. By providing an adequate notion of P\\l onka sum for logical matrices, we obtain soundness and completeness for the \\emph{containment companion} $\\vdash^{r}$ of an arbitrary (finitary) logic $\\vdash$, with respect to the P\\l onka sum of the matrix models of $\\vdash$.\n\n\nIn Section \\ref{sec:hilbert}, we focus on a specific (though very wide) class of logics, namely those  possessing a binary term called partition function (a property shared by the vast majority of known logics). We provide a method for obtaining an Hilbert-style axiomatization for a logic $\\vdash^{r}$ (Theorem \\ref{th: completezza calcolo Hilbert}) out of an axiomatization for (a finitary) logic $\\vdash$.\n\nFinally, in Section \\ref{sec: examples}, we put at work our machinery and characterize the axiomatization of containment companions of some well-known logics, namely of classical propositional logic, Belnap-Dunn and the Logic of Paradox.\n\n\n\\section{Preliminaries}\\label{sec: preliminari}\n\n\n\nFor standard background on universal algebra and abstract algebraic logic, we refer the reader, respectively, to \\cite{Be11g,BuSa00} and \\cite{Font16}. In this paper, algebraic languages are assumed not to contain constant symbols. \nMoreover, unless stated otherwise, we work within a fixed but arbitrary algebraic language. We denote algebras by $\\A, \\B, \\C\\dots$ respectively with universes $A, B, C \\dots$.\nLet $\\Fm$ be the algebra of formulas built up over a countably infinite set $\\Var$ of variables (which we indicate by $x,y,z,\\dots$). Given a formula $\\varphi\\in Fm$, we denote by $\\Var(\\varphi)$ the set of variables really occurring in $\\varphi$. Similarly, given $\\Gamma\\subseteq Fm$, we set\n\\[\n\\Var(\\Gamma)=\\bigcup \\{\\Var(\\gamma)\\colon \\gamma\\in\\Gamma\\}.\n\\]\nA \\emph{logic} is a substitution invariant consequence relation $\\vdash \\?\\? \\subseteq \\mathcal{P}(Fm) \\times Fm$ meaning that for every substitution $\\sigma \\colon \\Fm \\to \\Fm$,\n\\[\n\\text{if }\\Gamma \\vdash \\varphi \\text{, then }\\sigma [\\Gamma] \\vdash \\sigma (\\varphi).\n\\]\nGiven formulas $\\varphi, \\psi$, we write $\\varphi \\sineq \\psi$ as a shorthand for $\\varphi \\vdash \\psi$ and $\\psi \\vdash \\varphi$.\nA logic $\\vdash$ is \\emph{finitary} when for all $\\Gamma\\cup\\varphi\\subseteq Fm$:\n\\begin{align*}\n\\Gamma\\vdash\\varphi \\Longleftrightarrow \\exists \\Delta \\subseteq\\Gamma \\text{ such that } \\Delta \\text{ is finite and } \\Delta\\vdash\\varphi.\n\\end{align*}\n\n A \\emph{matrix} is a pair $\\langle \\A, F\\rangle$ where $\\A$ is an algebra and $F \\subseteq A$. In this case, $\\A$ is called the \\textit{algebraic reduct} of the matrix $\\langle \\A, F \\rangle$.\n \n \n Every class of matrices $\\mathsf{M}$ defines a logic as follows:\n\\begin{align*}\n\\Gamma \\vdash_{\\mathsf{M}} \\varphi \\Longleftrightarrow& \\text{ for every }\\langle \\A, F \\rangle \\in \\mathsf{M} \\text{ and homomorphism }h \\colon \\Fm \\to \\A,\\\\\n& \\text{ if }h[\\Gamma] \\subseteq F\\text{, then }h(\\varphi) \\in F.\n\\end{align*}\nWe say that a logic $\\vdash$ is \\emph{complete} with respect to a class of matrices $\\mathsf{M}$ when  $\\vdash_{\\mathsf{M}}\\;=\\;\\vdash$.\nSometimes, we will refer to such homomorphisms $h$ as \\emph{evaluations}.\n\n A matrix $\\langle \\A, F\\rangle$ is a \\emph{model} of a logic $\\vdash$ when\n\\begin{align*}\n\\text{if }\\Gamma \\vdash \\varphi, &\\text{ then for every homomorphism }h \\colon \\Fm \\to \\A,\\\\  \n&\\text{ if }h[\\Gamma] \\subseteq F\\text{, then }h(\\varphi) \\in F.\n\\end{align*}\nA set $F \\subseteq A$ is a (deductive) \\textit{filter} of $\\vdash$ on $\\A$, or simply a $\\vdash$-\\textit{filter}, when the matrix $\\langle \\A, F \\rangle$ is a model of $\\vdash$. We denote by $\\mathcal{F}i_{\\vdash}\\A$ the set of all filters of $\\vdash$ on $\\A$\n\n Although the present paper does not address the study of reduced models (for containment logics), in order to make it self-contained, we recall those notions, concerning reduced models, that will be used. Let $\\A$ be an algebra and $F \\subseteq A$. A congruence $\\theta$ of $\\A$ is \\emph{compatible} with $F$ when for every $a,b\\in A$,\n\\[\n\\text{if }a\\in F\\text{ and }\\langle a, b \\rangle \\in \\theta\\text{, then }b\\in F.\n\\]\nThe largest congruence of $\\A$ which is compatible with $F$ always exists, and is called the \\emph{Leibniz congruence} of $F$ on $\\A$. It is denoted by $\\Leibniz^{\\A}F$. \nThe \\emph{Suszko congruence} of $F$ on $\\A$, is defined as\n\\[\n\\@ifnextchar _  {\\mathchoice{\\tarskidsp\\kern-.07em}{\\tarskitxt\\kern-.07em} {\\tarskiscr\\kern-.07em}{\\tarskiscrscr\\kern-.07em}} {\\mathchoice{\\tarskidsp}{\\tarskitxt}{\\tarskiscr}{\\tarskiscrscr}}^{\\A}_{\\vdash}F \\coloneqq \\bigcap \\{ \\Leibniz^{\\A}G : F \\subseteq G \\text{ and }G \\in \\mathcal{F}i_{\\vdash}\\A \\}.\n\\]\n\n\n \n\n\n\n\nThe Leibniz and Suszko congruences allow to single out a distinguished class of models of logics. More precisely, given a logic $\\vdash$, we set\n\\begin{align*}\n\\Mod(\\vdash) & \\coloneqq \\{ \\langle \\A, F \\rangle : \\langle \\A, F \\rangle \\text{ is a model of }\\vdash \\};\\\\\n\\Modstar(\\vdash) & \\coloneqq \\{ \\langle \\A, F \\rangle \\in \\Mod(\\vdash) : \\Leibniz^{\\A}F \\text{ is the identity} \\};\\\\\n\\mathsf{Mod}^{\\textup{Su}}(\\vdash) & \\coloneqq \\{ \\langle \\A, F \\rangle \\in \\Mod(\\vdash) : \\@ifnextchar _  {\\mathchoice{\\tarskidsp\\kern-.07em}{\\tarskitxt\\kern-.07em} {\\tarskiscr\\kern-.07em}{\\tarskiscrscr\\kern-.07em}} {\\mathchoice{\\tarskidsp}{\\tarskitxt}{\\tarskiscr}{\\tarskiscrscr}}_{\\vdash}^{\\A}F \\text{ is the identity} \\}.\n\\end{align*}\nThe above classes of matrices are called, respectively, the classes of \\text{models}, \\textit{Leibniz reduced models} (or, simply reduced models), and \\textit{Suszko reduced models} of $\\vdash$.\n \n\n\nGiven a logic $\\vdash$, we set\n\\[\n\\Algstar(\\vdash) = \\{ \\A : \\text{ there is }F \\subseteq A \\text{ s.t. }\\langle \\A, F \\rangle \\in \\Modstar(\\vdash) \\},    \\text{ and}\n\\]\n\\[\n\\Alg(\\vdash) = \\{ \\A : \\text{ there is }F \\subseteq A \\text{ s.t. }\\langle \\A, F \\rangle \\in \\mathsf{Mod}^{\\textup{Su}}(\\vdash) \\}.\n\\]\n\n$\\Alg(\\vdash)$ is the class of algebraic reducts of matrices in $\\mathsf{Mod}^{\\textup{Su}}(\\vdash)$.\nThe class $\\Alg(\\vdash)$ is called the \\textit{algebraic counterpart} of $\\vdash$ as, for the vast majority of logics $\\vdash$, $\\Alg(\\vdash)$  is the class of algebras intuitively associated with $\\vdash$.\n\n\n\nTrivial matrices have a central role in the whole paper. We say that a matrix $\\langle \\A, F \\rangle$ is \\textit{trivial} if $F = A$. We denote by $\\langle \\boldsymbol{1}, \\{ 1 \\} \\rangle$ the trivial matrix, whose algebraic reduct $\\boldsymbol{1}$ is the trivial algebra.\nObserve that the latter matrix is a model (resp. reduced, Suszko reduced model) of every logic. Moreover, if $\\vdash$ is a logic and $\\langle \\A, F\\rangle \\in \\mathsf{Mod}^{\\textup{Su}}(\\vdash)$ is a trivial matrix, then $\\langle \\A, F\\rangle = \\langle \\boldsymbol{1}, \\{ 1 \\}\\rangle$.\n\nA set of models of a logic $\\vdash$ is said to be non trivial, if it does not contain trivial matrices. We indicate by $\\Mod_{+}(\\vdash)$ the set of non trivial models of a logic $\\vdash$.\n \n\n\n \n\\subsection*{P\\l onka sums}\n\n\nAs standard references on P\\l onka sums we mention \\cite{Plo67,Plo67a,Romanowska92}. A \\textit{semilattice} is an algebra $\\A = \\langle A, \\lor\\rangle$, where $\\lor$ is a binary associative, commutative and idempotent operation. Given a semilattice $\\A$ and $a, b \\in A$, we set\n\\[\na \\leq b \\Longleftrightarrow a \\lor b = b.\n\\]\nIt is easy to see that $\\leq$ is a partial order on $A$.\n\\begin{definition}\\label{Def:Directed system of algebras}\nA \\textit{direct system of algebras} consists of: \n\\benroman\n\\item a semilattice $I = \\langle I, \\lor\\rangle$;\n\\item a family of similar algebras $\\{ \\A_{i} : i \\in I \\}$ with pair-wise disjoint universes;\n\\item a homomorphism $f_{ij} \\colon \\A_{i} \\to \\A_{j}$, for every $i, j \\in I$ such that $i \\leq j$.\n\\eroman\nMoreover, $f_{ii}$ is the identity map for every $i \\in I$, and $f_{ik} = f_{jk} \\circ f_{ij}$, for $i \\leq j \\leq k$.\n\\end{definition}\n\nLet $X$ be a direct system of algebras as defined above. The \\textit{P\\l onka sum} of $X$, in symbols $\\PL(X)$ or $\\PL(\\A_{i})_{i \\in I}$\\footnote{When no confusion shall occur, we will write $\\PL(\\A_{i})$ instead of $\\PL(\\A_{i})_{i\\in I}$.}, is the algebra in the same type defined as follows: the universe of $\\PL(\\A_{i})_{i \\in I}$ is the union $\\bigcup_{i \\in I}A_{i}$. Moreover, for every $n$-ary basic operation $g$ and $a_{1}, \\dots, a_{n} \\in \\bigcup_{i \\in I}A_{i}$, we set\n\\[\ng^{\\PL(\\A_{i})_{i \\in I}}(a_{1}, \\dots, a_{n}) \\coloneqq g^{\\A_{j}}(f_{i_{1} j}(a_{1}), \\dots, f_{i_{n} j}(a_{n})),\n\\]\nwhere $a_{1} \\in A_{i_{1}}, \\dots, a_{n} \\in A_{i_{n}}$ and $j = i_{1} \\lor \\dots \\lor i_{n}$.\\ \n\nObserve that if in the above display we replace $g$ by any complex formula $\\varphi$ in $n$-variables, we still have that\n\\[\n\\varphi^{\\PL(\\A_{i})_{i \\in I}}(a_{1}, \\dots, a_{n}) = \\varphi^{\\A_{j}}(f_{i_{1} j}(a_{1}), \\dots, f_{i_{n} j}(a_{n})).\n\\]\n\\noindent \\textbf{Notation:} Given a formula $\\varphi$, we will often write $\\varphi^{\\PL}$ instead of $\\varphi^{\\PL(\\A_{i})_{i \\in I}}$ when no confusion shall occur. \n\n\\vspace{5pt}\n\n\n\nThe theory of P\\l onka sums is strictly related with a special kind of binary operation, called partition function.\n\n\\begin{definition}\\label{def: partition function}\nLet $\\A$ be an algebra of type $\\nu$. A function $\\cdot\\colon A^2\\to A$ is a \\emph{partition function} in $\\A$ if the following conditions are satisfied for all $a,b,c\\in A$, $ a_1 , ..., a_n\\in A $ and for any operation $g\\in\\nu$ of arity $n\\geqslant 1$.\n\\begin{enumerate}[label=\\textbf{P\\arabic*}., leftmargin=*]\n\\item $a\\cdot a = a$;\n\\item $a\\cdot (b\\cdot c) = (a\\cdot b) \\cdot c $;\n\\item $a\\cdot (b\\cdot c) = a\\cdot (c\\cdot b)$;\n\\item $g(a_1,\\dots,a_n)\\cdot b = g(a_1\\cdot b,\\dots, a_n\\cdot b)$;\n\\item $b\\cdot g(a_1,\\dots,a_n) = b\\cdot a_{1}\\cdot_{\\dots}\\cdot a_n $.\n\\end{enumerate}\n\\end{definition}\n\nDifferent definitions of partition function appeared in literature. We adopted the one which uses the minimal number of defining conditions (see \\cite{Romanowska92}). \n\nThe next result underlines the connection between P\\l onka sums and partition functions:\n\n\\begin{theorem}\\cite[Thm.~II]{Plo67}\\label{th: Teorema di Plonka}\nLet $\\A$ be an algebra of type $\\nu$ with a partition function $\\cdot$. The following conditions hold:\n\\begin{enumerate}\n\\item $A$ can be partitioned into $\\{ A_{i} : i \\in I \\}$ where any two elements $a, b \\in A$ belong to the same component $A_{i}$ exactly when\n\\[\na= a\\cdot b \\text{ and }b = b\\cdot a.\n\\]\nMoreover, every $A_{i}$ is the universe of a subalgebra $\\A_{i}$ of $\\A$.\n\\item The relation $\\leq$ on $I$ given by the rule\n\\[\ni \\leq j \\Longleftrightarrow \\text{ there exist }a \\in A_{i}, b \\in A_{j} \\text{ s.t. } b\\cdot a =b\n\\]\nis a semilattice order.\n\\item For all $i,j\\in I$ such that $i\\leq j$ and $b \\in A_{j}$, the map $f_{ij} \\colon A_{i}\\to A_{j}$, defined by the rule $f_{ij}(x)= x\\cdot b$ is a homomorphism. The definition of $f_{ij}$ is independent from the choice of $b$, since $a\\cdot b = a\\cdot c$, for all $a\\in A_i$ and $c\\in A_j$.\n\\item $Y = \\langle \\langle I, \\leq \\rangle, \\{ \\A_{i} \\}_{i \\in I}, \\{ f_{ij} \\! : \\! i \\leq j \\}\\rangle$ is a direct system of algebras such that $\\PL(Y)=\\A$.\n\\end{enumerate}\n\\end{theorem}\n\n\nThe statement of Theorem \\ref{th: Teorema di Plonka} displayed above relies on the assumption that the algebraic language contains no constant symbols\\footnote{When considering types containing constants, then additional conditions should be added to the definition of partition function. This results into a decomposition over a semilattice having a least element: constants of the P\\l onka sum will belong to the algebra whose index is the least element. For further details, see \\cite{plonka1984sum}.}. \nIt is worth remarking that the construction of Plonka sums preserves the validity of \\textit{regular identities}, i.e. identities of the form $ \\varphi \\thickapprox \\psi $ such that $\\Var(\\varphi) = \\Var(\\psi)$.\n\n\n\n\n\\section{Algebraic completeness}\\label{sec:compl}\n\nThe usual presentations of Kleene three-valued logics divide them into two families, depending on the meaning given to the connectives $\\land,\\lor$: \\emph{strong logics} -- including Strong Kleene and the Logic of Paradox \\cite{Priestfirst} -- and \\emph{weak logics} -- Bochvar logic ($\\mathrm{B_3}$) and paraconsistent weak Kleene (or Halld\\'en logic). Logics in each family differentiate upon the choice of the truth-set: $\\{1\\}$ in Strong Kleene and Bochvar, $\\{1,\\ant \\}$ in the logic of Paradox and paraconsistent weak Kleene.\n\nBochvar \\cite{BochvarBergmann} is the logic induced by the matrix $\\pair{\\WK,\\{1\\}}$ of the so-called weak Kleene tables\\footnote{In accordance with \\cite{Urquhart2001}, here we are actually considering Bochvar ``internal calculus'', which is only one of the two logics introduced in \\cite{BochvarBergmann}. The ``external calculus'' consists of a linguistic extension of the weak Kleene tables, with a connective $\\mathrm{t}$, interpreted (for every evaluation $h$) as $h(\\mathrm{t}\\varphi)= 1$ if and only if $h(\\varphi)=1$ (for further details, see \\cite{BochvarBergmann,KarpenkoTomova,FinnGrigolia}).} displayed in Figure \\ref{fig: tavole weak}. \n\n\\vspace{5pt}\n\\begin{figure}[h]\n\\begin{center}\\renewcommand{\\arraystretch}{1.25}\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}}\n   \\land & 0 & \\ant & 1 \\\\[.2ex]\n \\hline\n       0 & 0 & \\ant & 0 \\\\\n       \\ant & \\ant & \\ant & \\ant \\\\          \n       1 & 0 & \\ant & 1\n\\end{tabular}\n\\qquad\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}}\n   \\lor & 0 & \\ant & 1 \\\\[.2ex]\n \\hline\n     0 & 0 & \\ant & 1 \\\\\n     \\ant & \\ant & \\ant & \\ant \\\\          \n     1 & 1 & \\ant & 1\n\\end{tabular}\n\\qquad\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}}\n  \\lnot &  \\\\[.2ex]\n\\hline\n  1 & 0 \\\\\n  \\ant & \\ant \\\\\n  0 & 1 \\\\\n\\end{tabular}\n\n\\caption{The algebra $\\mathbf{WK}$ of weak Kleene tables.}\\label{fig: tavole weak}\n\\end{center}\n\\end{figure}\n\\vspace{10pt}\n\\noindent\nIt is not difficult to check that the algebra $\\mathbf{WK}$ is the P\\l onka sum of the two-element Boolean algebra $\\two$ and the trivial (Boolean) algebra $\\mathbf{\\ant}$ (over the index set given by the two-element semilattice)\\footnote{We refer the reader interested in further details directly to \\cite{Bonzio16}.}.\n\nBochvar logic can be equivalently presented as follows:\n\n\\begin{theorem}\\cite[Theorem 2.3.1]{Urquhart2001}\\label{teorema: Urquhart}\nThe following are equivalent: \n\\begin{enumerate}\n\\item $\\Gamma\\vdash_{\\mathrm{B_3}}\\varphi$;\n\\item $\\Gamma\\vdash_{\\ensuremath{\\mathsf{CL}}}\\varphi $ with $ \\Var(\\varphi)\\subseteq\\Var(\\Gamma)$ or $\\Gamma$ is an inconsistent set of formulas.  \n\\end{enumerate}\n\\end{theorem}\n \nIn words, Bochvar logic is the consequence relation obtained out of classical logics, imposing the constraint that variables of the conclusion (formula) shall be included into those of the (set of) premises. For this reason, $\\mathrm{B_{3}}$ is often referred to as a \\emph{containment logic}, see for e.g. \\cite{ferguson2017meaning,Parry}. \n\n\nThe following definition originates in \\cite{ThomasLavtesi}, but see also \\cite{CampercholiRaftery,JGR13}.\n\n\\begin{definition}\\label{def inconsistency terms}\nA set of formulas $\\Sigma$ is an antitheorem of a logic $\\vdash$ if\n$\\sigma[\\Sigma] \\vdash \\varphi$, for every substitution $\\sigma:\\Fm\\to\\Fm$ and formula $\\varphi$.\n\\end{definition}\n\nObserve that, if the set $\\Sigma(y_{1},\\dots,y_{n})$, where the variables $y_{1},\\dots,y_{n}$ really occur, is an antitheorem for $\\vdash$, then, by substitution, also $\\Sigma(x)$ (where only $x$ occurs) is an antitheorem for $\\vdash$. In other words, if a logic $\\vdash$ possesses an antitheorem $\\Sigma$, then it possesses an antitheorem in one variable only and we will write $\\Sigma(x)$.  \n\nThe most intuitive example one can keep in mind is the following: for any formula $\\varphi$, the set $\\{\\varphi,\\neg\\varphi\\}$ is an antitheorem of Intuitionistic, Classical and both local and global modal logics\n\n\n\nThe above characterization of Bochvar logic suggests that a logic $\\vdash^{r}$ satisfying an analogous criterion on the inclusion of variables as that of Theorem \\ref{teorema: Urquhart} can be associated to any arbitrary logic $\\vdash$. \n\n\\begin{definition}\\label{def: containment companion}\nLet $\\vdash$ be a logic. $\\vdash^{r}$ is the logic defined as \n\n\\[\n\\Gamma\\vdash^{r}\\varphi\\iff \\left\\{ \\begin{array}{ll}\n\\Gamma\\vdash\\varphi  \\ \\text{and} \\ \\Var(\\varphi)\\subseteq\\Var(\\Gamma)& \\text{or}\\\\\n\\Sigma(x)\\subseteq\\Gamma, & \\\\\n  \\end{array} \\right.  \n\\]\n\nwhere $\\Sigma(x)$ is an antitheorem of $\\vdash$.\n\\end{definition}\n\nWe will refer to $\\vdash^{r}$ as the \\emph{containment companion} of the logic $\\vdash$. It follows from the definition that $\\vdash^{r}$ and $\\vdash$ have the same antitheorems. \nBochvar logic is not the unique example of containment logic that can be found in literature. Indeed, the four-valued logic $\\mathrm{\\mathbf{S}_{fde}}$, introduced by Deutsch \\cite{Deutsch} (see also \\cite{BelikovPetrukhin, Szmuch2016, ciunilast}), can be counted as the containment companion of the Logic of Paradox (this follows from our analysis, see Subsection \\ref{Subsec: relevance}). Also one of the four-valued logics introduced by Tomova (see Example \\ref{ex: containment di PWK}) is a containment logic, more precisely the containment companion of \\ensuremath{\\mathsf{PWK}}.\n\n\nThe containment companion ($\\vdash^{r}$) of a logic $\\vdash$ is, somehow, ``opposite'' to the logic $\\vdash_{l}$ considered in \\cite{BonzioMorascoPrabaldi}, as the inclusion of variables works from conclusions to premises,  namely from ``right to left'', and not vice-versa (as for $\\vdash_{l}$).\n\\begin{lemma}\\label{lemma: finitarieta di r}\nLet $\\vdash$ be a finitary logic. Then $\\vdash^{r}$ is finitary.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $\\Gamma\\vdash^{r}\\varphi$. By definition of $\\vdash^{r}$, two cases are to be considered: \n\\begin{itemize}\n\\item[i)] $\\Gamma\\vdash\\varphi$ with $\\Var(\\varphi)\\subseteq\\Var(\\Gamma)$;\n\\item[ii)] $\\Gamma $ contains an antitheorem $\\Sigma(x)$ of $\\vdash$.\n\\end{itemize}\ni) Clearly, $\\mid\\Var(\\varphi)\\mid < \\infty$, hence there exist formulas $\\gamma_{1},\\dots,\\gamma_{n}\\in \\Gamma$ such that $\\Var(\\varphi)\\subseteq\\Var(\\gamma_{1})\\cup\\dots\\cup\\Var(\\gamma_{n})$. Since $\\vdash$ is finitary, then there exists a finite set $\\Gamma'\\subseteq\\Gamma$ such that $\\Gamma'\\vdash\\varphi$. If $\\Var(\\varphi)\\subseteq\\Var(\\Gamma')$, then also $\\Gamma'\\vdash^{r}\\varphi$, i.e. $\\vdash^{r}$ is finitary. So, suppose that $\\Var(\\varphi)\\not\\subseteq\\Var(\\Gamma')$. Consider $\\Delta=\\Gamma'\\cup\\{\\gamma_{1},\\dots,\\gamma_{n}\\}$. Obviously, $\\Delta$ is finite, $\\Var(\\varphi)\\subseteq\\Var(\\Delta)$ and $\\Delta\\subseteq\\Gamma$. By monotonicity of $\\vdash$, $\\Delta\\vdash\\varphi$, hence $\\Delta\\vdash^{r}\\varphi$, i.e. $\\vdash^{r}$ is finitary. \\\\\n\\noindent\nii) Since $\\Sigma(x)$ is an antitheorem for $\\vdash$, then $\\Sigma(x)\\vdash\\varphi$. Hence \\\\\n\\noindent\n$\\Sigma(x)\\vdash_{r}\\varphi$ and $\\Sigma(x)$ is finite (as $\\vdash$ is finite). \n\\end{proof}\n\n\n\n\nSince the algebra $\\WK$ is a P\\l onka sum (of Boolean algebras), it makes sense to ask whether the matrix $\\pair{\\WK,\\{1\\}}$ can be constructed as P\\l onka sum of (two) matrices. To the best of our knowledge, the construction of P\\l onka sums between matrices has been developed exclusively in \\cite{BonzioMorascoPrabaldi}. However, it is not difficult to check that the mentioned construction, when applied to the matrices $\\pair{\\two, \\{1\\}}$ and $\\pair{\\mathbf{\\ant}, \\emptyset}$ (where $\\two$ and $\\mathbf{\\ant}$ stand for the two-element Boolean algebra and the trivial algebra, respectively), does not result in  $\\pair{\\WK,\\{1\\}}$. This suggests that a different notion of direct system of logical matrices shall be introduced. \n\n\n\\begin{definition}\\label{Def:Directed-System-Matrices}\nAn \\textit{r-direct system} of matrices consists of: \n\\benroman\n\\item A semilattice $I = \\langle I, \\lor\\rangle$.\n\\item A family of matrices $\\{ \\langle\\A_{i},F_{i}\\rangle : i \\in I \\}$ such that \\\\ $I^{+}\\coloneqq\\{i\\in I: F_{i}\\neq\\emptyset\\} $ is a  sub-semilattice of $I$.\n\\item a homomorphism $f_{ij} \\colon \\A_{i} \\to \\A_{j}$, for every $i, j \\in I$ such that $i \\leq j$, satisfying also that: \n\\begin{itemize}\n\\item $f_{ii}$ is the identity map, for every $i \\in I$;\n\\item if $i \\leq j \\leq k$, then $f_{ik} = f_{jk} \\circ f_{ij}$;\n\\item if $F_{j}\\neq\\emptyset$ then $ f_{ij}^{-1}[F_{j}]= F_{i}$, for any $i\\leq j$.\n\\end{itemize}\n\\eroman\n\\end{definition}\n\nAs the nomenclature highlights, the above introduced notion of direct system of matrices is essentially different from the one in \\cite{BonzioMorascoPrabaldi}. The main difference concerns the interplay between homomorphisms of the system and matrices' filters.\n\n\nGiven a r-direct system of matrices $X$, we define a new matrix as\n\\[\n\\PL(X) \\coloneqq \\langle \\PL(\\A_{i})_{i \\in I}, \\bigcup_{i \\in I}F_{i}\\rangle.\n\\]\n\\vspace{5pt}\n\n\\noindent\nWe will refer to the matrix $\\PL(X)$ as the \\textit{P\\l onka sum} over the r-direct system of matrices $X$. Given a class $\\mathsf{M}$ of matrices, $\\PL(\\mathsf{M})$ will denote the class of all P\\l onka sums of r-direct systems of matrices in $\\mathsf{M}$.\n\n\n\nLet $h\\colon\\Fm\\to\\PL(\\A_i)$ be a homomorphism from the formula algebra into a generic P\\l onka sum of algebras. Then, for any formula $\\varphi\\in Fm$, we set \n\n\\[\n i_{h}(\\varphi)\\coloneqq\\bigvee\\{i\\in I:h(x)\\in A_{i}, x\\in\\Var(\\varphi)\\}.\n \\]\n \n \\vspace{2pt}\n\nIn words, $i_{h}(\\varphi)$ indicates the index where the formula $\\varphi$ is interpreted by the homomorphism $h$, into a P\\l onka sum. Moreover, for any $\\Gamma\\subseteq Fm$,\nwe set $i_{h}(\\Gamma)\\coloneqq\\bigvee \\{i_{h}(x)\\colon x\\in\\Var(\\Gamma)\\}$. \n\n\\begin{remark}\\label{rem: Gamma finito}\nNotice that the index $i_{h}(\\Gamma)$ is defined provided that the set $\\Var(\\Gamma)$ is finite. \nFor several results, we will assume that the logic $\\vdash$ is finite. Hence, by Lemma \\ref{lemma: finitarieta di r}, also $\\vdash^{r}$ is finitary, and this allows us to consider finite sets $\\Gamma\\subseteq Fm$, for which the existence of $i_{h}(\\Gamma)$ is assured.\nMoreover, observe that, for every homomorphism $h\\colon\\Fm\\to\\PL(X)$ from the formula algebra into a generic P\\l onka sum over an r-direct system of matrices $X$, and every $\\Gamma\\cup\\{\\varphi\\}\\subseteq Fm$, it is immediate to check that $\\Var(\\varphi)\\subseteq\\Var(\\Gamma)$ implies $i_{h}(\\varphi)\\leq i_{h}(\\Gamma)$.\n\\end{remark}\n\n\n\n\n\\begin{lemma}\\label{lemma: soundness}\nLet $X$ be an r-direct system of non trivial models of a finitary logic $\\vdash$. Then $\\PL(X)=\\langle\\PL(\\A_i),\\bigcup_{i\\in I}F_i\\rangle$ is a model of $\\vdash^{r}$.\n\\end{lemma}\n\n\\begin{proof}\n\nLet $X$ be an r-direct system of non trivial models of $\\vdash$. Assume $\\Gamma\\vdash^{r}\\varphi$. Since $\\vdash $ is finitary, so it is also $\\vdash^{r}$ (by Lemma \\ref{lemma: finitarieta di r}), there exists a finite subset $\\Delta\\subseteq\\Gamma $, such that $\\Delta\\vdash^{r}\\varphi$.\nWe distinguish the following cases:\n\n\\begin{itemize}\n\\item[(a)] $\\Sigma(x)\\subseteq\\Delta$, where $\\Sigma(x)$ is an antitheorem of $\\vdash$;\n\\vspace{3pt}\n\\item[(b)] $\\Delta\\vdash\\varphi$ with $\\Var(\\varphi)\\subseteq\\Var(\\Delta)$. \n\\end{itemize}\n\n\n\n\\noindent\nSince $X$ contains non-trivial models only, the case (a) easily follows by noticing that, for any homomorphism $h\\colon \\Fm\\to\\PL(\\A_i)$, $h[\\Sigma(x)]\\not\\subseteq F=\\bigcup_{i\\in I}F_i$.\nTherefore $\\Sigma(x)\\vdash_{\\PL(X)}\\varphi$, hence also $\\Delta\\vdash_{\\PL(X)}\\varphi$. \n\\vspace{5pt}\n\n\\noindent\nSuppose (b) is the case, i.e. $\\Delta\\vdash\\varphi$ with $\\Var(\\varphi)\\subseteq\\Var(\\Delta)$. Let $h\\colon\\Fm\\to\\PL(\\A_{i})$ be a homomorphism such that $h[\\Delta]\\subseteq F$.\nSince $\\Delta$ is a finite set, then we can fix $j\\coloneqq i_{h}(\\Delta)$ and, for any formula $\\delta\\in \\Delta$, we have $h(\\delta)\\in F_{i_{h}(\\delta)}$. This implies that each $i_{h}(\\delta)\\in I^{+}$ and, as $I^{+}$ forms a  sub-semilattice of $I$, we have that $j\\in I^{+}$.\n\nNow, define $g\\colon\\Fm\\to\\A_{j}$ as \n\\[\ng(x)\\coloneqq f_{i_{h}(x)j}\\circ h(x),\n\\]\nfor every $x\\in\\Var(\\Delta)$. For any $\\delta\\in \\Delta$, we have $g(\\delta)=f_{i_{h}(\\delta) j}\\circ h(\\delta)$, hence $g[\\Delta]\\subseteq F_{j}$. From the fact that $\\Delta\\vdash\\varphi$ and $\\langle\\A_{j},F_{j}\\rangle\\in\\Mod(\\vdash)$, it follows that $g(\\varphi)\\in F_{j}$. Setting $k\\coloneqq  i_{h}(\\varphi)$, by Remark \\ref{rem: Gamma finito}, we have $k\\leq j$ and this, together with the observation that $F_{j}\\neq\\emptyset$, implies $f_{kj}^{-1}[F_{j}]=F_{k}$. Moreover, we claim that $F_{k}\\neq\\emptyset$. Suppose, by contradiction, that $F_{k}=\\emptyset$. Then, by definition of r-direct system of matrices, we have that $f_{kj}^{-1}[F_{j}]=\\emptyset$, that is: there exists no $a\\in A_{k}$ such that $f_{kj}(a)\\in F_{j}$. On the other hand, since $\\Var(\\varphi)\\subseteq \\Var(\\Delta)$, then $g(\\varphi)=f_{kj}\\circ h(\\varphi)\\in F_{j}$, a contradiction.  \n\nFrom the fact that $g(\\varphi)\\in F_{j}$ together with $f_{kj}^{-1}[F_{j}]=F_{k}$, we conclude $h(\\varphi)\\in F_{k}$. This proves that $h(\\varphi)\\in F_{k}\\subseteq\\bigcup_{i\\in I}F_{i}$. \n\\end{proof}\n\n\\begin{remark}\\label{rem su inconsistency}\n\nObserve that the assumption on the non-triviality of models of the logic $\\vdash$ is crucial in Lemma \\ref{lemma: soundness}, as witnessed by the following example. Let $\\vdash$ be a theoremless logic possessing an anti-theorem $\\Sigma(x)$ (an example is the almost inconsistent logic). \n Set $X=\\pair{\\A\\oplus \\mathbf{1}, A}$ to be the r-direct system of models of $\\vdash$, consisting of the two algebras $\\mathbf{A}$ and $\\mathbf{1}$ with the unique homomorphism $f\\colon\\A\\to\\mathbf{1}$ (plus the identity homomorphisms).\nThen $\\Sigma(x)\\vdash y$, for an arbitrary variable $y$, and therefore $\\Sigma(x)\\vdash^{r} y$. However, $\\PL(X)$ is not a model of the latter inference (consider, for instance, an evaluation $v\\colon\\Fm\\to \\PL(\\A\\oplus \\mathbf{1})$ such that $v(x)=a\\in A$ and $v(y)=1$). \n\\end{remark}\n\nObserve that, if the logic $\\vdash$ does not possess an antitheorem, then the following holds:\n\\begin{corollary}\nLet $X$ be an r-direct system of models of a finitary logic  $\\vdash$ possessing no antitheorems. Then $\\PL(X)$ is a model of $\\vdash^{r}$.\n\\end{corollary}\n\n\n\nGiven a logic $\\vdash$ which is complete with respect to a class $\\mathsf{M}$ of matrices, we set  $\\mathsf{M^{\\emptyset}}\\coloneqq\\mathsf{M}\\cup\\langle\\A,\\emptyset\\rangle$, for any arbitrary $\\A\\in\\Alg(\\vdash)$.\n\n\n\n\\begin{theorem}\\label{completeness}\nLet $\\vdash$ be a finitary logic which is complete with respect to a class of non trivial matrices $\\mathsf{M}$. Then $\\vdash^{r}$ is complete with respect to $\\PL(\\mathsf{M^{\\emptyset}})$. \n\\end{theorem}\n\n\\begin{proof}\n\nWe aim at showing that $\\vdash^{r}=\\;\\vdash_{\\PL(\\mathsf{M^{\\emptyset}})}$. \\\\\n\\noindent\n($\\vdash^{r}\\;\\subseteq \\;\\vdash_{\\PL(\\mathsf{M^{\\emptyset}})}$). Consider $\\Gamma\\vdash^r\\varphi$ and a P\\l onka sum  $\\langle\\PL (\\A_i),\\bigcup_{i\\in I}F_{i}\\rangle$ of matrices in $\\mathsf{M^{\\emptyset}}$. Set $\\A=\\PL(\\A_i)$ The cases in which $\\Gamma$ is an antitheorem of $\\vdash$ or $\\langle\\A,\\emptyset\\rangle$ is a model of $\\vdash$ follow by Lemma \\ref{lemma: soundness}. \n\nSo, assume $\\langle\\A,\\emptyset\\rangle$ is not a model of $\\vdash$ and that $\\Gamma$ is not an antitheorem of $\\vdash$. \nLet $h\\colon Fm\\to \\A$ be a homomorphism such that $h[\\Gamma]\\subseteq \\bigcup_{i \\in I}F_{i}$. Suppose, in view of a contradiction, that $h(\\varphi)\\not\\in \\bigcup_{i \\in I}F_{i}$.\n Set $i_{h}(\\varphi) = j$ and $ i_{h}(\\Gamma) = k$; since $\\Var(\\varphi)\\subseteq\\Var(\\Gamma)$ then $j\\leq k$, by Remark \\ref{rem: Gamma finito}. We define a homomorphism $v\\colon \\Fm\\to \\A_k$, as follows\n\\[\nv(x)\\coloneqq f_{lk}\\circ h (x),\n\\]\nwhere $l=i_{h}(x)$. Clearly, $v[\\Gamma]=f_{kk}\\circ h[\\Gamma] = h[\\Gamma]\\subseteq F_{k}$ and $v(\\varphi)=f_{jk}\\circ h (\\varphi)\\in A_{k}\\smallsetminus F_k$, since $h(\\varphi)\\in A_{j}\\smallsetminus F_{j}$ and $F_{j}=f^{-1}_{jk}[F_k]$ (as we know that $F_k\\neq\\emptyset$). Therefore, we have $\\Gamma\\not\\vdash \\varphi$, which is a contradiction. \\\\\n\\noindent\n($\\vdash_{\\PL(\\mathsf{M^{\\emptyset}})}\\;\\subseteq\\;\\vdash^{r}$). By contraposition, we prove that $\\Gamma\\nvdash^{r}\\varphi $ implies $\\Gamma\\nvdash_{\\PL(\\mathsf{M^{\\emptyset}})}\\varphi$. To this end, assume $\\Gamma\\nvdash^{r}\\varphi$. If $\\Var(\\varphi)\\subseteq\\Var(\\Gamma)$, clearly $\\Gamma\\nvdash\\varphi$. Therefore there exists a matrix $\\langle\\A_{i},F_{i}\\rangle\\in\\mathsf{M}$ and a homomorphism $h\\colon\\Fm\\to\\A_{i}$ such that $h[\\Gamma]\\subseteq F_i$ and $h(\\varphi)\\notin F_i$. Upon considering the r-direct system $X=\\pair{\\langle\\A_{i},F_{i}\\rangle, \\{i\\}, id}$ and the homomorphism $h$, we immediately obtain $\\Gamma\\nvdash_{\\PL(\\mathsf{M^{\\emptyset}})}\\varphi$.\n\nThe only other case to consider is $\\Var(\\varphi)\\nsubseteq\\Var(\\Gamma)$. Preliminarily, observe that the assumption $\\Gamma\\nvdash^{r}\\varphi$ implies that $\\Gamma$ contains no anti-theorem $\\Sigma(x)$ for $\\vdash$. Therefore, since  $\\mathsf{M}$ is a class of models complete with respect to $\\vdash$, there exists a matrix $\\langle\\B,G\\rangle\\in\\mathsf{M}$ and a homomorphism $v\\colon\\Fm\\to\\B$ such that $v[\\Gamma]\\subseteq G$ and $v(\\varphi)\\notin G$. Consider any r-direct system formed by the matrices $\\langle\\B,G\\rangle$ and  $\\langle \\A,\\emptyset \\rangle$ for an appropriate $\\A\\in\\Alg(\\vdash)$  (the choice $\\A=\\boldsymbol{1}$ is always appropriate), with $\\langle\\A,\\emptyset\\rangle$  indexed as top element. Denote by $\\B \\oplus \\A^{\\emptyset}$ a P\\l onka sum over the r-direct system just described. \n\nThe homomorphism $g\\colon\\Fm\\to\\B \\oplus \\A^{\\emptyset}$ defined as\n\\[\ng(x)\\coloneqq \\left\\{ \\begin{array}{ll}\nv(x) \\ \\text{if} \\ \\ x\\in\\Var(\\Gamma),& \\\\\na \\ \\text{otherwise.} & \\\\\n  \\end{array} \\right.  \n\\]\nfor arbitrary $a\\in A$\n\\noindent\neasily witnesses $\\Gamma\\nvdash_{\\PL(\\mathsf{M^{\\emptyset}})}\\varphi$, as desired.\n\\end{proof}\n\n\\begin{example}\\label{ex: containment di PWK}\nLet $\\mathbf{K_{4}}=\\pair{\\{0,1,\\ant, n \\}, \\neg,\\land,\\vee}$ be the algebra given by the following tables \n\\vspace{5pt}\n\\begin{center}\\renewcommand{\\arraystretch}{1.25}\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}}\n  \\lnot &  \\\\[.2ex]\n\\hline\n  1 & 0 \\\\\n  \\ant & \\ant \\\\\n  n & n \\\\\n  0 & 1 \\\\\n\\end{tabular}\n\\qquad\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}}\n   \\land & 0 & \\ant & n & 1 \\\\[.2ex]\n \\hline\n       0 & 0 & \\ant & n &  0 \\\\\n       \\ant & \\ant & \\ant & n & \\ant \\\\\n       n & n & n & n & n \\\\          \n       1 & 0 & \\ant & n & 1 \n\\end{tabular}\n\\qquad\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}}\n   \\lor & 0 & \\ant & n & 1 \\\\[.2ex]\n \\hline\n       0 & 0 & \\ant & n &  1 \\\\\n       \\ant & \\ant & \\ant & n & \\ant \\\\\n       n & n & n & n & n \\\\          \n       1 & 1 & \\ant & n & 1 \n\\end{tabular}\n\n\\end{center}\n\\vspace{10pt}\n\nThe logic $\\mathbf{K^{w}_{4n}}= \\pair{\\mathbf{K_4},\\{1,\\ant\\}}$ is included among the four-valued regular logics counted by Tomova (see \\cite{Tomova,Petrukhin2}). \nObserve that $\\pair{\\mathbf{K_4},\\{1,\\ant\\}}$ is the P\\l onka sum (over the r-direct system) of the matrices $\\pair{\\mathbf{WK},\\{1,\\ant\\}}$ and $\\pair{\\mathbf{n},\\emptyset}$. Since \\ensuremath{\\mathsf{PWK}} is complete with respect to $\\pair{\\mathbf{WK},\\{1,\\ant\\}}$, then, it follows by Theorem \\ref{completeness}, that $\\mathbf{K^{w}_{4n}} =\\; \\vdash^{r}_{\\ensuremath{\\vDash_\\mathsf{pwk}}}$, i.e. $\\mathbf{K^{w}_{4n}}$ is the containment companion of \\ensuremath{\\mathsf{PWK}}. \n\\qed\n\\end{example}\n\nAs \\ensuremath{\\mathsf{PWK}} is the \\emph{left variable inclusion} companion of classical logic (see \\cite{BonzioMorascoPrabaldi}), the above example shows that the constructions yielding the (two) companions (left variable inclusion and containment) can actually be iterated, in alternation, starting from an arbitrary logic $\\vdash$ (for further details see \\cite{prabaldisub}). \n\n\\begin{remark}\nObserve that if $\\vdash$ is a logic which is complete with respect to a finite set of finite matrices, then so is $\\vdash^r$ (by Theorem \\ref{completeness}). This means that the containment companion of a logic $\\vdash$ preserves ``finite valuedness'', a notion introduced and studied in \\cite{CALEIROfinite}. \n\\end{remark}\n\n   \n\n\nTheorem \\ref{completeness} provides a complete class of matrices for an arbitrary logic of right variable inclusion. This class is obtained performing P\\l onka sums over r-direct systems of models of $\\vdash$ together with the matrices $\\langle \\A,\\emptyset\\rangle$ for any $\\A\\in\\Alg(\\vdash)$. Obviously, it is not generally the case that the matrix $\\langle \\A,\\emptyset\\rangle $ is a model of a logic $\\vdash$.\nFor this reason, it is not always true that P\\l onka sums over an r-direct systems of models of $\\vdash$ provide a complete matrix semantics for $\\vdash^{r}$. In this sense, the right variable inclusion companion of a logic is a logic of ``P\\l onka sums'' (of matrices) in weaker sense compared to the case of the left variable inclusion companion, fully described in \\cite{BonzioMorascoPrabaldi}.\nNonetheless, if $\\langle \\boldsymbol{1},\\emptyset\\rangle\\in\\Mod(\\vdash)$, the correspondence between $\\vdash^{r}$ and P\\l onka sums is fully recovered. This is actually the case of every theoremless logic, such as Strong Kleene Logic, $\\vdash_{\\ensuremath{\\mathsf{CL}}}^{\\land,\\lor}$.\n\n\\begin{corollary}\\label{cor completezza con incons}\nA finitary containment logic $\\vdash^{r}$ is complete w.r.t. any of the following classes of matrices:\n\\begin{center}\n$\\PL(\\Mod_{+}(\\vdash)\\cup\\langle \\A,\\emptyset\\rangle)$, $\\PL(\\Modstar_{+}(\\vdash)\\cup\\langle \\A,\\emptyset\\rangle)$, $\\PL(\\mathsf{Mod}^{\\textup{Su}}_{+}(\\vdash)\\cup\\langle \\A,\\emptyset\\rangle),$\n\\end{center}\nfor $\\A\\in\\Alg(\\vdash)$.\n\\end{corollary}\n\n\\noindent\nMoreover, observing that if $\\langle \\mathbf{1},\\emptyset\\rangle\\in\\Mod(\\vdash)$ then $\\langle \\mathbf{1},\\emptyset\\rangle\\in\\Modstar(\\vdash)$, the following holds\n\n\\begin{corollary}\nLet $\\vdash$ be a finitary logic such that $\\langle \\mathbf{1},\\emptyset\\rangle\\in\\Mod(\\vdash)$. Then $\\vdash^{r}$ is complete w.r.t. any of the following classes of matrices:\n\\begin{center}\n$\\PL(\\Mod_{+}(\\vdash))$, $\\PL(\\Modstar_{+}(\\vdash))$, $\\PL(\\mathsf{Mod}^{\\textup{Su}}_{+}(\\vdash)).$\n\\end{center}\n\\end{corollary}\n\nIn case $\\vdash$ does not possess anti-theorems, then the above corollaries can be restated as follows\n\n\\begin{corollary}\\label{cor da usare per calcolo hilbert no inc}\nLet $\\vdash$ be a finitary logic without antitheorems. Then $\\vdash^{r}$ is complete w.r.t. any of the following classes of matrices:\n\\begin{center}\n$\\PL(\\Mod(\\vdash)\\cup\\langle\\A,\\emptyset\\rangle)$, $\\PL(\\Modstar(\\vdash)\\cup \\langle\\A,\\emptyset\\rangle)$, $\\PL(\\mathsf{Mod}^{\\textup{Su}}(\\vdash)\\cup\\langle\\A,\\emptyset\\rangle),$\n\\end{center}\nfor any $\\A\\in\\Alg(\\vdash)$.\n\\end{corollary}\n\n\n\\begin{corollary}\nLet $\\vdash$ a finitary logic without antitheorems such that $\\langle \\boldsymbol{1},\\emptyset\\rangle\\in\\Mod(\\vdash)$, then $\\vdash^{r}$ is complete w.r.t. any of the following classes of matrices:\n\\begin{center}\n$\\PL(\\Mod(\\vdash))$, $\\PL(\\Modstar(\\vdash))$, $\\PL(\\mathsf{Mod}^{\\textup{Su}}(\\vdash)).$\n\\end{center}\n\\end{corollary}\n\n\n\n\n\n\n\n\n\\section{Hilbert style calculi (for logics with r-partition function)}\\label{sec:hilbert}\n\nPartitions functions, which have been defined for algebras (see Definition \\ref{def: partition function}), can be defined also for logics. \n\n\\begin{definition}\\label{def: logic with p-func.}\nA logic $\\vdash$ has a \\emph{r-partition function} if there is a formula $x\\ast y$, in which the variables $x$ and $y$ really occur, such that: \n\\benroman\n\\item$x,y\\vdash x\\ast y$, \n\\item$x\\ast y \\vdash x$,\n\\item $\\varphi(\\varepsilon,\\vec{z}\\?\\?)\\sineq\\varphi(\\delta,\\vec{z}\\?\\?)$,\n\\eroman\nfor every formula $\\varphi(v,\\vec{z}\\?\\?)$ and every identity of the form $\\varepsilon \\thickapprox \\delta$ in Definition \\ref{def: partition function}.\n\\end{definition}\n\n\nCondition (iii) in the Definition of r-partition function is actually equivalent to say that the term operation $\\ast$ is a partition function in every algebra $\\A\\in\\Alg(\\vdash)$. This is the consequence of the following (known) fact in abstract algebraic logic. \n\n\\begin{lemma}\\label{lemma su equazioni p-function}\nLet $\\vdash$ be a logic and $\\varepsilon,\\delta\\in Fm $. The following are equivalent:\n\\benroman\n\\item $\\Alg(\\vdash)\\vDash\\varepsilon\\approx\\delta$;\n\\item $\\varphi(\\varepsilon,\\vec{z}\\?\\?)\\sineq\\varphi(\\delta,\\vec{z}\\?\\?)$, for every formula $\\varphi(v,\\vec{z}\\?\\?)$.\n\\eroman\n\\end{lemma}\n\\begin{proof}\nSee \\cite[Lemma 5.74(1)]{Font16} and \\cite[Theorem 5.76]{Font16}.\n\\end{proof}\n\n\n\n\nFrom now on, we will denote both the formula $x\\ast y$ and the term operation $\\ast$ as r-partition functions with respect to a logic $\\vdash$.\n\nThe definition of partition functions for an arbitrary logic is introduced also in \\cite[Definition 16]{BonzioMorascoPrabaldi}. It shall be noticed that Definition \\ref{def: logic with p-func.} is essentially different (this also motivates the choice of the terminology $r$-partition function). Nevertheless, in most cases (for instance, all substructural logics, classical and modal logics) the very same formula plays both the role of a r-partition function and of a partition function in the sense of \\cite[Definition 16]{BonzioMorascoPrabaldi}. \n\n\\begin{example}\nLogics with a r-partition function abound in the literature. Indeed, the term $x\\ast y\\coloneqq x\\land(x\\lor y)$ is a partition function for every logic $\\vdash$ such that $\\Alg(\\vdash)$ has a lattice reduct. Such examples include all modal and substructural logics \\cite{GaJiKoOn07}. On the other hand, the term $x\\ast y\\coloneqq (y \\to y) \\to x$ as a r-partition function for all the logics $\\vdash$ whose class $\\Alg(\\vdash)$ possesses a Hilbert algebra (see \\cite{Di65}) or a BCK algebra (see \\cite{Iseki}) reduct.\n\\end{example}\n\n\\begin{remark}\\label{rem: vdash e vdashr hanno le stessa pf}\nIt is easily checked that a logic $\\vdash$ has r-partition function $\\ast$ if and only if $\\vdash^{r}$ has r-partition function $\\ast$.\n\\end{remark}\n\nIn the following, we extend P\\l onka representation theorem to r-direct systems of logical matrices.\n\n\n\n\n\\begin{theorem}\\label{th: plonka sum of r- matrices}\nLet $\\vdash$ be a logic with $r$-partition function $\\ast$, and $\\langle\\A,F\\rangle$ be a model of $\\vdash$ such that $\\A\\in\\Alg(\\vdash) $. Then Theorem \\ref{th: Teorema di Plonka} holds for $\\A$.\nMoreover, by setting $F_{i} \\coloneqq F\\cap A_{i}$ for every $i \\in I$, the triple\n\\[\nX=\\langle \\langle I,\\leq\\rangle , \\{ \\langle \\A_{i},F_{i}\\rangle\\}_{i\\in I}, \\{ f_{ij} \\! : \\! i \\leq j \\}\\rangle\n\\]\nis an r-direct system of matrices such that $\\PL(X)=\\langle\\A,F\\rangle$.\n\\end{theorem}\n\n\\begin{proof}\nTheorem \\ref{th: Teorema di Plonka} holds for $\\A$, by simply observing that $\\ast$ is a partition function for $\\A$. \nFor the remaining part, it will be enough to show: \n\\begin{itemize}\n\\item[(a)] for every $i,j\\in I$ such that $i\\leq j$, if $F_{j}\\neq\\emptyset$ then $f_{ij}^{-1}[F_{j}]=F_{i}$;\n\\item[(b)]  $I^{+}$ is a sub-semilattice of $I$.\n\\end{itemize}\n\n\nIn order to prove (a), consider $i,j\\in I$ such that $i\\leq j$ and let $F_{j}$ be non-empty. Assume, in view of a contradiction, that $f_{ij}^{-1}[F_{j}]\\neq F_{i}$. This implies that $F_{i}\\nsubseteq f_{ij}^{-1}[F_{j}]$ or that $f_{ij}^{-1}[F_{j}]\\nsubseteq F_{i}$. The first case immediately leads to the contradiction that $x,y\\nvdash x\\ast y$, while the second case contradicts $x\\ast y\\vdash x$. This proves (a).\n\n\nIn order to prove (b), consider $i,j\\in I^{+}$ and let $k= i\\lor j$, with $i,j,k\\in I$.  As $\\ast$ is a $r$-partition function  for $\\vdash$, $x,y\\vdash x\\ast y$. Since $i,j\\in I^{+}$, then $F_i$ and $F_j$ are non-empty, therefore there exist two elements $a\\in F_{i}$, $b\\in F_{j}$. We have $a\\ast^{\\A}b=f_{ik}(a)\\ast^{\\A_{k}}f_{jk}(b)\\in A_{k}$. This, together with the fact that $\\langle\\A,F\\rangle\\in\\Mod(\\vdash)$ implies $a\\ast b\\in F_{k}$, i.e. $F_{k}\\neq\\emptyset$. So $k\\in I^{+}$ and this proves (b).\n\\end{proof}\n\nGiven a logic $\\vdash$ with a r-partition function $\\ast$ and a model $\\langle\\A,F\\rangle$ of $\\vdash$ such that $\\A\\in\\Alg(\\vdash)$, we call \\emph{P\\l onka fibers} of $\\langle \\A, F \\rangle$ the matrices $\\{ \\langle \\A_{i},F_{i}\\rangle\\}_{i\\in I}$ given by the decomposition in Theorem \\ref{th: plonka sum of r- matrices}. From now on, when considering a model $\\pair{\\A,F}$ of a logic $\\vdash$ with $r$-partition function, we will assume that $\\pair{\\A, F}= \\PL(X)$, for a given direct system $\nX=\\langle \\langle I,\\leq\\rangle , \\{\\langle \\A_{i},F_{i}\\rangle\\}_{i\\in I}, \\{ f_{ij} \\! : \\! i \\leq j \\} \\rangle\n$, without explicitly mentioning the r-direct system $X$.\n\n\n\n\\begin{lemma}\\label{lemma su filtri non vuoti che sono modelli della logica iniziale}\nLet $\\vdash^{r}$ be a logic with $r$-partition function $\\ast$, and $\\langle\\A,F\\rangle\\in\\Mod(\\vdash^{r})$, with $\\A\\in\\Alg(\\vdash^{r})$. Then, the P\\l onka fibers $ \\langle\\A_{i},F_{i}\\rangle $, such that $i\\in I^{+}$, are models of $\\vdash$.\n\\end{lemma}\n\n\\begin{proof}\n\nLet $\\Gamma\\vdash\\varphi$ and suppose, by contradiction, that there exists a matrix $\\pair{\\A_j, F_j}$, with $j\\in I^{+}$, and a homomorphism $h\\colon\\Fm\\to\\A_{j}$ such that $h[\\Gamma]\\subseteq F_{j}$ and $h(\\varphi)\\notin F_{j}$. Preliminarily, observe that $\\Var(\\varphi)\\nsubseteq\\Var(\\Gamma)$ and, moreover, if $\\vdash$ has an antitheorem $\\Sigma(x)$, then $\\Sigma(x)\\nsubseteq\\Gamma$, for otherwise $\\Gamma\\vdash^{r}\\varphi$, which is in contradiction with our assumption that  $\\langle\\A,F\\rangle\\in\\Mod(\\vdash^{r})$. Denote by $X$ the (non-empty) set of variables occurring in $\\varphi$ but not in $\\Gamma$ and, for $\\gamma\\in \\Gamma$, let $X_{\\gamma}\\coloneqq\\{\\gamma\\ast x: x\\in X\\}$ and $\\Gamma^{-}_{\\gamma}\\coloneqq \\Gamma\\smallsetminus\\{\\gamma\\}$. Since $\\ast$ is a r-partition function for $\\vdash^{r}$, we have $\\gamma\\ast x\\vdash^{r} \\gamma$. Therefore $\\gamma\\ast x\\vdash \\gamma$ and $X_{\\gamma}\\vdash \\gamma$, which implies $X_{\\gamma},\\Gamma^{-}_{\\gamma}\\vdash\\varphi$, for any $\\gamma\\in\\Gamma$. Observe that $\\Var(\\varphi)\\subseteq\\Var (X_{\\gamma})\\cup\\Var(\\Gamma^{-}_{\\gamma})$, hence $X_{\\gamma},\\Gamma^{-}_{\\gamma}\\vdash^{r}\\varphi$.\n\nSince $h(\\gamma),h(\\varphi)\\in A_{j}$ and $x\\in\\Var(\\varphi)$, for every $x\\in X$, we have that $h(\\gamma\\ast x)=h(\\gamma)$, whence $h[X_{\\gamma}]= h(\\gamma)$. Now, for an arbitrary $a\\in A$, we define a homomorphism $g\\colon\\Fm\\to\\A$, as follows \n\\[\ng(x)\\coloneqq \\left\\{ \\begin{array}{ll}\nh(x) \\ \\text{if} \\ \\ x\\in\\Var(\\Gamma)\\cup\\Var(\\varphi)& \\\\\na \\ \\text{otherwise.} & \\\\\n  \\end{array} \\right.  \n\\]\n\nWe have $g[X_{\\gamma}] = h[X_{\\gamma}]= h(\\gamma)\\in F_j$, $g[\\Gamma^{-}_{\\gamma}]=h[\\Gamma^{-}_{\\gamma}]\\in F_j$ and $g(\\varphi)=h(\\varphi)\\not\\in F_j$. A contradiction.\n\\end{proof}\n\n\n\nThe following example provides a simple instance of the above Lemma \\ref{lemma su filtri non vuoti che sono modelli della logica iniziale}.\n\n\\begin{example}\\label{example lemma 24}\nConsider the matrix $\\pair{\\B, F}$ constructed as P\\l onka sum of the (non-trivial) Boolean algebras $\\A_{i},\\A_{j}\\A_{k},\\A_{s}$ over the r-direct system depicted below (the index set is the four-element Boolean algebra): circles indicate filters, consisting of the top elements $1_i, 1_j$ of $\\A_i,\\A_j$, respectively. Thus, $F = \\{1_{i},1_{j}\\}$. Dotted lines represent arbitrary P\\l onka homomorphisms.  \n\\begin{center}\n\n\\begin{tikzpicture}\n\\draw (-2,6) node {$\\bullet$};\n\\draw (-2,7) node {$ \\A_{s}$};\n\\draw (-2,8) node {$\\bullet$};\n\n\\draw (-4,5) node {$\\bullet$};\n\\draw (-4,4) node {$ \\A_{k}$};\n\\draw (-4,3) node {$\\bullet$};\n\n\\draw (0,5) node {\\circled{$\\bullet$}};\n\\draw (0,4) node {$ \\A_{j}$};\n\\draw (0,3) node {$\\bullet$};\n\n\\draw (-2,0) node {$\\bullet$};\n\\draw (-2,1) node {$ \\A_{i}$};\n\\draw (-2,2) node {\\circled{$\\bullet$}};\n\n\n\\draw (-2,6) edge[bend left=70] (-2,8) ;\n\\draw (-2,6) edge[bend right=70] (-2,8);\n\t\\draw (-4,5) edge[ dotted] (-2,8) ;\n\t\t\\draw (-4,3) edge[dotted] (-2,6) ;\t\n\t\t\\draw (0,5) edge[dotted] (-2,8) ;\n\t\t\\draw (0,3) edge[dotted] (-2,6) ;\n\t\t\n\\draw (-4,5) edge[bend left=70] (-4,3) ;\n\\draw (-4,5) edge[bend right=70] (-4,3) ;\n\t\\draw (-2,2) edge[dotted] (-4,5) ;\n\t\t\\draw (-2,0) edge[dotted] (-4,3) ;\n\n\\draw (0,5) edge[bend left=70] (0,3) ;\n\\draw (0,5) edge[bend right=70] (0,3) ;\n\n\\draw (-2,0) edge[bend left=70] (-2,2) ;\n\\draw (-2,0) edge[bend right=70] (-2,2) ;\n\t\\draw (-2,2) edge[dotted] (0,5) ;\n\t\t\\draw (-2,0) edge[dotted] (0,3) ;\n\n\\end{tikzpicture}\n\\end{center}\n\nBy Theorem \\ref{completeness}, $\\pair{\\B, F}$ is a model of Bochvar logic $\\mathsf{B_{3}}$. It can be easily checked that $\\B\\in \\Alg(\\mathsf{B}_3)$.\nMoreover, $I^+=\\{i,j\\}$ and clearly $\\langle\\A_i,1_i\\rangle,\\langle\\A_j,1_j\\rangle\\in\\Mod(\\ensuremath{\\mathsf{CL}})$.\n\\end{example}\n\n\nThe presence of a r-partition function yields an important syntactic consequence: it allows to adapt a Hilbert style calculus of a logic $\\vdash$ into a calculus, for its contaiment companion $\\vdash^{r}$. Despite $\\vdash^{r}$ is defined via a linguistic restriction constraint (on the inclusion of variables), the axiomatization that we obtain is free of any (linguistic) restriction. \n\nThroughout the remaining part of this section, we implicitly assume that the logic $\\vdash$ possesses an antitheorem. Our analysis can be easily adapted to the case where $\\vdash$ does not have antitheorems (see Remark \\ref{rem: Hilbert senza inconsistency terms}).\n\n\n In what follows, by a \\textit{Hilbert-style calculus with finite rules}, we understand a (possibly infinite) set of Hilbert-style rules, each of which has finitely many premises.\n\n\\begin{definition}\\label{def: hilbert calc per r}\nLet $\\mathcal{H}$ be a Hilbert-style calculus with finite rules, which determines a logic $\\vdash$ with a $r$-partition function $\\ast$ and an antitheorem $\\Sigma(x)$.  Let $\\mathcal{H}^{r}$ be the Hilbert-style calculus given by the following rules: \n\n\\begin{align}\nx\\ast \\varphi &\\rhd \\varphi \\tag{H0}\\label{Eq:Axiom0}\\\\\nx,y &\\rhd x\\ast y \\tag{H1}\\label{Eq:Axiom1}\\\\\nx\\ast y &\\rhd  x\\tag{H2}\\label{Eq:Axiom2}\\\\\n\\{\\gamma_{1},\\dots,\\gamma_{n}\\}\\smallsetminus\\{\\gamma_{i}\\}, \\gamma_{i}\\ast\\psi &\\rhd \\psi\\tag{H3}\\label{Eq:Axiom3} \\\\\n\\Sigma(x) &\\rhd \\alpha \\tag{H4}\\label{Eq:Axiom4}\\\\\n\\chi(\\delta, \\vec{z}\\?\\?) \\?\\?\\?\\lhd&\\rhd \\chi(\\varepsilon, \\vec{z}\\?\\?)\\tag{H5}\\label{Eq:Axiom5}\n\\end{align}\n\nfor every\n\\begin{enumerate}[(i)]\n\\item $\\rhd\\varphi$ axiom in $\\mathcal{H}$ ;\n\\item $\\gamma_{1},\\dots,\\gamma_{n}\\rhd \\psi$ rule in $\\mathcal{H}$ (and $\\gamma_i$ such that $i\\in\\{i,\\dots,n\\}$);\n\\item $\\delta \\thickapprox\\varepsilon$ equation in the definition of partition function, and formula $\\chi(v, \\vec{z}\\?\\?)$.\n\\end{enumerate}\n\\end{definition}\n\n\n\\begin{lemma}\\label{lemma: spezza dimostrazione completezza Hilbert}\nLet $\\vdash$ be a logic with a $r$-partition function $\\ast$, an antitheorem $\\Sigma(x)$ and let $\\langle\\A,F\\rangle\\in\\mathsf{Mod}^{\\textup{Su}}(\\vdash_{\\mathcal{H}^{r}})$. Then:\n\\benroman\n\\item $\\pair{\\A,F}=\\PL(X)$, where $X=\\pair{\\langle \\langle I,\\leq\\rangle, \\{ \\langle \\A_{i},F_{i}\\rangle\\}_{i\\in I}, \\{ f_{ij} \\! : \\! i \\leq j \\}}$ is an r-direct system of matrices;\n\\item if $X$ contains a trivial matrix then $\\A = \\mathbf{1}$.\n\\eroman\n\\end{lemma}\n\\begin{proof}\n(i) Since $\\langle\\A,F\\rangle\\in\\mathsf{Mod}^{\\textup{Su}}(\\vdash_{\\mathcal{H}^{r}})$, $\\A\\in\\Alg(\\vdash_{\\mathcal{H}^{r}})$. Moreover, observe that $\\ast $ is a $r$-partition function for $\\vdash_{\\mathcal{H}^{r}}$ (thanks to conditions \\eqref{Eq:Axiom1}, \\eqref{Eq:Axiom2}, \\eqref{Eq:Axiom5}). These facts, together with Theorem \\ref{th: plonka sum of r- matrices}, implies that $\\langle\\A,F\\rangle=\\PL(X)$, where $X=\\langle\\{ \\langle \\A_{i},F_{i}\\rangle\\}_{i\\in I}, \\{ f_{ij} \\! : \\! i \\leq j \\}, \\langle I,\\leq\\rangle\\rangle$ is an r-direct system of matrices.\\\\\n\\noindent\n(ii) Suppose that, for some $j\\in I$, $\\langle\\A_{j},F_{j}\\rangle$ is a trivial fiber of $\\langle\\A,F\\rangle$, i.e. $F_j = A_j$. Since $\\Sigma(x)$ is an anti-theorem (for $\\vdash$) and \\eqref{Eq:Axiom4}  is a rule of $\\mathcal{H}^{r}$, then, for every $i\\in I$, we have $A_{i}=F_{i}$, i.e. each fiber is trivial. Indeed, if there exists a non trivial fiber $\\langle\\A_{k},F_{k}\\rangle$ and an  element $c\\in A_{k}\\smallsetminus F_{k}$, then the evaluation $h\\colon\\Fm\\to\\A$, defined as $h(x)=a$, $h(y)=c$ (for an arbitrary $a\\in\\A_{j}$) is such that $h[\\Sigma(x)]\\subseteq F$ while $h(y)\\notin F$, against the fact that $\\Sigma(x)\\vdash_{\\mathcal{H}^{r}}y$. Moreover, the facts that each fiber is trivial and that $\\@ifnextchar _  {\\mathchoice{\\tarskidsp\\kern-.07em}{\\tarskitxt\\kern-.07em} {\\tarskiscr\\kern-.07em}{\\tarskiscrscr\\kern-.07em}} {\\mathchoice{\\tarskidsp}{\\tarskitxt}{\\tarskiscr}{\\tarskiscrscr}}^{\\A}F=id$ immediately implies $\\A=\\mathbf{1}$.\n\\end{proof}\n\n \n \n \\begin{theorem}\\label{th: completezza calcolo Hilbert}\n Let $\\vdash$ be a finitary logic with a $r$-partition function $\\ast$ and an antitheorem $\\Sigma(x)$. Let, moreover, $\\mathcal{H}$ be a Hilbert style calculus  with finite rules. If $\\mathcal{H}$ is complete for $\\vdash$, then $\\mathcal{H}^{r}$ is complete for $\\vdash^{r}$.\n \\end{theorem}\n \n \\begin{proof}\n \n Let us denote with $\\vdash_{\\mathcal{H}^{r}}$ the logic defined by $\\mathcal{H}^{r}$. We show that $\\vdash_{\\mathcal{H}^{r}}\\?=\\?\\vdash^{r}$.\n \n $(\\subseteq)$. It is immediate to check that every rule of $\\mathcal{H}^{r}$ holds in $\\vdash^{r}$.  \n \n $(\\supseteq)$. We now show that $\\mathsf{Mod}^{\\textup{Su}}(\\vdash_{\\mathcal{H}^{r}})\\subseteq\\Mod(\\vdash^{r})$. So let $\\langle\\A,F\\rangle\\in\\mathsf{Mod}^{\\textup{Su}}(\\vdash_{\\mathcal{H}^{r}})$. By Lemma \\ref{lemma: spezza dimostrazione completezza Hilbert}-(i), we know that $\\langle\\A,F\\rangle\\cong\\PL(X)$, where $X=\\langle\\{ \\langle \\A_{i},F_{i}\\rangle\\}_{i\\in I}, \\{ f_{ij} \\! : \\! i \\leq j \\}, \\langle I,\\leq\\rangle\\rangle$ is an r-direct system of matrices.\n \nThe fact that the matrix $\\langle\\A_{i},F_{i}\\rangle\\in\\Mod(\\vdash_{\\mathcal{H}})$ for each $i\\in I^{+}$ can be proved on the ground of \\eqref{Eq:Axiom0} and \\eqref{Eq:Axiom3} by adapting the proof of Lemma \\ref{lemma su filtri non vuoti che sono modelli della logica iniziale} to the calculus $\\mathcal{H}^{r}$. Recalling that $\\mathcal{H}$ is complete for $\\vdash$ we obtain that $\\langle\\A_{i},F_{i}\\rangle\\in\\Mod(\\vdash)$, for each $i\\in I^{+}$. \n\n\n  \n \n\\noindent \n\nMoreover, by Lemma \\ref{lemma: spezza dimostrazione completezza Hilbert}-(ii), we know that if $X$ contains a trivial matrix $\\pair{\\A_j, F_j}$, then $\\A = \\mathbf{1}$. \n\nTherefore, two cases may arise: (1) $\\A=\\mathbf{1}$, (2) $X$ contains no trivial fibers.\nIf (1) then clearly $\\langle\\A,F\\rangle\\in\\{\\langle\\mathbf{1},\\emptyset\\rangle, \\langle\\mathbf{1},\\{1\\}\\rangle \\}$. As $\\vdash^{r}$ is a theoremless logic \n$\\{\\langle\\mathbf{1},\\emptyset\\rangle, \\langle\\mathbf{1},1\\rangle \\}\\subseteq \\Mod(\\vdash^{r})$. If (2), then we can apply Lemma \\ref{lemma: soundness}, so $\\pair{\\A, F}=\\PL(X)\\in\\Mod(\\vdash^{r})$. \n \\end{proof}\n \n\n\n\n\n\n\n \n\n\n \n\\begin{remark}\\label{rem: Hilbert senza inconsistency terms}\nIt is easy to check that, if the logic $\\vdash$ does not possess antitheorems, then a Hilbert-style calculus for $\\vdash^{r}$ can be defined by simply dropping condition (\\ref{Eq:Axiom4}) from Definition \\ref{def: hilbert calc per r}. The completeness of $\\vdash^{r}$ with respect to such calculus can be proven by adapting the strategy in the proof of Theorem \\ref{th: completezza calcolo Hilbert}.\n\\end{remark} \n\n\n\\section{Examples of axiomatizations}\\label{sec: examples}\n\nIn this last section, we show how to obtain Hilbert-style axiomatizations of some containment logics.  \n\n\\subsection{Bochvar logic}\n\n\nBochvar logic is the containment companion of classical logic. Consider the following Hilbert-style axiomatization of classical propositional logic:\n\\benroman\n\\item[($\\mathbf{H}_{1}$)] $\\rhd\\varphi\\to\\varphi$\n\\item [($\\mathbf{H}_{2}$)]$\\rhd\\varphi\\to(\\psi\\to\\varphi)$\n\\item [($\\mathbf{H}_{3}$)]$\\rhd\\varphi\\to(\\psi\\to\\chi)\\to(\\varphi\\to\\psi)\\to(\\varphi\\to\\chi)$\n\\item [($\\mathbf{H}_{4}$)]$\\rhd(\\neg\\varphi\\to\\neg\\psi)\\to(\\psi\\to\\varphi)$\n\\item [($\\mathbf{H}_{5}$)]$\\varphi,\\varphi\\to\\psi\\rhd\\varphi$\n\\eroman\n\n Theorem \\ref{th: completezza calcolo Hilbert} allows to provide the following complete Hilbert style calculus for Bochvar logic $\\mathsf{B_{3}}$.\n\n\n\\begin{itemize}\n\\item[($\\mathbf{H}^{r}_{1}$)] $\\alpha\\ast(\\varphi\\to\\varphi)\\rhd\\varphi\\to\\varphi$\n\\item [($\\mathbf{H}^{r}_{2}$)]$\\alpha\\ast(\\varphi\\to(\\psi\\to\\varphi)\\rhd\\varphi\\to(\\psi\\to\\varphi)$\n\\item [($\\mathbf{H}^{r}_{3}$)]$\\alpha\\ast(\\varphi\\to(\\psi\\to\\chi)\\to(\\varphi\\to\\psi)\\to(\\varphi\\to\\chi))\\rhd\\varphi\\to(\\psi\\to\\chi)\\to(\\varphi\\to\\psi)\\to(\\varphi\\to\\chi)$\n\\item[($\\mathbf{H}^{r}_{4}$)] $\\alpha\\ast((\\neg\\varphi\\to\\neg\\psi)\\to(\\psi\\to\\varphi))\\rhd(\\neg\\varphi\\to\\neg\\psi)\\to(\\psi\\to\\varphi)$\n\\item [($\\mathbf{H}^{r}_{5}$)]$\\varphi\\ast\\psi,\\varphi\\to\\psi\\rhd\\psi$\n\\item [($\\mathbf{H}^{r}_{6}$)] $\\alpha,\\neg \\alpha\\rhd\\varphi$\n\\item[($\\mathbf{H}^{r}_{7}$)] $\\chi(\\delta, \\vec{z}\\?\\?) \\?\\?\\?\\lhd\\rhd\\chi(\\varepsilon, \\vec{z}\\?\\?)$ for every formula $\\chi(x,\\vec{z})$ and equation $\\delta\\thickapprox\\varepsilon$ in  Definition \\ref{def: partition function},\n\\end{itemize}\n\nwhere $\\varphi\\ast\\psi$ is an abbreviation for $\\varphi\\land(\\varphi\\lor\\psi)$.\n\n \n\\subsection{Belnap-Dunn logic}\n\nBelnap-Dunn is a four-valued logic $\\mathsf{B}$, originally introduced as \\emph{First Degree Entailment} within the research enterprise on relevance and entailment logic \\cite{AndersonBelnap,Belnap1977}.   \n\nConsider the algebraic language of type $1,2,2$, containing $\\neg,\\vee,\\land$. Recall that a \\emph{De Morgan lattice} is an algebra $\\A=\\pair{A,\\neg,\\vee,\\land}$ of type $1,2,2$ such that: \n\\benroman\n\\item $\\pair{A,\\land,\\vee}$ is a distributive lattice; \n\\item $\\neg $ satisfies the following equations: \n\\[\nx\\approx\\neg\\neg x, \\hspace{1cm} \\neg(x\\land y)\\approx \\neg x\\lor\\neg y, \\hspace{1cm} \\neg(x\\lor y)\\approx \\neg x\\land\\neg y.\n\\]\n\\eroman\n\nDe Morgan lattices, originally introduced by Moisil \\cite{Moisil} and, independently, by Kalman \\cite{Kalman58} (under the name of \\emph{distributive i-lattices}) form a variety, which is generated by the four element algebra $\\M=\\pair{\\{0,b,n,1\\},\\neg,\\vee,\\land}$, whose lattice reduct is displayed in Figure \\ref{fig: M4} and negation in the following table:\n\\begin{center}\\renewcommand{\\arraystretch}{1.25}\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}}\n  \\lnot &  \\\\[.2ex]\n\\hline\n  1 & 0 \\\\\n  b & b \\\\\n  n & n \\\\\n  0 & 1 \\\\\n\\end{tabular}\n\\end{center}\n\n\\begin{center}\n\\begin{figure}[h]\n\\begin{tikzcd}[row sep = tiny, arrows = {dash}]\n & 1 & \\\\\n & & \\\\\n & & \\\\\n b \\arrow[uuur, dash] & & n \\arrow[uuul] \\\\\n & & \\\\\n & & \\\\\n & 0\\arrow[uuul]\\arrow[uuur] &\n  \\end{tikzcd}\\caption{Hasse\n  diagram of the De Morgan lattice $\\M$.}\n \\end{figure}\\label{fig: M4}\n \n  \\end{center}\n\n$\\mathsf{B}$ is the logic induced by the matrix $\\pair{\\M,\\{1,b\\}}$ (or, equivalently, by $\\pair{\\M,\\{1,n\\}}$, see \\cite[Proposition 2.3]{font1997belnap}). $\\mathsf{B}$ is finitary and theoremless (purely inferential). Moreover, the class $\\Alg(\\mathsf{B})$ coincides with the variety of De Morgan lattices \\cite[Theorem 4.1]{font1997belnap}. Observe that the set $\\{\\varphi,\\neg\\varphi\\}$ is not an antitheorem of $\\mathsf{B}$ (indeed $\\varphi,\\neg\\varphi\\not\\vdash_{\\mathsf{B}}\\psi$). It is not difficult to check that $\\mathsf{B}$ does not possess antitheorems. \n\nRecall that a \\emph{lattice filter} of a De Morgan lattice $\\A$ is a subset $F\\subseteq A$ such that $x\\land y\\in F$ if and only if $x\\in F$ and $y\\in F$. The class of matrices \n\\[\n\\mathsf{M}=\\{\\pair{\\A, F} : \\; \\A \\text{ De Morgan algebra, } F\\subseteq A \\text{ lattice filter}\\}\n\\]\n\nis complete for $\\mathsf{B}$ \\cite[Corollary 2.6]{font1997belnap}. Observe that the matrices $\\pair{\\A,\\emptyset}\\in\\mathsf{M}$, for any De Morgan lattice $\\A$. It follows from our analysis (see Theorem \\ref{completeness}) that the containment companion $\\vdash^{r}_{\\mathsf{B}}$ of $\\vdash_{\\mathsf{B}}$ is complete with respect to $\\PL{(\\mathsf{M})}$, i.e. the class of all P\\l onka sums over r-direct systems of matrices in $\\mathsf{M}$.\n  \n\n\nWe present the Hilbert-style axiomatization for $\\mathsf{B}$ which is introduced in \\cite{font1997belnap} (and, independently in \\cite{Pynko95}). Since $\\mathsf{B}$ is theoremless, the calculus has no axioms and the following rules: \n\n\\benroman\n\\item[($\\mathsf{B}_1$)] $\\varphi \\land \\psi \\rhd \\varphi $;\n\\item [($\\mathsf{B}_{2}$)] $\\varphi \\land \\psi \\rhd \\psi$;\n\\item [($\\mathsf{B}_{3}$)]$\\varphi, \\psi\\rhd \\varphi  \\land \\psi $;\n\\item [($\\mathsf{B}_{4}$)]$\\varphi \\rhd \\varphi\\lor \\psi$;\n\\item [($\\mathsf{B}_{5}$)]$\\varphi\\lor \\psi\\rhd \\psi\\lor \\varphi$;\n\\item [($\\mathsf{B}_{6}$)]$\\varphi\\lor \\varphi \\rhd \\varphi$;\n\\item [($\\mathsf{B}_{7}$)]$\\varphi\\lor (\\psi\\lor\\chi) \\rhd (\\varphi\\lor\\psi)\\lor\\chi$;\n\\item [($\\mathsf{B}_{8}$)]$\\varphi\\lor (\\psi\\land\\chi) \\rhd (\\varphi\\lor\\psi)\\land(\\varphi\\lor\\chi)$;\n\\item [($\\mathsf{B}_{9}$)]$(\\varphi\\lor\\psi)\\land(\\varphi\\lor\\chi) \\rhd  \\varphi\\lor (\\psi\\land\\chi)$;\n\\item [($\\mathsf{B}_{10}$)]$\\varphi\\lor\\psi  \\rhd \\neg\\neg\\varphi\\lor\\psi$;\n\\item [($\\mathsf{B}_{11}$)]$\\neg\\neg\\varphi\\lor\\psi  \\rhd \\varphi\\lor\\psi$\n\\item [($\\mathsf{B}_{12}$)]$\\neg(\\varphi\\lor\\psi)\\lor\\chi  \\rhd (\\neg\\varphi\\land\\neg \\psi)\\lor\\chi$;\n\\item [($\\mathsf{B}_{13}$)]$ (\\neg\\varphi\\land\\neg \\psi)\\lor\\chi  \\rhd \\neg(\\varphi\\lor\\psi)\\lor\\chi$;\n\\item [($\\mathsf{B}_{14}$)]$\\neg(\\varphi\\land\\psi)\\lor\\chi  \\rhd (\\neg\\varphi\\lor\\neg \\psi)\\lor\\chi$;\n\\item [($\\mathsf{B}_{15}$)]$ (\\neg\\varphi\\lor\\neg \\psi)\\lor\\chi  \\rhd \\neg(\\varphi\\land\\psi)\\lor\\chi$;\n\\eroman\n\nA Hilbert-style axiomatization of $\\vdash^{r}_{\\mathsf{B}}$, (see Definition \\ref{def: hilbert calc per r} Theorem \\ref{th: completezza calcolo Hilbert}) is given by the following ($\\varphi\\ast\\psi$ is an abbreviation for $ \\varphi\\land(\\varphi\\lor\\psi)$): \n\n\\benroman\n\\item[($\\mathsf{B}^{r}_{1}$)] $\\varphi,\\psi \\rhd \\varphi \\ast \\psi $;\n\\item[($\\mathsf{B}^{r}_{2}$)] $\\varphi\\ast \\psi \\rhd \\psi $;\n\\item[($\\mathsf{B}^{r}_{3}$)] $(\\varphi \\land \\psi)\\ast\\varphi \\rhd \\varphi $;\n\\item [($\\mathsf{B}^{r}_{4}$)] $(\\varphi \\land \\psi)\\ast\\psi \\rhd \\psi$;\n\\item [($\\mathsf{B}^{r}_{5}$)]$\\varphi\\ast (\\varphi  \\land \\psi) , \\psi\\rhd \\varphi  \\land \\psi $;\n\\item [($\\mathsf{B}^{r}_{6}$)]$\\varphi, \\psi\\ast(\\varphi  \\land \\psi)\\rhd \\varphi  \\land \\psi $;\n\\item [($\\mathsf{B}^{r}_{7}$)]$\\varphi\\ast(\\varphi\\lor \\psi) \\rhd \\varphi\\lor \\psi$;\n\\item [($\\mathsf{B}^{r}_{8}$)]$(\\varphi\\lor \\psi)\\ast (\\psi\\lor \\varphi)\\rhd \\psi\\lor \\varphi$;\n\\item [($\\mathsf{B}^{r}_{9}$)]$(\\varphi\\lor \\varphi)\\ast\\varphi \\rhd \\varphi$;\n\n\\item [($\\mathsf{B}^{r}_{10}$)]$(\\varphi\\lor (\\psi\\lor\\chi))\\ast ((\\varphi\\lor\\psi)\\lor\\chi) \\rhd (\\varphi\\lor\\psi)\\lor\\chi$;\n\n\\item [($\\mathsf{B}^{r}_{11}$)]$\\varphi\\lor (\\psi\\land\\chi)\\ast((\\varphi\\lor\\psi)\\land(\\varphi\\lor\\chi)) \\rhd (\\varphi\\lor\\psi)\\land(\\varphi\\lor\\chi)$;\n\n\\item [($\\mathsf{B}^{r}_{12}$)]$((\\varphi\\lor\\psi)\\land(\\varphi\\lor\\chi))\\ast(\\varphi\\lor (\\psi\\land\\chi)) \\rhd  \\varphi\\lor (\\psi\\land\\chi)$;\n\n\\item [($\\mathsf{B}^{r}_{13}$)]$(\\varphi\\lor\\psi)\\ast (\\neg\\neg\\varphi\\lor\\psi)  \\rhd \\neg\\neg\\varphi\\lor\\psi$;\n\n\\item [($\\mathsf{B}^{r}_{14}$)]$(\\neg\\neg\\varphi\\lor\\psi)\\ast (\\varphi\\lor\\psi)\\rhd \\varphi\\lor\\psi$;\n\n\\item [($\\mathsf{B}^{r}_{15}$)]$(\\neg(\\varphi\\lor\\psi)\\lor\\chi)\\ast ((\\neg\\varphi\\land\\neg \\psi)\\lor\\chi)  \\rhd (\\neg\\varphi\\land\\neg \\psi)\\lor\\chi$;\n\n\\item [($\\mathsf{B}^{r}_{16}$)]$ ((\\neg\\varphi\\land\\neg \\psi)\\lor\\chi) \\ast (\\neg(\\varphi\\lor\\psi)\\lor\\chi)  \\rhd \\neg(\\varphi\\lor\\psi)\\lor\\chi$;\n\n\\item [($\\mathsf{B}^{r}_{17}$)]$(\\neg(\\varphi\\land\\psi)\\lor\\chi)\\ast ((\\neg\\varphi\\lor\\neg \\psi)\\lor\\chi)  \\rhd (\\neg\\varphi\\lor\\neg \\psi)\\lor\\chi$;\n\n\\item [($\\mathsf{B}^{r}_{18}$)]$ ((\\neg\\varphi\\lor\\neg \\psi)\\lor\\chi)\\ast (\\neg(\\varphi\\land\\psi)\\lor\\chi)  \\rhd \\neg(\\varphi\\land\\psi)\\lor\\chi$;\n\n\\item[($\\mathsf{B}^{r}_{19}$)] $\\chi(\\delta, \\vec{z}\\?\\?) \\?\\?\\?\\lhd\\rhd\\chi(\\varepsilon, \\vec{z}\\?\\?)$ for every formula $\\chi(x,\\vec{z})$ and equation $\\delta\\thickapprox\\varepsilon$ in  Definition \\ref{def: partition function}.\n\\eroman\n\n\n\n\\subsection{The Relevance logic $\\mathrm{\\mathbf{S}_{fde}}$}\\label{Subsec: relevance}\n\nThe logic $\\mathrm{\\mathbf{S}_{fde}}$ has been introduced by Deutsch \\cite{Deutsch}: it is induced the matrix $\\pair{\\mathbf{S}_{4}, \\{1,\\ant\\}}$, whose algebraic reduct $\\mathbf{S}_{4}=\\pair{\\{0,\\ant,m,1\\}, \\neg, \\land, \\vee}$ is given in the following tables.\n\\vspace{5pt}\n\\begin{center}\\renewcommand{\\arraystretch}{1.25}\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}}\n  \\lnot &  \\\\[.2ex]\n\\hline\n  1 & 0 \\\\\n  \\ant & \\ant \\\\\n  m & m \\\\\n  0 & 1 \\\\\n\\end{tabular}\n\\qquad\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}}\n   \\land & 0 & \\ant & m & 1 \\\\[.2ex]\n \\hline\n       0 & 0 & 0 & m &  0 \\\\\n       \\ant & 0 & \\ant & m & \\ant \\\\\n       m & m & m & m & m \\\\          \n       1 & 0 & \\ant & m & 1 \n\\end{tabular}\n\\qquad\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}}\n   \\lor & 0 & \\ant & m & 1 \\\\[.2ex]\n \\hline\n       0 & 0 & \\ant & m &  1 \\\\\n       \\ant & \\ant & \\ant & m & 1 \\\\\n       m & m & m & m & m \\\\          \n       1 & 1 & 1 & m & 1 \n\\end{tabular}\n\n\\end{center}\n\\vspace{10pt}\n\\noindent\nRecall that a \\emph{Kleene lattice} is a De Morgan lattice satisfying $x\\land\\neg x\\leq y\\lor\\neg y$. Kleene lattices form a variety ($\\mathsf{KL}$), generated by the 3-element algebra $\\mathbf{SK}=\\pair{\\{0,1,\\ant\\}, \\neg, \\vee, \\land}$, which is a subalgebra of $\\mathbf{S}_{4}$ (and also isomorphic to the two three-element subalgebras of $\\M$).\n\nThe logic of Paradox $\\mathsf{LP}$ (see \\cite{Priestfirst,BlRiVe01,Pynko}) is defined by the matrix $\\langle\\mathbf{SK},\\{1,\\ant\\}\\rangle$. The algebraic counterpart of $\\mathsf{LP}$ is exactly the variety of Kleene lattice, i.e. $\\mathsf{KL}=\\Alg(\\mathsf{LP})$.\nIt is immediate to check that the matrix $\\pair{\\mathbf{S}_{4}, \\{1,\\ant\\}}$ is the P\\l onka sum over the r-direct system of the two matrices $\\pair{\\mathbf{SK},\\{1,\\ant\\}}$ and $\\pair{\\mathbf{m},\\emptyset}$. Thus, it follows from Theorem \\ref{completeness} that $\\mathrm{\\mathbf{S}_{fde}}$ is the containment companion of the logic of Paradox. \n\nA finite Hilbert style calculus for $\\mathsf{LP}$ (see for instance \\cite{Superbelnap}) can be obtained by adding the axiom\n\n\\benroman\n\\item[$(\\mathsf{LP}_{1})$] $\\rhd \\varphi\\lor\\neg\\varphi$\n\\eroman\nto the calculus for the logic $\\mathsf{B}$ described above. Therefore, by Theorem \\ref{th: completezza calcolo Hilbert}, the calculus consisting of $(\\mathsf{B}_1^r)-(\\mathsf{B}_{19}^r)$ and\n\\benroman\n\\item [$(\\mathsf{LP}^r_{1})$] $\\alpha\\ast \\varphi\\lor\\neg\\varphi\\rhd \\varphi\\lor\\neg\\varphi $\n \n\\eroman\n\nis complete for $\\mathrm{\\mathbf{S}_{fde}}$.\n\n\n \n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nLet $(M,J)$ be a complex manifold of real dimension $2n$, $n \\geq 2$, equipped with a Hermitian metric $g$ with associated fundamental $2$-form $\\omega=g(J\\cdot,\\cdot)$. Its \\emph{Lee form}, defined by $\\theta=-d^*\\omega \\circ J$, is the unique $1$-form satisfying $d\\omega^{n-1}=\\theta \\wedge \\omega^{n-1}$.\n\nA fundamental class of Hermitian metrics is provided by \\emph{K\\\"ahler} metrics, satisfying $d\\omega=0$. In literature, many generalizations of the K\\\"ahler condition have been introduced: two of them are the \\emph{balanced} (or \\emph{semi-K\\\"ahler}) condition, characterized by $d^*\\omega=0$ (or equivalently $\\theta=0$ or $d\\omega^{n-1}=0$) and the \\emph{locally conformally K\\\"ahler} (LCK) condition, namely $(M,J)$ admits an open cover $\\{U_i\\}$ and smooth maps $f_i \\in C^{\\infty}(U_i)$ such that $e^{-f_i}g\\rvert_{U_i}$ is a K\\\"ahler metric on $(U_i,J\\rvert_{U_i})$, where $g$ denotes the LCK metric.\nThe LCK condition is equivalently characterized by \nthe conditions $d\\omega=\\frac{1}{n-1} \\theta \\wedge \\omega$, $d\\theta=0$. If $\\theta$ is parallel with respect to the Levi-Civita connection, the LCK metric is called \\emph{Vaisman}. For general results about LCK metrics, we refer the reader to \\cite{DO, Orn, OV, AO1}.\n\nA further weakening of both the balanced and the LCK conditions is given by the \\emph{locally conformally balanced} (LCB) condition, whose definition is analogous to the one for the LCK condition and which is equivalently defined by $d\\theta=0$. LCB metrics have been studied, for instance, in \\cite{AU, AU1, FT1, LY, OOS, Oti, Shi, Shi1, Yang}. When $n=2$, balanced metrics are K\\\"ahler and LCB metrics are LCK. \n\nRecall also that a Hermitian metric is called \\emph{strong K\\\"ahler with torsion} (SKT, also known as \\emph{pluriclosed}) if $\\partial \\overline\\partial \\omega=0$ or, equivalently, if the torsion of the associated \\emph{Bismut connection} vanishes. The Bismut connection $\\nabla^B$ of a Hermitian manifold $(M,J,g)$ is the unique linear connection on $M$ having totally skew-symmetric torsion and satisfying $\\nabla^Bg=0$, $\\nabla^BJ=0$ (see \\cite{Bismut,Gau}). Its associated \\emph{Bismut-Ricci form} $\\rho^B$ is the $2$-form locally defined by\n\\[\n\\rho^B(X,Y)=-\\frac{1}{2} \\sum_{i=1}^{2n} g(R^B(X,Y)f_i,Jf_i),\\quad X,Y \\in \\Gamma_{\\text{loc}}(TM),\n\\]\nwhere $\\{f_1,\\ldots,f_{2n}\\}$ is a local $g$-orthonormal frame and $R^{B}(X,Y)=[\\nabla^B_X,\\nabla^B_Y]-\\nabla^B_{[X,Y]}$ denotes the curvature of $\\nabla^B$.\n\nA \\emph{hypercomplex structure} on a smooth $4m$-dimensional manifold $M$ is given by a triple of (integrable) complex structures $(I_1,I_2,I_3)$ satisfying $I_1I_2I_3=-\\text{Id}_{TM}$. A Riemannian metric on $M$ is called \\emph{(locally conformally) hyperk\\\"ahler} (LCHK) if it is (locally conformally) K\\\"ahler with respect to the three complex structures and the three induced Lee forms coincide. Hypercomplex and hyperk\\\"ahler structures on Lie groups where studied for instance in \\cite{Bar}, where four-dimensional Lie groups admitting left-invariant hypercomplex structures are classified, and \\cite{BDF}, where, in particular, it is shown that left-invariant hyperk\\\"ahler metrics on Lie groups are flat.\n\nWe are interested in the case where $M$ is a simply connected almost abelian Lie group $G$ or a compact almost abelian \\emph{solvmanifold}, namely a quotient $\\Gamma \\backslash G$, with $G$ a simply connected almost abelian Lie group and $\\Gamma$ a lattice of $G$, i.e., a discrete subgroup of $G$. A connected (solvable) Lie group $G$ is called \\emph{almost abelian} if it admits an abelian normal subgroup of codimension one, or equivalently if the Lie algebra $\\mathfrak{g}$ of $G$ admits an abelian ideal $\\mathfrak{n}$ of codimension one, so that $\\mathfrak{g}$ is isomorphic to the semi-direct product $\\mathbb{R}^k \\rtimes_D \\mathbb{R}$ for some $D \\in \\mathfrak{gl}_k$. If $\\mathfrak{g}$ is non-nilpotent, such an ideal is unique and coincides with the nilradical of $\\mathfrak{g}$.\n\nA left-invariant Hermitian structure $(J,g)$ on $G$ or $\\Gamma \\backslash G$ descends to a structure on the Lie algebra $\\mathfrak{g}$ of $G$, so that one can speak of Hermitian structures on $\\mathfrak{g}$.\nWhen $\\mathfrak{g}$ is almost abelian of real dimension $2n$, as shown in \\cite{LV}, these can be fully characterized in terms of the matrix associated with $\\text{ad}_{e_{2n}}\\rvert_{\\mathfrak{n}}$ with respect to some fixed unitary basis $\\left\\{ e_1,\\ldots, e_{2n} \\right\\}$ \\emph{adapted} to the splitting $\\mathfrak{g}= J\\mathfrak{k} \\oplus \\mathfrak{n}_1 \\oplus \\mathfrak{k}$, where $\\mathfrak{k} \\coloneqq \\mathfrak{n}^{\\perp_g}$ and $\\mathfrak{n}_1 \\coloneqq \\mathfrak{n} \\cap J\\mathfrak{n}$, and such that $Je_i = e_{2n+1-i}$, $i=1,\\ldots,n$.\n\nK\\\"ahler, SKT, balanced and LCK almost abelian Lie algebras were studied in terms of the data $(a,v,A)$ in \\cite{LV,AL,FP,FP1,AO}. Six-dimensional almost abelian Lie algebras admitting SKT structures were classified in \\cite{FP}, and in \\cite{FS} the result was extended to a wider class of two-step solvable Lie algebras. For the classification of six-dimensional almost abelian Lie algebras carrying balanced structures, see \\cite{FP1}.\n\nIn Section \\ref{sec_lcb} we characterize LCB almost abelian Lie algebras in terms of the aforementioned algebraic data and in terms of the behaviour of the associated Bismut-Ricci form. \n\nIn the following section, we classify six-dimensional almost abelian Lie algebras admitting LCK structures and those admitting LCB structures, building on the classification of six-dimensional almost abelian Lie algebras admitting complex structures in \\cite{FP}, and remark which of the corresponding Lie groups admit compact quotients by lattices. \n\nIn \\cite{OOS}, the authors investigate the existence of two different types of special Hermitian metrics on a fixed compact complex nilmanifold (namely, the quotient of a simply connected nilpotent Lie group by a lattice): in Section \\ref{sec_comp}, we consider analogous questions for almost abelian solvmanifolds, highlighting similarities and differences with respect to the nilpotent setting.\n\nFinally, in Section \\ref{LCHK} we study LCHK structures on almost abelian Lie algebras, giving a classification result in every dimension.\n\n\\smallskip\n\\emph{Acknowledgements}. The author would like to thank Anna Fino for suggesting the subject of this paper and for many useful comments and discussions. The author is also grateful to an anonymous referee for useful comments. The author was supported by GNSAGA of INdAM.\n\n\\section{Locally conformally  balanced metrics} \\label{sec_lcb}\n\nLet $\\mathfrak{g}$ be a $2n$-dimensional almost abelian Lie algebra with a fixed abelian ideal $\\mathfrak{n}$ of codimension one. Assume $(J,g)$ is a Hermitian structure on $\\mathfrak{g}$ and denote by $\\mathfrak{n}_1 \\coloneqq \\mathfrak{n} \\cap J\\mathfrak{n}$ the maximal $J$-invariant subspace of $\\mathfrak{n}$, which does not depend on the metric $g$.\nThen, as shown in \\cite{LV}, with respect to a unitary basis $\\{e_1,\\ldots,e_{2n}\\}$ for $\\mathfrak{g}$ such that $\\mathfrak{n}=\\text{span}\\left<e_1,\\ldots,e_{2n-1}\\right>$, $\\mathfrak{n}_1=\\text{span}\\left<e_2,\\ldots,e_{2n-1}\\right>$, $Je_i=e_{2n+1-i}$, $i=1,\\ldots,n$, the matrix $B$ associated with $\\text{ad}_{e_{2n}}\\rvert_{\\mathfrak{n}}$ is of the form\n\\begin{equation} \\label{B}\n\tB=\\begin{pmatrix} a & 0 \\\\ v & A \\end{pmatrix}, \\quad a \\in \\mathbb{R},\\,v \\in \\mathfrak{n}_1,\\, A \\in \\mathfrak{gl}(\\mathfrak{n}_1,J_1),\n\\end{equation}\nwhere $J_1 \\coloneqq J \\rvert_{\\mathfrak{n}_1}$ and $\\mathfrak{gl}(\\mathfrak{n}_1,J_1)$ denotes endomorphisms of $\\mathfrak{n}_1$ commuting with $J_1$. We denote $\\mathfrak{k}\\coloneqq \\mathfrak{n}^{\\perp_g}=\\mathbb{R} e_{2n}$ and we say that the basis $\\{e_1,\\ldots,e_{2n}\\}$ is \\emph{adapted} to the splitting $\\mathfrak{g} = J\\mathfrak{k} \\oplus \\mathfrak{n}_1 \\oplus \\mathfrak{k}$.\nThe algebraic data $(a,v,A)$ fully characterizes the Hermitian structure $(J,g)$ and we we shall often denote the resulting Hermitian almost abelian Lie algebra by $(\\mathfrak{g}(a,v,A),J,g)$.\n\nBefore studying the LCB condition, we recall the known characterizations for special Hermitian almost abelian Lie algebras.\n\n\\begin{proposition}\n\t\\label{Herm-alm-ab}\nA Hermitian almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$ is\n\\begin{itemize}\n\t\\item \\makebox[1.35cm]{K\\\"ahler,\\hfill} if $v=0$, $A \\in \\mathfrak{u}(\\mathfrak{n}_1,J_1,g)$ (see \\cite{LV}),\n\t\\item {\\makebox[1.35cm]{LCK,\\hfill} if $v=0$, $A\\in \\mathbb{R}\\text{\\normalfont Id}_{\\mathfrak{n}_1}\\oplus \\mathfrak{u}(\\mathfrak{n}_1,J_1,g)$ or $n=2$, $A=0$ (see \\cite{AO}),}\n\t\\item {\\makebox[1.35cm]{balanced,\\hfill} if $v=0$, $\\operatorname{tr} A=0$ (see \\cite{FP1}),}\n\t\\item {\\makebox[1.35cm]{SKT,\\hfill} if $[A,A^t]=0$ and the eigenvalues of $A$ have real part $-\\frac{a}{2}$ or $0$ (see \\cite{AL}),}\n\\end{itemize}\nwhere $\\mathfrak{u}(\\mathfrak{n}_1,J_1,g)=\\mathfrak{so}(\\mathfrak{n}_1,g) \\cap \\mathfrak{gl}(\\mathfrak{n}_1,J_1)$.\n\\end{proposition}\n\nWe also recall that, in terms of an adapted unitary basis, the Lee form of a Hermitian almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$ is given by\n\\begin{equation} \\label{leeform}\n\\theta=(Jv)^{\\flat} - (\\operatorname{tr} A)e^{2n},\n\\end{equation}\nwhere the isomorphism $(\\cdot)^\\flat \\colon \\mathfrak{g} \\to \\mathfrak{g}^*$ is defined by $X^\\flat\\coloneqq g(X,\\cdot)$, $X \\in \\mathfrak{g}$. See \\cite{FP1} for details.\n\nWe are ready to prove the analogous characterization for LCB structures.\n\\begin{theorem} \\label{LCB_avA}\nA Hermitian almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$ is LCB if and only if \\mbox{$A^tv=0$}.\n\\end{theorem}\n\\begin{proof}\nObserve that, given any $1$-form $\\alpha \\in \\mathfrak{g}^*$, since\n$d\\alpha(X,e_{2n})=\\alpha([e_{2n},X])$, $X \\in \\mathfrak{g}$, one has\n\\[\nd\\alpha=(\\text{ad}_{e_{2n}}^* \\alpha) \\wedge e^{2n} = (a\\alpha(e_1)+\\alpha(v))\\,e^1 \\wedge e^{2n} + A^*(\\alpha\\rvert_{\\mathfrak{n}_1}),\n\\]\nwith respect to the fixed adapted unitary basis $\\{e_1,\\ldots,e_{2n}\\}$.\nThen the exterior derivative of the Lee form \\eqref{leeform} satisfies\n\\[\nd\\theta=g(v,Jv)\\,e^1 \\wedge e^{2n} + (A^tJv)^\\flat \\wedge e^{2n}=(A^tJv)^\\flat \\wedge e^{2n},\n\\]\nwhere $A^t \\in \\mathfrak{gl}(\\mathfrak{n}_1)$ is defined by $A^tX \\coloneqq (A^*(X^\\flat))^\\sharp$, $X \\in \\mathfrak{n}_1$, $(\\cdot)^\\sharp$ denoting the inverse of $(\\cdot)^\\flat$. Then $d\\theta$ vanishes if and only if $A^tJv=0$. $J_1$ commutes with $A$ and we have $J_1^t=-J_1$, so $J_1$ commutes with $A^t$ as well. The previous condition then reads $JA^tv=0$, which is equivalent to $A^tv=0$. \n\\end{proof}\n\nWe note that the condition $A^tv=0$ is equivalent to $g(v,AX)=0$ for all $X \\in \\mathfrak{n}_1$. In particular, when $v \\neq 0$, it implies $v \\notin \\operatorname{im} A$, so that $\\text{rank}(v\\lvert A)=\\text{rank}(A)+1$, where $v \\lvert A$ denotes the matrix obtained by juxtaposing $v$ and $A$.\n\nIn \\cite{AL}, the authors determined a formula for the Bismut-Ricci form of a Hermitian almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$, obtaining\n\\begin{equation}\\label{rhoB}\n\\rho^B=-\\left(a^2-\\tfrac{1}{2}a\\operatorname{tr}A+\\lVert v \\rVert^2\\right)\\, e^1 \\wedge e^{2n} - (A^tv)^{\\flat} \\wedge e^{2n},\n\\end{equation}\nin terms of the fixed adapted unitary basis $\\{e_1,\\ldots,e_{2n}\\}$ (cf. also \\cite{FP}).\n\nThe next result is a straightforward consequence of Theorem \\ref{LCB_avA} and formula \\eqref{rhoB}.\n\\begin{proposition}\nA Hermitian almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$ is LCB if and only if $\\rho^B$ is of type $(1,1)$ (namely, $J\\rho^B=\\rho^B$), or equivalently if $\\rho^B(X,Y)=0$ for every $X \\in \\mathfrak{n}_1$, $Y \\in \\mathfrak{g}$.\n\\end{proposition}\n\n\\section{Classification in dimension six}\n\nWe now focus on the six-dimensional case, with the goal of classifying almost abelian Lie algebras admitting LCB structures.\nAs recalled in the introduction, LCB structures generalize K\\\"ahler, balanced and LCK structures. Six-dimensional almost abelian Lie algebras carrying K\\\"ahler structures and balanced structures were classified in \\cite{FP} and \\cite{FP1} respectively. Therefore, before considering strictly LCB structures, we focus on the LCK condition.\n\nIn the following, we denote a Lie algebra via its structure equations: for example, the notation\n\\[\n\\mathfrak{g}_{4}=(f^{16},f^{26},f^{36},f^{46},0,0)\n\\] \nmeans that the Lie algebra $\\mathfrak{g}_{4}$ is determined by a fixed basis $\\{f_1,\\ldots,f_6\\}$ whose dual coframe $\\{f^1,\\ldots,f^6\\}$ satisfies $df^1=f^{16}$, $df^2=f^{26}$, $df^3=f^{36}$, $df^4=f^{46}$, $df^5=df^6=0$, where $f^{ij}$ is a shorthand for the wedge product $f^i \\wedge f^j$.\n\nIn \\cite{Saw}, it was proven that a nilpotent Lie algebra admits an LCK structure if and only if it is isomorphic to $\\mathfrak{h}_{2n+1} \\oplus \\mathbb{R}$, for some $n \\geq 1$, where\n\\[\n\\mathfrak{h}_{2n+1}=\\left(0,\\ldots,0,\\sum_{i=1}^{n} f^{2i-1} \\wedge f^{2i}\\right)\n\\]\ndenotes the $2n+1$-dimensional real Heisenberg algebra. In particular, the four-dimensional $\\mathfrak{h}_3 \\oplus \\mathbb{R} = (0,0,0,f^{12})$ is the only one which is also almost abelian and, by \\cite[Remark 3.4 (ii)]{AO} and \\cite[Remarks 2.1, 2.3]{AngO}, one of the only two almost abelian Lie algebras admitting non-K\\\"ahler Vaisman metrics, up to isomorphism, the other one being $\\mathfrak{aff}_2 \\oplus 2\\mathbb{R}$, where $\\mathfrak{aff}_2 =(0,f^{12})$ denotes the two-dimensional real affine Lie algebra. In fact, every Hermitian metric on $\\mathfrak{h}_3 \\oplus \\mathbb{R}$ and $\\mathfrak{aff}_2 \\oplus 2\\mathbb{R}$ is Vaisman.\n\n\\begin{theorem} \\label{LCK_class}\nLet $\\mathfrak{g}$ be a six-dimensional almost abelian Lie algebra. Then $\\mathfrak{g}$ admits an LCK structure $(J,g)$, but no K\\\"ahler structures, if and only if it is isomorphic to one of the following:\n\\begin{itemize}\n\t\\setlength{\\itemindent}{-1em}\n\t\\item[] $\\mathfrak{g}_{1}\\cm{=\\mathfrak{k}_1^{p,p}}=(f^{16},pf^{26},pf^{36},pf^{46},pf^{56},0)$,  \\,  $p \\neq 0$,\\smallskip\n\t\\item[] $\\mathfrak{g}_{2}\\cm{=\\mathfrak{k}_{8}^{p,q,q}}=(pf^{16},qf^{26},qf^{36},qf^{46}+f^{56},-f^{46}+qf^{56},0)$,  \\,  $pq \\neq 0$, \\smallskip\n\t\\item[] $\\mathfrak{g}_{3}\\cm{=\\mathfrak{k}_{11}^{p,q,q,s}}=(pf^{16},qf^{26}+f^{36},-f^{26}+qf^{36},qf^{46}+rf^{56},-rf^{46}+qf^{56},0)$,  \\,  $pq \\neq 0$,  $r \\neq 0$,\\smallskip\n\t\\item[] $\\mathfrak{g}_{4}\\cm{=\\mathfrak{k}_{20}^1}=(f^{16},f^{26},f^{36},f^{46},0,0)$,\\smallskip\n\t\\item[] $\\mathfrak{g}_{5}\\cm{=\\mathfrak{k}_{22}^{1,r}}=(f^{16},f^{26},f^{36}+rf^{46},-rf^{36}+f^{46},0,0)$, \\, $r \\neq 0$, \\smallskip\n\t\\item[] $\\mathfrak{g}_{6}\\cm{=\\mathfrak{k}_{25}^{p,p,r}}=(pf^{16}+f^{26},-f^{16}+pf^{26},pf^{36}+rf^{46},-rf^{36}+pf^{46},0,0)$, \\, $pr \\neq 0$.\n\\end{itemize}\nAmong these, only the indecomposable Lie algebras $\\mathfrak{g}_{1}^{p=-\\frac{1}{4}}$\\cm{$\\mathfrak{k}_1^{-\\frac{1}{4},-\\frac{1}{4}}$}, $\\mathfrak{g}_{2}^{p=-4q}$ \\cm{$\\mathfrak{k}_{8}^{p,-\\frac{1}{4}p,-\\frac{1}{4}p}$} and $\\mathfrak{g}_{3}^{p=-4q}$ \\cm{$\\mathfrak{k}_{11}^{p,-\\frac{1}{4}p,-\\frac{1}{4}p,s}$} are unimodular. None of the corresponding Lie groups admit compact quotients by lattices, by \\cite[Theorem 3.7]{AO}. \\end{theorem} \n\\begin{proof}\nLet $(J,g)$ be an LCK structure on $\\mathfrak{g}$. Let $\\{e_1,\\ldots,e_6\\}$ be a unitary basis of $(\\mathfrak{g},J,g)$ adapted to the splitting $\\mathfrak{g}=J\\mathfrak{k} \\oplus \\mathfrak{n}_1 \\oplus \\mathfrak{k}$, so that, by \\cite{AO}, the matrix $B$ associated with $\\text{ad}_{e_6}\\rvert_{\\mathfrak{n}}$ is of the form \\eqref{B}, with\n\\begin{equation} \\label{LCK}\nv=0, \\quad A= \\lambda\\,\\text{Id}_{\\mathfrak{n}_1} + U,\\; \\lambda \\in \\mathbb{R},\\,U \\in \\mathfrak{u}(\\mathfrak{n}_1,J_1,g).\n\\end{equation}\nSince $U$ is traceless, one must have $\\lambda=\\frac{\\operatorname{tr} A}{4}$.\nFollowing \\cite[Theorem 3.2]{FP}, up to taking a different basis $\\{e_2,\\ldots,e_5\\}$ for $\\mathfrak{n}_1$ and rescaling $e_6$, the fact that $A$ commutes with $J_1$ forces $A$ to be represented by a real $4 \\times 4$ matrix of one of the following types:\n\\begin{equation} \\label{A}\nA_1=\\left( \\begin{smallmatrix} p&0&0&0 \\\\ 0&p&0&0 \\\\ 0&0&q&0 \\\\ 0&0&0&q \\end{smallmatrix} \\right)\\!\\!, \\,\nA_2=\\left( \\begin{smallmatrix} p&1&0&0 \\\\ -1&p&0&0 \\\\ 0&0&q&0 \\\\ 0&0&0&q \\end{smallmatrix} \\right)\\!\\!, \\,\nA_3=\\left( \\begin{smallmatrix} p&1&0&0 \\\\ -1&p&0&0 \\\\ 0&0&q&r \\\\ 0&0&-r&q \\end{smallmatrix} \\right)\\!\\!, \\,\nA_4=\\left( \\begin{smallmatrix} p&1&0&0 \\\\ 0&p&0&0 \\\\ 0&0&p&1 \\\\ 0&0&0&p \\end{smallmatrix} \\right)\\!\\!, \\,\nA_5=\\left( \\begin{smallmatrix} p&1&-1&0 \\\\ -1&p&0&-1 \\\\ 0&0&p&1 \\\\ 0&0&-1&p \\end{smallmatrix} \\right)\\!\\!,\n\\end{equation}\n$p,q,r \\in \\mathbb{R}$, with $r \\neq 0$ to avoid redundancy.\nAll we need to do is determine which matrices $A_i$ in \\eqref{A} can be decomposed as $\\lambda \\operatorname{Id} + U$ for some $\\lambda \\in \\mathbb{R}$, $U \\in \\mathfrak{u}(\\mathfrak{n}_1,J_1,g)$. For each $i=1,2,3,4,5$, consider the matrix $U_i=A_i - \\frac{\\operatorname{tr} A}{4}\\operatorname{Id}$: for $i=4,5$, $U_i$ is never complex-diagonalizable (namely, diagonalizable as a complex matrix), so it cannot be skew-symmetric with respect to any metric; for $i=1,2,3$, the requirement that all the eigenvalues of $U_i$ should be pure imaginary imposes $p=q$, so that one is left with\n\\[\nU_1=\\left( \\begin{smallmatrix} 0&0&0&0 \\\\ 0&0&0&0 \\\\ 0&0&0&0 \\\\ 0&0&0&0 \\end{smallmatrix} \\right)\\!, \\quad\nU_2=\\left( \\begin{smallmatrix} 0&1&0&0 \\\\ -1&0&0&0 \\\\ 0&0&0&0 \\\\ 0&0&0&0 \\end{smallmatrix} \\right)\\!, \\quad\nU_3=\\left( \\begin{smallmatrix} 0&1&0&0 \\\\ -1&0&0&0 \\\\ 0&0&0&r \\\\ 0&0&-r&0 \\end{smallmatrix} \\right)\\!,\n\\]\nall of which are skew-symmetric with respect to the standard metric and commute with\n\\[\nJ_1=\\left( \\begin{smallmatrix} 0&-1&0&0\\\\1&0&0&0 \\\\ 0&0&0&-1 \\\\ 0&0&1&0  \\end{smallmatrix} \\right).\n\\]\nCompleting the corresponding $A_i$ to the full matrix\n\\[\nB=\\begin{pmatrix} a & 0 \\\\ 0 & A_i \\end{pmatrix}\n\\]\nrepresenting $\\text{ad}_{e_6}\\rvert_{\\mathfrak{n}}$ and assuming $\\operatorname{tr} A_i \\neq 0$ to discard the K\\\"ahler cases,\none can easily see which algebras can be obtained:\n\\begin{itemize}\n\\item[] $A_1$ yields $\\mathfrak{g}_{1}$ and $\\mathfrak{g}_{4}$,\n\\item[] $A_2$ yields $\\mathfrak{g}_{2}$ and $\\mathfrak{g}_{5}$,\n\\item[] $A_3$ yields $\\mathfrak{g}_{3}$ and $\\mathfrak{g}_{6}$. \\qedhere \n\\end{itemize}\n\\end{proof}\n\n\\begin{theorem} \\label{LCB_class}\nLet $\\mathfrak{g}$ be a six-dimensional almost abelian Lie algebra which does not admit balanced or LCK structures. If $\\mathfrak{g}$ is nilpotent, then it admits an LCB structure $(J,g)$ if and only if it is isomorphic to one of the following:\n\\begin{itemize}\n\\setlength{\\itemindent}{-1em}\n\\item[] $(0,0,0,0,0,f^{12})$,\n\\item[] $(0,0,0,f^{12},f^{13},f^{14})$.\n\\end{itemize}\nIf $\\mathfrak{g}$ is non-nilpotent, then it admits an LCB structure $(J,g)$ if and only if it is isomorphic to one of the following:\n\\begin{itemize}\n\\setlength{\\itemindent}{-1em}\n\\item[] $\\mathfrak{l}_{1} \\hspace{1.35mm} =(f^{16},pf^{26},pf^{36},qf^{46},qf^{56},0)$,  \\,  $pr \\neq 0$, $p \\neq \\pm q$, \\smallskip\n\\item[] $\\mathfrak{l}_{2} \\hspace{1.35mm} =(f^{16},pf^{26}+f^{36},pf^{36},pf^{46}+f^{56},pf^{56},0)$, \\, $p \\neq 0$, \\smallskip\n\\item[] $\\mathfrak{l}_{3} \\hspace{1.35mm} =(pf^{16},qf^{26},qf^{36},rf^{46}+f^{56},-f^{46}+rf^{56},0)$,  \\,  $pq \\neq 0$, $q \\neq \\pm r$, \\smallskip\n\\item[] $\\mathfrak{l}_{4} \\hspace{1.35mm} =(pf^{16},qf^{26}+f^{36},-f^{26}+qf^{36},rf^{46}+sf^{56},-sf^{46}+rf^{56},0)$, \\, $pqs \\neq 0$, $q\\neq \\pm r$,\\smallskip\n\\item[] $\\mathfrak{l}_{5} \\hspace{1.35mm} =(pf^{16},qf^{26}+f^{36}-f^{46},-f^{26}+qf^{36}-f^{56},qf^{46}+f^{56},-f^{46}+qf^{56},0)$, \\,  $pq \\neq 0$, \\smallskip\n\\item[] $\\mathfrak{l}_{6} \\hspace{1.35mm} =(f^{16},f^{26},0,0,0,0)$, \\smallskip\n\\item[] $\\mathfrak{l}_{7} \\hspace{1.35mm} = (f^{16},f^{26}+f^{36},f^{36},0,0,0)$, \\smallskip\n\\item[] $\\mathfrak{l}_{8} \\hspace{1.35mm} =(pf^{16}+f^{26},-f^{16}+pf^{26},0,0,0,0)$, \\, $p \\neq 0$, \\smallskip\n\\item[] $\\mathfrak{l}_{9} \\hspace{1.35mm} =(f^{16},pf^{26},pf^{36},0,0,0)$, \\,  $p \\neq 0$,\\smallskip\n\\item[] $\\mathfrak{l}_{10} =(pf^{16},qf^{26}+f^{36},-f^{26}+qf^{36},0,0,0)$,  \\, $pq \\neq 0$, \\smallskip\n\\item[] $\\mathfrak{l}_{11}                =(f^{16},f^{26},pf^{36},pf^{46},0,0)$, \\,  $p \\neq 0,\\pm1$,\\smallskip\n\\item[] $\\mathfrak{l}_{12}                =(f^{16},f^{26},f^{46},0,0,0)$,\\smallskip\n\\item[] $\\mathfrak{l}_{13}                =(f^{16},f^{26},qf^{36}+rf^{46},-rf^{36}+qf^{46},0,0)$, \\, $q \\neq \\pm 1$, $r \\neq 0$, \\smallskip\n\\item[] $\\mathfrak{l}_{14}                =(pf^{16}+f^{26},-f^{16}+pf^{26},f^{46},0,0,0)$, \\smallskip\n\\item[] $\\mathfrak{l}_{15}                =(f^{16}+f^{26},f^{26},f^{36}+f^{46},f^{46},0,0)$, \\smallskip\n\\item[] $\\mathfrak{l}_{16}                =(pf^{16}+f^{26},-f^{16}+pf^{26},qf^{36}+rf^{46},-rf^{36}+qf^{46},0,0)$, \\, $r \\neq 0$, $p^2+q^2 \\neq 0$, $p \\neq \\pm q$, \\smallskip\n\\item[] $\\mathfrak{l}_{17}                =(pf^{16}+f^{26}-f^{36},-f^{16}+pf^{26}-f^{46},pf^{36}+f^{46},-f^{36}+pf^{46},0,0)$, \\, $p \\neq 0$.\n\\end{itemize}\nAmong these, only $\\mathfrak{l}_1^{q=-\\frac{1}{2}-p}$,  $\\mathfrak{l}_{2}^{p=-\\frac{1}{4}}$, $\\mathfrak{l}_{3}^{r=-\\frac{p}{2}-q}$, $\\mathfrak{l}_{4}^{r=-\\frac{p}{2}-q}$, $\\mathfrak{l}_{5}^{q=-\\frac{p}{4}}$, $\\mathfrak{l}_{9}^{p=-\\frac{1}{2}}$, $\\mathfrak{l}_{10}^{q=-\\frac{p}{2}}$ and $\\mathfrak{l}_{14}^{p=0}$ are unimodular.  \\end{theorem}\n\\begin{proof}\nLet $(J,g)$ be an LCB structure on $\\mathfrak{g}$. As in Theorem \\ref{LCK_class}, we need to examine each matrix $A_i$ in \\eqref{A} to see whether they can satisfy the LCB condition $A_i^tv=0$ for some suitable metric and vector $v$. Of course $v=0$ is a sufficient condition and, in this case, after discarding the algebras admitting balanced or LCK structures (including the nilpotent $(0,0,0,0,f^{12},f^{13})$, which admits balanced structures, by \\cite{Uga}), we have that\n\\begin{itemize}\n\\item[] $A_1$ yields $\\mathfrak{l}_{1}$, $\\mathfrak{l}_{6}$, $\\mathfrak{l}_{9}$ and $\\mathfrak{l}_{11}$,\n\\item[] $A_2$ yields $\\mathfrak{l}_{3}$, $\\mathfrak{l}_{8}$, $\\mathfrak{l}_{10}$ and $\\mathfrak{l}_{13}$, \\item[] $A_3$ yields $\\mathfrak{l}_{4}$ and $\\mathfrak{l}_{16}$,\n\\item[] $A_4$ yields $\\mathfrak{l}_{2}$ and $\\mathfrak{l}_{15}$,\n\\item[] $A_5$ yields $\\mathfrak{l}_{5}$ and $\\mathfrak{l}_{17}$.\n\\end{itemize}\nTo complete the classification, we now assume $v \\neq 0$. Then $A^tv=0$ forces $A$ to be degenerate: we are then left with $A_1^{q=0}$, $A_2^{q=0}$ (both with $p$ possibly vanishing) and $A_4^{p=0}$. If $a=g([e_6,e_1],e_1)$ is not an eigenvalue of $A_i$, then $\\operatorname{im}(A-a\\,\\text{Id}_{\\mathfrak{n}_1})=\\mathfrak{n}_1$, so that $v=AX-aX$ for some $X \\in \\mathfrak{n}_1$ and the matrix $B$ corresponding to $\\text{ad}_{e_6}\\rvert_{\\mathfrak{n}}$ can be brought into the form\n\\[\nB=\\begin{pmatrix} a & 0 \\\\ 0 & A_i \\end{pmatrix},\n\\]\nsimply by replacing $e_1$ with $e_1^\\prime=e_1-X$, so that eventually we get some of the previously found Lie algebras. Otherwise, if $a$ is an eigenvalue of $A_i$, the algebraic multiplicity of $a$ as an eigenvalue of $B$ (namely, its multiplicity as a root of the characteristic polynomial) might exceed its algebraic multiplicity for $A_i$ by one: this happens exactly when $v \\notin \\operatorname{im}(A-a\\,\\text{Id}_{\\mathfrak{n}_1})$ and, in this case, $B$ is similar to a $5 \\times 5$ matrix obtained by taking $A_i$ and raising the rank of a Jordan block relative to the eigenvalue $a$ by one: this can occur for $A_1^{q=0}$, when $a=p$ or $a=0$, for $A_2^{q=0}$ when $a=0$ and for $A_4^{p=0}$, $a=0$.\n\nFor the cases $A_{i}^{q=0}$, $i=1,2$, with $a=0$, one can simply assume that the basis $\\{e_2,\\ldots,e_5\\}$ with respect to which $A_i$ is in the form \\eqref{A} is orthonormal, with $Je_2=e_3$, $Je_4=e_5$, and take $v=e_4$, for instance. \n\nFor $A_4^{p=0}$, assume again that $\\{e_2,e_3,e_4,e_5\\}$ is orthonormal, this time satisfying $Je_2=e_4$, $Je_3=e_5$ and take $v=e_3$, for example.\n\nFor the remaining case $A_1^{q=0}$, $a=p \\neq 0$, one can consider for example the Hermitian almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$ determined by the data\n\\[\na=p,\\quad v= \\left( \\begin{smallmatrix} 0\\\\0\\\\1\\\\0 \\end{smallmatrix} \\right), \\quad A=\\left( \\begin{smallmatrix} p&1&0&0\\\\0&0&0&0 \\\\ 0&0&0&0 \\\\ 0&0&1&p \\end{smallmatrix} \\right),\\,p \\neq 0,\n\\]\nwith respect to an adapted unitary basis $\\{e_1,\\ldots,e_6\\}$, $Je_i=e_{7-i}$, $i=1,2,3$. Then, it is easy to check that $A$ is similar to $A_1^{q=0}$ and that $A^tv=0$, $v \\notin \\operatorname{im}(A-p\\,\\text{Id}_{\\mathfrak{n}_1})$, so that the structure is LCB and the whole matrix $B$ is similar to\n\\[\n\\left(\\begin{smallmatrix} p&1&0&0&0 \\\\ 0&p&0&0&0 \\\\ 0&0&p&0&0 \\\\ 0&0&0&0&0 \\\\ 0&0&0&0&0 \\end{smallmatrix} \\right)\n\\]\nas desired.\n\nThese new cases with $v \\neq 0$ yield Lie algebras isomorphic to $\\mathfrak{l}_{7}$, $\\mathfrak{l}_{12}$, $\\mathfrak{l}_{14}$ or one of the two nilpotent Lie algebras of the statement, concluding the proof.\n\\end{proof}\n\n\n\\begin{remark}\nIt can be shown that, among the unimodular Lie groups whose Lie algebra appears in Theorem \\ref{LCB_class}, the ones with Lie algebra\n$\\mathfrak{l}_1^{q=-\\frac{1}{2}-p}$, $\\mathfrak{l}_{2}^{p=-\\frac{1}{4}}$, $\\mathfrak{l}_{3}^{r=-\\frac{p}{2}-q}$ and $\\mathfrak{l}_{9}^{p=-\\frac{1}{2}}$\ndo not admit any compact quotients by lattices.\n\nWe prove this only for $\\mathfrak{l}_1^{q=-\\frac{1}{2}-p}$, since the discussion for the other two Lie algebras is analogous.\nFollowing \\cite{Bock}, a co-compact lattice exists on such Lie groups if and only if there exists a non-zero $t_0 \\in \\mathbb{R}$ and a basis of $\\mathfrak{n}$ such that the matrix associated with $\\text{exp}(t_0\\text{ad}_{f_6})\\rvert_{\\mathfrak{n}}$ has integer entries. In the basis $\\{f_1,\\ldots,f_5\\}$ one easily computes \n\\begin{equation} \\label{exptad}\n\\text{exp}(t\\,\\text{ad}_{f_6})\\rvert_{\\mathfrak{n}}=\\text{diag}\\big(e^t,e^{pt},e^{pt},e^{-pt-\\frac{1}{2}t},e^{-pt-\\frac{1}{2}t}\\big).\n\\end{equation}\nIts minimal polynomial, namely the monic polynomial $P_t$ of least degree such that $P_t(\\text{exp}(t\\,\\text{ad}_{f_6})\\rvert_{\\mathfrak{n}})=0$, is of the form $P_t(x)=\\sum_{i=0}^3 a_i(t,p) x^i$, with coefficients\n\\[\na_0=-e^{\\frac{t}{2}},\\quad a_1=e^{t(1+p)}+e^{-\\frac{t}{2}}+e^{t\\left(\\frac{1}{2}-p\\right)},\\quad a_2=-e^{pt}-e^t-e^{-t\\left(\\frac{1}{2}+p\\right)},\\quad a_3=1.\n\\]\nIf \\eqref{exptad} is conjugate to an integer matrix for some $t_0$, then necessarily $P_{t_0}(x)$ is an integer polynomial, so that $a_0(t_0,p) \\in \\mathbb{Z}$ forces $t_0=2\\log k$, for some $k \\in \\mathbb{Z}_{>0}$. \nAssuming $a_2(t_0,p) \\in \\mathbb{Z}$, one computes\n\\[\nk^2\\left(k^2+a_2(t_0,p)\\right)+a_1(t_0,p)=\\tfrac{1}{k},\n\\]\nwhich is integer if and only if $k=1$, that is, $t_0=0$, a contradiction.\n\nInstead, for some choices of the parameters, the Lie groups with Lie algebra $\\mathfrak{l}_{4}^{r=-\\frac{p}{2}-q}$, $\\mathfrak{l}_{10}^{q=-\\frac{p}{2}}$ and $\\mathfrak{l}_{14}^{p=0}$ admit co-compact lattices (see \\cite{FP} and the references therein). Some results are known for the remaining Lie groups, namely the ones corresponding to $\\mathfrak{l}_{5}^{q=-\\frac{p}{4}}$ \\cm{$\\mathfrak{k}_{12}^{p,-\\frac{p}{4}}$} (see \\cite{CM}), but the existence of lattices on them is still an open problem.\n\\end{remark}\n\n\n\\section{Compatibility results between Hermitian metrics} \\label{sec_comp}\n\nIn this section, we ask whether a (unimodular) almost abelian Lie algebra endowed with a fixed complex structure may admit two different kinds of special Hermitian metrics.\n\nIn order to carry over the results to almost abelian solvmanifolds, we exploit the well-known ``symmetrization'' process. We summarize the results we need in the next lemma.\nRecall that a solvable Lie group is called \\emph{completely solvable} if all the eigenvalues of $\\text{ad}_X$ are real, for every $X$ in its Lie algebra.\n\\begin{lemma} {\\normalfont (\\cite{Bel, FG, Uga, Saw, AU})} \\label{symm}\nLet $\\Gamma \\backslash G$ be a compact solvmanifold endowed with a left-invariant complex structure $(J,g)$. Then, the existence of a balanced (resp.\\ SKT) metric implies the existence of a left-invariant balanced (resp.\\ SKT) metric.\nIf $G$ is completely solvable, the analogous results hold for LCK and LCB metrics.\n\\end{lemma}\n\n\\subsection{SKT and LCB} The SKT condition and the balanced condition are two ``transversal'' generalizations of the K\\\"ahler condition. Indeed, by \\cite{AI} a Hermitian metric which is both SKT and balanced is K\\\"ahler and it has been conjectured in \\cite{FV} that a compact complex manifold admitting an SKT metric and a balanced metric necessarily admits a K\\\"ahler metric as well. For almost abelian solvmanifolds, the conjecture was proven in \\cite{FP1}. \n\nThe same transversality no longer holds when considering the weaker LCB condition instead of the balanced condition, and the same Hermitian metric can even be SKT and LCB at the same time: in \\cite{FT1}, it was proven that\nevery non-K\\\"ahler compact homogeneous complex surface admits a compact torus bundle carrying an SKT and LCB metric; moreover, an example of compact nilmanifold in any even dimension admitting a left-invariant metric which is both SKT and LCB with respect to a fixed left-invariant complex structure was exhibited in \\cite{OOS}.\n\nIn addition, recalling that LCK metrics are particular instances of LCB metrics, it is easy to see that a non-K\\\"ahler LCK almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$ is also SKT if and only if it satisfies \\eqref{LCK} with $a \\neq 0$ and $\\lambda=-\\frac{a}{2}$ or if $n=2$, $\\mathfrak{g} \\cong \\mathfrak{h}_3 \\oplus \\mathbb{R}$ or $\\mathfrak{g} \\cong \\mathfrak{aff}_2 \\oplus 2\\mathbb{R}$ (cf. also \\cite{FP1}).\n\n\\begin{proposition}\nLet $\\mathfrak{g}$ be an almost abelian Lie algebra endowed with a complex structure $J$. If $(\\mathfrak{g},J)$ admits an SKT metric, then it admits an LCB metric as well.\n\\end{proposition}  \n\\begin{proof} Let $g$ denote the SKT metric. By \\cite{AL}, with respect a unitary basis $\\{e_1,\\ldots,e_{2n}\\}$ of $\\mathfrak{g}$ adapted to the splitting $\\mathfrak{g}=J\\mathfrak{k} \\oplus \\mathfrak{n}_1 \\oplus \\mathfrak{k}$, the matrix $B$ associated with $\\text{ad}_{e_{2n}}\\rvert_{\\mathfrak{n}}$ is of the form \\eqref{B}, with $[A,A^t]=0$ and the eigenvalues of $A$ having real part equal to $-\\frac{a}{2}$ or $0$.\n\nDecompose $v \\in \\mathfrak{n}_1=\\operatorname{im}(A-a\\,\\text{Id}_{\\mathfrak{n}_1}) \\oplus \\left(\\operatorname{im}(A-a\\,\\text{Id}_{\\mathfrak{n}_1})\\right)^{\\perp_g}$ as $v=AX-aX+v^\\prime$ for some $X \\in \\mathfrak{n}_1$ and $v^\\prime \\in \\left(\\operatorname{im}(A-a\\,\\text{Id}_{\\mathfrak{n}_1})\\right)^{\\perp_g}$, that is, $(A-a\\,\\text{Id}_{\\mathfrak{n}_1})^tv^\\prime=0$.\n\nConsider the new $J$-Hermitian metric $g^\\prime=g\\rvert_{\\mathfrak{n}_1} + ({e^1}^\\prime)^2 + ({e^{2n}}^\\prime)^2$, with $e_1^\\prime=e_1-X$, $e_{2n}^\\prime=Je_1^\\prime$. Then, the matrix $B^\\prime$ associated with $\\text{ad}_{e_{2n}^\\prime}\\rvert_{\\mathfrak{n}}$ with respect to the new adapted unitary basis $\\{e_1^\\prime,e_2,\\ldots,e_{2n-1}\\}$ for $\\mathfrak{n}$ is of the form\n\\[\nB^\\prime=\\begin{pmatrix} a & 0 \\\\ v^\\prime & A \\end{pmatrix},\n\\]\nwith $A$ as above and $(A-a\\,\\text{Id}_{\\mathfrak{n}_1})^tv^\\prime=0$. If $a \\neq 0$, $a$ is not an eigenvalue of $A$, so that $v^\\prime=0$. Instead, if $a=0$, we have $A^tv^\\prime=0$. In either case, the metric $g^\\prime$ is LCB.\n\\end{proof}\n\nUsing Lemma \\ref{symm}, we get\n\\begin{corollary}\nLet $\\Gamma \\backslash G$ be a compact almost abelian solvmanifold endowed with a left-invariant complex structure $J$. If $(\\Gamma \\backslash G,J)$ admits an SKT metric, then it admits an LCB metric as well.\n\\end{corollary}\n\n\\begin{example} \\label{exampleSKTLCB}\nWe now exhibit an example of compact almost abelian solvmanifold in any even dimension admitting a (left-invariant) Hermitian structure which is at the same time SKT and LCB. For any $n \\geq 2$, consider the $2n$-dimensional simply connected unimodular almost abelian Lie group $S_{2n}$ having indecomposable Lie algebra $\\mathfrak{s}_{2n}$ endowed with a fixed coframe $\\{e^1,\\ldots,e^{2n}\\}$ satisfying the structure equations\n\\begin{gather*}\nde^1=a\\,e^1 \\wedge e^{2n},\\quad de^2=-\\tfrac{a}{2} \\, e^2 \\wedge e^{2n} + e^3 \\wedge e^{2n},\\quad de^3=- e^2 \\wedge e^{2n} - \\tfrac{a}{2} \\, e^3 \\wedge e^{2n}, \\quad de^{2n}=0, \\\\\nde^{2i}=c \\, e^{2i+1} \\wedge e^{2n},\\quad de^{2i+1}=-c \\, e^{2i} \\wedge e^{2n},\\quad i=2,\\ldots,n-1,\n\\end{gather*}\nfor some $a,c \\in \\mathbb{R} - \\{0\\}$, with $c$ depending on $a$ in a way which we shall explain. \nNow, it is easy to check that the left-invariant Hermitian structure $(J,g)$ on $S_{2n}$ defined by\n\\[\nJe_1=e_{2n}, \\quad Je_{2i}=e_{2i+1},\\,i=1,\\ldots,n-1, \\quad g=\\sum_{i=1}^{2n} (e^i)^2,\n\\]\nis both SKT and LCB, satisfying in particular $v=0$.\n\t\nAs we shall now show, $S_{2n}$ admits compact quotients by lattices, for all $n$, for some values of $a$ and $c$: by \\cite{Bock}, this is equivalent to proving that there exists $t_0 \\in \\mathbb{R}-\\{0\\}$ such that $\\text{exp}(t_0B_{2n})$ is similar to an integer matrix, $B_{2n}$ being the $(2n-1) \\times (2n-1)$ matrix representing $\\text{ad}_{e_{2n}}\\rvert_{\\mathfrak{n}}$ in the fixed basis. The claim is true for $n=2$ for countably many values of $a \\in \\mathbb{R}-\\{0\\}$, with compact quotients of $S_4$ biholomorphic to Inoue surfaces (see \\cite[Section 3.2.2]{AO} for a detailed discussion). Fixing $a \\in \\mathbb{R}$ such that $S_4$ admits co-compact lattices, let $t_0\\in \\mathbb{R}-\\{0\\}$ be such that $\\text{exp}(t_0B_{4})$ is similar to an integer matrix and set $c \\coloneqq \\tfrac{2\\pi}{t_0}$, so that\n\\[\n\\text{exp}\\left(t_0 \\left( \\begin{smallmatrix} 0 & c \\\\ -c & 0 \\end{smallmatrix} \\right)\\right)=\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}\n\\]\nis an integer matrix. The claim then easily follows in any dimension by induction.\n\\end{example}\n\n\\subsection{Balanced and LCK}\nBy \\cite{AO}, almost abelian Lie groups which admit left-invariant LCK structures and compact quotients by lattices only exist in real dimension four. The resulting solvmanifolds are biholomorphic to primary Kodaira surfaces, Inoue surfaces, hyperelliptic surfaces or complex tori: out of these, the only ones admitting K\\\"ahler metrics (recall that K\\\"ahler is equivalent to balanced, in real dimension four) are\ncomplex tori or hyperellyptic surfaces, which, by \\cite{HK}, cannot admit non-K\\\"ahler LCK metrics. Thus, we phrase the next result only in terms of structures on Lie algebras and not on compact almost abelian solvmanifolds, where the situation is already completely understood.\n\nIt was proven in \\cite{OOS} that a nilpotent Lie algebra cannot admit a balanced metric and a non-K\\\"ahler LCK metric both compatible with the same complex structure. In the almost abelian setting, the situation is analogous, apart from one exception in the non-unimodular case.\n\\begin{proposition} \\label{prop_balLCK}\n\tLet $\\mathfrak{g}$ be an almost abelian Lie algebra endowed with a complex structure $J$. If $(\\mathfrak{g},J)$ admits a balanced metric, then it does not admit any non-K\\\"ahler LCK metrics, unless $\\mathfrak{g} \\cong \\mathfrak{aff}_2 \\oplus 2\\mathbb{R}$.\n\\end{proposition}\n\\begin{proof}\n\tAs we have recalled in Proposition \\ref{Herm-alm-ab}, an LCK almost abelian Lie algebra $(\\mathfrak{g}(a,v,A),J,g)$ can either satisfy\n\t\\eqref{LCK} or $n=2$, $A=0$, which corresponds to $\\mathfrak{g} \\cong \\mathfrak{h}_3 \\oplus \\mathbb{R}$ (if $a=0$, $v \\neq 0$), $\\mathfrak{g} \\cong \\mathfrak{aff}_2 \\oplus 2\\mathbb{R}$ (if $a \\neq 0$) or to $4\\mathbb{R}$ (if $a=0$, $v=0$).\n\t\n\tLet $g$ denote an LCK metric and assume \\eqref{LCK}. The result readily follows by observing that, in order to admit a balanced metric, $(\\mathfrak{g},J)$ must satisfy $\\operatorname{tr} A=\\operatorname{tr} \\text{ad}_X \\rvert_{\\mathfrak{n}_1}=0$ for all $X \\in \\mathfrak{g}$, so that $A \\in \\mathfrak{u}(\\mathfrak{n}_1,J_1,g)$. This implies that $g$ is K\\\"ahler.\n\t\n\tIf $n=2$ (in which case balanced implies K\\\"ahler) and we have $A=0$, note that $\\mathfrak{h}_3 \\oplus \\mathbb{R}$ does not admit K\\\"ahler structures, while all Hermitian structures on $4\\mathbb{R}$ are K\\\"ahler. On $\\mathfrak{aff}_2 \\oplus 2\\mathbb{R} = (f^{12},0,0,0)$, consider the complex structure defined by $Jf_1=f_2$, $Jf_3=f_4$. The Hermitian metric $g=\\sum_{i=1}^4 (f^i)^2$ is K\\\"ahler, while, denoting $f^i \\odot f^j=\\frac{1}{2}(f^i \\otimes f^j +f^j \\otimes f^i)$,\n\t\\[\n\tg^\\prime=2(f^1)^2+2(f^2)^2+(f^3)^2+(f^4)^2+2\\,f^1 \\odot f^3+2\\,f^2 \\odot f^4\n\t\\]\n\tis non-K\\\"ahler and LCK, with fundamental form $\\omega^\\prime=2f^{12}+e^{14}-e^{23}+e^{34}$ satisfying $d\\omega^\\prime=f^{124}=(f^2+f^4) \\wedge \\omega^\\prime$, $d(f^2+f^4)=0$.\n\\end{proof}\n\n\\subsection{Balanced and LCB}\nBalanced metrics are trivially LCB. One could ask whether there exist non-K\\\"ahler compact solvmanifolds endowed with a left-invariant complex structure admitting both balanced metrics and non-balanced LCB metrics. \n\nFor nilmanifolds, the answer is affirmative, as shown in \\cite{OOS}.\nAs a corollary of the next proposition, the analogous result is not true for completely solvable almost abelian solvmanifolds.\n\\begin{proposition} \\label{prop_balLCB}\nLet $\\mathfrak{g}$ be a unimodular almost abelian Lie algebra endowed with a complex structure $J$. If $(\\mathfrak{g},J)$ carries a balanced metric, then it cannot admit any non-balanced LCB metrics.\n\\end{proposition}\n\\begin{proof}\nBy the characterization of unimodular balanced almost abelian Lie algebras, we know that $[\\mathfrak{g},\\mathfrak{g}] \\subset \\mathfrak{n}_1$, since there exists an adapted unitary basis with respect to the balanced metric satisfying $a=0$, $v=0$. In particular, $\\text{rank}(\\text{ad}_X)=\\text{rank}(A)$ for all $X \\in \\mathfrak{g}$ transverse to $\\mathfrak{n}$. Assume a non-balanced LCB metric $g$ exists. Then, any adapted unitary basis for $(\\mathfrak{g},J,g)$ satisfies $a=0$, $\\operatorname{tr} A=0$, $A^tv=0$, with $v \\neq 0$ to ensure the metric is non-balanced. Now, this implies $\\text{rank}(\\text{ad}_X)=\\text{rank}(A)+1$ for all $X \\in \\mathfrak{g}$ transverse to $\\mathfrak{n}$, since $v \\notin \\operatorname{im} A$, a contradiction.\n\\end{proof}\n\nRecalling Lemma \\ref{symm}, we obtain\n\\begin{corollary}\nLet $\\Gamma \\backslash G$ be a completely solvable almost abelian solvmanifold endowed with a left-invariant complex structure $J$. If $(\\Gamma \\backslash G,J)$ carries a balanced metric, then it cannot admit any non-balanced LCB metrics.\n\\end{corollary}\n\n\\begin{remark}\nWe note that Proposition \\ref{prop_balLCB} is no longer true if one drops the hypothesis of unimodularity: this is clear from the example on the four-dimensional Lie algebra $\\mathfrak{aff}_2 \\oplus 2\\mathbb{R}$ in the proof of Proposition \\ref{prop_balLCK}, recalling that LCK implies LCB. However, one can easily find other examples of complex structures of higher-dimensional almost abelian Lie algebras admitting both balanced and non-balanced LCB metrics: consider the six-dimensional almost abelian Lie algebra (see \\cite{FP1})\n\\[\n\\mathfrak{b}_2=(f^{16},f^{36},0,f^{56},0,0),\n\\]\nendowed with the complex structure defined by $Jf_1=f_6$, $Jf_2=f_4$, $Jf_3=f_5$. On it, one has the balanced metric $g=\\sum_{i=1}^6 (f^i)^2$ and the non-balanced and non-LCK LCB metric\n\\[\ng^\\prime=3(f^1)^2+(f^2)^2+(f^3)^2+(f^4)^2+(f^5)^2+3(f^6)^2+2(f^1 \\odot f^2 + f^1 \\odot f^3 + f^4 \\odot f^6 + f^5 \\odot f^6),\n\\]\nwhose associated Lee form is the closed $1$-form $\\theta^\\prime=f^5+f^6$, as shown by a direct computation.\n\\end{remark}\n\n\n\\section{Locally conformally hyperk\\\"ahler metrics} \\label{LCHK}\nWe now turn our attention to the study of (locally conformally) hyperk\\\"ahler metrics on almost abelian Lie algebras.\n\nIn the nilpotent setting, these structures were studied in \\cite{OOS}, where it was proven that compact nilmanifolds never admit left-invariant LCHK structures, unless they are tori.\n\nIn the next theorem, we classify almost abelian Lie algebras admitting LCHK structures. Recall that the spectrum of a matrix (or an endomorphism) $D$, denoted by $\\operatorname{Spec}(D)$, is the set of its eigenvalues. Given $z \\in \\mathbb{C}$, we denote by $m_D(z)$ its algebraic multiplicity for $D$, namely its multiplicity as a root of the characteristic polynomial of $D$. When $D$ is complex-diagonalizable, $m_D(z)$ is also equal to the (complex) dimension of the corresponding eigenspace.\n\n\\begin{theorem}\nA $4m$-dimensional almost abelian Lie algebra $\\mathfrak{g}=\\mathbb{R}^{4m-1} \\rtimes_D \\mathbb{R}$ admits an LCHK structure if and only if $D \\in \\mathfrak{gl}_{4m-1}$ is complex-diagonalizable and\n\\begin{itemize}\n\t\\item[\\normalfont (i)] $\\operatorname{Spec}(D) \\subset a + \\mathbb{R} i$, for some $a \\in \\mathbb{R}$,\n\t\\item[\\normalfont (ii)] $m_D(a) \\geq 3$,\n\t\\item[\\normalfont (iii)] $m_D(a+ib) \\in 2 \\mathbb{Z}$, for every $b \\in \\mathbb{R}-\\{0\\}$.\n\\end{itemize}\nThe Lie algebra $\\mathfrak{g}$ admits a hyperk\\\"ahler structure if and only if the above holds, with $a=0$, in which case $\\mathfrak{g}$ is unimodular and decomposable ($\\mathfrak{g}=\\mathfrak{g}^\\prime \\oplus (m_D(0))\\mathbb{R}$, with $\\mathfrak{g}^\\prime$ indecomposable) and every LCHK structure on $\\mathfrak{g}$ is hyperk\\\"ahler. In particular, there do not exist unimodular almost abelian Lie algebras admitting non-hyperk\\\"ahler LCHK structures.\n\\end{theorem}\n\\begin{proof}\nAssume $\\mathfrak{g}$ admits an LCHK structure $(I_1,I_2,I_3,g)$. In particular, $(I_1,g)$ is an LCK structure, so that there exists an adapted $(I_1,g)$-unitary basis $\\{e_1,\\ldots,e_{4m}\\}$ of $\\mathfrak{g}$ such that the matrix $B$ associated with $\\text{ad}_{e_{4m}}\\rvert_{\\mathfrak{n}}$ is of the form \\eqref{B} with the conditions \\eqref{LCK} or $m=1$, $A=0$, by \\cite{AO}. Note that, up to conjugation, we obtain the same $B$ when considering a basis adapted to $(I_2,g)$ or $(I_3,g)$.\n\nAssume \\eqref{LCK} holds. It follows that $B$ is complex-diagonalizable, hence $D$ is. Moreover, $\\operatorname{Spec}(B) \\subset \\{a\\} \\cup \\lambda + \\mathbb{R} i$. We claim that $\\lambda=a$: if this were not the case, in order for $(I_2,g)$ to be LCK, one should have that $I_1(e_{4m})=\\pm I_2(e_{4m})$, implying $I_3(e_{4m})=I_1I_2(e_{4m})= \\pm e_{4m}$, contradicting $I_3^2=-\\text{Id}_{\\mathfrak{g}}$. For the same reason, it follows that $m_{B}(a) \\geq 3$, to accommodate for the fact that $I_1(e_{4m})$, $I_2(e_{4m})$ and $I_3(e_{4m})$ should be eigenvectors for $B$ with real eigenvalue, hence equal to $a$.\n\nNow, denote by $V_z \\subset \\mathfrak{n} \\otimes \\mathbb{C}$ the eigenspace for $B$ corresponding to the eigenvalue $z \\in \\mathbb{C}$, and define \n\\[\n\\mathfrak{m} \\coloneqq \\mathfrak{n} \\cap I_1\\mathfrak{n} \\cap I_2 \\mathfrak{n} \\cap I_3 \\mathfrak{n}=\\text{span}\\left<e_{4m},I_1(e_{4m}),I_2(e_{4m}),I_3(e_{4m})\\right>^{\\perp_g}.\n\\]\nWe note that $I_1$, $I_2$, $I_3$ must preserve $W_z \\coloneqq (V_{z}+\\overline{V_{z}}) \\cap \\mathfrak{m}$, for all $z \\in \\text{Spec}(B)$, since $\\text{ad}_{e_{4m}}\\rvert_{\\mathfrak{m}}$ commutes with the restriction of each of the three complex structures on $\\mathfrak{m}$. Note that, when $z$ is not real, we have that $W_z$ is the set of real elements of $V_{z}\\oplus \\overline{V_{z}}$. It follows that $W_z$ inherits a hyperhermitian structure, so that its real dimension is a multiple of four, which implies that the complex dimension of $V_z$ is a multiple of $2$, when $z$ is not real. Up to rescaling $B$ to recover $D$, points (i), (ii) and (iii) of the statement follow. \n\nAssume now that $B$ satisfies $A=0$, with $m=1$. In particular we have that $\\mathfrak{g}$ is four-dimensional, with $\\dim [\\mathfrak{g},\\mathfrak{g}]=1$, so, by \\cite[Proposition 3.2]{Bar}, $\\mathfrak{g}$ does not admit hypercomplex structures.\n\nConversely, assume $\\mathfrak{g}=\\mathbb{R}^{4m-1} \\rtimes_D \\mathbb{R}$ satisfies $D$ being complex-diagonalizable and requirements (i), (ii) and (iii). It follows that, up to a change of basis $\\{e_1,\\ldots,e_{4m-1}\\}$ of $\\mathbb{R}^{4m-1}$, $D$ is of the form\n\\begin{equation} \\label{hyperD}\nD=\\text{diag}\\left(C_{1},C_{2},\\ldots,C_{m-1},a,a,a\\right),\n\\end{equation}\nwhere, for $i=1,\\ldots,m-1$, $C_i$ is a $4 \\times 4$ matrix of the form\n\\[\nC_i= \\left(\\begin{smallmatrix} \na & b_i & 0 & 0 \\\\\n-b_i & a & 0 & 0 \\\\\n0 & 0 & a & -b_i \\\\\n0 & 0 & b_i & a\n\\end{smallmatrix}\\right),\n\\]\nfor some (possibly vanishing) $b_i \\in \\mathbb{R}$. Denoting by $e_{4m}$ the generator of the extra $\\mathbb{R}$, an explicit LCHK structure on $\\mathfrak{g}$ is given by $(I_1,I_2,I_3,g)$, with $g=\\sum_{i=1}^{4m} (e^i)^2$ and, with respect to the fixed basis, $I_i=\\text{diag}(K_i,\\ldots,K_i)$, $i=1,2,3$, with\n\\[\nK_1=\\left( \\begin{smallmatrix}\n\t0&-1&0&0\\\\\n\t1&0&0&0\\\\\n\t0&0&0&-1 \\\\\n\t0&0&1&0\n\\end{smallmatrix} \\right), \\quad\nK_2=\\left( \\begin{smallmatrix}\n\t0&0&-1&0\\\\\n\t0&0&0&1\\\\\n\t1&0&0&0 \\\\\n\t0&-1&0&0\n\\end{smallmatrix} \\right), \\quad\nK_3=\\left( \\begin{smallmatrix}\n\t0&0&0&-1\\\\\n\t0&0&-1&0\\\\\n\t0&1&0&0 \\\\\n\t1&0&0&0\n\\end{smallmatrix} \\right).\n\\]\nThe three induced Lee forms are all equal to the closed $1$-form $\\theta=-(4m-2)ae^{4m}$.\n\nThe part of the claim regarding hyperk\\\"ahler structures easily follows from the fact that the K\\\"ahler condition on almost abelian Lie algebras corresponds to \\eqref{LCK}, with $\\lambda=0$. In particular, in this case, we note that, if $D$ is of the form \\eqref{hyperD}, with $m_D(0)=3+4h$, we can assume $b_i=0$, $i=m-h,\\ldots,m-1$, so that $\\text{span}\\left<e_{4(m-h)-3},\\ldots,e_{4m-1}\\right>$ is an abelian subalgebra of dimension $m_D(0)=3+4h$, while its complement, $\\text{span}\\left<e_1,\\ldots,e_{4(m-h-1)},e_{4m}\\right>$, is an indecomposable almost abelian Lie algebra.\n\\end{proof}\n\nThe previous theorem can be used to get a more precise list of almost abelian Lie algebras admitting hyperk\\\"ahler or LCHK structures: in the next proposition, we cover dimensions $4$, $8$ and $12$.\n\n\\begin{proposition}\nLet $\\mathfrak{g}$ be a $4m$-dimensional almost abelian Lie algebra.\n\\begin{itemize}[leftmargin=3em, itemindent=-1em]\n\\item\nIf $m=1$, $\\mathfrak{g}$ admits a hyperk\\\"ahler structure if and only if $\\mathfrak{g}=4\\mathbb{R}$, while it admits a non-hyperk\\\"ahler LCHK structure if and only if it is isomorphic to $(f^{14},f^{24},f^{34},0)$. \\smallskip\n\\item\nIf $m=2$, $\\mathfrak{g}$ admits a hyperk\\\"ahler structure if and only if it is isomorphic to one among \\smallskip\n\\begin{itemize}\n\t\\setlength{\\itemindent}{-1em}\n\t\\item[] $8\\mathbb{R}=(0,0,0,0,0,0,0,0)$,\\smallskip\n\t\\item[] $(f^{28},-f^{18},f^{48},-f^{38},0,0,0,0)$, \\smallskip\n\\end{itemize}\nwhile it admits a non-hyperk\\\"ahler LCHK structure if and only if it is isomorphic to \\smallskip\n\\begin{itemize}\n\t\\setlength{\\itemindent}{-1em}\n\t\\item[] $(f^{18},f^{28},f^{38},f^{48},f^{58},f^{68},f^{78},0)$,\\smallskip\n\t\\item[] $(f^{18},f^{28},f^{38},f^{48}+pf^{58},-pf^{48}+f^{58},f^{68}+pf^{78},-pf^{68}+f^{78},0)$, \\,$p \\neq 0$.\n\\end{itemize}\\smallskip\n\\item\nIf $m=3$, $\\mathfrak{g}$ admits a hyperk\\\"ahler structure if and only if it is isomorphic to one among \\smallskip\n\\begin{itemize}\n\t\\setlength{\\itemindent}{-1em}\n\t\\item[] $12\\mathbb{R}=(0,0,0,0,0,0,0,0,0,0,0,0)$,\\smallskip\n\t\\item[] $(f^{2,12},-f^{1,12},f^{4,12},-f^{3,12},0,0,0,0,0,0,0,0)$,\\smallskip\n\t\\item[] $(f^{2,12},-f^{1,12},f^{4,12},-f^{3,12},pf^{6,12},-pf^{5,12},pf^{8,12},-pf^{7,12},0,0,0,0)$, \\,$p \\neq 0$, \\smallskip\n\\end{itemize}\nwhile it admits a non-hyperk\\\"ahler LCHK structure if and only if it is isomorphic to \\smallskip\n\\begin{itemize}\n\\setlength{\\itemindent}{-1em}\n\t\\item[] $(f^{1,12},f^{2,12},f^{3,12},f^{4,12},f^{5,12},f^{6,12},f^{7,12},f^{8,12},f^{9,12},f^{10,12},f^{11,12},0)$,\\smallskip\n    \\item[] $(f^{1,12},f^{2,12},f^{3,12},f^{4,12},f^{5,12},f^{6,12},f^{7,12},f^{8,12}+pf^{9,12},-pf^{8,12}+f^{9,12},f^{10,12}+pf^{11,12},\\\\-pf^{10,12}+f^{11,12},0)$, \\,$p \\neq 0$,\\smallskip\n    \\item[] $(f^{1,12},f^{2,12},f^{3,12},f^{4,12}+pf^{5,12},-pf^{4,12}+f^{5,12},f^{6,12}+pf^{7,12},-pf^{6,12}+f^{7,12},f^{8,12}+qf^{9,12},\\\\-qf^{8,12}+f^{9,12},f^{10,12}+qf^{11,12},-qf^{10,12}+f^{11,12},0)$, \\,$pq \\neq 0$.\n\\end{itemize}\n\\end{itemize}\n\\end{proposition}\n\n\\medskip \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCryptocurrencies like Bitcoin \\cite{nakamoto2008bitcoin} have gained popularity as an alternative method of payment. Blockchain, a cryptographically secure, tamper proof ledger, forms the backbone of such decentralized network, guaranteeing pseudonymity of participant. The records stored in this distributed ledger can be verified by anyone in the network. Consensus algorithms like Proof-of-Work \\cite{nakamoto2008bitcoin}, \\cite{o2014bitcoin}, \\cite{bano2017consensus}, Proof-of-Stake \\cite{king2012ppcoin}, \\cite{li2017securing}) are used for reaching agreement on state change in the ledger across the network participants. However,  computation time taken by such consensus algorithm is the major bottleneck in scalability of blockchain based transaction \\cite{croman2016scaling}, \\cite{poon2016bitcoin}. To be at par with traditional methods of payment like Visa, PayPal, scaling blockchain transactions is an important concern which needs to be addressed, without compromising on the privacy.\n\nSeveral solutions like sharding \\cite{luu2016secure}, \\cite{gencer2016service}, alternate consensus architecture \\cite{kiayias2017ouroboros}, \\cite{miller2014permacoin}, \\cite{buterin2017casper}, \\cite{park2018spacemint}, \\cite{eyal2016bitcoin}, \\cite{pass2017hybrid}, side-chains \\cite{back2014enabling} have been proposed in Layer-one. But this requires revamping the trust assumptions of the base layer and changing the codebase. A more modular approach is exploring scalability in Layer-two \\cite{gudgeon2019sok}. It massively cuts down data processing on the blockchain by running computations off-chain. The amount of data storage on Layer-one is minimized. Taking transactions off the base layer, while still anchored to it, would free up processing resources to do other things. Also Layer-two relies on Layer-one for security. Several solutions like \\cite{decker2018eltoo}, \\cite{decker2015fast},  \\cite{luu2016secure} have been proposed. \\textit{Payment Channel} \\cite{decker2015fast} stands out as a practically deployable answer to the scalability issue. \n\nAny two users, with mutual consent, can open a payment channel by locking their funds in a deposit. Users can perform several off-chain payments between each other without recording the same on blockchain. This is done by locally agreeing on the new deposit balance, enforced cryptographically by smart contracts \\cite{poon2016bitcoin}, key based locking \\cite{malavoltamulti} etc. Whenever one of the party wants to close the payment channel, it broadcasts the transaction on blockchain with the final balance. None of the parties can afford to cheat by claiming payment for an older transaction. Opening of new payment channel between parties which are not connected directly has its overhead where funds get locked for a substantial amount of time. This can be avoided by leveraging on the set of existing payment channels for executing a transaction, proving beneficial in terms of resource utilization. These set of payment channels form the \\textit{Payment Channel Network (PCN)}. Several problem such as routing, security and interoperability needs to be addressed in such a network.\n\n\n\n\n\n   \nThe major challenge in designing any protocol for PCN is to ensure privacy of payer and payee and hiding the payment value transferred. No party, other than the payer and payee, should get any information about the transaction. Thus any routing algorithm designed for such a network must be decentralized, where individual nodes take decision based on the information received from its neighbourhood. Several distributed routing algorithms exists but they suffer from various disadvantages - Elias et al. \\cite{elias} requires a single node to maintain list of active vertices for executing push relabel algorithm on single source-sink pair, Flare  \\cite{prihodko2016flare} requires intermediate users to reveal the current capacity of their payment channels to the sender for computation of the maximum possible value to be routed through a payment path, Canal  \\cite{viswanath2012canal} entrusts a single node for computing maximum flow in a graph. Landmark-based routing algorithms \\cite{silentwhispers}, \\cite{speedymurmur} decide the number of landmarks by trial and error. If the total number of landmarks is $k$, then the payment value is split into $k$ microtransactions randomly without considering the nature of the graph. Such a myopic approach for routing each microtransaction may result in failure as it does not allow optimal utilization of the available capacities present across multiple paths.\n\nIt was first mentioned in Elias et al. \\cite{elias} that push relabel fits better as a routing algorithm for PCN  because it proceeds locally, taking into account the residual capacity of each payment channel. However, the push relabel algorithm used for single source-sink pair \\cite{elias} is not decentralized in nature. A distributed version of the same was implemented in their paper for multiple source-sink pair but it is not well defined. It is not clear how many payment transfer can be allowed at a time through a channel. Further, it was assumed that each payment value for a source-sink pair is unsplittable. This assumption does not work in real life since the payment value might be higher than the bottleneck capacity of a single path. Deciding feasible routes even for a single payment transfer is an involved process in a distributed network. This motivated us to design a new routing algorithm for PCN which is privacy-preserving, efficient as well as scalable.\n\n\n\n\n\n\\vspace{-0.2cm}\n\\subsection{\\textbf{Our Contributions}}\nThe following contributions have been made in this paper :\n\\begin{itemize}[leftmargin=*]\n\n\\vspace{-0.1cm}\n\\item We have proposed a privacy preserving distributed routing algorithm, \\textit{HushRelay}, in payment channel network. \n\n\\item We have implemented the scheme and its performance has been compared with SpeedyMurmur \\cite{speedymurmur} in terms of \\textit{success ratio} and \\textit{time taken to route (TTR)} a payment. Testing was done on real instances of Ripple Network and Lightning Network \\footnote{In the absence of widespread PCN, we use the statistics of such real instances to create the network} and it is observed that \\emph{HushRelay} attains a success ratio of 1 in both the cases. However \\emph{SpeedyMurmur} attained a maximum success ratio of 0.9815 and 0.907 respectively, when number of landmarks is 6. The time taken to execute the routing algorithm in Ripple like Network and Lightning Network are 2.4s and 0.15189s for \\emph{HushRelay} but it takes 4.736s and 1.937s for \\emph{SpeedyMurmur}. These statistics justify our claim of the algorithm being efficient and scalable. The code is given in \\cite{Code}.\n\n\\item The proposed routing algorithm is modular and it can be combined with any other privacy preserving payment protocol.\n\\end{itemize}\n \n\\subsection{\\textbf{Organization}}\nSection \\ref{2} discusses the state-of-the-art in PCN. Section \\ref{3} gives a brief overview of the preliminaries. Section \\ref{4} defines the problem statement and Section \\ref{5} provides discusses \\emph{HushRelay} with Section \\ref{route} dealing with Generic Construction and Section \\ref{correct} providing the proof of correctness. Performance analysis of each subprotocol of \\textit{HushRelay} has been stated in Section \\ref{6} and Section \\ref{7} concludes the paper.\n\\vspace{-0.2cm}\n\\section{Related Work}\n\\label{2}\nA payment channel network is a peer-to-peer, path based transaction (PBT) network, where each party operates independent of other parties. Several P2P path-based transaction networks such as such as the Lightning Network for Bitcoin \\cite{poon2016bitcoin}, the Raiden Network for Ethereum \\cite{raiden}, SilentWhispers \\cite{silentwhispers}, InterLedger \\cite{thomas2015protocol}, Atomic-swap \\cite{atomic}, TeeChain \\cite{lind2017teechain} etc. have been developed over the years. Perun \\cite{dziembowski2017perun} proposes a more efficient network structure which is built around payment hubs. An extension of payment channel, State Channel Network \\cite{dziembowski2018general}, not only supports off chain payment but allows execution of complex smart contract. Spider network \\cite{sivaraman2018routing} adheres to a packet-switched architecture for payment channel network. Payment is split into several transaction units and it is transmitted over time across different paths. However the split does not take into account the bottleneck capacity of each path which might lead to failure of payment. BlAnC \\cite{panwar2019blanc}, a fully decentralized blockchain-based network, has been proposed which transfers credit between a sender and receiver on demand. \n\nTill date, the routing algorithms proposed for payment channel network are as follows : Canal \\cite{viswanath2012canal} - uses a centralized server for computing the path, Flare \\cite{prihodko2016flare} - requires intermediate nodes to inform source node about their residual capacity, SilentWhispers \\cite{silentwhispers} - a distributed PBT network without using any public ledger, SpeedyMurmur \\cite{speedymurmur} - a privacy preserving embedded based routing, extending Voute \\cite{roos2016voute}, depending on presence of landmark nodes. SpeedyMurmur is the most relevant privacy preserving distributed routing algorithm. However, it makes use of repeated trials to figure out a suitable split of the total transaction value across multiple paths. Elias et al. \\cite{elias} proposed an extended push relabel for finding payment flow in the payment network. They are the first to point out the flaw in assumption of considering transaction \\textit{unsplittable} for existing routing techniques. In real life, splitting of fund across multiple path is inevitable since the bottleneck capacity of a single path may be lower than the total value of fund transfer. Later, a distributed approach for PCN routing, CoinExpress \\cite{yu2018coinexpress}, was proposed for finding routes that fulfill payment with higher success ratio. A routing algorithm based on swarm intelligence, ant colony optimization \\cite{grunspan2018ant} has been explored. Hoenisch et al. \\cite{hoenisch2018aodv} proposed an adaptation of an Ad-hoc On-demand Distance Vector (AODV)-based routing algorithm which supports different cryptocurrencies allowing transactions across multiple blockchains. We observe that none of the past works provide an efficient and secure routing algorithm. It is either susceptible to leaking of sensitive information or there exist a central entity controlling the routing algorithm.\n\n\n\n\n\n\\section{Background}\n\\label{3}\nIn this section, we provide the required background on payment channel network. \nThe terms source\/payer means the sender node. Similarly, sink\/payee\/destination means the receiver node and transaction means payment transfer.\n\n\\subsection{\\textbf{Payment Channel Network}}\n\\begin{definition}\n\\label{basic}\nA Payment Channel Network (PCN) \\cite{malavolta} is defined as a bidirected graph $G:=(V,E)$, where $V$ is the set of accounts dealing with cryptocurrency and $E$ is the set of payment channels opened between a pair of accounts. A PCN is defined with respect to a blockchain. Only opening and closing of payment channel gets recorded on blockchain apart from disputed transactions where settlement is done by broadcasting the transaction on blockchain.\n\\end{definition}\n\nBasic operations of PCN \\cite{malavolta}-  \n\\begin{itemize}[leftmargin=*]\n\\item \\texttt{openPaymentChannel$(v_1,v_2,\\alpha,t,m)$} : For a given pair of accounts  $v_1,v_2 \\in V$, channel capacity $\\alpha$ (initial balance escrowed), timeout value of $t$ and processing fee charged $m$, \\texttt{openPaymentChannel} creates a new payment channel $(id_{(v_1,v_2)},\\alpha,t,m) \\in E$, where $id_{(v_1,v_2)}$ is the channel identifier, provided both $v_1$ and $v_2$ has authorized to do so and the funds contributed by each of them sum up to value $\\alpha$. \n\n\\item \\texttt{closePaymentChannel$(id_{(v_1,v_2)},\\tilde{\\alpha})$} : Given a channel identifier $id_{(v_1,v_2)}$ with balance $\\tilde{\\alpha}$, \\texttt{closePaymentChannel} removes the channel from $G$ provided it is authorized to do so by both $v_1,v_2 \\in V$. The balance $\\tilde{\\alpha}$ gets written on blockchain and this amount is distributed between $v_1$ and $v_2$ as per the net balance recorded.\n\n\\item \\texttt{payVal$(p(s,r),val)$} : $p(s,r)$ denotes a path between sender $s$ and receiver $r$. It is defined by a set of identifiers $id_{(s,v_1)},id_{(v_1,v_2)},\\ldots,id_{(v_n,r)}$, $s,v_1,v_2,\\ldots,v_n,r \\in V$, having enough credit to allow transfer of $val$ from $s$ to $r$, if for each payment channel denoted by $id_{(v_i,v_{i+1})}$ has capacity of at least $\\beta  \\geq val_i', val_i'=val+\\Sigma_{j=i+1}^{n} fee(v_j),   0 \\leq i \\leq n, v_0=s \\ and \\ v_{n+1}=r$, where $fee(v_j)$ is the processing fee charged by each intermediate node $v_j$ in $p(s,r)$. A successful \\texttt{payVal} operation leads to a decrease of capacity of each payment channel $id_{(v_i,v_{i+1})}$ by $val_i'$. Else the capacity of the channel remains unaltered. \n\\end{itemize}   \n\n\n\\subsection{\\textbf{Payment Flow problem}}\n\\label{flowpay}\nConsider a directed graph $G:=(V,E) : \\  n=|V|, m=|E|, m \\geq n-1$, having two distinguished vertices, source $s \\in V$, sink $r \\in V, s \\neq r$, as a \\textit{flow network.} For a pair of vertices $v,w$, distance from $v$ to $w$ in graph $G$ is defined by $d_G(v,w)$, the minimum number of edges on the path from $v$ to $w$; if there is no path from $v$ to $w$, $d_G(v,w)=\\infty$. A positive real-valued capacity $c(v,w)$, defined by $c : E \\rightarrow \\mathbb{R}$, is the amount of funds that can be\ntransferred between two nodes sharing an edge. For every edge $(v,w) \\in E$ ; if $(v,w) \\not \\in E$, then $c(v,w)=0$. A flow $f$ on $G$ is a real-valued function  on vertex pairs satisfying the constraints \\cite{tarjan}, \\cite{thuy2005distributed} :\n\\begin{equation}\n\\label{eq1}\n\\begin{matrix}\nf(v,w) \\leq c(v,w), \\ \\forall (v,w) \\in V \\times V \\ \\textrm{(capacity)}, \\\\\nf(v,w)=-f(w,v), \\ \\forall (v,w) \\in V \\times V \\ \\textrm{(antisymmetry)}, \\\\\n\\Sigma_{u \\in V} f(u,v)=0 \\ \\forall v \\in V-\\{s,r\\} \\ \\textrm{(flow-conservation)}, \\\\\n\\end{matrix}\n\\end{equation}\nThe net flow into the sink is given by $f$, where:\n\\begin{equation}\nf=\\Sigma_{v \\in V} f(v,r)\n\\end{equation}\nA payment channel network can be mapped to flow network with channels forming the edges and funds locked on each channel can be considered as the edge capacity. Finding the maximum flow value from source to sink for a flow network is termed as the \\textit{Maximum Flow problem}. In the context of PCN, given a payment value $val$, one has to find a feasible flow from payer to payee, which is termed here as \\textit{Payment Flow problem}. Any max-flow algorithm with subtle modifications can be applied here, taking into account the preflow $f$ of each vertex (except the source and sink) on the network. A preflow is a real-valued function on a vertex pair which satisfies the first two constraints of Eq. \\ref{eq1} and a weaker form of the third constraint :\n\\begin{equation}\n\\Sigma_{u \\in V} f(u,v)\\geq 0, \\ \\forall v \\in V-\\{s,r\\} \\ \\textrm{(non-negativity constraint)},\n\\end{equation}\n\nA residual capacity of an edge $(v,w) \\in E$ is the amount of capacity remaining after the preflow $f$, i.e. $c(v,w)-f(v,w)$ and it is denoted by $r_f(v,w)$. A residual graph $G_f=(V,E_f)$ for a preflow $f$ is the graph whose vertex set is $V$ and edge set $E_f$ is the set of residual edges $(v,w) \\in E \\ : \\ r_f(v,w)>0$. \nThe \\textit{flow excess e(v)} of a vertex $v$ is the net balance of funds in node $v$ denoted by $\\Sigma_{u \\in V} f(u,v)$. The algorithm ends with all vertices except $s$ and $r$ having zero excess flow. If sink is unreachable or if the network does not have adequate capacity for transferring the amount $val$, then the excess value is pushed back to source $s$. \n\n\n\n \n\n\n\n\n\\section{Problem Statement}\n\\label{4}\nIt is not always possible to route the transaction across a single path as the value may be quite high compared to minimum capacity of the designated path. Hence it is better to find set of paths such that the total amount to be transferred is split across each such path. We define the problem as follows -\n\\begin{problem}\n\\textit{Given a payment channel network $G(V,E)$, a transaction request ($s,r,val$) for a source-sink pair $(s,r)$, the objective is to find a set of paths $p_1,p_2,\\ldots,p_m$ for transferring the fund from $s$ to $r$ such that $p_1$ transfers $val_1$, $p_2$ transfers $val_2,\\ldots,$ $p_m$ transfers $val_m : val=\\Sigma_{i=1}^m val_i$ without violating \\textit{transaction level privacy} i.e. neither the sender nor the receiver of a particular transaction must be identified as well as hiding the actual transaction value from intermediate parties.}\n\\end{problem}\n\n\n\\section{Our Proposed Construction}\n\\label{5}\nIn this section we provide a detailed overview of the routing algorithm, \\textit{HushRelay}. The payment network comprises set of payment channels denoted by channel identifier $id_{(i,j)}, (i,j)\\in E$.\nWe describe state the model and the assumptions made. \n\n\\subsection*{\\textbf{Network Model and its Assumptions}}\n\\begin{itemize}[leftmargin=*]\n    \\item The network is static i.e. no opening of new payment channel or closing of existing payment channel is considered.\n    \\item The topology of the network is known by any node in the network since any opening or closing of channel is recorded on the blockchain.\n    \\item Atmost one timelock contract is allowed to be established on a payment channel at a time.\n    \\item Sender of a payment chooses set of paths to the receiver according to her own criteria.\n    \\item  The current value on each payment channel is not published but instead kept locally by the users sharing a payment channel. \n    \\item Pairs of users sharing a payment channel communicate through secure and authenticated channels\n(such as TLS).\n\n    \n\\end{itemize}  \n\n\\subsection{Generic Algorithm}\n\\label{route}\nSince we consider PCN as a flow network, for solving the \\textit{payment flow problem} in the given network for executing a transaction request $(s,r,val)$, we propose a routing algorithm inspired from distributed push relabel algorithm stated in \\cite{tarjan}, \\cite{thuy2005distributed}. The algorithm proceeds locally by exchange of messages between neighbouring nodes. No single entity controls the flow in the network.\n\nBefore discussing the algorithm, we briefly describe the \\textit{Push Relabel} algorithm for a single source-sink pair (as stated in \\cite{tarjan}):\n\\begin{itemize}[leftmargin=*]\n\\item The instruction \\textit{push} redirects the excess flow of a vertex to the sink via its neighbouring vertices. The amount of excess flow that can be \\textit{pushed} from a vertex $v$ to one of its neighbouring vertex $w$ is $\\delta=min(e(v),r_{f(v,w)})$, where $r_{f(v,w)}$ is the residual capacity of edge $(v,w)$. The value $\\delta$ is added to the preflow value $f(v,w)$ (subtracted from $f(w,v)$) and subtracted from $e(v)$. Any push which results in zero residual capacity of the edge is said to be \\textit{saturating}. \n\n\\item A valid labeling function $d : V \\rightarrow I^{+}\\cup\\{0,\\infty\\}$ is used for estimating the distance of a vertex $v$ from sink $r$. $d(s)=n, d(r)=0$ and $d(v)\\leq d(w)+1$ for every residual edge $(v,w)$. The label $d(v) < n$ forms the lower bound on the actual distance from $v$ to $r$ in the residual graph $G_f$ and if $d(v)\\geq n$, then $d(v)-n$ is a lower bound on the actual distance from $r$ in the residual graph. \n\\item A relabeling operation is initiated when a vertex with excess flow has a label less than or equal to that of the neighbouring vertex. Once relabeling is done, it can initiate a push operation. So one can think labels to denote the potential level, where flow can occur from a region of higher potential to a region of lower potential.\n\n\\item A vertex $v$ is defined as active $v \\in V - \\{s,r\\}, d(v) < \\infty$, and $e(v)>0$. The maximum-flow algorithm is initialized with preflow value $f$, which is summation of the edge capacities of all edges incident from the source vertex $s$ and rest all other edges have zero flow.  \n\\end{itemize}\n\nFor our distributed algorithm, \\textit{HushRelay}, the basic operations is \\textit{Push} and \\textit{Relabel}, with all the nodes acting as individual processing unit in parallel. The network model considered for payment channel network is asynchronous. \nSynchronization across all the nodes is achieved via use of \\textit{acknowledgements} \\cite{tarjan}. A vertex $v$ tries to push excess flow to one of its neighbouring vertex $w$ if and only if, as per the information maintained by $v$, label $d(v)=d(w)+1$. It first sends a request message with the information $(v,\\delta,d(v),e(v))$. Vertex $w$ can either accept the push by sending an acknowledgement or it may reject it by sending a negative acknowledgement $(NAK)$. If $d(v)=d(w)+1$, then $w$ sends to $v$ a message of the form $(accept,w,\\delta,d(w))$ and $v$ initiates the push. Otherwise, if $d(w)\\geq d(v)$ or $d(v) < d(w)+1$, then it sends a message $(reject,w,\\delta,d(w))$ where $d(w)$ is the updated distance label of $w$. A reject message will cause $v$ to update the value of $d(w)$. When a distance label of the vertex increases, it sends the information of new label to all its neighbouring nodes. \n\nAs seen in \\textit{Push Relabel} algorithm \\cite{tarjan}, the label initially set for source and sink node reveals the identity of payer and payee. To obfuscate their identity from other intermediate nodes in the network, we use a dummy source vertex $s'$ for node $s$ and a dummy sink vertex $r'$ for node $r$. Note that the existence of dummy node is known only by the source and sink. In the initialization phase of \\textit{HushRelay}, a directed virtual edge from $s'$ to $s$ and from $r$ to $r'$ is established. Since $s',r'$ are virtual entities, introduction of edge $(s',s)$ and $(r,r')$ is not recorded in the blockchain. The capacity is initialized to $c(s',s)=val, c(r,r')=val$  and the label is set as $d(s')=n+2, d(s)=0, d(r)=0$ and $d(r')=0$. The flow $f(s',s)$ is set to $val$, $f(r,r')=0$, excess flow $e(s)=val$, $e(r)=e(r')=0$. For all vertices $v \\in V-\\{s,r\\}$, $d(v)=0,e(v)=0$, $f(w,v)=0, (w,v) \\in E, w,v \\in V-\\{s,r\\}$. We mention the procedure of \\textit{Push}, \\textit{Push-request} and \\textit{Relabel} for a vertex in Procedure \\ref{algo:v4}, \\ref{algo:vpr} and \\ref{algo:v5} respectively.\n\\begin{proc}[!ht]\n    \\SetKwInOut{Input}{Input}\n    \\SetKwInOut{Output}{Output}\n\n     \\Input{ Active vertex $v \\in V, e(v)>0$, vertices $w: (v,w) \\in E$ }\n      \n    \\caption{Push(v,w,d)}\n        \\label{algo:v4}\n     \\If{ $v\\neq r'$ and $v \\neq s'$}\n     {\n         Set $find\\_neighbour$=0 \\\\\n         \\While{neighbour $w$ of $v$ : $e(v)>0, r_f(v,w)>0 \\ and \\ d(w) < d(v)$}\n         {\n         \t$v$ generates a push of the value $\\delta=min(e(v),r_f(v,w)$ \\\\\n         \t$f(v,w)=f(v,w)+\\delta$\\\\\n         \t$e(v)=e(v)-\\delta$\\\\\n         \t$r_f(v,w)=c(v,w)-f(v,w)$\\\\\n         \n         \tSend \\textit{Push-request(w,v,$f(v,w),\\delta$)} to node $w$ \\\\\n         \t\\If{\\textit{NAK} received}\n         \t{\n         \tUpdate information $d(w)$ \\\\\n         \t$f(v,w)=f(v,w)-\\delta$\\\\\n         \t$e(v)=e(v)+\\delta$\\\\\n         \t$r_f(v,w)=c(v,w)+f(v,w)$\\\\\n         \t}\n         \t\\Else\n         \t{\n         \t Set $find\\_neighbour=1$\\\\\n         \t \n         \t}\n        \n         }\n         \\If{$find\\_neighbour=0$}\n          {\n             Call \\textit{Relabel} function.\n          }\n     }\n\\end{proc}\n\n\n\n\n\\begin{proc}[!ht]\n    \\SetKwInOut{Input}{Input}\n    \\SetKwInOut{Output}{Output}\n\n     \\Input{ Active vertex $w \\in V, e(w)>0$, vertex $v: (w,v) \\in E$ }\n      \n    \\caption{Push-request(v,w,$f(w,v)$,$\\delta$)}\n        \\label{algo:vpr}\n     \\If{ $d(v)<d(w)$}\n     {\n         $r_f(v,w)=f(w,v)$\\\\\n         $e(v)=e(v)+\\delta$ \\\\\n         \\If{$v\\neq t'$ and $v \\neq s'$}\n         {\n         \t  send \\textit{Push} request to its neighbouring nodes.\\\\\n         \t  \n         \t  \n         }\n         Send \\textit{push request accepted} message to node $w$.\n     }\n     \\Else{\n          Send \\textit{negative acknowledgement (NAK)} message and current value of $d(v)$ to node $w$.\n     }\n\\end{proc}\n\n\n\\begin{proc}[!ht]\n    \\SetKwInOut{Input}{Input}\n    \\SetKwInOut{Output}{Output}\n\n     \\Input{ Active vertex $v \\in V, e(v)>0$, vertices $w: (v,w) \\in E, r_f(v,w)>0, d(v)\\leq d(w)$ }\n      \n    \\caption{Relabel(v,w,d)}\n        \\label{algo:v5}\n     \\begin{enumerate}\n     \\item Update $d(v)=min(d(w) ,(v,w) \\in E) +1$\n     \\item Inform all the neighbours of vertex $v$ about the updated label $d(v)$.\n     \\end{enumerate}\n\\end{proc} \n\n \n\n\n\nThe algorithm terminates when there are no active vertex left (except the dummy source and dummy sink) in the graph. The number of messages exchanged (\\textit{for push request, push accepted\/NAK, height updation}) is also bounded. The communication complexity, termination condition of the algorithm and an upper bound on the $d$ value of a node in the given graph is stated in \\cite{thuy2005distributed}. $\\mathcal{O}(n^2m)$ messages are exchanged in the asynchronous implementation with $\\mathcal{O}(n^2)$ runtime. The overhead lies in the interprocessor communication between a vertex and its neighbours.\n\n\\begin{example}\nConsider a network given in Fig. \\ref{fig:example1}. Sender $S$ intends to make a payment of 15 units to receiver R. Dummy vertices $S'$ and $R'$ is added to the network with edges $(S',S)$ and $(R,R')$. The edge capacities are as follows : $c(S',S)=15, c(S,A)=10, c(S,B)=10, c(A,C)=10, c(B,C)=15, c(C,R)=20$ and $c(R,R')=15$. \\textit{HushRelay} is implemented on the network to obtain a feasible flow of value 15 from source node S to sink node R. The initial state is given in Fig. \\ref{fig:example1} (a). \n\\begin{figure}[!ht]%\n    \\centering\n    \\label{img1}\n\n    \\subfloat[Initial state]{{\\includegraphics[width=6.3cm]{fig1} }}%\n    \\end{figure}\n\nIn the initialization phase, each nodes is assigned a label of 0 except dummy vertex $S'$ where d(S') is the count of the number of nodes (except S') in the network. As given in Fig. \\ref{fig:example1} (b), S' sends a push request of 15 units to S.  \n\n\\begin{figure}[!ht]\\ContinuedFloat\n\n    \\centering \n               \\subfloat[state]{{\\includegraphics[width=6.3cm]{fig02} }}%\n    \\end{figure}\n In Fig. \\ref{fig:example1} (c), S accepts the push request as $d(S)<d(S')$ and an excess flow of 15 units is assigned to S. S changes its label calling \\emph{relabel} function and changes $d=1$. Now S sends a push request of 10 units to A, bounded by the capacity of payment channel SA.     \n\\begin{figure}[!ht]\\ContinuedFloat\n    \\centering\n\n        \\subfloat[state]{{\\includegraphics[width=6.3cm]{fig2} }}%\n    \\end{figure}\n \\\\In Fig. \\ref{fig:example1} (d), A accepts the push request as $d(A)<d(S)$ and gets an excess flow of 5 units. $d(A)$ is changed to 1. S still has an excess flow of 5. It sends a push request to B. Simultaneously A sends a push request of 10 units to C. \n\\begin{figure}[!ht]\\ContinuedFloat\n    \\centering\n\n    \\subfloat[state]{{\\includegraphics[width=6.3cm]{fig3} }}%\n            \n\\end{figure}\nIn Fig. \\ref{fig:example1} (e), B accepts the push request as $d(B)<d(S)$ and has excess flow of 5 units. $d(B)$ is changed to 1. Similarly C accepts push request as $d(C)<d(A)$ and has an excess flow of 10 units. $d(C)$ is changed to 1. B sends a push request to C and C sends a push request to R. In Fig. \\ref{fig:example1} (f), upon receipt of request from B, C finds that $d(C)=d(B)$ and hence it sends a NAK to B. R accepts the push request from C and gets an excess flow of 10. It changes its label to $d(R)=1$ and consequently, it sends a push request of 10 units to R'. In Fig. \\ref{fig:example1} (g), B changes its label to $d(B)=2$. R' accepts the push request.\n\\begin{figure}[!ht]\\ContinuedFloat\n    \\centering\n    \\subfloat[state]{{\\includegraphics[width=6.3cm]{fig4} }}%\n \\end{figure}\n \n\n \\begin{figure}[!ht]\\ContinuedFloat\n    \\centering\n            \\subfloat[state]{{\\includegraphics[width=6.3cm]{fig5} }}%\n\\end{figure}\n\n \\begin{figure}[!ht]\\ContinuedFloat\n    \\centering\n\n    \\subfloat[Initial state]{{\\includegraphics[width=6.3cm]{fig6} }}%\n\\end{figure}\nIn Fig. \\ref{fig:example1} (h), B sends a push request of 5 units to C.\n\n \\begin{figure}[!ht]\\ContinuedFloat\n    \\centering\n            \\subfloat[Initial state]{{\\includegraphics[width=6.3cm]{fig7} }}%\n\\end{figure}\n In Fig. \\ref{fig:example1} (i), C accepts the push request and changes it label to 1. It sends a push request of 5 units to R.\n\n \\begin{figure}[!ht]\\ContinuedFloat\n    \\centering\n        \\subfloat[Initial state]{{\\includegraphics[width=6.3cm]{fig8} }}%\n    \\end{figure}\n\n In Fig. \\ref{fig:example1} (j), R finds that $d(R)=d(C)$ and hence it sends a NAK to C. \n\n \\begin{figure}[!ht]\\ContinuedFloat\n    \\centering\n            \\subfloat[Initial state]{{\\includegraphics[width=6.1cm]{fig9} }}%\n        \\end{figure}\nIn Fig. \\ref{fig:example1} (k), C undergoes a relabeling operation and $d(C)$ is changed to 2.\n\\begin{figure}[!ht]\\ContinuedFloat\n    \\centering\n            \\subfloat[Initial state]{{\\includegraphics[width=6.3cm]{fig10} }}%\n\\end{figure}\n   In Fig. \\ref{fig:example1} (l), C again sends a push request of 5 units to R. \n\\begin{figure}[!ht]\\ContinuedFloat\n    \\centering\n\n        \\subfloat[Initial state]{{\\includegraphics[width=6.3cm]{fig11} }}%\n\\end{figure}\n\nIn Fig. \\ref{fig:example1} (m), R accepts the push request and it sends a push request of 5 units to R'.\n\\begin{figure}[!ht]\\ContinuedFloat\n    \\centering\n\n  \\subfloat[Initial state]{{\\includegraphics[width=6.1cm]{fig12} }}%\n\\end{figure}\n   In Fig. \\ref{fig:example1} (n), the algorithm terminates transferring 15 units to R'\n\\begin{figure}[!ht]\\ContinuedFloat\n    \\centering\n\n\\subfloat[After addition of dummy vertices S' and R', execution of the routing algorithm]{{\\includegraphics[width=6.1cm]{fig25} }\n\n\\caption{Execution of HushRelay}%\n\\label{fig:example1}%\n\\end{figure}\n\\end{example}\n\n\n\n\n\\subsection{Proof of correctness of the HushRelay}\n\\label{correct}\nWe state the following lemmas which justifies the correctness of our routing algorithm.\n\\begin{lemma}\n\\label{lm1}\nIf $val \\leq \\textrm{maximum flow}$ in $G$, then $s$ can successfully transfer funds to $r$.\n\\end{lemma}\n\\textbf{Proof of Lemma \\ref{lm1}}. Given that $\\tilde{d} < \\textrm{maximum flow}$, let us assume that transaction from $s$ to $r$ fails due to non existence of sufficient capacity from sender to receiver. Now we execute the distributed push relabel algorithm which will return the maximum flow in the graph. Let us denote it by $f_m$. But our algorithm was able to find augmenting path for flow till $\\tilde{d} - \\gamma$, where $\\gamma >0$ is an integral value, since our transaction failed. If there exists no more augmenting paths, then $f_m = \\tilde{d} - \\gamma$, which implies $f_m < \\tilde{d}$. This contradicts the fact that $f_m > \\tilde{d}$. Hence our assumption was wrong.     \n\n\\begin{lemma}\n\\label{lm2}\nFor $ v \\in V\\setminus \\{s',t'\\}, e(v)=0$ on termination.\n\\end{lemma}\n\n\n\\textbf{Proof of Lemma \\ref{lm2}}. Assume that there exist one vertex $\\hat{v} \\in V\\setminus \\{s',t'\\} : e(\\hat{v})>0$ after termination. But since termination condition has been reached, it means there vertex $s$ is not reachable from this vertex $\\hat{v}$. Let set of vertices not reachable from $\\hat{v}$ be denoted by $V'$ and those reachable from $\\hat{v}$ be $V-V'$.\n\\begin{equation}\n\\begin{matrix}\n\n\ne(v)=\\Sigma_{k \\in V, (v,k) \\in E} f(k,v) \\\\\n    = \\Sigma_{k \\in V',(v,k) \\in E} f(k,v) + \\Sigma_{k \\in V-V',(v,k) \\in E} f(k,v) \\\\\n    = \\Sigma_{k \\in V',(v,k) \\in E} f(k,v) \\\\ (\\because \\Sigma_{k \\in V-V',(v,k) \\in E} f(k,v)=0 , \\textrm{flow conservation constraint})\n\\end{matrix}\n\\end{equation}   \nBut since $e(v)>0$, then $\\Sigma_{k \\in V',(v,k) \\in E} f(k,v)>0$, which means there still exists some augmenting path from $s$ to $v$ ($s \\in V'$). Hence it contradicts the assumption of termination. \n\n\\begin{lemma}\n\\label{lm3}\nFor all edges $(v,w) \\in E, v,w \\in V$, $f(v,w)\\leq c(v,w)$.\n\\end{lemma}\n\n\\textbf{Proof of Lemma \\ref{lm3}}. In the algorithm \\textit{Push}, the flow value $\\delta$ for a given edge $(v,w) \\in E$, the flow value $f(v,w)$ from vertex $v$ to vertex $w$ is decided by $\\min(e(v),r_f(v,w))$. Since $r_f(v,w)=c(v,w)-f(v,w)$,  $f(v,w)\\leq c(v,w)$ and $e(v) \\leq \\tilde{d}$. Flow value will be bounded by $\\tilde{d}$, if $\\tilde{d} < c(v,w)$ or $c(v,w)$ otherwise.   \n\n\n\n\\subsubsection{\\textbf{Propagating the flow information to source node}}\n\\label{hide}\nEach edge $e \\in E$ involved in transfer of payment from source to sink will generate temporary key $k_e$ for encrypting the flow message to be propagated back to the source node. The sink node, on termination, generates a key $k_{sink}$ as well some random message $rm$, equivalent to size of the packet or its multiple. Each such packet contains flow information which is shared with a predecessor node. It constructs a message $m'$ containing the information of identity of preceding vertex $w$, the non negative flow  $f_{wv}>0$ along with key $k_{wv}$. It encrypts the packet with $k_{sink}$, $E'=Enc_{k_{sink}}(w,f_{wv},k_{wv})$, and concatenates the randomly generated message $rm$ with the encrypted packet to construct message $E'||rm$. It shares this information with $w$. If $w$ is honest, it will construct a similar message $m'$, for its neighbour say $u$ containing the identity of $u$, flow $f_{uw}$, key $k_{uw}$. It is encrypted with $k_{wv}$ to get $E''$. The encrypted message is concatenated with the message received from its successor i.e. $E''||E'||rm$. This continues till all the packets reach the source vertex $s$. The sink vertex shares $k_{sink}$ and set of randomly generated message $rm$ with source vertex $s$ via secure communication channel. $s$ discards $rm$ from the received message and starts decrypting, beginning with the message encrypted by sink. On decryption, it retrieves the flow information, identity of vertex and key with which it will decrypt the next encrypted packet. All duplicate information on flow is discarded and the remaining one is used for reconstructing the flow across the network. This is the routing information denoted by $\\mathcal{P}$.\n\n\n\n \\begin{table*}[!ht]\n\\begin{center}\n\n\n \\caption{SpeedyMurmur vs HushRelay - Performance Analysis on Real Instances }\n   \\label{tab:1}\n  \\scalebox{0.9}[0.9]{\n\n\n  \\begin{tabular}{|p{3cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|} \n  \\hline\n\\multirow{4}{*}{Network\/Algorithm} &\n      \\multicolumn{8}{c|}{SpeedyMurmur} &\n      \\multicolumn{2}{c|}{HushRelay} \\\\\n      \\cline{2-11}\n \n     & \\multicolumn{4}{c|}{Success Ratio} &\\multicolumn{4}{c|}{Time taken}\n     & Success Ratio &Time taken\\\\\n\n &\n      \\multicolumn{4}{c|}{Number of Landmarks} &\n      \\multicolumn{4}{c|}{Number of Landmarks} & &  \\\\\n      \\cline{2-9}\n      &1 &2 &4 &6 &1 &2&4 &6 & &\\\\\n\\hline\nRipple Network  &0.38 &0.7 &0.92 &0.98 &1.66s &2.2s &3.23s &4.74s &1 &2.4s\\\\\n\\hline\nLightning Network  &0.42 &0.65 &0.83 &0.91 &0.61s &0.69s &0.83s &1.94s &0.99 &0.15s\\\\\n\\hline\n \\end{tabular}\n\n}\n\\end{center}\n\\end{table*}\n\n\n\\section{Performance Analysis of \\textit{HushRelay}}\n\\label{6}\n\\subsubsection*{Experimental Setup}\nIn this section, we define the experimental setup. The code for \\textit{HushRelay} is available in \\cite{Code}. System configuration used is : \\texttt{Intel Core i5-8250U CPU, Kabylake GT2 octa core processor, frequency 1.60 GHz}, OS : \\textit{Ubuntu-18.04.1 LTS} (64 bit). The programming language used is C, compiler - gcc version 5.4.0 20160609. The library \\textit{igraph} was used for generating random graphs of size ranging from 50 to 25000, based on Bar\\'{a}basi-Albert model \\cite{albert2002statistical}, \\cite{barabasi2003scale}. Payment Channel Network follows the scale free network where certain nodes function as hub (like central banks), having higher degree compared to other nodes \\cite{javarone2018bitcoin}. For implementing the cryptographic primitives, we use the library \\textit{Libgcrypt}, version-1.8.4 \\cite{libgcrypt}, which is based on code from GnuPG. \n\\begin{figure}[h]\n\\centering\n\\subfloat[Success Ratio vs Number of Nodes]{{\\includegraphics[width=8cm]{success1}}}%\n\\qquad\n\\subfloat[Time To Route vs Number of Nodes]{{\\includegraphics[width=8cm]{time1} }\n\\qquad\n\\caption{Analysis of \\textit{HushRelay} and \\textit{SpeedyMurmur}}%\n\\label{fig:example2}%\n\\end{figure}\n\\subsubsection*{Evaluation}\nFollowing metrics are used to compare the performance of the routing algorithm, \\textit{HushRelay} with \\textit{SpeedyMurmur} \\cite{speedymurmur} \n\\begin{itemize}[leftmargin=*]\n\\item Success Ratio : It is the ratio of number of successful payment to the total number of payment transfer request submitted in an epoch. \n\\item TTR \\textit{(Time Taken to Route)} : Given a payment transfer request, it is the time taken from the start of routing protocol till its completion (returning the set of feasible paths).\n\\end{itemize}\nWe allow just one trial (i.e. $a=1$) of \\textit{SpeedyMurmur} since \\textit{HushRelay} executes just once. The number of landmarks is varied as 1,2,4 and 6.\n\n\\begin{itemize}[leftmargin=*]\n\\item Real Instances - \\emph{HushRelay} and \\emph{SpeedyMurmur} has been executed on real instances - Ripple Network \\cite{malavolta}, Lightning Network \\cite{seres2019topological}. The results are tabulated in Table \\ref{tab:1}.\n\\item Simulated Instances - The capacity of each payment channel is set between 20 to 100 and each transaction value ranges from 10 to 80. For each synthetic graph, we have executed a set of 2000 transactions, with original state of the graph being restored after a transaction gets successfully executed. The source code for SpeedyMurmur is available in \\cite{crysp}.  It is written in Java and makes use of the graph analysis tool GTNA\\footnote{https:\/\/github.com\/BenjaminSchiller\/GTNA}.  From the graphs plotted in Fig. \\ref{fig:example2} a) and b), it is seen that as the number of landmarks increases, SpeedyMurmur gives better success ratio but at the cost of delayed routing. On the other hand, our routing algorithm, which is independent of any landmark, achieves a better success ratio in less time. \n\\end{itemize}\nFrom the results, we can infer that random splitting of capacity without any knowledge of residual graph may lead to failure in spite of presence of routes with the required capacity. \n\n\\section{Conclusion}\n\\label{7}\nIn this paper, we have proposed a novel privacy preserving routing algorithm for payment channel network, \\textit{HushRelay} suitable for simultaneous payment across multiple paths. From the results, it was inferred that our proposed routing algorithm outperforms landmark based routing algorithms in terms of success ratio and the time taken to route. Currently all our implementations assume that the network is static. In future, we would like to extend our work for handling dynamic networks as well. Our algorithms have been defined with respect to a transaction between a single payer and payee but it can extended to handle multiple transaction by enforcing blocking protocol or non blocking protocol to resolve deadlocks in concurrent payments. \\cite{malavolta}.\n\n\n\n\n\\bibliographystyle{IEEEtranS}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecently, Ancona proved the numerical Hodge standard conjecture for abelian fourfolds \\cite{Ancona}. \nIn fact, he proved a general theorem for certain rank 2 pure motives in mixed characteristic \\cite{Ancona}*{8.1}, and showed that this general theorem is applicable to abelian fourfolds over finite fields. In this paper, we point out some other cases where the general theorem can be applied. (See \\cite{Ancona}*{A.9} for another example.) \n\nIn Ancona's work, the main cases are\n\\begin{enumerate}\n    \\item an absolutely simple abelian fourfold, and\n    \\item the product of a simple abelian threefold and an elliptic curve. \n\\end{enumerate}\n(See \\cite{Ancona}*{A.8} for some discussion.)\nIn this paper, we generalize the second case as follows: \n\n\\begin{thm}\\label{main:example}\nLet $A$ be a simple abelian variety over a field $k$. Assume either\n\\begin{itemize}\n\\item $\\dim A$ is \\emph{prime}, or\n\\item some specialization of $A$ to a finite field is absolutely simple and \\emph{almost ordinary}\\footnote{This means its Newton polygon is the same as the one of the product of a supersingular elliptic curve and $\\dim A-1$ ordinary elliptic curves. Such a simple abelian variety exists \\cite{LO}.}. \n\\end{itemize}\nLet $E$ be an elliptic curve. The numerical Hodge standard conjecture holds for $A\\times A$ and $A\\times E$. \n\\end{thm}\n\nAs in \\cite{Ancona}*{1.6}, this combined with \\cite{Clozel} implies\n\n\\begin{cor}\nThe numerical equivalence on $A\\times A, A\\times E$ coincides with the $\\ell$-adic homological equivalence on $A\\times A, A\\times E$ for infinitely many $\\ell$. \n\\end{cor}\n\n\\begin{rem}\nAssume that $k$ is a finite field. \nIn both cases, the Tate conjecture for $A$ is known, and, all algebraic classes come from the intersection of divisors \\cites{Tate, Tankeev:prime, LZ}. Therefore, the numerical Hodge standard conjecture holds for $A$ itself and the numerical equivalence on $A$ coincides with the $\\ell$-adic homological equivalence on $A$ for every $\\ell$ \\cite{Milne:polarization}*{3.7}, \\cite{Ancona}*{Section 5}. \nHowever, if $\\dim A\\geq 3$, $A^2$ and $A \\times E$ may have an exotic Tate class in the middle degree, i.e., a class that cannot be written using Tate classes of degree 2, and the Tate conjecture is not known except the case of the product of a simple threefold and an ordinary elliptic curve \\cite{Milne2022}\\footnote{This relies on the work of Markman \\cite{Markman} on Weil classes.}. \n\\end{rem}\n\nWe prove a slightly more general statement. Let $A$ be an absolutely simple abelian variety of dimension $g$ over a finite field $\\bF_q$. Let $\\alpha_1, \\dots, \\alpha_{2g}$ be the Frobenius eigenvalues of the first cohomology so that $\\overline{\\alpha}_i=\\alpha_{i+g}$. \nSet $\\beta_{i}\\coloneqq q\/ \\alpha_i^2, 1\\leq i\\leq g$. Let $\\Gamma'$ denote the multiplicative group generated by $\\beta_i, 1\\leq i \\leq g$ inside $\\bQ(\\alpha_1, \\dots, \\alpha_{2g})$. The rank of $\\Gamma'$ has been studied, e.g., \\cites{Zarhin:nonsimple, Zarhin:eigenvalues}. Following \\cite{DKZB}, we call it the \\emph{angle rank} of $A$. The angle rank is always less than or equal to $g$. If the angle rank is $g$ or $A$ is a supersingular elliptic curve\\footnote{The angle rank is $0=g-1$ in this case.}, all the Tate classes on $A^n$ for a positive integer $n$ can be written using Tate classes of degree 2, and the Tate conjecture holds for $A^n$.  \n(This is the case for all abelian surfaces and elliptic curves.)\nThe converse is also true. \nRecall that such a Tate class is called \\emph{Lefschetz} and a Tate class is \\emph{exotic} if it is not Lefschetz. \nWe are interested in the easiest case with possible exotic Tate classes:\n\n\\begin{thm}\\label{main:general}\nIf the angle rank of $A$ is $g-1$ or $g$ and $\\dim A >1$ is odd, then the numerical Hodge standard conjecture holds for $A\\times A$ and $A\\times E$, where $E$ is an elliptic curve. \n\\end{thm}\n\n\\begin{rem}\nTankeev \\cite{Tankeev:prime}*{p.332} showed that the angle rank is $g-1$ or $g$ if $g=\\dim A$ is an odd prime\\footnote{Tankeev excludes the case $g=3$, but the same argument actually works.}. \nLenstra and Zarhin \\cite{LZ} showed that that if $A$ is almost ordinary, the angle rank is $g-1$ when $g$ is odd and $g$ when $g$ is even; see \\cite{LZ}*{6.7} (and \\cite{DKZB}*{1.5}) for a slightly more general case. \n\\end{rem}\n\n\\begin{rem}\nIf the angle rank of $A$ is $g-1$ and $E$ is ordinary, then $A\\times E$ has no exotic Tate classes; see Corollary \\ref{ordinary}. The same holds trivially if the angle rank of $A$ is $g$ and $E$ is supersingular. \n\\end{rem}\n\n\\begin{rem}\\label{even}\nIf we assume instead that $g$ is even and the angle rank is $g-1$ (or $g$), then we can show that the numerical Hodge standard conjecture holds for $A$ itself. This partly generalizes the case of absolutely simple abelian fourfolds in \\cite{Ancona} because one can show that the angle rank is $\\geq 3$ for an absolutely simple abelian fourfold if its Frobenius generates a CM field of degree $8$. \n\\end{rem}\n\nNow, Theorem \\ref{main:general} clearly implies Theorem \\ref{main:example}, so we will focus on Theorem \\ref{main:general}.  \nWe shall show that $A\\times A$ may have an exotic Tate class only in the middle degree, and they form a 2-dimensional space so that we can apply \\cite{Ancona}*{8.1}. The case of $A\\times E$ is similar. \n\nFinally, let us mention that we study the Tate conjecture and the Hodge standard conjecture for self-products of K3 surfaces in \\cite{IIK}. \n\n\\section{A lemma on the Hodge standard conjecture}\nLet $A$ be an abelian variety of dimension $g$ over a field with a polarization $L$. \nLet $\\cZ^{n}_{\\num}(A)_{\\bQ}$ denote the space of algebraic cycles of codimension $n$ modulo numerical equivalence.  \nRecall that the Lefschetz standard conjecture holds for $A$ and $L$, and we define the primitive part $\\cZ^{n, \\prim}_{\\num}(A)_{\\bQ}$ of $\\cZ^{n}_{\\num}(A)_{\\bQ}$. \n\n\\begin{conj}[The numerical Hodge standard conjecture]\nFor a nonnegative integer $n\\leq g\/2$, \nThe pairing\n\\[\n\\langle -, -\\rangle_n \\colon \\cZ^{n, \\prim}_{\\num}(A)_{\\bQ} \\times \\cZ^{n, \\prim}_{\\num}(A)_{\\bQ} \\to \\bQ; (\\alpha, \\beta) \\mapsto\n(-1)^{n} \\alpha \\cdot \\beta \\cdot L^{g-2n}\n\\]\nis positive definite. \n\\end{conj}\n\nLet us say that a class in $\\cZ^n_{\\num}(A)_{\\bQ}$ is \\emph{exotic} if it cannot be written as the intersection of divisors. \n\n\\begin{lem}\\label{independence}\nIf $A$ has exotic classes only in the middle degree, then the numerical Hodge standard conjecture is independent of $L$. \n\\end{lem}\n\n\\begin{proof}\nLet $\\cL^{n}_{\\num}(A)_{\\bQ}$ denote the subspace of $\\cZ^n_{\\num}(A)_{\\bQ}$ spanned by the intersections of divisors. \nThe numerical Hodge standard conjecture is known for $\\cL^{n}_{\\num}(A)_{\\bQ}$ by specializing to a finite field \\cite{Milne:polarization}*{3.7}, \\cite{Ancona}*{Section 5}. \nIn particular, only the middle degree is a problem. There is an orthogonal decomposition with respect to $\\langle -,- \\rangle_n$\n\\[\n\\cZ^{g\/2}_{\\num}(A)_{\\bQ}=\\cL^{g\/2}_{\\num}(A)_{\\bQ}\\oplus \\cE^{g\/2}_{\\num}(A)_{\\bQ}, \n\\]\nwhere $\\cE^{g\/2}_{\\num}(A)_{\\bQ}$ is the space of exotic classes. This decomposition is independent of $L$. \nThe numerical Hodge conjecture holds for $A$ and $L$ if and only if $\\langle -, - \\rangle_{g\/2}$ is positive definite on $\\cE^{g\/2}_{\\num}(A)_{\\bQ}$, and the latter statement is independent of $L$.   \n\\end{proof}\n\n\\section{Exotic Tate classes}\nWe assume that $A$ is an absolutely simple abelian variety of dimension $g >2$ defined over a finite field $\\bF_q$ of characteristic $p$. \nWe use the notation $\\alpha_i, \\beta_i$ as in the introduction. \nSuppose first that the angle rank of $A$ is $g-1$. This implies that $\\End (A)\\otimes \\bQ$ is a number filed of degree $2g$ generated by Frobenius. \n\n\\begin{lem}\nAssume that the angle rank is $g-1$. \nAfter replacing $\\alpha_i$ by $\\alpha_{i+g}$ if necessary, the only relation among $\\beta_1, \\dots \\beta_g$ has the form of\n\\[\n(\\beta_1 \\cdots \\beta_g)^N = 1\n\\]\nfor some $N$. \n\\end{lem}\n\n\\begin{proof}\nLet $\\beta_1^{\\bZ} \\cdots \\beta_g^{\\bZ}$ denote the free abelian group of rank $g$ with the basis $\\beta_1, \\dots, \\beta_g$, and let $\\Gamma_1$ be the kernel of the natural map\n\\[\n\\beta_1^{\\bZ} \\cdots \\beta_g^{\\bZ} \\to \\bQ(\\alpha_1, \\dots, \\alpha_{2g})\\setminus\\{0\\}. \n\\]\nBy assumption, $\\Gamma_1$ is a free abelian group of rank 1. \nSo, the Galois group of $\\bQ(\\alpha_1, \\dots, \\alpha_{2g})$ acts naturally on $\\Gamma_1$ via $\\{\\pm 1\\}\\subset \\Aut (\\Gamma_1)$. Note that the Galois group acts on $\\{\\{\\beta_1^{\\pm 1}\\}, \\dots, \\{\\beta_g^{\\pm 1}\\}\\}$ by permutation and the action is transitive, and the Galois group contains the complex conjugation so that $\\overline{\\beta}_i=\\beta_i^{-1}$. \nThis implies that a generator of $\\Gamma_1$ has the form of\n\\[\n\\beta_1^{\\pm N} \\beta_2^{\\pm N} \\cdots \\beta_g^{\\pm N}\n\\]\nfor some $N$. \n\\end{proof}\n\n\\begin{cor}\nLet $\\ell$ be a prime different from $p$. \nIf $g$ is odd (resp. even), any exotic $\\ell$-adic Tate class of $A\\times A$, $A\\times E$ (resp. $A$) is in the middle degree. If an exotic Tate class exists, then the space of exotic Tate classes is two-dimensional for $A\\times A$ (resp. $A$) and four-dimensional for $A\\times E$. \n\\end{cor}\n\n\\begin{cor}\\label{ordinary}\nIf $g$ is odd and $E$ is ordinary, then $A\\times E$ has no exotic Tate classes. \n\\end{cor}\n\nA similar argument shows the following:\n\n\\begin{lem}\nSuppose the angle rank of $A$ equals $g$ and $g$ is odd, then any exotic $\\ell$-adic Tate class of $A\\times E$ is in the middle degree. If an exotic Tate class exists, then $E$ is ordinary and the space of exotic Tate classes is two-dimensional. \n\\end{lem}\n\nNext, we construct a motivic counterpart of possible exotic Tate classes using complex multiplication. \nLet us first recall some facts about the motive of $A$ \\cite{Ancona}*{Section 4, Section 6}. \nSet $B\\coloneqq \\End (A)\\otimes\\bQ$ and write $L\\subset \\overline{\\bQ}$ for the Galois closure of $B$ with $\\Sigma\\coloneqq \\Hom (B, L)$. As in \\cite{Ancona}*{6.6}, there is the following decomposition in the category of Chow motives with coefficients in $L$:\n\\[\n[H^1 (A)]=\\bigoplus_{\\sigma \\in \\Sigma} M_{\\sigma}\n\\]\nthat induces \\cite{Ancona}*{6.7 (1)}\n\\[\n[H^g (A)]=\\bigoplus_{I\\subset \\Sigma, \\# I =g} M_I,   \n\\]\nwhere $M_I=\\otimes_{i \\in I} M_i$.  \nThis further induces the following decomposition \\cite{Ancona}*{6.7 (2)}, in the category of Chow motives with coefficients in $\\bQ$, \n\\[\n[H^g (A)]=\\bigoplus M_{[I]}, \n\\]\nwhere $[I]$ denotes the Galois orbit of $I$ and $M_{[I]}$ is the direct sum of $M_?$ over the Galois orbit.  \nThis decomposition is orthogonal as numerical motives with respect to $\\langle -, -\\rangle^{\\otimes g}_{1, \\textnormal{mot}} $ defined in \\cite{Ancona}*{3.6}. \nSimilarly, $[H^{2g}(A\\times A)]$ has such a decomposition and we have summands like\n\\[\nM_{I^2}\\coloneqq M_I\\otimes M_{I}, \\quad M_{[I^2]}. \n\\]\n\n\\begin{prop}\nAssume that the angle rank is $g-1$. \n\\begin{enumerate}\n    \\item If $g$ is even, there exists at most one $[I]$ such that the $\\ell$-adic realization of $M_{[I]}$ is exotic. The numerical algebraic classes in $M_{[I]}$ is zero or two-dimensional. \n    \\item If $g$ is odd and $E$ is supersingular, there exists at most one $[I]$ such that the $\\ell$-adic realization of $M_{[I]}\\otimes H^1 (E)$ is exotic. The numerical algebraic classes in $M_{[I]}$ is zero or four-dimensional. \n    \\item If $g$ is odd, there exists at most one $[I^2]$ such that the $\\ell$-adic realization of $M_{[I^2]}$ is exotic. The numerical algebraic classes in $M_{[I^2]}$ is zero or two-dimensional.  \n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nThis follows from the description of exotic Tate classes and \\cite{Ancona}*{6.8}. The key claim here is that the relevant Galois orbit only has two elements, and it controls the dimension of numerical algebraic classes.  \n\\end{proof}\n\nWe call $M_{[I]}, M_{[I]}\\otimes H^1(E), M_{[I^2]}$ \\emph{exotic} if it has a nonzero numerical algebraic class. \nIf it is the case, their $\\ell$-adic realizations are the only exotic Tate classes. \nBy \\cite{Ancona}*{5.3} and Lemma \\ref{independence}, the numerical Hodge standard conjecture for $A, A\\times A, A\\times E$ reduces to the corresponding problem on $M_{I}, M_{[I]}\\otimes H^1(E), M_{[I^2]}$ respectively, with respect to $\\langle -, -\\rangle^{\\otimes g}_{1, \\textnormal{mot}}, \\langle -, -\\rangle^{\\otimes g+1}_{1, \\textnormal{mot}}. \\langle -, -\\rangle^{\\otimes 2g}_{1, \\textnormal{mot}}$ for some polarization. \n\nA similar construction makes sense for $A \\times E$ if $g$ is odd, the angle rank is $g$, and $E$ is ordinary.  \n\nFinally, when $g$ is odd and $E$ is supersingular, an exotic $M_{[I]}\\otimes H^1(E)$ has a decomposition into rank 2 motives\n\\[\nM_{[I]}\\otimes H^1(E) =M_1 \\oplus M_2\n\\]\northogonal with respect to $\\langle -, -\\rangle^{\\otimes g+1}_{1, \\textnormal{mot}}$. \nMore precisely, the Galois action on $[I]$ gives rise to an imaginary quadratic field $F$ inside $B$ and there is an embedding $F\\hookrightarrow \\End (E_{\\overline{\\bF}_q})\\otimes \\bQ$ by exactly the same argument as in the proof of \\cite{Ancona}*{7.16}. The actions of $F$ on $M_{[I]}$ and $H^1(E)$ induce the above decomposition. \n\n\\section{Ancona's theorem for rank 2 motives}\nTo conclude the proof of Theorem \\ref{main:general}, we recall Ancona's theorem and then use CM liftings to apply it. \n\nLet $K$ be a $p$-adic field with the ring of integers $O_K$ with residue field $k$. \nFix an embedding $\\sigma \\colon K\\hookrightarrow \\bC$. \nWe shall use the language of relative Chow motives over $O_K$, equipped with base changes to $\\bC$ via $\\sigma$ and to $k$ via the specialization.   \nFor a relative Chow motive $M$ over $O_K$, we write $V_B$ for the Betti realization of $M_{\\bC}$. \nLet $V_Z$ denote the space of numerical algebraic cycles in $M_k$, i.e., homomorphisms from $\\mathbbm{1}$ modulo numerical equivalences. Both $V_B$ and $V_Z$ are $\\bQ$-vector spaces. \nIf $M$ has a quadratic form\n\\[\nq\\colon \\Sym^2 (M) \\to \\mathbbm{1}, \n\\]\nthen it induces ($\\bQ$-valued) quadratic forms $q_B, q_Z$ on $V_B, V_Z$ respectively. \n\n\\begin{thm}[Ancona \\cite{Ancona}*{8.1}]\\label{rank 2}\nLet $M$ be a relative Chow motive over $O_K$ with a quadratic form $q$. \nAssume that\n\\begin{itemize}\n    \\item $\\dim_{\\bQ} V_B=\\dim_{\\bQ} V_Z=2$, and \n    \\item $q_B\\colon V_B\\times V_B \\to \\bQ$ is a polarization of Hodge structures. \n\\end{itemize}\nThen, $q_Z$ is positive definite. \n\\end{thm}\n\n\\begin{proof}[Proof of Theorem \\ref{main:general}]\nFollowing the proof of \\cite{Ancona}*{3.18}, we use CM liftings to prove Theorem \\ref{main:general}. \nLet $A$ be as in Theorem \\ref{main:general}. \nSet $B\\coloneqq \\End (A_{\\bF_q})\\otimes \\bQ$.  \nAfter enlarging $\\bF_q$, we can find a finite extension $O_K$ of $W(\\bF_q)$ and an abelian scheme $\\cA$ over $O_K$ with $B\\to \\End (\\cA)$ such that the reduction $\\cA_{\\bF_q}$ is $B$-isogenous to $A$. \nWe may replace $A$ by $\\cA_{\\bF_q}$ and assume that a polarization on $\\cA_{\\bF_q}$ lifts to a polarization on $\\cA$. \n\nIf $A^2$ has no exotic classes, there is nothing to prove. So, assume some $M_{[I^2]}$ is exotic. \nBy \\cite{Ancona}*{5.3}, it suffices to show the the paring $\\langle -, - \\rangle^{\\otimes 2g}_{1, \\textnormal{mot}}$ is positive definite on the exotic $M_{[I^2]}$. By the construction of $M_{[I^2]}$ and the paring, it lifts to a relative Chow motive with a quadratic form over $O_K$ (cf. \\cite{Ancona}*{4.1, 4.2} and references therein, and the proof of \\cite{Ancona}*{3.18}). \nBy the definition of the exotic $M_{[I^2]}$, this lift satisfies the assumption of Theorem \\ref{rank 2}. \nSo, $\\langle -, - \\rangle^{\\otimes 2g}_{1, \\textnormal{mot}}$ is positive definite on $M_{[I^2]}$. \n\nThe case of $A\\times E$ is similar as in \\cite{Ancona}. Let us consider the case $E$ is supersingular. The decomposition\n\\[\nM_{[I]}\\otimes H^1(E) =M_1 \\oplus M_2\n\\]\nis constructed using the action of the imaginary quadratic field $F\\subset B$ on $E$, and it may also lifts by taking a lift of $E$ with the action of $F$. \n\\end{proof}\n\n\\begin{rem}\nConsider the case $A\\times A$. \nThe Hodge type of the Betti realization of the lifts of the exotic classes have the form of $(2a, 2b), (2b. 2a)$ with $a+b=g$. In particular, it is never $(g,g)$ and any exotic class cannot be lifted to an algebraic class of $\\cA_{\\bC} \\times \\cA_{\\bC}$. Therefore, Ancona's theorem is essential. \n\\end{rem}\n\nRemark \\ref{even} can be proved in the same way. \n\n\\begin{bibdiv}\n\\begin{biblist}\n\\bib{Ancona}{article}{\n   author={Ancona, Giuseppe},\n   title={Standard conjectures for abelian fourfolds},\n   journal={Invent. Math.},\n   volume={223},\n   date={2021},\n   number={1},\n   pages={149--212},\n   issn={0020-9910},\n   review={\\MR{4199442}},\n   doi={10.1007\/s00222-020-00990-7},\n}\n\\bib{Clozel}{article}{\n   author={Clozel, L.},\n   title={Equivalence num\\'{e}rique et \\'{e}quivalence cohomologique pour les\n   vari\\'{e}t\\'{e}s ab\\'{e}liennes sur les corps finis},\n   language={French},\n   journal={Ann. of Math. (2)},\n   volume={150},\n   date={1999},\n   number={1},\n   pages={151--163},\n   issn={0003-486X},\n   review={\\MR{1715322}},\n   doi={10.2307\/121099},\n}\n\\bib{DKZB}{article}{\n   author={Dupuy, Taylor},\n   author={Kedlaya, Kiran S.}, \n   author={Zureick-Brown, David},\n   title={Angle ranks of abelian varieties},\n   eprint={https:\/\/arxiv.org\/abs\/2112.02455}, \n}\n\\bib{IIK}{misc}{\n    author={Ito, Kazuhiro},\n    author={Ito, Tetsushi},\n    author={Koshikawa, Teruhisa},\n    note={in preparation},\n}\n\\bib{LO}{article}{\n   author={Lenstra, Hendrik W., Jr.},\n   author={Oort, Frans},\n   title={Simple abelian varieties having a prescribed formal isogeny type},\n   journal={J. Pure Appl. Algebra},\n   volume={4},\n   date={1974},\n   pages={47--53},\n   issn={0022-4049},\n   review={\\MR{354686}},\n   doi={10.1016\/0022-4049(74)90029-2},\n}\n\\bib{LZ}{article}{\n   author={Lenstra, Hendrik W., Jr.},\n   author={Zarhin, Yuri G.},\n   title={The Tate conjecture for almost ordinary abelian varieties over\n   finite fields},\n   conference={\n      title={Advances in number theory},\n      address={Kingston, ON},\n      date={1991},\n   },\n   book={\n      series={Oxford Sci. Publ.},\n      publisher={Oxford Univ. Press, New York},\n   },\n   date={1993},\n   pages={179--194},\n   review={\\MR{1368419}},\n}\n\\bib{Markman}{article}{\n    author={Markman, Eyal}, \n    title={The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians}, \n    eprint={https:\/\/arxiv.org\/abs\/1805.11574}, \n}\n\\bib{Milne:polarization}{article}{\n   author={Milne, J. S.},\n   title={Polarizations and Grothendieck's standard conjectures},\n   journal={Ann. of Math. (2)},\n   volume={155},\n   date={2002},\n   number={2},\n   pages={599--610},\n   issn={0003-486X},\n   review={\\MR{1906596}},\n   doi={10.2307\/3062126},\n}\n\\bib{Milne2022}{article}{\n    author={Milne, J. S.}, \n    title={The Tate and standard conjectures for certain abelian varieties}, \n    eprint={https:\/\/arxiv.org\/abs\/2112.12815},\n}\n\\bib{Tankeev:prime}{article}{\n   author={Tankeev, S. G.},\n   title={Cycles of abelian varieties of prime dimension over finite and\n   number fields},\n   language={Russian},\n   journal={Izv. Akad. Nauk SSSR Ser. Mat.},\n   volume={47},\n   date={1983},\n   number={2},\n   pages={356--365},\n   issn={0373-2436},\n   review={\\MR{697300}},\n}\n\\bib{Tate}{article}{\n   author={Tate, John},\n   title={Endomorphisms of abelian varieties over finite fields},\n   journal={Invent. Math.},\n   volume={2},\n   date={1966},\n   pages={134--144},\n   issn={0020-9910},\n   review={\\MR{206004}},\n   doi={10.1007\/BF01404549},\n}\n\\bib{Zarhin:nonsimple}{article}{\n   author={Zarhin, Yu. G.},\n   title={The Tate conjecture for nonsimple abelian varieties over finite\n   fields},\n   conference={\n      title={Algebra and number theory},\n      address={Essen},\n      date={1992},\n   },\n   book={\n      publisher={de Gruyter, Berlin},\n   },\n   date={1994},\n   pages={267--296},\n   review={\\MR{1285371}},\n}\n\\bib{Zarhin:eigenvalues}{article}{\n   author={Zarhin, Yuri G.},\n   title={Eigenvalues of Frobenius endomorphisms of abelian varieties of low\n   dimension},\n   journal={J. Pure Appl. Algebra},\n   volume={219},\n   date={2015},\n   number={6},\n   pages={2076--2098},\n   issn={0022-4049},\n   review={\\MR{3299720}},\n   doi={10.1016\/j.jpaa.2014.07.024},\n}\n\\end{biblist}\n\\end{bibdiv}\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:1}\n\nThe substantial homogeneity of Type~Ia supernovae (SNe~Ia), together with the fact\nthat the peak luminosity of individual objects can be observationally determined\nthrough comparisons with well-calibrated samples of nearby SNe~Ia, has propelled \nthem to ``gold standard'' status among extragalactic distance indicators. Precise \ndistance measurements out to $z \\approx 1.7$ have been made \\citep{Riess01,Riess04},\nrevealing the surprising cosmological result that the expansion rate of the\nuniverse is currently accelerating \\citep{Riess98,Perlmutter99}; see\n\\citet{Filippenko04a,Filippenko05} for extensive reviews. Although the\nprogenitor systems have not yet been conclusively identified, the general \nconsensus is that SNe~Ia arise from carbon-oxygen white dwarfs (CO WDs) that \naccrete matter through some mechanism until they achieve a density of $\\sim 3 \n\\times 10^9 {\\rm\\ g\\ cm}^{-3}$ in their centers, leading to a runaway thermonuclear \nreaction that incinerates the star \\citep{Woosley86}.  This occurs, coincidentally, \nwhen a CO WD's mass is nearly the Chandrasekhar limit of $\\sim$1.4~$M_\\odot$.\nSimulations of exploding CO WDs are able to reproduce the main spectral and\nphotometric characteristics of SNe~Ia (e.g., \\citealt{Leibundgut00}, and\nreferences therein), lending theoretical support to this scenario.\n\nHowever, many questions remain concerning the SN~Ia progenitors and \nexplosion mechanism.  What is the nature of the ``donor'' that is\nresponsible for providing the material that the WD accretes?  Is there more\nthan one channel by which the accretion can take place?  Where does the\nthermonuclear runaway begin inside the WD, and how does it propagate throughout\nthe star?  As with so many things, the ``devil is in the details,'' and there\nis a growing sentiment that the answers to these fundamental questions may come\nfrom careful study of the {\\it differences} seen among SNe~Ia, rather than from\nthe similarities alone.\n\nSpectropolarimetry offers the only direct probe of early-time SN geometry, and\nthus is an important diagnostic tool for discriminating among SN~Ia progenitor\nsystems and theories of the explosion physics.  The essential idea is this: A\nhot, young SN atmosphere is dominated by electron scattering, which by its\nnature is highly polarizing.  For an unresolved source that has a spherical\ndistribution of scattering electrons, the directional components of the\nelectric vectors of the scattered photons cancel exactly, yielding zero net\nlinear polarization.  Any asymmetry in the distribution of the scattering\nelectrons, or of absorbing material overlying the electron-scattering\natmosphere, results in incomplete cancellation, and produces a net polarization\n\\citep[see, e.g.,][]{Leonard11}.\n\nInitial broad-band polarimetry studies found SNe~Ia to possess zero or, at\nmost, very weak ($< 0.2\\%$) intrinsic polarization \\citep{Wang96}, suggesting a\nhigh degree of symmetry for their scattering atmospheres.  More recent\nspectropolarimetric studies capable of resolving individual line features,\nhowever, are revealing a complex picture, with both continuum and line\npolarization now convincingly established for at least a subset of the SN~Ia\npopulation.\n\nTo date, three SNe~Ia have been examined in detail with spectropolarimetry at\nearly times: SN~1999by \\citep{Howell01}, SN~2001el \\citep{Kasen03,Wang03}, and,\nmost recently, SN~2004dt \\citep{Wang05}.  In this paper we present single-epoch\nspectropolarimetry of four SNe~Ia: SN~1997dt, SN~2002bf, SN~2003du, and\nSN~2004dt, obtained about $21$, 3, 18, and 4 days (respectively) after maximum\nlight. \n\nA particular motivation for this multi-object investigation is to attempt to\nlink spectral and photometric peculiarities of individual SNe~Ia with their\nspectropolarimetric characteristics.  Accordingly, the four events span a range\nof properties: SN~1997dt is likely somewhat subluminous, SN~2002bf and\nSN~2004dt exhibit unusually high-velocity absorption lines (in the case of\nSN~2002bf, the highest ever seen in an SN~Ia for the epochs considered), and\nSN~2003du is slightly overluminous.\n\nThis paper is organized as follows.  We briefly review the present photometric,\nspectroscopic, and spectropolarimetric state of knowledge of SNe~Ia in\n\\S~\\ref{sec:2}, focusing particular attention on objects sharing\ncharacteristics with the SNe~Ia included in our spectropolarimetric survey.  We\ndescribe and present the spectropolarimetry in \\S~\\ref{sec:3}; to assist in the\nclassification, we also include optical photometry and additional spectroscopy\nfor two of the events, SN~2002bf and SN~2003du.  We analyze the data in\n\\S~\\ref{sec:4}, and present our conclusions in \\S~\\ref{sec:5}.  Note that\npreliminary discussions of the spectropolarimetry of SN~1997dt and SN~2002bf\nhave been given by \\citet{Leonard1} and \\citet{Filippenko04}, respectively.\n\n\\section{Background}\n\\label{sec:2}\n\\subsection{Photometric Properties of SNe~Ia}\n\\label{sec:2.1}\n\nType Ia SNe typically rise to peak $B$-band brightness in about 20 days,\ndecline by about $3$ mag over the next $35$ days, and then settle into a nearly\nconstant descent of $\\sim$1.55 mag (100\\ day)${^{-1}}$ for the next year.\nHowever, it is now confirmed beyond doubt that deviations exist from the\ncentral trend; see, for example, \\citet{Phillips99}.  Some events rise and fall\nmore slowly, producing ``broader'' light curves (e.g., SN~1991T), whereas\nothers rise and fall more quickly, yielding ``narrower'' light curves (e.g.,\nSN~1991bg).  The width of the light curves near peak correlates strongly with\nluminosity, in the sense that broader light curves generally indicate\nintrinsically brighter objects (but see \\citealt*{Jha05}, for exceptions), with\na total spread of about a factor of sixteen in absolute peak $B$-band\nbrightness among the population \\citep{Altavilla04}.\n\nThe ability to individually determine the luminosity of SNe~Ia through\nexamination of their light-curve shapes has been the primary driver for\ncosmological applications.  There are several photometric calibration\ntechniques in common use that correlate the observed light-curve shape, or\nsupernova color, with luminosity (see \\citealt{Wang05a}, and references\ntherein). The simplest calibration technique involves measuring the decline in\n$B$ during the first 15 days after maximum $B$-band brightness, $\\Delta\nm_{15}(B)$. A typical value is $\\Delta m_{15}(B) \\approx 1.1$ mag, which\ncorresponds to $M_B = -19.26 \\pm 0.05$ mag according to the calibrations of\n\\citet{Hamuy96}.  Underluminous, SN 1991bg-like objects yield values up to\n$\\Delta m_{15}(B) = 1.94$ mag, while overluminous, SN 1991T-like objects \ngo as low as $\\Delta m_{15}(B) = 0.81$ mag \\citep{Altavilla04}.  There is a\ncontinuum of values in between the two extremes.\n\nFor the two SNe with optical photometry presented in this paper, we shall use\nthe ``multicolor light-curve shape'' method \\citep[MLCS; e.g.,][]{Riess96},\nwhich has been revised (and hereafter referred to as MLCS2k2) by \\citet{Jha02}\nand S. Jha et al. (in preparation) to include $U$-band light curves from\n\\citet{Jha05}, a more self-consistent treatment of extinction, and an improved\ndetermination of the unreddened SN~Ia color.  This method is based on\ndetermining $\\Delta$ (a dimensionless number related to magnitude that \nparameterizes the light-curve shape), $t_0$ (the time of $B$-band maximum light), \n$\\mu_0$ (the distance modulus for an assumed value of $H_0$, here taken to be \n$65\\ {\\rm km\\ s\\ }^{-1}{\\rm\\ Mpc}^{-1}$), and $A_V^0$ (the visual extinction at $t_0$).\n\nThe mechanism that produces the dispersion in SN~Ia luminosity is not known;\npresent speculations range from different progenitor systems to differences in\nthe explosion mechanism and flame propagation (e.g., deflagration, detonation,\ndelayed-detonation, or, most recently, gravitationally confined detonation; see,\nrespectively, \\citealt*{Nomoto84,Arnett69,Khokhlov91,Plewa04}).\nSome have further speculated that global asymmetries in the expanding ejecta\nmay result in viewing-angle dependent luminosity \\citep{Wang03}.  One\nobservational fact that all theories must confront is that the most luminous\nSNe~Ia have thus far only been seen in late-type galaxies; the faintest\nobjects tend to prefer elliptical galaxies, but have also been found in spirals\n\\citep[e.g., ][and references therein]{Howell01a,Benetti05}.\n\n\\subsection{Spectroscopic Properties of SNe~Ia}\n\\label{sec:2.2}\n\nEarly-time spectra of SNe~Ia (see \\citealt{Filippenko97} for a review)\ntypically exhibit lines of intermediate-mass elements (IMEs), such as\nmagnesium, silicon, sulfur, and calcium, with some contribution from iron-peak\nelements, especially at near-ultraviolet wavelengths.  As time progresses, and\nthe photosphere recedes deeper into the ejecta, lines of Fe come to dominate\nthe spectrum.  This spectral evolution suggests a burning front that\nincinerates some of the progenitor's carbon and oxygen all the way to the\niron peak deep inside the ejecta, but then leaves the outer layers only\npartially burned.  It is interesting to note that overluminous SNe~Ia show\nenhanced Fe features at early times, whereas subluminous ones show weak\nearly-time Fe features.\n\nAt early times (e.g., $-20 {\\rm\\ d} < t < 20 {\\rm\\ d}$ from the date of maximum\n$B$ brightness, $B_{\\rm max}$), the spectra of typical SNe~Ia evolve rapidly\nand with such uniformity that it is possible to determine the age of an event\nrelative to the date of $B_{\\rm max}$ to within $\\sim$2 days from a single\nspectrum alone \\citep{Riess97,Foley05}. The spectroscopic peculiarities of subluminous\nand overluminous events currently preclude their ``spectral feature ages'' from\nbeing derived accurately through comparison with average SN~Ia spectra.  For\ninstance, pre-maximum spectra of overluminous events lack the strong\n\\ion{Si}{2} $\\lambda 6355$ absorption that is so prominent in normal and\nunderluminous SN~Ia spectra \\citep{Filippenko92a}.\n\nAnother point of distinction among SN~Ia spectra comes from the blueshifts of\nthe spectral lines.  Increasing attention is being paid to a small but growing\ngroup of ``high-velocity'' (HV) SNe~Ia, whose spectra around maximum light are\ncharacterized by unusually broad and highly blueshifted absorption troughs in\nmany line features, indicating optically thick ejecta moving about\n4000--5000~km s$^{-1}$\\ faster than is typically seen for SNe~Ia\n\\citep{Branch87,Benetti05}. Well-studied examples of HV~SNe~Ia include\nSN~1983G \\citep[][and references therein]{Branch93}, SN~1984A\n\\citep{Branch87,Barbon89}, SN~1997bp \\citep{Anupama97}, SN~1997bq \\citep[][and\nreferences therein]{Lentz01}, SN~2002bo \\citep{Benetti04}, SN~2002dj\n\\citep{Benetti04}, and SN~2004dt \\citep{Wang05}.  \\citet{Wang05} note that in a\nspectrum of one HV~SN~Ia, SN~2004dt, some line features do not possess\nabnormally high velocity, such as those identified with \\ion{S}{2}.  Since two\nof our objects, SN~2002bf and SN~2004dt, are HV~SNe~Ia, we briefly review the\nsalient features that are known about this class of objects.\n\nThe most thorough and recent study of the diversity of SN~Ia expansion\nvelocities is that of \\citet{Benetti05}, who apply a statistical treatment to\ndata from 26 SNe~Ia and find that HV SNe~Ia indeed make up a kinematically\ndistinct group.  Its members have normal peak luminosity (based on their\n$\\Delta m_{15}(B)$ values: $1.09 \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} \\Delta m_{15}(B) \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 1.37$ mag),\nreside in all types of host galaxies (e.g., both ellipticals and spirals, but\nwith a preference for later types), and have very strong \\ion{Si}{2} features.\nThey are further distinguished by a large temporal velocity gradient,\n$\\dot{v}$\\ $ > 70\\ {\\rm km\\ s}^{-1} {\\rm\\ day}^{-1}$, where $\\dot{v}$ is defined to be the\naverage daily rate of decrease of the expansion velocity between maximum light\nand the time the \\ion{Si}{2} $\\lambda 6355$ feature disappears.  HV~SNe~Ia\nappear to have similar photometric characteristics to ordinary Type~Ia events,\nalthough there is some indication of subtle differences in their color\nevolution \\citep{Benetti04}.\n\nWhether HV SNe~Ia represent the extreme end of a continuum of more typical\nSNe~Ia, or require a different explosion mechanism, progenitor system, or\nexplosion physics, remains controversial.  What is certain is that any model\nmust have significant optical depth in the IMEs at high velocities at early\ntimes.  \\citet{Branch93} originally proposed that HV~SNe~Ia may simply result\nfrom more energetic explosions.  However, \\citet{Benetti04} show that the Fe\nnebular lines have velocities comparable to those of normal SNe~Ia, which tends\nto weaken the argument for higher overall kinetic energy.  There is also\nevidence from early-time spectra of SN~2002bo that HV~SNe~Ia possess very\nlittle unburned carbon in their outer layers, in contrast with more typical\nSNe~Ia like SN~1994D \\citep{Benetti04}.  The high velocities of the IMEs\ncoupled with the lack of primordial carbon may indicate that burning to Si\npenetrates to much higher layers in HV~SNe~Ia than it does in more normal\nevents.  This is a feature of some delayed-detonation models, such as those\nstudied by \\citet{Lentz01}.  However, \\citet{Wang05} argue that a strong\ndetonation wave is unlikely to generate the clumpy and asymmetrically\ndistributed silicon layer that is inferred from pre-maximum spectropolarimetric\nobservations of SN~2004dt.  As another alternative, \\citet{Benetti04} propose\nthat IMEs produced at deeper layers may simply be more efficiently mixed\noutward in HV~SNe~Ia than is typical.  A prediction of this mechanism is that\nwhile the IME products of explosive carbon burning will be mixed in with the\nprimordial C and O in the outer layers, the C and O should also be mixed inward\nwithin these events, and exist at lower velocities than are normally seen.\n\nIt is appropriate at this point to mention the recently proposed\ngravitationally confined detonation (GCD) model of \\citet{Plewa04}, which\nstands as an intriguing alternative to the standard deflagration or\ndelayed-detonation scenarios.  The mechanism involves the slightly off-center\nignition of a deflagration that produces a buoyancy-driven bubble of material\nthat reaches the stellar surface at supersonic speeds, where it laterally\naccelerates the outer stellar layers.  This material, gravitationally confined\nto the white dwarf, races around the star and, in $\\sim$2~s, converges at a\npoint opposite to the location of the bubble's breakout, creating conditions\ncapable of igniting the nuclear fuel and triggering a detonation that can\nincinerate the white dwarf and result in an energetic explosion.  \n\nOf interest to the present study is the recent work of \\citet{Kasen05}, who\ninvestigate the spectral, and spectropolarimetric, consequences of the GCD\nmodel.  They focus their investigation on the interaction of the expanding\nejecta with an ellipsoidal, metal-rich extended atmosphere formed from the\nbubble of deflagration products (taken to be 57\\% Si, 27\\% S, 7.1\\% Fe, and\n2.7\\% Ca, plus smaller amounts of other metals; see \\citealt{Khokhlov93}), and\nfind that a dense, optically thick pancake of metal-rich material is formed at\npotentially large velocity on the side of the ejecta where the bubble emerged.\nFor low atmosphere masses (e.g., resulting from a bubble of mass $0.008\nM_\\odot$), the pancake of material spans the velocity range 17,000--28,000\nkm s$^{-1}$\\ and is geometrically detached from the bulk of the SN ejecta.  This might\nexplain the detached, high-velocity \\ion{Ca}{2} near-infrared (IR) triplet\nabsorption seen in pre-maximum spectra of some SNe~Ia (e.g., SN~2001el, see\n\\S~\\ref{sec:2.3.3}).  For larger bubble and, hence, atmosphere masses ($m_{\\rm\natm} \\gtrsim 0.016 M_\\odot$), the absorbing pancake moves at lower velocities\n(e.g., 10,000--21,000 km s$^{-1}$\\ for $m_{\\rm atm} = 0.08 M_\\odot$) and could blend\nwith the region of IMEs in the SN ejecta and potentially increase the blueshift\nof several of the spectral features.  This could provide an\norientation-dependent explanation for the origin of HV~SNe~Ia, whereby sight\nlines in which the pancake more completely blocks the photosphere produce the\nanomalously large velocities.  The spectropolarimetric consequences of this\nmodel are discussed in \\S~\\ref{sec:2.3.1}.\n\n\\subsection{Spectropolarimetric Properties of SN~Ia}\n\\label{sec:2.3}\n\\subsubsection{Supernova Polarization Mechanisms}\n\\label{sec:2.3.1}\n\nThe first definitive proof that some SNe are polarized came from\nobservations of the Type II SN~1987A \\citep{Cropper88}, which exhibited a\nmodest temporal increase in continuum polarization during the photospheric\nphase, as well as sharp polarization modulations across strong P-Cygni flux\nlines \\citep{Jeffery91a}.  While intrinsic polarization has now been\nestablished in over a dozen SNe (for recent reviews, see \\citealt{Wheeler00};\n\\citealt{Filippenko04}; \\citealt{Leonard11}), including at least three SNe~Ia, \nthe exact origin of both continuum and line polarization remains controversial.\n\nThere are essentially two mechanisms by which supernova continuum polarization\nis thought to be produced: (1) a globally aspherical photosphere and\nelectron-scattering atmosphere\n\\citep[e.g.,][]{Shapiro82,Hoflich91,Jeffery91,Leonard2,Wang01}, and (2)\nionization asymmetry produced by the decay of asymmetrically distributed\nradioactive $^{56}{\\rm Ni}$, perhaps flung out into \\citep{Chugai92,Hoflich01}\nor beyond \\citep{Kawabata02,Leonard8} the expanding ejecta in clumps.  The\nsimplest, and most well-studied, globally aspherical geometry is that of an\nellipsoid, and we shall make frequent reference to the ``ellipsoidal model'' in\nthe following discussion.  In the second model, in which an ionization asymmetry\nis present, continuum polarization is generated by light from the (either\nspherical or aspherical) photosphere scattering off of asymmetrically\ndistributed free electrons that exist in clouds surrounding clumps of\nradioactive $^{56}{\\rm Ni}$.  Variations on both of these polarization\nmechanisms are also possible.  For instance, an aspherical distribution of\n$^{56}{\\rm Ni}$ could also result in SN polarization by providing an asymmetry\nin a source of optical photons (produced by the thermalization of\n$\\gamma$-rays) relative to the scattering medium.\n\nTo explain polarization modulations seen across spectral lines, it is important\nto differentiate between the emission peaks and blueshifted absorption troughs\nthat are characteristic of P-Cygni profiles in total-flux spectra of SNe.  Line\npeaks have usually been assumed to consist of intrinsically unpolarized\nphotons.  This is because although resonance scattering by a line is an\ninherently polarizing process \\citep{Jeffery91}, directional information for\nscattered photons is lost in an SN atmosphere since the timescale for\nrandomizing collisional redistribution of the relative level populations within\nthe fine structure of the atomic levels of a line transition is much shorter\nthan the characteristic timescale for absorption and reemission in a line\n\\citep{Hoflich96}.  The assumption of intrinsically unpolarized emission lines\nis also commonly used to derive the interstellar polarization (ISP;\n\\S~\\ref{sec:2.3.2}).  Note, though, that if ionization asymmetry exists above\nthe photosphere (from, e.g., clumps of radioactive $^{56}{\\rm Ni}$), then even\nintrinsically unpolarized emission-line photons may become polarized by\nelectron scattering within the SN atmosphere.\n\nFor SNe~Ia, line-blanketing, due largely to Fe, is particularly severe at\nwavelengths below $\\sim$5000~\\AA, where theoretical models \\citep{Howell01}\nsuggest that nearly complete depolarization of any ``continuum'' light may be\nassumed.  Conversely, the broad spectral region $6800 \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} \\lambda\n\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 7800$~\\AA\\ is largely devoid of line opacity \\citep{Kasen04,Howell01},\nat least near maximum light, and thus may give a good indication of true\ncontinuum polarization level.\n\nThe explanation of polarization changes in absorption troughs is controversial.\nOne well-studied hypothesis, which follows logically from the ellipsoidal\nmodel, is that selective blocking of more forward-scattered and, hence, less\npolarized, light in P-Cygni absorption troughs results in trough polarization\nincreases \\citep{Jeffery91,Leonard1,Leonard3}, potentially producing\n``inverted'' P-Cygni polarization profiles when coupled with emission peak\ndepolarization.  Some SN modelers have claimed that this ``geometrical\ndilution'' mechanism, however, is rather poor at polarizing SN~Ia light, and\nthat polarization {\\it decreases} should actually be seen in absorption\ntroughs, since much of the light reaching the observer in those spectral\nregions has been absorbed and reemitted by the line \\citep{Howell01}.  \n\nA key point is that, in either case, to produce line trough polarization changes\nunder the ellipsoidal model, some continuum polarization must exist.  In fact,\n\\citet{Leonard4} show that the strength of a line trough polarization feature\ncan be used to place a lower bound on the true intrinsic continuum polarization\nlevel under the ellipsoidal model according to\n\n\\begin{equation}\np_{cont} \\geq \\frac{\\Delta p_{tot}} {(I_{cont}\/I_{trough}) - 1},\n\\label{eqn:1}\n\\end{equation}\n\n\\noindent \nwhere $I_{cont}$ is the interpolated value of the continuum flux\nat the location of the line trough, $I_{trough}$ is the total flux at the\nline's flux minimum, and $\\Delta p_{tot}$ is the total polarization change\nobserved in the line trough.  Thus, a critical test of the ellipsoidal model is\nto see whether the continuum polarization observed in the spectral region\n6800--7800~\\AA\\ is sufficient to explain observed line trough polarization\nchanges.  Lack of significant continuum polarization in an object with strong\nline features argues against the ellipsoidal model.  An additional prediction\nis that {\\it no} rotation of the polarization angle (PA) should be seen across\nthe line, since the continuum and line-forming regions share the same geometry.\n\nA final way to produce a polarization change in a line trough is through\nasymmetry in the distribution of elements in the ejecta material located above\nthe photosphere along the line-of-sight (l-o-s), hereafter referred to as the\n``clumpy ejecta'' model.  Asymmetry in the distribution of material with\nsignificant optical depth may unevenly block the underlying photospheric light,\nthereby producing a net polarization change and\/or PA rotation in a line\ntrough, even when the photosphere is spherical. The GCD model discussed in\n\\S~\\ref{sec:2.2} provides a natural mechanism by which to generate line-trough\npolarization through this model, since it predicts an optically thick pancake\nof high-velocity, metal-rich material overlying the photosphere on the side in\nwhich the bubble emerged.  In particular, if the GCD model provides the correct\nmechanism by which to produce HV~SNe~Ia, then polarization changes in the\nabsorption troughs of the strong, HV metal lines in these objects should be\nparticularly pronounced since the l-o-s necessarily intercepts a substantial\nfraction of the pancake.\n\nFor SNe~Ia, an interesting alternative to the simple ellipsoidal model for\nproducing {\\it both} line and continuum polarization is presented by\n\\citet{Kasen04}, who explore the polarization consequences of a conical hole in\nthe ejecta due to the interaction with a companion star, which we shall refer\nto as the ``ejecta-hole'' model.  By considering various viewing angles and\nhole sizes, \\citet{Kasen04} demonstrate that both continuum and line\npolarization and PA changes can be generated.  In general, viewing angles\nalmost directly down the hole yield low continuum polarization with\npolarization increases in strong line troughs, whereas sight lines more nearly\nperpendicular to the hole result in larger continuum polarization (prominently\nseen in the spectral region 6800--7800~\\AA) and ``inverted P-Cygni'' line\npolarization profiles; this latter case is nearly indistinguishable from the\npredictions of the simple ellipsoidal model.\n\nIonization asymmetry is also capable of generating both continuum polarization\nas well as PA and polarization level changes through line features, since both\ncontinuum and line photons will scatter off of concentrations of free electrons\nin (or beyond) the ejecta.  An impressive example of this mechanism potentially\nbeing at work is given by \\citet{Chugai92} for the case of SN~1987A, in which\nthe spectropolarimetry data, as well as asymmetries in the flux line profiles,\nare convincingly reproduced by the effects of two clumps of $^{56}{\\rm Ni}$ in\nthe far (receding) hemisphere of the ejecta.\n\n\\subsubsection{Removing Interstellar Polarization}\n\\label{sec:2.3.2}\n\nA problem that plagues interpretation of all SN polarization measurements is\nproper removal of the ISP.  Since directional extinction resulting from\naspherical interstellar dust grains aligned by some mechanism along the l-o-s\nto an SN can contribute a large polarization to the observed signal, an attempt\nmust be made to remove it prior to analyzing SN spectropolarimetry data. This\nis notoriously difficult, although a number of different techniques have been\nadvanced over the years, which we here summarize.\n\nAn excellent way to derive Galactic ISP is through observations of distant,\nintrinsically unpolarized, ``probe stars'' close to (within $\\sim$0.5$\\ensuremath{^{\\circ}}$)\nthe l-o-s to the SN \\citep[e.g.,][]{Leonard8,Leonard7}.  Deriving the total\nISP, which includes the contribution from dust in the host galaxy, however, is\nmore difficult.\n\nThe most basic technique is to {\\it place limits on the ISP from reddening\nconsiderations.}  Since the same dust that polarizes starlight should redden it\nas well, it seems logical to expect a correlation between reddening and ISP.\nBecause the alignment of dust grains is not total (or has multiple preferred\norientations due to non-uniformity of the magnetic field along the l-o-s), and\ngrains are probably only moderately elongated particles, it is not surprising\nthat, through the analysis of thousands of reddened Galactic stars, only an\nupper bound on the polarization efficiency of Galactic dust can be derived\n\\citep{Serkowski75}: ${\\rm ISP} \/ E_{B-V} < 9.0\\%\\ {\\rm mag}^{-1}$.  However,\nthe polarization efficiency of the dust in external galaxies is not well\nstudied, and in one of the few investigations carried out to date,\n\\citet{Leonard7} find compelling evidence for polarization efficiency well in\nexcess of the empirical Galactic limit for dust in NGC~3184 along the l-o-s to\nthe Type II-P SN~1999gi.  It is not clear at this point whether meaningful\nconstraints can thus be placed on the ISP of extragalactic SNe that are\nsignificantly reddened by host-galaxy dust.  Nonetheless, total reddening\narguments are still often used to set ``reasonable'' limits on the ISP.\n\nAnother method to get a handle on the ISP is to assume axisymmetry for any SN\nasphericity, a situation that reveals itself through a straight-line\ndistribution of points when the spectropolarimetry is plotted in the $q$--$u$\nplane.  If axisymmetry exists, the ISP is constrained to lie along the axis\ndefined by the line \\citep{Howell01}.  Its absolute value, however, is\nuncertain without additional input.\n\nA more precise technique relies on the theoretical expectation that {\\it\nunblended emission lines consist of unpolarized light}, and that any\npolarization observed in emission-line photons comes from the ISP\n\\citep{Jeffery91a,Tran97,Wang96,Leonard3}.  This assumption is thought to be\nmost valid at early times, when any $^{56}{\\rm Ni}$ concentrations are likely\nto be below the photosphere.  Note, though, that to use this method, care must\nbe taken to isolate the emission-line photons from the underlying continuum\nlight \\citep[e.g.,][]{Tran97}, which may be intrinsically polarized.\n\nA related method is to assume that a particular spectral region is\nintrinsically {\\it completely} unpolarized, with all of the observed\npolarization therefore coming from ISP.  Some empirical support exists that, in\nsome objects, this may be the case for very strong emission lines\n\\citep{Kawabata02,Leonard8,Wang04a}.  For SNe~Ia in particular, it has\nsometimes been assumed that the far blue spectral region (e.g., typically $<\n5000$~\\AA) satisfies this criterion due to the effect of the heavy\nline-blanketing and multiple, depolarizing, line scatters, largely due to\niron-group elements.  This qualitative expectation is demonstrated\nquantitatively by \\citet{Howell01}, who present model polarization spectra\nresulting from delayed-detonation models in realistic SN Ia atmospheres.  More\nrecently, \\citet{Wang05} update this approach by choosing only specific\nspectral regions at blue wavelengths ($\\lambda < 5000$~\\AA) that are not\ndominated by obvious individual line features in either flux or polarization\nfor the ISP determination, rather than the arbitrary blue edge of the spectrum, as \nadopted by \\citet{Howell01}. The central idea here is that while strong individual line \nfeatures might impart their own polarization signature (e.g., through geometrical \ndilution in ellipsoidal models, or through blocking of the parts of the photosphere \nin clumpy-ejecta models), the spectral regions in between specific features\nprobably decrease any effective continuum polarization by the numerous\noverlapping spectral lines at blue wavelengths.\n\nFinally, if polarimetry is obtained after the electron-scattering\noptical depth of the atmosphere has dropped well below unity, then it\nmay be adequate to assume the observed polarization to be due entirely\nto ISP across the whole spectrum \\citep[e.g.,][]{Wang03}.\n\nImproperly removed, ISP can increase or decrease the derived intrinsic\npolarization, and it can change ``valleys'' into ``peaks'' (or vice\nversa) in the polarization spectrum. Since it is so difficult to be\ncertain of accurate removal of ISP, it is generally safest to focus on\n(a) temporal changes in the polarization with multiple-epoch data, (b)\ndistinct line features in spectropolarimetry having high\nsignal-to-noise ratio (S\/N), and (c) continuum polarization unlike the\ncharacteristic ``Serkowski-law'' wavelength dependence imparted by\ndust \\citep[e.g.,][]{Whittet92}.\n\n\\subsubsection{Previous SN~Ia Polarimetry Studies}\n\\label{sec:2.3.3}\n\nEvidence for intrinsic polarization in SNe~Ia was initially difficult to find,\nas the first investigations detected only marginally significant polarization\namong normal-brightness events observed near maximum, $p < 0.2$\\%\n\\citep[e.g.,][]{Wang96,Wang97}.  Broad-band polarimetry of one overluminous\nevent observed over a month after maximum also found no intrinsic polarization\ndown to a level of $p \\approx 0.3\\%$ \\citep{Wang96}.  Significant advances have\nbeen made in the last few years. \\citet{Leonard1} reported the first\nconvincing, albeit weak, features in the polarization of an SN~Ia, SN~1997dt,\nwhich was likely a subluminous event; these data are presented and analyzed in\nmore detail in the present study.\n\nThe first thorough spectropolarimetric study of an SN~Ia is that by\n\\citet{Howell01}.  In spectropolarimetry obtained at maximum light, the\nsubluminous SN~1999by exhibits a polarization change of $\\sim$0.8\\% from\n4800~\\AA\\ to 7100~\\AA, and a sharp polarization modulation of $\\sim$0.4\\%\nacross the strong \\ion{Si}{2} $\\lambda6355$ absorption.  These features are\nexplained within the context of an ellipsoidal model with a global asphericity\nof $\\sim$20\\%, observed equator-on.  This physical picture was achieved by\nassuming the ISP to be the observed polarization at the blue edge of the\nspectrum, which was about $0.2\\% {\\rm\\ at\\ } 4800$~\\AA.  With this choice of\nISP, the inferred intrinsic polarization rises from $0\\%$ to about $0.8\\%$ from\nblue to red, with a sharp depolarization from $0.4\\%$ to near $0\\%$ across the\n\\ion{Si}{2} $\\lambda 6355$ feature.  An argument supporting both the\nellipsoidal model as well as the ISP choice is that after ISP removal the PA\nbecomes nearly independent of wavelength.  \\citet{Howell01} note, on the other hand, \nthat a wavelength-independent PA also results if one assumes the ISP to be the\nobserved polarization of the far {\\it red} edge of the spectrum; in this case,\nthe polarization modulation across the \\ion{Si}{2} $\\lambda 6355$ becomes a\npolarization increase.  Such a scenario could result from selective blocking of\nforward-scattered light, as described in \\S~\\ref{sec:2.3.1}.  However,\n\\citet{Howell01} find the theoretical arguments supporting the former ISP to be\nmore compelling.  With this ISP, the redward rise in intrinsic polarization is\nattributed to the decreasing importance of line opacities, and the increased\ninfluence of continuum electron scattering at longer wavelengths.  The\ndepolarization across the \\ion{Si}{2} $\\lambda6355$ absorption is also\nattributed to the depolarizing effect of line scattering.  Despite the\ndetection of intrinsic polarization for SN~1997dt and SN~1999by, their unusual\n(subluminous) nature nevertheless left some doubt about intrinsic polarization in {\\it\nnormal} SNe~Ia.\n\n   That doubt has recently been put to rest with the work of \\citet{Kasen03} on\nSN~2001el.  For this normal-luminosity event, the percent polarization changed\nfrom blue to red by $\\sim$0.4\\% in spectropolarimetry obtained 1 week before\nmaximum brightness.  However, the extraordinary feature here is the existence\nof distinct high-velocity \\ion{Ca}{2} near-infrared (IR) triplet absorption\n($v$ = 18,000--25,000 km s$^{-1}$) in addition to the usual, lower-velocity\n\\ion{Ca}{2} feature.  A similar, but much weaker, high-velocity feature had\nbeen previously observed in SN~1994D, and perhaps in other SNe~Ia as well; the\nnumber of pre-maximum spectra covering the near-IR spectral range is small.\nThe polarization is seen to change dramatically in this feature, by $\\sim$0.4\\%.  \nThe \\ion{Ca}{2} near-IR feature is examined by \\citet{Kasen03} in an\nelegant study, which concludes that it is likely due to photospheric\nobscuration by a clumped shell of high-velocity material.  Using multi-epoch\ndata, \\citet{Wang03} demonstrate that the nature of the polarization changes\nover the course of two weeks following this early epoch, becoming nearly\nundetectable a week after maximum brightness, further solidifying the case for\nintrinsic polarization at early times.  Although not commented on by either\nstudy, it appears that the \\ion{Si}{2} $\\lambda 6355$ line also shows a\npolarization modulation in the earliest epoch, and with a different PA from the\n\\ion{Ca}{2} near-IR feature.  Although it does not affect the main results, a\ncautionary note on the difficulty of ISP determination is set by the fact that\nthe two studies arrive at quite different values: \\citet{Kasen03} derive an ISP\nof very nearly $0\\%$ by assuming the blue edge of the spectrum to be\nunpolarized, whereas \\citet{Wang03} obtain an ISP of $\\sim$0.6\\% by\nattributing the observed polarization $38$ days after maximum light, when\nSN~2001el is argued to be in the nebular phase, entirely to ISP.\n\nMost recently, \\citet{Wang05} analyze a single epoch of pre-maximum\nspectropolarimetry of SN~2004dt, an HV~SN~Ia for which an ISP of $q_{\\rm ISP} =\n0.2 \\pm 0.1\\%$ and $u_{\\rm ISP} = -0.2 \\pm 0.1\\%$ is derived from the observed\npolarization of a handful of narrow, blue spectral regions in which no single\nspectral feature dominates in the total-flux spectrum.  This results in rather\nlow continuum polarization, $p \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 0.4\\%$, but very strong modulations\nacross spectral lines.  The polarization spikes reach $2\\%$ in the deep troughs\ndue to \\ion{Si}{2} $\\lambda 4130$ and $\\lambda 6355$; lesser peaks are observed\nin features identified with \\ion{Mg}{2} $\\lambda 4471$, a blend of \\ion{Si}{2}\n$\\lambda\\lambda 5041, 5056$ and \\ion{Fe}{2} $\\lambda\\lambda 4913, 5018, 5169$,\nand the \\ion{Ca}{2} near-IR triplet.  All line polarization has similar directional\nbehavior in the $q$--$u$ plane, suggesting a common origin.  Interestingly,\nwhereas other strong line features in the total-flux spectrum are characterized\nby strong polarization modulations, \\ion{O}{1} $\\lambda7774$ shows no\npolarization signature.  These features are explained in terms of optically\nthick bubbles of IMEs, the result of partial burning, that are asymmetrically\ndistributed within an essentially spherical oxygen substrate that remains from\nthe progenitor. Note that \\citet{Wang97a} find polarization\nvariation at a level of $> 0.5\\%$ across the strong \\ion{Si}{2}\n$\\lambda 6355$ feature of another HV~SN~Ia, SN~1997bp, in unpublished\nspectropolarimetry obtained near maximum light.\n\nFrom this small sample, the observations thus far suggest that normal and,\nperhaps, overluminous events are weakly polarized ($p \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 0.4\\%$), with\nsubluminous ones possessing somewhat larger values ($p \\approx 0.8\\%$).  HV~SNe~Ia\nhave modest continuum polarization, but possess the highest line polarization\nof all, achieving polarizations of up to $2\\%$ in the strongest features.  We\nassess the robustness of these tentative trends with the present study of\nfour SNe~Ia.\n\n\\subsection{The Type Ia Supernovae 1997dt, 2002bf, 2003du, and 2004dt}\n\\label{sec:2.4}\n\nSN~1997dt was discovered \\citep{Qiao97} by the Beijing Astronomical Observatory\nSupernova Survey \\citep{Li96} on 1997 November 22.44 (UT dates are used\nthroughout this paper) at an unfiltered magnitude of $\\sim$15.3 in the Sbc\ngalaxy NGC~7448. Images of the same field taken eight days earlier show no star\nat the position of the SN to a limiting unfiltered magnitude of about 18.5.  An\noptical spectrum obtained immediately (0.06 days) after discovery showed it to\nbe a Type~Ia event \\citep{Qiao97}; a subsequent examination by \\citet{Li01a}\nestimates the age at discovery to be $-7 \\pm 5$ days relative to maximum light,\nwhich places the date of maximum at very roughly 1997 November 29.\n\\citet{Tonry03} report that SN~1997dt suffers from a host extinction of $A_V =\n0.46 {\\rm\\ mag}$, the median of the values derived from MLCS and ``Bayesian\nAdapted Template Match Method'' \\citep[J. L. Tonry et al. 2005, in\npreparation;][]{Tonry03} analyses of its unpublished light curves.  An MLCS2k2\nfit (S. Jha et al., in preparation) suggests that it is subluminous, with\n$\\Delta = 0.94$ (corresponding to a $\\Delta m_{15}(B) \\approx 1.8$ mag; see\n\\S~\\ref{sec:2.1}), although there is a long shoulder of probability to lower\n$\\Delta$ values, indicating that there is a wide range of light-curve\nshapes that can fit the sparse number of photometric points (2 in $B$, 3 in\n$V$, and 2 in $I$).  From a pre-maximum spectrum posted at the Center for\nAstrophysics' Recent Supernovae\nPage,\\footnote{\\url{http:\/\/cfa-www.harvard.edu\/cfa\/oir\/Research\/supernova\/RecentSN.html}}\nwe estimate $\\cal R$(\\ion{Si}{2}) $\\approx 0.3$ (see \\citealt{Nugent95}), which\nimplies $\\Delta m_{15}(B) \\approx 1.33$ mag from the correlation derived by\n\\citet{Benetti04}, also consistent with a somewhat subluminous classification.\n\nSN~2002bf was discovered \\citep{Martin02} by the Lick Observatory and Tenagra\nObservatory Supernova Searches \\citep[LOTOSS;][]{Schwartz00} on 2002 February\n22.30 at an unfiltered magnitude of $\\sim$17, very close to the nucleus of the\nSb galaxy PGC~029953.  An image of the same field taken twenty days earlier\nshowed nothing at the position of SN~2002bf to a limiting unfiltered magnitude\nof $\\sim$19. Optical spectra obtained on 2002 March 6.21 by \\citet{Matheson02}\nand on 2002 March 7.41 by \\citet{Filippenko02} identified it as a Type~Ia\nevent.  Both groups noted that the expansion velocity derived from the\nabsorption minimum of the \\ion{Si}{2} $\\lambda 6355$~\\AA\\ line was\nsignificantly greater than normal for an SN~Ia near maximum light, indicating\nthat it may be an HV~SN~Ia.  A ``spectral feature age'' \\citep{Riess97} of $0\n\\pm 2$ days was derived from the March 6 spectrum \\citep{Matheson02}.  The\nMLCS2k2 analysis of the light curves presented in \\S~\\ref{sec:4.2.1} yields\n2002 March $4.37 \\pm 0.50$ as the date of maximum $B$ light, consistent with\nthe age on March 6 derived by \\citet{Matheson02}.\n\nSN~2003du was discovered \\citep{Schwartz03} by LOTOSS on 2003 April 22.4 at an\nunfiltered magnitude of $\\sim$15.9 in the SBd galaxy UGC~9391.  An image of the\nsame field taken fifteen days before discovery showed nothing at the position\nof SN~2003du to a limiting unfiltered magnitude of $\\sim$19. An optical\nspectrum obtained shortly thereafter, on 2003 April 24.06, identified it as a\nType~Ia event roughly two weeks before maximum light \\citep{Kotak03}.  The\nMLCS2k2 analysis of the light curves presented in \\S~\\ref{sec:4.2.1} yields\n2003 May $6.12 \\pm 0.50$ as the date of maximum $B$ light, consistent with the\nearlier spectral age estimate.  It also agrees with the epochs of maximum\ndetermined by the recent photometric studies of SN~2003du by \\citet{Anupama05}\nand \\citet{Gerardy04}.\n\nSN~2004dt was discovered \\citep{Moore04} by the Lick Observatory Supernova\nSearch \\citep{Filippenko03a} at an unfiltered magnitude of $\\sim$16.1 in the\nSBa galaxy NGC 799 on 2004 August 11.48.  An image of the same field taken ten\ndays earlier showed nothing at the position of SN~2004dt to a limiting\nunfiltered magnitude of $\\sim$18.  Spectra taken within 2 days of discovery by\n\\citet{Galyam04}, \\citet{Patat04}, and \\citet{Salvo04} confirmed it to be an\nSN~Ia before maximum light.  \\citet{Patat04} noted that several absorption\nlines showed high expansion velocities, a result confirmed by the recent study\nby \\citet{Wang05}, suggesting that, like SN~2002bf, SN~2004dt is an HV~SN~Ia.\nA preliminary analysis of the light curves of SN~2004dt shows that maximum\nlight occurred near 2004 August 20 (W. Li, personal communication).  A series\nof {\\it Hubble Space Telescope (HST)} UV spectral observations was obtained as\npart of program GO-10182 (P.I. Filippenko), and will be analyzed in a future\npaper.\n\n\n\\section{Observations and Reductions}\n\\label{sec:3}\n\nWe obtained single-epoch spectropolarimetry of SN~1997dt, SN~2002bf, SN~2003du,\nand SN~2004dt on days 21, 3, 18, and 4 (respectively) after maximum light.  We\nalso obtained additional optical spectroscopy and \\bvri\\ photometry of\nSN~2002bf and SN~2003du.  For SN~2002bf, our photometry samples $-10$ to $57$\ndays from the time of maximum light, with one additional flux spectrum taken on\nday 9 after maximum.  For SN~2003du, our ground-based photometry covers $-5$ to\n$113$ days from maximum, with one additional epoch on day 436 taken using the\nHigh Resolution Channel (HRC) of the Advanced Camera for Surveys (ACS) on board\n{\\it HST}.  Five additional spectral epochs sample its development from days\n$24$ to $82$ after maximum.\n\n\n\\subsection{Photometry}\n\\label{sec:3.1}\n\n\\subsubsection{Ground-Based Photometry of SN 2002bf and SN 2003du}\n\\label{sec:3.1.1}\n\nAll ground-based photometric data were obtained using either the 0.76-m Katzman\nAutomatic Imaging Telescope \\citep[KAIT; ][]{Filippenko01,Li03} or the Nickel 1\nm reflector \\citep{Li01}, both located at Lick Observatory.\nFigures~\\ref{fig:1} and \\ref{fig:2} show KAIT images of PGC~029953 and\nUGC~9391, the host galaxies of SN~2002bf and SN~2003du, respectively.  Also\nlabeled in the KAIT images are the ``local standards'' in both fields that were\nused to measure the relative SN brightness on non-photometric nights.  We\nobtained 12 epochs of Johnson-Cousins \\bvri\\ photometry (\\citealt[][for\n$BV$]{Johnson66}; \\citealt[][for $RI$]{Cousins81}) for SN 2002bf, all taken\nwith KAIT, and 38 epochs of \\bvri\\ photometry for SN~2003du, 33 of them taken\nwith KAIT and five with the Nickel telescope.  We also obtained three\npre-maximum unfiltered observations of SN~2002bf with KAIT (approximating the\n$R$ band, see \\citealt{Li03}), and one additional epoch of $RI$ photometry with\nKAIT of SN~2003du.  \n\nFor the photometry we employed the usual techniques of galaxy ``template''\nsubtraction \\citep[][and references therein]{Li00}, point-spread function\nfitting \\citep[][and references therein]{Stetson91}, and using ``local\nstandards'' to determine the \\bvri\\ brightnesses of the SNe on non-photometric\nnights; in general, we closely followed the technique detailed by\n\\citet{Leonard6}.  We note that the galaxy subtraction procedure for SN~2002bf\nwas particularly challenging since it is only $4\\farcs1$ from its host galaxy's\ncenter.\n\nThe absolute calibration of the SN~2002bf field was accomplished on the\nphotometric nights of 2002 May 14 and 2004 March 17 with the Nickel telescope,\nand 2003 February 3 and 2004 March 18 with KAIT, by observing several fields of\nLandolt (1992) standards over a range of airmasses in addition to the SN~2002bf\nfield.  The absolute calibration of the SN~2003du field was similarly derived\nfrom data taken on the photometric nights of 2003 May 31, June 1, June 26, and\nAugust 27 with the Nickel telescope, and of 2003 May 22 and 2004 March 18 with\nKAIT.  The color terms used to transform the filtered instrumental magnitudes\nto the standard Johnson-Cousins system are those of \\citet{Foley03}.  We list\nthe measured \\bvri\\ magnitudes and the $1\\sigma$ uncertainties, taken as the\nquadrature sum of a typical photometric error and the $1\\sigma$ scatter of the\nphotometric measurements from all of the photometric nights, of the local\nstandard stars in Tables~\\ref{tab:1} and \\ref{tab:2}.\n\nAfter deriving the \\bvri\\ magnitudes of the SNe based on a comparison with each\nof the local standards, we took the weighted mean of the individual estimates\nas the final standard magnitude of the SNe at each epoch in each filter.  The\nresults of our ground-based photometric observations are given in\nTables~\\ref{tab:3} and \\ref{tab:4} and shown in\nFigures~\\ref{fig:3} and \\ref{fig:4}.  The reported uncertainties come from the\nquadratic sum of the photometric errors (reported by DAOPHOT) and the\ntransformation errors.  For SN~2002bf, the uncertainty produced by the\ndifficult galaxy-subtraction process contributed the majority of the error on\nnights with low S\/N.\n\n\\subsubsection{{\\it Hubble Space Telescope} Photometry of SN 2003du}\n\\label{sec:3.1.2}\n\nWe obtained {\\it HST}\\ images during the course of two orbits of a\n$29^{\\prime\\prime} \\times 26^{\\prime\\prime}$ field of view centered on\nSN~2003du on 2004 July 15, 434 days after $B_{\\rm max}$, with the ACS\/HRC\ndetector through filters F435W, F555W, F625W, and F814W (hereafter referred to\nas $B, V, R, {\\rm\\ and\\ } I$, respectively), as part of our Snapshot survey\nprogram (GO-10272; P.I. Li) to investigate the late-time photometric behavior and\nenvironment of nearby SNe.  SN~2003du was detected in all images.  Total\nexposure times in \\bvri\\ were, respectively, $1680 {\\rm\\ s}$ (data archive\ndesignation j8z441011\/3011), $960 {\\rm\\ s}$ (j8z442011\/4011), $720 {\\rm\\ s}$\n(j8z441021\/3021), and $1440 {\\rm\\ s}$ (j8z442021\/4021).  \n\nTo derive the {\\it HST} photometry, we followed as closely as possible the\nprocedure detailed by \\citet{Sirianni05}, including correction for the effects\nof SN light contaminating the background region, aperture corrections, the\n``red-halo'' effect (for the $I$-band), and CTE degradation \\citep{Riess03}.\nWe translated the resulting instrumental magnitudes to the standard\nJohnson-Cousins \\bvri\\ system by using the coefficients and color corrections\ntabulated by \\citet{Sirianni05}.  The final results of our {\\it HST} photometry\nare included in Table~\\ref{tab:4} and displayed in Figure~\\ref{fig:4}.\n\n\\subsection{Spectropolarimetry and Spectroscopy}\n\\label{sec:3.2}\n\nWe obtained single epochs of spectropolarimetry for SN~2002bf and SN~2003du on\n2002 March 7 and 2003 May 24, respectively, with the Low-Resolution Imaging\nSpectrometer \\citep{Oke95} in polarimetry mode (LRISp; Cohen\n1996)\\footnote{Instrument manual available at\n\\url{http:\/\/www2.keck.hawaii.edu\/inst\/lris\/pol\\_quickref.html}.} at the\nCassegrain focus of the Keck-I 10-m telescope.  We observed SN~1997dt on 1997\nDecember 20 with LRISp using the Keck II 10-m telescope, and SN~2004dt on 2004\nAugust 24 with the Kast double spectrograph \\citep{Miller93} with polarimeter\nat the Cassegrain focus of the Shane 3-m telescope at Lick Observatory.  We\nreduced the polarimetry data according to the methods outlined by\n\\citet{Miller88} and detailed by \\citet{Leonard3} and \\citet{Leonard4}.  \n\nThe polarization angle offset between the half-wave plate and the sky\ncoordinate system was determined by observing the following polarized standard\nstars from the list of \\citet{Schmidt92a} and setting the observed $V$-band\npolarization position angle (i.e., $\\theta_V$, the debiased, flux-weighted\naverage of the polarization angle over the wavelength range 5050--5950~\\AA; see\n\\citealt{Leonard3}) equal to the cataloged value: BD $+64^\\circ106$ (1997\nDecember 20), BD $+59^\\circ389$ (2002 March 7), and HD 161056 (2003 May 24).\nOn the night of 2004 August 24, we averaged the polarization angle offsets\nderived from observations of three polarized standards from the\n\\citet{Schmidt92a} list, HD 204827, BD $+59^\\circ389$, and HD 19820; the\nindividual offsets were internally consistent to within $1^\\circ$.  To check\nfor instrumental polarization, the following null standards taken from the\nlists of \\citet{Turnshek90}, \\citet{Mathewson70}, \\citet{Schmidt92a}, and\n\\citet{Berdyugin95}, were also observed: HD 94851 (1997 December 20), HD 57702\n(2002 March 7), HD 109055 and BD $+32^\\circ3739$ (2003 May 24), and HD 212311\n(2004 August 24).  All stars were measured to be null to within $0.1\\%$, which\nis also our estimate of the systematic uncertainty of a continuum polarization\nmeasurement made with either the Keck or Lick spectropolarimeters\n\\citep[e.g.,][]{Leonard3}.\n\nAdditional specifics of the observations of SN~2002bf taken on 2002 March 7,\nincluding an investigation of the potential impact of second-order light\ncontamination and instrumental polarization (both shown to be minimal) in the\nsetup used on this night, are given by \\citet{Leonard8}.  Second-order light\ncontamination is not a concern for our observation of SN~1997dt due to its\nlimited spectral range. For SN~2003du, the use of a dichroic (D560) to split\nthe beam near 5600~\\AA\\ eliminates second-order light contamination on the\nred side.  Our spectropolarimetric observation of SN~2004dt was taken in a\nsetting that included the use of an order-blocking filter (GG455) to prevent\ncontamination by second-order light at red wavelengths.  Beyond $\\sim$9000~\\AA, \nhowever, second-order contamination may exist, but for reasons similar to\nthose discussed by \\citet{Leonard8} we believe it to have minimal\nimpact for this particular object.\n\nTo derive the total-flux spectra, we extracted all one-dimensional\nsky-subtracted spectra optimally \\citep{Horne86} in the usual manner.  Each\nspectrum was then wavelength and flux calibrated, and was corrected for\ncontinuum atmospheric extinction and telluric absorption bands\n\\citep{Wade88,Bessell99,Matheson00}.  With the exception of the\nspectropolarimetric observations of SN~1997dt and SN~2002bf, all spectra were\ntaken near the parallactic angle \\citep{Filippenko82}, so the spectral shape\nshould be quite accurate.  Table~\\ref{tab:5} lists the spectropolarimetric and\nspectral observations for all four SNe.  Figures~\\ref{fig:5}--\\ref{fig:8}\nshow the observed spectropolarimetry data of the four objects, and\nFigures~\\ref{fig:9} and \\ref{fig:10} show the complete series of spectra\nobtained for SN~2002bf and SN~2003du, respectively.\n\n\\section{Analysis}\n\\label{sec:4}\n\n\\subsection{Spectroscopy}\n\\label{sec:4.1}\n\nOur spectrum of SN~1997dt, taken $\\sim$21 days after maximum, shows typical\nfeatures for an SN~Ia at this phase (Fig.~\\ref{fig:5}a).  Similarly, our\nspectral sequence of SN~2003du (Fig.~\\ref{fig:10}) closely follows the\nevolution of the normal-luminosity SN~Ia~1994D \\citep{Patat96,Filippenko97} in\nterms of the strengths and blueshifts of line features.  This is consistent\nwith the analyses of \\citet{Anupama05} and \\citet{Gerardy04}, in which spectra\ntaken before and shortly after maximum light were also examined.  This\nconvincingly establishes SN~2003du as a spectroscopically ``typical'' SN~Ia\n\\citep{Branch93a}.\n\nAs discussed in \\S~\\ref{sec:2.4}, the spectra of both SN~2002bf and SN~2004dt\nare peculiar in one regard: the blueshifts of many of the spectral lines, most\nnoticeably \\ion{Si}{2} $\\lambda 6355$, occur at significantly higher velocity\nthan is typical for an SN~Ia at this phase, and indicate that these are both\nHV~SNe~Ia.\n\nFor comparison, we have measured the velocities of the \\ion{Si}{2}\n$\\lambda6355$ line in a number of other HV~SNe~Ia from our database, along with\nthe spectroscopically normal SN~1994D, the subluminous SN~1991bg\n\\citep[e.g.,][]{Filippenko92b}, and the overluminous SN~1991T\n\\citep[e.g.,][]{Filippenko92a}.  We present the results in Table~\\ref{tab:6}\nand Figure~\\ref{fig:11}, from which it is clear that both SN~2002bf and\nSN~2004dt belong to the class of HV SNe~Ia; indeed, SN~2002bf is the most\nextreme HV~SN~Ia yet observed for its epochs.  The figure also suggests that\nline velocity is not strongly correlated with luminosity, although the\nexpansion velocity of the subluminous SN~1991bg is somewhat lower than typical\nvalues.\n\nFigure~\\ref{fig:12} presents a spectral comparison near maximum light of three\nHV~SNe~Ia (SN~2002bf, SN~2002bo, and SN~2004dt) with the spectroscopically\nnormal SN~1994D.  The excessive blueshift of the \\ion{Si}{2} $\\lambda 6355$\ntrough is obvious for the three HV~SN~Ia compared with SN~1994D.  While it must\nbe cautioned that the spectra span a range of ages of about five days near\nmaximum light, a time when significant spectral development occurs, the\nblueshift differences are much greater than can be explained by age differences\nalone.  In addition, the \\ion{Si}{2} line is significantly stronger in the\nHV~SNe~Ia compared with SN~1994D, with equivalent widths of $\\gtrsim 140$~\\AA\\\ncompared with $\\sim$100~\\AA\\ for SN~1994D.  Conversely, the \\ion{O}{1}\n$\\lambda 7774$ absorption is relatively weaker in the HV~SN~Ia events, with\nequivalent widths of $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 100$~\\AA\\ in the HV~SNe~Ia compared with\n$125$~\\AA\\ for SN~1994D.  The fact that the \\ion{Si}{2} line is significantly\nstronger in the HV~SNe~Ia sample, and the \\ion{O}{1} line relatively weaker, is\nconsistent with the scenario in which a greater fraction of C and O is burned\nto IMEs in HV~SN~Ia than in more typical events; it also follows naturally from\nthe GCD model, since the obscuring pancake is formed from the products of\noxygen burning.\n\nWhile HV~SNe~Ia share many spectral characteristics, it is clear that they also\nexhibit spectral diversity.  For instance, whereas the absorption trough of the\n\\ion{Ca}{2} near-IR triplet is significantly blueshifted for SN~2002bf and\nSN~2002bo relative to SN~1994D ($16,300\\ {\\rm km\\ s}^{-1}$ and $14,500\\ {\\rm km\\ s}^{-1}$ for SN~2002bf\nand SN~2002bo, respectively, compared with $\\sim$10,900~km s$^{-1}$\\ for SN~1994D,\nwhere we have assumed $\\lambda_0 = 8579$ for the \\ion{Ca}{2} near-IR triplet, a\nvalue derived using the prescription given by \\citealt{Leonard5}), it has a\nmore normal blueshift ($v = 10,900$~km s$^{-1}$) in SN~2004dt.  The equivalent widths\nof the \\ion{Ca}{2} near-IR lines also show a suggestive trend, with both\nSN~2002bf and SN~2002bo having widths of $> 200$~\\AA, while both SN~2004dt and\nSN~1994D have equivalent widths of $\\sim$100~\\AA.  This may indicate that the\nexplosive nucleosynthesis did not proceed up to Ca as far out in the atmosphere\n(or in the bubble in the GCD scenario) of SN~2004dt as it did in the other two\nHV~SNe.\n\nFinally, it is clear from Figure~\\ref{fig:12} that not all lines share the\nextreme velocities seen in the \\ion{Si}{2} $\\lambda 6355$ feature. For\ninstance, as first noticed by \\citet{Wang05} in a pre-maximum spectrum of\nSN~2004dt, the \\ion{S}{2} ``W'' feature in the spectra of SN~2002bo and SN~2004dt\nindicates velocities comparable to those in SN~1994D; for SN~2002bf they\nare somewhat larger, but still not near the velocity of the \\ion{Si}{2}\n$\\lambda 6355$ line.  \\citet{Wang05} propose that this may indicate that sulfur\nis more confined to the lower-velocity, inner region.\n\n\\subsection{Photometry}\n\\label{sec:4.2}\n\n\\subsubsection{Ground-Based Photometry}\n\\label{sec:4.2.1}\n\nThe results of the MLCS2k2 application to the photometry of SN~2002bf and\nSN~2003du are given in Table~\\ref{tab:7}.  For details of the MLCS2k2 procedure\nused, see S. Jha et al. (in preparation); an overview of the technique is\nprovided by \\citet{Riess05}.  The MLCS2k2 analysis finds SN~2002bf to be of\ntypical luminosity.  SN~2003du is slightly overluminous, although its\npre-maximum spectral evolution \\citep{Anupama05} demonstrates that it is {\\it\nnot} a SN~1991T-like event, as the strength of the \\ion{Si}{2} $\\lambda 6355$\nfeature is comparable to that seen in spectra of normal SNe~Ia.\n\nFor SN~2002bf, $B_{\\rm max}$ occurred on 2002 March $4.37 \\pm 0.50$, which is\n$10$ days after discovery and 5 days before our filtered observations\ncommenced.  For SN~2003du, $B_{\\rm max}$ occurred on 2003 May $6.12 \\pm 0.50$,\nwhich is $14$ days after discovery, and the same day that our \\bvri\\\nobservations began.  Our derived date of maximum light for SN~2003du agrees\nwith those found by \\citet{Anupama05} and \\citet{Gerardy04} using independent\ndata sets.\n\n\\subsubsection{{\\it HST} Photometry of SN~2003du}\n\\label{sec:C1c}\n\nLate-time photometry (e.g., $t > 200 {\\rm\\ d}$) of SNe~Ia exists for only a\nhandful of objects \\citep[see, e.g., ][and references therein]{Milne01}.  From\nthe small sample, there are two main features of note.  First, SN~Ia decline\nrates are typically much faster than the decay slope of $^{56}{\\rm Co}\n\\rightarrow$ $^{56}{\\rm Fe}$ of $0.98\\ {\\rm mag\\ (100\\ d)}^{-1}$ predicts.\nThis decay mechanism is thought to be primarily responsible for powering the\nluminosity from the early nebular phase out to $\\sim$1000 days for SNe of all\ntypes.  Essentially, the $^{56}{\\rm Co}$ decays release most of their energy in\nthe form of $\\gamma$-rays, which, given enough optical depth, can become\ntrapped in the ejecta and Compton scatter off free electrons.  The energetic\nelectrons generate optical photons primarily through ionization and excitation\nof atoms, and the ejecta are transparent to these photons.  A small portion\n($\\sim$3.5\\%, see \\citealt*{Arnett79}) of the total $^{56}{\\rm Co}$ decay\nenergy comes in the form of positrons, which may deposit their kinetic energy\nin the ejecta and then annihilate with electrons, producing two $\\gamma$-ray\nphotons of energy $E_\\gamma = m_e c^2$.  The steeper decline seen in the\nlate-time photometry of SNe~Ia has been explained by significant transparency\nof the ejecta to $\\gamma$-ray photons \\citep{Milne99} and positron escape\n\\citep{Milne01}.\n\nA quantitative measure of the late-time decline is given by\n\\citet{Cappellaro97}, who investigate how the decline from maximum to 300 days\nafter peak $V$ brightness, denoted $\\Delta m_{300}(V)$, correlates with\nintrinsic SN brightness as derived from the $\\Delta m_{15}(B)$ parameter.\nAlthough limited by small sample size (5 objects), they find a convincing\ncorrelation, with $\\Delta m_{300}(V)$ going from 6.7 mag to 8.4 mag as the\nsample runs from overluminous (SN~1991T) to subluminous (SN~1991bg) events,\nwith the normal-brightness SN~1994D characterized by $\\Delta m_{300}(V) = 7.3$\nmag.  A second feature that has thus far been seen in only two SN~Ia events\n(SN~1991T and SN~1998bu; see \\citealt*{Schmidt94a} and \\citealt*{Cappellaro01},\nrespectively) is a sudden flattening of the late-time optical light curves,\nwhich has been attributed to the contribution of a light echo from foreground\ndust clouds.\n\nOur photometric data from {\\it HST}, taken $436$ days after $B_{\\rm max}$,\nallow us to investigate the late-time photometric behavior of SN~2003du.  From\nthe inset of Figure~\\ref{fig:4}, it is clear that SN~2003du, like all SNe~Ia\nanalyzed before it, declines significantly faster than the $^{56}{\\rm Co}\n\\rightarrow$ $^{56}{\\rm Fe}$ decay rate.  In fact, we measure the average decay\nrate in $V$ (shown by \\citealt*{Milne01} to track the bolometric luminosity of\nan SN~Ia quite accurately) of SN~2003du between our last two photometric epochs\non days $113$ and $436$ to be $\\Delta V = 1.47 \\pm 0.02 {\\rm\\ mag\\ (100\\\nd)}^{-1}$.  The slope also appears to have been rather constant throughout the\nperiod between our two last epochs, as indicated by the good agreement of the\nlate-time data taken from \\citet{Anupama05} near day 300 and the decay slope\ndetermined from our data alone.  Using the \\citet{Anupama05} data point and our\nestimate of $V_{\\rm max}$ (Table~\\ref{tab:7}), we derive $\\Delta m_{300}(V) =\n6.74$, which is most consistent with the values found by \\citet{Cappellaro97}\nfor overluminous events.  That SN~2003du may be somewhat overluminous was also\nsuggested by its peak $V$ magnitude of $-19.67 \\pm 0.02$ mag, which is 0.17 mag\nbrighter than the fiducial template used in the MLCS2k2 procedure\n(Table~\\ref{tab:2}).  There is no evidence from our data of any contribution to\nthe SN brightness from a light echo, although we note that the major\nindications of additional contributions to the apparent brightness of SNe~1991T\nand 1998bu did not become obvious until epochs $\\gtrsim 500 {\\rm\\ d}$.\n\n\\subsection{Reddening}\n\\label{sec:4.3}\n\n\\subsubsection{Techniques to Estimate SN Ia Reddening}\n\\label{sec:4.3.1}\n\nAccurate determination of SN reddening is crucial both for deriving intrinsic\nSN properties as well as interpreting spectropolarimetry, since the same dust\nthat reddens SN light can also polarize it as discussed in \\S~\\ref{sec:2.3.2}.\nWhen multi-band photometry is available, the MLCS2k2 technique can accurately\nestimate the total extinction (see \\citealt{Riess05} for discussion).  When\nthis is lacking, other methods must be used.\n\nGalactic extinction along the l-o-s is accurately estimated by the dust maps of\n\\citet[][hereafter SFD]{Schlegel98} to an estimated precision of $\\sim$15\\%.\nFor host-galaxy extinction, a rather crude approach, to which many\ninvestigators resort in the absence of photometry, is to employ the rough\ncorrelation found between the total equivalent width ($W_\\lambda$) of the\ninterstellar (IS) \\ion{Na}{1} D doublet ($\\lambda\\lambda 5890, 5896$) and\nreddening \\citep{Barbon90}.  The \\citeauthor{Barbon90} correlation has\nbeen subsequently improved upon by \\citet{Munari97}, who derive a more precise\nrelation that uses just the equivalent width of the \\ion{Na}{1} D2 ($\\lambda\n5890$) line.  Both relations, however, warrant healthy degrees of skepticism\nsince sodium is known to be only a fair tracer of the hydrogen gas column\n(especially in dense environments, where sodium may be heavily depleted; e.g.,\n\\citealt*{Cohen73}), from which the dust column is then estimated.  The\ndust-to-gas ratio also varies significantly among galaxies (e.g.,\n\\citealt*{Issa90}), and it could well be that in the case of SNe,\ncircumstellar, rather than interstellar, dust is present, for which the\ndust-to-gas ratio (or the extinguishing properties of the dust itself) could be\nunusual.  A final complication is that, at the resolution typical of most\noptical spectra, individual absorption components along the l-o-s, whose\ncontributions should be considered separately to determine the total reddening,\nare blended into a single profile.  In such situations, the \\citet{Munari97}\nrelation formally yields an {\\it upper limit} to the reddening, and caution\nmust be used since the derived value could seriously overestimate the actual\nreddening.  Nonetheless, the \\citet{Munari97} relation is still frequently used\nto get approximate reddening values, or upper limits, especially in the case of\nnull detections of IS \\ion{Na}{1} D lines.\n\n\n\\subsubsection{The Reddening of the Four SNe Ia}\n\\label{sec:4.3.2}\n\nFor SN~2002bf and SN~2003du we shall adopt the MLCS2k2 values of $E\\BminusV = 0.08\n\\pm 0.04$ mag and $E\\BminusV = 0.01 \\pm 0.01$ mag (respectively) reported in\nTable~\\ref{tab:7} (for $R_V = 3.1$).  For SN~1997dt, \\citet{Tonry03} report a\nhost-galaxy reddening value of $E\\BminusV_{\\rm Host} = 0.15$ mag (assuming $R_V =\n3.1$), the median of the values derived from MLCS2k2 and ``Bayesian Adapted\nTemplate Match Method'' analyses.  The SFD dust maps predict $E\\BminusV_{\\rm\nSFD}^{\\rm MW} = 0.06$ mag for SN~1997dt, for a total estimate of $E\\BminusV = 0.21$\nmag.\n\nOur spectrum of SN~1997dt also exhibits strong \\ion{Na}{1} D IS absorption at\nboth the redshift of the host galaxy and the MW.  The resolution of our\nspectrum, $\\sim$5~\\AA\\ (Table~\\ref{tab:5}), while sufficient to deblend the\n\\ion{Na}{1} D doublet, is not fine enough to resolve the individual absorption\ncomponents that likely contribute to the D1 and D2 profiles.  For host-galaxy\nabsorption, we measure $W_\\lambda^{\\rm tot} = 0.77$~\\AA, with $W_\\lambda^{\\rm\nD2} = 0.42$~\\AA\\ and $W_\\lambda^{\\rm D1}= 0.35$~\\AA.  This translates to\n$E\\BminusV_{\\rm NaID}^{\\rm Host} \\leq 0.21$ mag from the \\citet{Munari97} relation.\nFor the MW, we measure $W_\\lambda^{\\rm tot} = 0.76$~\\AA, with $W_\\lambda^{\\rm\nD2} = 0.44$~\\AA\\ and $W_\\lambda^{\\rm D1} = 0.32$~\\AA, yielding $E\\BminusV_{\\rm\nNaID}^{\\rm MW} \\leq 0.23$ mag from the \\citeauthor{Munari97} relation.  The\nvalues given by SFD and \\citet{Tonry03} are consistent with the upper limits\nderived from the sodium relation.  The purported accuracy of the SFD Galactic\nreddening value and the upper limit set by the \\citeauthor{Munari97} relation\nfor the host reddening would indicate an upper reddening limit of $E\\BminusV_{\\rm\ntotal} \\leq 0.27$ mag.  This limit will prove to have important implications\nwhen we examine the spectropolarimetry of SN~1997dt in \\S~\\ref{sec:4.4.2}.  The\nfact that the \\citeauthor{Munari97} relation overpredicts the reddening for\nboth the host galaxy and, especially, the MW, may indicate that multiple,\nunresolved components make up the IS line profiles.  Given the upper reddening\nlimit, we conclude $E\\BminusV = 0.21 \\pm 0.06$ mag for SN~1997dt.\n\nFor SN~2004dt, $E\\BminusV_{\\rm SFD}^{\\rm MW} = 0.03$ mag, and \\citet{Wang05} report\na range of $E\\BminusV_{\\rm total} = 0.14$ to 0.2 mag from analysis of the color of\ntheir pre-maximum spectrum.  However, they also note that SN~2004dt does not\nshow noticeable \\ion{Na}{1} D IS lines at the redshift of NGC~799.  We confirm\nthe lack of \\ion{Na}{1} D lines in our spectrum as well, although its poor\nresolution ($\\sim$18~\\AA, see Table~\\ref{tab:5}) makes deriving even an upper\nlimit to the strength of the \\ion{Na}{1} D IS lines difficult.  Formally, our\nprocedure yields $W_\\lambda {\\rm (3\\sigma)} = 0.1$~\\AA\\ for host-galaxy\n\\ion{Na}{1} D, which translates to a predicted upper limit of $E\\BminusV_{\\rm\nNaID}^{\\rm Host} < 0.02$ mag.  We thus confirm the discrepancy noted by\n\\citet{Wang05} between the lack of IS sodium absorption and reddening inferred\nfrom other methods. It could be that the spectroscopic peculiarities of HV\nSNe~Ia produce photometric irregularities that affect the general reddening\nrelations.  As mentioned in \\S~\\ref{sec:2.2}, \\citet{Benetti04} do find\nphotometric peculiarities, albeit minor, in their study of another HV~SN~Ia,\nSN~2002bo. However, since the \\ion{Na}{1} D relation, especially for the host\ngalaxy, is also prone to error, we have no convincing way to decide between the\ntwo estimates.  We thus take the simple average of the low and high reddening\nestimates, and incorporate the disparity into our final estimate's uncertainty.\nThis yields $E\\BminusV = 0.11 \\pm 0.06$ mag as our best reddening estimate for\nSN~2004dt.\n\n\\subsection{Spectropolarimetry of Four SNe Ia}\n\\label{sec:4.4}\n\nTo summarize: SN~1997dt is likely a somewhat subluminous event that is thought\nto be reddened by $E\\BminusV = 0.21 \\pm 0.06$ mag, with an upper limit of $0.27$\nmag.  SN~2002bf and SN~2004dt are HV~SNe~Ia, with SN~2002bf being the most\nextreme example yet observed.  Finally, SN~2003du is a slightly overluminous\nevent that is minimally reddened.  At maximum light, a previous subluminous\nevent has been found to be moderately polarized ($p \\approx 0.8\\%$), whereas\nnormal to overluminous examples have been less so ($p \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 0.4\\%$).  The\ntwo HV~SNe~Ia that have reported polarization measurements show the greatest\npolarization features yet observed for any SN~Ia type, with values approaching\n$2\\%$ in the strongest lines of SN~2004dt (\\S~\\ref{sec:2.3.3}).\n\nWe now examine our single-epoch spectropolarimetry of SN~1997dt, SN~2002bf, SN\n2003du, and SN~2004dt obtained on days 21, 3, 18, and 4, respectively, after\nmaximum light.  For each SN, we shall first discuss the observed polarization\nand then attempt to remove the ISP through the technique of \\citet{Wang05},\nwhich assumes that spectral regions lacking strong individual flux or\npolarization features at blue wavelengths ($\\lambda < 5000$~\\AA) are\nintrinsically unpolarized.  Since the choice of the specific spectral regions\nto use for ISP determination is admittedly somewhat subjective, we shall be\ncareful to point out how our conclusions would change with other ISP choices.\n\n\n\nFollowing ISP removal and examination of the data in the $q$--$u$ plane, we\ncalculate the intrinsic polarization of the SN by rotating the $q$--$u$ axes\nthrough a single angle, $\\theta$, that places the greatest degree of\npolarization change across the spectrum along the rotated $q$ axis.  The angle\nby which the axes are rotated is determined through a uniform-weight,\nleast-squares fit to the ISP-subtracted data in the $q$--$u$ plane.  The\npolarization degree measured along the rotated $q$ axis is then referred to as\nthe rotated Stokes parameter \\citep[RSP;][]{Tran95,Leonard3}, whereas that\nmeasured along the rotated $u$ axis is denoted URSP.\\footnote{Note that RSP\nand URSP, as defined here, are identical to the \\citet{Wang01} definitions of\nthe polarization along the ``dominant'' and ``orthogonal'' axes, $P_d$ and\n$P_o$, respectively.  That is, by definition, RSP shows polarization strength\nvariation along the chosen polarization angle (i.e., the dominant axis),\nwhereas the URSP shows polarization strength variation along the axis that is\northogonal to the dominant axis in the $q$--$u$ plane.  The main difference\nbetween our construction of the RSP in the {\\it observed} polarization plots\n(Figures \\ref{fig:5}--\\ref{fig:8}) and the {\\it ISP-subtracted} data\n(Figures \\ref{fig:15}, \\ref{fig:17}, and \\ref{fig:19}) is that in\nthe former the polarization angle about which the RSP is determined is a\nsmoothly varying function of wavelength (i.e., it is the polarization angle,\n$\\theta$, smoothed over many bins; see \\citet{Leonard3} for more details on the\nprocedure), whereas in the latter, RSP and URSP are determined with respect to\na single, unchanging polarization angle.  For consistency with\nspectropolarimetry work in other fields, as well as our own prior studies, we\nshall continue to use the RSP and URSP designations, rather than the $P_d$ and\n$P_o$ designations of \\citet{Wang01}.}\n\nWe shall then analyze the resulting spectropolarimetry within the context of\nthe models described in \\S~\\ref{sec:2.3.1}.  By construction, the greatest\ndegree of polarization change, especially from the continuum regions, should\noccur in the RSP.  Examination of URSP is especially important in the line\nfeatures, however, as it can be used to discriminate between the predictions of\nthe ellipsoidal and clumpy-ejecta models.  In particular, the simple\nellipsoidal model demands the existence of a single value of $\\theta$ capable\nof placing all line polarization changes along the RSP (i.e., URSP should show\nno polarization features).\n\n\\subsubsection{Two HV~SNe~Ia:  SN 2002bf and SN 2004dt}\n\\label{sec:4.4.1}\n\nWe begin by considering the spectropolarimetry of the two HV~SNe~Ia, SN~2002bf\nand SN~2004dt, which are displayed in Figures~\\ref{fig:6} and \\ref{fig:8},\nrespectively.  Both objects show very low levels of observed polarization, with\n$p_V = 0.03\\%$, $\\theta_V = 62^\\circ$ for SN~2002bf and $p_V = 0.25\\%$,\n$\\theta_V = 146\\ensuremath{^{\\circ}}$ for SN~2004dt, where $p_V$ and $\\theta_V$ approximate\nthe rest-frame $V$-band polarization and polarization angle derived by\ncalculating the debiased, flux-weighted averages of $q$ and $u$ over the\ninterval 5050--5950~\\AA\\ \\citep[see][]{Leonard3}.  The polarization of\nSN~2004dt may exhibit an upward trend with wavelength, increasing from about\n$0.3\\%$ at blue wavelengths to $\\sim$1.0\\% at the red edge ($\\lambda =\n9600$~\\AA); a similar but smaller trend exists in the data for SN~2002bf.\nNeither object shows significantly different polarization in the ``continuum''\nregion 6800--7800~\\AA\\ from that observed in the heavily line-blanketed\nregion below 5000~\\AA\\ (\\S~\\ref{sec:2.3.1}), suggesting that the intrinsic\ncontinuum polarization is quite low for both objects.\n\nThe salient features of both data sets are the extraordinarily large\npolarization modulations ($\\sim$2\\%) across certain P-Cygni lines, most\nnotably the \\ion{Ca}{2} near-IR triplet for SN~2002bf and the \\ion{Si}{2}\n$\\lambda 6355$ line for SN~2004dt.  Smaller features may be discerned across\nother lines in both objects.\n\nBefore attempting ISP removal, it is instructive to compare our SN~2004dt data\nwith those obtained by \\citet{Wang05} taken eleven days earlier, when the SN\nwas about seven days before maximum light.  The interval separating the two\nobservations is one marked by rapid spectral and photometric evolution.  For\nSN~2004dt, the velocity of the minimum of the high-velocity \\ion{Si}{2}\n$\\lambda 6355$ line recedes by about $3700\\ {\\rm km\\ s}^{-1}$ (from $17,200$ to $13,500\\\n{\\rm km\\ s}^{-1}$), and the weaker, lower-velocity lines such as \\ion{S}{2} $\\lambda\\lambda\n5612, 5654$ ($\\lambda_0 = 5635$~\\AA) decrease by about $2100\\ {\\rm km\\ s}^{-1}$ (from\n$11,000$ to $8900\\ {\\rm km\\ s}^{-1}$).  Remarkably, in spite of the great changes that\noccur in the flux spectrum between the two epochs, we find the\nspectropolarimetry to be virtually unchanged.  The overall polarization, in\nboth magnitude and polarization angle, is consistent with the earlier epoch,\nand the detected line-polarization features, most notably those at \\ion{Si}{2}\n$\\lambda 6355$ and the \\ion{Ca}{2} near-IR triplet, are also very similar.  As\nwith the \\citeauthor{Wang05} data, our observations do not show significant\npolarization modulation across the important \\ion{O}{1} $\\lambda 7774$ line,\nalthough this line is now much weaker in the total-flux spectrum than it was\nearlier.  Weaker polarization features, which are clearly detected in the\nearlier epoch, are difficult to confirm in our lower S\/N data, but our data are\nnot inconsistent with the earlier measurements, within the errors.\n\nAs discussed in \\S~\\ref{sec:2.3.3}, \\citet{Wang05} explain the\nspectropolarimetry of SN~2004dt in terms of optically thick bubbles of IMEs\nthat are asymmetrically distributed within an essentially spherical oxygen\nsubstrate that remains from the progenitor material.  \\citet{Wang05} further\nnote that the polarization behavior of the high-velocity lines is similar to\nthe low-velocity lines in the early epoch, implying that the structures that\nobscure the photosphere have great radial extent. Within this context, then, it\nmay not be surprising that the polarization features have not evolved\nsignificantly between the two epochs, despite the great evolution of line\nvelocity.\n\nIn order to directly compare the spectropolarimetry of SN~2002bf and SN~2004dt\nwith each other, we must attempt to remove the ISP.  To establish the ISP, we\napply the technique of \\citet{Wang05} and choose the spectral regions indicated\nin Figures~\\ref{fig:6}a and \\ref{fig:8}a.  For SN~2002bf, this yields $(q_{\\rm\nISP}, u_{\\rm ISP}) = (0.01\\%, 0.05\\%)$, or ISP$_{\\rm max} = 0.05\\%$ at\n$\\theta_{\\rm ISP} = 39\\ensuremath{^{\\circ}}$ at an assumed peak wavelength of the ISP of\n$\\lambda_{\\rm max} = 5500$~\\AA.  For SN~2004dt we derive $(q_{\\rm ISP}, u_{\\rm\nISP}) = (0.3\\%, -0.2\\%)$, which is within the uncertainty of the ISP found by\n\\citet{Wang05} through this same approach, $(q_{\\rm ISP}, u_{\\rm ISP}) =\n(0.2\\%, -0.2\\%)$.  In order to facilitate direct comparisons between our data\nand those of \\citet{Wang05}, we shall adopt their ISP value for our study of\nSN~2004dt as well.\n\nAfter removal of the small amount of ISP for SN~2002bf, we derive a\nbest-fitting axis with ${\\rm PA\\ } = 123\\ensuremath{^{\\circ}}$ (Fig.~\\ref{fig:13}), about\nwhich we calculate the intrinsic RSP and URSP shown in Figure~\\ref{fig:15}.  In\na similar way, for SN~2004dt we derive a best-fitting axis of ${\\rm PA\\ } =\n146\\ensuremath{^{\\circ}}$, which may be compared with the PA of $\\sim$150$\\ensuremath{^{\\circ}}$ found by\n\\citet{Wang05}.  Again, since the two values do not differ significantly, we\nadopt the \\citeauthor{Wang05} direction for ease of comparison between the two\ndata sets.  The axis and ISP choices for SN~2004dt are indicated in\nFigure~\\ref{fig:14}, and the resulting RSP and URSP are shown in\nFigure~\\ref{fig:15}.\n\nFrom examination of Figure~\\ref{fig:15}, it is clear that the two events have\nspectropolarimetric similarities. Both show low overall polarization and\nmodulations across the \\ion{Si}{2} $\\lambda 6355$ and \\ion{Ca}{2} near-IR\ntriplet absorptions.  The detailed character of the line polarizations do\ndiffer, however.  For SN~2002bf there is a $2\\%$ polarization change across the\n\\ion{Ca}{2} feature, with a more modest ($\\sim$0.4\\%) feature detected in the\n\\ion{Si}{2} line.  (When discussing overall ``polarization change'' across a\nline feature without specific reference to either the RSP or URSP, we\neffectively mean the quadrature sum of the changes seen in the two parameters.)\nFor SN~2004dt, the situation is reversed.  In the case of the \\ion{Ca}{2} line,\nthis may be related to its relative strength and velocity in the two spectra,\nas both the equivalent width and velocity of the feature in the flux spectrum\nare much greater in SN~2002bf than in SN~2004dt.  As discussed earlier\n(\\S~\\ref{sec:4.1}), it is possible that burning to Ca occurred more extensively\nin SN~2002bf than it did in SN~2004dt.  The explanation for the \\ion{Si}{2}\nline disparity, however, is not so obvious, as the lines have similar strengths\nand velocities.\n\nClearly, strong line and weak continuum polarization levels disfavor the simple\nellipsoidal model.  Applying Eq.~(\\ref{eqn:1}) to the data for SN~2002bf and\nSN~2004dt yields lower bounds on the expected continuum polarization of $p_{\\rm\ncont} \\geq 0.7\\%$ and $p_{\\rm cont} \\geq 1.3\\%$, respectively. For both\nobjects, however, we measure $p < 0.4\\%$ in both the observed and\nISP-subtracted data for the spectral region 6800--7800~\\AA, which is\nlargely devoid of line opacity in SNe~Ia near maximum (\\S~\\ref{sec:2.3.1}).\nThe ellipsoidal model therefore appears to be ruled out as the explanation for\nthe polarization characteristics of these objects.\n\nThe clumpy-ejecta model, on the other hand, provides a natural explanation for\nhigh line and low continuum polarization levels (\\S~\\ref{sec:2.3.1}).  In\nparticular, the GCD scenario can successfully explain (1) the high line\nvelocities, (2) the large polarization change in the line troughs, (3) the\ngreat radial extent of the obscuring material (i.e., for $m_{\\rm atm} = 0.08\nM_\\odot$, the obscuring pancake spans the range 10,000--21,000 km s$^{-1}$), and (4)\nthe lack of significant continuum polarization. A remaining challenge is to\nexplain the lack of any polarization change across the \\ion{O}{1} $\\lambda\n7774$ line, especially in the earlier \\citet{Wang05} data for SN~2004dt, when\nthis line is extremely strong in the total-flux spectrum.  Recent\nnucleosynthesis calculations based on multi-dimensional (2D and 3D)\nhydrodynamical simulations of the thermonuclear burning phase in SNe~Ia show\nthat as much as $40\\%$ to $50\\%$ of the ejected matter in SNe~Ia is unburned\ncarbon and oxygen \\citep{Travaglio04}.  This has led \\citet{Wang05} to propose\nthat the primordial oxygen is nearly spherically distributed, within which\nasymmetrically distributed IME clumps, or an absorbing pancake in the GCD\nmodel, are embedded.\n\nWith this in mind, an important prediction of the clumpy-ejecta model\n(including the GCD scenario) is that the \\ion{O}{1} $\\lambda 7774$ line should,\nin fact, show a polarization change with a polarization angle that differs by\n$90^\\circ$ from what is observed in the \\ion{Si}{2} and \\ion{Ca}{2}\nlines since its distribution is essentially the inverse of these IMEs.\nHowever, some oxygen is probably also contained in the clumps or pancake, since\nit is produced by explosive carbon burning in small quantities and should be\npresent when such burning products as magnesium exist, whose spectral signature\nis unequivocally seen in the spectra.  This would tend to reduce the\npolarization level in the \\ion{O}{1} $\\lambda 7774$ line.  Neither our data,\nnor those of \\citet{Wang05}, are of sufficiently high S\/N to detect such a\nchange, and it must be left to future, higher-S\/N studies focused especially at\nearly times when the \\ion{O}{1} $\\lambda 7774$ feature is strong.  Finding such\na PA change would further strengthen the case for the clumpy-ejecta and\/or GCD\nscenario.\n\nWe note that the ejecta-hole model of \\citet{Kasen04} is also capable of\nproducing large line polarization with weak continuum polarization for\nsight-lines near to the hole (\\S~\\ref{sec:2.3.1}).  Arguing against this\nmodel in these cases, though, is that it offers no natural explanation for why\nhigh line velocities should be associated with sight lines near to the hole.\nIn fact, the \\citeauthor{Kasen04} models predict {\\it lower} absorption\nvelocities when viewing down the hole.\n\nIn all, then, our study finds that HV~SNe~Ia are, as a group, characterized by\nmuch stronger line-polarization features than are seen in other SN~Ia\nvarieties.  The ellipsoidal model is incapable of explaining the polarization\ncharacteristics of these objects, whereas the clumpy-ejecta and ejecta-hole\nmodels are more successful.  On balance, the case for clumpy ejecta appears to\nbe the most convincing explanation, with the GCD model investigated by\n\\citet{Kasen05} able to reproduce many of the observed spectral and\nspectropolarimetric features.\n\n\\subsubsection{SN 1997dt}\n\\label{sec:4.4.2}\n\nWe next turn to the likely subluminous SN~1997dt, which was observed about\nthree weeks past maximum light.  Figure~\\ref{fig:5} reveals an extraordinarily\nhigh level of observed polarization, $p_V = 3.46\\%$ at $\\theta_V = 112^\\circ$.\nThis is by far the largest polarization yet observed for an SN~Ia.  A distinct\npolarization feature is detected in the \\ion{Fe}{2} $\\lambda 4555$ trough and\nprobably also in the \\ion{Si}{2} $\\lambda 5972$ + \\ion{Na}{1} D and \\ion{Si}{2}\n$\\lambda 6355$ lines.  The degree of change in the \\ion{Fe}{2} $\\lambda 4555$\nfeature reaches nearly $1\\%$ in the $q$ parameter.\n\nThis amount of observed continuum polarization is surprising, given that our\nbest total reddening estimate predicts an upper bound on the ISP of only\n$1.89\\%$ from the \\citet{Serkowski75} relation (see \\S~\\ref{sec:2.3.2}).  If we\ntrust both the reddening estimate and the upper ISP bound, then an intrinsic SN\npolarization of at least $1.57\\%$ must exist to explain the observed\npolarization, far higher than has been indicated for any previous SN~Ia.\nHowever, the technique of \\citet{Wang05} suggests a much higher ISP level:\n$(q_{\\rm ISP}, u_{\\rm ISP}) = (2.53\\%, 2.68\\%)$, or ISP$_{\\rm max} = 3.60\\%$,\n$\\theta_{\\rm ISP} = 113\\ensuremath{^{\\circ}}$, for $\\lambda_{\\rm max} = 6500$~\\AA, the\nwavelength of maximum ISP that yields the most convincing Serkowski-law fit to\nthe observed polarization (Fig.~\\ref{fig:5}d).  If this truly is the ISP, then\nit implies an extraordinarily high polarization efficiency for the dust along\nthe l-o-s to SN~1997dt: ${\\rm ISP} \/ E_{B-V} \\approx 18\\%\\ {\\rm mag}^{-1}$,\nwhich is double the observed Galactic limit of $9\\%\\ {\\rm mag}^{-1}$.  Of\ncourse, some of this discrepancy could be removed if the true reddening were\ngreater than we suspect.  However, even allowing the reddening to equal the\nupper limit of $E\\BminusV_{\\rm total} < 0.27$ mag set in \\S~\\ref{sec:4.3.2} still\nrequires the dust-polarization efficiency to exceed the Galactic limit.\n\nIt thus appears that we face a stark choice: either SN~1997dt has the highest\nintrinsic polarization of any SN~Ia yet observed, or the dust along the l-o-s\nhas an exceptionally high polarization efficiency.  The epoch of our\nobservation of SN~1997dt is unique for a subluminous SN~Ia, so there is no\nempirical database from which to draw expectations and help decide between\nthese two options.\n\nThe situation for SN~1997dt, while perplexing, in fact is not unique: a similar\npalette of possibilities presented themselves in our previous study of a single\nepoch of spectropolarimetry of SN~1999gi, an SN~II-P that also had a low\nreddening and large observed polarization \\citep{Leonard4,Leonard6}.  Like\nSNe~Ia, SNe~II-P as a group have historically shown very low intrinsic\ncontinuum polarization.  After considering many polarization production\nmechanisms, including polarization due to newly formed dust in the SN ejecta\nand dust reflection by one or more off-center dust blobs external to the SN, we\nconcluded that the most likely explanation for the polarization of SN~1999gi\nwas that the host-galaxy dust along the l-o-s possesses a very high\npolarization efficiency, ${\\rm ISP}\/E\\bminusv = 31^{+22}_{-9}\\% {\\rm\\ mag}^{-1}$,\nwhich remains the largest value yet inferred for a single sight line in either\nthe MW or an external galaxy \\citep{Leonard7}.\n\nThere are arguments favoring a similar explanation here.  First, a Serkowski\nlaw reasonably fits the observed continuum polarization of SN~1997dt\n(Fig.~\\ref{fig:5}d).  Further, at 21 days past maximum, SN~1997dt may be\nnearing the end of its photospheric phase, a time when spectropolarimetry may\nbe losing its efficacy as an asymmetry probe due to a lack of electrons\navailable to scatter the light.  If we believe this to be the case, then we\nshould not expect large polarization, even if the SN is highly aspherical.  If\nwe accept the large ISP, then we naturally would like to know whether it is due\nto dust in the MW or NGC~7448.  Unfortunately, we have not observed any distant\nGalactic ``probe'' stars (\\S~\\ref{sec:2.3.2}) near to the l-o-s to estimate the\nGalactic ISP.  There are, however, reasons to suspect that it is low.  First,\n20 stars within $10\\ensuremath{^{\\circ}}$ of the l-o-s are listed in the agglomeration of\nstellar polarization catalogs by \\citet{Heiles00}, and the greatest observed\npolarization is only $0.3\\%$.  Second, the great majority of the reddening is\ndue to host-galaxy dust, since $E\\BminusV_{\\rm Host} = 0.15$ mag while $E\\BminusV_{\\rm\nMW} = 0.06$ mag (\\S~\\ref{sec:4.3.2}).  As was the case with SN~1999gi, we would\nagain conclude that it is the dust within the host galaxy that must have the\nextraordinarily high polarization efficiency.\n\nThere are arguments to oppose this, however.  Previous SN~Ia polarization\nstudies have found intrinsic polarizations increasing toward red wavelengths\n(\\S~\\ref{sec:2.3.3}), which could certainly mimic a Serkowski law over the\nlimited wavelength band covered by our observations.  Further, the line\npolarization features demonstrate that the SN must possess at least some\nintrinsic polarization.  Finally, the relatively late phase of the observation,\ninvoked previously to argue {\\it against} high intrinsic polarization, can also\nbe used to argue {\\it in favor} of it: at the stage immediately before an SN\nbegins the transition to the nebular phase, the deepest layers of the ejecta\nare revealed.  Although in need of confirmation by detailed modeling, for this\nepoch one can plausibly argue that an optical photosphere still exists with\nsufficient optical depth to electron scattering, perhaps even reaching the\nsingle-scattering limit, which is the most polarizing atmosphere\n\\citep{Hoflich91}.  If the explosion mechanism itself is asymmetric, the\nlargest imprint of the asymmetry would presumably be in the innermost layers,\nwhich could lead to a very large intrinsic polarization that reveals itself\njust at this late phase. \n\nIn fact, such an effect is seen in the\nspectropolarimetry of the SN~II-P 2004dj as it transitions to the nebular phase\n(D. C. Leonard et al., in preparation). However, in the case of SN~2004dj, the\nstrong increase is observed primarily in the ``continuum'' region\n6800--7800~\\AA\\ (and in a few strong line troughs), not in the overall level\nacross the whole spectrum.  Although in need of confirmation by detailed\nmodeling, depolarizing line blanketing may also be significant for SNe~Ia at\nblue wavelengths at this epoch, which would again argue for a large ISP as the\nexplanation of the high observed polarization.  One possibility that\ncircumvents this difficulty is that asymmetrically distributed concentrations\nof radioactive Ni, recently uncovered in the thinning ejecta at this relatively\nlate epoch, are responsible for the large polarization.\n\nCuriously, the single, high-polarization observation of SN~1999gi\noccurred at a similar stage of its evolution, right at the end of the\noptical plateau that characterizes the photospheric phase in SNe~II-P.\nIt is thus unfortunate that no other spectropolarimetric epochs were\nobtained for either event (SN 1999gi and SN 1997dt) to serve as a\nbasis for comparison.  Multi-epoch data covering the transition from\nthe photospheric to the nebular phases of SNe of all types will\ncertainly help reveal more definitively the physical explanation for\nsuch high observed polarizations at these late times.\n\nIt may be possible to gain insight into the cause of the observed polarization\nfrom examination of the spectropolarimetry data in the $q$--$u$ plane, shown in\nFigure~\\ref{fig:16}.  ISP originating from a single source (i.e., characterized\nby a single PA) will spread intrinsically unpolarized data points along a line\nin a direction that intersects the origin in the $q$--$u$ plane.  Finding that\ndata lie predominantly along such a line, and exhibit a Serkowski law spectral\nshape, provides compelling evidence that a large, single source of ISP\ndominates the observed signal.  Such was the case for SN~II-P 1999gi\n\\citep{Leonard7}.  Similarly, for SN~1997dt, an elongation of the data points\nfrom blue to red wavelengths in a direction that roughly points back toward the\norigin also exists. \n\nWhen the ISP derived earlier (ISP$_{\\rm max} = 3.60\\%$, $\\theta_{\\rm ISP} =\n113\\ensuremath{^{\\circ}}$, for $\\lambda_{\\rm max} = 6500$~\\AA) is removed, the wavelength\ndependence of the polarization largely disappears (Fig.~\\ref{fig:16}),\nstrengthening the argument for a single, dominant ISP source.  The data are\nthen seen to lie along an axis with a fairly well-defined PA of $58\\ensuremath{^{\\circ}}$\n(Fig.~\\ref{fig:16}), against which we calculate the RSP and URSP shown in\nFigure~\\ref{fig:17}.  Although of individually low to moderate significance,\nthe polarization features in the \\ion{Fe}{2} $\\lambda 4555$ and \\ion{Si}{2}\n$\\lambda 6355$ troughs, seen in both RSP and URSP, do seem to prefer the same\ngeneral direction, with most of the modulation occurring along the URSP axis\n(e.g., the direction perpendicular to the main axis in the $q$--$u$ plane).\nTaken at face value, consistent line-trough polarization changes in a direction\ndiffering from that preferred by the continuum favors an origin in the\nselective blocking of photospheric light by clumpy and asymmetrically\ndistributed intermediate and iron-peak elements overlying the photosphere as\nopposed to an ellipsoidal scenario.  \n\nOther plausible ISP choices, however, yield different conclusions.  For\ninstance, a smaller ISP of $2.6\\%$ at the same PA yields a more nearly\nspherical constellation of points centered near $(q, u) = (-0.7\\%, -0.07\\%)$,\nwith the line excursions now pointing back toward the origin.  Since some\nmodelers predict that, in SNe~Ia with an ellipsoidal asphericity, polarization\n{\\it decreases} may exist in absorption troughs (\\S~\\ref{sec:2.3.1}), the\nellipsoidal model cannot therefore be ruled out in this case as the cause of\nthe polarization.  Given the marginal significance of the weaker features,\nadditional interpretation of the line-trough polarization degrees and\ndirections is probably not warranted with these data.\n\nIn conclusion, we find evidence for a large ISP contribution to the observed\npolarization of SN~1997dt, probably in the range $2.6\\% \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} {\\rm ISP}\n\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 3.6\\%$.  This implies a polarization efficiency for the dust along the\nl-o-s in NGC~7448 that exceeds the Galactic limit.  We do, however, also find\nevidence for polarization intrinsic to the object, most convincingly in\nspecific line features and, perhaps, in the continuum as well.  A combination\nof asymmetrically distributed radioactive Ni and synthesized IMEs overlying the\nphotosphere may provide the simplest explanation for the line and potential\ncontinuum polarization at this late phase, although the ellipsoidal model \ncannot be definitively ruled out.\n\n\\subsubsection{SN 2003du}\n\\label{sec:4.4.3}\n\nOur spectropolarimetry of SN~2003du represents the highest S\/N data of our\nstudy, and permits a more detailed analysis of line features than has been\npossible for the other objects.  SN~2003du presents a low level of observed\npolarization across most of the spectrum, with $p_V = 0.04\\%$, $\\theta_V =\n17^\\circ$ (Fig.~\\ref{fig:7}).  The polarization exhibits an increasing trend\nwith wavelength, rising from nearly zero at blue wavelengths to $\\sim$0.2\\% at\nthe red edge of the spectrum.  There are distinct and significant polarization \nchanges across several absorption features, including the \\ion{Ca}{2} near-IR \ntriplet, the \\ion{Si}{2} $\\lambda 6355$ line, probably a few weaker lines such \nas \\ion{Fe}{2} $\\lambda 4924$, and very tentatively the \\ion{Ca}{2} H \\& K\nabsorption.\n\nApplying the technique of \\citet{Wang05}, and choosing the regions indicated in\nFigure~\\ref{fig:7}a to estimate the ISP, yields $(q_{\\rm ISP}, u_{\\rm ISP}) =\n(-0.02\\%, 0.0\\%)$, or ISP$_{\\rm max} = 0.02\\%$, $\\theta_{\\rm ISP} = 90\\ensuremath{^{\\circ}}$,\nfor an assumed $\\lambda_{\\rm max} = 5500$~\\AA.  This very low ISP is\nconsistent with the negligible reddening found earlier.  Furthermore, on the\nsame night the SN~2003du data were taken, we observed the\ndistant Galactic star BD~$+50^\\circ1593$ (spectral type F8, $V = 10.64$ mag),\nlocated just $0.28^\\circ$ from the l-o-s of SN~2003du, and found it to be null\nto within $0.1\\%$.  From its spectroscopic parallax, we estimate BD~$+50^\\circ1593$\nto be at least $190$ pc away which, at a Galactic latitude of $53^\\circ$,\nsatisfies the criterion of \\citet{Tran95} that a good MW ``probe'' star be more\nthan 150 pc from the Galactic plane.  We thus have multiple reasons to suspect\nlittle ISP contaminating the data.\n\nAfter removal of the minimal ISP, a well-defined axis with ${\\rm PA\\ } =\n107\\ensuremath{^{\\circ}}$ is derived (Fig.~\\ref{fig:18}), about which we calculate the\nintrinsic RSP and URSP, shown in Figure~\\ref{fig:19}.  Compared with what was\nseen in the HV~SNe~Ia, and even SN~1997dt, the line-polarization features are\nnot large, amounting to no more than $0.3\\%$.  However, the very high S\/N of\nthese data makes the detections unequivocal, and establishes intrinsic\npolarization in a spectroscopically and photometrically normal SN~Ia at the\nlatest phase yet observed.\n\nThe behavior of the line-trough polarization for the \\ion{Ca}{2} near-IR\ntriplet and \\ion{Si}{2} $\\lambda 6355$ are very similar.  Both show sharp\nincreases of $\\sim$0.2\\% in RSP, as well as overall increases in URSP.  Modest\nRSP increases of $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 0.1\\%$ may also be discerned in the \\ion{Fe}{2}\n$\\lambda 4924$ and \\ion{Ca}{2} H \\& K absorptions.  At this phase, the\n\\ion{O}{1} $\\lambda 7774$ feature in SNe~Ia is quite weak and, coupled with\ntelluric A-band contamination, makes definitive detection of polarization\nmodulation in this important region difficult; the observed changes are at\nabout the level of the statistical noise.\n\nThe commonality between the polarization behavior of the Si and Ca lines argues\nfor similar origins. Given the high S\/N of the data, we can examine rather\nclosely the basic predictions of the ellipsoidal model that (a) the\npolarization angle should be independent of wavelength, and (b) the overall\npolarization should increase from blue to red wavelengths, with an expectation\nof $p \\rightarrow 0\\%$ at $\\lambda \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 5000$ \\AA\\ (if the line opacity\nremains strong in these regions at the epoch of observation;\n\\S~\\ref{sec:2.3.1}).  \n\nWith our initial choice of ISP, the first criterion is clearly not satisfied,\nas there are obvious excursions in URSP throughout the spectrum, with especially\nsharp modulations occurring across the strongest lines.  In fact, for ISP\nvalues constrained to lie along the symmetry axis we are unable to find any\nvalue that convincingly satisfies both criteria of the ellipsoidal model:\nvalues in the upper-right quadrant (e.g., $[q_{\\rm ISP}, u_{\\rm ISP}] = [0.3\\%,\n0.18\\%]$) produce a polarization {\\it decrease} with wavelength across the\nspectrum, while ISP values in the lower-left quadrant that satisfy the\ncriterion of $p \\rightarrow 0\\%$ at blue wavelengths (e.g., $[q_{\\rm ISP},\nu_{\\rm ISP}] = [-0.1\\%, -0.07\\%]$) result in strong PA changes across the\nspectral lines.  Choosing an ISP point far beyond the constellation of data\npoints, which may serve to make the PA changes less objectionable (although\nstill statistically significant), has the unfortunate consequence of straining\nthe limits implied from the very low reddening, at least for dust with normal\npolarizing efficiency.  \n\nIn addition, examining the URSP behavior of the \\ion{Si}{2} $\\lambda 6355$ and\n\\ion{Ca}{2} near-IR lines in Figure~\\ref{fig:19} more closely, we see that the\ngenerally increasing trends in URSP across the lines show sharp decreases right\nat the locations of peak RSP modulation, although the statistical significance\nof the modulations, especially for \\ion{Ca}{2}, is not high.  If the abrupt\nchanges in URSP are real, such structure is readily explained under the\nclumpy-ejecta model by variations in the distribution of the IMEs as a function\nof radius in the expanding ejecta.  \\citet{Chugai92} also demonstrates that\nsuch sharp changes in line features can be produced by excitation asymmetry.\nOn the other hand, there is no obvious mechanism to produce such an effect in\nthe simple ellipsoidal models.  We thus conclude that it is difficult to\nreconcile the basic predictions of the ellipsoidal model with the data for\nSN~2003du.\n\nWe therefore suspect either clumps in the ejecta overlying the photosphere or\nionization asymmetry as the cause of the inferred line and, perhaps, continuum\npolarization, and are led again to disfavor the ellipsoidal model as the cause\nof the intrinsic polarization of this SN.\n\n\n\\section{Conclusions}\n\\label{sec:5}\n\nWe present post-maximum single-epoch spectropolarimetry of four SNe~Ia,\nbringing to six the number of SNe~Ia thus far examined in detail with\nspectropolarimetry during the early phases.  The four objects span a range of\nspectral and photometric properties, yet all are demonstrated to be\nintrinsically polarized.  This suggests that asphericity and\/or asymmetry may\nbe a ubiquitous characteristic of SNe~Ia in the first weeks after maximum\nlight.  The nature and degree of the polarization varies considerably within\nthe sample, but in a way that is consistent with, and extends, previously\nsuspected trends.  Our main spectropolarimetry results are as follows:\n\n\\begin{enumerate}\n\n\\item SN~2002bf and SN~2004dt, both HV~SNe~Ia observed shortly after maximum\nbrightness, exhibit the largest polarization features yet seen definitively for\nany subtype of SN~Ia, with modulations of up to $\\sim$2\\% in the troughs of the\nstrongest lines.  The overall polarization level of both objects is minimal,\nat least at blue wavelengths; there is a possible trend of increasing\npolarization with wavelength for both objects, though neither shows significant\npolarization in the ``continuum'' region 6800--7800~\\AA.  The\nISP contamination is not thought to be large in either object.\n\n\\item SN~1997dt, believed to be a somewhat subluminous event, has the highest\nobserved overall polarization of any SN~Ia yet studied, $p_V = 3.46\\%$, at 21\ndays past maximum light.  This demands either an extraordinarily large\npolarization efficiency for the dust along the l-o-s in NGC~7448, the largest\nintrinsic SN~Ia polarization thus far found, or perhaps some combination of the\ntwo.  The observed polarization rises by about $0.5\\%$ from blue ($\\lambda =\n4300$~\\AA) to red ($\\lambda = 6700$~\\AA) wavelengths, approximating a\nSerkowski-law ISP curve rather convincingly, albeit one with a somewhat unusual\npeak wavelength ($\\lambda_{\\rm max} \\approx 6500$~\\AA).  A polarization\nmodulation of nearly $1\\%$ in the strong \\ion{Fe}{2} $\\lambda 4555$ absorption,\nand a more modest change of $\\sim 0.3\\%$ in the \\ion{Si}{2} $\\lambda 6355$\nline, demonstrate that the SN does possess intrinsic polarization features.\nHowever, we conclude that ISP is responsible for the bulk of the overall\npolarization that is observed, with $2.6\\% \\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} {\\rm ISP}\n\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 3.6\\%$, and that the polarization efficiency of the dust along the\nl-o-s in NGC~7448 likely exceeds the empirical Galactic limit.\n\n\\item SN~2003du is a slightly overluminous SN~Ia.  Our spectropolarimetry,\n  taken 18 days after maximum light, is the highest S\/N data obtained for our\n  sample of objects.  It reveals a low continuum polarization that increases by\n  $\\sim$0.3\\% from blue to red wavelengths, with distinct changes of $\\sim$0.2\\% \n  detected in the \\ion{Si}{2} $\\lambda 6355$ and \\ion{Ca}{2} near-IR\n  triplet lines; smaller changes may be detected in weaker lines.  The very\n  similar behavior of the polarization in the two strongest lines, in both\n  magnitude and direction in the $q$--$u$ plane, suggests a common polarization\n  origin.  ISP is thought to be minimal.\n\n\n\\end{enumerate}\n\nOrdered by increasing strength of line-polarization features in SNe~Ia, we find\nas follows: ordinary\/overluminous $<$ subluminous $<$ HV~SNe~Ia, with the\nstrength of the line-polarization features increasing from $0.2\\%$ in the\nslightly overluminous SN~2003du to $2\\%$ in both HV~SNe~Ia in our study.\nAbsolute continuum polarization levels are more difficult to establish, due\nlargely to uncertainties in the ISP, but there are compelling reasons to\nbelieve that at least three of our objects possess very little intrinsic\npolarization in spectral regions outside of specific line features.  The\n\\citet{Howell01} study of SN~1999by and our data on SN~1997dt provide some\nevidence that continuum polarization may be higher in subluminous objects than\nin other types.\n\nThere are a number of alternatives from which to choose for the origin of\npolarization in SNe~Ia, including global asphericity (e.g., the ellipsoidal\nmodel), ionization asymmetry, and clumps in the ejecta overlying the\nphotosphere.  The small, redward rise in the {\\it overall} polarization level\nthat is discerned in at least three of our objects can be reproduced by models\npossessing either global asphericity or an ionization asymmetry.  Under the\nellipsoidal model, the levels of continuum polarization for SN~2002bf,\nSN~2003du, and SN~2004dt imply minor to major axis ratios of around $0.9$ if\nviewed equator-on \\citep{Hoflich91,Wang03}.  This level of asphericity would\nproduce a luminosity dispersion of about 0.1 mag for random viewing\norientations \\citep{Hoflich91}, which could explain some of the dispersion seen\nin the brightness-decline relation of SNe~Ia.  If the proposed global\nasphericity is more complicated, then the luminosity of a single SN~Ia may\ndepend on viewing angle in a non-trivial way such that, even for a large sample\nof objects, an overall bias to slightly higher or lower values may result.\n\nThe potential for significantly greater continuum polarization, perhaps of\n$\\sim$1\\%, in the likely subluminous SN~1997dt observed $\\sim$21 days after\nmaximum would imply a more severe distortion, of at least $20\\%$, from the\nmodels of \\citet{Hoflich91}.  Since most cosmological applications of SNe~Ia\nrely on data acquired closer to maximum light, such a late-time asphericity,\neven if common, would not seriously affect the utility of SNe~Ia as distance\nindicators.\n\nTo explain the ubiquitous line polarization, the simple ellipsoidal model is\neffectively ruled out for three of our objects, including, most convincingly,\nthe two HV events.  From a number of lines of reasoning, the most convincing\nexplanation is partial obscuration of the photosphere by clumps of newly\nsynthesized IMEs forged in the explosion.\n\nFor the HV~SNe~Ia, in particular, the GCD model studied by \\citet{Kasen05}\nprovides a plausible explanation for many of the observed spectral and\nspectropolarimetric characteristics.  It predicts the existence of an optically\nthick pancake of material with significant radial extent that partially\nobscures the optical photosphere, producing larger line velocities and\nequivalent widths for many spectral features, and stronger line-trough\npolarization than is seen in more typical events.  These qualitative\nexpectations are borne out by our data: The line features of SN~2002bf and\nSN~2004dt possess the strongest polarization modulations and greatest\nequivalent widths of our sample.  The astonishing similarity between our epoch\nof spectropolarimetry of SN~2004dt, $\\sim$4 days after maximum, and that\npresented by \\citet{Wang05} from 11 days earlier, provides compelling evidence\nthat the obscuring material also possesses great radial extent in the thinning\nejecta.\n\nThat SNe~Ia may be separable into different groups based on their {\\it\nspectropolarimetric} characteristics yields one more clue to assist in\nnarrowing down progenitor possibilities and\/or models for the physics of the\nexplosion.  The assertion that some fraction of the IMEs in the ejecta of\nSNe~Ia may be confined to bubbles or filaments is, however, a rather blunt\ndiscriminatory tool: At the present stage of theoretical modeling,\ndeflagration, delayed-detonation, off-center delayed-detonation, and GCD models\ncan all plausibly be argued to produce clumps in the ejecta \\citep[e.g.,][and\nreferences therein]{Wang05}.  The specific predictions of the GCD model in\nparticular need to be further examined, preferably in full three-dimensional\nsimulations, to test whether the explosion mechanism itself is viable, and, if\nso, whether it can quantitatively reproduce the observed characteristics of at\nleast some SNe~Ia in detail.  On the observational front, higher S\/N data,\npreferably obtained at multiple epochs, will help to narrow down the\npossibilities as well.  With the steady advances being made in the theoretical\nunderstanding of these events, and the growing rate of SNe~Ia studied in detail\nwith spectropolarimetry, prospects for improving our understanding of these\nevents are bright.\n\n\\acknowledgments \n\nWe thank Aaron Barth, Louis-Benoit Desroches, Mohan Ganeshalingam, Deborah\nHutchings, Ed Moran, and Karin Sandstrom for assistance with some of the\nobservations and data reduction, and Saurabh Jha for producing MLCS2k2 fits for\ntwo of our objects. We thank an anonymous referee for useful suggestions that\nresulted in an improved manuscript.  This research has made use of the\nNASA\/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion\nLaboratory, California Institute of Technology, under contract with NASA.  The\nwork of A.V.F.'s group at UC Berkeley is supported by National Science Foundation\n(NSF) grant AST-0307894. D.C.L. is supported by an NSF Astronomy and\nAstrophysics Postdoctoral Fellowship under award AST-0401479.  Additional\nfunding was provided by NASA grants GO-9155, GO-10182, and GO-10272 from the\nSpace Telescope Science Institute, which is operated by the Association of\nUniversities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.\nA.V.F. is grateful for a Miller Research Professorship at UC Berkeley, during\nwhich part of this work was completed.  Some of the data presented herein were\nobtained at the W. M. Keck Observatory, which is operated as a scientific\npartnership among the California Institute of Technology, the University of\nCalifornia, and NASA; the Observatory was made possible by the generous\nfinancial support of the W. M. Keck Foundation.  KAIT was made possible by\ngenerous donations from Sun Microsystems, Inc., the Hewlett-Packard Company,\nAutoScope Corporation, Lick Observatory, the NSF, the\nUniversity of California, and the Sylvia \\& Jim Katzman Foundation. The\nassistance of the staffs at Lick and Keck Observatories is greatly appreciated.\n\n\\clearpage\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{0pt}{18pt plus 4pt minus 2pt}{18pt plus 2pt minus 2pt}\n\tThe intricate interplay between magnetism and nematicity in different families of iron-based superconductors has attracted great interest in the past few years \\cite{Fernandes2014,PDai2015,Si2016}. In iron pnictides, magnetism and nematicity are tightly coupled; the antiferromagnetic transition is always coincidental with, or closely preceded by, a tetragonal-to-orthorhombic structural transition. The proximity of the two transitions can be naturally explained within the spin-nematic scenario, where the structural transition is driven by a vestigial nematic order arising from fluctuations associated with the antiferromagnetic stripe transition (see Fig. \\ref{fig:Fig1}(b))\\cite{CFang2008,Xu08,Fernandes2012}. In iron chalcogenides, the coupling between magnetism and nematicity is less obvious. FeSe undergoes a nematic phase transition without any long-range magnetic order \\cite{McQueen2009,Bohmer2017}, which has been interpreted as evidence that the nematic order in FeSe is of orbital origin \\cite{Baek2015}. Nevertheless, spin stripe fluctuations do develop below the nematic transition \\cite{Wang2016}, and static stripe order can be induced by hydrostatic pressure \\cite{Kothapalli16,Matsuura2017}.\n\t\\begin{figure}\n\t\t\\includegraphics[trim={0 0.1cm 0cm 0.2cm},clip,width=0.5\\textwidth]{Fig1.png}\n\t\t\\caption{(a-b) Schematic spin configurations of the (a) double-stripe phase, with wave-vector $\\mathbf{Q} = (\\pi, 0)$, and (b) single-stripe phase, with $\\mathbf{Q} = (\\pi, \\pi)$. (c-d) Schematic diagrams of the Montgomery method for the elastoresistivity measurement in (c) $B_{1g}$ and (d) $B_{2g}$ configuration. (e-f) The anisotropic resistivity $(\\rho_{xx} - \\rho_{yy})$ as a function of anisotropic strain $(\\epsilon_{xx} - \\epsilon_{yy})$ for Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ ($0 \\leq x \\leq 0.53$) at $T = 20$K in (e) $B_{1g}$ and (f) $B_{2g}$ channels. The $B_{1g}$ elastoresistivity coefficient $m_{11} - m_{12}$ and $B_{2g}$ elastoresistivity coefficient $2m_{66}$ can be obtained by fitting the linear slope of resistivity versus strain. The samples with high doping concentrations ($x = 0.38, 0.45, 0.53$) show predominantly a $B_{2g}$ response while the low doping ones ($x = 0, 0.12$) show comparable $B_{1g}$ and $B_{2g}$ responses.}\n\t\t\\label{fig:Fig1}\n\t\\end{figure}\n\t\\begin{figure*}\n\t\\includegraphics[width=1\\textwidth]{FIG2.eps}\n\t\\caption{Temperature and doping dependence of nematic fluctuations of annealed Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ in the (a) $B_{1g}$ channel, with elastoresistivity coefficient $m_{11} - m_{12}$, and (b) $B_{2g}$ channel, with  $2m_{66}$. For clarity, the elastoresistivity data for each doping are offset by 15 and 20 in (a) and (b), respectively. (c-e) Temperature dependence of (c) $m_{11} - m_{12}$ for $x = 0$, (d) $2m_{66}$ for $x = 0.45$ and (e) $-2m_{66}$ for Ba(Fe$_{0.93}$Co$_{0.07}$)$_{2}$As$_{2}$. Lower panels show the inverse. Solid black curves are Curie-Weiss fits. The optimal fitting range is determined by the greatest corresponding adjusted R-square value. Shaded gray regions indicate the range of temperatures where the elastoresistivity coefficients follow a Curie-Weiss law. }\n\t\\label{fig:fig2}\n\t\\end{figure*}\n\t\\\\\n\t\\indent While there are ongoing debates on the mechanism by which nematicity forms without static magnetism in FeSe \\cite{Glasbrenner2015,Wang2015,Yu2015,Khodas16,Kontani17}, Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ provides another platform to approach this problem. As selenium is replaced by tellurium (i.e. $x$ is changed from $1$ to $0$), the nematic phase transition is suppressed \\cite{Terao2019}, and inelastic neutron scattering experiments revealed a complex evolution of the spin correlations associated with different magnetic patterns \\cite{Lumsden2010,Liu2010,Zaliznyak10316,Xu2018}. In particular, close to optimal doping ($x \\sim 0.5$), the wave-vector of spin fluctuations at low temperatures is $(\\pi, \\pi)$ [in the crystallographic Brillouin zone], identical to the antiferromagnetic order in the iron pnictides. As the tellurium concentration increases, both superconductivity and the $(\\pi, \\pi)$ spin fluctuations disappear. The latter are replaced by short-range magnetic correlations near $(0, \\pi)$ that eventually condense into the static double-stripe phase in Fe$_{1+y}$Te \\cite{Li2009} (Fig. \\ref{fig:Fig1}(a)). Previous elastoresistivity measurements revealed a diverging $B_{2g}$ nematic susceptibility in optimally doped Fe$_{1+y}$Te$_{1-x}$Se$_{x}$, consistent with the existence of $(\\pi, \\pi)$ spin fluctuations \\cite{Kuo958}. This finding suggests that nematic and magnetic fluctuations remain strongly intertwined even in the absence of static nematic and magnetic orders. Nevertheless, in contrast to the magnetic sector, the behavior of nematic fluctuations for doping concentrations beyond optimal is still poorly characterized. The compositional dependence of the nematic susceptibility in Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ would therefore constitute an important step in the effort to elucidate the relationship between nematicity and magnetism.\n\t\\\\\n\t\\indent Another motivation to study Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ is to understand the influence of orbital selectivity on the nematic instability. Orbital selectivity (or orbital differentiation) refers to the fact that different orbitals are renormalized differently by electronic correlations, a characteristic property of Hund's metals that appears to be much more prominent in the iron chalcogenides in comparison with the pnictides \\cite{Yin2011,Lanata2013,Medici2014}. Experimentally, recent scanning tunneling microscopy (STM) measurements revealed the impact of orbital differentiation on the superconducting state \\cite{Sprau75}. Theoretically, it has been suggested that orbitally selective spin fluctuations may be the origin of nematicity without magnetism in FeSe \\cite{Fanfarillo2018}. Nematic order was also proposed to enhance orbital selectivity by breaking the orbital degeneracy, leading to asymmetric effective masses in different $d$-orbitals \\cite{Yu2018}. The effect of orbital differentiation becomes even more extreme as selenium is replaced by tellurium. In Fe$_{1+y}$Te$_{1-x}$Se$_{x}$, angle resolved photoemission spectroscopy (ARPES) revealed a strong loss of spectral weight of the $d_{xy}$ orbital at high temperatures, which was interpreted in terms of proximity to an orbital-selective Mott transition \\cite{Yi2015}. Similar drastic changes were also observed as a function of doping \\cite{Liu2015}, mimicking the evolution of spin fluctuations. Nevertheless, the impact of orbital incoherence on nematicity remains little explored \\cite{Bascones17}.\n\t\\\\\n\t\\indent In this report, we present systematic measurements of both the $B_{1g}$ and $B_{2g}$ nematic susceptibilities of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ ($0 \\leq x \\leq 0.53$) using the elastoresistivity technique. We demonstrate that the doping dependence of the two nematic susceptibilities closely track the evolution of the corresponding spin fluctuations. In particular, a diverging $B_{1g}$ nematic susceptibility is observed in the parent compound Fe$_{1+y}$Te, suggesting that the spin-nematic paradigm also applies to the double-stripe AFM order \\cite{Zhang2017, Moreo17, Borisov2019}. A diverging $B_{2g}$ nematic susceptibility is observed over a wide range of doping ($0.17 \\leq x \\leq 0.53$), and its magnitude is strongly enhanced by both Se doping and annealing. In addition, the temperature dependence of the $B_{2g}$ nematic susceptibility shows significant deviation from Curie-Weiss behavior above 50K. This is in sharp contrast to the iron pnictides, where the Curie-Weiss temperature dependence extends all the way to 200K. This unusual temperature dependence is captured by a theoretical calculation that includes the loss of spectral weight of the $d_{xy}$ orbital, revealing its importance for $B_{2g}$ nematic instability.\n\t\\\\\n\t\\indent Single crystals of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ were grown by the modified Bridgeman method. The electrical, magnetic and superconducting properties of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ are known to sensitively depend on $y$, the amount of excess iron. To study these effects, crystals were annealed in selenium vapor to reduce the amount of excess iron. By symmetry, the $B_{1g}$ and $B_{2g}$ nematic susceptibilities are proportional to the elastoresistivity coefficients $m_{11} - m_{12}$ and $2m_{66}$, respectively. We performed the elastoresistivity measurements in the Montgomery geometry, which enables simultaneous determination of the full resistivity tensor, hence the precise decomposition into different symmetry channels, as illustrated in Fig. \\ref{fig:Fig1}(c) and (d). Details of the Montgomery elastoresistivity measurements can be found elsewhere \\cite{Kuo958}. The crystal orientation is determined by polarization resolved Raman spectroscopy. Representative data of anisotropic resistivity as a function of anisotropic strain at 20K in $B_{1g}$ and $B_{2g}$ channels are shown in Fig. \\ref{fig:Fig1}(e) and (f). The $B_{1g}$ elastoresistivity coefficient $m_{11} - m_{12}$ and the $B_{2g}$ one,  $2m_{66}$, can be obtained by fitting the linear slope of resistivity versus strain. Samples with high doping concentrations ($x = 0.38, 0.45, 0.53$) show predominantly a $B_{2g}$ response while the low doping ones ($x = 0, 0.12$) show comparable $B_{1g}$ and $B_{2g}$ responses.\n\t\\\\\n\t\\indent Figs. \\ref{fig:fig2}(a) and (b) show the temperature dependence of $m_{11} - m_{12}$ and  $2m_{66}$ of annealed Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ for $0 \\leq x \\leq 0.53$. For $0.28 \\leq x \\leq 0.53$, $2m_{66}$ shows a strong temperature dependence that grows continuously as temperature decreases. For $x = 0.45$, $2m_{66}$ reaches a value of $\\sim$ 100, comparable to optimally doped pnictides. While preserving a similar diverging temperature dependence, the maximum absolute value of $2m_{66}$ decreases rapidly as selenium concentration decreases, from 100 for $x = 0.45$ to 8 for $x = 0.17$. On the other hand, $m_{11} - m_{12}$ shows a diverging response when $x$ is below $0.17$, which is in the vicinity of the double-stripe AFM order. As selenium concentration increases, $m_{11} - m_{12}$ evolves to a temperature independent response, with small kinks at low temperatures likely coming from contamination of $2m_{66}$ due to misalignment. Overall, our observation of the doping dependence of $2m_{66}$ and $m_{11} - m_{12}$ is consistent with the evolution of low-temperature spin fluctuations from predominantly $(\\pi,0)$ at small $x$ to predominantly $(\\pi,\\pi)$ at optimal doping $x\\sim 0.5$ \\cite{Lumsden2010,Liu2010,Zaliznyak10316,Xu2018}.\n\t\\\\\n\t\\indent To gain more insight, we fit the $2m_{66}$ and $m_{11} - m_{12}$ to a Curie-Weiss temperature dependence:\n\t\\begin{eqnarray}\n\tm=m^{0}+\\frac{\\lambda}{a(T-T^{*})}\n\t\\end{eqnarray}\n\tFor the parent compound Fe$_{1+y}$Te, $m_{11} - m_{12}$ can be well fitted to a Curie-Weiss behavior in the temperature range just above the double stripe AFM ordering temperature $T_{\\rm mag}$ = 71.5K (Fig. \\ref{fig:fig2}(c)). The fitted Curie-Weiss temperature $T^{*}$ is slightly smaller than $T_{\\rm mag}$. Despite the smaller absolute value ($\\sim$ 10 at maximum), the behavior of $m_{11} - m_{12}$ is reminiscent of the $2m_{66}$ in the parent phase of iron pnictides, suggesting that the spin-nematic mechanism is still at play here, in agreement with theoretical expectations \\cite{Zhang2017, Moreo17, Borisov2019}. \n\t\\\\\n\t\\begin{figure}\n\t\t\\includegraphics[trim={0 1cm 0 0cm},clip,width=0.48\\textwidth]{FIG3.eps}\n\t\t\\caption{Comparison with the Hall coefficient $R_{H}$. (a) Doping dependence of the absolute value of the $B_{2g}$ elastoresistivity coefficient $2m_{66}$ (red squares) and of $R_{H}$ (black diamonds) at 16K. Dashed lines are guide to the eyes. (b) Colormap of the negative Hall coefficient -$R_{H}$, (c) of $m_{11} - m_{12}$ (d) and of $|2m_{66}|$ as a function of temperature and doping. The double-spin stripe and the superconducting transition temperatures are denoted as blue squares and yellow triangles, respectively.}\n\t\t\\label{fig:fig3}\n\t\\end{figure}\n\t\\indent Fig. \\ref{fig:fig2}(d) shows the Curie-Weiss fitting of $2m_{66}$ for the $x = 0.45$ sample. The fitting of $2m_{66}$ only works at low temperatures, as can be seen in the linear temperature dependence of $|2m_{66} -2m_{66}^{0}|^{-1}$ below 50K. It shows a significant deviation for temperature greater than 50K. The $T^{*}$ obtained from the low-temperature fitting is close to 0K.  Intriguingly, the $T^{*}$ extracted from the Curie-Weiss fitting is approximately zero for all $0.17 \\leq x \\leq 0.53$, while the Curie constant $\\lambda\/a$ decreases with $x$ (SOM). While the number of doping concentrations studied in the current work is insufficient to support a power law analysis, $2m_{66}$ at constant $T = 16$K appears to be diverging as $x$ increases from 0.17 to 0.45 (Fig. \\ref{fig:fig3}(a)). Both the doping dependence and the near zero $T^{*}$ are consistent with the existence of a putative nematic quantum critical point at $x \\sim 0.5$. Interestingly, recent work doping FeSe with Te suggests that the $90$K nematic transition of FeSe is continuously suppressed and extrapolates to $0$ at $x \\sim 0.5$ \\cite{Terao2019}.\n\t\\\\\n\t\\indent This deviation from Curie-Weiss at high temperatures is very unusual. In the iron pnicitides, such a deviation was only observed at low temperatures in transition-metal doped BaFe$_{2}$As$_{2}$ (Fig. \\ref{fig:fig2}(e)) and LaFeAsO. This unusual temperature dependence of $2m_{66}$ appears to echo the coherent-incoherent crossover observed by ARPES \\cite{Yi2015}, where the spectral weight of the $d_{xy}$ orbital is strongly suppressed as the temperature increases or as the selenium concentration decreases. To further confirm this correlation, we measured the Hall coefficient $R_H$, which has been demonstrated to be a good indicator of this incoherent-to-coherent crossover \\cite{Ding2014,Otsuka2019,Liu2015}. The recovery of the $d_{xy}$ spectral weight is generally correlated with a sign-change of $R_H$ \\cite{Otsuka2019} from positive to negative. Fig. \\ref{fig:fig3}(a) shows the low-temperature $R_H$ and $2m_{66}$ as a function of doping, whereas Fig. \\ref{fig:fig3}(b-d) contain the full temperature and doping dependence of $R_H$,  $m_{11} - m_{12}$, and $|2m_{66}|$, respectively. These plots reveal the strong correlation between a negative $R_H$ and an enhancement of $2m_{66}$.\n\t\\\\\n\t\t\\begin{figure}\n\t\t\\includegraphics[width=0.38\\textwidth]{fig4.eps}\n\t\t\\caption{The effect of annealing on the nematic susceptibility of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$. (a-c) Temperature dependence of (a) normalized in-plane resistivity ($R\/R_{300 \\rm{K}}$), (b) Hall coefficient $\tR_{H}$ and (c) elastoresistivity coefficient $2m_{66}$ of as-grown (black) and annealed (red) samples for $x = 0.45$. The vertical grey line marks the temperature below which the behavior of annealed and as-grown samples starts to deviate from each other. Inset of (a) shows the temperature dependence of the zero field cooling (ZFC) magnetic susceptibility measured at 100Oe (H $\\parallel$ ab). The superconducting volume fraction is significantly enhanced for the annealed sample. }\n\t\t\\label{fig:Fig4}\n\t\\end{figure}\n\t\\indent The properties of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ also depend on the amount of excess iron, which can only be removed by annealing \\cite{Sun_2019}. Taking $x = 0.45$ as an example, the resistance of the annealed sample is metallic for temperatures below 150K (Fig. \\ref{fig:Fig4}(a)). As Fig. \\ref{fig:Fig4}(b) shows, at around 40K the Hall coefficient of the annealed sample turns from positive to negative, which is a signature of incoherent to coherent crossover. In contrast, the resistance of the as-grown sample shows a weakly insulating upturn at low temperatures (Fig. \\ref{fig:Fig4}(a) black dashed curve), and the Hall coefficient remains positive at all temperatures (Fig. \\ref{fig:Fig4}(b) black circles), indicating that the $d_{xy}$ orbital is still incoherent at low temperatures. Interestingly, at the same temperature where the resistance and the Hall coefficient of the as-grown and annealed samples depart from each other, the elastoresistivity coefficient $2m_{66}$ shows a pronounced enhancement for the annealed sample (Fig. \\ref{fig:Fig4}(c)). Such an enhancement was observed in all annealed samples (SOM), providing further evidence of the correlation between the enhancement of the nematic susceptibility and the coherence of the $d_{xy}$ orbital.\n\t\\\\\n\t\\indent The doping and annealing dependences of $2m_{66}$ presented above strongly suggest that the $B_{2g}$ nematic susceptibility also have an orbitally-selective character.\n\tIndeed, previous theoretical works have highlighted the impact of orbital degrees of freedom on spin-driven nematicity \\cite{Fanfarillo15, Khodas16, Kontani16, Yu2018, Chubukov18, Fanfarillo2018}. Using a slave-spin approach, Ref. \\cite{Bascones17} found a suppression of the orbital-nematic susceptibility due to orbital incoherence. To model our data, we employ the generalized random phase approximation (RPA) of Ref. \\cite{Christensen2016} to compute the spin-driven nematic susceptibility for the five-orbital Hubbard-Kanamori model (details in the SM). For fully coherent orbitals, it was found that the largest contribution to the nematic susceptibility $\\chi_{\\rm nem}$ comes from the $d_{xy}$ orbital. Thus, one expects that $\\chi_{\\rm nem}$ would be suppressed if the $d_{xy}$ orbital were to become less coherent.\n\t\\\\\n\t\t\\begin{figure}\n\t\t\\includegraphics[trim={0 0cm 0cm 0cm},clip,width=0.5\\textwidth]{FIG5.eps}\n\t\t\\caption{Comparison with theoretical calculations. (a-b) Temperature dependence of (a) $|2m_{66}|$ and (b) $|2m_{66} - 2m_{66}^{0}|^{-1}$ of optimally doped Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$ (red circles) and BaFe$_{2}$(As$_{0.66}$P$_{0.34}$)$_{2}$ (black squares). The red and black lines show Curie Weiss fittings. The data for BaFe$_{2}$(As$_{0.66}$P$_{0.34}$)$_{2}$ follows a Curie-Weiss behavior all the way up to $200$K, whereas for Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$, it deviates from Curie-Weiss behavior above $\\sim$ 50K. (c-d) Theoretical calculation of the normalized nematic susceptibility $\\chi_{\\rm nem}$ and its inverse, plotted as a function of the relative temperature with respect to the theoretical nematic transition temperature ($T-T_{\\rm nem}$) for different spectral weight $0.7 \\leq Z_{xy} \\leq 1$.}\n\t\t\\label{fig:fig5}\n\t\\end{figure}\n\t\\indent To verify this scenario, we calculated how $\\chi_{\\rm nem}$ changes upon suppressing the spectral weight $Z_{xy}$ of the $d_{xy}$ orbital. For our purposes, the reduction in $Z_{xy}$ acts phenomenologically as a proxy of the incoherence of this orbital, similarly to \\cite{Sprau75}, but its microscopic origin is not important. Fig. \\ref{fig:fig5}(c)-(d) contrasts the nematic susceptibility for $ 0.7 \\leq Z_{xy} \\leq 1$. We note two main trends arising from the suppression of $d_{xy}$ spectral weight: first, as anticipated, the nematic susceptibility (and the underlying nematic transition temperature, which is non-zero in the model) is reduced (Fig. \\ref{fig:fig5}(c)). Second, its temperature dependence changes from a Curie-Weiss-like behavior over an extended temperature range to a behavior in which the inverse nematic susceptibility quickly saturates and strongly deviates from a linear-in-T dependence already quite close to the nematic transition (Fig. \\ref{fig:fig5}(d)). These behaviors are remarkably similar to those displayed by the elastoresistance data shown in Fig. \\ref{fig:fig5}(a)-(b), with $Z_{xy} = 1$ mimicking the behavior of optimally P-doped BaFe$_{2}$As$_{2}$ and $Z_{xy} < 1$, of optimally doped Fe$_{1+y}$Te$_{1-x}$Se$_x$. Interestingly, the susceptibility associated with $(\\pi, \\pi)$ fluctuations is also suppressed by the decrease in $Z_{xy}$, in qualitative agreement with the neutron scattering experiments \\cite{Xu2012} (for a more detailed discussion, see SM). Of course, since $Z_{xy}$ in our model is an input, and not calculated microscopically, our model is useful to capture tendencies, but not to extract the experimental value of $Z_{xy}$. Furthermore, note that in our calculation $Z_{xy}$ is temperature-independent, while in the experiment it changes with temperature.\n\t\\\\\n\t\\indent In summary, our results reveal the close connection between nematic fluctuations and spin fluctuations in Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ for both $B_{1g}$ and $B_{2g}$ channels. Additionally, the unusual temperature dependence of the $B_{2g}$ nematic susceptibility can be attributed to the coherent-to-incoherent crossover experienced by the $d_{xy}$ orbital, providing direct evidence for the orbital selectivity of the nematic instability. Our work presents Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ as an ideal platform to study the physics of intertwined orders in a strongly correlated Hund's metal.\n\n\\begin{acknowledgments}\nWe thank Ming Yi for fruitful discussion. The work at UW was supported by NSF MRSEC at UW (DMR-1719797). The material synthesis was supported by the Northwest Institute for Materials Physics, Chemistry, and Technology (NW IMPACT) and the Gordon and Betty Moore Foundation's EPiQS Initiative, Grant GBMF6759 to J.-H.C.  J.H.C. acknowledge the support of the David and Lucile Packard Foundation, the Alford P. Sloan Foundation and the State of Washington funded Clean Energy Institute. Theory work (M.H.C. and R.M.F.) was supported by the U. S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Award No. DE-SC0020045. The Raman measurement is partially supported by Department of Energy, Basic Energy Sciences, Materials Sciences and Engineering Division (DE-SC0012509).\n\\end{acknowledgments}\n\n\n\\section{0pt}{18pt plus 4pt minus 2pt}{18pt plus 2pt minus 2pt}\n\t\\section{S1 Polarized Raman spectroscopy on Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$}\n\tPolarized Raman spectroscopy was used to determine the crystal orientation. Representative data of the Raman spectra are shown in FIG. S\\ref{fig:FigS1}. Measurements were made on freshly cleaved surfaces of annealed Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$ single crystals in a backscattering configuration at 20K using a 532nm laser, a 1200 groove mm$^{-1}$ grating spectrometer (Princeton Acton 2500i) and a liquid-nitrogen cooled charge-coupled device (CCD) detector. The laser power was set to 100$\\mu$W and the laser spot was focused on a $\\sim$ 2 $\\times$ 2 $\\mu m^{2}$ surface. The (ab) configuration corresponds to the polarization of the incident (scattered) light along the crystallographic a (b) axis, while the (xy) configuration corresponds to the polarization of the incident (scattered) light along the x (y) direction, which is the direction rotated by 45$^{\\circ}$ from the a (b) axis in the ab plane. The presence and absence of $B_{1g}$ phonon mode at 208 cm$^{-1}$ in the (xy) and (ab)  configuration unambiguously confirm the determination of crystal orientation. A weak $A_{1g}$  phonon mode at 159 cm$^{-1}$ was observed in both (ab) and (xy) configurations due to a leakage of other polarization components as observed elsewhere. \n\t\\begin{figure*}\n\t\\renewcommand{\\figurename}{Fig. S}\n\t\\includegraphics[trim={0 0.1cm 0cm 0.2cm},clip,width=1\\textwidth]{FigS1.eps}\n\t\\caption{Polarized Raman spectra on  Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$ in the (ab) and (xy) configurations at 20K. The $B_{1g}$ mode at 208 cm$^{-1}$ distinguishes the (xy) configuration from the (ab) configuration.}\n\t\\label{fig:FigS1}\n\\end{figure*}\n\\section{S2 Annealing effects of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$}\nAs grown crystals are known to host excess iron which can be removed by annealing in Te or Se vapor. Samples with and without excess Fe show drastically different electrical, magnetic, and superconducting properties. To study these effects, crystals were cleaved into thin slices ($\\sim$ 1 mm), loaded in a crucible with another crucible of an appropriate amount of selenium powder beneath it, sealed in quartz tubes, and annealed at 500$^\\circ$C for a week. Further annealing in a Se vapor causes significant hardship in cleaving sizable samples. The average compositions of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ samples were determined on at least 4 regions of the crystals by the energy-dispersive x-ray (EDXS) method using the Sirion XL30 scanning electron microscope. We only use EDS measurement to determine x (Se\/Te) ratio. For the measurement of y, even though we consistently obtained a smaller value of y after annealing, the determination of the absolute value of y is inconclusive. Similar difficulty to accurately determined y has also been reported previously \\cite{Rinott17}. By comparing the transport data and the position and the width of transition, we estimate the upper limit of y is below 0.06 for x $<$ 0.4 and and 0.02 for x $>$ 0.4. \n\\\\\n\\indent The annealing significantly changed the electrical transport behavior, as shown in FIG. S\\ref{fig:figS3} insets. After annealing, the resistivity changes from a weakly insulating behavior for as-grown crystals (black curves) to a metallic behaivor for annealed samples (red curves). The superconducting transition also becomes sharper with an increased T$_{c}$ after annealing. Fig. S\\ref{fig:figS3} shows the elastoresistivity coefficient in the B$_{2g}$ channel, $2m_{66}$, for both as-grown and annealed Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ (x = 0.17 - 0.45). The elastoresistivity coefficient $2m_{66}$ is enhanced for the annealed samples (red squares), suggesting the correlation between the enhanced nematic susceptibility and the coherence of the $d_{xy}$ orbital as discussed in the main text. We notice that the $2m_{66}$ changes sign as x decreases, and the x = 0.28 even shows a sign changing before and after annealing. This behavior might be related to the sensitivity of the sign of resistivity anisotropy to the shape of Fermi surfaces and spin fluctuations, which warrants future investigation.\n\\begin{figure*}\n\t\\includegraphics[trim={0 0cm 0 0.2cm},clip,width=1\\textwidth]{FigS3.eps}\n\t\\renewcommand{\\figurename}{Fig. S}\n\t\\caption{Temperature dependence of $2m_{66}$ for as-grown (black squares) and annealed (red squares) (a) Fe$_{1+y}$Te$_{0.83}$Se$_{0.17}$ (b) Fe$_{1+y}$Te$_{0.72}$Se$_{0.28}$ (c) Fe$_{1+y}$Te$_{0.62}$Se$_{0.38}$ and (d) Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$. The black and red lines underneath the square data show the Curie Weiss fittings for the $2m_{66}$ of as-grown and annealed samples respectively. The fitting parameters are listed in Table S\\ref{tab:tabs2}. For as-grown crystals, $2m_{66}$ for x = 0.17 and 0.28 are negative. After annealing, $2m_{66}$ for x = 0.28 flips sign from negative to positive and the $2m_{66}$ for x = 0.17 remains negative. Insets of (a)-(d) are the temperature dependences of normalized resistances for as-grown (black) and annealed (red) Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ (x = 0.17 0.28 0.38 0.45) respectively.}\n\t\\label{fig:figS3}\n\\end{figure*}\n\\section{S3 Elastoresistivity measurement}\n\\begin{figure}\n\t\\includegraphics[trim={0 0cm 0cm 0cm},clip,width=0.45\\textwidth]{FIGS4.png}\n\t\\renewcommand{\\figurename}{Fig. S}\n\t\\caption{Temperature dependence of (a) the isotropic ($A_{1g}$) elastoresistivity response to the tensile strain and (b) the anisotropic ($B_{2g}$) elastoresistivity response to the anisotropic strain, of a single crystal of Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$ prepared along the Fe-Fe bonding direction. The zero-strain point is determined by the intersection of the strain dependence of $\\rho_{xx}$ and $\\rho_{yy}$ at each temperature. }\n\t\\label{fig:FigS4}\n\\end{figure}\n\\indent The elastoresistivity measurement is conducted by gluing the square shape samples on a piezoelectric stack with the Montgomery contact geometry. Square edge of the samples are cut along the Fe-Fe (Fe-Ch) bonding direction for $B_{2g}$ ($B_{1g}$) elastoresistivity measurements, with electrical contacts at the four corners of the square. With this method, we are able to measure the resistivity along two perpendicular directions on the same piece of sample, and therefore eliminate the problems caused by variations of the samples and the transmitted strain. By gluing the sample on the sidewall of the piezoelectric stack and applying voltages, we apply an in situ tunable anisotropic strain on the sample. The magnitude of the strain is calibrated by measuring the resistance change of a strain gauge glued on the other side of the piezo stack. In this approach, one can decompose the shear strain into different symmetry channels and precisely determine the elastoresistivity coefficients. In the configuration shown in the inset of FIG. S\\ref{fig:FigS4}(a), the strain can be decomposed into the anisotropic $B_{2g}$ strain and the isotropic $A_{1g}$ strain. FIG. S\\ref{fig:FigS4}(a) and (b) show the elastoresistivity responses in the $A_{1g}$ and $B_{2g}$ symmetry channels. The linear slope at zero strain point of the elastoresistivity in the $A_{1g}$ and $B_{2g}$ channels are the first order elastoresistivity coefficients $m_{A_{1g}}^{A_{1g}}$ and $m_{B_{2g}}^{B_{2g}}$, respectively. One may also notice nonlinear responses in the elastoresistivity of annealed Fe$_{1+y}$Te$_{0.55}$Se$_{0.45}$ are present in both the $B_{2g}$ and $A_{1g}$ channels. According to the symmetry constraints, the nonlinear term in the $A_{1g}$ channel is most likely due to the second-order $B_{2g}$ strain:\n\\begin{eqnarray}\n(\\frac{\\Delta\\rho}{\\rho_{0}})_{A_{1g}}=m_{A_{1g}}^{A_{1g}}\\epsilon_{A_{1g}}+m_{A_{1g}}^{B_{2g},B_{2g}}(\\epsilon_{B_{2g}})^{2}+m_{A_{1g}}^{A_{1g},A_{1g}}(\\epsilon_{A_{1g}})^{2}+O(\\epsilon^{3})\n\\end{eqnarray}\n\\indent While for the $B_{2g}$ response (FIG. S\\ref{fig:FigS4} (b)), the nonlinearity in the $B_{2g}$ channel can be caused by either a mixed-in isotropic resistivity due to the sample's deviation from a perfect square along the Fe-Fe bonding direction, or a second-order strain $\\epsilon_{B_{2g}}\\epsilon_{A_{1g}}$. \n\\begin{eqnarray}\n(\\frac{\\Delta\\rho}{\\rho_{0}})_{B_{2g}}^{m}=m_{B_{2g}}^{B_{2g}}\\epsilon_{B_{2g}}+m_{A_{1g}}^{B_{2g},B_{2g},mixed}(\\epsilon_{B_{2g}})^{2}+m_{B_{2g}}^{B_{2g},A_{1g}}\\epsilon_{B_{2g}}\\epsilon_{A_{1g}}+O(\\epsilon^{3})\n\\end{eqnarray}\t\n\\indent Here, $(\\frac{\\Delta\\rho}{\\rho_{0}})_{B_{2g}}^{m}$ is the measured value of resistivity anisotropy, which contains the mixed-in isotropic component. In this measurement, we mainly focus on the anisotropy term $m_{B_{2g}}^{B_{2g}}$, also known as $2m_{66}$ in the Voigt notation, which is proportional to the nematic susceptibility in the $B_{2g}$ symmetry channel of $D_{4h}$ symmetry group. By finding the anisotropic strain neutral point and fitting the anisotropic elastoresistivity response quadratically, we are able to remove the mixed-in error and extract the first order elastoresistivity coefficient $2m_{66}$. \n\\section{S4 Curie-Weiss fitting of elastoresistivity coefficient $2m_{66}$ and $m_{11} - m_{12}$}\n\\begin{figure*}\n\t\\includegraphics[trim={0cm 2cm 0cm 1cm},clip,width=1\\textwidth]{FIGS5.eps}\n\t\\renewcommand{\\figurename}{Fig. S}\n\t\\caption{Curie Weiss fitting of the $B_{2g}$ elastoresistivity coefficient $2m_{66}$ of both as-grown and annealed Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ (x =  0.28, 0.38, 0.45, 0.53). (a) - (d) as-grown Fe$_{1+y}$Te$_{1-x}$Se$_{x}$. (e) - (g) annealed Fe$_{1+y}$Te$_{1-x}$Se$_{x}$. Upper panels show $|2m_{66}|$, whereas lower panels show $|2m_{66} - 2m_{66}^{0}|^{-1}$ (left axes of lower panels, blue symbols) and  $|2m_{66} - 2m_{66}^{0}|(T-T^{*})$ (right axes of lower panels, yellow circles). Black (upper panels) and red (lower panels) lines show the fits to Curie-Weiss behavior of $|2m_{66}|$ and  $|2m_{66} - 2m_{66}^{0}|^{-1}$respectively. Grey horizontal lines (lower panels) shows the average values of $|2m_{66} - 2m_{66}^{0}|(T-T^{*})$ in the fitting temperature range. The optimal fitting range (grey shaded) is determined by the greatest corresponding adjusted R-Square value. For (a),  $2m_{66}$ is negative. For (b) - (h),  $2m_{66}$ is positive.}\n\t\\label{fig:FIGS5}\n\\end{figure*}\n\\indent FIG. S\\ref{fig:FIGS5} shows the Curie Weiss fitting of the $B_{2g}$ elastoresistivity coefficient $2m_{66}$ for both as-grown and annealed Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ (x = 0.28 - 0.53). The optimal fitting range is illustrated by the grey shaded area. The divergent temperature dependence of $2m_{66}$ can be fitted by the Curie Weiss law well below 50K but deviates from the Curie Weiss law at higher temperatures. Other fitting parameters are listed in Table S\\ref{tab:tabs2}. The bare nematic critical temperature $T^{*}$ is negative for all as-grown samples. With the Se concentration increases, the bare nematic critical temperature $T^{*}$ increases and approaches zero for x = 0.53. For annealed samples, the $T^{*}$ extracted from the Curie-Weiss fitting is approximately zero, while the Curie constant $\\lambda\/a$ decreases with x. Near x = 0, in the proximity of the double spin stripe order, a diverging $B_{1g}$ nematic susceptibility was also observed. The Curie Weiss fitting parameters from the fit of $B_{1g}$ elastoresistivity coefficient $m_{11} - m_{12}$ are listed in Table S\\ref{tab:tabs3}. \n\\squeezetable\n\\begin{table}\n\t\t\\renewcommand{\\tablename}{TABLE S}\n\t\\caption{\\label{tab:tabs2}Curie-Weiss fitting parameters from the Fit of $2m_{66}$ for as-grown and annealed Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ (x = 0.17 - 0.53)}\n\t\\begin{ruledtabular}\n\t\t\\begin{tabular}{ccccccc}\n\t\t\tMaterials&  Fitting Range (K) & $2m_{66}^{0}$ & $\\lambda\/a_{0} (K)$ & $T^{*} (K)$ &  Sample Dimension ($\\mu m$) & Adjust R-square\\\\\n\t\t\t\\hline\n\t\t\t&&&As-grown&&&\\\\\n\t\t\t\\hline\n\t\t\tFe$_{1+y}$Te$_{0.83}$Se$_{0.17}$ & $11-140$ & $3.2\\pm0.4$ & $-397\\pm16$ & $-44.6\\pm7.7$ & $530\\times480\\times20$ & $0.9904$\\\\\n\t\t\tFe$_{1+y}$Te$_{0.72}$Se$_{0.28}$ & $17-96$ & $7.0\\pm0.6$ & $-811\\pm10$ & $-24.5\\pm2.9$ & $1210\\times1150\\times10$ & $0.9982$\\\\\n\t\t\tFe$_{1+y}$Te$_{0.62}$Se$_{0.38}$ & $15-50$ & $-6.8\\pm1.0$ & $592\\pm55$ & $-9.0\\pm2.1$ & $870\\times740\\times40$ & $0.9984$\\\\\n\t\t\tFe$_{1+y}$Te$_{0.55}$Se$_{0.45}$ & $20-68$ & $-15.22\\pm3.0$ & $1594\\pm222$ & $-9.3\\pm1.2$ & $1080\\times1070\\times30$ & $0.9966$\\\\\n\t\t\tFe$_{1+y}$Te$_{0.47}$Se$_{0.53}$ & $20-93$ & $-16.7\\pm2.0$ & $1752\\pm159$ & $-0.8\\pm1.7$ & $670\\times570\\times60$ & $0.9981$\\\\\n\t\t\t\\hline\n\t\t\t&&&Annealed&&&\\\\\n\t\t\t\\hline\n\t\t\tFe$_{1+y}$Te$_{0.83}$Se$_{0.17}$ & $25-200$ & $2.2\\pm0.2$ & $-243\\pm10$ & $-1.9\\pm1.9$ & $460\\times440\\times10$ & $0.9965$\\\\\n\t\t\tFe$_{1+y}$Te$_{0.72}$Se$_{0.28}$ & $17-49$ & $-5.3\\pm0.6$ & $359\\pm32$ & $-0.4\\pm1.2$ & $840\\times800\\times50$ & $0.9991$\\\\\n\t\t\tFe$_{1+y}$Te$_{0.62}$Se$_{0.38}$ & $19-46$ & $-24.4\\pm2.3$ & $1583\\pm135$ & $-2.9\\pm1.5$ & $380\\times360\\times30$ & $0.9995$\\\\\n\t\t\tFe$_{1+y}$Te$_{0.55}$Se$_{0.45}$ & $18-56$ & $-28.6\\pm3.1$ & $2004\\pm186$ & $1.5\\pm1.2$ & $410\\times360\\times30$ & $0.9988$\n\t\t\\end{tabular}\n\t\\end{ruledtabular}\n\\end{table}\n\\squeezetable\n\\begin{table}\n\t\\renewcommand{\\tablename}{TABLE S}\n\t\\caption{\\label{tab:tabs3}Curie-Weiss fitting parameters from the Fit of $m_{11} - m_{12}$ for Fe$_{1+y}$Te}\n\t\\begin{ruledtabular}\n\t\t\\begin{tabular}{ccccccc}\n\t\t\tMaterials&  Fitting Range (K) & $m_{11}^{0}-m_{12}^{0}$ & $\\lambda\/a_{0} (K)$ & $T^{*} (K)$ &  Sample Dimension ($\\mu m$) & Adjust R-square\\\\\n\t\t\t\\hline\n\t\t\tFe$_{1+y}$Te & $71-104$ & $0.29\\pm0.06$ & $13\\pm4$ & $70.2\\pm0.1$ & $443\\times440\\times24$ & $0.9992$\\\\\n\t\t\\end{tabular}\n\t\\end{ruledtabular}\n\\end{table}\n\n\n\\section{S5 RPA calculation of the nematic susceptibility}\n\nTo capture the effect of orbital degrees of freedom on the nematic susceptibility, we begin by introducing the multi-orbital Hubbard-Kanamori Hamiltonian (momentum sums are implicit):\n\\begin{eqnarray}\n\t\\mathcal{H} &=& \\sum_{\\mu} \\left[\\epsilon_{\\mu\\nu}(\\mbf{k}) - \\mu \\delta_{\\mu\\nu}\\right]c^{\\dagger}_{\\mbf{k}\\mu\\sigma}c_{\\mbf{k}\\nu\\sigma} \\nonumber + U \\sum_{\\mu} n_{\\mbf{q}\\mu\\uparrow} n_{-\\mbf{q}\\mu\\downarrow} + U' \\sum_{\\substack{\\mu < \\nu \\\\ \\sigma \\sigma'}}n_{\\mbf{q}\\mu\\sigma}n_{-\\mbf{q}\\nu\\sigma'} \\\\ &+& \\frac{J}{2}\\sum_{\\substack{\\mu\\neq\\nu \\\\ \\sigma\\sigma'}}c^{\\dagger}_{\\mbf{k}+\\mbf{q}\\mu\\sigma}c_{\\mbf{k}\\nu\\sigma}c^{\\dagger}_{\\mbf{k}'-\\mbf{q}\\nu\\sigma'}c_{\\mbf{k}'\\mu\\sigma} + \\frac{J'}{2}\\sum_{\\substack{\\mu\\neq\\nu \\\\ \\sigma}}c^{\\dagger}_{\\mbf{k}+\\mbf{q}\\mu\\sigma}c^{\\dagger}_{\\mbf{k}'-\\mbf{q}\\mu\\bar{\\sigma}}c_{\\mbf{k}'\\nu\\bar{\\sigma}}c_{\\mbf{k}\\nu\\sigma}\\,.\n\\end{eqnarray}\nHere $\\mu$ and $\\nu$ are orbital indices, $\\sigma$ labels spin and we assume $U'=U-2J$ and $J'=J$. $\\epsilon^{\\mu\\nu}(\\mbf{k})$ denotes the dispersion, in this case obtained from a tight-binding fit to DFT. In the results shown in the main text, we used the band structure parameters presented in Ref.~[\\onlinecite{ikeda10}], which give three hole-like Fermi pockets and two electron-like Fermi pockets. While this tight-binding parametrization is not intended to model specifically optimally doped FeTe$_{1-x}$Se$_x$, it offers a solid framework to elucidate, on general grounds, the tendencies of how the nematic susceptibility is affected by the suppression of $d_{xy}$ orbital spectral weight.\n\nThe interactions can be conveniently expressed as elements of the same tensor:\n\\begin{eqnarray}\n\tU^{\\mu\\mu\\mu\\mu} = U\\,, \\quad U^{\\mu\\nu\\nu\\mu} = U'\\,, \\quad U^{\\mu\\nu\\mu\\nu} = J'\\,, \\quad U^{\\mu\\mu\\nu\\nu} = J\\,.\n\\end{eqnarray}\nThe RPA expression for the spin-driven nematic susceptibility of this model was derived previously by two of us in Ref. \\cite{christensen16} (see also Ref. \\cite{fanfarillo18}). Here, we simply quote the result from the former paper, derived in the approximation that the antiferromagnetic order parameter is diagonal in orbital space (as shown for instance in Ref. \\cite{christensen17}). The nematic susceptibility in this case is a rank-4 tensor given by:\n\\begin{eqnarray}\n\t\\chi_{\\rm nem}^{\\mu\\nu\\rho\\lambda}\/N = \\left( \\int_q \\chi^{\\rho \\iota}(q)\\chi^{\\lambda \\kappa}(q) \\right) \\left(\\delta_{\\mu \\kappa}\\delta_{\\nu \\iota} - g^{\\mu\\nu \\gamma \\phi} \\int_q \\chi^{\\gamma \\iota }(q)\\chi^{\\phi \\kappa}(q) \\right)^{-1}\\,,\\label{eq:nem_susc}\n\\end{eqnarray}\nwhere $\\int_q \\equiv T \\sum_{\\omega_m}\\int \\mathrm{d}^2 \\mbf{q}$, $N=3$ is the number of components of the magnetic order parameter, and the Einstein summation convention is used. Here\n\\begin{eqnarray}\n\t\\chi^{\\mu\\nu}(q) = \\left[ (U_{\\mu\\nu})^{-1} - \\chi_0^{\\mu\\nu}(q) \\right]^{-1}\n\\end{eqnarray}\nis the RPA magnetic susceptibility, which is a rank-2 tensor. The bare magnetic susceptibility is given by:\n\\begin{eqnarray}\n\t\\chi^{\\mu\\nu}_{0}(q) = - \\sum_k \\mathcal{G}^{\\mu\\nu}(k+q)\\mathcal{G}^{\\nu\\mu}(k)\\,,\n\\end{eqnarray}\nwhere repeated indices are not summed and $\\mathcal{G}^{\\mu\\nu}$ denotes the bare multi-orbital Green's function:\n\\begin{eqnarray}\n\t\\mathcal{G}^{\\mu\\nu}(k) = \\sum_{m} \\frac{a^{m}_{\\mu}(\\mbf{k})a^{m}_{\\nu}(\\mbf{k})^{\\ast}}{i\\omega_n - \\xi^{m}(\\mbf{k})}\\,,\n\\end{eqnarray}\nwith $k=(i\\omega_m,\\mbf{k})$. For notational convenience, we introduce:\n\\begin{eqnarray}\n\t\\mathcal{G}_X^{\\mu\\nu} \\equiv \\mathcal{G}^{\\mu\\nu}(\\mbf{k}+\\mbf{Q}_1,i\\omega_m)\\,,\\quad \n\t\\mathcal{G}_Y^{\\mu\\nu} \\equiv \\mathcal{G}^{\\mu\\nu}(\\mbf{k}+\\mbf{Q}_2,i\\omega_m)\\,.\n\\end{eqnarray}\nHere, $\\mbf{Q}_1 = (\\pi, 0)$ and $\\mbf{Q}_2 = (0, \\pi)$ (in the 1-Fe unit cell). Finally, the nematic coupling constant $g$ in Eq. (\\ref{eq:nem_susc}) is also a rank-4 tensor obtained from convolutions of the Green's functions:\n\\begin{eqnarray}\n\tg^{\\rho\\nu\\eta\\mu} &=& -\\frac{1}{16}\\sum_{k}\\Big( 2\\mathcal{G}^{\\mu\\rho}\\mathcal{G}^{\\rho\\nu}_{X} \\mathcal{G}^{\\nu\\eta}\\mathcal{G}^{\\eta\\mu}_{X} - \\mathcal{G}^{\\mu\\rho}\\mathcal{G}^{\\rho\\eta}_{X} \\mathcal{G}^{\\eta\\nu}\\mathcal{G}^{\\nu\\mu}_{X} - \\mathcal{G}^{\\mu\\rho}\\mathcal{G}^{\\rho\\nu}_{X} \\mathcal{G}^{\\nu\\eta}\\mathcal{G}^{\\eta\\mu}_{Y} \\nonumber \\\\ && \\qquad - \\mathcal{G}^{\\nu\\rho}\\mathcal{G}^{\\rho\\mu}_{X} \\mathcal{G}^{\\mu\\eta}_{X+Y}\\mathcal{G}^{\\eta\\nu}_{X} + \\mathcal{G}^{\\mu\\rho}\\mathcal{G}^{\\rho\\eta}_{X} \\mathcal{G}^{\\eta\\nu}_{X+Y}\\mathcal{G}^{\\nu\\mu}_{Y} \\Big) + (X \\leftrightarrow Y)\\,,\n\\end{eqnarray}\nwhere repeated indices are not summed.\n\nTo elucidate how orbital differentiation affects the nematic susceptibility within our framework, we follow the approach outlined in Ref.~\\cite{kreisel17} and modify the Green's function by:\n\\begin{eqnarray}\n\t\\mathcal{G}^{\\mu \\nu} \\rightarrow \\sqrt{Z_{\\mu}Z_{\\nu}}\\mathcal{G}^{\\mu\\nu}\\,,\n\\end{eqnarray}\nwhere we stress that, once again, the Einstein convention is not assumed. Since both $g^{\\mu\\nu\\rho\\lambda}$ and $\\chi^{\\mu\\nu}(q)$ depend on the Green's functions, a reduced $Z$ factor will have an intricate impact on the nematic susceptibility. As discussed in the main text, this is a phenomenological way to mimic the complicated effect of incoherence and loss of spectral weight on the response function, whose validity requires that the system remains in a metallic state.\n\nThe nematic susceptibility plotted in the main text for different values of $Z_{xy}$ corresponds to the largest eigenvalue $\\lambda_{\\rm nem}$ of the equation:\n\\begin{equation}\n\\chi_{\\rm nem}^{\\mu\\nu\\rho\\lambda} \\Phi^{(n)}_{\\mu \\nu} = \\lambda^{(n)}_{\\rm nem} \\Phi^{(n)}_{\\rho \\lambda} \n\\end{equation}\nIn Fig. S\\ref{fig:nematic_susc_weight}, we show how the corresponding eigenvector $\\Phi^{(n)}_{\\mu \\nu}$ changes for decreasing $Z_{xy}$. Here we used $U=1.2$ eV and $J=U\/6$. To highlight the impact of $Z_{xy}$ we fixed the filling to $5.9$ electrons per site. As expected, the main contribution to the nematic susceptibility, signaled by the brightest squares in the figure, shifts from intra $xy$-orbital processes to intra $xz\/yz$-orbital processes. \n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{nem_weight.pdf}\n\\renewcommand{\\figurename}{Fig. S}\n\\caption{\\label{fig:nematic_susc_weight} $\\Phi^{(1)}_{\\mu\\nu}$ for $Z_{xy}=1$ and $Z_{xy}=0.7$ at a temperature immediately prior to the nematic instability of each case. For $Z_{xy}=1$, the dominant contribution to the nematic susceptibility arises from the $xy$ orbital. As $Z_{xy}$ is reduced, the $xz$ and $yz$ orbitals become the dominant ones.}\n\\end{figure}\n\n\nA direct consequence of the reduction of the nematic susceptibility is a suppression of the nematic transition temperature, which in our model is manifested as a divergence of the largest eigenvalue $\\lambda_{\\rm nem}$. In Fig. S\\ref{fig:temp_as_Z}, we plot both the nematic transition temperature $T_{\\rm nem}$ and the bare (i.e. non-renormalized by nematic order) magnetic transition temperature $T_{\\rm mag}$ as a function of $Z_{xy}$. Note that not only are both transition temperatures strongly suppressed, but their separation also decreases significantly for decreasing $Z_{xy}$. As noted in Ref. \\cite{christensen16}, these RPA transition temperatures are, not surprisingly, overestimated with respect to the actual transition temperatures. For this reason, and to be able to compare the temperature dependencies of the nematic susceptibilities of systems with very different values of $T_{\\rm nem}$, in the main text we plot $\\lambda_{\\rm nem}$ as a function of $T - T_{\\rm nem}$.\n\\begin{figure}\n\\includegraphics[width=0.4\\textwidth]{Temp_as_Z.pdf}\n\\renewcommand{\\figurename}{Fig. S}\n\\caption{\\label{fig:temp_as_Z} Nematic (blue) and bare magnetic (red) transition temperatures as a function of $1-Z_{xy}$. Both are reduced by reducing $Z_{xy}$, along with their relative separation.  At $Z_{xy}=0.7$ the separation between the two vanishes within our temperature resolution ($\\delta T < 0.2$ meV).}\n\\end{figure}\n\nThe suppression of $T_{\\rm mag}$ for decreasing $Z_{xy}$ is also manifested in the suppression of the overall magnitude of the bare magnetic susceptibility calculated at the spin-stripe wave-vector, $\\chi_{\\rm mag}$, as shown in Fig. S\\ref{fig:magnetic_susceptibility}.\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{inverse_mag_susc_abs_T.png}\n\\renewcommand{\\figurename}{Fig. S}\n\\caption{\\label{fig:magnetic_susceptibility} Inverse magnetic susceptibility for $ 0.7 \\leq Z_{xy}\\leq 1$ as a function of temperature. $Z_{xy}$ has a pronounced effect on the intensity of the magnetic susceptibility for a fixed temperature, and reducing $Z_{xy}$ reduces the overall intensity. On the other hand, the temperature dependence of the susceptibility is little affected by the change in $Z_{xy}$, and the magnetic susceptibility remains Curie-Weiss like, albeit the slope changes. Here we show the leading eigenvalue $\\lambda_{\\rm mag}$ as defined in Eq.~\\eqref{eq:mag_susc_def}.}\n\\end{figure}\nHere, $\\chi_{\\rm mag}$ was calculated in a manner similar to the nematic susceptibility:\n\\begin{equation}\n\t\\chi^{\\mu\\nu}(\\mathbf{q}) \\Psi^{(n)}_{\\nu}(\\mathbf{q}) = \\lambda_{\\rm mag}^{(n)}(\\mathbf{q})\\Psi^{(n)}_{\\mu}(\\mathbf{q})\\,,\\label{eq:mag_susc_def}\n\\end{equation}\nwhere, in Fig. S\\ref{fig:magnetic_susceptibility}, $\\mathbf{q}=\\mathbf{Q}_1$ or $\\mathbf{q}=\\mathbf{Q}_2$ and we show the leading eigenvalue. Note that, in contrast to the nematic susceptibility, the temperature dependence of $\\chi_{\\rm mag}$ is not strongly affected by the reduction of $d_{xy}$ spectral weight, as seen clearly from Fig. S\\ref{fig:magnetic_susceptibility}. However, for a fixed temperature, there is a strong suppression of $\\chi_{\\rm mag}$ with decreasing $Z_{xy}$. This last feature is in qualitative agreement with neutron scattering experiments, which showed that, in optimally-doped FeTe$_{1-x}$Se$_x$, the low-energy stripe-type magnetic fluctuations are suppressed with increasing temperature \\cite{xu12}. According to our analysis in the main text, increasing the temperature promotes a less coherent $d_{xy}$ orbital. Note, however, that Ref. \\cite{xu12} also reported that, as temperature was increased, besides a suppression of intensity at the stripe wave-vector, an incommensurate peak appeared in the momentum-resolved magnetic susceptibility at low energies. We did not observe such incommensurate peaks in our energy integrated magnetic susceptibility, i.e. the peak remains at $\\mathbf{q}=\\mathbf{Q}_1$ (and at $\\mathbf{q}=\\mathbf{Q}_2$), suggesting that this effect cannot be captured phenomenologically by a constant $Z_{xy}$ factor.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Background}\nIn the last decade or so there have been significant advances in developing new\nhigh throughput technologies such as microarray, ChiP-Chip or ChiP-seq~\\cite{lee2002}\nto uncover large scale cellular networks. Understanding these\nnetworks has been facilitated by decomposing these networks into smaller\nparts, so-called motifs, with known structure and function~\\cite{milo2002}.\nTwo major strategies currently exist for accomplishing this task. The\nfirst one, so called top-down analysis involves mining databases\nfor recurring patterns of interacting genes~\\cite{milo2002, shen2002, yeger2004}, or in the\ncase of signaling networks, proteins-protein interactions~\\cite{gerstein2006}. The emerging motifs are\nanalyzed for function, especially the specific roles they might serve in\na particular network. One interesting result from this work is the discovery of an enrichment in\nso-called feedforward motifs~\\cite{mangan2003a}, Fig. 1 (a), a three gene network.\nAlon and colleagues~\\cite{mangan2003a,mangan2003b,kalir2005,zaslaver2006}\nsubsequently demonstrated the functionality of these motifs using both\ntheory and experiment.\n\nAnother approach to understanding biological networks is the\nbottom-up approach~\\cite{guido2006}. Here knowledge of specific\nprotein-DNA interactions and signaling proteins has allowed\nresearchers to synthetically design gene regulatory networks. The\ndesign of these networks has resulted in the construction of a wide\nvariety of network motifs such as oscillators, switches, and logic\ngates~\\cite{andrianantoandro2006, kaernweissbook, entus2007}. One of the goals\nof synthetic biology is to design simple networks that can behave\nas modules. In recent years\nthere has been a resurgence of interest in the quantitative\nproperties of gene regulatory networks~\\cite{bolouri2002,kaern2003,alon2007}.\nIn addition, the ability to carry out relatively precise\nmeasurements of gene activity using synthetic biology\ntechniques~\\cite{andrianantoandro2006} and single cell\nmeasurements~\\cite{lahav2004,gevazatorsky2006} has led to considerable progress\nin this field.\n\nThe top-down and bottom-up approaches are two ways to generate\nfunctional modules. Based on a few network\nmotifs motivated by bottom-up approaches, we generated several novel functional network\nmotifs by exploring additional transcriptional control mechanisms and combinations of feedforward\nnetworks.\n\nOne of the principles of gene regulation is regulated recruitment,\nwhere transcription factors bind to the promoter regions of a gene,\nrecruiting other transcription factors and RNA Polymerase\n(RNAP)~\\cite{ptashne}. Furthermore, signaling molecules often\nactivate some of these transcription factors through binding\nor phosphorylation.\nThere are multiple ways by which these interactions\ntake place, such as blocking the promoter region, twisting\nthe DNA, Histone modification, and DNA looping, {\\it\netc}.~\\cite{bintu2005a}. In addition, the combinatorial complexity\nof multiple transcription factors that bind to the same promoter region\nwill also determine the transcription rate ~\\cite{buchler2003}.\nGiven all this complexity, a very useful approach to model these\nkinds of interactions in gene regulatory networks is the\nShea-Ackers approach~\\cite{sheaackers1982}. In this work, we used this method to\nconstruct several different kinds of network motifs.\n\nThe network motifs we will discuss can be broadly classified into\nthree categories. (1) Feedforward related motifs:\nobtained by additional feedbacks on the basic incoherent (type I)\nfeedforward network \\cite{milo2002}. (2) Tunable motif: This\nsubnetwork can either behave as an oscillator or a bistable switch\ndepending upon the concentration of a transcription factor. (3)\nAdjustable gates: This motif can switch between an AND and an OR gate\ndepending upon the concentration of a transcription factor. All\nthree network motifs arise from multiple transcription factor regulation\nat a particular gene. In designing these {\\it in silico} networks,\nwe didn't concern ourselves very much with exact mechanism such as DNA\nlooping {\\it etc.}, but rather focused on the architecture of the resulting\nnetwork.\n\nIn the next section, we briefly review the Shea-Ackers approach, which is used\nto compute the transcription rate of genes. Then, we will describe three different kinds\nof models, amplitude filters,\ntunable motifs, and adjustable gates. The latter two behave functionally\nvery differently depending upon the concentration of an input transcription\nfactor. A discussion and summary can be found in the last section.\n\n\n\n\\section*{Results and Discussion}\n\n\\subsection*{Modeling Gene Regulatory Networks Using Shea-Ackers Method}\n\nA common method for modeling protein-gene interactions is by using\nMichaelis-Menten or Hill function kinetics. However, in\nthe models we will describe in this paper, all kinetics will be\nbased on the Shea-Ackers\nformalism\\cite{sheaackers1982,bintu2005b}. This method estimates\nthe probabilities for different transcription states from\nwhich an overall rate of transcription is derived. The approach is\nless empirical and more flexible than the standard methods\nand enables one to easily incorporate repressers and activators\ninto the equation. In addition the formalism has a relatively\nstraightforward relationship to the stochastic representations of\ngene expression models. Finally, the free energy terms in the\nformalism can be easily changed by altering the promoter region\nwhich enables the models based on Shea-Ackers to be tested\nexperimentally.\n\nWe assume that the occupancy of the binding sites on promoters is\ngoverned by equilibrium statistical thermodynamics\nprobabilities~\\cite{sheaackers1982,bintu2005b,hill1960}. The\nprobabilities arise from the binding free energies, the free\nenergies of interaction between transcription factors bound to\nadjacent sites, and the concentrations of the participating\ntranscription factors and the RNA polymerase. This assumption\nholds when binding and unbinding\nto the promoter is rapid~\\cite{bintu2005b}. Slow binding\nhas been shown to result in stochasticity due to operator\noccupancy fluctuations~\\cite{kepler2001, hornos2005}, whereas for\nsmall concentrations, the individual birth and death of proteins\nintroduce noise into the system~\\cite{kaern2005}.\n\nWe illustrate the Shea-Ackers method with a given\ngene regulated by two transcription factors, $T_1$ and $T_2$.\nWe assume that these three proteins bind at three distinct sites on\nthe promoter, providing nine possible combinations of binding. Each combination\nis associated with a free energy that\nis proportional to the probability of this state\nand can be represented by the following equation ~\\cite{bintu2005b}:\n\\begin{equation}\nf = \\exp (-\\Delta G_f\/{k_B T})\\ {[T_1]}^{n_1} {[T_2]}^{n_2} [R]\n\\end{equation}\nwhere $\\Delta G_f$ is the free energy difference of the\nbound and unbound states, and the factors $[T_1]$ and [$T_2]$ are\nthe concentrations of transcription factors, $R$ is the RNAP\nconcentration and $n_1$ and $n_2$ are the number of monomers\nwhich combine into higher order multimers. $k_B$ and T are the\nBoltzmann constant and absolute temperature respectively. From\nEq (1) the normalized probability of a given state is then\ngiven by the ratio\n\\begin{equation}\n  Z_f = \\frac{\\exp (-\\Delta G_f\/{k_B T}) {[T_1]}^{n_1} {[T_2]}^{n_2} [R]}{\\sum_1^{9} \\exp (-\\delta G_f\/{k_B T}) \\ {[T_1]}^{n_1} {[T_2]}^{n_2} [R]}\n\\end{equation}\nIf we assume that the rate of transcription is proportional to the\nrelative probability when the polymerase is bound to the gene, then\nwe can partition the states into the\npolymerase bound state ($Z_{on}$) and the\npolymerase unbound state ($Z_{off}$)~\\cite{buchler2003}. The probability of gene expression is then:\n\\begin{equation}\nP=\\frac{Z_{\\mbox{\\scriptsize on}}}{(Z_{on}+Z_{\\mbox{\\scriptsize off}})}.\n\\end{equation}\nIf a transcription factor is an activator, its free energy to\nbind with the RNAP will be very low, and hence this\ninteraction will be favored. Whereas for an inhibitor, it would\nbe highly improbable for RNAP to be recruited for transcription.\nTherefore, the interactions between the transcription factors and RNAP\ndetermine the regulatory rules, which we will explore in the next\nsection for several different kinds of dynamical networks.\n\n\n\\subsection*{Feedforward Related Motifs}\n\nFirst we assume that the\ndynamics lumps together transcription and translation into one\nprocess. Although explicit modeling and experiment has been shown\nto give rise to interesting effects such as protein\nbursts~\\cite{mcadams1998, kaern2005} or oscillations as a result of transciption delay~\\cite{Hasty2008}, we believe that the main\nfeatures of our models are captured by protein-DNA interactions.\n\n\\subsubsection*{Incoherent Type I Feedforward Related Networks}\n\n\nHigh throughput approaches have uncovered several re-occurring\nmotifs termed the feedforward motif~\\cite{milo2002,shen2002}.\nFig. 1(a) shows a common transcription\nfactor for two genes, where the third gene is regulated in a\nfeedforward fashion.\nFig. 1(b) shows a\nprotein-protein interaction in addition to this scheme.\nFeedforward motifs can lead to two types of dynamics depending on\nthe nature of the two signals that converge on the third\ngene. Fig. 2 illustrates a functional\nrepresentation of the schematic shown in\nFig. 1(a)~\\cite{mangan2003a, mangan2003b}.\nIn this representation there are three different kinds of\ninteractions: protein degradation, protein synthesis and gene\nregulation. In Fig. 2 the input transcription\nfactor $p_1$ modulates the activity of a target gene (G2)\ndirectly and indirectly through the gene product $p_2$ of another\ngene (G1), which is also a transcription factor. The\ninteraction between $p_1$ and $p_2$ at the promoter region of G2\nand their ability to recruit RNAP determines the rate of\ntranscription of Gene G2. This type of architecture has been\nshown~\\cite{mangan2003b,kalir2005,zaslaver2006} to lead to two types\nof dynamics depending on the nature of the regulation that occurs\nat the target gene G2. If $p_1$, $p_2$ are both activators, the\ngene circuit acts as a low pass filter, i.e.\\ it is able to filter\nout transient signals and transcribe only when the input signal is\nlong lived~\\cite{mangan2003a}. If $p_1$, $p_2$ regulate G2 as an\nactivator and repressor respectively then the system can act as a\nbandpass filter, since the delayed response of $p_1$ through $p_2$\ntends to suppress activity of G2~\\cite{mangan2003b}. This has been\nsuggested to be a general mechanism for speeding up response times in\ntranscriptional networks~\\cite{mangan2003a, zaslaver2006}. Recently,\nit has also been argued that the steady state characteristics of\nsuch incoherent feedforward loops could be very important in\nestablishing spatial stripes and pulsed temporal expression profiles\nof transcription factors involved in developmental\nprocesses~\\cite{ishihara2005}.\n\n\n\n\\subsubsection*{Simple Feedforward}\n\nWe assume that $p_1$ activates G1 and G2, $p_2$, the\ngene product of G1, is recruited by $p_1$, and the protein complex\n$p_1 p_2$ acts as a repressor of G2 (we assume that $p_2$ cannot\nbind to G2 by itself). These assumptions lead to the following\nrate equations,\n\\begin{eqnarray}\n\\label{simpleFeedforwardEquation}\n\\frac {d [p_2]}{dt}&=& \\frac{a_1 + a_2 [p_1] }{1+ a_3 + a_4 [p_1]}-\\gamma_1 [p_2] \\\\ \\nonumber\n\\frac {d [p_3]}{dt}&=&\\frac{b_1 + b_2 [p_1] }{1+ b_3 + b_4 [p_1] + b_5 [p_1][p_2]}-\\gamma_2 [p_3],\n\\end{eqnarray}\nwhere the transcriptional rates, and the ``lumped'' parameters are\nderived in the appendix. $a_1, b_1$ represent leaky\ntranscription, which is due to the probability that RNAP can bind to\nthe operator region of the gene in the absence of a recruiting\ntranscription factor. At steady state, before G1 is saturated with\n$p_1$, $p_2$ is proportional to $p_1$. The transcription rate\nfor G2 (assuming negligible leaky transcription) can be approximated\nas,\n\\begin{equation}\nT \\propto \\frac{{b_1}^{\\prime} [p_1] }{1+ {b_2}^{\\prime} [p_1]+\n{b_3}^{\\prime} {[p_1]}^2}\n\\end{equation}\nThe transcriptional rate rises in proportion to $p_1$ for small $p_1$, and\nfalls in proportion to $1\/p_1$ for large values of $p_1$. Hence it\nreaches a maximum at some intermediate value of $p_1$.\n\n\nIn Fig. 3, we show the steady state values of\n$p_2$ and $p_3$ with respect to the input $p_1$. We note that the\nconcentration of $p_3$ has a maximum for a given value of $p_1$.\n\nAt low input concentration, $p_1$ transcribes G2 and G1, and hence\nas its input level increases, $p_2$ tends to grow. Recruitment of\n$p_2$ by $p_1$ at G2 makes it possible for the $p_1 p_2$ complex\nto halt further transcription of G2. This module is aptly named an\n``amplitude filter'' (originally called a ``band detector'',\n\\cite{basu2004, basu2005, kaernweissbook}, since its output is\nmaximal for a specific range of input. Such biphasic response has\nalso been discussed in other systems~\\cite{wolf2003, mayya2005}.\nRecently, Ishihara et. al.~\\cite{ishihara2005} discussed the band properties of\nsuch networks and used this to explain pulsed behavior and\npatterning in {\\it Drosophila} developmental processes.\n\n\\textsl{\\\\The effect of dimerization} \\pb\n\nIn this section we will consider the effect of dimerization of\n$p_2$, before it binds to the regulatory region of G2.\nDimerization has been shown to be important to\nreduce the effects of stochastic fluctuations~\\cite{bundschuh2003}\nin a feedforward regulatory scheme. Here we discuss the steady\nstate behavior of a network regulated by $p_1$, which\nrecruits a dimer of $p_2$, and this complex is a repressor at G2.\nThe only changes that need to be made to\nEq.~\\ref{simpleFeedforwardEquation} are to include the\ndimerization equations. These modified equations are,\n\n\\begin{eqnarray}\n\\frac {d [p_2]}{dt}&=& \\frac{e_1 + e_2 [p_1] }{1+ e_3 + e_4 [p_1]} -\\gamma_1 [p_2] -2 (k_{d1} {[p_2]}^2-k_{d2} [p_d]) \\\\ \\nonumber\n\\frac {d [p_3]}{dt}&=&\\frac{d_1 + d_2 p_1 }{1+ d_3 + d_4 p_1 +  d_5 p_1 p_d}-\\gamma_2 p_3 \\\\ \\nonumber\n\\frac {d p_d}{dt}&=& k_{d1} {[p_2]}^2 -k_{d2} [p_d] -k_{d3} [p_d] \\\\ \\nonumber\n\\end{eqnarray}\n\nwhere in addition to the formation and dissociation of the dimer\ncomplex, the dimer can also degrade. With these equations it is\neasy to compute the steady state values of $p_3$ with respect to\nthe input $p_1$. In Fig. 4,\nwe plot the steady state value of $p_3$ as a function of\n$p_1$ in the upper panel, which shows a much steeper fall off of the amplitude filter\ncurve, compared to the simple feedforward case discussed earlier.\nThe amplitude filter in this case, has a narrower bandwidth, and\nhence its filtering capabilities of the input transcription factor\n$p_1$ are much more enhanced. The sharp fall off is due to the\nquadratic dependence of the amount of dimer with respect to the\ninput; as the input $p_1$ increases the amount of available dimer\n$p_d$ to be recruited by $p_1$ at G2 increases, thus increasing\nthe amount of repression at G2. Dimerization of $p_1$ has the\neffect of increasing both the width and shifting the peak of the\namplitude filter curve.\n\n\n\\textsl{\\\\Effects of mutations at G1 and G2} \\pb\nWe now discuss the effects of two types of mutations at the binding\nsites at G1 and G2, for the simple feedforward with dimerization of\n$p_2$. Similar qualitative results can be obtained for the simple\nfeedforward and mixed feedforward. In general, a mutation at the binding\nsite tends to change the free energy of binding of the transcription factor,\ngenerally decreasing the binding affinity. In\nFig. 4(a), a mutation at G1 reduces the ability of $p_1$ to bind to it. Hence larger\namounts of $p_1$ are required to achieve the same transcription rate\nthereby shifting the amplitude filter peak to the right. In\nFig. 4(b), a mutation at G2 reduces the ability of $p_2$ to be recruited to G2.\nThis leads to a slower fall off, since repression does not\ntake place very efficiently. The two types of mutations can be used\nto engineer the shape of the amplitude filter, by changing its\nbandwidth and peak value~\\cite{entus2007}.\n\n\\subsubsection*{Mixed Feedforward Motifs}\n\nIn Fig. 5, we show a slightly different\nfeedforward scheme, which is a functional interpretation of\nFig. 1b. The gene product $p_2$ is\nshown as an activator of G2, whereas $p_1$ is a repressor of G2.\nAlso indicated in the figure is a protein-protein interaction,\nwhereby the input $p_1$ binds to $p_2$, and targets it for\ndegradation.\n\n\nWe assume that $p_1$ activates G1 leading to the production of\n$p_2$; $p_2$ can individually bind to G2 and act as an activator.\nFurthermore $p_1$ can be recruited by $p_2$ at the operator region\nof G2, and together this complex acts as a repressor. $p_1$ binds\nto $p_2$ and actively degrades it. One possible mechanism by which this can\noccur is if $p_1$ labels $p_2$ with ubiquitin molecules for\nproteolytic degradation~\\cite{alberts}. The above assumptions lead\nto the following rate equations,\\\n\\begin{eqnarray}\n\\frac {d [p_2]}{dt}&=& \\frac{c_1 + c_2 [p_1] }{1+ c_3 + c_4\n[p_1]}-\\gamma_c [p_1] [p_2] -\\gamma_1 [p_2] \\\\ \\nonumber \\frac {d\n[p_3]}{dt}&=&\\frac{d_1 + d_2 [p_2] }{1+ d_3 + d_4 [p_1] + d_5\n[p_2] + d_6 [p_1] [p_2]}-\\gamma_2 [p_3],\n\\end{eqnarray}\nwhere the transcriptional rates, and the ``lumped'' parameters are\nderived in the appendix. In the equation for $p_2$, the extra\ndegradation term is due to the protein-protein interaction between\n$p_1$ and $p_2$. For this system of equations the plots shown in\nFig. 6 show the behavior of the\nsteady state value of $p_3$, demonstrating the amplitude filter\neffect.\n\n\n\nAs $p_1$ increases, $p_2$ begins to grow and transcribe G2;\nhowever, two key factors prevent G2 from being continually\ntranscribed with further increases in $p_1$. The first is that\n$p_2$ is targeted by $p_1$ for degradation, and the second is that\n$p_2$ binds to G2 and recruits $p_1$, which turns off the\ntranscription.\n\nIn both the simple incoherent Type I feedforward, with\/without\ndimerization, as well as the mixed feedforward, we have shown how\nit is possible to obtain a basic amplitude filter. The property of\nsuch an amplitude filter can now be further exploited to generate\nnew types of networks when additional feedbacks are added to this\nbasic motif.\n\n\\textsl{\\\\Application 1: Time Ordering}\\pb\n\nThe amplitude filter or ''concentration detector'' has been shown\nto perform temporal processing functions such as pulse\ngeneration~\\cite{basu2004}. Pulse-like behavior has been simulated\nin~\\cite{ishihara2005}, with a series of cascaded feedforward\nloops, which in fact use the amplitude filter property of the\nnetworks to generate pulsatile behavior. We use a similar idea\nwhere a single transcription factor could serve as an\ninput to several amplitude filters, each of which has a different\nshape. In particular we assume that for each amplitude filter\nmodule, the peaks of $p_3$ occur at different\ninput transcription factor ($p_{1opt}$) concentration values.\nThen, as the input concentration crosses $p_{1opt}$, the amplitude filters get\nactivated in a sequence, depending on how far apart $p_{1opt}$ are\nfor different amplitude filters~\\cite{mangan2003b}.\n\n\nA single input can therefore activate several genes in a sequence. One\nexample could be the sequential release of different\nproteins required in a certain developmental process.\n\n\nFig. 8a plots $p_3$ as a function of the\ninput transcription factor concentration, showing a shifted peak for\nthe two amplitude filters with respect to the same input\ntranscription factor. In Fig. 8b, the\ntemporal profile for the outputs are shown assuming that the input\nconcentration is ramped up as a function of\ntime. This leads to an ordered protein production in time.\n\n\n Kashtan et. al.~\\cite{kashtan2004}, have explored the\nconsequences of multi-output networks regulated by\nfeedforward networks. They show through simulations that in some of\nthese cases feedforward loops with a common input can regulate\ngenes in a temporal order. Such temporal order can occur in\nmulti-output feedforward loop systems such as in the {\\it E. coli}\nflagellar synthesis regulation systems, where proteins need to\nassembled in a timed fashion to make up the flagellar basal-body\nmotor~\\cite{kalir2001}. It has also been found\nin~\\cite{eichenberger2004}, that the logic of the program of gene\ntranscription during differentiation in {\\it Bacillus Subtilis}\nsporulation involves a series of feedforward loops that generate\ngene transcription in a pulse like manner.\n\n\n\\textsl{\\\\Application 2: Homeostatic Networks}\\pb\n\nHomeostatic networks are important in several biological systems.\nOne example in which homeostasis has been shown to occur through\nintegral feedback control is in the chemotaxis network in {\\it E.\ncoli} (\\cite{yidoyle2000}). Our motivation was to design a network\nusing the amplitude filter which would display homeostasis to\ninput perturbations.\n\nUsing the output of the amplitude filter module as an input to\nitself, one can construct the motif shown in\nFig. 9(a), where the filter module's output\n$p_3$ participates as a transcription factor for gene G1, whose\nprotein product $p_1$ serves as the input\nto the filter module. The feedback could be positive or negative, depending on whether the\namplitude filter output $p_3$ is an activator or repressor of G1.\nIn addition, an external input protein A can bind as an\nactivator to gene G1. Depending on the interaction between\nthe input control A, the feedback $p_3$, and RNAP, we obtain\ndifferent types of behavior. Consider $p_3$ as an activator of\nG1 in Fig. 9(a), we assume that the\ninteraction between $A$, $p_3$ and RNAP is such that we obtain an\n{\\bf AND} gate at G1, {\\it i.e.} G1 expresses only when both $A$\nand $p_3$ are present. The equation that describes the\nadditional variable, $p_1$, is given by\n\\begin{eqnarray}\n\\frac{d p_1}{d t}&=&\\frac{u_1+u_2 [A][p_3]}{1+u_3 + u_4 [A]+u_5\n[p_3]+u_6 [A][p_3]}-\\gamma_3 p_1\n\\end{eqnarray}\nwhich is the equation for the input of the amplitude filter, and the\nequation for $p_3$, the output of the amplitude filter, is the same as in\nEq 6. The steady state plot for the output $p_3$ as a function of the\ninput A, is shown in Fig. 10(a).\n\n\n\n\nFigure 10 shows that the steady state values of\n$p_3$ are constant even for large inputs. As the input A of\nthe filter module increases, the output $p_3$ decreases (assuming\nthat $p_3$ is maximal at the initial value of A). This is because\nthe input transcription factor concentration moves away from\n$p_{1opt}$, at which the maximal value of transcription occurs.\nTherefore, this decreases the transcription of the filter module,\nand since its output feeds back as an activator to G1, its input\nlevel tends to decrease the transcription of G1. Essentially the\nfilter module balances the increase in the input A to G1 by\ndecreasing its output. As seen in the lower plot of\nFig. 10(a), $p_3$ stabilizes to an almost\nfixed value even though the input control $A$ increases in time.\nThis is an example of a homeostatic gene network which fixes its\nresponse to input transcription factor concentrations. The other\ncase shown in the scheme in Fig. 10(b),\nwhere $p_3$ is a repressor for G1.\nIn the regulation at G1, A as an activator and $p_3$ is a repressor. The\nnegative feedback of $p_3$ into G1 suppresses the input to the\namplitude filter. If A decreases, the input to the\nfilter module $p_1$ also decreases, this reduces the transcription rate\n(assuming that at steady state the value of $p_3$ is maximal).\nThis then lifts the repression from G1, and the input to the\nfilter module increases, thereby balancing the effect due to the\nreduced input A, and achieving the same steady state as before.\nThe above two circuits produce a fixed amount of output proteins,\neven though the input might vary by a large amount. In the first\ncase it is homeostatic to an increase in the concentration value\nof the input transcription factor, in the second case it is\nhomeostatic with respect to a decrease in the input concentration.\n\n\n\\subsection*{Tunable Motifs}\n\n\nIn this section we discuss an example of a gene network that\nexhibits oscillatory dynamics at small values of an input\ntranscription factor, and bistability for high input values.\nThis network therefore implements two\ndifferent types of motif functionality, depending upon the\nconcentration of an input transcription factor. Recently Voigt et.\nal.~\\cite{voigt2005} discussed a model of a network in the\nBacillus sporulation pathway,  which was shown to exhibit one of\ntwo alternatives, i.e.\\ either a bistable switch or oscillatory\nbehavior, depending on the environmental conditions.  A\nsynthetically constructed network exhibiting multifunctionality in\n{\\it E. coli}~\\cite{atkinson2003} was demonstrated to be able to flip\nfunction from an oscillator to a switch, by removing a particular\ninteraction.\n\nThere are now several examples of synthetically designed genetic\nnetworks such as switches~\\cite{gardner2000,becskei2001} and\noscillators~\\cite{elowitz2000,andrianantoandro2006} that use the\ncommon rules of mutual inhibition and activation to achieve a\ndesired functional behavior. The network we will describe is\nmotivated by the work of Gardner et. al.~\\cite{gardner2000}, where\nthe authors designed a toggle switch. In ~\\cite{hasty2001}, it was\ndiscussed how a toggle switch could be converted into a relaxation\noscillator by suitably manipulating the basic toggle switch\nnetwork by adding extra regulation. We further extend this design\nby introducing a new type of regulation, which involves an\nexternal transcription factor whose concentration can flip the\nsystem function between an oscillator and a switch.\n\nFig. 11 shows two mutually repressing genes, G1 and G2.\nThe repression is assumed to occur through\ntetramer binding of each of their gene products, $p_1$ and $p_2$,\nthe gene products of G1 and G2, ($p_1$ binds to\nG2, and $p_2$ binds to G1). Hence, if these were the only\ninteracting genes in the network, the system could be in one\nof two stable states, i.e., G1 is fully expressed, and G2 is\nsilent or {\\it vice versa}. Gene G3 is activated by $p_2$, and\nits product $p_3$ further activates G1. The feedback of $p_3$ to\nG1 has the effect of turning the bistable switch into a relaxation\noscillator~\\cite{hasty2001}. Assuming that initially G2 is {\\bf ON},\nwhich causes G3 to get transcribed and $p_3$ to grow, $p_3$ then\nactivates gene G1, leading to the growth of $p_1$. Thus G1\nswitches to a high state, which ultimately shuts down gene G2 due\nto its repressive effects. This leads to a decrease in $p_3$ which\nsubsequently turns G1 {\\bf OFF}, thereby completing one cycle of\nrelaxation oscillations.\n\nConsider now an extra piece of regulation, an external input $A$,\nas an activator of gene G1. $A$ can bind to the promoter region and\nactivate G1, but can also cooperatively bind with the tetramer of\n$p_2$, which represses G1. $p_3$ can also bind cooperatively\nwith the tetramer of $p_2$, which has the effect of repressing G1.\nThe two activators $A$ and $p_3$, however, are assumed to be\nmutually exclusive, i.e.\\ both $A$, and $p_3$ cannot bind\ntogether, but each can individually bind to the DNA. As described\nin the Appendix, these regulatory rules make G1 behave like an {\\bf\nOR} gate with respect to the inputs $A$ and $p_3$. From the above\nregulatory mechanisms, the following equations for the protein\ndynamics emerge,\n\n\\begin{eqnarray}\n\\frac {dp_1}{dt}&=& \\frac{m_1 [A]+m_2 [p_3]}{1+m_3 [A]+m_4\n[p_2]^4+m_5 [A][p_2]^4+m_6 [p_3] +m_7 [A] [p_3]} -\\gamma_1 p_1 \\\\ \\nonumber\n\\frac {dp_2}{dt}&=&\\frac{n_1}{1+n_2+n_3 [p_1]^4} -\\gamma_2 p_2, \\\\ \\nonumber\n\\frac{dp_3}{dt}&=&\\frac{o_1+o_2 [p_2]}{1+ o_3 + o_4 [p_2]}\n-\\gamma_3 p_3,\n\\end{eqnarray}\n\nWe now consider approximating the transcription rate for G1 for\nsmall and large values of the control transcription factor, $A$. For\nsmall $A$,\n\\begin{equation}\nT \\simeq \\frac{m_2 [p_3]}{1+m_4 [p_2]^4 + m_6 [p_3] +m_7 [p_2]^4\n[p_3]}\n\\end{equation}\nG1 is activated by $p_3$, and repressed by $p_2$. As\ndiscussed earlier $p_3$ toggles the bistable switch\nformed between the gene products of G1 and G2. For large $A$,\n\\begin{equation}\nT \\simeq \\frac{1}{\\frac{m_3}{m_1}+\\frac{m_5}{m_1}[p_2]^4}\n\\end{equation}\nwhich is the transcription rate, one would obtain a toggle\nswitch, between G1 and G2.  When $A$ is large, it is more likely\nfor $A$ to bind to the DNA than its competitor $p_3$ and the\nsystem is put into the bistable regime. The input $A$ can\ntherefore be used to tune the system into a relaxation\noscillator, or a switch. Fig. 12 shows the\nbifurcation plot for the steady state values of $p_1$ as a\nfunction of the input transcription factor $A$. The plot shows a\nsubcritical Hopf bifurcation~\\cite{strogatz,kuznetsov} for $p_1\n\\simeq 9$, and a saddle-node bifurcation at $p_1 \\simeq 35$. As\ndiscussed earlier, the system exhibits oscillations for $A< \\simeq\n11$ and bistability for $A> \\simeq 35$. In the right panel of\nFig. 12, the time series plots for $p_1$,\n$p_2$, $p_3$, are displayed, which show oscillations, for the case\nwhere $A=1$. In the lower right panel of\nFig. 12 the bistable behavior of $p_1$ is\ndisplayed (for the input $A=50$), taking one of two values,\ndepending on the initial conditions.\n\n\n\\subsection*{Adjustable Gates}\n\nThe transcriptional interaction between two transcription factors\nand RNAP has previously been shown to generate several instances of\nlogic such as {\\bf AND, OR, XOR}, {\\it etc.}~\\cite{buchler2003}.\nMultiple transcription factors can regulate the gate properties of a\nnetwork through their input concentrations. Alon and\ncolleagues~\\cite{setty2003} have studied the gate properties of the\nregulation of the {\\it lacZYA} operon in {\\it E. coli}; and by using a\nmathematical model, their work shows that the regulation can be made to behave\nas a fuzzy AND, a pure AND, and an OR logic gate. In this section we\nconsider gene regulation by three transcription factors which have the\nproperty of switching between two different logical functions,\ndepending on one of the input transcription factor concentrations.\nFurther we propose that such a circuit with appropriate feedback can\nbe made to exhibit other kinds of functionality such as homeostasis\nor oscillatory behavior.\n\nConsider gene G1 transcribed by the interaction of three\ntranscription factors, $p_1$, $p_2$ and $p_3$. $p_3$ is the control\nthat determines the output logic. We make the following\nassumptions: binding of all three transcription factors and\nRNAP to the gene is unlikely; individual binding of $p_1, p_2,\np_3$ and RNAP is also unfavorable; we also assume that\ntranscription is not leaky. With these assumptions, transcription\noccurs due to the binding of the following complexes $p_1 p_2 P,\np_1 p_3 P, p_2 p_3 P$. The transcription rate, takes the following\nform,\n\\begin{equation}\nT=\\frac{r_1 p_1 p_2 + r_2 p_1 p_3 +r_3 p_2 p_3}{1+ r_4 p_1+ r_5\np_2 + r_6 p_3  +r_7 p_1 p_2 + r_8 p_1 p_3 +r_9 p_2 p_3+r_{10} p_1\np_2 p_3}\n\\end{equation}\nThe above formula can be simplified for two cases, i.e.\\ high\nand low values of the concentration of the control transcription\nfactor, $p_3$. For low values of $p_3$,\n\\begin{equation}\nT \\simeq \\frac{r_1 p_1 p_2}{1+ r_4 p_1+ r_5 p_2+r_7 p_1 p_2 }\n\\end{equation}\nwhich implies that the transcription is activated only when both\n$p_1$ and $p_2$ are present, implementing an {\\bf AND}\ngate. For high values of $p_3$, the transcription rate is,\n\\begin{equation}\nT=\\frac{{r_2}^{\\prime} p_1  +{r_3}^{\\prime}p_2}{1  +{r_8}^{\\prime}\np_1 +{r_9}^{\\prime} p_2 +{r_{10}}^{\\prime} p_1 p_2}\n\\end{equation}\nwhere ${r_i}^{\\prime}=\\frac{r_i}{r_6}$, for $i=2,3,8,9,10$. Here the\ntranscription is activated when either $p_1$ or $p_2$ is present,\nimplementing an {\\bf OR} gate. The control $p_3$ is able to switch from an\n{\\bf AND } to {\\bf OR} gate. In Fig. 13 we plot the\ntranscription rates as a function of the input transcription factors, $p_1$ and\n$p_2$.\n\n\nThe ability to switch from one kind of logical function to another\nby varying the control $p_3$ opens up the possibility to use such\na motif in a gene network with other interacting genes. Consider\nFig. 14(i), a regulatory circuit\nwith G1 having three inputs $p_1, p_2$ and the control $p_3$,\nwhich emerges from a long negative feedback loop from its gene\nproduct $p_5$ in the following way: The gene G2 constitutively\nproduces protein $p_4$, which forms a protein complex with $p_5$.\n$p_4$ is also a transcription factor for the gene G3, which\nproduces protein $p_3$, and hence this is how a feedback into G1\nis achieved. We further assume that $p_4$ binds to the regulatory\nregion of G3 as a tetramer. The transcription factors $p_1$ and\n$p_2$ are assumed to be external to the system, and can be\neither set to a constant or a time dependent value.\nIn our case we shall fix the value of $p_2$, but allow\n$p_1$ to fall to low levels, starting at some fixed value.\nHowever, the dynamics of the network is symmetric with respect to\n$p_1$ and $p_2$, and hence we could have equally well chosen\nto vary $p_2$ and keep $p_1$ fixed. Fig. 15 shows\nthe logical structure for the adjustable gate network. Note that\nthe output from the gate determines its behavior. The\ndifferential equations for the rates of production of the various\nspecies are,\n\\begin{eqnarray}\n\\frac{d[p_5]}{dt} &=& \\frac{r_1 [p_1] [p_2] + r_2 [p_1] [p_3] +r_3\n[p_2] [p_3]}{1+ r_4 [p_1]+ r_5 [p_2] + r_6 [p_3]  +r_7 [p_1] [p_2]\n+ r_8 [p_1] [p_3] +r_9 [p_2]\n[p_3]+r_{10} [p_1] [p_2] [p_3]} \\\\ \\nonumber \\\\ \\nonumber\n&-& k_1 [p_4] [p_5] + k_2 [C] - \\gamma_{p_5} [p_5] ,\\\\ \\nonumber \\\\ \\nonumber \\frac{d[C]}{dt}\n&=& k_1 [p_4] [p_5] - k_2 [C] ,\\\\ \\nonumber \\frac{d [p_4]}{dt} &=&\nc_0 - k_1 [p_4] [p_5] + k_2 [C] -\\gamma_{p_4} [p_4],\\\\ \\nonumber\n\\frac{d [p_3]}{dt} &=&\n\\frac{v_2 [p_4] ^4}{1 + v_4 [p_4] ^4} - \\gamma_{p_3} [p_3], \\\\\n\\nonumber\n\\end{eqnarray}\nFor gene G1, the input $p_2$ is held constant, but the input $p_1$\nis made to decay from some initial value.\nGene G2 produces $p_4$ constitutively, and\nis sequestered by the output of G1, {\\it i.e.} $p_5$, into the\ncomplex $C$. Since $p_4$ is a transcription factor for G3, its\nsequestration away from G2 results in a lower value of the protein\n$p_3$. Hence the control $p_3$ is changed, the gate properties\nof G1 switch between {\\bf AND}\/{\\bf OR}. The negative feedback\narising from G1 onto itself is inhibitory, due to the complex\nformation. In Fig. 14(ii), the upper plot shows\nthe steady states of the value of the control $p_3$ as a function\nof the input $p_1$. A supercritical Hopf bifurcation is seen to\noccur at $p_1 \\simeq 2.5$. Fig. 14(ii) lower\nright-hand plot shows steady oscillations of $p_3$. The oscillations\narise due to the gate properties of G1. Initially, when the system\nis at steady steady state, and the inputs $p_1$ and $p_2$ are fixed,\n$p_3$ has a small value which makes G1 behave as an {\\bf AND} gate.\nsince both inputs are present, the output $p_5$ is high. There\nis thus considerable sequestration of $p_4$ due to the complex\nformation. This implies a reduced production of $p_3$, which is\nconsistent with G1 being in the {\\bf AND} state. Now if one of the\ninputs to the system is removed ({\\it e.g} by making $p_1$ decay),\nsince initially the gate is in the {\\bf AND} state, the output $p_5$\ndecreases. This results in the release of $p_4$, which begins\nincreasing the transcription of G3; then $p_3$ increases, and the\ngate G1 switches to the {\\bf OR} state. But in this state, G1 can\nbe transcribed by $p_2$, and the output $p_5$ increases. This once\nagain results in sequestration of  $p_4$, and finally reduces\nthe control $p_3$, and in this way we complete one cycle. The system\ntherefore switches back and forth between the two states and hence\nthis leads to oscillations.\n\n\nWe now describe another application of an adjustable gate. However\nin this case, the gate properties are reversed, i.e., the gate\nimplements an {\\bf OR} gate for low input control transcription\nfactor concentration, and an {\\bf AND} gate for high input control\ntranscription factor concentration. We discuss the nature of the\nregulation and its consequences without simulations, since this\ncase is very similar to the previously described model. For this\ncase we assume the following regulatory rules: all three\ntranscription factors and RNAP bind to the gene;\neach of the transcription factors $p_1, p_2$ and\nRNAP bind individually to the gene; and the complex $p_1$ $p_2$ and\nRNAP can bind to the gene. These assumptions lead to the following\nrate law,\n\\begin{equation}\nT=\\frac{s_1 p_1 p_2+ s_2  p_1 +s_3  p_2 +s_4 p_1 p_2 p_3}{1+ s_5\np_1+ s_6  p_2 +s_7 p_3  +s_8 p_1 p_2 +s_9 p_1 p_3+s_{10} p_2\np_3+s_{11} p_1 p_2 p_3}\n\\end{equation}\nBy inspection it is clear that for low $p_3$, G1 behaves like an\n{\\bf OR} gate with respect to $p_1$ and $p_2$, whereas for high $p_3$, it\nbehaves like an {\\bf AND} gate. If we now consider the same network\nas described above but substitute this motif into G1, then the\nsystem shows an almost homeostatic behavior with respect to a change\nin its input. This is easy to understand since initially, when both\ninputs are present, the output $p_5$ must be large. Due to the\nnature of the feedback, this determines the value of $p_3$ to be at\na low level. The system is therefore initially in the {\\bf OR}\nstate. If now one of the inputs is suddenly decreased, since the\nsystem is initially in an {\\bf OR} state, the output would continue\nto be high. There would be a small transient due to the sudden\nchange in input, but the system would reach a steady state as\nbefore. This then allows this network to be fairly unperturbed to\nchanges in one of its input transcription factors.\n\n\nWhat could be the function of such networks with adjustable gates?\nIn the first case (with {\\bf AND} gate properties for low $p_3$),\na sudden drop in a transcription factor concentration could set up\noscillatory patterns in the network which would then signal the\nnext program to be carried out by the genetic network. In the\nsecond case (with the gate properties reversed), it is clearly\nuseful to have a homeostatic network which works in such a way so\nas to counteract any sudden changes in input transcription factor\nconcentration levels.\n\nAlthough we have not found explicit examples for many of the\nnetworks we have discussed, which display such complex behavior,\nwe believe that such an endeavor is worth exploring.\n\n\n\n\n\\section*{Conclusions}\n\nAs a first step towards recognizing and understanding large\ncomplicated pathways, we have discussed in this work the modular\ndesign of several functional network motifs. Each of the networks\nconsists of genes which are regulated by multiple transcription\nfactors. The combinatorial regulation was explored in each case, and\nthe networks which emerged were found to have very distinct\nproperties. Our modeling procedure used the Shea-Ackers method~\\cite{bintu2005b}, which allowed us to derived the rates of\ntranscription which were then used to explore the network dynamics.\nThe networks could broadly be classified into: networks which are\nderived from incoherent feedforward motifs; and networks which can\nchange their gating properties based upon an external input.\n\nWe first discussed the steady state properties of feedforward\nnetworks, which can be used as amplitude filters. We found that both\nthe Type I simple and mixed feedforward networks led to a similar\ndesign, i.e.\\ filtering out the input transcription factor\nconcentration, although both networks worked through different types\nof interactions. Furthermore, we discussed the effects of\ndimerization for a simple Type I feedforward, which has the effect\nof narrowing the bandwidth of the amplitude filter. To study the\nfilter characteristics of these networks, we simulated the effects\nof mutations which would change the protein-DNA binding strengths.\nThese generally have the effect of shifting the amplitude filter\ncurve and modulating its bandwidth. Furthermore we described how\nthese motifs can be applied to a biological setting. By having a\ncommon transcription factor as the input to two amplitude filter\nmodules, but with shifted filter characteristics, it was possible to\nobtain a time ordered response of protein production. Homeostatic\nnetworks emerged if the output of the amplitude filter was fed back\nto itself. This network was found to be resilient in its output to\neither increasing or decreasing values of an external input\ntranscription factor, depending on whether the feedback was assumed\nto be positive or negative respectively.\n\nWe next described a tunable motif network, where the regulation at\none of the genes made it possible for the networks to exhibit\nbistability or oscillatory behavior if one of the external input\ntranscription factor concentration was made to increase\/decrease\nrespectively. Finally we discussed gate properties of regulation at\na gene, which can be made to switch between an {\\bf AND\/OR}\ndepending on the one of the input transcription factor\nconcentrations.\n\n\n\\section*{Methods}\n\n\\subsection*{Simulations}\n\nAll simulations were carried out using the Systems Biology Workbench\n(SBW) tools~\\cite{Sauro:Omics}: the network designer,\nJDesigner, the simulation engine Jarnac~\\cite{Sauro:2000}.\nBifurcation diagrams were computed using SBW with an interface to\nMATLAB~\\cite{Cameron:MATLAB}, and a bifurcation discovery\ntool~\\cite{ChickarmaneBifTool}. Bifurcation plots were also computed\nand cross checked using\nOscill8~\\footnote{http:\/\/sourceforge.net\/projects\/oscill8}, an\ninteractive bifurcation software package which is linked to\nAUTO~\\cite{Do81}, and SBW~\\cite{Sauro:Omics}. In all our simulations\nthe species concentrations are regarded as dimensionless, whereas\nthe kinetic constants have dimensions of inverse time, with\ndimensionless Michaelis-Menten constants. All models are available\nas Jarnac scripts (supplied in the supplement) which can be easily translated to\nSBML~\\cite{hucka:2002d} using the JarnacLite tool that is part of\nthe SBW suite~\\cite{Sauro:Omics}.\n\n\n\n\n\\section*{Author contributions}\n    HS helped conceive and fund the project; SY contributed to manuscript preparation and final data analysis.\n\n\n\\section*{Acknowledgements}\n  \\ifthenelse{\\boolean{publ}}{\\small}{}\nThis work was supported by grants from the National Science Foundation (0432190 and FIBR 0527023) to HMS. The authors wish to thank Carsten Peterson for useful discussions and most significantly to Vijay Chickarmane for technical assistance in carrying out the simulations and deriving the transcription factor binding kinetics. VC declined to be a co-author on the manuscript but was also supported by NSF FIBR 0527023.\n\n\n\n\n\n{\\ifthenelse{\\boolean{publ}}{\\footnotesize}{\\small}\n \\bibliographystyle{Fundamental_Dynamic_Units_Sauro} \n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s1}\nA $\\ensuremath{\\mathcal{PT}}$-symmetric quantum theory is defined by a Hamiltonian that is invariant\nunder combined space reflection (parity) $\\ensuremath{\\mathcal{P}}$ and time reversal $\\ensuremath{\\mathcal{T}}$. An early\nclass of non-Hermitian $\\ensuremath{\\mathcal{PT}}$-symmetric Hamiltonians that has been studied in\ndepth is $H=p^2+x^2(ix)^\\varepsilon$ ($\\varepsilon$ real). These Hamiltonians are $\\ensuremath{\\mathcal{PT}}$\ninvariant because $x\\to-x$ under $\\ensuremath{\\mathcal{P}}$ and $i\\to-i$ under $\\ensuremath{\\mathcal{T}}$. It was observed\nin Refs.~\\cite{R1,R2} that the eigenvalues of this class of Hamiltonians are\nreal, discrete, and positive for all $\\varepsilon\\geq0$ and the reality of these\neigenvalues was attributed to the $\\ensuremath{\\mathcal{PT}}$ symmetry of $H$. Subsequently, the\nreality of the spectrum was established at a mathematically rigorous level in\nRefs.~\\cite{R3,R4}.\n\nThe eigenvalues of a $\\ensuremath{\\mathcal{PT}}$-symmetric Hamiltonian are either real or come in\ncomplex-conjugate pairs. If the eigenvalue spectrum is entirely real, the $\\ensuremath{\\mathcal{PT}}$\nsymmetry of the Hamiltonian is said to be {\\it unbroken}, but if some of the\neigenvalues are complex, the $\\ensuremath{\\mathcal{PT}}$ symmetry of the Hamiltonian is said to be\n{\\it broken}. Many studies of model quantum systems whose Hamiltonians are\n$\\ensuremath{\\mathcal{PT}}$ invariant have been published (see Refs.~\\cite{R5,R11}). $\\ensuremath{\\mathcal{PT}}$-symmetric\nHamiltonians often exhibit a transition from a parametric region of unbroken\n$\\ensuremath{\\mathcal{PT}}$ symmetry to a region of broken $\\ensuremath{\\mathcal{PT}}$ symmetry. This $\\ensuremath{\\mathcal{PT}}$ transition\noccurs in both the classical and in the quantized versions of a $\\ensuremath{\\mathcal{PT}}$-symmetric\nHamiltonian \\cite{R2} and this transition has been observed in numerous\nlaboratory experiments \\cite{R5,R11}.\n\nMany papers on $\\ensuremath{\\mathcal{PT}}$-symmetric Hamiltonians having discrete spectra have been\npublished, but there have been only very few studies of $\\ensuremath{\\mathcal{PT}}$-symmetric\nHamiltonians having continuous spectra. Therefore, in this paper we consider\nHamiltonians $H=p^2+V(x)$ whose potentials possess continuous spectra. The\npotentials $V(x)$ that we discuss here decay to $0$ as $|x|\\to\\infty$. Thus, it\nis not surprising that we find that in general the real part of the spectrum is\ncontinuous and that these eigenvalues range continuously from 0 to $+\\infty$.\nSince the potential $V(x)$ is {\\it odd} under $x\\to-x$ and is pure imaginary,\n$V(x)$ is $\\ensuremath{\\mathcal{PT}}$ invariant, and hence complex eigenvalues must appear as\ncomplex-conjugate pairs. On the basis of our study we believe that short-range\npotentials typically have a finite number of discrete complex eigenvalues and\nthat long-range potentials have an infinite number of discrete complex\neigenvalues.\n\nTo be specific, in this paper we study five one-dimensional Hamiltonians of the\nform $H_n=p^2+V_n(x)$, where\n\\begin{eqnarray}\nV_1(x)&=&iA_1\\,{\\rm sech}(x)\\tanh(x),\\nonumber\\\\\nV_2(x)&=&iA_2\\,x\/(1+x^4),\\nonumber\\\\\nV_3(x)&=&iA_3\\,x\/(1+|x|^3),\\nonumber\\\\\nV_4(x)&=&iA_4\\,{\\rm sgn}(x)\\theta(2.5-|x|),\\nonumber\\\\\nV_5(x)&=&iA_5\\,x\/(1+x^2),\n\\label{e1}\n\\end{eqnarray}\nand the strength parameters $A_n$ are real. In all cases $V_n(x)$ is odd in $x$\nand is pure imaginary, so the $H_n$ are all $\\ensuremath{\\mathcal{PT}}$ symmetric. Furthermore, these\npotentials all vanish as $|x|\\to\\infty$. In all these cases the Hamiltonians $H_n$\nhave real eigenvalues that range continuously from $0$ to $+\\infty$. However,\nthe universal property of the five Hamiltonians studied here is that in addition\nto the real continuous part of the spectra, there are complex-conjugate pairs of\ndiscrete eigenvalues for sufficiently large strength parameters $A_n$. Thus, in\nall these cases the $\\ensuremath{\\mathcal{PT}}$ symmetry of these Hamiltonians $H_n$ is broken.\n\nThe five potentials $V_n(x)$ vanish at different rates for large $|x|$: The\nScarf-II potential $V_1(x)$ decays exponentially for large $|x|$\n\\cite{R6,R7,R8,R9}, the rational potentials $V_2(x)$ and $V_3(x)$ decay\nalgebraically like $|x|^{-3}$ and $|x|^{ -2}$ for large $|x|$, and the\nstep-function potential $V_4(x)$ has compact support \\cite{R10}. These are all\n{\\it short-range} potentials and we find that these potentials confine a {\\it\nfinite} number of discrete complex bound states. The number and size of these\ncomplex eigenvalues increase as the strength parameters $A_n$ increase.\n\nThe potential $V_5(x)$ is special; it vanishes slowly like $1\/|x|$ for large\n$|x|$. Because it vanishes slowly,\/and to\nit is a {\\it long-range} potential like the\nCoulomb potential. Even though this potential is bounded, the property that it\nis long range allows it to confine {\\it infinitely many} discrete complex bound\nstates. Like the Balmer series for the hydrogen atom, the sequence of\ncomplex-conjugate pairs of eigenvalues converges to a limit point, which happens\nto be at $0$, and the $k$th pair of eigenvalues approaches $0$ like $1\/k^2$.\n\nIn previous studies of $\\ensuremath{\\mathcal{PT}}$-symmetric Hamiltonians it was found that accurate\nnumerical calculations of real discrete eigenvalues could be done by using the\nshooting method \\cite{R1}. However, if the discrete eigenvalues are complex, the\nshooting method becomes unwieldy. Therefore, alternative techniques based on the\nfinite-element and variational methods were used. A numerical technique known as\nthe {\\it Arnoldi algorithm} was used in Ref.~\\cite{R12}.\n\nLet us summarize the numerical technique used in this paper: For numerical\ncalculations of eigenvalues one cannot work directly on the infinite $x$ axis,\nso one reduces the problem to solving the Schr\\\"odinger equation on a large but\n{\\it finite} interval. Consequently, the numerical techniques used to calculate\neigenvalues can only return discrete values and one must determine whether a\ngiven eigenvalue belongs to a discrete or a continuous part of the spectrum. To\ndistinguish between these two possibilities we examine the associated\neigenfunctions and observe how they satisfy the boundary conditions. As\nexplained in detail in Ref.~\\cite{R12}, the eigenfunctions associated with\ndiscrete eigenvalues are localized in space (like bound states) and decay to 0\nsmoothly and exponentially as $x$ approaches the boundary points of the\ninterval. However, the eigenfunctions for eigenvalues that belong to the\ncontinuous part of the spectrum drop abruptly to $0$ at one or both endpoints\nof the finite interval.\n\nThe technique used here to compute the eigenvalues of $H_n$ is called {\\it\nChebyshev spectral collocation}. This technique relies on the properties of\nChebyshev polynomials and Chebyshev series and is explained in detail in\nRef.~\\cite{R13}. To calculate the spectra of the Hamiltonians $H_n$ by using\nChebyshev spectral collocation we replace the infinite $x$ axis by the finite\ninterval $-L\\leq x\\leq L$. We then decompose the interval $[-L,L]$ into $N$\nsubintervals bounded by grid points at $x_j$, where $j=0,~1,~2,~3,~\\dots,~N$.\nThese subintervals are not of equal length; rather, the subintervals shorten\nas we approach the endpoints at $x=-L$ and $x=L$. To determine the positions of\nthe grid points, we construct a semicircle of radius $L$ centered at the origin\n$x=0$ and divide the circle into equal sectors. We then project onto the $x$\naxis. Thus, the grid points are located at $x_j=L\\cos(\\pi j\/N)$. For all\ncomputations done in this paper we take $N=2^{14}-1$. The first and last grid\npoints lie at $x=\\pm L$, but since $N$ is odd there is no grid point at the\norigin. We make this choice because the potential $V_4(x)$ is discontinuous at\n$x=0$. To find the eigenvalues we impose homogeneous boundary conditions at the\nendpoints $\\pm L$ and finally let $L$ tend to infinity. In all cases we take\n$L=10$, $100$, and $1000$, and we find that the eigenvalues converge rapidly to\ntheir $L=\\infty$ values.\n\nThis paper is organized very simply. In Secs.~\\ref{s2}-\\ref{s6} we describe\nin turn the spectra of $V_n(x)$ for $n=1$, 2, 3, 4, and 5, and then in\nSec.~\\ref{s7} we give some concluding remarks.\n\n\\section{Eigenvalues for the Scarf-II potential $V_1(x)$}\n\\label{s2}\nFor the potential $V_1(x)$ in (\\ref{e1}) we have chosen $A_1=30$. In\nFig.~\\ref{F1} we plot the eigenvalues in the complex plane for $L=10$ in panel\n(a) and for $L=100$ in panel (b). We observe two kinds of eigenvalues,\nbound-state eigenvalues, which are indicated by circles, and continuum\neigenvalues, which are indicated by crosses in (a) and by dots in (b). As we\nwill see in Fig.~\\ref{F4}, we can distinguish between bound-state and continuum\neigenvalues by examining the corresponding eigenfunctions. This plot of the\nabsolute values of the eigenfunctions as functions of $x$ shows that the\neigenfunctions for bound-state eigenvalues decay smoothly and exponentially to 0\nas $x$ approaches the boundaries at $\\pm L$ while the continuum eigenfunctions\nabruptly drop to 0 at one or both boundaries. Note that as $L$ increases from\n$10$ in (a) to $100$ in (b), the positions of the bound-state eigenvalues\nstabilize rapidly but the continuum eigenvalues approach the real axis.\n\n\\begin{figure}[b!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig1a.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig1b.png}\n\\end{center}\n\\caption{[Color online] Eigenvalues in the complex plane for $V_1(x)$ in\n(\\ref{e1}) with the strength parameter $A_1=30$ for the case $L=10$ in panel\n(a) and for $L=100$ in panel (b). Continuum eigenvalues are indicated by red\ncrosses in panel (a) and by green dots in panel (b). Discrete bound-state\neigenvalues are indicated by black circles. Note that as $L$ increases, the\nlocations of the bound states stabilize but the continuum eigenvalues all\ncollapse onto the real axis. As they do so, a new complex-conjugate pair of \nbound states is uncovered. Observe that the continuum eigenvalues come in\ncomplex-conjugate pairs until the real parts of these eigenvalues exceeds about\n$28$. Above this critical value the continuum eigenvalues are all real.}\n\\label{F1}\n\\end{figure}\n\nWe observe two kinds continuum eigenvalues in Fig.~\\ref{F1}. Above a critical\nvalue near $28$ the continuum eigenvalues are real, but below this critical\nvalue the continuum eigenvalues come in complex-conjugate pairs that lie\nslightly above and below the real axis. These pairs of eigenvalues approach the\nreal axis as $L$ increases but the position of the critical value near $x=28$\ndoes not change. With increasing $L$ each member of the complex-conjugate pair\nof eigenvalues approaches the real axis vertically, but as they reach the real\naxis, one member of the pair moves slightly rightward and the other moves\nslightly leftward along the real axis, thus doubling the density of points.\n(This behavior of the continuum eigenvalues is in exact analogy to the motion of\nthe roots of the famous Wilkenson polynomial \\cite{R14}.) For $L=1000$, the\ncontinuum eigenvalues below $x=28$ are extremely close to the real axis, but the\npositions of the complex bound states do not move, as we can see in\nFig.~\\ref{F2}.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig2.png}\n\\end{center}\n\\caption{[Color online] Eigenvalues in the complex plane for $V_1(x)$ in\n(\\ref{e1}) with the strength parameter $A_1=30$ for the case $L=1000$. The\ndiscrete bound-state eigenvalues (black circles) have not moved but the\ncontinuum eigenvalues (blue dots) are now very close to the real axis.}\n\\label{F2}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig3.png}\n\\end{center}\n\\caption{[Color online] Logarithmic plot of the eigenvalue data in\nFigs.~\\ref{F1} and \\ref{F2}. This plot presents dramatic evidence that the\ntransition from slightly complex continuum eigenvalues to exactly real\ncontinuum eigenvalues near $x=28$ is sharp. There is a jump at the dashed line\nof about {\\it ten orders of magnitude} at this transition point. Observe\nthat the location of the transition does not change as $L$ is increased from\n$10$ to $100$ to $1000$.}\n\\label{F3}\n\\end{figure}\n\nFigure~\\ref{F3} is a logarithmic plot of the eigenvalues shown in Figs.~\\ref{F1}\nand \\ref{F2}. Observe that the critical point at $x=28$ at which the continuum\neigenvalues jump from being complex-conjugate pairs to real numbers does not\nmove as $L$ is increased.\n\nTo distinguish between discrete and continuum eigenvalues we investigate the\nbehavior of the associated eigenfunctions. Six possible behaviors of the\neigenfunctions are displayed in Fig.~\\ref{F4}. In general, the eigenfunctions of\ndiscrete eigenvalues decay smoothly to 0 at the endpoints of the interval but\nthe eigenfunctions of continuum eigenvalues abruptly drop to 0 at one or both\nendpoints.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig4a.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig4b.png}\\\\\n\\includegraphics[scale=0.50]{Fig4c.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig4d.png}\\\\\n\\includegraphics[scale=0.50]{Fig4e.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig4f.png}\n\\end{center}\n\\caption{[Color online] Plots of the absolute value of the eigenfunctions\nassociated with some eigenvalues in Fig.~\\ref{F1}. Panels (a), (b), and (c)\ndisplay the eigenfunctions of the three discrete bound-state eigenvalues in\npanel (b) in Fig.~\\ref{F1}: Panel (a) shows the eigenfunction for the eigenvalue\n$2.374999999702702+12.272301129148877\\,i$ for $L=10$; panel (b) shows the\neigenfunction for the eigenvalue $5.875000000021835 +6.817945071620461\\,i$ for\n$L=10$; panel (c) shows the eigenfunction for the eigenvalue $7.374999997301000+\n1.363589013462076\\,i$ for $L=100$. The next three plots show the behavior of the\neigenfunctions for some continuum eigenvalues: Panel (d) shows the eigenfunction\nfor the continuum eigenvalue $7.361943725638523+2.501634415578858\\,i$ for $L=\n10$; panel (e) shows the eigenfunction for the continuum eigenvalue\n$0.277713365597523+0.009073644654185\\,i$ for $L=100$; panel (f) shows the\neigenfunction for the continuum eigenvalue $30.234638465149410+0.000000000526705\n\\,i$ for $L=10$. Bound-state eigenfunctions decay smoothly and exponentially as\n$x$ approaches the endpoints but the continuum eigenfunctions abruptly and\nsharply drop to $0$ at one or both endpoints.}\n\\label{F4}\n\\end{figure}\n\n\\section{Eigenvalue behavior of $V_2(x)$}\n\\label{s3}\nWhile the potential $V_1(x)$ decays exponentially for large $|x|$, the\npotential $V_2(x)$ decays algebraically like $|x|^{-3}$ for large $|x|$.\nNevertheless, the spectral properties of $V_2(x)$ are strikingly similar to\nthose of $V_1(x)$. For $V_2(x)$ the analog of Fig.~\\ref{F1} is Fig.~\\ref{F5}.\nAgain, we have taken the strength parameter $A_2$ to be $30$ and we observe one\ncomplex-conjugate pair of bound-state eigenvalues for $L=10$ in panel (a) and\ntwo complex-conjugate pairs of bound-state eigenvalues for $L=100$ in panel (b).\nThe new pair of bound-state eigenvalues is uncovered as the continuum\neigenvalues collapse towards the real axis. \n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig5a.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig5b.png}\n\\end{center}\n\\caption{[Color online] Energy spectra for $V_2(x)$ with $A_2=30$. Panel (a)\nshows the eigenvalues for $L=10$ and panel (b) shows the eigenvalues for\n$L=100$. Like the case for the Scarf-II potential $V_1(x)$, the continuum\npart of the spectrum is slightly complex until the critical point near $21$,\nafter which the continuum eigenvalues are real. One pair of bound-state\neigenvalues (black circles) is visible in panel (a) but as $L$ is increased\nto $100$ in panel (b), a new pair of bound-state eigenvalues is uncovered.}\n\\label{F5}\n\\end{figure}\n\nIf we increase $L$ to $1000$, we observe in Fig.~\\ref{F6} that the bound-state\neigenvalues do not move. However, the continuum eigenvalues lie very close to\nthe real axis.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig6.png}\n\\end{center}\n\\caption{[Color online] Energy eigenvalues for the $V_2(x)$ potential for\n$A_2=30$. For this calculation $L=1000$. Note that the discrete bound-state \nenergies have not changed from their values in Fig.~\\ref{F5}(b). However, the \ncontinuum part of the spectrum has moved closer to the real axis.}\n\\label{F6}\n\\end{figure}\n\nAs with the Scarf-II potential $V_1(x)$, there is a transition in the continuum\npart of the spectrum that for this model occurs near $21$. To examine this\ntransition, we plot the eigenvalues on a logarithmic scale in Fig.~\\ref{F7}.\nObserve that near $21$ the continuum eigenvalues undergo an abrupt jump in\ntheir imaginary parts of 10 orders of magnitude.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig7.png}\n\\end{center}\n\\caption{[Color online] Logarithmic plot of the energy eigenvalues for the\npotential $V_2(x)$ plotted for $L=10$, $100$, and $1000$. Like the eigenvalues\nin Fig.~\\ref{F3}, the continuum eigenvalues here undergo an abrupt jump at the\ndashed line near the critical value close to $21$, where the imaginary parts of\nthe continuum eigenvalues suddenly drop by about 10 orders of magnitude.\nThe location of this line is insensitive to the value of $L$.}\n\\label{F7}\n\\end{figure}\n\n\\section{Eigenvalue behavior of $V_3(x)$}\n\\label{s4}\nThe spectral structure associated with the potential $V_3(x)$ is qualitatively\nsimilar to that of $V_2(x)$. We take the strength parameter $A_3=30$ and\nplot the eigenvalues for $L=10$ in Fig.~\\ref{F8}, panel (a), and for $L=100$\nin Fig.~\\ref{F8}, panel (b).\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig8a.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig8b.png}\n\\end{center}\n\\caption{[Color online] Energy eigenvalues for $V_3(x)$ with $A_3=30$. In panel\n(a) we display the data for $L=10$ and in panel (b) we display the data for\n$L=100$. Observe that as $L$ increases, the continuous eigenvalues approach the\nreal axis. However the discrete complex-conjugate bound-state eigenvalues\n(black circles) remain fixed.}\n\\label{F8}\n\\end{figure}\n\nObserve that as $L$ is increased from $10$ to $100$, the discrete bound-state\neigenvalues do not move but the continuum part of the spectrum rapidly\napproaches the real axis. In Fig.~\\ref{F9} we increase the value of $L$ to 1000.\nThis higher-accuracy calculation shows that the continuum eigenvalues are\nextremely close to the real axis. However, there is still a critical point where\nthe continuum eigenvalues go from having a small imaginary part to a vanishing\nimaginary part. This transition point is near $27$ and the transition is\nindicated in Fig.~\\ref{F10} by a dashed line. Once again, the logarithmic plot\nshows that at the transition the imaginary parts of the continuum eigenvalues\nabruptly drop by about ten orders of magnitude.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig9.png}\n\\end{center}\n\\caption{[Color online] Energy eigenvalues for $V_3(x)$ with $A_3=30$\ncalculated at $L=1000$.}\n\\label{F9}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig10.png}\n\\end{center}\n\\caption{[Color online] Logarithmic plot of the data in Figs.~\\ref{F8} and\n\\ref{F9}. As in Figs.~\\ref{F3} and \\ref{F7} we see once again that there is a\ntransition point, in this case near 27, at which the imaginary parts of the\ncontinuum eigenvalues suddenly drop by about ten orders of magnitude. The\nlocation of this transition, which is indicated by a dashed line, appears to be\nindependent of the choice of $L$.}\n\\label{F10}\n\\end{figure}\n\n\\section{Eigenvalue behavior of $V_4(x)$}\n\\label{s5}\nThe pattern of eigenvalues associated with $V_4(x)$ is similar to that of\n$V_1(x)$, $V_2(x)$, and $V_3(x)$. For this potential we take the strength\nparameter $A_4=3$ and plot the spectra for $L=10$ and $L=100$ in Fig.~\\ref{F11}\nand for $L=1000$ in Fig.~\\ref{F12}. These figures show no qualitatively new\nfeatures.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig11a.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig11b.png}\n\\end{center}\n\\caption{[Color online] Energy eigenvalues associated with $V_4(x)$ for $L=10$\nin panel (a) and for $L=100$ in panel (b). The strength parameter $A_4=3$. One\npair of bound-state energies (black circles) can be seen in panel (a) but a\nnew pair becomes visible in Panel (b).}\n\\label{F11}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig12.png}\n\\end{center}\n\\caption{[Color online] Energy eigenvalues for $V_4(x)$ with $A_4=3$ for\n$L=1000$. Note that the bound-state eigenvalues have not changed from their\nposition from those in panel (b) of Fig.~\\ref{F11}.}\n\\label{F12}\n\\end{figure}\n\nA logarithmic plot of the eigenvalue data in Figs.~\\ref{F11} and \\ref{F12} is \nshown in Fig.~\\ref{F13}. Once again we see a transition, in this case near\n$9.5$, at which the continuum eigenvalues abruptly drop in magnitude by about\nten orders of magnitude. The location of the transition is again insensitive to\nthe value of $L$.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig13.png}\n\\end{center}\n\\caption{[Color online] Logarithmic plot of the eigenvalue data in\nFigs.~\\ref{F11} and \\ref{F12}. The data in this figure, like the data in\nFigs.~\\ref{F3}, \\ref{F7}, and \\ref{F10}, indicates that the sharp transition,\nwhich in this case is close to $9.5$ is not sensitive to the value of $L$.}\n\\label{F13}\n\\end{figure}\n\n\\section{Eigenvalue behavior of $V_5(x)$}\n\\label{s6}\nThe most interesting and surprising results that we have obtained concern the \neigenspectrum associated with $V_5(x)$. For this potential we take the strength\nparameter $A_5=10$. Because this is a long-range potential, it is not easy to\nobtain accurate and trustworthy numerical results, and we have had to do the\n$L=1000$ calculation in {\\it quadruple} precision (for all other results in\nthis paper double precision is sufficient). In Fig.~\\ref{F14} we plot the\neigenvalues for $L=10$ in panel (a) and $L=100$ in panel (b). There is one pair\nof bound-state eigenvalues in panel (a). When we increase the size of the\ninterval, we see in panel (b) that the continuum spectrum has dropped much\ncloser to the real axis and has uncovered three new pairs of bound-state\neigenvalues.\n\nFigure~\\ref{F14} reveals two new effects that we have not observed in our\nstudies of short-range potentials. First, the sequence of bound-state\neigenvalues has {\\it turned around} and is heading backward towards the origin.\nIn Figs.~\\ref{F1}, \\ref{F5}, \\ref{F8}, and \\ref{F11} the real parts of the\neigenvalues are increasing, not decreasing. Second, the transition in the\ncontinuum part of the spectrum at which the eigenvalues become real is no\nlonger a fixed point on the real axis; rather, the transition point is moving\nup the real axis as $L$ increases. In panel (a) the transition is near\n$16$ but in panel (b) it is near $28$.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig14a.png}\\hspace{-.1cm}\n\\includegraphics[scale=0.50]{Fig14b.png}\n\\end{center}\n\\caption{[Color online] Energy eigenvalues associated with $V_5(x)$ with\nstrength parameter $A_5=10$. In panel (a) we take $L=10$ and we observe one\ncomplex-conjugate pair of bound-state eigenvalues; in panel (b) we take $L=100$\nand observe three new pairs of complex bound-state eigenvalues. Unlike the\nresults for short-range potentials, this figure shows that the sequence of\nbound-state eigenvalues is turning around and heading back towards the\norigin. Also, the transition points in the continuum part of the spectrum\nare not fixed but are moving up the real axis from about $16$ in panel (a)\nto about $28$ in panel (b).}\n\\label{F14}\n\\end{figure}\n\nIf we increase $L$ to $1000$, Fig.~\\ref{F15} shows that there are now {\\it\nnine} complex-conjugate pairs of bound-state eigenvalues (which are not easy\nto see clearly). This sequence of bound-state eigenvalues is tending towards the\norigin. To observe the bound-state eigenvalues more clearly we have replotted in\nFig.~\\ref{F16} the data in Fig.~\\ref{F15} on a log-log plot. We can see on\nthis plot that the sequence of bound-state eigenvalues is becoming linear. \n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig15.png}\n\\end{center}\n\\caption{[Color online] Energy eigenvalues associated with $V_5(x)$ with\nstrength parameter $A_5=10$. In this figure we have have increased $L$ from \n$100$ in Fig.~\\ref{F14} (right panel) to $1000$ and there are now nine complex\npairs of bound-state energies (not easy to distinguish). The numerical\ncalculations needed to produce this figure required quadruple precision.}\n\\label{F15}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig16.png}\n\\end{center}\n\\caption{[Color online] Plot of the eigenvalue data in Fig.~\\ref{F15} on a\nlog-log graph. In this plot we can now easily see nine bound-state eigenvalues.}\n\\label{F16}\n\\end{figure}\n\nTo observe the transition points in the continuum part of the eigenspectrum,\nwe have plotted the data in Figs.~\\ref{F14} and \\ref{F15} on a logarithmic graph\nin Fig.~\\ref{F17}. Note that the transition point in the continuum eigenvalues\nfrom complex to real is moving up the real axis; it is no longer fixed as it was\nfor finite-range potentials. Observe that at the transition near $40$ there is a\ndrop of nearly $20$ (and not $10$) orders of magnitude. To see this effect\nrequires that we use quadruple and not double precision in our numerical\ncalculations.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.50]{Fig17.png}\n\\end{center}\n\\caption{[Color online] Logarithmic plot of the eigenvalue data in\nFigs.~\\ref{F14} and \\ref{F15}. Observe that the transition point in the\ncontinuum part of the spectrum is moving up the real axis as $L$ increases\nfrom $L=10$ to $L=100$ to $L=1000$.}\n\\label{F17}\n\\end{figure}\n\nThe most interesting aspect of the long-range potential $V_5(x)$ is that it\nappears to confine an infinite number of bound states and the complex\nbound-state energies appear to be approaching $0$. To verify this we use\nRichardson extrapolation \\cite{R14} to study the behavior of the sequence of\nbound-state energies.\n\nRichardson extrapolation enables one to find the limit of the sequence $\\{a_k\\}$\nas $k\\to\\infty$ if the limit is a finite number. Given such a sequence we can\ncalculate more and more accurate Richardson extrapolants, which converge faster\nto the limiting value. The formulas for the first five Richardson extrapolants\nare given by\n\\begin{eqnarray}\nR_k^{(1)}&=&(k+1)a_{k+1}-ka_k,\\nonumber\\\\\nR_k^{(2)}&=&\\left[(k+2)^2a_{k+2}-2(k+1)^2a_{k+1}+k^2a_k\\right]\/2!,\\nonumber\\\\\nR_k^{(3)}&=&\\left[(k+3)^3a_{k+3}-3(k+2)^3a_{k+2}+3(k+1)^3a_{k+1}-k^3a_k\\right]\/\n3!,\\nonumber\\\\\nR_k^{(4)}&=&\\left[(k+4)^4a_{k+4}-4(k+3)^4a_{k+3}+6(k+2)^4a_{k+2}\\right.\n\\nonumber\\\\\n&&\\left.\\qquad\\qquad-4(k+1)^4a_{k+1}+k^4a_k\\right]\/4!,\\nonumber\\\\\nR_k^{(5)}&=&\\left[(k+5)^5a_{k+5}-5(k+4)^5a_{k+4}+10(k+3)^5a_{k+3}\\right.\n\\nonumber\\\\\n&&\\left.\\qquad\\qquad-10(k+2)^5 a_{k+2}+5(k+1)^5a_{k+1}+k^5a_k\\right]\/5!.\n\\label{e2}\n\\end{eqnarray}\n\nFrom our numerical analysis, we have determined that the $k$th bound-state\nenergy $E_k$ has the asymptotic form \n\\begin{equation}\nE_k\\sim\\frac{\\alpha}{k^2}\\pm i\\frac{\\beta}{k^3}\\qquad(k\\gg1),\n\\label{e3}\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ are real numbers. This shows that the bound-state\neigenvalues associated with $V_5(x)$ share many of the quantitative features of\nthe Balmer series for the hydrogen atom. As indicated in the tables \\ref{t1} and\n\\ref{t2}, we have determined that the numerical value of $\\alpha$ is about $25$\nand the numerical value of $\\beta$ is about $61$. To obtain these results we\nhave multiplied the real part of $E_k$ by $k^2$ and the imaginary part of\n$E_k$ by $k^3$ and computed the first five Richardson extrapolations.\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n$k$ & ${\\rm Re}\\,E_k$ & $k^2\\,{\\rm Re}\\,E_k$ & $R_k^{(1)}$ & $R_k^{(2)}$ &\n$R_k^{(3)}$ & $R_k^{(4)}$ & $R_k^{(5)}$ \\\\\n\\hline\n1& 0.83298288 & 0.83298 & 10.2355 & 26.6628 & 29.8431 & 23.8084 & 24.2927\\\\\n2& 1.38356468 & 5.53426 & 21.1871 & 29.0481 & 25.0154 & 24.2120 & 25.5106\\\\\n3& 1.19465086 & 10.7519 & 25.1176 & 26.6284 & 24.4798 & 25.1396 & 25.2280\\\\\n4& 0.89645517 & 14.3433 & 25.7219 & 25.5541 & 24.8568 & 25.1949 & 25.0599\\\\\n5& 0.66476032 & 16.6190 & 25.6660 & 25.2553 & 25.0258 & 25.1199 &    --- \\\\\n6& 0.50352322 & 18.1268 & 25.5486 & 25.1692 & 25.0676 &    ---  &    --- \\\\\n7& 0.39157331 & 19.1871 & 25.4538 & 25.1354 &    ---  &    ---  &    --- \\\\\n8& 0.31203794 & 19.9704 & 25.3830 &    ---  &    ---  &    ---  &    --- \\\\\n9& 0.25397318 & 20.5718 &    ---  &    ---  &    ---  &    ---  &    --- \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{t1} Real parts of the first nine eigenvalues $E_k$ $(1\\leq k\\leq\n9)$ associated with $V_5(x)$ and the first five Richardson extrapolants\nconstructed from the sequence $\\{k^2\\,{\\rm Re}\\,E_k\\}$. Evidently, the real\nparts of the eigenvalues vanish like $\\alpha k^{-2}$, where $\\alpha$ is roughly\n$25$.}\n\\end{table}\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n$k$ & ${\\rm Im}\\,E_k$ & $k^3\\,{\\rm Im}\\,E_k$ & $R_k^{(1)}$ & $R_k^{(2)}$ &\n$R_k^{(3)}$ & $R_k^{(4)}$ & $R_k^{(5)}$ \\\\\n\\hline\n1& 3.90859038 & 3.90859 & 29.8484 & 63.7004 & 62.5280 & 53.5920 & 61.0166\\\\\n2& 2.10981263 & 16.8785 & 52.4164 & 62.8211 & 55.3792 & 59.7791 & 62.5903\\\\\n3& 1.06386938 & 28.7245 & 57.6188 & 58.3560 & 58.3125 & 61.7871 & 61.3005\\\\\n4& 0.56168819 & 35.9480 & 57.9136 & 58.3342 & 60.2980 & 61.4830 & 60.9267\\\\\n5& 0.32272930 & 40.3412 & 58.0538 & 59.1758 & 60.8905 & 61.1739 &    --- \\\\\n6& 0.20043182 & 43.2933 & 58.3744 & 59.8188 & 61.0165 &    ---  &    --- \\\\\n7& 0.13250064 & 45.4477 & 58.7355 & 60.2180 &    ---  &    ---  &    --- \\\\\n8& 0.09200917 & 47.1087 & 59.0650 &    ---  &    ---  &    ---  &    --- \\\\\n9& 0.06644330 & 48.4372 &    ---  &    ---  &    ---  &    ---  &    --- \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{t2} Imaginary parts of the first nine eigenvalues $E_k$ $(1\\leq\nk\\leq9)$ associated with $V_5(x)$ and the first five Richardson extrapolants\nconstructed from the sequence $\\{k^3\\,{\\rm Im}\\,E_k\\}$. The imaginary parts of\nthe eigenvalues vanish like $\\beta k^{-3}$, where $\\beta$ is roughly $61$.}\n\\end{table}\n\n\\section{Conclusions}\n\\label{s7}\n\nIn this paper we have studied numerically the five $\\ensuremath{\\mathcal{PT}}$-symmetric potentials\nin (\\ref{e1}) that have continuous spectra. Each of these potentials is pure\nimaginary and vanishes as $|x| \\to\\infty$. The interesting feature of these\npotentials is that even though they vanish at $\\pm\\infty$, they still confine\nbound states. Of course, an imaginary potential can confine bound states. For\nexample, the $ix^3$ potential has an infinite number of bound states \\cite{R1}.\nHowever, this potential becomes {\\it stronger} as $|x|\\to\\infty$. We emphasize\nthat the five potentials that we have studied here {\\it decay} and become {\\it\nweaker} as $|x|\\to\\infty$. Even more remarkable is the fact that the potential\n$V_5(x)$, which decays very slowly as $|x|=\\infty$, binds an {\\it infinite}\nnumber of bound states and that the sequence of bound-state energies\nasymptotically approaches the Balmer series for the hydrogen atom.\n\n\\vspace{0.5cm}\n\\footnotesize\n\\noindent\nCMB thanks the Alexander von Humboldt Foundation for partial financial support.\n\\normalsize\n\n\\vspace{0.5cm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Sec:Introduction}\n\nThe propagation of nonlinear waves in inhomogeneous media has captured much recent interest in physics and biophysics \\cite{Ivancevic2013}. Nonlinear\npulse excitations regulate the functioning in many physical models of biological systems, such as the Purkinje's fibres of the cardiac muscles\n\\cite{Aslanidi1999, Mornev2000}, neural fibres \\cite{Scott1975}, DNA chains \\cite{Polozov1988, Yakushevich2004} and muscular networks\n\\cite{Gojkovic2016}. The study of the effects that can be produced on these regulatory signals by the presence of spatial inhomogeneities, such as\nlocalised injuries and necrotic sites, has gained major interest in recent works \\cite{Mornev2000, Grinevich2015, Grinevich2016}. For instance,\ninjuries in the heart muscular tissue may produce annihilation, transmission or reflection of incident waves, depending on the geometrical size and\nshape of the defects. These disturbances may produce a disorder in the rhythmic contractions of the muscular walls,\nleading to cardiac arrhythmias and different pathologies \\cite{Mornev2000}. A similar situation occurs in the propagation of topological\none-dimensional solitons in DNA chains, which has been considered as a suitable model to explain the formation of open states or {\\it bubbles} in the\ndouble helix \\cite{Grinevich2015}. This bubble formation is an essential mechanism for the regulation and control of gene transcription\n\\cite{Polozov1988, Yakushevich2004}. Structure heterogeneities in the DNA chain and the presence of boundary interphase walls may affect the\npropagation regime of these regulatory signals during replication processes.\n\nIn the scattering of waves by localised defects, a crucial point to consider is that nonlinear waves exhibit a highly complex behaviour. Transmission,\nreflection, and annihilation are far from being the only possible phenomena that may occur. Indeed, solitons with an internal structure, such\nas sine-Gordon (sG) kinks and breathers, may produce many remarkable phenomena when colliding with localised inhomogeneities. The main phenomenon is\nthe internal mode instability \\cite{Gonzalez2002, Gonzalez2003}, which may cause the breaking of solitons, the formation of multikinks, the creation of\nkink-antikink pairs and the formation of localised structures \\cite{GarciaNustes2012, GarciaNustes2017}. The formation of such structures in sG\nsystems may have important implications not only in biological systems \\cite{Ivancevic2013}, but also in physical systems such as\nJosephson Junctions (JJs) \\cite{Barone1982} and ferromagnetic materials \\cite{Mikeska1978}.\n\nIn this article, we investigate the complex phenomena exhibited by a two-dimensional sG system with localised inhomogeneities. Formation of localised structures with bubble-like and drop-like shape are produced in two situations: (a) for line solitons trapped by a single localised\ninhomogeneity, and (b) for travelling line solitons colliding with an array of localised inhomogeneities. The structure formation is understood using\nthe one-dimensional theory of activation of internal modes in sG solitons. The outline of the article is as follows: Section \\ref{Sec:Theory} gives an overview of the one-dimensional theory of activation of internal modes of sG kinks under the action of inhomogeneous external forces. In Sec.\n\\ref{Sec:StationarySolitons} the two-dimensional model is introduced, and the internal mode instabilities of line solitons are studied. To avoid any scattering effect, we\nconsider steady line solitons. In Sec. \\ref{Sec:Arrays}, the scattering of line solitons with a spatial\narray of localised inhomogeneities is investigated. Finally, Sec. \\ref{Sec:Conclusions} gives some concluding remarks.\n\n\n\n\\section{Preliminaries}\n\\label{Sec:Theory}\n\nIn this section, an overview of well-known preliminary results of sG kinks under the action of external forces is given. These results provide\nthe theoretical framework for the understanding of the phenomena reported in the subsequent sections.\n\n\\subsection{Kinks driven by homogeneous external forces}\n\nThe driven and damped two-dimensional sG system is a particular case of the more general nonlinear Klein-Gordon (KG) system\n\\begin{equation}\n \\label{Eq01}\n  \\partial_{tt}\\phi(\\mathbf{r},t)-\\nabla^2\\phi(\\mathbf{r},t)+\\gamma\\partial_t\\phi(\\mathbf{r},t)-G(\\phi)=f,\n\\end{equation}\nwhere $\\mathbf{r}=(x,y)$, $\\gamma$ is a linear-damping coefficient, $f$ is a constant external force and $G(\\phi)=-\\partial U(\\phi)\/\\partial\\phi$,\nwith the nonlinear potential $U(\\phi)$ possessing at least two minima separated by a barrier.\nIt is well known that these two minima are fixed points of the\nassociated dynamical system, and that kink solutions are heteroclinic trajectories joining such fixed points \\cite{vanSaarloos1990}. For sG systems,\nthe potential function is given by $U(\\phi)=1-\\cos\\phi$ \\cite{Peyrard2004}, and the equation (\\ref{Eq01}) for $f=0$ has the well known stationary\nkink-antikink solutions (line solitons), given by $ \\phi_{\\pm}(\\mathbf{r})=4\\arctan\\exp\\left(\\pm x\\right)$. Here, $\\phi_+$ and $\\phi_-$ reads\nfor the kink and antikink, respectively.\n\nA space independent force with a constant value $f$ leads to an effective potential of the form $U_{eff}(\\phi)=U(\\phi)-f\\phi$. In this case, kink\nsolutions exist for equation (\\ref{Eq01}) only if this effective potential still have at least two minima separated by a barrier. That condition ceases to\nbe fulfilled for a certain critical value $f_c$ of the external force. When $f>|f_c|$ kink solutions do not exist, which means that an initially at\nrest kink will be destroyed by the external force \\cite{Gonzalez1992, Gonzalez2007}. The exact value of $f_c$ depends indeed on the particular\npotential function of the system. For instance, using the sG potential, minima $\\phi_j$ $(j=0,1\\ldots)$ of the effective potential satisfy the\nequation $\\sin\\phi_j=f$, which has no real solutions for $\\phi_j$ if $|f|>1$. Thus, the critical value is $f_c=1$ for sG systems. \n \nIf $f<|f_c|$, kink solutions in perturbed sG systems exists, and its dynamics in the neighbourhood of fixed points and separatrices can be\ninvestigated using the so-called qualitative theory of dynamical systems \\cite{Guckenheimer1986, McLaughlin1978, Kivshar1989, Sanchez1998,\nChacon2008, Gonzalez2007-II}. Based on this, it is possible to generalise the results to other equations that are topologically equivalent to those with the exact\nsolutions \\cite{Gonzalez1996, Gonzalez2003}.\n\n\\subsection{Soliton breaking by inhomogeneous forces}\n\nIn the sense of the qualitative theory of dynamical systems, for a general $x$-dependent force $F(\\mathbf{r})=f(x)$, it is known that a zero of\n$f(x)$ at $x=x_*$ is an equilibrium point for the centre-of-mass position of the kink \\cite{Gonzalez1992, Gonzalez1996, Gonzalez2003, Gonzalez2008}. The\nequilibrium is stable if $\\left.df(x)\/dx\\right|_{x=x_*}>0$, and unstable if $\\left.df(x)\/dx\\right|_{x=x_*}<0$. If the kink is on a stable\nequilibrium point, the shape and position of the kink will be recovered for any initial perturbation \\cite{Gonzalez2003}. On the contrary, if the\nkink is in an unstable equilibrium position, it is stretched by the inhomogeneous force that is acting on its body in opposite directions.\nThe value $f_c$ is the limit of $|f(x)|$ that the kink can resist the stretching without being destroyed \\cite{Gonzalez2007}. If $|f(x)|>f_c\\forall x$, the force will destroy the line soliton.\n\nNotwithstanding, there is a more subtle mechanism for the kink destruction when $|f(x)|>f_c$ only in a localised region in space. This mechanism\nis the internal mode instability \\cite{Gonzalez2007-II}. In one-dimensional systems,\nthe stability analysis of internal modes of sG kinks has been solved exactly \\cite{Gonzalez2002, Gonzalez2003}. The procedure starts by introducing\nthe ansatz\n\\begin{equation}\n \\label{Eq02}\n\\phi(x)=4\\arctan\\exp\\left(\\pm Bx\\right), \n\\end{equation}\nwhere $B$ is a parameter that controls the width of the kink. Solving an inverse problem, one obtains that equation (\\ref{Eq02}) is a solution of the \none-dimensional sG equation if the external force is given by\n\\begin{equation}\n \\label{Eq02Force}\nF(x) = 2(B^2 -1)\\mbox{sinh}(Bx)\\mbox{sech}^2(Bx),\n\\end{equation}\nwhich is an antisymmetric spatial function that vanishes exponentially for $x\\to\\pm\\infty$. For increasing (decreasing) values\nof $B$, the force is less (more) localised in space and has decreasing (increasing) extreme values. Thus, the parameter $B$ also controls the extreme\nvalues and the extension of $F(x)$ in space.\nInteractions of DNA special sites with ligands or perturbation of fluxons by dipole currents in JJs are typical examples of strong and local\nexternal influences with the same topological form as equation (\\ref{Eq02Force}) \\cite{Polozov1988, Malomed2004}.\nFrom the stability analysis follows that the first stable internal shape mode arises for $1\/6<B^2<1\/3$, while for $B^2<1\/6$ many other internal modes can\nappear. For $B^2<2\/[\\Lambda_*(\\Lambda_*+1)]$, where $\\Lambda_*=(5+\\sqrt{17})\/2$, the first internal mode becomes unstable, leading to soliton\nbreakups and shape instabilities \\cite{Gonzalez1992,Gonzalez2002, Gonzalez2003}. These instabilities play a central role in the formation of\nstructures reported in the following sections.\n\n\n\\section{Stationary line solitons in the presence of a single localised inhomogeneity}\n\\label{Sec:StationarySolitons}\n\n\\begin{figure}[t]\n \\begin{center}\n  \\scalebox{0.27}{\\includegraphics{fig01}}\\scalebox{0.27}{\\includegraphics{fig02}}\\\\\n  \\scalebox{0.27}{\\includegraphics{fig03}}\\scalebox{0.27}{\\includegraphics{fig04}}\n  \\caption{(Colour online) (a) Inhomogeneous force of equation (\\ref{Eq04}) for $B=0.1$ and $\\sigma=15$. The inset shows the static line soliton\n  used as initial condition in the numerical simulations of section \\ref{Sec:StationarySolitons}. (b) A drop-like structure after $10\\,000$ time\n  iterations for $B=0.28$ and $\\sigma=8.9$. The inset shows the elliptic-like structure that precedes the drop formation. The value of the field\n  $\\phi$ is indicated with the same colour scale. (c) A bubble-like structure after $10\\,000$ time iterations for $B=0.27$ and $\\sigma=10$.\n  The inset shows the elliptic-like structure that precedes the bubble formation. (d) A multi-structure formed after $8\\,300$ time iterations for\n  $B=0.1$ and $\\sigma=6$.}\n  \\label{fig01}\n \\end{center}\n\\end{figure}\n\n\nFollowing the one-dimensional system, consider the two-dimensional perturbed sG equation\n\\begin{equation}\n \\label{Eq03}\n \\partial_{tt}\\phi(\\mathbf{r},t)=\\nabla^2\\phi(\\mathbf{r},t)-\\sin\\phi(\\mathbf{r},t)-\\gamma\\partial_t\\phi(\\mathbf{r}, t)+F(\\mathbf{r}),\n\\end{equation}\nwhere the inhomogeneous force $F(x,y)$ is given by \\cite{GarciaNustes2017}\n\\begin{equation}\n\\label{Eq04}\nF(x, y)=2(B^2 -1)\\mbox{sech}(Bx)\\mbox{tanh}(Bx)e^{-y^2\/\\sigma^2},\n\\end{equation}\nand a real parameter $\\sigma$ has been introduced to control the spread of the force in the $y$-direction.\nFigure \\ref{fig01}(a) shows the force of equation (\\ref{Eq04}), which represents a single localised inhomogeneity centred at the origin. The point\n$x_*=0$ will be an equilibrium point for two-dimensional kinks. The external force of equation (\\ref{Eq04}) is physically realisable in real\nexperiments. Indeed, the sG equation (\\ref{Eq03}) with this kind of external perturbation models the propagation of fluxons in two-dimensional JJs\nwith a current dipole device inserted into one of the superconducting electrodes \\cite{Ustinov2002, Malomed2004, GarciaNustes2017}.\n\nThis section shows how the inhomogeneity of equation (\\ref{Eq04}) affects the dynamics of a line soliton, having a special focus on the dynamics of\nthe soliton internal structure. To avoid any scattering effect, consider an initially at rest kink with its centre-of-mass located at the equilibrium point. This initial condition is depicted in the inset of figure \\ref{fig01}(a) and is given by\n$\\phi(x,y,0)=4\\arctan\\,\\exp(x\/L)$ with $\\partial_t{\\phi}(x,y,0)=0$, where $L$ is an arbitrary parameter.\nWe have used an explicit finite difference scheme of the second order of\naccuracy for the numerical solution of equation (\\ref{Eq03}), with $\\gamma=0.01$ and homogeneous Neumann boundary conditions. The used regular mesh size is $1\\,200\\times 1\\,200$. The space intervals are $\\Delta x=\\Delta y=0.1$, and the time increment is \\hbox{$\\Delta t=0.02$}.\n\nFigure \\ref{fig01} shows the main results obtained from numerical simulations.\nFigure \\ref{fig01}(b) shows a drop-like structure created for $B=0.28$ and $\\sigma=8.9$. The local instability of the shape mode leads to a\nlocal break-up of the initially at rest line soliton. This initial breaking consists in a local creation of a pair kink-antikink, giving an\nelliptic-like structure that grows in time [see the inset of figure \\ref{fig01}(b)]. This kink zone propagates until the localised force is\nweak enough, and the contribution of curvature effects and kink interactions are important. The elliptic form eventually collapses due to the attractive interaction forces between the kink and\nantikink, developing a final stable drop-like structure in the negative region of $x$. Notice that the drop is sustained by the positive part of the external force. A kink is still present in the system due to the conservation of the topological charge and propagates\ntowards the negative $x$-direction with a velocity $\\mathbf{v}$, as showed in figure \\ref{fig01}(b).\n\nIf we proceed to diminish the value of $B$, the preferred structure is one with the reverse topological charge: a bubble-like\nstructure. Figure \\ref{fig01}(c) shows a bubble-like structure created for $B=0.27$ and $\\sigma=10$. In this case, the force is extended enough in\nthe $x$-direction to prevent the collapse and sustain the walls of the elliptic form, forming a bubble sustained by the negative part of the force. The mechanism of bubble formation\nis similar to the drop formation, giving that both are preceded by an elliptic-like form that grows in time [see inset of figure \\ref{fig01}(c)].\nOver a critical value of the elliptical area, the kink-antikink interaction is too weak to produce the collapse of the bubble. In this case, the external force supports the structure. Otherwise, the external force cannot sustain the walls of the bubble and the structure collapses by kink-antikink interaction, triggering the formation of a drop-like structure. Therefore, the interplay of both parameters $B$ and $\\sigma$ will determine which kind of structure will be formed.\nIndeed, further numerical simulations have show that these instabilities in sG kinks can be activated varying any of both parameters $B$ or $\\sigma$\n\\cite{GarciaNustes2017}.\n\nIf we continue further diminishing the values of parameters $B$ and $\\sigma$, it is expected that the line soliton will exhibit more complex\nphenomena due to the local destabilisation of more internal modes. Indeed, there is a coexistence region in the parameter space where a bubble\nand a drop merge in a stable structure, like the bubble-drop bound state depicted in figure \\ref{fig01}(d) for $B=0.1$ and $\\sigma=6$. From the\none-dimensional theory, it is known that for $B\\lesssim0.2564$ the formation of two-kink solitons occurs \\cite{GarciaNustes2012}. The later explains the\ntwo-kink profile of the bubble showed in figure \\ref{fig01}(d). More complex multi-structures can be observed for smaller values of $B$ and\n$\\sigma$, such as multi-kink bubbles and drops.\n\nNotice that the localised structures showed in Fig.~\\ref{fig01} are called \\emph{bubbles} and \\emph{drops} giving that the equation (\\ref{Eq03}) also appears in the description of structural phase transitions \\cite{GarciaNustes2017, Gonzalez1999, Gonzalez2006}. The results of this article can\nbe interpreted as the driving of a system to a new phase by the activation of a local instability.\n\n\n\\section{Scattering of solitons in an array of localised inhomogeneities}\n\\label{Sec:Arrays}\n\nIn order to simulate some real situations in physical and biological systems, this section deals with travelling line solitons that collide with a\ndistribution of localised inhomogeneities. Consider in equation (\\ref{Eq03}) the following spatial distribution of inhomogeneous forces\n\\begin{equation}\n \\label{Eq05}\nF(x,y)= F_o+2\\sum_{i=1}^4(B_i^2-1)\\mbox{sech}\\left[B_i(x-x_i)\\right]\\mbox{tanh}\\left[B_i(x-x_i)\\right]e^{-(y-y_i)^2\/\\sigma^2},\n\\end{equation}\nwhere $F_o=0.1$ is a constant homogeneous contribution to the total external force, $(x_i,y_i)$ is the coordinate-position of the core of the $i$-th inhomogeneity, $B_i$ is its\nrespective control parameter, $\\sigma=10$, and $i\\in\\left\\{1,\\,2,\\,3,\\,4\\right\\}$. Notice that the inhomogeneous contribution of force (\\ref{Eq05})\nis given by the superposition of forces of the same form as equation (\\ref{Eq04}).\n\nThe position of the inhomogeneities are fixed for all simulations in this section, and are given by $\\mathbf{r}_1=(-15, 25)$, $\\mathbf{r}_2=(15, 25)$, $\\mathbf{r}_3=(-15, -25)$ and\n$\\mathbf{r}_4=(15, -25)$. With these coordinates, the inhomogeneities are placed at the vertices of a rectangle centred at the origin.\nThe initial condition is given by $\\phi(x,y,0)=4\\arctan\\,\\exp[(x-xo)\/L]$ with $\\partial_t\\phi(x,y,0)=0$, which is an initially at rest line soliton\nplaced some distance away from the array of inhomogeneities. The initial $x$-coordinate of the centre-of-mass of the kink is given by $x_o=50$. Due to the homogeneous part of the force (\\ref{Eq05}) and the damping term of equation (\\ref{Eq03}), the line soliton accelerates towards the negative $x$-direction before reaching an asymptotically constant velocity. When the soliton collides with the inhomogeneities, many remarkable phenomena may occur due to possible internal mode\ninstabilities.\n\n\\begin{figure}[t]\n \\begin{center}\n  \\scalebox{0.32}{\\includegraphics{fig05}}\\scalebox{0.32}{\\includegraphics{fig06}}\\\\\n  \\scalebox{0.32}{\\includegraphics{fig07}}\n  \\caption{(Colour online) Scattering of travelling line solitons after collisions with a distribution of localised inhomogeneities. (a) Generation\n  of a bubble induced by the collapse of an echo-wave, for $B_1=0.27$, $B_2=3.0$, $B_3=B_4=0.3$, and $\\gamma=0.5$. (b) Kink transmission for\n  $\\gamma=0.1$, $B_1=B_4=0.27$ and $B_2=B_3=1.1$. A bubble and a drop remains trapped at the inhomogeneities after the transmission of the soliton.\n  (c) Kink transmission and formation of complex interacting structures (bubbles and drops) for $\\sigma=10$, $\\gamma=0.01$, and\n  $B_i=0.27$ $(i=1,2,3,4)$.}\n  \\label{fig02}\n \\end{center}\n\\end{figure}\n\nFigure \\ref{fig02}(a) shows the results from numerical simulations for $B_1=0.27$, $B_2=3.0$, $B_3=B_4=0.3$, and $\\gamma=0.5$. For this combination of\nparameters, the $1$st inhomogeneity activates the internal modes of kinks, while the $2$nd inhomogeneity tends to trap them at position\n$\\mathbf{r}_2$. When the line soliton reaches the first line of inhomogeneities, part of the kink is deformed due to the collision with the $1$st\ninhomogeneity, while another part is trapped by the $2$nd inhomogeneity. Notice that this initial bent of the soliton is the so-called \\emph{echo effect} reported in\nbiophysical systems and is considered as responsible for certain types of disorders and cardiac arrhythmias \\cite{Mornev2000}. This echo-wave is eventually stabilised in size, and forms a stable bubble-like structure sustained by the $1$st inhomogeneity. Notice that the echo-waves observed\nin this section have some resemblance with the elliptic-like structures observed in section \\ref{Sec:StationarySolitons} [see insets of figure\n\\ref{fig01}(b) and (c)].\n\nFor smaller values of $\\gamma$ the soliton reaches the inhomogeneities with higher terminal speeds, and we would expect the\nenhancing of the transmission of the soliton across the inhomogeneities. Figure \\ref{fig02}(b) shows the results from numerical simulations for $\\gamma=0.1$, $B_1=B_4=0.27$ and\n$B_2=B_3=1.1$. In this case, the $1$st and $4$th inhomogeneities activate the internal modes of the soliton. The $2$nd and $3$rd\ninhomogeneities tend weakly to trap the soliton. Indeed, when the kink collides with the first pair of inhomogeneities, the $2$nd cannot capture the\nkink, while the $1$st inhomogeneity produces an echo-effect. This echo-wave eventually collapse and forms a stable drop, as showed in figure\n\\ref{fig02}(b). A similar mechanism occurs when the line soliton reaches the second pair of inhomogeneities. However, the speed of the line\nsoliton has diminished due to previous collisions. The echo-wave generated at the $4$th inhomogeneity partially collapse, as indicated in figure\n\\ref{fig02}(b), but is eventually stabilised and forms a bubble. Thus, the relative speed of collision also plays an important role in determining\nwhich kind of structure will be formed. \n\nA highly complex dynamics is observed diminishing the parameters $B_i$ in the array of inhomogeneities. Figure \\ref{fig02}(c) shows the\nresults from numerical simulations when $B_i=0.27$ for all $i$. After the collision, there is an echo effect followed by the formation of bubbles.\nThe inhomogeneities cannot sustain such bubbles, and they collapse to form drops eventually. However, during this drop formation, another pair of\nbubbles are created in the neighbour inhomogeneities. In figure \\ref{fig02}(c) is clearly seen repulsive interactions between these new bubbles and\nthe previously formed drops. Notice that these drops also collide, enhancing the formation of new structures with the topological charge of three around\nthe origin. More complex dynamics can be observed diminishing the values of $B_i$ or $\\sigma$.\n\n\\section{Conclusion}\n\\label{Sec:Conclusions}\n\nIn this article, the formation of localised structures in a two-dimensional sG system has been reported considering two different scenarios: (a) the\npresence of a single localised inhomogeneity, and (b) the presence of an array of localised inhomogeneities. These inhomogeneities are modelled by a family of topologically equivalent external forces. The localised structures are formed with a bubble-like and drop-like shape and are\nsustained by the same inhomogeneities that create them. The structure creation is entirely due to the dynamics of the internal modes of the soliton,\nand not to any scattering effects. However, the internal mode instabilities can also occur during scattering processes, and exhibit a highly complex\ndynamics.\n\nAs a final remark, notice that the reported phenomena may have important implications in biological systems. The formation of bubbles, and the \ntrapping and breaking of such structures by localised inhomogeneities may be related to different functional disorders in similar systems.\nMoreover, the study of the stability and transport of such bubbles may give some light on how to introduce shock perturbations, and produce the\ndesired effects in a controlled way. Gene transcription in DNA chains can also be regulated by the insertion of this kind of external perturbations.\n\n\n\\ack\n\nThis article was written after a conference at the XX Symposium of the Physical Society of Chile (SOCHIFI), held in Santiago de Chile on November 2016.\nThe author thanks the organising committee for this occasion. The author also thanks Prof. M\\'onica A. Garc\\'ia-\\~Nustes and Prof. Jorge A.\nGonz\\'alez for many fruitful discussions and the active collaboration on the subject of this article. This work was supported by CONICYT doctorado nacional No.\n21150292.\n\n\n\\section*{References}\n\n\\providecommand{\\noopsort}[1]{}\\providecommand{\\singleletter}[1]{#1}%\n\\providecommand{\\newblock}{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:one}\n\n\nThe microlensing phenomenon occurs by the gravity of lensing objects regardless of their \nluminosity. Due to this property, microlensing provides an important tool to detect \nvery faint and even dark objects that cannot be observed by other methods. However, \nit is difficult to conclude the faint\/dark nature of the lens just based on the event \ntimescale $t_{\\rm E}$, which is the only observable related to the lens mass for general \nlensing events,  because the timescale is related to not only the lens mass $M$ but \nalso the relative lens-source proper motion $\\mu$ and distance to the lens $D_{\\rm L}$ \nand source $D_{\\rm S}$ by\n\\begin{equation}\nt_{\\rm E} = {\\theta_{\\rm E}\\over \\mu};\\qquad\n\\theta_{\\rm E} = \\sqrt{\\kappa M \\pi_{\\rm rel}};\\qquad\n\\pi_{\\rm rel}={\\rm au} \\left( {1\\over D_{\\rm L}}-{1\\over D_{\\rm S}}\\right),\n\\label{eq1}\n\\end{equation}\nwhere $\\kappa=4G\/(c^2{\\rm au})$ and $\\theta_{\\rm E}$ is the angular Einstein radius.\nIn order to reveal the nature of lenses, their masses should be determined.\n\n\nFor the unique determination of the lens mass, it is required to measure two additional \nobservables. These observables are the angular Einstein radius $\\theta_{\\rm E}$ and the \nmicrolens parallax $\\pi_{\\rm E}$. They are related to the lens mass by \\citep{Gould2000}\n\\begin{equation}\nM={\\theta_{\\rm E}\\over \\kappa\\pi_{\\rm E}}.\n\\label{eq2}\n\\end{equation}\nWith $\\pi_{\\rm E}$ and $\\theta_{\\rm E}$, the distance to the lens is also determined by\n\\begin{equation}\nD_{\\rm L} = {{\\rm au}\\over \\pi_{\\rm E}\\theta_{\\rm E} + \\pi_{\\rm S}},\n\\label{eq3}\n\\end{equation}\nwhere $\\pi_{\\rm S}={\\rm au}\/D_{\\rm S}$.\n\n\nThe angular Einstein radius is determined by detecting deviations in lensing light \ncurves caused by finite-source effects.  Deviations induced by finite-source effects \narise due to the differential magnification in which different parts of the source \nsurface are magnified by different amount.  For events produced by single mass \nobjects, finite-source effects can be detected in the special case in which the \nlens passes over the surface of the source \\citep{Witt1994, Gould1994a, Nemiroff1994}.  \nHowever, the ratio of the angular source radius $\\theta_*$ to the angular Einstein \nradius $\\theta_{\\rm E}$ is very small, $\\rho~\\sim 1\/1000$ for events associated with \nmain-sequence source stars and $\\sim~1\/100$ even for events involved with giant \nsource stars, and thus the rate of source-crossing events is accordingly very low. \nThe chance to detect finite-source effects is relatively much higher for events \nproduced by binary objects. This is because a binary lens forms caustics that can \nextend over an important portion of the Einstein ring, and the event with a source \ncrossing the caustic exhibits a light curve affected by the finite-source effect \nduring the caustic crossing.\n\n\nIn the early-generation lensing surveys that were conducted with a $>1$ day cadence, \nmeasuring the angular Einstein radius by detecting finite-source effects was \nobservationally a challenging task. This is because the duration of the deviation \ninduced by finite-source effects is, in most cases, $< 1$ day and thus it was difficult \nto detect the deviation.  The unpredictable nature of caustic crossings also made it \ndifficult to cover crossings from high-cadence follow-up observations \\citep{Jaroszynski2001}.  \nHowever, with the inauguration of lensing surveys using globally distributed multiple \ntelescopes equipped with wide-field cameras, the observational cadence has been dramatically \nincreased to $< 1$ hour, making it possible to measure $\\theta_{\\rm E}$ for a greatly increased \nnumber of lensing events.  \n\n\nOne channel to measure the microlens parallax is simultaneously observing lensing \nevents from the ground and in space: `space-based microlens parallax' \n\\citep{Refsdal1966, Gould1994b}. The physical size of the Einstein radius for a \ntypical lensing event is of the order of au. Then, if space observations are \nconducted using a satellite in a heliocentric orbit, e.g., {\\it Deep Impact} \n\\citep{Muraki2011} spacecraft or {\\it Spitzer Space Telescope} \\citep{Dong2007, \nCalchi2015, Udalski2015b}, \nthe lensing light curve obtained from the satellite observation will be substantially \ndifferent from that obtained from the ground-based observation.  For events with well \ncovered light curves from both the ground and in space, then, the microlens parallax \ncan be precisely measured by comparing the two light curves. \n\n\n\nAnother channel to measure $\\pi_{\\rm E}$ is analyzing deviations induced by \nmicrolens-parallax effects in lensing light curves obtained from ground-based \nobservations: `annual microlens parallax' \\citep{Gould1992}. In the single frame \nof Earth, such deviations occur due to the positional change of an observer caused \nby the orbital motion of Earth around the Sun.  For typical lensing events produced \nby low-mass stars, however, the event timescale is several dozen days, which comprises \na small fraction of the orbital period of Earth, i.e., year, and thus deviations \ninduced by the annual microlens-parallax effects are usually very minor.  As a \nresult, it is difficult to detect the parallax-induced deviations for general events, \nand even for events with detected deviations, the uncertainties of the measured $\\pi_{\\rm E}$ \nand the resulting lens mass can be considerable.  \n\n\nIn this work, we analyze the binary-lensing event OGLE-2016-BLG-0156.  The light \ncurve of the event, which is characterized by three distinctive widely-separated \npeaks, exhibits pronounced parallax-induced deviations, from which we precisely \nmeasure the microlens parallax.  All the peaks are densely covered from continuous \nand high-cadence survey observations and the analysis of the peaks leads to the \nprecise measurement of the angular Einstein radius.  We characterize the lens by \nmeasuring the masses of the lens components from $\\pi_{\\rm E}$ and $\\theta_{\\rm E}$.\n\n\n\n\n\\section{Observation and Data}\n\nThe source star of the lensing event OGLE-2016-BLG-0156 is located in the bulge \nfield with equatorial coordinates \n$({\\rm R.A.},{\\rm decl.})_{\\rm J2000}=(17:56:36.63, -31:04:40.7)$.  The corresponding \ngalactic coordinates are $(l,b)=(359.4^\\circ, -3.14^\\circ)$. The event was found in \nthe very early part of the 2016 bulge season by the Optical Gravitational Lensing \nExperiment \\citep[OGLE:][]{Udalski2015a} survey.  The OGLE lensing survey was conducted \nusing the 1.3 m telescope of the Las Campanas Observatory, Chile.  The source brightness \nhad remained constant, with a baseline magnitude of $I_{\\rm base}\\sim 18.74$,   until \nthe end of the 2015 season since it was monitored by the survey in 2009.  When the \nevent was found, the source brightness was already $\\sim 0.5$ magnitude brighter than \nthe baseline magnitude, indicating that the event started during the $\\sim 4$ month \ntime gap between the 2015 and 2016 seasons when the bulge field could not be observed \nduring the passage of the field behind the Sun.  The OGLE observations were conducted \nat a $\\sim 1$--2 day cadence and data were acquired mainly in $I$ band with some $V$ \nband data obtained for the source color measurement.\n\n\nTwo other microlensing survey groups of the Microlensing Observations in Astrophysics \n\\citep[MOA:][]{Bond2001, Sumi2003} and the Korea Microlensing Telescope Network \n\\citep[KMTNet:][]{Kim2016} independently detected the event. The MOA survey observed \nthe event with a $\\sim 0.5$ hour cadence in a customized broad $R$ band using the 1.8 m \ntelescope of the Mt.~John University Observatory, New Zealand. The event was entitled \nMOA-2016-BLG-069 in the list of MOA transient \nevents.\\footnote{http:\/\/www.massey.ac.nz\/~iabond\/moa\/alert2016\/alert.php} \nThe KMTNet observations were \nconducted using 3 identical 1.6 m telescopes that are located at the Siding Spring \nObservatory, Australia, Cerro Tololo Interamerican Observatory, Chile, and the South \nAfrican Astronomical Observatory, South Africa. We refer to the individual KMTNet \ntelescopes as KMTA, KMTC, and KMTS, respectively. The event, dubbed as KMT-2016-BLG-1709\nin the 2016 KMTNet event list\\footnote{http:\/\/kmtnet.kasi.re.kr\/ulens\/2016\/}, \nwas located in the KMTNet BLG01 field toward which observations were conducted with a \n$\\sim 0.5$ hour cadence. The field almost overlaps the BLG41 field that was additionally \ncovered to fill the gaps between the CCD chips of the BLG01 field. The source happens to \nbe located in the gap between the chips of the BLG41 field and thus no data was obtained \nfrom the field. KMTNet observations were conducted mainly in $I$ band and occasional $V$ \nband observations were carried out to measure the source color.\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f1.eps}\n\\caption{\nLight curve of OGLE-2016-BLG-0156. The colors of data points correspond to those of \ntelescopes, marked in the legend, used for observations. The times marked by arrows \nindicate the centers of the peaks at ${\\rm HJD}^\\prime={\\rm HJD}-2450000\\sim 7462.4$ \n($t_1$), 7493.7 ($t_2$), and 7512.1 ($t_3$).\n\\vskip0.2cm\n}\n\\label{fig:one}\n\\end{figure}\n\n\n\nPhotometry of the individual data sets are processed using the codes of the individual \nsurvey groups. All the photometry codes utilize the difference imaging technique \ndeveloped by \\citet{Alard1998}. For the KMTC data set, we additionally conduct \npyDIA photometry\\footnote{The pyDIA code is a python package for performing \ndifference imaging and photometry developed by \\citet{Albrow2017}. The difference-imaging \npart of this software implements the algorithm of \\citet{Bramich2013} with extended delta \nbasis functions, enabling independent control of the degrees of spatial variation for \nthe differential photometric scaling and differential PSF variations between images.} \nfor the source color measurement.  For the use of the multiple data \nsets reduced by different codes, we normalize the error bars of the individual data \nsets using the method described in \\citet{Yee2012}.\n\n\nFigure~\\ref{fig:one} shows the light curve of OGLE-2016-BLG-0156. The light \ncurve is characterized by three distinct peaks centered at \n${\\rm HJD}^\\prime={\\rm HJD}-2450000 \\sim 7462.4$ ($t_1$), 7493.7 ($t_2$), \nand 7512.1 ($t_3$). The peaks are widely separated with time gaps \n$\\Delta t_{1-2}\\sim 31.3$ days between the first and second peaks and \n$\\Delta t_{2-3}\\sim 18.4$ days between the second and the third peaks. \n\n\nIn Figure~\\ref{fig:two}, we present the enlarged views of the individual peaks. \nIt is found that the source became brighter by $\\gtrsim 3$ magnitudes during very \nshort periods of time, indicating that the peaks were produced by the source \ncrossings over the caustic. Caustics produced by a binary lens form closed curves \nand thus caustic crossings usually occur in multiples of two.  The region between \nthe second and third peaks shows a U-shape pattern, which is the characteristic \npattern appearing when the source passes inside a caustic, suggesting that the \npair of the peaks centered at $t_2$ and $t_3$ were produced when the source star \nentered and exited the caustic, respectively. On the other hand, the first peak \nhas no counterpart peak. Such a single-peak feature can be produced when the \nsource crosses a caustic tip in which the gap between the caustic entrance and \nexit is smaller than the source size.\n\n\nWe note that all the peaks were densely covered. The first peak was covered by the \ncombined data sets obtained using the three KMTNet telescopes, the second peak was \nresolved by the OGLE+KMTA data sets, and the last peak was covered by the MOA+KMTS \ndata sets. The dense and continuous coverage of all the caustic crossings were \npossible thanks to the coordination of the high-cadence survey experiments using \nglobally distributed telescopes.\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f2.eps}\n\\caption{\nEnlarged view of the three peaks in the lensing light curve.\nThe locations of the individual peaks in the whole light curve \nare marked by $t_1$, $t_2$, and $t_3$ in Fig.~\\ref{fig:one}.\n\\vskip0.3cm\n}\n\\label{fig:two}\n\\end{figure}\n\n\n\n\\begin{figure*}\n\\epsscale{0.82}\n\\plotone{f3.eps}\n\\caption{\nThe top two panels show the best-fit model light curve, the curve superposed on the data \npoints, obtained by considering both the microlens-parallax and lens-orbital effects. \nThe lower four panels compare the residuals of the 4 tested models. The `parallax' \nand `orbit' models are obtained by separately considering the microlens-parallax and \nlens-orbital effects. The `standard' model consider neither of these higher-order \neffects.\n\\vskip0.3cm\n}\n\\label{fig:three}\n\\end{figure*}\n\n\n\\section{Modeling Light Curve}\n\n\\subsection{Model under Rectilinear Relative Lens-Source Motion}\n\nThe caustic-crossing features in the observed light curve indicates that the event is \nlikely to be produced by a binary lens and thus we conduct binary-lens modeling of \nthe light curve.  We begin by searching for the sets of the lensing parameters that \nbest explain the observed light curve under the assumption of the rectilinear \nlens-source motion, wherein the lensing light curve is described by 7 principal parameters. \nThe first three parameters $(t_0, u_0, t_{\\rm E})$ are identical to those of a \nsingle-lens events, describing the time of the closest lens-source approach, the \nseparation at that time, and the event timescale, respectively. Because a binary lens \nis composed of two masses, one needs a reference position for the lens. We set the \nbarycenter of the binary lens as the reference position.  Due to the binary nature \nof the lens, one needs another three parameters $(s, q, \\alpha)$, indicating the \nprojected binary separation (normalized to $\\theta_{\\rm E}$), the mass ratio between the \nlens components, and the angle between the binary axis and the source trajectory, \nrespectively. The last parameter $\\rho$, which represents the ratio of the angular \nsource radius $\\theta_*$ to the angular Einstein radius, i.e., $\\rho=\\theta_*\/\\theta_{\\rm E}$ \n(normalized source radius), is needed to describe the deviation of lensing \nmagnifications caused by finite-source effects during caustic crossings.\n\n\nModeling the light curve is done through a multiple-step process. In the first \nstep, we conduct a grid search for the parameters $s$ and $q$ and, for a given \nset of $s$ and $q$, the other parameters are searched for using a downhill approach \nbased on the Markov Chain Monte Carlo (MCMC) method. We identify local solutions \nfrom the $\\Delta\\chi^2$ maps obtained from this preliminary \nsearch. In the second step, we refine the individual local solutions first by \ngradually narrowing down the parameter space and then allowing all parameters \n(including the grid parameters $s$ and $q$) to vary. If a satisfactory solution is \nnot found from these searches, we repeat the process by changing the initial values \nof the parameters. We refer to the model based on these principal parameters as \n`standard model'.\n\n\nFrom these searches, we find that it is difficult to find a lensing model that \nadequately describes the observed light curve.\nIn the bottom panel of Figure~\\ref{fig:three} labeled as `standard', we \npresent the residual of the standard model. It shows that the model fit to the \nfirst peak is very poor although the model relatively well describes the second \nand third peaks.\n\n\n\\subsection{Model with Higher-order Effects}\n\nThe difficulty in finding a lensing model that fully explains all the features \nin the observed light curve under the assumption of the rectilinear lens-source \nmotion suggests that the motion may not be rectilinear. This possibility is \nfurther supported by the long duration of the event.\n\n\nTwo major effects cause accelerations in the relative lens-source motion. One is \nthe microlens-parallax effect. The other is the orbital motion of the lens: \nlens-orbital effect. We therefore conduct additional modeling considering \nthese higher-order effects. \n\n\nIncorporating the microlens-parallax effect into lensing modeling requires \ntwo additional parameters of $\\pi_{{\\rm E},N}$ and $\\pi_{{\\rm E},E}$. They represent the two components \nof the microlens-parallax vector $\\mbox{\\boldmath $\\pi$}_{\\rm E}$ directed to the north and east, \nrespectively. The microlens-parallax vector is related to \n$\\pi_{\\rm rel}$, $\\theta_{\\rm E}$, and the relative lens-source proper motion vector \n$\\mbox{\\boldmath $\\mu$}$ by\n\\begin{equation}\n\\mbox{\\boldmath $\\pi$}_{\\rm E}={\\pi_{\\rm rel}\\over \\theta_{\\rm E}}{\\mbox{\\boldmath $\\mu$} \\over \\mu}.\n\\label{eq4}\n\\end{equation}\n\n\nConsidering the lens-orbital effect also requires additional parameters. Under \nthe approximation that the positional changes of the lens components induced by \nthe lens-orbital effect during the event is small, the effect is described by \ntwo parameters of $ds\/dt$ and $d\\alpha\/dt$. They represent the change rates of \nthe binary separation and the source trajectory angle, respectively.\n\n\nWe conduct a series of additional modeling runs considering the higher-order effects. \nIn the `parallax' and `orbit' modeling runs, we separately consider the microlens-parallax \nand lens-orbital effects, respectively. In the `orbit+parallax' modeling run, we \nsimultaneously consider both the higher-order effects. For solutions considering \nmicrolens-parallax effects, it is known that there may exist a pair of degenerate \nsolutions with $u_0>0$ and $u_0<0$ due to the mirror symmetry of the source trajectory \nwith respect to the binary axis \\citep{Smith2003, Skowron2011}.  \nWe inspect this `ecliptic degeneracy' \nwhen microlens-parallax effects are considered in modeling. \n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f4.eps}\n\\caption{\nCumulative distribution of $\\chi^2$ difference between the `parallax' and \n`orbit+parallax' models. The light curve in the upper panel is presented to show \nthe region of the fit improvement.\n}\n\\label{fig:four}\n\\end{figure}\n\n\n\\begin{deluxetable}{lcc}\n\\tablecaption{Comparison of models\\label{table:one}}\n\\tablewidth{240pt}\n\\tablehead{\n\\multicolumn{2}{c}{Model}       &\n\\multicolumn{1}{c}{$\\chi^2$}  \n}\n\\startdata                                              \nStatic            &            &  34436.4   \\\\\nOrbit             &            &  20019.4   \\\\\nParallax          & $u_0>0$    &  5749.4   \\\\\n--                & $u_0<0$    &  6026.4   \\\\\nOrbit + parallax  & $u_0>0$    &  5215.4   \\\\\n--                & $u_0<0$    &  5221.2     \n\\enddata                            \n\\end{deluxetable}\n\n\n\nIn Table~\\ref{table:one}, we list the results of the individual modeling runs in terms \nof $\\chi^2$ values of the fits. In order to visualize the goodness of the fits, we also \npresent the residuals of the individual models in the lower panels of Figure~\\ref{fig:three}. \nFor the pair of solutions with $u_0>0$ and $u_0<0$ obtained considering microlens-parallax \neffects, we present the residuals of the solution yielding a better fit.\n\n\nWe compare the fits to judge the importance of the individual higher-order effects. \nFrom this, it is found that the major features of the light curve, i.e., the three \npeaks, still cannot be adequately explained by the orbital effect alone, although \nthe effect improves fit by \n$\\Delta\\chi^2\\sim 14407.0$ \nwith respect to the standard \nmodel. See the residual labeled as `orbit' in Figure~\\ref{fig:three}. For the parallax \nmodel, on the other hand, the fit greatly improves, by \n$\\Delta\\chi^2\\sim 28677.0$, \nand \nall the three peak features are approximately described.  See the residual labeled as \n`parallax' in Figure~\\ref{fig:three}. We also find that the fit further improves, by \n$\\Delta\\chi^2\\sim 534.0$ \nwith respect to the parallax model, by additionally considering \nthe lens-orbital effects. This indicates that although the lens-orbital effect is not the \nprime higher-order effects, it is important to precisely describe the light curve.  Due \nto the relatively minor improvement, it is not easy to see the additional fit improvement \nby the lens-orbital effect from the comparison of the residuals of the `parallax' and \n`orbit+parallax' models.  We, therefore, present the cumulative distribution of $\\Delta\\chi^2$ \nbetween the two models as a function of time in Figure~\\ref{fig:four}. \nIt is found that the fit improves throughout the event and major improvement occurs at the \nfirst and the second peaks and after the third peak. \nThis indicates that the widely separated \nmultiple peak features in the lensing light curve help to constrain the subtle higher-order \neffects. \nTo check the consistency of the fit improvement, we \nalso plot the distributions for the individual data sets.\nFrom the distributions, one finds that the $\\chi^2$ improvement shows up in all data sets \n(OGLE, MOA, KMTC, and KMTS) except for the KMTA data set. \nWe judge that the marginal orbital signal in the KMTA data set is \ncaused by the relatively lower photometry quality than the other \nKMTNet data sets and the resulting smaller number of data points (626 points \ncompared to 975 and 1214 points of the KMTS and KMTC data sets, respectively).\nWe find the ecliptic degeneracy is quite severe although the model with $u_0>0$ is \npreferred over the model with $u_0<0$ by \n$\\Delta\\chi^2\\sim 5.8$.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f5.eps}\n\\caption{\nTriangular diagram showing the \n$\\Delta\\chi^2$ distributions of MCMC points in the planes of the pair of the \nhigher-order lensing lensing parameters $\\pi_{{\\rm E},N}$, $\\pi_{{\\rm E},E}$, $ds\/dt$, and $d\\alpha\/dt$.\nPoints marked in red, yellow, green, cyan, and blue \nrepresent those with $1\\sigma$, $2\\sigma$, $3\\sigma$, $4\\sigma$, and $5\\sigma$, respectively.\n}\n\\label{fig:five}\n\\end{figure}\n\n\n\n\\begin{deluxetable}{lcc}\n\\tablecaption{Best-fit lensing parameters\\label{table:two}}\n\\tablewidth{240pt}\n\\tablehead{\n\\multicolumn{1}{c}{parameter}       &\n\\multicolumn{1}{c}{$u_0>0$}      & \n\\multicolumn{1}{c}{$u_0<0$}  \n}\n\\startdata                                              \n$t_0$ (${\\rm HJD}^\\prime$) &     7504.730 $\\pm$ 0.012       &          7504.644 $\\pm$ 0.015     \\\\\n$u_0$                      &     0.083 $\\pm$ 0.001          &          -0.085 $\\pm$ 0.001       \\\\\n$t_{\\rm E}$ (days)         &     68.19 $\\pm$ 0.18           &          67.56 $\\pm$ 0.39         \\\\\n$s$                        &     0.727 $\\pm$ 0.001          &          0.731 $\\pm$ 0.002        \\\\\n$q$                        &     0.869 $\\pm$ 0.006          &          0.841 $\\pm$ 0.006        \\\\\n$\\alpha$ (rad)             &     1.408 $\\pm$ 0.003          &          -1.408 $\\pm$ 0.002       \\\\\n$\\rho$ ($10^{-3}$)         &     0.615 $\\pm$ 0.013          &          0.620 $\\pm$ 0.012        \\\\\n$\\pi_{{\\rm E},N}$          &     0.334 $\\pm$ 0.008          &          -0.347 $\\pm$ 0.003       \\\\\n$\\pi_{{\\rm E},E}$          &     -0.335 $\\pm$ 0.011         &          -0.406 $\\pm$ 0.012       \\\\\n$ds\/dt$ (yr$^{-1}$)        &     0.168 $\\pm$ 0.010          &          0.198 $\\pm$ 0.013        \\\\\n$d\\alpha\/dt$ (yr$^{-1}$)   &     -0.958 $\\pm$ 0.039         &          1.059 $\\pm$ 0.011        \\\\\n$I_{s,{\\rm OGLE}}$         &      19.37 $\\pm$ 0.01          &          19.37 $\\pm$ 0.01         \\\\   \n$I_{b,{\\rm OGLE}}$         &      19.77 $\\pm$ 0.01          &          19.77 $\\pm$ 0.01    \n\\enddata                            \n\\tablecomments{${\\rm HJD}^\\prime={\\rm HJD-2450000}$. }\n\\end{deluxetable}\n\n\n\nIn Table~\\ref{table:two}, we present the lensing parameters of the best-fit models. Because \nthe ecliptic degeneracy is severe, we present both the $u_0>0$ and $u_0<0$ \nsolutions.  Also presented are the $I$-band magnitudes of the source, $I_{s,{\\rm OGLE}}$, \nand the blend, $I_{b,{\\rm OGLE}}$, estimated based on the OGLE data.  We note that the \nlensing parameters of the two solutions are roughly in the relation \n$(u_0, \\alpha, \\pi_{{\\rm E},N}, d\\alpha\/dt)\\leftrightarrow -(u_0, \\alpha, \\pi_{{\\rm E},N}, d\\alpha\/dt)$\n\\citep{Skowron2011}. \nSeveral facts should noted for the obtained lensing parameters. First, the event timescale, \n$t_{\\rm E}\\sim 68$ days, is substantially longer than typical lensing events with \n$t_{\\rm E} \\sim 20$ days. Second, the binary parameters $(s,q)\\sim (0.73, 0.87)$ indicate \nthat the lens is comprised of two similar masses with a projected separation slightly smaller \nthan $\\theta_{\\rm E}$. Third, the normalized source radius $\\rho\\sim 0.62\\times 10^{-3}$ is smaller \nby about a factor $\\sim 2.5$ than the value of an event typically occurring on a star with a \nsimilar stellar type to the source of OGLE-2016-BLG-0156.  \nSince $\\rho=\\theta_*\/\\theta_{\\rm E}$, the \nsmall $\\rho$ value suggests that the angular Einstein radius is likely to be big. Finally, \nthe parameters describing the higher-order effects, i.e., $\\pi_{{\\rm E},N}$, $\\pi_{{\\rm E},E}$, $ds\/dt$, and \n$d\\alpha\/dt$, are precisely determined with fractional uncertainties $\\sim 2.4\\%$, $3.3\\%$, \n$6.3\\%$, and $4.1\\%$, respectively. In Figure~\\ref{fig:five}, we present the $\\Delta\\chi^2$ \ndistributions of MCMC points in the planes of the pair of the higher-order lensing parameters.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f6.eps}\n\\caption{\nLens system configuration. The curve with an arrows represents the source trajectory with \nrespect to the caustic (closed curve composed of concave segments).  To show the variation \nof the caustic caused by the lens orbital motion, we present caustics at 3 moments \ncorresponding to the times of the three peaks in the lensing light curve. We mark the \npositions of source crossings for the individual peaks occurring at $t_1$, $t_2$, and $t_3$. \nThe small dots marked by $M_1$ and $M_2$ present the positions of the binary lens components. \nLengths are scaled to the angular Einstein radius corresponding to the total mass of the lens.\n}\n\\label{fig:six}\n\\end{figure}\n\n\n\nIn Figure~\\ref{fig:three}, we present the model light curve, which is plotted over the \ndata points, of the best-fit solution, i.e., the orbit+parallax model with $u_0>0$. \nIn Figure~\\ref{fig:six}, we present the corresponding lens-system configuration, showing \nthe trajectory of the source with respect to the caustic. When the binary separation $s$ \nis close to unity, caustics form a single closed curve, `resonant caustic', and as the \nseparation becomes smaller, the caustic becomes elongated along the direction perpendicular \nto the binary axis and eventually splits into 3 segments, in which one 4-cusp central \ncaustic is located around the center of mass and the other two triangular caustics are \nlocated away from the center of mass.   \\citep{Erdl1993, Dominik1999}. For OGLE-2016-BLG-0156, \nthe caustic topology corresponds to the boundary between the single closed-curve \n(`resonant') and triple closed-curve (`close') topologies.  The source moved approximately \nin parallel with the elongated caustic, crossing the caustic 3 times at the positions marked \nby $t_1$, $t_2$, and $t_3$. The first peak was produced by the source crossing over the slim \nbridge part of the caustic connecting the 4-cusp central caustic and one of the triangular \nperipheral caustics.  The peak could in principle have been produced by the source star's \napproach to the the right cusp of the upper triangular caustic.  We check this possibility \nand find that it cannot explain the light curve in the region around the first peak.  The \nsecond and third peaks were produced when the source passed the upper and lower right \nparts of the central caustic, respectively.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f7.eps}\n\\caption{\nModel light curve (curve superposed on data points) obtained under the assumption \nof the rectilinear relative lens-source motion. The lower panel shows the \nresidual from the model. The inset in the upper panel shows the lens system \nconfiguration corresponding to the model.\n\\vskip0.2cm\n}\n\\label{fig:seven}\n\\end{figure}\n\n\n\nWe note that the well-covered 3-peak feature in the lensing light curve provides a very \ntight constraint on the source trajectory, and thus on the higher-order effects.  To \ndemonstrate the high sensitivity of the light curve to the slight change of the source \ntrajectory induced by the higher-order effects, in Figure~\\ref{fig:seven}, we present \nthe model fit of the standard solution and the corresponding lens system configuration. \nOne finds that the straight source trajectory without higher-order effects can describe \nthe second and third peaks by crossing similar parts of the central caustic to those of \nthe solution obtained considering the higher-order effects. However, the extension of \nthe trajectory crosses the upper triangular caustic, resulting in a light curve that \ndiffers greatly from the observed one. The importance of well-covered multiple peaks \nin determining $\\mbox{\\boldmath $\\pi$}_{\\rm E}$ was first pointed out by \\citet{An2001} and a good \nexample was presented by \\citet{Udalski2018} for the quintuple-peak lensing event \nOGLE-2014-BLG-0289.\n\n\n\n\n\n\n\\section{Physical Lens Parameters}\n\n\\subsection{Angular Einstein Radius}\n\nFor the unique determinations of the mass and distance to the lens, one needs to \ndetermine $\\theta_{\\rm E}$ as well as $\\pi_{\\rm E}$. See Equations (\\ref{eq2}) and (\\ref{eq3}). \nThe angular Einstein radius is estimated from the combination of the normalized \nsource $\\rho$ and the angular source radius $\\theta_*$ by $\\theta_{\\rm E}=\\theta_*\/\\rho$. \nThe $\\rho$ value is determined from the light curve modeling. Then, one needs to \nestimate $\\theta_*$ for the determination of $\\theta_{\\rm E}$.  \n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f8.eps}\n\\caption{\nKMTC $I$ and $V$-band data sets processed using the pyDIA photometry code.\nThe data sets are used for the source color measurement.\n}\n\\label{fig:eight}\n\\end{figure}\n\n\nWe estimate the angular source radius from the de-reddened color, $(V-I)_0$, \nand brightness, $I_0$.  For this, we first measure the instrumental (uncalibrated) \nsource color and brightness from the KMTC $V$ and $I$-band data sets processed using the pyDIA \nphotometry. \nWe estimate the source color using the regression of the $V$ and $I$-band data sets.\nThe color can also be estimated by using the model and we find that the source \ncolor estimated by both ways are consistent.\nFigure~\\ref{fig:eight} shows the KMTC $I$ and $V$-band data.  In the \nsecond step, following the method of \\citet{Yoo2004}, we calibrate the color and \nbrightness of the source using the centroid of the red giant clump (RGC) in the \ncolor-magnitude diagram as a reference.  In Figure~\\ref{fig:nine}, we mark the \npositions of the source, with \n$(V-I, I)=(1.99\\pm 0.01, 19.31\\pm 0.01)$, \nand the RGC centroid, \n$(V-I, I)_{\\rm RGC}=(2.43,15.98)$, in the instrumental color-magnitude diagram.  \nWith the known de-reddened color and brightness of the RGC centroid, \n$(V-I,I)_{{\\rm RGC},0}=(1.06,14.46)$ \\citep{Bensby2011, Nataf2013}, combined with \nthe measured offsets in color, $\\Delta(V-I)=-0.44$, and brightness, $\\Delta I=3.33$, \nbetween the source and the RGC centroid, we estimate that the de-reddened color and \nbrightness of the source star are \n$(V-I,I)_{0}=(V-I,I)_{0,{\\rm RGC}}+\\Delta (V-I,I)=(0.62\\pm 0.01, 17.78\\pm 0.01)$, \nindicating that \nthe source is a turn-off star.  In the last step, we convert the measured $V-I$ \ninto $V-K$ using the color-color relation of \\citet{Bessell1988} and then estimate \nthe angular source radius using the relation between the color and surface \nbrightness of \\citet{Kervella2004}.  It is estimated that the angular source \nradius is\n\\begin{equation}\n\\theta_*=0.79 \\pm 0.06\\ \\mu{\\rm as}.\n\\label{eq5}\n\\end{equation} \nIn addition to the measurement error, the source color estimation is further \naffected by the uncertainty in determining RGC centroid and the differential \nreddening of the field.\n\\citet{Bensby2013} showed that for lensing events in the fields with \nwell defined RGCs, the typical error in the source color estimation is \nabout 0.07 mag. \nWe, therefore, estimate the errorbar of $\\theta_*$ by considering this \nadditional error.\n\n\nThe estimated angular Einstein radius is \n\\begin{equation}\n\\theta_{\\rm E}= 1.30 \\pm 0.09\\ {\\rm mas}.\n\\label{eq6}\n\\end{equation}\nFor a typical lensing event produced by a low-mass star ($\\sim 0.3~M_\\odot$) \nlocated halfway between the source and observer ($D_{\\rm L}\\sim 4~{\\rm kpc}$), \nthe angular Einstein radius is \n$\\theta_{\\rm E}=\\sqrt{\\kappa M \\pi_{\\rm rel}}\n\\sim 0.55~{\\rm mas}~(M\/0.3~M_\\odot)^{1\/2}$.\nThen the estimated angular Einstein radius is $\\gtrsim 2$ times \nbigger than the \nvalue of a typical lensing event.  This is expected from the small value of the \nnormalized source radius. Combined with the event timescale, the relative \nlens-source proper motion in the geocentric frame is estimated by\n\\begin{equation}\n\\mu_{\\rm geo} = {\\theta_{\\rm E}\\overt_{\\rm E}} =\n6.94 \\pm 0.69\\ {\\rm mas}\\ {\\rm yr}^{-1}.\n\\label{eq7}\n\\end{equation}\nThe corresponding proper motion in the heliocentric frame is estimated by\n\\begin{equation}\n\\mu_{\\rm helio} =\n\\left\\vert\n\\mu_{\\rm geo} {\\mbox{\\boldmath $\\pi$}_{\\rm E}\\over \\pi_{\\rm E}} +\n{\\bf v}_{\\oplus,\\perp}{\\pi_{\\rm rel} \\over {\\rm au}}\\right\\vert =\n5.94 \\pm 0.43\\ {\\rm mas}\\ {\\rm yr}^{-1}.\n\\label{eq8}\n\\end{equation}\nHere ${\\bf v}_{\\oplus,\\perp}=(v_{\\oplus,\\perp,N}, v_{\\oplus,\\perp,E})=\n(3.1,17.5)\\ {\\rm km}\\ {\\rm s}^{-1}$ denotes the \nprojected velocity of Earth at $t_0$. \n\n\nIn Table~\\ref{table:three}, we summarize the estimated values of the angular \nEinstein radius, relative lens-source proper motion (in both geocentric and \nheliocentric frames), and the direction of the relative motion, i.e., \n$\\phi=\\tan^{-1} (\\mu_{{\\rm helio},E}\/\\mu_{{\\rm helio},N})$.  We also present \nthe quantities resulting from the $u_0<0$ solution.  The obtained quantities are \nslightly different from those of the $u_0>0$ solution due to the slight differences \nin $\\rho$, $\\pi_{{\\rm E},N}$, and $\\pi_{{\\rm E},E}$.\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{f9.eps}\n\\caption{\nLocations of the source and the centroid of red giant clump (RGC) in the instrumental \ncolor-magnitude diagram of stars around the source.\nThe diagram is constructed using the pyDIA photometry of \nKMTC $I$ and $V$-band data.\n\\vskip0.5cm\n}\n\\label{fig:nine}\n\\end{figure}\n\n\n\n\\begin{deluxetable}{lcc}\n\\tablecaption{Einstein radius and Proper Motion\\label{table:three}}\n\\tablewidth{240pt}\n\\tablehead{\n\\multicolumn{1}{c}{Quantity}       &\n\\multicolumn{1}{c}{$u_0>0$}        &\n\\multicolumn{1}{c}{$u_0<0$}         \n}\n\\startdata                                              \n$\\theta_{\\rm E}$                            &   $1.30 \\pm 0.09$  &   $1.28 \\pm 0.09$    \\\\\n$\\mu_{\\rm geo}$ (mas yr$^{-1}$)      &   $6.94 \\pm 0.50$  &   $6.91 \\pm 0.50$    \\\\\n$\\mu_{\\rm helio}$ (mas yr$^{-1}$)    &   $5.94 \\pm 0.43$  &   $4.88 \\pm 0.35$    \\\\\n$\\phi$                               &   $334^\\circ$      &   $215^\\circ$        \n\\enddata                            \n\\end{deluxetable}\n\\smallskip\n\n\n\n\n\\subsection{Mass and Distance}\n\n\nWith the measured $\\pi_{\\rm E}$ and $\\theta_{\\rm E}$, the masses of the individual lens components are \ndetermined as\n\\begin{equation}\nM_1=\n0.18 \\pm 0.01\\ M_\\odot,\n\\label{eq9}\n\\end{equation}\nand\n\\begin{equation}\nM_2=qM_1=\n0.16 \\pm 0.01\\ M_\\odot.\n\\label{eq10}\n\\end{equation}\nIt is estimated that the lens is located at a distance\n\\begin{equation}\nD_{\\rm L} = \n1.35\\pm 0.09\\ {\\rm kpc}.\n\\label{eq11}\n\\end{equation}\nThe determined masses and distance indicates that the lens is a binary \ncomposed of two M dwarfs located in the disk.  The projected separation \nbetween the lens components is\n\\begin{equation}\na_\\perp = s\\theta_{\\rm E}D_{\\rm L} =  \n1.28 \\pm 0.09\\ {\\rm au}.\n\\label{eq12}\n\\end{equation}\nWe note that the $u_0<0$ solution yields similar lens parameters.\nIn Table~\\ref{table:four}, we list the physical lens parameters for both \nthe $u_0>0$ and $u_0<0$ solutions.\n\n\n\nWe check the validity of the solution by estimating the projected kinetic-to-potential \nenergy ratio. We compute the ratio from the physical lens parameters of $M=M_1+M_2$ \nand $a_\\perp$ and the measured lensing parameters of $s$, $\\alpha$, $ds\/dt$, \nand $d\\alpha\/dt$ by\n\\begin{equation}\n\\left( {{\\rm KE}\\over {\\rm PE}}\\right)_\\perp =\n{ (a_\\perp\/{\\rm au})^3\\over 8\\pi^2(M\/M_\\odot) }\n\\left[\n\\left( {1\\over s}{ds\/dt\\over {\\rm yr}^{-1}} \\right)^2 +\n\\left( {d\\alpha\/dt \\over {\\rm yr}^{-1}}\\right)^2 \n\\right].\n\\label{eq13}\n\\end{equation}\nIn order for the lens system to be a gravitationally bound system, \nthe solution should satisfy the condition of\n$({\\rm KE}\/{\\rm PE})_\\perp \\leq {\\rm KE}\/{\\rm PE} \\leq 1.0$,\nwhere ${\\rm KE}\/{\\rm PE}$ denotes the intrinsic energy ratio.\nThe estimated ratio $({\\rm KE}\/{\\rm PE})_\\perp \\sim 0.08$ satisfies\nthis condition. The low value of the ratio suggests that the binary\ncomponents are aligned along the line of sight.\n\n\n\n\\subsection{Lens Brightness}\n\n\nAlthough the lens components are M dwarfs, they are located at a close distance, \nand the flux from the lens can comprise a significant portion of the blended flux, \ne.g., OGLE-2017-BLG-0039 \\citep{Han2018}.  To check this possibility, we estimate \nthe expected brightness of the lens.  The stellar types of the lens components are \nabout M4.5V and M5.0V with absolute $I$-band magnitudes of $M_{I,1}\\sim 10.5$ and \n$M_{I,2}\\sim 11.0$ for the primary and companion, respectively, resulting in the \ncombined magnitude $M_I\\sim 10.0$.  With the known distance to the lens, the \nde-reddened $I$-band magnitude is then $I_{L,0}=M_I + 5\\logD_{\\rm L} -5 \\sim 20.6$.  \nFrom the OGLE extinction map \\citep{Nataf2013}, the total $I$-band extinction \ntoward the source is $A_{I,{\\rm tot}}\\sim 1.48$.  Assuming that about half of \nthe total extinction is caused by the dust and gas located in front of the lens, \ni.e., $A_I\\sim 0.7$, the expected brightness of the lens is\n\\begin{equation}\nI_{\\rm L} = I_{{\\rm L},0}+A_I \\sim \n21.3.\n\\label{eq14}\n\\end{equation}  \nCompared to the brightness of the blend, $I_{b,{\\rm OGLE}}\\sim 19.8$,\nit is found that the the flux from the lens comprises an important fraction,\n$\\sim 25\\%$, of the blended light. \nWe note that the color constraint of the blended light cannot be used because the \nuncertainty of the $V$-band blend flux measurement is bigger than the flux itself.\n\n\n\n\\begin{deluxetable}{lcc}\n\\tablecaption{Physical lens parameters\\label{table:four}}\n\\tablewidth{240pt}\n\\tablehead{\n\\multicolumn{1}{c}{Parameter}      &\n\\multicolumn{1}{c}{$u_0>0$}        &\n\\multicolumn{1}{c}{$u_0<0$}  \n}\n\\startdata                                              \n$M_1$ ($M_\\odot$)             &   $0.18 \\pm 0.01$  &  $0.16 \\pm 0.01$    \\\\ \n$M_2$ ($M_\\odot$)             &   $0.16 \\pm 0.01$  &  $0.13 \\pm 0.01$    \\\\  \n$D_{\\rm L}$ (kpc)                   &   $1.35 \\pm 0.09$  &  $1.24 \\pm 0.08$    \\\\ \n$a_\\perp$    (au)             &   $1.28 \\pm 0.09$  &  $1.16 \\pm 0.08$    \\\\   \n$({\\rm KE}\/{\\rm PE})_\\perp$   &   0.08             &  0.08\n\\enddata                            \n\\end{deluxetable}\n\\smallskip\n\n\nThe bright nature of the lens combined with the high relative lens-source proper \nmotion suggests that the lens can be directly observed from high-resolution \nfollow-up observations.  \nFor the case of the lensing event OGLE-2005-BLG-169, \nthe lens was resolved from the source on the Keck AO images when they were separated \nby $\\sim 50$ mas after $\\sim 8$ years after the event \\citep{Batista2015}.  By applying \nthe same criterion, the lens and source of OGLE-2016-BLG-0156 can be resolved if \nsimilar follow-up observations are conducted $\\sim 6$ years after the event, i.e., \nafter 2022. \nFor the case of the another lensing event OGLE-2012-BLG-0950, \n\\citet{Bhattacharya2018} resolved the source and lens using Keck and the \n{\\it Hubble Space Telescope} when they were separated by $\\sim 34$ mas. \nAccording to this criterion, then, \nthe source and lens of this event would be resolved in 2022.\n\n\nBecause follow-up observations are likely to to be conducted in near infrared bands, we estimate \nthe expected $H$-band brightness of the lens.  The absolute $H$-band magnitudes of \nthe individual lens components are \n$M_H\\sim 8.2$ and 8.7 resulting in the combined brightness of $M_{H,0}\\sim 7.7$.  \nWith $A_I=A_{I,{\\rm tot}}\/2\\sim 0.7$ and \n$E(V-I)=E_{\\rm tot}(V-I)\/2\\sim 0.6$ from \\citet{Nataf2013} and adopting the relation \n$A_H\\sim 0.108 A_V\\sim 0.14$ of \\citet{Nishiyama2008}, we estimate that $H$-band \nbrightness of the lens is \n\\begin{equation}\nH_{\\rm L} = M_{H,0} + A_H + 5\\log D_{\\rm L} -5 \n\\sim 17.9.\n\\label{eq15}\n\\end{equation}\nThe $H$-band brightness of the source is \n\\begin{equation}\nH_{\\rm S}=I_{\\rm S}-E(I-H)-(I-H)_0 \\sim 17.5,\n\\label{eq16}\n\\end{equation}\nwhich is similar to that of the the lens.\nWhen the lens brightness is similar to the brightness of the source,\nthe lens and source can be better resolved as demonstrated for events \nOGLE-2005-BLG-169 \\citep{Bennett2015}, \nMOA-2008-BLG-310 \\citep{Bhattacharya2017}, and\nOGLE-2012-BLG-0950 \\citep{Bhattacharya2018}.\n\n\n\n\n\n\\section{Conclusion}\n\nWe analyzed a binary microlensing event OGLE-2016-BLG-0156.  We found that \nthe light curve of the event exhibited pronounced deviations induced by \nhigher-order effects, especially the microlens effect. It is found that the \nmultiple-peak feature provided a very tight constraint on the microlens-parallax \nmeasurement.  In addition, the good coverage of all the peaks from the combined \nsurvey observations allowed us to precisely measure the angular Einstein radius.  \nWe uniquely determined the physical lens parameters from the measured values of \n$\\pi_{\\rm E}$ and $\\theta_{\\rm E}$ and found that the lens was a binary composed of two M dwarfs \nlocated in the disk.  \nWe also found that the flux from the lens comprises an important fraction \nof the blended flux.\nThe bright nature \nof the lens combined with the high relative lens-source motion suggested that the lens \ncould be directly observed from high-resolution follow-up observations. \n\n\n\\acknowledgments\nWork by CH was supported by the grant (2017R1A4A1015178) of National Research Foundation of Korea.\nWork by AG was supported by US NSF grant AST-1516842.\nWork by IGS and AG were supported by JPL grant 1500811.\nAG received support from the\nEuropean Research Council under the European Union's\nSeventh Framework Programme (FP 7) ERC Grant Agreement n.~[321035].\nThe MOA project is supported by JSPS KAKENHI Grant Number JSPS24253004,\nJSPS26247023, JSPS23340064, JSPS15H00781, and JP16H06287.\nYM acknowledges the support by the grant JP14002006.\nDPB, AB, and CR were supported by NASA through grant NASA-80NSSC18K0274. \nThe work by CR was supported by an appointment to the NASA Postdoctoral Program at the Goddard \nSpace Flight Center, administered by USRA through a contract with NASA. NJR is a Royal Society \nof New Zealand Rutherford Discovery Fellow.\nThe OGLE project has received funding from the National Science Centre, Poland, grant\nMAESTRO 2014\/14\/A\/ST9\/00121 to AU.\nThis research has made use of the KMTNet system operated by the Korea\nAstronomy and Space Science Institute (KASI) and the data were obtained at\nthree host sites of CTIO in Chile, SAAO in South Africa, and SSO in\nAustralia.\nWe acknowledge the high-speed internet service (KREONET)\nprovided by Korea Institute of Science and Technology Information (KISTI).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nQuantum theory (QT) is in conflict with noncontextual realism \\cite{Specker60,Bell66,KS67}, that is, with the assumption that the results of measurements reveal pre-existing properties that are not affected by compatible measurements. Noncontextual realism is a legitimate assumption in so far as the statistics of the measurement outcomes are not perturbed by other measurements, or, in other words, whenever measurements cannot be used to communicate information. No signaling holds when the measurements are spacelike separated. If this is the case, noncontextual realism is called local realism. It is well known that QT is also in conflict with local realism \\cite{Bell64}.\n\nHowever, most quantum violations of noncontextual realism do not occur in scenarios in which parties perform spacelike separated measurements. Still, they can be reveled in experiments \\cite{Cabello08}. This shows that contextuality, besides being ``a class of `paradoxes' which result from counterfactual logic'' \\cite{Peres02}, is a phenomenon that can be experimentally tested \\cite{Szangolies15}.\n\nAmong the violations of noncontextual realism that cannot be observed in Bell-like tests there are many cases of interest: Qutrit contextuality \\cite{KCBS08}, quantum-state-independent contextuality \\cite{Cabello08,BBCP09,YO12}, simple fully contextual correlations \\cite{Cabello13b}, contextuality associated to universal quantum computation via magic states \\cite{HWVE14,DGBR15,RBDOB15}, and absolute maximal contextuality \\cite{ATC15}. This variety of examples motivates the challenge of whether there is a unified universal way to test any form of contextuality.\n\nCertifying contextuality requires observing the violation of a noncontextuality (NC) inequality \\cite{KCBS08,Cabello08}, which is an inequality for correlations satisfied by any noncontextual hidden-variable theory (NCHVT), i.e., for any theory assuming noncontextual realism. Testing a NC inequality entails considerable difficulty and requires solving some problems.\n\n\n\\begin{figure}[tb]\n\\vspace{-2.4cm}\n \\includegraphics[width=0.48 \\textwidth]{Fig1.pdf}\n \\vspace{-2.6cm}\n \\caption{(color online). (a) A typical contextuality experiment requires preparing individual quantum systems in a quantum state $|\\psi\\rangle$ and performing a sequence of projective measurements on each of them. (b) Contextuality experiments can be substantially simplified by exploiting that any form of contextuality can be tested by measuring sequences of only two compatible binary observables. The problem is that in most quantum systems sequential projective measurements are unfeasible. In these cases, a possible test of contextuality (not without drawbacks) is one based on the scheme in panel (c). It requires replacing the projective measurement of $i$ in panel (b) by a demolition measurement of $i$ followed, in those trials with outcome $i=1$, by a preparation of the eigenstate of $i$ with eigenvalue $1$ or, in those trials with outcome $i=0$, by a preparation of the quantum state predicted by L\\\"uders's rule. \\label{Fig1}}\n\\end{figure}\n\n\nThe first difficulty is related to the fact that in a test of a NC inequality every observable must be measured using the same device in any context. An alternative would be, before conducting the experiment, certifying that different devices measure the same observable since they produce the same distributions of probability for any possible state. However, this is unfeasible using a finite sample of states. Without such certification it seems unreasonable to assume that measurement outcomes do not depend on the context when measurements themselves do \\cite{ABBCGKLW13}. This problem can be avoided by designing devices that exactly measure each of the required observables in any possible context.\n\nThe second difficulty comes from the fact that, except for the case of Bell inequalities (which are NC inequalities), testing other NC inequalities requires performing sequential measurements of quantum observables \\cite{LKGC11}, as illustrated in Fig.\\ \\ref{Fig1} (a). This is challenging. Recall that the measurement of a quantum observable, as defined by von Neumann \\cite{vonNeumann32} (i.e., a projective measurement), should not only produce an outcome (that should be ``indelibly recorded'' \\cite{MW84} before any further measurement) but also transform the quantum state according to L\\\"uders's rule \\cite{Luders51}. The problem is that only a few experiments have successfully implemented sequential projective measurements (e.g., Ref.\\ \\cite{KZGKGCBR09}). The typical measurements on quantum systems are actually demolition measurements which absorb the system, preventing subsequent measurements, or fail to transform the state of the system according to L\\\"uders's rule.\n\nNevertheless, the main difficulty is achieving the no-signaling condition with sequential measurements. Recall that in a contextuality test each measurement should not affect the probabilities of the outcomes of any subsequent measurement. That is, if the correlations between $A$ and $B$ and between $A'$ and $B$ both appear in the NC inequality and $B$ is measured in the second place, then $\\varepsilon \\equiv |\\sum_a P(a,b|A,B)-\\sum_{a'} P(a',b|A',B)|$ must be zero for any $b$. The problem is that it is impossible to have $\\varepsilon=0$ in actual experiments since the number of trials is finite, long-lasting experiments suffer drifts, and there are inevitable imperfections \\cite{Szangolies15,GKCLKZGR10}. One could argue that the condition of perfect no signaling is not even achieved in experiments with spacelike separated measurements (e.g., $\\varepsilon = 0.07 \\pm 0.10$ in Ref.\\ \\cite{Hensen15}). However, in these experiments no signaling can be assumed on physical grounds and a nonzero $\\varepsilon$ can be attributed to statistical fluctuations. In contrast, in experiments with sequential measurements, a nonzero $\\varepsilon$ should be taken into account in the analysis \\cite{Szangolies15,Winter14,DKL15}. A practical issue, however, is that these analysis are in general difficult or are not well defined beyond the case of two-point correlations.\n\nDespite these problems, little effort has been made for designing contextuality tests with small $\\varepsilon$ that are easy to analyze. From this perspective, a main challenge is to have tests in which the influence of past measurements on future measurements is experimentally indistinguishable from the influence of future measurements on past measurements. More precisely, experiments in which $\\varepsilon$ is zero within the same experimental error that $\\varepsilon' \\equiv |\\sum_b P(a,b|A,B)-\\sum_{b'} P(a,b'|A,B')|$. Assuming causality, $\\varepsilon'$ is, by definition, zero. However, when calculated from experimental data, this zero will be inevitably affected by an error. Only if the experimental value of $\\varepsilon$ is compatible with zero and is affected by a similar error can one reasonably assume that no signaling is satisfied within the experimental precision. In this case, the violation of the NC inequality can be taken as a compelling evidence of contextuality.\n\nAnother difficulty comes from the infeasibility of implementing projective sequential measurements on most quantum systems, particularly on high-dimensional quantum systems \\cite{MTT07,PB11,DBSBLC14,CEGSXLC14,CAEGCXL14} which are necessary for some forms of quantum contextuality \\cite{Cabello08,Cabello13b,ATC15,CDLP13}. One solution is implementing sequentially the unitary transformations corresponding to the desired measurements and detect the systems afterwards. This is the approach adopted in some experiments with photons \\cite{MWZ00,LLSNRWZ11}, neutrons \\cite{HLBBR03,BKSSCRH09}, and molecular nuclear spins \\cite{MRCL09}. Other solution, closer to the ideal contextuality experiment in Fig.\\ \\ref{Fig1} (a), consists of encoding the outcome of each measurement in an extra degree of freedom (e.g., extra spatial modes \\cite{ARBC09}, time bin \\cite{AACB13}, or polarization \\cite{MANCB14}) before performing the next one. However, this solution has one disadvantage: (I) The complexity of the setup grows exponentially with the number of sequential measurements. This does not only entail practical problems but also introduces a conceptual issue since these extra degrees of freedom could provide the memory classical systems need to simulate quantum contextuality \\cite{KGPLC11,FBVFLLFC15}. Moreover, both solutions have other disadvantage: (II) The observers cannot delay the choice of the next measurement to the moment in which the result of the previous measurement is indelibly recorded. As a result, the system is not really forced to answer before the context is fixed \\cite{PR88}. This indicates that other interesting challenge is designing contextuality tests for high dimensional quantum systems and free of disadvantages (I) and (II).\n\nIn this paper we show how to reveal any form of quantum contextuality in an experiment requiring sequences of only two measurements and allowing a better control on the experimental imperfections and a simple analysis. In addition, we show how to design contextuality tests for quantum systems in which sequential projective measurements are unfeasible and free of disadvantages (I) and (II).\n\n\n\\section{Main result}\n\n\nIn this section we show that two-point correlation experiments between binary compatible observables are sufficient to test any form of quantum contextuality. In the next section we will explain why this is important.\n\nTo understand what we mean by ``any form of quantum contextuality,'' we start by recalling that any given NC inequality can be associated to graph \\cite{CSW14}. This association requires converting the linear combination of probabilities in the initial NC inequality into a positive linear combination $S$ of probabilities of yes-no tests (represented in QT by rank 1 projectors). $S$ can then be associated to a graph $G$ in which each probability is represented by a vertex while edges connect vertices corresponding to probabilities of exclusive events (i.e., corresponding to alternative outcomes of a sharp measurement \\cite{CY14}). The maximum of $S$ for NCHVTs and QT are two characteristic numbers of $G$: The independence number, $\\alpha(G)$, and the Lov\\'asz number, $\\vartheta(G)$, respectively \\cite{CSW14}. Reciprocally, any graph $G$ can be converted into a NC inequality such that the bound for NCHVTs is $\\alpha(G)$ and the maximum in QT is $\\vartheta(G)$ \\cite{CSW14}. This establishes a one-to-one connection between any possible quantum violation of any possible NC inequality and a graph. It is in this sense that any form of quantum contextuality can be represented by a graph.\n\nHere we show that any NC inequality represented by a graph $G$ can be further transformed into a NC inequality involving only two-point correlations, while still having $\\alpha(G)$ as upper bound for NCHVTs and $\\vartheta(G)$ as maximum in QT.\n\n{\\em Theorem.} For any graph $G$ with vertex set $V(G)$ and edge set $E(G)$ the following inequalities are tight:\n\\begin{equation}\n\\begin{split}\n\\label{main}\n{\\cal S} \\equiv\n\\sum_{i \\in V(G)} P(1|i) - \\sum_{(i,j) \\in E(G)} P(1,1|i,j)\n& \\stackrel{\\mbox{\\tiny{NCHVTs}}}{\\leq} \\alpha(G) \\\\\n& \\stackrel{\\mbox{\\tiny{QT}}}{\\leq} \\vartheta(G),\n\\end{split}\n\\end{equation}\nwhere $P(1|i)$ is the probability of obtaining result $1$ when observable $i$ is measured and $P(1,1|i,j)$ is the joint probability of obtaining result $1$ for $i$ and result $1$ for $j$.\n\nThe quantum maximum $\\vartheta(G)$ can be attained by preparing a quantum state $|\\psi\\rangle$ and measuring a set of rank 1 projectors $\\{i=|i\\rangle \\langle i|\\}$ such that each projector is associated to a vertex of $G$, adjacent vertices are associated orthogonal projectors, and $\\sum_i |\\langle i | \\psi \\rangle |^2 = \\vartheta(G)$. The set $|\\psi \\rangle \\cup \\{|i\\rangle\\}$ is called a Lov\\'asz-optimum-orthogonal representation of $\\overline{G}$ \\cite{CDLP13}. Its existence is guaranteed by the definition of $\\vartheta(G)$.\n\n\n\\begin{figure}[tb!]\n\\includegraphics[trim = 3.6cm 5.2cm 3.6cm 5.4cm,clip,width=8cm]{Fig2.pdf}\n \\caption{(color online). Simplest nontrivial example of how the graph $G$ is related to the graph $G'$. (a) For any graph $G$ with vertex set $V(G)$, there is a NC inequality of the type $S = \\sum_{i \\in V} P(1,0,\\ldots,0|i,i_1,\\ldots,i_{n(i)}) \\stackrel{\\mbox{\\tiny{NCHVTs}}}{\\leq} \\alpha(G)$, which is violated by QT up to $\\vartheta(G)$; the set $\\{i_j\\}_{j=1}^{n(i)}$ contains the observables corresponding to vertices adjacent to vertex $i$. The probabilities in this inequality are well defined both in NCHVTs and in QT but, in general, involve incompatible tests. This is a problem for designing experiments. (b) By replacing the probabilities $P(1,0,\\ldots,0|i,i_1,\\ldots,i_{n(i)})$ in $G$ by $P(1|i)$ and every edge $(i,j) \\in G$ by the vertices corresponding to $P(0,0|i,j)$, $P(0,1|i,j)$, and $P(1,0|i,j)$ and adding edges between vertices corresponding to exclusive events, we construct a new graph $G'$ which is associated to a new NC inequality, given in Eq.\\ (\\ref{main}), which has the same bounds for NCHVTs and QT that the NC inequality associated to $G$, but involves only compatible observables and can be tested by measuring only two-point correlations. \\label{Fig2}}\n\\end{figure}\n\n\n{\\em Proof.} We can rewrite ${\\cal S}$ as\n\\begin{equation}\n\\begin{split}\n\\label{S}\n\\sum_{i \\in V(G)} P(1|i) - |E(G)| + \\sum_{(i,j) \\in E(G)} \\left[P(0,0|i,j) \\right.\\\\\n\\left. +P(0,1|i,j)+P(1,0|i,j)\\right] \\equiv {\\cal S'} - |E(G)|,\n\\end{split}\n\\end{equation}\nwhere $|E(G)|$ is the cardinal of $E(G)$. Then,\n\\begin{equation}\n{\\cal S'} \\stackrel{\\mbox{\\tiny{NCHVTs}}}{\\leq} \\alpha(G') \\stackrel{\\mbox{\\tiny{QT}}}{\\leq} \\vartheta(G'),\n\\end{equation}\nwhere $G'$ is the graph of the events in ${\\cal S'}$. Therefore,\n\\begin{equation}\n{\\cal S} \\stackrel{\\mbox{\\tiny{NCHVTs}}}{\\leq} \\alpha(G') - |E(G)| \\stackrel{\\mbox{\\tiny{QT}}}{\\leq} \\vartheta(G') - |E(G)|.\n\\end{equation}\nNow we have to prove that $\\alpha(G')=\\alpha(G)+|E(G)|$ and $\\vartheta(G')=\\vartheta(G)+|E(G)|$.\n\nThe maximum of ${\\cal S'}$ for any NCHVT is always attained by assigning probability $1$ to some events and $0$ to the others. For every $(i,j) \\in E(G)$, there are five vertices in $G'$: those corresponding to the probabilities of the events $1|i$, $1|j$, $0,0|i,j$, $0,1|i,j$, and $1,0|i,j$. See Fig.\\ \\ref{Fig2} (b). If $P(1|i)=1$ and $P(1|j)=1$, then $P(0,0|i,j)=0$, $P(0,1|i,j)=0$, and $P(1,0|i,j)=0$. Thus if we do this assignment for all edges in $G$ we obtain ${\\cal S'}=|V(G)|$ (i.e., the number of vertices of $G$), which is less than or equal to $\\alpha(G)+|E(G)|$. On the other hand, if $P(1|i)=0$ and $P(1|j)=0$, then $P(0,0|i,j)=1$, $P(0,1|i,j)=0$, and $P(1,0|i,j)=0$. Thus if we do this assignment for all edges in $G$, we obtain ${\\cal S'}=|E(G)|$, which is strictly smaller than $\\alpha(G)+|E(G)|$. If $P(1|i)=1$ and $P(1|j)=0$, then $P(0,0|i,j)=0$, $P(0,1|i,j)=0$, and $P(1,0|i,j)=1$. However, in general, we cannot do this assignment for all edges in $G$. An assignment maximizing ${\\cal S'}$ for NCHVTs is one maximizing ${\\cal S}$ (recall that ${\\cal S'} = {\\cal S} + |E(G)|$) and assigning probability $1$ to at most one of the vertices that are adjacent in $G$. This is exactly accomplished by any assignment leading to ${\\cal S}=\\alpha(G)$. Then, for every edge $(i,j)$ in $G$ with probabilities $1$ and $0$, the sum of the probabilities of the corresponding five-vertex subgraph of $G'$ is $2$, while for every $(i,j) \\in E(G)$ with probabilities both $0$, the sum of the probabilities of the corresponding five-vertex subgraph of $G'$ is $1$. Therefore, the sum of the probabilities of the events in $G'$ is $\\alpha(G)+|E(G)|$. This proves the upper bound of ${\\cal S}$ for NCHVTs.\n\nFor the maximum of ${\\cal S'}$ in QT consider the assignment of probabilities to the vertices of $G'$ corresponding, via Born's rule, to a Lov\\'asz-optimum-orthogonal representation of $\\overline{G}$. Since, for every edge $(i,j)$ of $G$, $P(1|i, 1|j)=0$, then $P(0,0|i,j)+P(0,1|i,j)+P(1,0|i,j)=1$. To prove that this is actually the quantum maximum notice that ${\\cal S}$ (and ${\\cal S'}$) contains only probabilities which are well defined within QT (irrespective of the order in which measurements are performed) only if all pairs $(i,j) \\in E(G)$ correspond to compatible observables. Then, maximizing $\\sum_{i\\in V} |\\langle i | \\psi \\rangle|^2-\\sum_{(i,j) \\in E(G)} | \\langle j | i \\rangle \\langle i | \\psi \\rangle |^2$ is equivalent to maximizing $\\sum_{i\\in V(G)} |\\langle i | \\psi \\rangle|^2$ under the restriction that $\\langle j | i \\rangle=0$ for $(i,j) \\in E(G)$. Consequently, $\\vartheta(G')=\\vartheta(G)+|E(G)|$. \\hfill \\vrule height6pt width6pt depth0pt\n\nFor clarity's sake we have not addressed the case of vertex-weighted graphs. However, the result presented in this section can be extended to vertex-weighted graphs with weights that are natural numbers (i.e., to the graphs corresponding to inequalities with rational coefficients) by noticing that any of these graphs can be converted into a standard graph by copying $n$ times each vertex with weight $n$.\n\n\n\\section{Discussion}\n\n\nThe observation that any NC inequality can be converted into a NC inequality of the form (\\ref{main}) may change the way contextuality is tested in the laboratory. This is so because testing (\\ref{main}) only requires measuring sequences of two observables, which allows us to simultaneously overcome many obstacles that, so far, had prevented experiments on most forms of contextuality. More specifically:\n\n(i) On one hand, the fact that only two sequential measurements are required will help to improve the quality of the experimental results by reducing the imperfections and making easier to identify and correct them.\n\n(ii) On the other hand, the fact that only two-point correlations are required allows us to apply existing tools for analyzing contextuality experiments with imperfections which are only defined in this case \\cite{DKL15}.\n\n(iii) Furthermore, it allows us to design tests free of problems (I) and (II) using systems in which sequential nondemolition measurements are not feasible. If only sequences of two measurements are needed, then the complexity of setup for testing each sequence is constant and thus problem (I) is solved. On the other hand, the first measurement can be simulated by a demolition measurement followed by a preparation that depends on the outcome. For example, the projective measurement of $i$ in Fig.\\ \\ref{Fig1} (b) can be simulated by a demolition measurement of $i$ followed by a preparation of a new system in state $|i\\rangle$ if the outcome of $i$ is $1$, or in the state predicted by L\\\"uders's rule for the state $|\\psi\\rangle$ if the outcome of $i$ is $0$; see Fig.\\ \\ref{Fig1} (c). Actually, only the first part is needed for obtaining the probabilities in EQ.\\ (\\ref{main}), which can then be expressed as $P_{|\\psi\\rangle}(1,1|i,j)=P_{|\\psi\\rangle}(1|i) P_{|i\\rangle}(1|j)$, where $P_{|\\phi\\rangle}(\\ldots)$ denotes the probability $P_{|\\phi\\rangle}(\\ldots)$ for the quantum state $|\\phi\\rangle$. Then, the choice of the second measurement is made after the result of the first measurement is indelibly recorded, thus solving problem (II).\n\nNevertheless, it is important to point out that the experiments using the scheme in Fig.\\ \\ref{Fig1} (c) present some conceptual disadvantages with respect to those using the schemes in Figs.\\ \\ref{Fig1} (a) and \\ref{Fig1} (b). For example, the scheme in Fig.\\ \\ref{Fig1} (c) provides an operational justification for the assumption of outcome noncontextuality, but only in experiments with sequences of two measurements on $|\\psi\\rangle$. There, it is possible to check that, when $|\\psi\\rangle$ is prepared, $i$ does not affect the outcome probabilities for any $j$ that is measured afterwards. However, this is not guaranteed for other possible preparations because the way the measurement of $i$ in Fig.\\ \\ref{Fig1} (c) transforms the state depends on $|\\psi\\rangle$ (and, in particular, fails to imitate a projective measurement when the outcome is $i=0$). In contrast, experiments using the schemes in Figs.\\ \\ref{Fig1} (a) and \\ref{Fig1} (b) provide a much more compelling operational justification of the assumption of outcome noncontextuality, since they allow us to check repeatability and no signaling for any preparation and any sequence of compatible measurements.\n\nNotice also that the experiments in Figs.\\ \\ref{Fig1} (b) and \\ref{Fig1} (c) are not thermodynamically equivalent. The measurement of $i$ in Fig.\\ \\ref{Fig1} (c), a demolition measurement followed by a preparation, dissipates more heat than the projective measurement of $i$ in Fig.\\ \\ref{Fig1} (b). Since classical simulations of quantum contextuality can be distinguished from truly quantum contextuality by the fact that the former produces more heat \\cite{CGGLW15}, then to certify quantum contextuality it is more convenient to use tests like those in Fig.\\ \\ref{Fig1} (b) than those in Fig.\\ \\ref{Fig1} (c).\n\nIn addition to the advantages for designing better experiments, the fact that any form of contextuality can be revealed in experiments measuring only two compatible observables also has some conceptual advantages:\n\n(iv) It allows for a novel black box approach to contextuality, extending the black box approach to nonlocality \\cite{NGHA15}. The basic elements would be pairs of boxes with respective inputs $x,y \\in V(G)$ such that $(x,y) \\in E(G)$ and respective outputs $a,b\\{0,1\\}$. Unlike the case of nonlocality, here we are not limited by the condition that each box should be attributable to a party spacelike separated from the others. Still, we can use tools like postselection, composition, and wiring of boxes. This approach may help to understand why certain sets of nonlocal correlations \\cite{NGHA15} are not realizable.\n\n(v) It suggests a way for developing a resource theory of contextuality in analogy to the resource theory of nonlocality \\cite{deVicente14}.\n\n(vi) It permits translating any NC inequality into a communication complexity scenario involving only two-point communication. This may help in elucidating the connection between the principle that constraints quantum contextuality \\cite{Cabello13} and information-theoretic principles that are essentially bipartite \\cite{PPKSWZ09}.\n\n(vii) It allows for building theory-independent sets of events with exclusivity relations leading to noncontextual and quantum bounds on demand. For any graph $G$, one can assign unit vectors to the vertices of $G$ such that adjacent vectors are orthogonal. This means that, according to QT, adjacent vertices correspond to alternative outcomes of a sharp measurement and, therefore, to exclusive events. However, these exclusivities cannot be justified in a theory-independent way unless we identify each of the vertices in $G$ with an operational procedure that explains why adjacent vertices correspond to exclusive events. The construction used in inequality (\\ref{main}) avoids the problem of finding such procedures and allows us to construct theory-independent sets of events with noncontextual and quantum bounds equal to those of $G$. Notice that the only assumption in the transition from $G$ to $G'$ is that edges in $G$ correspond to jointly measurable observables.\n\n(viii) The recent observation that contextuality is a necessary resource for fault-tolerant universal quantum computation via magic states \\cite{HWVE14,DGBR15,RBDOB15} depends on a definition of contextuality based on the violation of inequalities involving probabilities of noncompatible observables (following Ref.\\ \\cite{CSW14}). The fact that any of these inequalities can be converted into a NC inequality of the type (\\ref{main}) allows us to reformulate the contextuality-computation connection in terms of the traditional definition of contextuality as the violation of inequalities involving probabilities of compatible observables \\cite{Peres02}.\n\n\n\\begin{acknowledgments}\nWe thank L.\\ Aolita, M.\\ Ara\\'ujo, G.\\ Ca\\~{n}as, J.\\ Chen, V.\\ D'Ambrosio, E.\\ S.\\ G\\'omez, J.-\\AA.\\ Larsson, G.\\ Lima, A.\\ J.\\ L\\'opez-Tarrida, S.\\ P\\'adua, J.\\ R.\\ Portillo, F.\\ Sciarrino, S.\\ van Dam, and A.\\ Winter for useful conversations. This work was supported by the FQXi large grant project ``The Nature of Information in Sequential Quantum Measurements'' and Project No.\\ FIS2014-60843-P, ``Advanced Quantum Information'' (MINECO, Spain), with FEDER funds.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\IEEEPARstart{T}{rough} capturing both intensities and directions from sampled light rays, light field achieves high-quality view synthesis without the need of complex and heterogeneous information (e.g., geometry and texture). More importantly, benefit from the light field rendering technology~\\cite{LFrendering}, light field is capable of producing photorealistic views in real-time, regardless of the scene complexity or non-Lambertian effect. This high quality rendering usually requires light fields with disparities between adjacent views to be less than one pixel, i.e., the so-called densely-sampled light field (DSLF). However, typical DSLF capture either suffers from a long period of acquisition time (e.g., DSLF gantry system~\\cite{LFrendering,LFrig}) or falls into the well-known resolution trade-off problem, i.e., the light fields are sampled sparsely in either the angular or the spatial domain due to the limitation of the sensor resolution~\\cite{Lytro}.\n\nRecently, a more promising way is the fast capturing of a sparsely-sampled (angular domain) light field followed by a direct reconstruction or a depth-based view synthesis~\\cite{DoubleCNN,shi2020learning} with advanced deep learning techniques. On the one hand, typical learning-based reconstruction methods~\\cite{LFCNN,WuEPICNN2018,YeungECCV2018} employ multiple convolutional layers to map the low angular resolution light field to the DSLF. But due to the limited perceptive range of convolutional filters~\\cite{Long2014Do}, the networks will fail to collect enough information among the correspondences when dealing with large disparities, leading to aliasing effects in the reconstructed light field. On the other hand, depth-based view synthesis methods address the large disparity problem through plane sweep (depth estimation), and then synthesize novel views using learning-based prediction~\\cite{DoubleCNN,DeepStereo,mildenhall2019local}. However, such methods require depth consistency along the angular dimension, and thus, often fail to solve the depth ambiguity caused by the non-Lambertian effects.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{.\/materials\/teaser.pdf}\n\\end{center}\n\\vspace{-4mm}\n   \\caption{We propose a spatial-angular attention module embedded in a multi-scale reconstruction structure for learning-based light field reconstruction. The network perceives correspondence pixels in a non-local manner, and is able to produce high quality reconstruction using sparse input. In the bottom results, the input light fields are upsampled by using nearest interpolation for better demonstration. Light fields courtesy of Moreschini~\\textit{et al.}~\\cite{ICME2018} and Adhikarla~\\textit{et al.}~\\cite{kiran2017towards}.}\n\\label{fig:Teaser}\n\\end{figure}\n\nIn this paper, we propose a Spatial-Angular Attention Network, termed as SAA-Net, to achieve DSLF reconstruction from a sparse input. The proposed SAA-Net perceives correspondences in the Epipolar Plane Image (EPI) in a non-local manner, solving the aforementioned non-Lambertian effect and large disparity in an unified framework (Sec. \\ref{Sec:Network}). Specifically, the SAA-Net is composed by two parts, a spatial-angular attention module and an U-net backbone. Motivated by the non-local attention mechanism in~\\cite{wang2018non-local, zhang2018self}, for each pixel in the input light field, the Spatial-Angular Attention Module (termed as SAAM for short) computes the responses from all the positions in its epipolar plane, and produces an attention map that records the correspondences along the angular dimension, as shown in Fig. \\ref{fig:Teaser} (top). This correspondence information in the attention map is then applied to driven the reconstruction in the angular dimension via multiplication and deconvolution.\n\nTo efficiently perform the non-local attention, we propose a convolutional neural network with multi-scale reconstruction structure. The network follows the basic architecture of the U-net, i.e., an encoder-decoder structure with skip connections. The encoder compresses the input light field in the spatial dimensions and removes redundancy information for the SAAM. Rather than simply reconstruct the light field at the end of the network, we propose a multi-scale reconstruction structure by performing deconvolution along the angular dimension in each skip connection branch, as shown in Fig. \\ref{fig:Teaser} (top). The proposed multi-scale reconstruction structure maintains the view consistency in the low spatial scale while preserving fine details in the high spatial scales.\n\nFor the network training, we propose a spatial-angular perceptual loss that is specifically designed for the high-dimensional light field data (Sec. \\ref{Sec:training}). Rather than computing the high-level feature loss~\\cite{dosovitskiy2016generating, Johnson2016Perceptual} by feeding each view in the light field to a 2D CNN (e.g., the commonly-used VGG~\\cite{simonyan2015very}), we pretrain a 3D auto-encoder that considers the consistency in both the spatial and angular dimensions of the light field. We demonstrate the superiority of the  SAA-Net by performing extensive evaluations on various light field datasets. The proposed network presents high-quality DSLF on challenge cases with both non-Lambertian effects and large disparities, as illustrated in Fig. \\ref{fig:Teaser} (bottom). In summary, we make the following  contributions\\footnote{We will release the source code of this work upon acceptance.}:\n\\begin{itemize}\n    \\item A spatial-angular attention module that perceives correspondences non-locally in the epipolar plane;\n    \\item A multi-scale reconstruction structure for efficiently performing the non-local attention in the low spatial scale while also preserving the high frequencies;\n    \\item A spatial-angular perceptual loss specifically designed for the high-dimensional light field data.\n\\end{itemize}\n\n\\section{Related Work}\n\\subsection{Light Lield Reconstruction}\nFirst, we will give a brief review on researches of light field view synthesis (or view synthesis) depending on whether the depth information is explicitly used.\n\n\\textbf{Depth image-based view synthesis.} Typically, these kind of approaches first estimate the depth of a scene~\\cite{Tao,Occ,huang2017robust,zhang2016robust}, then warp and blend the input views to synthesize a novel view~\\cite{DeepStereo,DoubleCNN,soft3D,shi2020learning}. Conventional light field depth estimation approaches follow the pipeline of stereo matching~\\cite{scharstein2002taxonomy}, i.e., cost computation, cost aggregation (or cost volume filtering) and post refinement. The main different is that light field converts the disparity from the discrete space into a continuous space~\\cite{Wanner}, deriving various depth cues specifically for a light field, e.g., structure tensor-based local direction estimation~\\cite{Wanner}, depth from defocus~\\cite{Tao,Occ}. Also, some learning-based approaches incorporate the depth estimation pipeline described above with 2D convolution-based feature extraction, 3D convolution-based cost volume refinement and depth regression~\\cite{kendall2017end,tsaiattention} For novel view synthesis, input views are warped to the novel viewpoints with sub-pixel accuracy using bilinear interpolation and blended in different manners, e.g., total variation optimization~\\cite{Wanner}, soft blending~\\cite{soft3D} and learning-based synthesis~\\cite{Zheng2018ECCV}.\n \nRecently, researchers mainly focus on the studies for maximizing the quality of synthesized views based on the deep learning technique. Flynn \\textit{et al.}~\\cite{DeepStereo} proposed a learning-based method to synthesize novel views with predicted probabilities and colors for each depth plane. Kalantari \\textit{et al.}~\\cite{DoubleCNN} further employed a sequential network to infer depth (disparity) and color, and optimized the model via end-to-end training. Shi \\textit{et al.}~\\cite{shi2020learning} developed a convolutional network that fuses low-level pixels and high-level features in an unified framework. Zhou \\textit{et al.}~\\cite{zhou2018stereo} introduced a learning-based MultiPlane Image (MPI) representation that infers a novel view by the alpha blending of different images. Mildenhall \\textit{et al.}~\\cite{mildenhall2019local} further proposed to use multiple MPIs to synthesize a local light field. \n\n\\textbf{Reconstruction without explicit depth.} These kind of approaches treat the problem of light field reconstruction as the approximation of plenoptic function. In the Fourier domain, the sparse sampling in the angular dimension produces overlaps between the original spectrum and its replicas, leading to aliasing effect. Classical approaches~\\cite{chai2000plenoptic,zhang2003spectral} consider a reconstruction filter (usually in a wedge shape) to extract the original signal while filtering the aliasing high-frequency components. For instance, Vagharshakyan \\textit{et al.}~\\cite{Shearlet} utilized an adapted discrete shearlet transform in the Fourier domain to remove the high-frequency spectra that introduce aliasing effects. Shi \\textit{et al.}~\\cite{LFfourier} performed DSLF reconstruction as an optimization for sparsity in the continuous Fourier domain.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{.\/materials\/RF.pdf}\n\\end{center}\n\\vspace{-4mm}\n   \\caption{Analysis of reconstruction quality in terms of the network receptive field and disparity range of the scene. For a scene with small disparities, both networks (a) with small receptive field ($27\\times27$ pixels) and (b) with large receptive field ($53\\times53$ pixels) are able to reconstruct high-quality light field. However, for a scene with large disparities, network (c) with small receptive field suffers from severe aliasing effects. While network (d) with large receptive field can still produce plausible results. We show the sparely-sampled inputs on the top row and the reconstructed on the bottom. The receptive field of each network is visualized with green box. The input EPIs are stretched along the angular dimension fore better demonstration.}\n\\label{fig:RF}\n\\end{figure*}\n\nIn recent years, some learning-based approaches were also proposed for depth-independent reconstruction~\\cite{LFCNN,WuEPICNN2018,YeungECCV2018,wang2018end}. Zhu \\emph{et al}.~\\cite{zhu2019revisiting} proposed an auto-encoder that combines convolutional layers and convLSTM layer~\\cite{shi2015convolutional}. For explicitly addressing the aliasing effects, Wu \\emph{et al}.~\\cite{WuEPICNN2018} took advantage of the clear texture structure of the EPI and proposed a ``blur-restoration-deblur'' framework. However, when applying a large blur kernel for large disparities, the approach tends to fail at recovering the high-frequency details, and thus leading to the blur effect. Wang~\\cite{wang2018end} further proposed to apply a 3D CNN that takes a 3D slice as the input. Yeung \\textit{et al.}~\\cite{YeungECCV2018} directly fed the entire 4D light field into a 4D convolutional network, and applied a coarse-to-fine model to iteratively refine the spatial and angular dimensions of the light field. Wu \\emph{et al}.~\\cite{wu2019learning} proposed an evaluation network for EPIs with different shear amount, termed as sheared EPI structure. In this structure, the depth information is implicitly used to select a well reconstructed EPI. However, the performance of the network is limited due to the finite perceptive field of the convolutional neurons, especially when handling the large disparity problem.\n\n\\subsection{Attention Mechanism}\nAttention was first built to imitate the mechanism of human perception that mainly focuses on the salient part~\\cite{itti1998model,rensink2000dynamic,corbetta2002control}. Vaswani \\textit{et al.}~\\cite{vaswani2017attention} indicated that the attention mechanism is able to solve the long term dependency problem even without being embedded in the backbone of a recurrent or CNN. Therefore, the attention mechanism is recently developed to enabling the non-local perception in the spatial or temporal dimension~\\cite{zhang2019deep}. \n\nTo achieve the feature of non-local perception, Hu \\textit{et al.}~\\cite{hu2019senet} and Woo \\textit{et al.}~\\cite{woo2018cbam} proposed to use a global pooling (max-pooling or average-pooling) followed by a multi-layer perceptron to aggregate the entire information in the spatial dimension. Tsai \\textit{et al.}~\\cite{tsaiattention} introduced an attention module in the angular dimension to weight the contribution of each view in a light field. Vaswani \\textit{et al.}~\\cite{vaswani2017attention} proposed to use a weighted average of the responses from all the positions with respect to a certain position in the latent space, which is called self-attention (also known as intra-attention). Alternatively, Wang \\textit{et al.}~\\cite{wang2018non-local} achieved the self-attention by using matrix multiplication between reshaped feature maps, and is termed as non-local attention. For a high-dimensional task like video classification, the proposed module reshapes the 4D tensor (time, height, width and channel) into a 2D matrix. Zhang \\textit{et al.}~\\cite{zhang2018self} further extended this idea into a Generative Adversarial Network (GAN). Rather than using the non-local attention mechanism, Wang \\textit{et al.}~\\cite{wang2019learning} proposed a parallax attention module to compute the response across two stereo images. For each epipolar line in the stereo images (feature maps), the 2D matrices (width and channel) are multiplied to produce a sparse attention map that implies correspondences. In this paper, we extend the non-local attention mechanism to high dimensional light field data. For each pixel in the input 3D light field, the attention is computed in the 2D epipolar plane, rather than the epipolar line in~\\cite{wang2019learning} or the entire 3D data space in~\\cite{wang2018non-local}.\n\n\\section{Problem Analysis}\\label{Sec:Problem}\nIn the following analysis, we empirically show that the performance of a learning-based light field method is closely related to the perception range of its neuron (or convolutional filter), especially when addressing the large disparity problem. Deep neural network is proved to be a powerfull technique in solving ill-posed inverse problems~\\cite{jin2017deep}. In the light field reconstruction problem, the performance of a deep neural network mainly depends on two factors: disparity range of the scene (input light field) and network structure. Since the first factor is normally unalterable once the light field is acquired, typical deep learning-based approaches pursue a more appropriate architecture for higher performance~\\cite{LFCNN,DoubleCNN,WuEPICNN2018,YeungECCV2018}. Among those deep learning-based approaches, the depth-based view synthesis methods convert the feature maps into a physically meaningful depth map, while depth-independent methods directly map them to novel views. Essentially, both of two kind of approaches employ convolutional filter to generate responses (feature maps) between correspondence pixels.\n\nTo quantify the measurement of the capability to capture correspondences, we apply the concept of receptive field introduced in~\\cite{Long2014Do,zhou2015object}. The receptive field measures the number of pixels that are connected to a particular filter in the CNN, i.e., the number of correspondence pixels perceived by the convolutional filter. \n\nWe analyse the reconstruction qualities of two networks with the same structure but different receptive field sizes, as illustrated in Fig. \\ref{fig:RF}. For a scene with small disparity (about 3 pixels in the demonstrated example), both networks with small receptive field ($27\\times27$ pixels) and large receptive field ($53\\times53$ pixels) can reconstruct high angular resolution light fields (EPIs) with view consistency, as shown in Fig. \\ref{fig:RF}(a) and Fig. \\ref{fig:RF}(b). However, for a scene with large disparity (about 9 pixels), the network with small receptive field is not able to collect enough information from the corresponding pixels of its center point, as shown clearly at the top of Fig. \\ref{fig:RF}(c). Note that the actual size of the receptive field can be smaller than its theoretical size~\\cite{zhou2015object}. Consequently, the reconstructed result suffers from severe aliasing effects, as shown at the bottom of Fig. \\ref{fig:RF}(c). In comparison, the network with large receptive field can produce high quality result. In this example, the input EPIs are stretched along angular dimension for better demonstration.\n\n\\begin{figure*}\n\t\\begin{center}\n\t\t\\includegraphics[width=1\\linewidth]{.\/materials\/CNN.pdf}\n\t\\end{center}\n\t\\vspace{-4mm}\n\t\\caption{Architecture of the proposed Spatial-Angular Attention Network (SAA-Net). The SAA-Net is composed of (a) a multi-scale reconstruction structure, and (b) a Spatial-Angular Attention Module (SAAM). The input is a 3D slice ($L(u,v,s)$ or $L(v,u,t)$) of the light field.The batch and channel dimensions are omitted in the figure.}\n\t\\label{fig:CNN}\n\\end{figure*}\n\nDue to the limitation of parameter amount, it is intractable to expand the receptive field by stubbornly deepening the network or enlarging the filter size. The fundamental idea of the proposed approach is to design a light field reconstruction network that catches the correspondences non-locally across the spatial and angular dimensions of the light field via non-local perception. We achieve this with two aspects: 1) a spatial-angular attention module inspired by the non-local attention mechanism~\\cite{wang2018non-local, zhang2018self}; and 2) an encoder-decoder network that can reduce the redundancies in the light field so that the non-local perception can be implemented efficiently.\n\n\\section{Spatial-Angular Attention Network}\\label{Sec:Network}\nIn this section, we first introduce the overall architecture of the proposed Spatial-Angular Attention Network for light field reconstruction, which is termed as SAA-Net. We then present the proposed spatial-angular attention module that is specifically designed for disentangling the disparity information with a non-local perception. The input of the SAA-Net is a 3D light field slice with two spatial dimensions and one angular dimension, i.e., $L(x,y,s)$ (or $L(y,x,t)$). By splitting light field into 3D slices, the proposed network can be adopted for not only 3D light fields from a single-degree-of-freedom gantry system but also 4D light fields from plenoptic camera and camera array system.\n\nFor a 4D light field $L(x,y,s,t)$, we adopt a hierarchical reconstruction strategy similar with that in~\\cite{WuEPICNN2018}. The strategy first reconstruct 3D light fields using slices $L_{t^*}(x,y,s)$ and $L_{s^*}(y,x,t)$, then use the 3D light fields from the synthesized views to generate the final 4D light field.\n\n\\subsection{Network Architecture}\nWe propose a multi-scale reconstruction structure to maintain the view consistency (i.e., continuity in the angular dimension) in the low spatial scale while preserving fine details in the high spatial scales. The backbone of the proposed SAA-Net follows the encoder-decoder structure with skip connections, also known as U-net, as shown in Fig. \\ref{fig:CNN}(a). But the proposed SAA-Net has two particular differences: 1) We use deconvolution along the angular dimension in each skip connection branch before the concatenation in the decoder part; 2) We apply convolution layers with stride specifically in the spatial dimensions of the light field. Table~\\ref{table:Archi} provides the detailed configuration of the proposed SAA-Net.\n\nThe \\textbf{encoder} part of the SAA-Net construct multi-scale light field features and reduces the redundant information in the spatial dimension to alleviate the computational and GPU memory costs for the non-local perception in the spatial-angular attention module. We use two convolutional layers (3D) with stride $[2, 2]$ and $[2, 1]$ to downsample the spatial resolution of the light field with ratio 4 and 2 along the width and height dimension, respectively. Before each downsampling, two 3D convolutional layers with filter sizes $3\\times1\\times3$ and $1\\times3\\times3$ (width, height and angular) are employed to take place of a single convolutional layer with filter size $3\\times3\\times3$, reducing $1\/3$ parameters without performance degradation.\n\nThe \\textbf{skip connections} are fed with the feature layers before the downsampling in each encoder level, as shown in Fig. \\ref{fig:CNN}(a), which have full and half spatial resolutions. For each skip connection, a deconvolution layer (also known as transpose convolution layer) is then applied to upsample the feature map in the angular dimension, followed by a $1\\times1\\times1$ convolution. For the reconstruction of a 3D light field $L(x,y,s)$, the angular information can be mainly extracted from the 2D EPI $E(x,s)$, therefore, the filter size in each deconvolution layer is set to $3\\times1\\times7$.\n\nThe \\textbf{decoder} part of the SAA-Net upsamples the feature map from the spatial-angular attention module by using two deconvolution layer with stride $[2, 1]$ and $[2, 2]$ in the spatial dimensions (width and height). The decoder also receives information from the skip connections by concatenating the them along the channel dimension~\\cite{Eilertsen2017HDR}, as shown in Fig. \\ref{fig:CNN}(a). We then use two 3D convolutional layers with filter sizes $3\\times1\\times3$ and $1\\times3\\times3$ to compress the channel numbers in each level of the decoder. This can be considered as the blending of the light field features from different reconstruction scale. Note that all the reconstructions (upsampling operations) in the angular dimension are performed by the skip connections or the spatial-angular attention module, where latter will be introduced in the following subsection.\n\n\\begin{table}\n\\caption{Detail configuration of the proposed SAA-Net, where $k$ denotes the kernel size, $s$ the stride, $chn$ the number of channels, Conv the 3D convolution layer, Deconv the 3D deconvolution layer and Concat the concatenation.}\n\\label{table:Archi}\n\\begin{center}\n\\begin{tabular}{l|cccc}\nLayer & $k$ & $s$ & $chn$ & Input\\\\\n\\hline\n\\multicolumn{5}{c}{Encoder}\\\\\n\\hline\nConv1$\\_1$ & $3\\times1\\times3$ & - & 1\/24 & $L(x,y,s)$\\\\\nConv1$\\_2$ & $1\\times3\\times3$ & - & 24\/24 & Conv1$\\_1$\\\\\nConv1$\\_3$ & $3\\times3\\times1$ & $[2,2,1]$ & 24\/48 & Conv1$\\_2$\\\\\nConv2$\\_1$ & $3\\times1\\times3$ & - & 48\/48 & Conv1$\\_3$\\\\\nConv2$\\_2$ & $1\\times3\\times3$ & - & 48\/48 & Conv2$\\_1$\\\\\nConv2$\\_3$ & $3\\times1\\times1$ & $[2,1,1]$ & 48\/96 & Conv2$\\_2$\\\\\nConv3$\\_1$ & $1\\times1\\times1$ & - & 96\/48 & Conv2$\\_3$\\\\\nConv3$\\_2$ & $3\\times1\\times3$ & - & 48\/48 & Conv3$\\_1$\\\\\nConv3$\\_3$ & $1\\times3\\times3$ & - & 48\/48 & Conv3$\\_2$\\\\\nConv3$\\_4$ & $3\\times1\\times3$ & - & 48\/48 & Conv3$\\_3$\\\\\nConv3$\\_5$ & $1\\times3\\times3$ & - & 48\/48 & Conv3$\\_4$\\\\\n\\hline\n\\multicolumn{5}{c}{Skip connection}\\\\\n\\hline\nDeconv4$\\_1$ & $3\\times1\\times7$ & $[1,1,\\alpha_a]$ & 24\/24 & Conv1$\\_2$\\\\\nConv4$\\_2$ & $1\\times1\\times1$ & - & 24\/24 & Deconv4$\\_1$\\\\\nDeconv5$\\_1$ & $3\\times1\\times7$ & $[1,1,\\alpha_a]$ & 48\/48 & Conv2$\\_2$\\\\\nConv5$\\_2$ & $1\\times1\\times1$ & - & 48\/48 & Deconv5$\\_1$\\\\\n\\hline\n\\multicolumn{5}{c}{SAAM}\\\\\n\\hline\n\\multicolumn{5}{c}{Decoder}\\\\\n\\hline\nConv6$\\_1$ & $1\\times1\\times1$ & - & 48\/96 & SAAM\\\\\nDeconv6$\\_2$ & $4\\times1\\times1$ & $[2,1,1]$ & 96\/48 & Conv6$\\_1$\\\\\nConcat1 & - & - & -  & Conv6$\\_1$;Conv4$\\_2$\\\\\nConv6$\\_3$ & $3\\times1\\times3$ & - & 48\/48 & Concat1\\\\\nConv6$\\_4$ & $1\\times3\\times3$ & - & 48\/48 & Conv6$\\_3$\\\\\nDeconv7$\\_1$ & $4\\times4\\times1$ & $[2,2,1]$ & 48\/24 & Conv6$\\_4$\\\\\nConcat2 & - & - & -  & Conv7$\\_1$;Conv5$\\_2$\\\\\nConv7$\\_2$ & $3\\times1\\times3$ & - & 24\/24 & Concat2\\\\\nConv7$\\_3$ & $1\\times3\\times3$ & - & 24\/24 & Conv7$\\_2$\\\\\nConv8 & $3\\times3\\times3$ & - & 24\/1 & Conv7$\\_3$\\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\\subsection{Spatial-Angular Attention Module}\nInspired by the non-local attention mechanism in \\cite{wang2018non-local,zhang2018self}, we propose a Spatial-Angular Attention Module (SAAM) to disentangling the disparity information in light field. The main differences between the proposed SAAM and the previous non-local attention~\\cite{wang2018non-local,zhang2018self} are as follows: 1) Since the disparity information is encoded in the EPI, the non-local attention mechanism is performed in the 2D epipolar plane rather than the entire 3D space; 2) Taking advantage of the non-local perception of the EPI, we embed light field reconstruction in the spatial-angular attention module.\n\nA straightforward choice of performing spatial-angular attention is to embed the attention module in each resolution scale of the U-net. However, implementing non-local perception in the full resolution light field (feature map) is intractable in terms of computation complexity and GPU memory. Alternatively, we insert the proposed SAAM between the encoder and decoder as shown in Fig. \\ref{fig:CNN}(b). \n\nIn a 3D convolutional network, the feature layer will be a 5D tensor $\\phi\\in\\mathbb{R}^{B\\times W\\times H\\times A\\times C}$ (i.e., batch, width, hight, angular and channel). We first apply two convolution layers with kernel size $1\\times1\\times1$ to produce two feature layers $\\phi_a$ and $\\phi_b$ with size of $B\\times W\\times H\\times A\\times C'$. The channel number $C'$ is set to be $\\frac{C}{8}$ (i.e., $C'=6$) for computation efficiency. Then the feature layers $\\phi_a$ and $\\phi_b$ are reshaped into 3D tensors $\\phi'_a$ and $\\phi'_b$ of shapes $BH\\times WA\\times C'$ and $BH\\times C'\\times WA$, respectively. In this way, we merge the angular and width dimensions ($s$ and $x$ or $t$ and $y$ in a light field) together to implement the non-local perception in the epipolar plane. \n\nWe apply batch-wise matrix multiplication between $\\phi'_a$ and $\\phi'_b$ and a softmax function to produce a attention map $M$ as illustrated in Fig. \\ref{fig:CNN}(b). The attention map is composed of $BH$ matrices with shape $WA\\times WA$. Each matrix can be considered as a 2D expansion map of a 4D tensor $M'\\in\\mathbb{R}^{W\\times A\\times W\\times A}$ (the batch and height dimensions are neglected). The point $M'(x_0,s_0,x_1,s_1)$ indicates the response of light field position $L(x_0,y,s_0)$ to position $L(x_1,y,s_1)$ in the latent space. In other words, the attention map is able to capture correspondence among all the views in the input 3D light field. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{.\/materials\/att_map.pdf}\n\\end{center}\n\\vspace{-4mm}\n   \\caption{Visualization of the attention map. (a) An EPI with a foreground point $A$ and a background point $B$; (b) The corresponding high spatial-angular resolution EPI; (c) Three sub-maps extracted from the attention map. A point will have a strong response at the location of its correspondence in the attention map.}\n\\label{fig:att_map}\n\\end{figure}\n\nWe demonstrate the non-local perception of the proposed SAAM by visualizing a part of the attention map as shown in Fig. \\ref{fig:att_map}. In this example, there are two points $A$ and $B$ with remarkable visual features as shown in Fig. \\ref{fig:att_map}(a). And their corresponding points in other views are marked as $A'$ ($A''$) and $B'$ ($B''$). As the viewpoint changes along the angular dimension, the background point $B$ will be occluded by the foreground point $A$, which is demonstrated more obviously in Fig. \\ref{fig:att_map}(b). Fig. \\ref{fig:att_map}(c) shows three sub-maps extracted from the attention map $M'$ with $s_0=1$ and $s_1=1,2$ and 3, respectively. It can be clearly seen that a point will have the strongest response at the location of its correspondence in the attention map. For instance, the response $R(A,A')$ at the location $M'(8,1,6,2)$ for the corresponding patch $(A,A')$ (the middle sub-figure of Fig. \\ref{fig:att_map}(c)), and the response $R(A,A'')$ at the location $M'(8,1,4,3)$ for the corresponding patch $(A,A'')$ (the right sub-figure of Fig. \\ref{fig:att_map}(c)). For the occluded point $B$, the location of the maximum response changes from $M'(7,1,7,1)$ to $M'(7,1,4,3)$\\footnote{Due to the subpixel disparity of point $B$, the actual location of the maximum response could be $M'(7,1,5.5,2)$ in the middle sub-figure of Fig. \\ref{fig:att_map}(c).}. In this case, the attention module is able to locate the occluded point $B''$ through the surrounding pixels.\n\nSimilar with $\\phi'_a$ and $\\phi'_b$, $\\phi'_c$ is obtained by another $1\\times1\\times1$ convolution using input tensor $\\phi$, followed by the reshape operation. The main difference is that the channel number of the feature layer is $C''=\\frac{C}{2}$, i.e., $C''=24$ in our implementation. Another batch-wise matrix multiplication is applied between the attention map $M$ and $\\phi'_c$, resulting a 3D tensor $\\phi'_d\\in\\mathbb{R}^{BH\\times WA\\times C''}$. We then reshape $\\phi'_d$ into a 5D tensor $\\phi_d\\in\\mathbb{R}^{B\\times W\\times H\\times A\\times C''}$ and adopt a $1\\times1\\times1$ convolution to expand the channel dimension from $C''$ to $C$, generating a 5D tensor (or feature layer) $\\phi_e\\in\\mathbb{R}^{B\\times W\\times H\\times A\\times C}$. We further multiply the feature layer $\\phi_e$ by a trainable scale parameter (initialized as 0) and add back the input feature layer.\n\nWe implicitly adopt the non-local similarities or correspondences captured by the spatial-angular attention in the latent space and perform light field reconstruction by using deconvolution in the angular dimension, as shown in Fig. \\ref{fig:CNN}. The output of the SAAM is a 5D tensor $\\phi_f\\in\\mathbb{R}^{B\\times W\\times H\\times (\\alpha_a (A-1)+1)\\times C}$. By combining the proposed SAAM with the feature maps in the skip connections, the network is able to reconstruct light field with view consistency while also preserving the high frequency components. Detailed parameter setting of SAAM is listed in Table \\ref{table:Archi_SAAM}.\n\n\\begin{table}\n\\caption{Detail configuration of the proposed Spatial-Angular Attention Module (SAAM), where MatMul denotes the matrix multiplication and Add the element-wise addition.}\n\\label{table:Archi_SAAM}\n\\begin{center}\n\\begin{tabular}{l|cccc}\nLayer & $k$ & $s$ & $chn$ & Input\\\\\n\\hline\nConv1 & $1\\times1\\times1$ & - & 48\/6 & Encoder\\\\\nConv2 & $1\\times1\\times1$ & - & 48\/6 & Encoder\\\\\nConv3 & $1\\times1\\times1$ & - & 48\/6 & Encoder\\\\\nReshape1 &- & - & - & Conv1 \\\\\nReshape2 &- & - & - & Conv2 \\\\\nReshape3 &- & - & - & Conv3 \\\\\nMatMul1 & - & - & - & Reshape1;Reshape2 \\\\\nSoftmax & - & - & - & MatMul1\\\\\nMatMul2 & - & - & - & Softmax;Reshape3 \\\\\nReshape4 &- & - & - & MatMul2 \\\\\nConv4 & $1\\times1\\times1$ & - & 24\/48 & Reshape4\\\\\nAdd & - & - & 48\/48  & Encoder;Conv4\\\\\nDeconv & $3\\times1\\times7$ & $[1,1,\\alpha_a]$ & 48\/48 & Add\\\\\nConv$\\_6$ & $1\\times1\\times1$ & - & 48\/48 & Deonv\\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\section{Network Training}\\label{Sec:training}\n\\subsection{Spatial-Angular Perceptual Loss}\nTypical learning-based light field reconstruction or view synthesis methods optimize the network parameters by formulating a pixel-wise loss between the inferred image and the desired view (or EPI~\\cite{WuEPICNN2018}). Recently, researches~\\cite{zhou2018stereo,zhang2019image,meng2019high,mildenhall2019local} show that formulating the loss function in the high-level feature space will motivate the restoration of high-frequency components. This high-level feature loss, also known as perceptual loss, can be computed from part of the feature layers in the autologous network~\\cite{dosovitskiy2016generating} or other pre-trained networks~\\cite{Johnson2016Perceptual}, such as the commonly-used VGG network~\\cite{simonyan2015very}.\n\nIn this paper, we propose a spatial-angular perceptual loss that is specifically designed for the high-dimensional light field data. Existing approaches~\\cite{mildenhall2019local,meng2019high} for light field reconstruction apply perceptual loss between 2D sub-aperture images, neglecting the view consistency constraint in the angular dimension. Alternatively, we propose to use a 3D light field encoder to map the 3D light fields into high-dimensional feature tensors (width, height, angular and feature channel). To achieve this, we design another 3D encoder-decoder network (auto-encoder)\\footnote{The architecture of the 3D auto-encoder for the perceptual loss is different with that of the SAA-Net.} optimized by using unsupervised learning, i.e., the network is trained by inferring (compress and restore) the input light field. We then employ the encoder part to extract the high-level feature for the proposed spatial-angular perceptual loss. Note that the auto-encoder can be also generalized to 4D form. But given that some light field datasets have only one angular dimension (e.g., light fields from gantry system in~\\cite{ICME2018}) and the proposed SAA-Net also takes 3D light field as input, we only adopts 3D convolution in the encoder and decoder.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{.\/materials\/AE.pdf}\n\\end{center}\n\\vspace{-4mm}\n   \\caption{Architecture of the 3D encoder-decoder network designed for the proposed spatial-angular perceptual loss.}\n\\label{fig:AE}\n\\end{figure}\n\nFig. \\ref{fig:AE} demonstrates the computation process of the proposed spatial-angular perceptual loss as well as the designed auto-encoder. We use 3D convolutional layers with kernel size $3\\times3\\times3$ to encode the 3D light field (the SAA-Net output or the ground truth) into latent representations. The encoder applies stride convolutional layers with stride 2 in each dimension to compress the light field from low-level pixel space into high-level feature space. The decoder employs bilinear upsampling and convolutional layers to restore the light field from the latent representations. Detailed configuration of each layer is shown in Fig. \\ref{fig:AE}. The loss function $\\mathcal{L}_{AE}$ for optimizing the auto-encoder is as\n$$\\mathcal{L}_{AE}(L_{HR})=\\Vert f_{AE}(L_{HR})-L_{HR} \\Vert_1,$$\nwhere $f_{AE}$ denotes the auto-encoder.\n\nWith the unsupervised learning, the encoder part is trained to extract high-frequency features in different scales. In this paper, we use the second, fourth and sixth layers in the encoder to form the spatial-angular perceptual loss\n$$\n\\mathcal{L}_{feat}(\\hat{L}_{HR},L_{HR})=\\sum\\limits_{l=2,4,6}\\lambda_{feat}^{(l)}\\Vert\\phi_{ae}^{(l)}(\\hat{L}_{HR})-\\phi_{ae}^{(l)}(L_{HR})\\Vert_1,\n$$\nwhere $\\phi_{ae}^{(l)}(\\cdot)$ $(l={2,4,6})$ denotes the feature layers in the encoder, and $\\lambda_{feat}=0.2, 0.2, 0.1$ is a set of hyperparameters for the proposed spatial-angular perceptual loss, $\\hat{L}_{HR}$ is the light field reconstructed by the SAA-Net and $L_{HR}$ is the desired high-angular resolution light field.\n\nTo prevent the potential possibility that different light field patches are mapped to the same feature vector~\\cite{dosovitskiy2016generating}, our loss function also contains a pixel-wise term $\\mathcal{L}_{pix}$ using Mean Absolute Error (MAE) between $\\hat{L}_{HR}$ and $L_{HR}$, i.e.,\n$$\\mathcal{L}_{pix}(\\hat{L}_{HR},L_{HR})=\\Vert \\hat{L}_{HR}-L_{HR} \\Vert_1.$$\nThen the final loss function $\\mathcal{L}_{SAA}$ for training the SAA-Net is defined as\n\\begin{equation}\\label{eq:loss}\n\\mathcal{L}_{SAA}=\\mathcal{L}_{pix}+\\mathcal{L}_{feat}.\n\\end{equation}\nThe two terms are weighted by the set of hyperparameters $\\lambda_{feat}$ in the perceptual loss.\n\n\\subsection{Training Data}\nWe use light fields from the Stanford (New) Light Field Archive~\\cite{StanfordLFdatasets} as the training dataset, which contains 12 light fields\\footnote{The light field \\textit{Lego Gantry Self Portrait} is excluded from the training dataset since the moving camera may influence the reconstruction performance.} with $17\\times17$ views. Since the network input is 3D light fields, we can extract 17 $L(x,y,s)$ and 17 $L(y,x,t)$ in each 4D light field set. Similar with the data augmentation strategy proposed in~\\cite{zhu2019revisiting}, we augment the extracted 3D light fields using shearing operation~\\cite{Ng2005Fourier}\n$$L_d(x,y,s)=L(x+(s-\\frac{S}{2})\\cdot d,y,s),$$\nwhere $S$ is the angular resolution of the 3D light field $L(x,y,s)$ and $L_d(x,y,s)$ is the resulting 3D light field with shear amount $d$. $L_d(y,x,t)$ can be obtained similarly. In practice, we use two shear amounts $d\\pm2$. The shearing-based data augmentation increases the number of training examples by 2 times. More importantly, the disparity effects in the augmented light field will be more obvious as shown in Fig. \\ref{fig:data_aug}, enabling the network to address the large disparity problem.\n\nTo accelerating procedure the training and insure the same resolution between the input examples in the meantime, the extracted 3D light fields are cropped into sub-light fields of spatial resolution $64\\times24$ (width and height for $L(x,y,s)$ or height and width for $L(y,x,t)$) with stride 40 pixels. About $6.7\\times10^5$ examples can be extracted from the 3D light fields (original and augmented).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{.\/materials\/data_aug.pdf}\n\\end{center}\n\\vspace{-4mm}\n   \\caption{An illustration of training data augmentation using shearing operation. For clear display, one of the spatial dimension in the 3D light field is ignored.}\n\\label{fig:data_aug}\n\\end{figure}\n\n\\subsection{Implementation Details}\nTwo models with reconstruction factors (upsampling scale in the angular dimension) $\\alpha_a=3,4$ are trained. The input\/output angular resolution of the training samples for these two models are 5\/17 and 6\/16, respectively. Although the reconstruction factor of the network is fixed, we can achieve a flexible upsampling rate through network cascade. The training is performed on the Y channel (i.e., the luminance channel) of the YCbCr color space. We initialize the weights of both convolution and deconvolution layers by drawing randomly from a Gaussian distribution with a zero mean and standard deviation $1\\times10^{-3}$, and the biases by zero. The network is optimized by using ADAM solver~\\cite{Kingma2014Adam} with learning rate of $1\\times10^{-4}$ ($\\beta_1=0.9$, $\\beta_2=0.999$) and mini-batch size of 28. The training model is implemented using the \\emph{Tensorflow} framework~\\cite{TensorFlow}. The network converges after $8\\times10^5$ steps of backpropagation, taking about 35 hours on a NVIDIA Quadro GV100.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{.\/materials\/Result2.pdf}\n\\end{center}\n\\vspace{-4mm}\n   \\caption{Comparison of the results on the light fields from the CIVIT Dataset~\\cite{ICME2018} ($16\\times$ upsampling). The results show one of our reconstructed view, EPIs extracted from light fields reconstructed by each methods.}\n\\label{fig:Result2}\n\\end{figure*}\n                                                                                                                                                                                                                                                                                                                                                                                                                                                                           \n\\section{Evaluations}\nIn this section, we evaluate the proposed SAA-Net on several datasets, including light fields from gantry system, light fields from plenoptic camera (Lytro Illum~\\cite{Lytro}). We mainly compare our approach with three state-of-the-arts learning-based methods by Kalantari~\\textit{et al.}~\\cite{DoubleCNN} (depth-based), Wu~\\textit{et al.}~\\cite{WuEPICNN2018} (without explicit depth) and Yeung~\\textit{et al.}~\\cite{YeungECCV2018} (without explicit depth). To empirically validate the effectiveness of the proposed schemes, we perform ablation studies of our approach by training our network without the SAAM, without the multi-scale reconstruction structure and without the spatial-angular perceptual loss, respectively. The quantitative evaluations is reported by measuring the average PSNR and SSIM~\\cite{SSIM} values over the synthesized views of the luminance channel. For more quantitative and qualitative evaluations, please see the submitted video.\n\n\\begin{table*}\n\\caption{Quantitative results (PSNR\/SSIM) of reconstructed light fields on the light fields from the CIVIT Dataset~\\cite{ICME2018}.}\n\\label{table:Result2}\n\\vspace{-3mm}\n\\begin{center}\n\\begin{tabular}{p{2.5cm}|c|p{1.9cm}<{\\centering} p{1.9cm}<{\\centering} p{1.9cm}<{\\centering} p{1.9cm}<{\\centering} p{1.9cm}<{\\centering} p{1.9cm}<{\\centering}}\n& Scale & \\textit{Seal \\& Balls} & \\textit{Castle} & \\textit{Holiday} & \\textit{Dragon} & \\textit{Flowers} & Average\\\\\n\\hline\nKalantari~\\textit{et al.}~\\cite{DoubleCNN}& \\multirow{4}*{$8\\times$} & 46.83 \/ 0.990 & 39.14 \/ 0.973 & 36.03 \/ 0.979 & 43.97 \/ 0.989 & 39.00 \/ 0.989 & 40.99 \/ 0.984\\\\\nWu~\\textit{et al.}~\\cite{WuEPICNN2018}& & 49.01 \/ 0.997 & 37.67 \/ 0.984 & 40.46 \/ 0.995 & 48.38 \/ 0.997 & 45.85 \/ 0.998 & 44.27 \/ 0.994\\\\\nYeung~\\textit{et al.}~\\cite{YeungECCV2018} & & 49.83 \/ 0.997 & 40.84 \/ 0.993 & 41.16 \/ 0.996 & 48.61 \/ 0.997 & 47.83 \/ 0.997 & 45.65 \/ 0.996\\\\\nOur proposed & &\\textbf{51.05 \/ 0.998} & \\textbf{43.15 \/ 0.994} & \\textbf{42.27 \/ 0.997} &\\textbf{49.68 \/ 0.998} &\\textbf{48.35 \/ 0.998} &\\textbf{46.90 \/ 0.997}\\\\\n\\hline\nKalantari~\\textit{et al.}~\\cite{DoubleCNN}& \\multirow{7}*{$16\\times$} & 43.13 \/ 0.985 & 36.03 \/ 0.965 & 32.44 \/ 0.961 & 39.50 \/ 0.985 & 35.21 \/ 0.973 & 37.26 \/ 0.974\\\\\nWu~\\textit{et al.}~\\cite{WuEPICNN2018}& & 45.21 \/ 0.994 & 35.20 \/ 0.977 & 35.58 \/ 0.987 & 46.39 \/ 0.997 & 41.60 \/ 0.995 & 40.80 \/ 0.990\\\\\nYeung~\\textit{et al.}~\\cite{YeungECCV2018} & & 44.38 \/ 0.992 & 37.86 \/ 0.989 & 36.06 \/ 0.988 & 45.52 \/ 0.997 & 42.30 \/ 0.994 & 41.22 \/ 0.992\\\\\nw\/o SAAM &   & 46.85 \/ 0.995 & 37.78 \/ 0.989 & 36.17 \/ 0.988 & 47.10 \/ 0.998 & 42.98 \/ 0.996 & 42.18 \/ 0.993\\\\\nw\/o MSR structure & & 46.53 \/ 0.995 & 38.33 \/ 0.990 & 36.94 \/ 0.989 & 46.92 \/ 0.997 & 43.01 \/ 0.996 & 42.35 \/ 0.993\\\\\nw\/o SAP loss & & 49.02 \/ 0.996 & 40.69 \/ 0.992 & 38.97 \/ 0.992 & 48.23 \/ 0.997  & 44.46 \/ 0.997 & 44.27 \/ 0.995\\\\\nOur proposed & &\\textbf{49.35 \/ 0.997} & \\textbf{40.85 \/ 0.992} & \\textbf{39.21 \/ 0.993} & \\textbf{48.54 \/ 0.998}  & \\textbf{44.67 \/ 0.997} & \\textbf{44.52 \/ 0.995}\\\\\n\\end{tabular}\n\\end{center}\n\\vspace{-4mm}\n\\end{table*}\n\n\\subsection{Evaluations on Light Fields from Gantry Systems}\nA gantry system capture a light field by mounting a conventional camera on a mechanical gantry. Typical gantry system takes minutes to hours (depending on the angular density) to take a light field. With a high quality DSLF reconstruction \/ view synthesis approach, the acquisition time will be considerably reduced. In this experiment, we use light fields from the CIVIT Dataset~\\cite{ICME2018} ($1\\times193$ views of resolution $1280\\times720$) and the MPI Light Field Archive~\\cite{kiran2017towards} ($1\\times101$ views of resolution $960\\times720$) with upsampling scales $8\\times$ and $16\\times$. Since the vanilla version of the network by Yeung et al.~\\cite{YeungECCV2018} was specifically designed for 4D light fields, we modify the convolutional layers for the 3D input while remain the network architecture unchanged. The networks by Kalantari~\\textit{et al.}~\\cite{DoubleCNN} and Yeung et al.~\\cite{YeungECCV2018} are re-trained using the same training dataset as the proposed network. In this experiment, the performances in terms of both angular sparsity and non-Lambertian are taken into consideration.\n\nFig. \\ref{fig:Result2} shows the reconstruction results on three light fields, \\textit{Castle}, \\textit{Holiday} and \\textit{Flowers}, from the CIVIT Dataset~\\cite{ICME2018} with upsampling scale $16\\times$ (disparity range $d_{\\min}-d_{\\max}=14$px). The first case and the third case have thin structures with complex occlusions. The depth and learning-based approach by Kalantari~\\textit{et al.}~\\cite{DoubleCNN} fail to estimate depth maps accurately enough to warp the input images, and the color CNN is not able to correct the misaligned views, producing ghosting artifacts as shown in the figure. For the second case, we demonstrate reconstructed EPIs in a highly non-Lambertian region, as shown in the figure. Caused by the depth ambiguity, the approach by Kalantari~\\textit{et al.}~\\cite{DoubleCNN} produces choppiness artifacts along the angular dimension. Due to the limited receptive field of the networks, the results by Wu~\\textit{et al.}~\\cite{WuEPICNN2018} and Yeung et al.~\\cite{YeungECCV2018} show aliasing effects in various degrees. Table \\ref{table:Result2} lists the quantitative measurements on the light fields from the CIVIT Dataset~\\cite{ICME2018} with upsampling scale $8\\times$ and $16\\times$.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{.\/materials\/Result3.pdf}\n\\end{center}\n\\vspace{-4mm}\n   \\caption{Comparison of the results on the light fields from the MPI Light Field Archive~\\cite{kiran2017towards} ($16\\times$ upsampling).}\n\\label{fig:Result3}\n\\end{figure*}\n                                                                                                                                                                                                                                                                                                                                                                                                                                                                           \n\\begin{table*}\n\\caption{Quantitative results (PSNR\/SSIM) of reconstructed light fields on the light fields from the MPI Light Field Archive~\\cite{kiran2017towards}.}\n\\label{table:Result3}\n\\vspace{-3mm}\n\\begin{center}\n\\begin{tabular}{p{2.5cm}|c|p{1.9cm}<{\\centering} p{1.9cm}<{\\centering} p{1.9cm}<{\\centering} p{1.9cm}<{\\centering} p{1.9cm}<{\\centering} p{1.9cm}<{\\centering}}\n& Scale & \\textit{Bikes} & \\textit{FairyCollection} & \\textit{LivingRoom} & \\textit{Mannequin} & \\textit{WorkShop} & Average\\\\\n\\hline\nKalantari~\\textit{et al.}~\\cite{DoubleCNN}& \\multirow{4}*{$8\\times$} & 34.83 \/ 0.969 & 36.66 \/ 0.977 & 46.35 \/ 0.991 & 40.62 \/ 0.983 & 38.66 \/ 0.986 & 39.42 \/ 0.981\\\\\nWu~\\textit{et al.}~\\cite{WuEPICNN2018}& & 38.39 \/ 0.990 & 40.32 \/ 0.992 & 45.48 \/ 0.996 & 43.26 \/ 0.995 & 41.55 \/ 0.995  & 41.80 \/ 0.994\\\\\nYeung~\\textit{et al.}~\\cite{YeungECCV2018} & & 39.55 \/ 0.993 & 40.25 \/ 0.993 & 47.32 \/ 0.997 & 44.49 \/ 0.996 & 43.17 \/ 0.996 & 42.96 \/ 0.995\\\\\nOur proposed & &\\textbf{40.53 \/ 0.995} & \\textbf{42.23 \/ 0.995} & \\textbf{47.96 \/ 0.997} &\\textbf{45.02 \/ 0.996} &\\textbf{45.29 \/ 0.997}  &\\textbf{44.21 \/ 0.996}\\\\\n\\hline\nKalantari~\\textit{et al.}~\\cite{DoubleCNN}& \\multirow{4}*{$16\\times$} & 30.67 \/ 0.935 & 32.39 \/ 0.952 & 41.62 \/ 0.973 & 37.15 \/ 0.970  & 33.94 \/ 0.971 & 35.15 \/ 0.960\\\\\nWu~\\textit{et al.}~\\cite{WuEPICNN2018}& & 31.22 \/ 0.951 & 30.33 \/ 0.942 & 42.43 \/ 0.991 & 39.53 \/ 0.989 & 33.49 \/ 0.977 & 35.40 \/ 0.970\\\\\nYeung~\\textit{et al.}~\\cite{YeungECCV2018} & & 32.67 \/ 0.967 & 31.82 \/ 0.969 & 43.54 \/ 0.993 & 40.82 \/ 0.992 & 37.21 \/ 0.988 & 37.21 \/ 0.982\\\\\nOur proposed & &\\textbf{36.01 \/ 0.985} & \\textbf{36.13 \/ 0.982} & \\textbf{46.45 \/ 0.997} & \\textbf{41.08 \/ 0.993}  & \\textbf{39.11 \/ 0.992} & \\textbf{39.76 \/ 0.990}\\\\\n\\end{tabular}\n\\end{center}\n\\vspace{-4mm}\n\\end{table*}\n\nFig. \\ref{fig:Result3} shows the reconstruction results on three light fields, \\textit{Bikes}, \\textit{FairyCollection} and \\textit{WorkShop}, from the MPI Light Field Archive~\\cite{kiran2017towards} with upsampling scale $16\\times$ (disparity range up to 33.5px). The first case has complex structure occluded by the foreground bikes as shown in the top row of Fig. \\ref{fig:Result3}. The baseline methods fail to reconstruct the complex structure in the background. Among them, the depth and learning-based approach~\\cite{DoubleCNN} fail to estimate a proper occlusion relation between the bikes and the background. The second scene is a non-Lambertian case, i.e., a refractive glass before the toys. The approach by Kalantari~\\textit{et al.}~\\cite{DoubleCNN} cannot reconstruct the refractive object. And the reconstructed EPIs by the baseline methods~\\cite{WuEPICNN2018,YeungECCV2018} appear severe aliasing effects. Table \\ref{table:Result3} lists the quantitative measurements on the light fields from the MPI Light Field Archive~\\cite{kiran2017towards} with upsampling scale $8\\times$ and $16\\times$.\n\n\\textbf{Ablation studies.} We empirically validate the proposed approach by performing the following ablation studies. First, we replace the proposed SAAM with a simple transpose convolution layer, denoted as ``w\/o SAAM'' for short. As show by the quantitative result in Table \\ref{table:Result2}, the average PSNR value decreases about 2.3dB without the SAAM. In the second ablation study, we use a typical 3D U-net as the backbone and remove the transpose convolution layer in the SAAM, denoted as ``w\/o MSR structure'' for short (without the Multi-Scale Reconstruction structure). The angular reconstruction is simply realized by using transpose convolution at the end of the network. The performance of the network decreases about 2.2dB in terms of PSNR. In the last ablation study, we train the proposed SAA-Net simply by using the pixel-wise term (MAE loss) without the proposed spatial-angular perceptual loss, denoted as ``w\/o SAP loss'' for short. The performance (PSNR) decreases about 0.3dB.\n\n\\subsection{Evaluations on Light Fields from Lytro Illum}\nWe evaluate the proposed approach using three Lytro light field datasets (113 light fields in total), the \\textit{30 Scenes} dataset by Kalantari~\\textit{et al.}~\\cite{DoubleCNN}, and the \\textit{Reflective} and \\textit{Occlusions} categories from the Stanford Lytro Light Field Archive~\\cite{StanfordLytro}. In this experiment, we reconstruct a $7\\times7$ light field from $3\\times3$ views ($3\\times$ upsampling) and a $8\\times8$ light field from $2\\times2$ views ($7\\times$ upsampling). Since the vanilla versions of the networks by Kalantari~\\textit{et al.}~\\cite{DoubleCNN} and Yeung et al.~\\cite{YeungECCV2018} are trained on Lytro light fields, we use their original parameters without re-training. Note that the proposed network is not fine-tuned on any Lytro light field datasets, and the results are produced by the same set of network parameters for both upsampling scales $3\\times$ and $7\\times$.\n\nWe demonstrate two cases with relatively large disparities (maximum disparity up to 13px), \\textit{IMG1743} from the \\textit{30 Scenes}~\\cite{DoubleCNN} and \\textit{Occlusions 23} from the \\textit{Occlusions} category~\\cite{StanfordLytro}, as shown in Fig. \\ref{fig:Result1}. In both cases, the reconstruction results by Wu~\\textit{et al.}~\\cite{WuEPICNN2018} and Yeung et al.~\\cite{YeungECCV2018} show ghosting artifacts around the region with large disparity (background in the \\textit{IMG1743} case, and foreground in the \\textit{Occlusions 23} case), which we believe are caused by the limited receptive field of their networks. The depth and learning-based approach by Kalantari~\\textit{et al.}~\\cite{DoubleCNN} produces plausible result in the first case, but appears tearing artifacts near the occlusion boundary as marked by the red arrow in the EPI. In the second case, the approach by Kalantari~\\textit{et al.}~\\cite{DoubleCNN} fail to estimate proper depth information, introducing misalignment as shown by the EPI. In comparison, the proposed SAA-Net provides reconstructed light fields with higher view consistency (as shown in the demonstrated EPIs). Table \\ref{table:Result1} lists the quantitative results on the evaluated Lytro light fields. The PSNR and SSIM values are averaged over the light fields in each dataset.\n\n\\begin{figure*}\n\t\\begin{center}\n\t\t\\includegraphics[width=1\\linewidth]{.\/materials\/Result1.pdf}\n\t\\end{center}\n\t\\vspace{-4mm}\n\t\\caption{Comparison of the results on the light fields from Lytro Illum. The results show the error map (absolute error of the grey-scale image) and the EPIs at the location marked by red lines. Light fields are from the \\textit{30 Scenes}~\\cite{DoubleCNN} and the \\textit{Occlusions} category~\\cite{StanfordLytro}.}\n\t\\label{fig:Result1}\n\\end{figure*}\n\n\\begin{table}\n\\caption{Quantitative results (PSNR\/SSIM) of reconstructed views on the light fields from Lytro Illum~\\cite{Lytro}. The \\textit{30 Scenes} dataset courtesy of Kalantari~\\textit{et al.}~\\cite{DoubleCNN}, and the \\textit{Reflective} and \\textit{Occlusions} categories are from the Stanford Lytro Light Field Archive~\\cite{StanfordLytro}.}\n\\label{table:Result1}\n\\begin{center}\n\\begin{tabular}{l|c|ccc}\n& Scale & \\textit{30 Scenes}  &\\textit{Reflective} &\\textit{Occlusions}\\\\\n\\hline\n\\scriptsize{Kalantari~\\textit{et al.}}~\\cite{DoubleCNN}& \\multirow{4}*{$3\\times$} & 39.62\/0.978 & 37.78\/0.971 & 34.02\/0.955 \\\\\nWu~\\textit{et al.}~\\cite{WuEPICNN2018}& & 41.85\/0.992 & 41.76\/0.986 & 38.52\/0.970 \\\\\nYeung~\\textit{et al.}~\\cite{YeungECCV2018}& & 44.53\/0.990 & 42.56\/0.975  &39.27\/0.945\\\\\nOur proposed & & \\textbf{44.69}\/\\textbf{0.996} & \\textbf{43.99}\/\\textbf{0.991} & \\textbf{40.33}\/\\textbf{0.969} \\\\\n\\hline\n\\scriptsize{Kalantari~\\textit{et al.}}~\\cite{DoubleCNN}& \\multirow{4}*{$7\\times$} & 38.21\/0.974 & 35.84\/0.942 & 31.81\/0.895 \\\\\nWu~\\textit{et al.}~\\cite{WuEPICNN2018}& & 36.74\/0.969 & 36.55\/0.964 & 33.11\/0.939 \\\\\nYeung~\\textit{et al.}~\\cite{YeungECCV2018}& & \\textbf{39.22}\/0.977 & 36.47\/0.947  & 32.68\/0.906\\\\\nOur proposed & & 39.09\/\\textbf{0.983} & \\textbf{37.47}\/\\textbf{0.977} & \\textbf{33.77}\/\\textbf{0.952} \\\\\n\\end{tabular}\n\\end{center}\n\\vspace{-4mm}\n\\end{table}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{.\/materials\/att_map1.pdf}\n\\end{center}\n\\vspace{-4mm}\n   \\caption{Additional results of attention map on scenes with (a) large disparity and (b) non-Lambertian effect.}\n\\label{fig:att_map1}\n\\end{figure*}\n\n\\subsection{Further Analysis}\n\\textbf{Spatial-angular attention map.} We visualize additional attention maps on scenes with large disparity and non-Lambertian effect as shown in Fig. \\ref{fig:att_map1}. We demonstrate the spatial-angular attention on a scene with large disparities in Fig. \\ref{fig:att_map1}(a). In this case, the disparity between neighbouring views are about 16 pixels. Due to the spatial downsampling in the SAA-Net, the disparity of the light field (feature map) fed to the SAAM is about 4 pixels, see the top left figure in Fig. \\ref{fig:att_map1}(a). Part of the attention map $M'(x_0,s_0,x_1,s_1), s_0=1, s_1=1,2,3$ is visualized in the bottom of Fig. \\ref{fig:att_map1}(a). We can clearly see the the response moves from $R(A,A)$ at the positon $M'(13,1,13,1)$ to $R(A,A'')$ at the position $M'(13,1,5,1)$ along the angular dimension.\n\nFig. \\ref{fig:att_map1}(b) demonstrates the spatial-angular attention on a scene with non-Lambertian effect. In this case, the positional relation of the corresponding points $B$, $B'$ and $B''$ does not follow their depth, as clearly shown in the top right figure of Fig. \\ref{fig:att_map1}(b). We visualize part of the attention map $M'(x_0,s_0,x_1,s_1), s_0=3, s_1=1,2,3$ in the bottom of Fig. \\ref{fig:att_map1}(b). The result shows that the proposed SAAM is able to catch the correspondences even for regions with non-Lambertian effects.\n\n\\textbf{Computational time.} For a 3D light field, the network takes about 53 seconds to reconstruct a $1\\times97$ light field from $1\\times7$ views of spatial resolution $960\\times720$ ($16\\times$ upsampling), i.e., 0.54s per view. For a 4D light field from Lytro Illum, it takes about 18 seconds to reconstruct a $7\\times7$ light field from $3\\times3$ views of spatial resolution $536\\times376$ ($3\\times$ upsampling), i.e., less than 0.36s per view. And the reconstruction of a $8\\times8$ Lytro light field from $2\\times2$ views ($7\\times$ upsampling) takes less than 30 seconds, i.e., less then 0.5s per view. The above evaluations are performed on an Intel Xeon Gold 6130 CPU @ 2.10GHz with a NVIDIA Quadro GV100.\n\n\\textbf{Limitations.} Repetitive patterns in the input light field can cause multiple plausible responses in the non-local attention, leading to misalignments in the reconstructed light fields. A possible solution is to introduce a smooth term in the attention map to penalize the multiple responses. \n\nAlthough we have proposed a multi-scale reconstruction structure to alleviate the GPU memory cost, the SAAM will still eat up the GPU memory when dealing with an input light field with high spatial-angular resolution. For example, when reconstructing light fields from the CIVIT Dataset~\\cite{ICME2018} ($1\\times25$ input views of resolution $1280\\times720$), we have to disassemble the 3D data into sub-light fields of resolution $25\\times1280\\times6$. Our investigation shows that the disassembling is harmful to the reconstruction quality. Decomposing a large tensor into the combination of small tensors~\\cite{kolda2009tensor} will solve this problem in a more essential way.\n\n\\section{Conclusions}\nWe have proposed a spatial-angular attention module in a 3D U-net backbone to capture correspondence information non-locally in the light field reconstruction problem. The introduced Spatial-Angular Attention Module (termed as SAAM) is designed to compute the responses from all the positions in the epipolar plane for each pixel in the light field and produce a spatial-angular attention map that records the correspondences. The attention map is then applied to driven the light field reconstruction via deconvolution in the angular dimension. We further propose a multi-scale reconstruction structure based on the 3D U-net backbone that implements the SAAM efficiently in the low spatial scale, while also preserving fine details in the high spatial scales by using decovlution-based reconstruction in each skip connenction. For the network training, a spatial-angular perceptual loss is designed specifically for the high-dimensional light field data by pretraining a 3D auto-encoder. The evaluations on light fields with challenging non-Lambertian effects and large disparities have demonstrated the superiority of the proposed spatial-angular attention network.\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1}Introduction}\n\nRandomness plays a prominent role in quantum computing, quantum information processing and physics in general. In particular, random unitaries chosen from the Haar measure \\cite{4} on the unitary group U(N)  find  applications  in  randomized benchmarking \\cite{6} , noise estimation \\cite{5}, quantum metrology \\cite{OAC+16}, as well as modeling thermalization \\cite{7} and even black hole physics \\cite{8}. \nUnfortunately, genuine Haar distributed unitaries are hard to create as the scaling required is exponential in the number of qubits \\cite{9}. \nOn  the other  hand   efficient  substitutes of  Haar distributed  unitaries   were  shown to  exist \\cite{1}, \\cite{10}, \\cite{11}, \\cite{12}, \\cite{13}, \\cite{3}, \\cite{15}. These substitutes are known as unitary t-designs \\cite{10} - ensembles over subsets of U(N) which mimic  exactly \\cite{10}, \\cite{11} , \\cite{15} or approximately \\cite{1}, \\cite{12}, \\cite{13},\\cite{3},\\cite{15} choosing from the Haar measure up to order t in the statistical moments. \n\nFollowing \\cite{15} our approach is to harness the quantum randomness arising from applying a measurement based (MB) scheme  to produce approximate designs.\nIn MB computation \\cite{14}, unitary determinsitic computation is achieved by making sequential, adaptive measurements on an entangled multipartite state, known as a graph state \\cite{20}. Without these adaptive feedforward corrections, the inherent randomness of the measurements effectively samples from ensembles of unitaries.\nIn \\cite{15} it was shown that starting with a fixed graph state, applying fixed angle measurements (with no need for feedforward corrections), effectively samples from an approximate t-design. \nFurthermore this process is efficient in the number of qubits, preperation and measurements, following from the efficiency of the construction of Brandao et al. \\cite{1}. \nIndeed the construction of the graph state essentially mimics the random circuit construction of Brandao et al. \\cite{1}.\nHowever, in doing so, the graph itself is rather complicated, and moreover is not a simple regular lattice. \nA natural question is then, can simple, regular lattices (such as those useful for universal measurement based computation \\cite{14}, \\cite{19})   applied to generate t-designs?\nAs well as being more convenient from a practical point of view (in terms of generating the graph state), this connects the question of optimal generation of ensembles to standard measurement based quantum computation. Furthermore it requires a new proof that it is an approximate t-design (though the techniques also follow along the lines of \\cite{1} it does not follow directly from their results).\n\nIn this  work we  show that it is possible. In particular we show that running fixed measurement MB scheme on a regular graph with poly-log (in n,  t and $\\dfrac{1}{\\varepsilon}$  ) number of qubits, with no feed-forward, results in an ensemble of random unitaries which forms a $\\varepsilon$-approximate t-design ensemble. \nThe graph we use is very similar to the brickwork graph known to be a universal resource for MB quantum computation \\cite{19}. The proofs presented here rely principally on the G-local random circuit construction (GLRC) of Brandao et al. \\cite{1}, the  detectability lemma (DL) of Aharonov et al. \\cite{2} as well as a theorem \\cite{33}, \\cite {3}, \\cite{1} on the equivalence between tensor product expanders (TPE's) and approximate t-designs.\\\\\n\nThis paper is divided as follows: section II defines some preliminary notions, section III provides a brief statement of the results, section IV contains a detailed proof of the results, finally section V discusses briefly some potential applications.\n\n\\section{\\label{sec:level2}Preliminaries}\n\n\n\n\n\\subsection{\\label{subsec:level1} MBQC}\n\nThe model of computation used throughout this work is the measurement based quantum computation (MBQC) model. This model was first proposed by Raussendorf and Briegel \\cite{14} as an alternative to the gate model of quantum computing \\cite{16}.\nComputation is carried out by first preparing a large entangled state (a graph state), followed by single qubit measurements. \nCrucially, in order to deterministically perform a desired unitary, the measurements must be done adaptively - to counter the inherent randomness arising from the measurements the measurement angles are corrected by feedforward process using previous measurement results.  However, instead of doing these corrections we will use this randomness as a resource to sample from an ensemble of unitaries with the desired structure - namely that they are a t-design.\n\nA \\emph{graph state} \\cite{20} is a pure entangled quantum state of $n$ qubits in one to one correspondence with a graph $G=\\{E,V\\}$ of $n$ vertices  and edges. Each vertex $i \\in V$ of the graph is associated with a qubit, and each edge ${i,j \\in E}$ represents a preperation entanglement\n\\begin{eqnarray}\n|G\\rangle = \\prod_{i,j \\in E} CZ_{i,j} |+...+\\rangle_V \\nonumber.\n\\end{eqnarray}\nFor computations on quantum inputs, a set of vertices $I \\subset  V$ are assigned as the input vertices, with initial input state $|\\psi_{in}\\rangle_I$, and the associated \\emph{open graph state} is defined as\n\\begin{eqnarray} \\label{eq Open Graph State}\n|G(\\psi)\\rangle = \\prod_{i,j \\in E} CZ_{i,j} |\\psi_{in}\\rangle_I|+...+\\rangle_{V\\setminus I}.\n\\end{eqnarray}\nWe also identify the set of output qubits by the vertices $O\\subset V$. Computation is then carried out by sequentially measuring all the non output qubits, in our case in a basis on the $X-Y$ plane. Each measurement is represented by an angle $\\alpha$, corresponding to measureing in the basis \\newline $\\{|\\pm\\alpha\\rangle := (|0\\rangle \\pm e^{i \\alpha}|1\\rangle)\\}$.\nIn standard MBQC the correction strategy is given by the gflow \\cite{21}- a partial order over the graph as well as a function, which give a time order for the measurements and the dependency respectively. In this work, following \\cite{15},  we do not adapt the measurement angles so that for each measurement result a potentially different unitary is performed.\n\nFollowing the convention of \\cite{21}  (see also \\cite{15}) we represent these resources as graphs where the input vertices have squares on them and the measured qubits have an angle inside representing the measurement angle, hence the quantum outputs $O$ are empty circles (note that in \\cite{21} measured qubits are simply coloured black).\nFigure \\ref{fig1} illustrates this for a simple example. Following equation (\\ref{eq Open Graph State}), for input state $|\\psi_{in}\\rangle = a|0\\rangle+b|1\\rangle$ Fig. \\ref{fig1} corresponds to an initial open graph state \n\\begin{eqnarray}\n|G_\\psi\\rangle & =& a |0\\rangle |+\\rangle + b |1\\rangle |-\\rangle \\nonumber \\\\\n& = &\\frac{1}{\\sqrt{2}}\\left(|+\\alpha\\rangle HZ(\\alpha) |\\psi_{in}\\rangle +|-\\alpha\\rangle H Z Z(\\alpha) |\\psi_{in}\\rangle \\right)\\nonumber,\n\\end{eqnarray}\nwhere $H$ is the Hadamard gate, $Z$ is the Pauli Z gate, $Z(\\alpha):=e^{-i \\alpha Z\/2}$ is a rotation by angle $\\alpha$ around the Z axis. Denoting $m$ as the binary measurement outcome, associating $m=0$ to outcome $+\\alpha$ and $m=1$ to $-\\alpha$, it is clear that measuring the first qubit is equivalent to applying the random unitary\n\\begin{eqnarray}\nU(m) = H Z^m Z(\\alpha), \n\\end{eqnarray}\nwith equal probabilty for $m=0$ and $m=1$.\nAs in \\cite{15}, the same idea applied to larger graphs, with more inputs and outputs is the source of the random unitary ensembles we will study in this work.\n\n\n\\begin{figure}[h]\n\\begin{center}\n\n\n\\graphicspath{}\n\n\\includegraphics[trim={0 10cm 0 0cm}, scale=0.18]{fig1corr.pdf}\n\\caption{MB scheme on a 2 qubit cluster state, the input (squared) qubit when measured at an angle $\\alpha$ in the XY plane results in propagation of the input state to output along with application of a random unitary $U$ =$ H $$Z^{m}$ $Z$($\\alpha$).}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\n\\subsection{\\label{subsec:level2}t-designs and Tensor Product Expanders}\nUnitary t-designs \\cite{10}, \\cite{12}, \\cite{13}  are subsets of the unitary group U(N) , sampled with some probability distribution, which mimic either exactly or approximately choosing from the Haar measure on U(N) up to order t in the statistical moments. Our concern in this work is with the latter, known as $\\varepsilon$-approximate t-designs. \nMore formally, let H =($C ^{ 2} )^{ \\otimes n} $  be a Hilbert space on n qubits, define the density matrix $\\rho$=$\\ket{\\phi}\\bra{\\phi}$ with $\\ket{\\phi}$  a unit vector in H (n-qubit state). Then let \\{ $p_{i}$ , $U_{i}$ \\} be a collection of unitaries in U(N)= U($2^{n}$) (the n-qubit unitary group), where we sample each $ U_{i}$ with probability $p_{i}$ . Let $\\mu_{H}$ denote the Haar measure \\cite{4}  on U($2^{n}$ ). Consider now t-copies of our n-qubit system, an ensemble $\\{p_i, U_i\\}$ is an  \\emph{exact t-design} if it satisfies\n\\begin{equation}\n\\begin{aligned}\n\\label{eq1}\n\\sum_{i}p_{i} U_{i} ^{\\otimes t} \\rho^{t} U_{i} ^{\\dagger \\otimes t}=\\int_{ U(2^{n})}U ^{\\otimes t}\\rho^{t}U ^{\\dagger \\otimes t}\\mu_{H}(dU),\n\\end{aligned}\n\\end{equation}\nfor all $\\rho^{t} \\in \\mathcal{B}(H^{\\otimes t})$, where $\\int_{ U(2^{n})}U ^{\\otimes t}\\rho^{t}U ^{\\dagger \\otimes t}\\mu_{H}(dU)$ represents averaging over the Haar measure. \n\n\n$\\varepsilon$-approximate t-designs are defined similarly as ensembles which satisfy\n\\begin{equation}\n\\begin{aligned}\n\\label{eq2}\n(1-\\varepsilon)\\int_{ U(2^{n})}U ^{\\otimes t}\\rho^{t}U ^{\\dagger \\otimes t}\\mu_{H}(dU) \\leq \\sum_{i}p_{i} U_{i} ^{\\otimes t} \\rho^{t} U_{i} ^{\\dagger \\otimes t} \\\\ \\leq (1+\\varepsilon) \\int_{ U(2^{n})}U ^{\\otimes t}\\rho^{t}U ^{\\dagger \\otimes t}\\mu_{H}(dU).\n\\end{aligned}\n\\end{equation\n\nApproximate t-designs can also be defined using various norms \\cite{1}, \\cite{13}, \\cite{3}.\nA useful concept equivalent to a t-design is a tensor product expander (TPE) \\cite{22} . We say that a couple \\{ $p_{i}$ , $U_{i}$ \\}  is an ($\\eta$,t)-TPE if the following equation holds \\cite{3}, \\cite{22} :\n\n\\begin{equation}\n\\begin{aligned}\n\\label{eq3}\ng(t,\\mu):=\\mid\\mid\n\\sum_{i }p_i U_i^{\\otimes t,t} - \\int_{U(2^{n}) } U^{\\otimes t,t}\\mu_{H}(dU) \\mid\\mid_\\infty\\leq\\eta,\n\\end{aligned}\n\\end{equation}\nwhere $U^{\\otimes t,t} =U^{\\otimes t} \\otimes U^{\\star \\otimes t}$,  $\\star$ denotes the complex conjugate and $\\mu$ represents the probability measure on U(d) which results in choosing $U_i$ with probability $p_i$. \nIn our main proof we will rely on the following theorem \\cite{33}, \\cite {3}, \\cite{1} on the equivalence between TPE's and $\\varepsilon$-approximate t-designs:\n\n\\newtheorem{theorem}{Theorem}\n\\begin{theorem} \\cite{33}, \\cite {3}, \\cite{1}\\\\\n Let \\{ $p_{i}$ , $U_{i}$ \\} be an ($\\eta$,t)-TPE with $\\eta$  < 1.Denote by $\\mathcal{U}_{i}$ the set of all possible unitaries $U_{i}$.  Then iterating this TPE k times ( i.e. obtaining the product $U$=$\\prod_{j=1,...,k}$$U_{\\pi (j)}$  with the $U_{\\pi (j)}$'s independently chosen from the ensemble  \\{$p_{i}$ , $U_{i}$\\} ,\\\\ $\\pi$(j) $\\in$ $\\{1,\u2026,|\\mathcal{U}_{i}|\\}$ with k $\\ge$ $\\dfrac{1}{log(\\dfrac{1}{\\eta})}log(\\dfrac{d^{t}}{\\varepsilon})$ results in an ensemble\\{ {$p_U$  , $U$=$\\prod_{j=1,...,k}$$U_{\\pi (j)}$  }\\} which is an $\\varepsilon$-approximate t-design. Here $d$ is the dimension of the unitary group.\n\\end{theorem}\n\n\nFor convenience we define the t'th \\emph{moment super operator} of  \\{$p_{i}$ , $U_{i}$\\}  (or simply moment super operator) as follows:\n\\begin{equation}\n\\begin{aligned}\n\\label{momop}\n  M_t [\\mu]:=\\sum_{i} p_i U_i^{\\otimes t,t}.\n\\end{aligned}\n\\end{equation}\nIts role in the TPE condition (\\ref{eq3}) means  $M_t [\\mu]$  plays a major role in our proofs. Indeed, for any ensemble such that sampling it many times does eventually lead to the Haar measure, then $g(t,\\mu)$ is equal to the second highest eigenvalue of $M_t[\\mu]$ \\cite{12,1}.\nWe will then follow the techniques of  \\cite{1}, \\cite{12}, \\cite{13} in connecting the calculation of this to gaps of Hamiltonians, which will allow us to prove the connection to t-designs via theorem 1.\n\n\n\n\n\\subsection{\\label{subsec:level3}Many body Physics and t-designs}\n\nIt has been known for some time \\cite{1}, \\cite{12}, \\cite{13}  that the problem of estimating the scaling rate (number of iterations needed to reach a desired accuracy $\\varepsilon$) of an $\\varepsilon$- approximate t-design can be reduced to a problem of finding the spectral gap (the difference of energy between the ground and first excited state) of some many-body Hamiltonian. \nHere we give an overview of these techniques, in particular as used in \\cite{1}.\n\n\nAn extensive body of research (\\cite{23}, \\cite{24}, \\cite {25}, \\cite{26} and many others) has been devoted to the case of 1D spin chains, with local Hamiltonians (we assume a finite interaction range) with translational symmetry (see Fig.\\ref{fig2}). \nWe will focus exclusively on this case, and more precisely on a type of 1D Hamiltonian (the one we use in our proof) consisting of local terms acting on nearest neighbor spins $ i$ and $i$+1 with translational symmetry, which are frustration free (the entire Hamiltonian can be minimized by minimizing each of its local terms individually) as well as verifying the Nachtergaele criterion (\\cite{23}, condition C.3). \nThis family of  Hamiltonians was used by Brandao et al. \\cite {1}  to study their local random circuit (LRC) construction, and later the so-called G-local random circuits (GLRC). \nWe will briefly define these families of circuits and review these proofs.\n\n\\begin{figure}[h]\n\\begin{center}\n\n\n\\graphicspath{}\n\\includegraphics[trim={0 12cm 0 16cm}, scale=0.08]{Figurefornewarticlecorr.pdf}\n\\caption{Example of a 4 spins 1D system. The local Hamiltonians $h_{i,i+1}$  have a range of 2 (i.e. act on nearest neighbor spins). For example $h_{1,2}$ acts on spins 1 and 2. Translational invariance means that any $ h_{i,i+1}$ has the same form on all 2 qubit systems ($i$, $i$+1). In this case the total Hamiltonian of the system of 4 spins can be written as a sum of local Hamiltonians, i.e. H=$h_{1,2}$+$h_{2,3}$+$h_{3,4}$. }\n\\label{fig2}\n\\end{center}\n\\end{figure}\n\nThe local random circuits (LRC) in \\cite{1} generate random circuits on n-qubits as follows.\nFor each run of the LRC, a unitary $ U$ $\\in$ U(4) is chosen from the Haar measure on U(4), then an index $i$  is chosen uniformly at random from the set \\{1,\u2026., n-1\\} , finally $U$ is applied to qubits $i$ and $ i$+1. \nThe LRC defines a couple \\{$\\mu_{LRC}$,$\\mathcal{U}$\\} where $\\mu_{LRC}$ is the probability measure induced by one LRC run, and $\\mathcal{U}$ is the set of  all the possible unitaries which can be generated by one LRC run. \nWe arrive at the following moment super operator associated to one run of the LRC\n\\begin{equation}\n\\begin{aligned}\n\\label{eq4}\nM_{t}[\\mu_{LRC}]=\\dfrac{1}{n-1}\\sum_{i=1}^{n-1}\\int_{U(4)}U_{i,i+1}^{\\otimes t,t}\\mu_{H}(dU)=\\dfrac{1}{n-1}\\sum_{i=1}^{n-1}P_{i,i+1},\n\\end{aligned}\n\\end{equation}\n\\\\ where $ U_{i,i+1}$=$ 1^{\\otimes i-1}\\otimes U \\otimes1^{\\otimes n-i-1}$, $U$ $ \\in$ U(4), $P_{i,i+1}:=\\int_{U(4)}U_{i,i+1}^{\\otimes t,t}\\mu_{H}(dU)$ and $\\mu_{H}$ is the Haar measure on U(4). \n\n Since each of the $P_{i,i+1}$'s is Hermitian, then $M_t [\\mu_{LRC} ]$  is itself Hermitian.\nNow consider the Hamiltonian \n\\begin{equation}\nH=\\sum_{i}h_{i,i+1},\n\\end{equation}\nwhere $ h_{i,i+1}$=$1- P_{i,i+1}$. Then\n\\begin{equation}\nM_{t}[\\mu_{LRC}] = I-\\dfrac{H}{n-1}\n\\end{equation} \nThe ground space of H has an eigenvalue of 0 and the gap between its ground and first excited spaces gives the second highest eigenvalue of $M_{t}[\\mu_{LRC}]$.\nThis $H$ is a 1D spin chain Hamiltonian with nearest neighbor local terms which are translationally invariant. It is also frustration free by construction because the Hamiltonian can be minimized simply by minimizing all of its local terms (taking their ground state of energy 0) individually.\nBrandao et al. also proved that this Hamiltonian verifies the Nachtergaele criterion, then also bounded the Nachtergaele Bound using path coupling techniques \\cite{1}. \nIn this way they show that the spectral gap $\\Delta H$  of $H$ admits the following (polynomial in t) bound for n $\\ge$ $\\lfloor 2.5log_{2}(4t) \\rfloor$ :\n\\begin{equation}\n\\begin{aligned}\n\\label{eq5}\n\\Delta H \\ge (1700. \\lfloor log_{2} (4t) \\rfloor^{2}. t^{5}. t^{\\dfrac{3.1}{log(2)}})^{-1},\n\\end{aligned}\n\\end{equation}\nwhere $\\lfloor x \\rfloor$ denotes the floor function acting on variable $x$.\n\n\nWe now move to G-local random circuits, which are the finite set counter parts of LRC.\nOne run of the GLRC follows exactly as the LRC case, but instead of choosing a unitary $U$ from the Haar measure of U(4), we choose with uniform probability from a finite set G of SU(4) which is universal and contains inverses. One can show from the beautiful result of Bourgain and Gamburd \\cite{26} that the Hamiltonian\n$H_{GLRC}$=$\\sum_{i} h^{'}_{i,i+1}$=$\\sum_{i}(1-P^{'}_{i.i+1})$ with \n$P^{'}_{i.i+1}$ =$\\dfrac{1}{|G|} \\sum_{U \\in G}(1^{\\otimes i-1}\\otimes U \\otimes1^{\\otimes n-i-1})^{\\otimes t,t}$  admits the following bound for its spectral gap:\n\\begin{equation}\n\\begin{aligned}\n\\label{eq6}\n\\Delta H_{GLRC} \\ge \\alpha.\\Delta H,\n\\end{aligned}\n\\end{equation}\nwith $\\alpha$ a constant and $\\Delta H$ the spectral gap of the LRC Hamiltonian.\n\nNote that because the set G contains unitaries and their inverses and samples them uniformly, then $ \\mu_{G }$($U$)=$\\mu_{G}$ ($U^{\\dagger}$),  for all $U$ ,$ U^{\\dagger}$ $\\in$ G. This means that $P^{'}_{i,i+1}$ (hence $ H_{GLRC}$ ) a Hermitian operator, and the above definition of a GLRC Hamiltonian makes sense.\nWe can rewrite equation (\\ref{eq6}) as follows:\n\\begin{equation}\n\\begin{aligned}\n\\label{eq7}\n\\Delta H_{GLRC} \\ge (C. \\lfloor log_{2} (4t)\\rfloor^{2}. t^{5}. t^{\\dfrac{3.1}{log(2)}})^{-1}= P_{GLRC}\n\\end{aligned}\n\\end{equation}\n C being a constant depending on the gate set G. \n\nThese spectral gaps directly give the second highest eigenvalue of the corresponding moment super operators, equal to the $g(t, \\mu)$ in equation (\\ref{eq3}) confirming the TPE conditions, which through theorem 1 then allow statements about their efficiency as t-designs \\cite{1}.\n\n\n\\section{\\label{sec:level4} Main Results}\n\nIn order to state the main results, we define some simple graph states for two input qubits. We will  hence forth refer to a graph state with an MB scheme applied to it as a gadget. These will act as the building blocks for our construction. \nConsider the 2, 5-column 2-row brickwork states with a fixed angle MB scheme in Fig.\\ref{fig3} and Fig.\\ref{fig4} which we call $S_{I_{1}}$  and $S_{I_{2}}$ .\n\n\\begin{figure}[h]\n\\begin{center}\n\n\n\\graphicspath{}\n\\includegraphics[trim={0 12cm 0 0cm}, scale=0.4]{fig3acorr.pdf}\n\\caption{The 2-row, 5-column brickwork state gadget giving rise to   $S_{I_{1 }}$ }\n\\label{fig3}\n\\end{center}\n\\end{figure}\n\\begin{figure}[h]\n\\begin{center}\n\n\n\\graphicspath{}\n\\includegraphics[trim={0 12cm 0 0cm}, scale=0.4]{fig3bcorr.pdf}\n\\caption{The 2-row, 5-column brickwork state gadget giving rise to   $S_{I_{2 }}$ }\n\\label{fig4}\n\\end{center}\n\\end{figure}\n\n$S_{I_{1} }$  and $ S_{I_{2} }$ give rise respectively to 2 MB ensembles \\{$\\dfrac{1}{2^{8}}$,$U_{1_{i}}$\\} and \\{$\\dfrac{1}{2^{8}}$,$U_{2_{i}}$\\}. \nThe number of unitaries generated by the MB scheme on the 2-row, 5-column brickwork state is $2^{8}$\n\nIt can be easily checked that each unitary of the ensembles $S_{I_{1} }$  and $ S_{I_{2} }$  contains (up to a global phase) an inverse in the ensemble. \nThat is, denoting $\\mathcal{U}_{S_{I_1}}$ as the set of unitaries generated by $S_{I_1}$ (and similarly for $S_{I_2}$),  for all $U_{1_{i}}$$\\in \\mathcal{U}_{S_{1_i}}$\n there exists $U_{1_{j}}$$\\in \\mathcal{U}_{S_{I_1}}$ such that $U_{1_{i}}=U^{\\dagger}_{1_{j}}$. Simlarly for $S_{I_2}$.\n\nConsider now a 13-column brickwork gadget: B= $S_{I_{1}} \\circ S_{I_{2 }} \\circ S_{I_{1} }$, where we mean by W $\\circ$ V   a concatenation which identifies the output of graph W as an input of graph V.  We are now in a position to state our main results:\\\\ \\\\\n\n\\begin{theorem}\n  The gadget B gives rise to an ensemble of unitaries which is \\\\\n  (i) universal on SU(4) \\\\\n  (ii) contains elements and their inverses, and\\\\\n  (iii) is sampled with a uniform probability. \\\\\n\\end{theorem}\n\nThis theorem means that the set of unitaries generated by B (call it $\\mathcal{U}_B$) satisfies the conditions necessary to form a GLRC. Though this is not exactly how our construction works, it will be important in proving our construction works in the proof of theorem 3.\n\nNow consider the gadget on n-qubits given in Fig.\\ref{fig5} which we call $G_{n}$. \nThe horizontal line with a circle in the middle means a direct link between output and input performed only on the 1st and last rows.  \nThe square with the letter B  is our 13-column brickwork gadget B, and the empty 3 sided square means that there is no vertical entanglement.    \n\n\\begin{figure}[h]\n\\begin{center}\n\n\n\\graphicspath{}\n\\includegraphics[trim={0 5cm 0 0cm},scale=0.4]{fig4corr.pdf}\n\\caption{The graph gadget $G_{n}$ pictured here for even n (the odd n case follows straightforwardly) }\n\\label{fig5}\n\\end{center}\n\\end{figure}\n\n\n\nThe first and last rows  of $G_{n}$ are made up of 13 qubits, and all rows in between are made up of 25 qubits. This gives rise in total to a graph composed of  26$+$25(n$-$2) = 25n$-$24 qubits. We now state our second main result. \\\\ \\\\\n\n\\begin{theorem}\n The k(n,t,$\\varepsilon$)-fold concatenation of $G_{n}$,  \\newline $E_{n}$ = $G_{n} \\circ G_{n} \\circ$... , results in an ensemble of unitaries which forms an $\\varepsilon$-approximate t-design on n-qubits ( n $\\ge$ $\\lfloor 2.5log_{2}(4t) \\rfloor$ ), with: \\\\  k(n,t,$\\varepsilon$)  $\\ge$  $\\dfrac{3}{log_{2}(1+\\dfrac{P_{GLRC}}{2})}(nt+log_{2}(\\dfrac{1}{\\varepsilon}))$\n\\end{theorem}\n\n\n\\section{\\label{sec:level5}Proofs}\n\\subsection{\\label{subsec:level6}Proof of Theorem 2}\nBefore going on to universality, let us briefly explain why the set of unitaries generated by B,  $\\mathcal{U}_B$ contains inverses ($(ii)$ in Theorem 2). \nAny element $U$ $\\in$ $\\mathcal{U}_B$ may be written as: $U$=$U_{1}.U_{2}.U_{1}^{'}$, where $U_{1}$  , $U_{1}^{'}$ $\\in$ $\\mathcal{U}_{S_{I_1}}$and $U_{2}$ $\\in$ $\\mathcal{U}_{S_{I_2}}$. \nSince $\\mathcal{U}_{S_{I_1}}$and $\\mathcal{U}_{S_{I_2}}$contain unitaries and their inverses, we can always find $ U^{\\dagger}$=$U_{1}^{ ' \\dagger}$.$U_{2}^{\\dagger}$.$U_{1}^{\\dagger}$  $\\in$ $\\mathcal{U}_B$. Furthermore, each unitary $U_{\\{m\\}} \\in \\mathcal{U}_B$ associated to a specific binary string $\\{m\\}$ is sampled with a uniform probability of $\\dfrac{1}{|\\mathcal{U}_B|}$, proving $(iii)$ in Theorem 2.\n\n\nThe remainder of this subsection is devoted to proving universality ($(i)$ in Theorem 2), and we will use the approach outlined in \\cite{27} and \\cite{34} for doing so. Following  \\cite{27} and \\cite{34}, one can show that the group generated by the set of unitaries \\{$A$,$A^{\\dagger}$,$C$,$C^{\\dagger}$,$E$,$E^{\\dagger}$,$F$,$F^{\\dagger}$\\} is dense (universal) on U(4) if the following conditions are satisfied:\\\\ \\\\\n$C_{1}$ :   $H_{1}$:=$\\dfrac{log(A)}{i}$ , $H_{2}$:=$\\dfrac{log(C)}{i}$, $H_{3}$:=$\\dfrac{log(E)}{i}$ and $H_{4}$:=$\\dfrac{log(F)}{i}$  and their commutators form a set of 16 linearly independent Hamiltonians which span the Lie algebra \\cite{28} of U(4).\\\\ \\\\\n$C_{2}$  :  $H_{1}$, $ H_{2}$, $ H_{3}$ and $ H_{4}$ have eigenvalues that are irrationally related to $\\pi$ .\\\\\n\nWe first consider $C_{1}$. We found 4 distinct unitaries $A$=$U_{\\{m\\}}$, $C$=$U_{\\{m^{'}\\}}$, $E$=$U_{\\{m^{''}\\}}$, and $F$=$U_{\\{m^{'''}\\}}$ .  Where $U_{\\{m\\}}$, $U_{\\{m^{'}\\}}$, $U_{\\{m^{''}\\}}$ and $U_{\\{m^{'''}\\}}$ $\\in$ $\\mathcal{U}_B$  are associated to the binary strings \\{$m$\\}=\\{0,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0,0,1,1,0,1,1,1,0\\} , \\{$m^{'}$\\}=\\{0,0,0,1,1,1,1,0,1,0,1,0,1,1,1,1,0,1,1,1,0,1,0,0\\}, \\{$m^{''}$\\}=\\{0,0,0,1,1,1,1,0,1,0,1,0,1,1,1,1,0,1,1,1,0,1,0,0\\}, and \\{$m^{'''}$\\}=\\{0,1,1,1,0,1,1,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,1,0\\}. We adopt the convention that  the first 12 binary numbers appearing  in a given binary string represent the measurement results on qubits of the first row of B from left to right (input towards output), and the last 12 binaries represent the measurements performed on the qubits of the second row of B from left to right. \n\n\nWe then construct 16 Hamiltonians $H_{1}$ ,\u2026, $H_{16}$ as follows:\\\\ \\\\\n$H_{1}=\\dfrac{log(A)}{i}$\\\\\n$H_{2}=\\dfrac{log(C)}{i}$\\\\\n$H_{3}=\\dfrac{log(E)}{i}$\\\\\n$H_{4}=\\dfrac{log(F)}{i}$\\\\\n$H_{5}=i.[H_{1},H_{2}]$\\\\\n$H_{6}=i.[H_{1},H_{3}]$\\\\\n$H_{7}=i.[H_{1},H_{4}]$\\\\\n$H_{8}=i.[H_{2},H_{3}]$\\\\\n$H_{9}=i.[H_{2},H_{4}]$\\\\\n$H_{10}=i.[H_{2},H_{5}]$\\\\\n$H_{11}=i.[H_{2},H_{6}]$\\\\\n$H_{12}=i.[H_{3},H_{4}]$\\\\\n$H_{13}=i.[H_{3},H_{5}]$\\\\\n$H_{14}=i.[H_{3},H_{6}]$\\\\\n$H_{15}=i.[H_{4},H_{5}]$\\\\\n$H_{16}=i.[H_{4},H_{6}]$\\\\\n\nAfter that, we expand each of the 16 Hamiltonians in the basis:\n P=\\{$P_{ij}$\\} $i$,$j$=0,..,3, where  P is a basis of the Lie algebra of U(4) . \n In other words, we write each: $H_{k}$=$a_{k}^{ij}$.($P_{ij}$) (Einstein summation convention adopted over i and j), where the  $a_{k}^{ij}$  ' s are real numbers. \n Since P is a basis of the Lie algebra of U(4) (over the field of real numbers ),  proving linear independence of the 16 Hamiltonians \\{$H_{k}$\\} $k$=1,..16  in the basis P means that the set \\{$H_{k}$\\} is itself a basis of the Lie algebra of U(4). \n The linear independence of the 16 generators \\{$H_k$\\} is equivalent to the non-vanishing of the determinant of a 16 by 16 matrix M, where each of the 16 columns of M are made up of the 16 coefficients\\{$a_{k}^{ij}$\\} for a given $ k$. \nWe found that the 16 Hamiltonians of our above constructed scheme give rise to a matrix M with non-vanishing determinant \\footnote{this was done numerically however well within numerical precision}, thus this scheme forms a basis of the Lie algebra of U(4) and $C_{1}$ is verified for a subset of $\\mathcal{U}_B$ (and hence for $\\mathcal{U}_B$ itself ). \n\n \n \nProving $C_{2}$ requires the use of a result in algebraic number theory called Lehmer's theorem \\cite{30}. Its context is described in the following Lemma:\n\\newtheorem{lem}{Lemma}\n\\begin{lem} \\cite{30} \n  If n>2 and k and n are coprime integers, then 2cos($\\dfrac{2k\\pi}{n}$) is an algebraic integer.\n\\end{lem}\nAn algebraic number  is a complex number which is a solution of a polynomial equation with integer coefficients. The minimal polynomial of an algebraic number $z$ is the polynomial of lowest degree with integer coefficients for which $z$ is a solution. An algebraic integer is an algebraic number whose minimal polynomial is monic (that is, the coefficient in front of the highest degree variable is 1) \\cite{31}. \n\nLehmer's theorem states that angles $\\alpha$ = $\\dfrac{2k\\pi}{n}$ which are rationally related to $\\pi$ must have $2cos(\\alpha)$  an algebraic integer. So, if we can find instances of angles $\\alpha$ in which $2cos(\\alpha)$ is not an algebraic integer, then $\\alpha$ has to be irrationally related to $\\pi$  as a consequence of Lehmer's theorem. \nEach of the eigenvalues $\\lambda$ of  $A$,  $C$, $E$ or $F$ is a complex number with unit norm (because they are unitary matrices). Thus, $\\lambda$=$e^{i\\theta}$. We  calculated the expression $ 2cos(\\theta)$\nand constructed its minimal polynomial. \nWe found that for each of the eigenvalues $\\lambda$ of $A$ , $C$, $E$ and $F$, $ 2cos(\\theta)$  does not verify a monic minimal polynomial (not an algebraic integer) \nand thus all the $\\theta$'s are irrationally related to $\\pi$ by Lehmer's theorem. Further, because $A$, $C$, $E$ and $F$ are diagonal in the same basis as their Hamiltonians $H_{1}$, $H_{2}$, $H_{3}$ and $H_{4}$ \\cite{27},\n  the $\\theta$'s we calculated are the eigenvalues of these Hamiltonians. Hence, the eigenvalues of the Hamiltonians are irrationally related to $\\pi$ which proves $C_{2}$. \n \n\nProving $C_{1}$ and $C_{2}$ means that the subset  \\{$A$,$A^{\\dagger}$,$C$,$C^{\\dagger}$,$E$,$E^{\\dagger}$,$F$,$F^{\\dagger}$\\} of $\\mathcal{U}_B$ is universal on U(4), and thus so is $\\mathcal{U}_B$. But $(i)$ further requires that the set be on SU(4). Fortunately, the moment super operator of a set sampled from U(4) can always be thought of as a sampling from SU(4). This can be seen by noting that for all $U$ $\\in$ U(4) we have $det(U)$ $\\neq$ 0, hence $U^{\\otimes t,t }$=$|det(U)|^{\\dfrac{t}{2}}$.$U^{ ' \\otimes t,t}$=$U^{ ' \\otimes t,t}$,  where $U^{'}$ $\\in$ SU(4). \n\n\n\\subsection{\\label{subsec:level7}Proof of Theorem 3}\nOur approach for proving Theorem 3 can be summarized by 2 steps.  \nIn the first step, we prove that the ensemble generated by the gadget $G_{n}$ is an ($\\eta$,t)-TPE with $\\eta$=poly(t) <1 . We do so by using Aharonov et al.'s  detecability lemma \\cite{2}. \nThe second step uses Theorem 1 to establish the bound on $k(n,t,\\varepsilon)$.\n\nConsider a GLRC n-qubit Hamiltonian\n\\begin{eqnarray}\nH_{GLRC} &=& \\sum_{i}h^{'}_{i,i+1} \\nonumber \\\\\n&=&\\sum_{i}(1-P^{'}_{i,i+1})\n\\end{eqnarray}\nwith G = $\\mathcal{U}_B$, and  \\\\ $P^{'}_{i,i+1}$=$\\dfrac{1}{|\\mathcal{U}_B|} \\sum_{U\\in \\mathcal{U}_B}(1^{\\otimes i-1}\\otimes U\\otimes1^{\\otimes n-i-1})^{\\otimes t,t}$. Define $ P_{odd}$=$P^{'}_{1,2} .P^{'}_{3,4}$\u2026\nand  $P_{even}$=$P^{'}_{2,3} . P^{'}_{4,5}$\u2026.$ P_{odd}$ and $ P_{even}$ can be considered as projectors onto the \"odd\" and \"even\" ground spaces of  $H_{GLRC}$. Let $P_{0}$ be the projector onto the entire ground space of  $H_{GLRC}$. Further, because $H_{GLRC}$ is constructed from  universal sets on U(4), then its ground space projector is nothing but the  t'th  Haar moment super operator \\cite{12}, \nIn other words $P_{0}$=$\\int_{U(2^{n})}$$U^{\\otimes t,t }$$ \\mu_{H}(dU)$  ;   $U$ $\\in$ U($2^{n}$) and $\\mu_{H}$ being the Haar measure on U($2^{n}$) . \n\n\nThe statement of the detecability lemma is the following :\n\\begin{lem} \\cite{2} \\\\ $\\mid\\mid P_{even}.P_{odd} - P_{0} \\mid\\mid_{\\infty}$  $\\leq$  $(1+\\dfrac{\\Delta H_{GLRC}}{2} )^{-\\dfrac{1}{3}}$\n\\end{lem}\n\nTo relate this to the ensemble generated by the gadget $ G_{n}$ we prove the following Lemma:\n\\begin{lem}\n $ M_{t} [\\mu_{G_{n} } ]$=$P_{even}.P_{odd}$\n\\end{lem}\n\\textbf{Proof of Lemma 3}  :\\\\ \\\\ We first note that because all unitaries are drawn independently, we can think of the moment super operator as being composed of 2 layers (an odd layer (left part of the gadget of  Fig.\\ref{fig5}) and an even layer (right part of the gadget of  Fig.\\ref{fig5} ), this is similar to reasoning found in \\cite{3}). Then:\\\\ \t\n\n$M_{t} [\\mu_{G_{n} } ]$=($\\dfrac{1}{|\\mathcal{U}_B|})^{\\delta_{even}}\\sum_{U_{23} \\in \\mathcal{U}_B,U_{45} \\in \\mathcal{U}_B,...} (U_{23}\\otimes U_{45} \\otimes \u2026)^{\\otimes t,t}$. ($\\dfrac{1}{|\\mathcal{U}_B|})^{\\delta_{odd}}\\sum_{U_{12} \\in \\mathcal{U}_B,U_{34} \\in \\mathcal{U}_B,...} (U_{12} \\otimes  U_{34} \\otimes \u2026)^{\\otimes t,t}$, \\\\  \nwhere  $\\delta_{odd}$=\\{ $\\dfrac{n}{2}$  if  n$mod$2=0 or $\\dfrac{n-1}{2}$  if n$ mod$2 =1 \\} \\\\  and  $\\delta_{even}$=\\{ $\\dfrac{n}{2}-1$  if n$mod$2=0 or $\\dfrac{n-1}{2}$  if n$mod$2=1 \\}.\\\\\n\n\nNote that since the $U_{ii+1}$ ' s are independently drawn from $\\mathcal{U}_B$, one can rewrite this as:\\\\ \n\n$M_{t} [\\mu_{G_{n} } ]$= ($\\dfrac{1}{|\\mathcal{U}_B|}  \\sum_{U_{23} \\in \\mathcal{U}_B}(1 \\otimes U_{23} \\otimes 1^{\\otimes n-3})^{\\otimes t,t}$ .\\\\$\\dfrac{1}{|\\mathcal{U}_B|}(\\sum_{U_{45} \\in \\mathcal{U}_B}(1^{\\otimes 3}\\otimes U_{45}\\otimes 1^{\\otimes n-5})^{\\otimes t,t}$...) .\\\\($\\dfrac{1}{|\\mathcal{U}_B|} (\\sum_{U_{12} \\in \\mathcal{U}_B}(U_{12} \\otimes 1^{\\otimes n-2})^{\\otimes t,t}$.\\\\$\\dfrac{1}{|\\mathcal{U}_B|}\\sum_{U_{34} \\in \\mathcal{U}_B}(1^{\\otimes 2} \\otimes U_{34} \\otimes 1^{\\otimes n-4})^{\\otimes t,t}$...)\\\\=$(P^{'}_{2,3}.P^{'}_{4,5}...)$.$(P^{'}_{1,2}.P^{'}_{3,4}...)$\\\\=$P_{even}$. $P_{odd}$. $\\Box$ \\\\ \\\\  \n\nThen, as a direct consequence of the detectibility lemma we obtain:\n\\begin{equation}\n\\begin{aligned}\n\\label{dl}\ng(t,\\mu_{G_{n} }) = \\mid\\mid M_{t} [\\mu_{G_{n} } ] - P_{0} \\mid\\mid_{\\infty}\\leq(1+\\dfrac{\\Delta H_{GLRC}}{2} )^{-\\dfrac{1}{3}}\n\\end{aligned}\n\\end{equation}\n \nAll that remains now is to bound the RHS of Equation (\\ref{dl}). Using Equation (\\ref{eq7}) one directly obtains:\n                                                                                       \n\\begin{equation}\n\\begin{aligned}\n\\label{eq8}\n(1+\\dfrac{\\Delta H_{GLRC}}{2})^{-\\dfrac{1}{3}}\\leq(1+\\dfrac{P_{GLRC}}{2})^{-\\dfrac{1}{3}} \n\\end{aligned}\n\\end{equation}\nEquation (\\ref{dl}) along with Equation (\\ref{eq8}) directly leads to the following Corollary:\n\\newtheorem{corollary}{Corollary}\n\\begin{corollary}\nThe ensemble $\\{\\dfrac{1}{|\\mathcal{U_{G}}|} , \\mathcal{U_{G}}\\}$ generated by the gadget $G_{n}$ is an ($\\eta$,t)-TPE with :\\\\\n $\\eta$=$(1+\\dfrac{P_{GLRC}}{2})^{-\\dfrac{1}{3}}$= poly(t) < 1.\n\\end{corollary}\nPlugging Corollary 1 into Theorem 1 with: \\\\ \\{ $p_{i}$, $U_{i}$ \\}=$\\{\\dfrac{1}{|\\mathcal{U_{G}}|} , \\mathcal{U_{G}}\\}$ ,  $d$=$2^{n}$ and $\\eta$=$(1+\\dfrac{P_{GLRC}}{2})^{-\\dfrac{1}{3}}$ , then multiplying and dividing the bound of  $k$ in Theorem 1 by $log(2)$  allows one to obtain Theorem 3.\n\n\\section{\\label{sec:level8}Conclusions and discussion}\nWe have found a simple n-qubit graph gadget which implements an $\\varepsilon$-approximate t-design under repeated concatenations with fixed measurement and no feedforward.\nThe number of concatenations $k(n,t,\\varepsilon)$=$\\Omega(nt+ log(\\dfrac{1}{\\varepsilon}))$ required is linear in both the qubit number and order t of the design. Also, because the number of qubits in the  graph gadget scales linearly with  n, we thus   only require $\\Omega(n^{2}t + nlog(\\dfrac{1}{\\varepsilon}))$ qubits in total to implement the gadget $E_{n}$=$G_{n} \\circ G_{n} \\circ...$. Furthermore, the choice of the 2-qubit gadget B is not at all unique. In fact, $G_{n}$ could be made even more practical provided simpler (less number of qubits, less needed entanglements,...) 2-qubit  gadgets possessing the properties of B can be found. \n\n\nOur construction is very similar to the brickwork state, which is a universal resource for MBQC \\cite{19} - it is basically the brickwork state but with regular holes.\nIn MBQC these holes would simply teleport the inputs through, so that the proofs of universality of \\cite{19} easily extend to our graph - that is, concatenations of the graph used in  $G_n$ is also a universal resource for MBQC.\nIn addition to being pleasing from a practical point of view, this opens the door to applications of techniques for delegation of ensemble generation, as done for computation \\cite{19,barz2013experimental}, and indeed the possibility to hide whether one is sampling unitaries or performing some deterministic computation.\\\\ \\\\ \\\\\n\n\n\n\\section{\\label{sec:level8}Acknowledgements}\n\nWe thank Y. Nakata for fruitful discussions and  P. Turner for comments. R. Mezher Acknowledges funding from the Lebanese PhD grant $CNRS-L$\/$UL$.\nDM acknowledges support from ANR grant COMB.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSemi-analytic models of galaxy formation (SAMs) are well established tools for\nexploring galaxy formation scenarios in their cosmological context. The problem of\nhow galaxies form and evolve is described by a set of coupled differential equations\ndealing with well-defined astrophysical processes. These are driven by dark\nmatter halo merger trees that determine the source terms in the equation network\n\\citep[for reviews see e.g.][]{BensonReview,Baugh2006,SomervilleDave2015}.\nDue to the approximate nature of the methods used in these simulations, and the\nuncertainties in the physical process that are modelled, these models include a large\nnumber of uncertain parameters. While order of magnitude estimates for these\nparameters can be made, their precise values must be determined by comparison\nto observational data.\n\nTraditionally, parameter values have been set through a trial-and-error approach,\nwhere the galaxy formation modeller varies an individual parameter developing\nintuition about its effects on the model predictions for a particular observable and\nthen uses this understanding to select a parameter set that gives a good description\nof the observations.\nDespite its simplicity, and obvious limitations, this procedure has led to\nsubstantial progress in the field.\nRecently, however, several papers have employed more rigorous statistical methods to\nexplore the\nhigh dimensional parameter space systematically\n\\citep{Kampakoglou2008,Henriques2009,Bower2010,Henriques2013,Benson2014,Lu2014,\nHenriques2015}. Such approaches\nprovide a richer analysis, and seek to identify the regions of\nparameter space that are in agreement with observational data, and not just to find\noptimal parameter values. This therefore informs as to the uniqueness of the\nparameter choices, and provides understanding of the degeneracies between different\nparameters.\n\nIn this work we study which constraints are imposed on the semi-analytic model\n\\textsc{galform}\\xspace by the observations of galaxy stellar mass function (GSMF).  We first consider\nthe constraints imposed by local observations and then investigate how the\nparameters are further constrained by the introduction of high\nredshift data. This makes powerful\nuse of  the iterative emulator technique described by \\citet{Bower2010}, which\nprovides an efficient way of probing a high dimensional parameter space.\nImportantly, the method allows additional constraints to be added in post-processing.\nThus, we start by finding the region in the parameter space which contains models\nthat produce a good match to the local Universe GSMF.\nThis region is, then, further probed to check whether a match to higher\nredshift data is possible.\nBy analysing 2D projections of the plausible models sub-volume and performing a\nprincipal component analysis of it, we are able to study the degeneracies and\ninteractions between the most constrained parameters. We note that typical\napproaches to analysing comparable models using Bayesian MCMC require millions of\nmodel runs (at least), while the approach used here, which utilises Bayesian\nemulation, only required tens of thousands of model runs, representing a substantial\nimprovement in efficiency.\n\n\n\n\n\n\nClosely reproducing the observed high-redshift galaxy mass function\n\\citep{Cirasuolo2010, Henriques2013} is problematic for many galaxy formation models.\n\\citet{Henriques2013}, for example, concludes that the effectiveness of galaxy\nfeedback (specifically the re-incorporation time of expelled gas) must depend on\n\\green{the virial mass of the dark matter halo\non the basis of a Monte Carlo exploration of the parameter space of their model.\nThis is not fully satisfactory, however, since one would expect the re-incorporation\ntime to be physically related to the halo dynamical time and not the halo mass.\n}\n\nIn this paper, we explore an appealing and well-motivated alternative. Observations\nof galaxy winds \\citep[eg.,][]{heckman1990, martin2012} suggest that the effective\nmass loading is strongly dependent on the surface density of star formation. It\nappears that efficient outflows are more readily generated when star formation\noccurs in dense bursts than when the star formation occurs in a smooth\nand quiescent disk.  These observations motivate a more careful exploration of the\ntreatment of galaxy winds from starburst and quiescent disks, and in this paper we\nparametrize the mass loading of the wind independently in these two cases. This may\nnaturally resolve the difficulty presented by observations of the high redshift GSMF\nsince the cosmic star formation rate density may be more dominated by starbursts at\nhigh redshift, while it is dominated by quiescent star formation at low redshift\n\\citep{Malbon2007}.\n\nThis paper is organized as follows.\nIn \\S\\ref{sec:model} we describe the galaxy formation model, specifying\nwhich parameters were varied and briefly reviewing the physical meaning of the\nmost relevant of them.\nIn \\S\\ref{sec:emulator} the iterative history matching methodology is\nreviewed.\nIn \\S\\ref{sec:z0} we present our results for the matching to the local\nGSMF.\nIn \\S\\ref{sec:high_z} we examine the effects of including higher redshift\ndata.\nIn \\S\\ref{sec:subspace}, 2D projections of the parameter space are\nanalysed.\nIn \\S\\ref{sec:pca}, the results of a principal component analysis of the\nnon-implausible volume are shown.\nFinally, in \\S\\ref{sec:summary}, we summarize our conclusions.\n\n\n\\section{Galaxy formation model}\n\\label{sec:model}\n\nThe basis of this paper is the semi-analytic model \\textsc{galform}\\xspace, first introduced by\n\\citet{Cole2000}. Our starting point is the model discussed by \\citet[][hereafter\nGP14]{Gonzalez-Perez2014}, which re-calibrates the version described in\n\\citet{Lagos2012}\n to match observational data taking into account the\nbest-fitting cosmological parameters obtained by WMAP7 \\citep{WMAP7}.\nThe model of \\citet{Lagos2012} is itself a development of the version presented\nby \\citet{Bower2006} -- which introduced AGN feedback and disc instabilities to\nthe original \\textsc{galform}\\xspace model -- introducing a modified prescription for star formation\nin galaxy discs (\\S\\ref{sec:BR_starformation}, see \\citealt{Lagos2011sfl} for an\nin-depth discussion).\n\nWe note that there is now a more modern variant of the \\textsc{galform}\\xspace model which differs from\nthe base model used here. This model, described {comprehensively}\nby \\citet{Lacey2016}, assumes two initial mass functions (IMFs), one for quiescent\nstar formation and a different one for star bursts\n{-- an approach which improves the model predictions for number counts and\nredshift distribution of sub-millimetre galaxies.\nThe model presented here assumes a universal IMF, which considerably simplifies\ncomparison with the galaxy stellar mass function. A universal IMF is compatible with\ndirect observational measurements: see \\citet{Bastian2010} and \\citet{Smith2015} for\na recent discussion.}\n\n\n\\subsection{Differences from GP14}\n\nAlthough the model we use here is based on GP14, there are a number of\nsmall, but important, differences.\nFirstly, the merger trees in the present study were constructed using the Monte Carlo\nalgorithm described by \\citet{Parkinson2008}, which is based on the\nExtended Press-Schechter theory \\citep{Bower1991,LaceyCole1993}, while\nGP14 uses merger trees extracted from a Millenium-class N-body\nsimulation \\citep{Guo2013}.\n\\green{-- t}he use of Monte Carlo merger trees allows \\textsc{galform}\\xspace to run significantly faster since\nit is possible to control the number of haloes with a given final mass in the\nsimulation, whereas in the case of the N-body trees, most of the computational time\nis spent on over-represented small mass haloes.\nIn GP14, ram-pressure stripping is modelled by\ncompletely and instantaneously removing the hot gas halo when a galaxy becomes a\nsatellite. Here we follow the same prescription as \\citet{Font2008}, which uses the\n\\citet{McCarthy2008} ram-pressure stripping model that is based on hydrodynamic\nsimulations \\green{-- a} similar update to the model is used in \\citet{Lagos2014}.\n\\green{\nFinally, the present model adopts the IMF obtained by \\citet{ChabrierIMF}, while GP14\nuses a \\citet{KennicuttIMF} IMF.\n}\n\\subsection{Varied parameters}\n\\label{sec:parameters}\n\n\\begin{table*}\n\\begin{center}\n\\caption{\nParameters varied in this work, the physical processes  and their ranges.\nFor reference, values of these parameters used in GP14 are shown.\n}\n\\input{parameters_table.tex}\n\\label{tab:parameters}\n\\end{center}\n\\end{table*}\n\nThe semi-analytic approach to the problem of galaxy formation relies on a large\nnumber of parameters which codify the\nuncertainties associated with the many astrophysical processes involved.\nSince the emulator technique allows us to survey a parameter space of high\ndimensionality both quickly and at a relatively low computational cost, we are\nable to vary parameters simultaneously. One should bear in mind\nthat varying a larger number of parameters in the present approach corresponds\nto a more \\emph{conservative} choice, since it requires less \\textit{a priori}\nassumptions about the role of each parameter.\n\nWe varied 20 parameters, all of which are listed, together with their ranges,\nin table \\ref{tab:parameters}. We outline the physical meaning of\nparameters related to star formation and feedback in the subsections\nbelow, for further details, we refer the reader to the original\npapers, and to \\citet{Lacey2016}.\n\nFor the purposes of sampling, computations of volumes and principal\ncomponents analysis, the parameters were rescaled to $[-1,1]    $ within the\ninitial range, either linearly,\n\\begin{equation}\n\\label{eq:scale_lin}\n p^{(s)} = 2\\left(\\frac{p-p_\\text{min}}{p_\\text{max}-p_\\text{min}}\\right)-1\\,,\n\\end{equation}\nor logarithmically,\n\\begin{equation}\n\\label{eq:scale_log}\n p^{(l)} =\n2\\left[\\frac{\\log_{10}(p\/p_\\text{min})}{\\log_{10}(p_\\text{max}\/p_\\text{min})}\n\\right]-1\\,.\n\\end{equation}\nThe scaling used is also listed in table \\ref{tab:parameters}.\n\n\\subsubsection{Quiescent star formation}\n\nIt is assumed in the model that the surface density of the star formation rate is set by the surface density of molecular gas (see \\citealt{Lagos2011sfl} and references\ntherein),\n\\label{sec:BR_starformation}\n\\begin{equation}\n\\dot \\Sigma_\\star = \\nu_{0,\\text{sf}} \\,\\Sigma_\\text{mol} =\\nu_{0,\\text{sf}}\nf_\\text{mol} \\Sigma_\\text{gas}\\label{eq:BR_starformation}\\,,\n\\end{equation}\nwhere \\green{$\\Sigma_\\text{gas}$ is the surface density of cold gas in the disc and}\nthe fraction of molecular hydrogen,\n$f_\\text{mol}=R_\\text{mol}\/(R_\\text{mol}+1)$, is computed using the pressure\nrelation of \\citet{BlitzRosolowsky2006}\n\\begin{equation}\n R_\\text{mol} = \\left(\\frac{P_\\text{ext}}{P_\\text{sf}}\\right)^{\\beta_\\text{sf}}\n \\label{eq:BR_Rmol}\\,,\n\\end{equation}\nwith\n\\begin{equation}\n P_\\text{ext}=\\frac{\\upi}{2} G \\Sigma_\\text{gas}\\left[ \\Sigma_\\text{gas} +\\left(\n\\frac{\\sigma_\\text{gas}}{\\sigma_\\star}\\right)\\Sigma_\\star \\right]\\,.\n\\end{equation}\n\n\n\\subsubsection{Star formation bursts}\n\nDuring a starburst the star formation rate is set to\n\\label{sec:burst_starformation}\n\\begin{equation}\n \\text{SFR}_\\text{burst} = \\frac{M_\\text{gas,bulge}}{\\tau_{\\star, \\text{burst}}}\n\\end{equation}\nwith\n\\begin{equation}\n \\tau_{\\star, \\text{burst}} = \\max\\left( f_\\text{dyn}\\tau_\\text{dyn},\\,\\,\n            \\tau_{\\text{min,burst}}  \\right)\\label{eq:burst}\n\\end{equation}\nwhere $\\tau_\\text{dyn}$ is the dynamical time of the newly formed spheroid and\n$f_\\text{dyn}$ and $\\tau_{\\text{min,burst}}$ are model parameters.\n\n\n\n\\subsubsection{Supernovae feedback}\n\\label{sec:SN_feedback}\nThe outflow of gas from the disc or the bulge of a galaxy is modelled\nusing\n\\begin{equation}\\label{eq:alphahot}\n\\dot M_\\text{out,disc\/burst} = \\beta \\times \\text{SFR}_\\text{disc\/burst}\n\\end{equation}\nwhere $\\text{SFR}_\\text{disc\/burst}$ are the total star formation rates in the\nquiescent and starburst cases and $\\beta$ is the \\emph{mass loading}, given by\n\\begin{equation}\n\\beta = \\beta_{0,\\text{disc\/burst}}\n\\left(\\frac{V_\\text{disc\/bulge}}{200\\,\\text{km}\\,\\text{s}^{-1}}\n\\right)^{-\\alpha_\\text{hot}}\n\\label{eq:beta}\n\\end{equation}\nwhere $V_\\text{disc\/bulge}$ are the circular velocity associated with\nwith the disc (in the quiescent case) or with the newly formed spheroid\n(bulge) component (in a starburst).\n\nIn previous \\textsc{galform}\\xspace works the mass loadings associated with discs and bursts were\nassumed to share the same normalization, i.e. $\\beta_{0,\\text{burst}}=\\beta_{0,\\text{disc}} = \\beta_0$.\nThis assumption was relaxed in the present work. The notation in previous works was\nalso slightly different: the equivalent parameter\n\\begin{equation}\nV_\\text{hot}~\\equiv~(200\\,\\text{km}\\,\\text{s}^{-1})~\\times~\\beta_0^{\n-1\/\\alpha_\\text{hot}}\n\\end{equation}\nwas used instead.\n\nThe outflowed gas is assumed to be once more available to cool and form stars\non a time-scale\n\\begin{equation}\n t_\\text{reinc} = \\frac{\\tau_\\text{halo}}{\\alpha_\\text{reheat}},\n\\label{eq:reheat}\n\\end{equation}\nwhere $\\tau_\\text{halo}$ is the dynamical time of the halo.\n\\green{The amount of cold gas available (as well as the amount of stars formed) is\ndetermined by simultaneously solving for both the star formation rate and the\noutflow rate.}\n\n\n\\subsubsection{AGN feedback}\n\\label{sec:AGN_feedback}\n\nThe model assumes the cooling of gas from the hot gas halo can be disrupted by\nthe injection of energy by the AGN. This is assumed to happen only at haloes\nunder `quasi-hydrostatic equilibrium', defined by\n\\begin{equation}\n t_\\text{cool}(r_\\text{cool}) > \\alpha_\\text{cool}^{-1} t_\\text{ff}(r_\\text{cool})\n\\label{eq:tcool}\n\\end{equation}\nwhere $t_\\text{cool}$ and $r_\\text{cool}$ are the cooling time and radius and\n$t_\\text{ff}$ is the free fall time. Thus, the parameter $\\alpha_\\text{cool}$ determines\nthe halo mass at which AGN feedback is effective (i.e. lower values of\n$\\alpha_\\text{cool}$ implies AGN feedback active in smaller mass haloes).\n\nThe cooling of gas from the hot gas halo is interrupted if a galaxy satisfies\nequation \\eqref{eq:tcool}, and\n\\begin{equation}\n L_\\text{cool} < \\epsilon_\\text{edd}L_\\text{edd}\n\\end{equation}\nwhere $L_\\text{edd}$ is the Eddington luminosity of the central galaxy's black hole.\n\n\\subsubsection{Disc stability}\n\\label{sec:stability}\n\nDiscs are considered stable if they satisfy\n\\begin{equation}\n\\frac{V_\\text{max}}{\\sqrt{{1.68} \\,G M_\\text{disc}\/r_\\text{disc}}}<  f_{\\rm stab}\n\\end{equation}\nwhere $f_{\\rm stab}$ is a model parameter close to 1. If at any timestep this\ncriterion is not satisfied, it is assumed that the disc is quickly converted\ninto an spheroid due to a disc instability and a starburst is triggered\n\\green{-- i.e. all the gas and stars are instantaneously moved into the spheroid\ncomponent where the star formation follows equation~\\eqref{eq:burst}.}\n\n\n\\section{Bayesian Emulation Methodology}\n\\label{sec:emulator}\n\nThe use of complex simulation models, such as \\textsc{galform}\\xspace, is now widespread across many\nscientific areas.\nSlow simulators with high dimensional input and\/or output spaces give rise to several\nmajor problems, the most ubiquitous being that of\nmatching the model to observed data, and the subsequent global parameter\nsearch that such a match entails.\n\nThe general area of Uncertainty Analysis has been developed within the Bayesian\nstatistical community to solve the corresponding problems associated with slow\nsimulators~\\citep{Craig97_Pressure,Kennedy01_Calibration}. A core part of this area\nis the use of emulators: an emulator is a\nstochastic function that mimics the \\textsc{galform}\\xspace model but which is many orders of\nmagnitude faster to evaluate, with specified prediction uncertainty that varies\nacross the input space \\citep{OHagan06_Tutorial,Vernon2010,galf_stat_sci}.\nAny subsequent calculation one wishes to do with \\textsc{galform}\\xspace can\ninstead be performed far more efficiently using an emulator \\citep{Higdon09_Coyote2}.\nFor example, an emulator can be used within an MCMC algorithm to greatly speed up\nconvergence \\citep{Kennedy01_Calibration,Higdon04_prediction, Henderson:2009aa}.\nThis is especially useful as for scenarios possessing moderate to high numbers of\ninput parameters,\nMCMC algorithms often require vast numbers (billions, trillions or more) of model\nevaluations to adequately explore the input space and\nreach convergence: see for example the excellent discussion in\n\\cite{geyer2011introduction}. Such numbers of evaluations are clearly\nimpractical for models that possess substantial run time, such as \\textsc{galform}\\xspace. Another\nmajor issue with MCMC is that of pseudo convergence:\nan MCMC algorithm may after a large number of iterations {\\it appear} to have\nconverged and hence pass every convergence test, but continued running would\neventually reveal a sudden and substantial  change in chain location, showing that\nthe chain had not in fact reached equilibrium at all \\citep{geyer2011introduction}.\n\nHence, although we fully support the Bayesian paradigm, we do not use an MCMC\nalgorithm here, due both to the reasons discussed above, and to the fact that a\nBayesian MCMC approach requires a full joint probabilistic specification across all\nuncertain\nquantities, that is often hard to make and hard to justify.\nWe instead outline a more efficient and robust approach known as iterative history\nmatching using Bayesian emulation~\\citep{Vernon2010}. Here the set of all inputs\ncorresponding to acceptable matches to the\nobserved data is found, by iteratively removing unacceptable regions of the input\nspace in waves. History matching naturally incorporates Bayesian emulation and has\nbeen\nsuccessfully employed in a range of scientific disciplines\nincluding galaxy formation~\\citep{Vernon2010,Vernon10_CS_rej,Bower2010,\ngalf_stat_sci}, epidemiology~\\citep{Yiannis_HIV_1,Yiannis_HIV_2}, oil reservoir\nmodelling~\\citep{Craig96_Pressure,Craig97_Pressure,JAC_Handbook,JAC_sma_samp},\nclimate modelling~\\citep{Williamson:2013aa}\nand environmental science~\\citep{asses_mod}. History matching can be viewed as a\nuseful precursor to a fully Bayesian analysis\nthat is often in itself sufficient for model checking and model development.\nHere we use it within a Bayes Linear framework, a simpler, more tractable\nversion of Bayesian statistics, where only expectations, variances and covariances\nneed to be\nspecified \\citep{Goldstein_99,Goldstein07_BayesLinearBook}. However, if one is\ncommitted to  a full Bayesian MCMC approach, performing an a priori history match\ncan dramatically improve the subsequent efficiency of the MCMC by first removing the\nvast regions of input parameter space that would have extremely low\nposterior probability.\n\n\n\\subsection{Emulator Construction}\n\\label{sec:emconstr}\n\nWe now outline the core emulator methodology \\citep[see][for\nfurther description]{Vernon2010,Bower2010}.\nWe represent the \\textsc{galform}\\xspace model as a function $f(x)$, where\n$x=~\\!(\\nu_{0,\\text{sf}},P_\\text{sf}\/k_\\text{B},\\dots,\\epsilon_\\text{strip},\\alpha_\\text{rp})$\nis a\nvector\ncomposed of the 20 input parameters given in table~\\ref{tab:parameters}, and $f$ is a\nvector containing all \\textsc{galform}\\xspace outputs of interest, specifically the GSMF at various\nmass bins and redshifts. To construct an emulator we generally perform an initial\nspace filling set of wave 1 runs, using a maximin latin hypercube design over the\nfull 20 dimensional input space\n\\citep[see][for details]{Bower2010,SWMW89_DACE,Santner03_DACE,Currin91_BayesDACE}.\nFor each output $f_i(x)$, $i=1\\dots q$, a Bayesian emulator can be structured as\nfollows:\n\\begin{equation}\n\\label{eq_emulator}\nf_i(x) = \\sum_j \\beta_{ij}  g_{ij}(x_{A_i}) + u_i(x_{A_i}) + v_i(x)\n\\end{equation}\nHere $\\beta_{ij}$, $u_i(x_{A_i})$ and $v_i(x)$ are uncertain quantities to be\ninformed by the current set of runs.\nThe active variables $x_{A_i}$ are a subset of the inputs that are found to be most\ninfluential for output $f_i(x)$. The $g_{ij}$ are known deterministic\nfunctions of $x_{A_i}$, with a common choice being low order polynomials, and the\n$\\beta_{ij}$ are unknown regression coefficients. $u_i(x_{A_i})$ is a Gaussian\nprocess with, for example, zero mean and possible covariance function:\n\\begin{equation}\n\\label{eq_corr}\n{\\rm Cor}(u_i(x_{A_i}),u_i(x'_{A_i})) = \\sigma^2_{u_i} {\\rm exp}\\left\\{-\n\\|x_{A_i}-x_{A_i}'\\|^2 \/ \\theta_i^2 \\right\\}\n\\end{equation}\nwhere $\\sigma^2_{u_i}$ and $\\theta_i$ are the variance and correlation length of\n$u_i(x_{A_i})$ which must be specified,\nand $v_i(x)$ is an uncorrelated nugget with expectation zero and ${\\rm Var} (v_i(x))\n=  \\sigma^2_{v_i}$, that represents the effect of the remaining inactive input\nvariables, and\/or any stochasticity exhibited by the model \\citep{Vernon2010}.\n\nWe could employ a fully Bayesian approach by specifying joint prior distributions for\nall uncertain quantities in equation~(\\ref{eq_emulator}), and subsequently updating\nbeliefs about $f_i(x)$ in light of the wave~1 runs via Bayes theorem.\nHere instead we prefer to use the more tractable Bayes Linear approach, a version of\nBayesian statistics that requires only expectations, variances and covariances for\nthe prior specification, and which uses only efficient matrix calculations, and no\nMCMC\n\\citep{Goldstein07_BayesLinearBook}.\nTherefore if we are prepared to specify ${\\rm E}(\\beta_{ij})$, ${\\rm\nVar}(\\beta_{ij})$, $\\sigma^2_{u_i}$, $\\sigma^2_{v_i}$ and $\\theta_i$, we can obtain\nthe\ncorresponding Bayes Linear priors for $f_i(x)$ namely ${\\rm E}(f_i(x)), {\\rm\nVar}(f_i(x))$ and\n${\\rm Cov}(f_i(x),f_i(x'))$ using equations~(\\ref{eq_emulator}) and (\\ref{eq_corr}).\n\nThe initial wave of $n$ runs is performed at input locations $x^{(1)},\nx^{(2)},\\dots,x^{(n)}$ which give model output values $D_i = (f_i(x^{(1)}),\nf_i(x^{(2)}),\\dots,f_i(x^{(n)}))$, where $i$ labels the model output. We obtain the\nBayes Linear adjusted expectation ${\\rm E}_{D_i}(f_i(x))$ and variance ${\\rm\nVar}_{D_i}(f_i(x))$ for $f_i(x)$ at new input point $x$ using:\n\\begin{align}\n {\\rm E}_{D_i}(f_i(x)) =&\\; \\label{eq_BLE}\n{\\rm E}(f_i(x)) \\nonumber\\\\\n&+\\, {\\rm Cov}( f_i(x), D_i) {\\rm Var}(D_i)^{-1} (D_i - {\\rm E}(D_i))\n\\\\\n{\\rm Var}_{D_i}(f_i(x)) =&\\;  \\label{eq_BLV}\n{\\rm Var}(f_i(x)) \\nonumber\\\\\n            &-\\, {\\rm Cov}( f_i(x), D_i) {\\rm Var}(D_i)^{-1} {\\rm Cov}(D_i,f_i(x))\n\\end{align}\nThe emulator thus provides a prediction ${\\rm E}_{D_i}(f_i(x))$ for the behaviour of\nthe \\textsc{galform}\\xspace model at new input point $x$ along with a corresponding $x$ dependent\nuncertainty ${\\rm Var}_{D_i}(f_i(x))$. It is the later feature that strongly\ncontributes to emulators being more advanced than interpolators. These two quantities\n${\\rm E}_{D_i}(f_i(x))$ and ${\\rm Var}_{D_i}(f_i(x))$ are used directly in the\nimplausibility measures that form the basis of the global parameter search described\nbelow.\n\n\n\\subsection{Simple 1-dimensional Example}\n\\label{sec:1demul}\n\n\nTo clarify the above description we outline the construction of a simple\n1-dimensional emulator of the function\n\\begin{equation}\nf(x) \\;\\;=\\;\\; 3 \\,x \\sin\\left(\\frac{5\\upi(x-0.1)}{ 0.4}\\right)\n\\end{equation}\nfor which we perform a set of $n=10$ equally spaced wave 1 runs at locations\n$x^{(j)}=0.1,\\dots,0.5$ giving rise to run data\n\\begin{equation}\\label{eq_Di_sim}\nD = (f(x^{(1)}), f(x^{(2)}),\\dots,f(x^{(n)}))\n\\end{equation}\nwhere we have dropped the $i$ subscript as the output is only 1-dimensional.\n\nFor simplicity we reduce the emulator's regression terms $\\beta_{ij} g_{ij}(x_A)$, in\nequation~(\\ref{eq_emulator}), to a constant $\\beta_0$ and remove the\nnugget $v_i(x)$ as there are no inactive inputs. The emulator\nequation~(\\ref{eq_emulator}) therefore reduces to:\n\\begin{equation}\\label{eq_sim_em}\nf(x) \\;\\;=\\;\\; \\beta_0 + u(x)\n\\end{equation}\nA possible prior specification is to treat the constant or mean term $\\beta_0$ as\nknown, with ${\\rm E}(\\beta_0)=0.1$ and hence ${\\rm Var}(\\beta_0)=0$.\nWe also set $\\sigma_{u}=0.6$ and $\\theta = 0.06$: a choice that represents curves of\nmoderate\nsmoothness.\nWe can now calculate all terms on the rhs of equations~(\\ref{eq_BLE}) and\n(\\ref{eq_BLV}) using equations~(\\ref{eq_sim_em}), (\\ref{eq_corr}) and\n(\\ref{eq_Di_sim}), for example:\n\\begin{eqnarray}\n{\\rm E}(f(x)) &=& \\beta_0 \\\\\n{\\rm Var}(f(x)) &=& \\sigma_{u}^2 \\\\\n{\\rm E}(D) &=& (\\beta_0, \\dots, \\beta_0)^T\n\\end{eqnarray}\nwhile ${\\rm Cov}( f(x), D)$ is now a row vector of length $n$ with $j$th component\n\\begin{eqnarray}\n{\\rm Cov}( f(x), D)_j &=& {\\rm Cov}( u(x), u(x^{(j)})) \\\\\n&=& \\sigma^2_{u} {\\rm exp}\\left\\{- \\|x-x^{(j)}\\|^2 \/  \\theta^2 \\right\\} \\nonumber\n\\end{eqnarray}\nand ${\\rm Var}(D)$ is an $n\\times n$ matrix with $(j,k)$ element\n\\begin{eqnarray}\n{\\rm Var}(D)_{jk} &=& {\\rm Cov}( u(x^{(j)} ), u(x^{(k)})) \\\\\n&=& \\sigma^2_{u} {\\rm exp}\\left\\{- \\|x^{(j)}-x^{(k)}\\|^2 \/  \\theta^2 \\right\\}\n\\nonumber\n\\end{eqnarray}\nWe can now construct the emulator by calculating the adjusted expectation and\nvariance ${\\rm E}_{D}(f(x))$ and ${\\rm Var}_{D}(f(x))$ from equations~(\\ref{eq_BLE})\nand (\\ref{eq_BLV}) respectively, for any new input point $x$.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{.\/simple_emul4_a.pdf}\n \\caption{\nThe 1-dimensional emulator as constructed in \\S\\ref{sec:1demul}. The\n\\green{dashed (blue)}\nline is the emulator prediction ${\\rm E}_{D}(f(x))$ as a function of $x$, and\nthe credible\ninterval ${\\rm E}_{D}(f(x)) \\pm 3 \\sqrt{{\\rm Var}_{D}(f(x))}$ is given by the\n\\green{dotted (red)}\nlines. The true function $f(x)$ is shown as the black solid line, and the 10 model\nruns that make up the vector $D$ used to build the emulator are given as the red\npoints.\n         }\n \\label{fig:1demul}\n\\end{figure}\n\nFig.~\\ref{fig:1demul} shows the 1-dimensional emulator where ${\\rm E}_{D}(f(x))$ as\na function of $x$ is given by the \\green{dashed} blue line, and the credible interval\n${\\rm E}_{D}(f(x)) \\pm 3 \\sqrt{{\\rm Var}_{D}(f(x))}$ by the \\green{dotted} red lines. We can see\nthat ${\\rm E}_{D}(f(x))$ precisely interpolates the known runs at outputs $D$, with\nzero uncertainty (as the red lines touch at these points): a desirable feature as\nhere $f(x)$ is a deterministic function. The\ncredible regions get wider the further we are from known runs, appropriately\nreflecting our lack of knowledge in these regions.\nThe true function $f(x)$ is given by the solid black line which lies within the\ncredible region for all $x$, only getting close to the boundary for $x>0.5$.\nThis demonstrates the power of an emulator: using only a small number of runs we can\nsuccessfully mimic relatively complex functions to a known accuracy, a feature that\nscales well in higher dimensions due to the chosen form of the emulator. The speed of\nBayesian emulators is also crucial\nfor global parameter searches where we may need to evaluate the emulator a huge\nnumber of times to fully explore the input space.\nNote that the\nemulator calculation is extremely fast because it only requires matrix multiplication\nfor each new $x$. The inverse ${\\rm Var}(D_i)^{-1}$ that features in\nequations~(\\ref{eq_BLE}) and (\\ref{eq_BLV}) is independent of\n$x$ (and indeed of $D_i$) and hence can be performed only once, offline and in\nadvance of even the run evaluations $D_i$.\n\n\n\n\n\\subsection{Emulating in Higher Dimensions}\\label{sec:em_high_dim}\n\nWhen emulating functions possessing high input dimension, the polynomial regression\nterms\n$\\beta_{ij} g_{ij}(x_{A_i})$ in the emulator equation~(\\ref{eq_emulator}) become more\nimportant, as they efficiently capture many of the more global features often present\nin the physical model~\\citep{Vernon2010,Vernon10_CS_rej}. Prior specifications for\nthe $\\beta_{ij}$ can be given,\nbased say on structural knowledge of the model, or on past experience running a\nfaster but simpler previous version of the model \\citep{JAC_Handbook}. However, if no\nstrong prior\nknowledge is available and the number of runs performed is reasonably high, a vague\nprior limit can be taken in the Bayes Linear update equations~(\\ref{eq_BLE}) and\n(\\ref{eq_BLV}), resulting in the adjusted expectation and variance of the\n$\\beta_{ij}$ terms tending toward their Generalised Least Squares (GLS) estimates.\nFor space filling runs, such as those from a maximin latin hypercube, the GLS\nestimates can be accurately approximated by the\ncorresponding Ordinary Least Squares (OLS) estimates, which can also be used to\nestimate\n$\\sigma_{u_i}^2$, providing further efficiency gains \\citep{Vernon2010}.\n\n\nIn addition, the choice of active input variables $x_{A_i}$ and the choice of the\nspecific regression terms $\\beta_{ij} g_{ij}(x_{A_i})$ that feature in the emulator,\ncan both be made using linear model selection techniques based on AIC or BIC\ncriteria. For example, these can be simply employed using the lm() and step()\nfunctions in R \\citep{Vernon2010,R-Core-Team:2015aa}. The use of active variables\n$x_{A_i}$ can lead to substantial\ndimensional reduction of the input space of each of the outputs, and hence convert a\nhigh dimensional\nproblem into a collection of low dimensional problems, which is often far easier to\nanalyse \\citep[see][for further discussion of this benefit]{Vernon10_CS_rej}. It is\nworth noting that reasonably accurate emulators can often be constructed just using\nsuch regression models. This can be a sensible first step\n\\cite[see][]{Yiannis_HIV_3}, before one attempts the construction of a full emulator\nof the form given in equation~\\eqref{eq_emulator}.\n\n\\subsection{Iterative History Matching via Implausibility}\n\\label{sec:HM}\n\nWe now describe the powerful iterative global search method known as History\nMatching \\citep{Craig96_Pressure,Craig97_Pressure}, which\nnaturally incorporates the use of Bayesian emulators, and which has been\nsuccessfully applied across a variety of scientific disciplines.\nIt aims to identify the set  $\\mathcal{W}$ of all inputs $x$ that would give rise\nto an acceptable match between the \\textsc{galform}\\xspace outputs $f(x)$ and the corresponding vector\nof observed data $w$, and proceeds iteratively, discarding regions of input space\nwhich are deemed {\\it implausible} based on information from the emulators. For more\ndetail on the contents of this section see\n\\citep{Vernon2010,Vernon10_CS_rej}.\n\nFor an output $f_i(x)$ we define the implausibility measure:\n\\begin{equation}\\label{eq_imp1}\nI_i^2(x,w_i) \\;\\;=\\;\\; \\frac{({\\rm E}_{D_i}(f_i(x)) - w_i)^2}\n{ {\\rm Var}_{D_i}(f_i(x)) + \\sigma^2_{\\epsilon_i} + \\sigma^2_{e_i} }\n\\end{equation}\nwhich takes the distance between the emulator's prediction of the $i$th output\n${\\rm E}_{D_i}(f_i(x))$ and the actual observed data $w_i$ and standardises\nit with respect to the variances of the three major uncertainties: the emulator\nuncertainty ${\\rm Var}_{D_i}(f_i(x))$, the model discrepancy $\\sigma^2_{\\epsilon_i}$\nand the observation error $\\sigma^2_{e_i}$.\n\nThe least familiar of these is the model discrepancy $\\sigma^2_{\\epsilon_i}$ which\nis an upfront acknowledgement of the deficiencies of the \\textsc{galform}\\xspace model\nin terms of assumptions used, missing physics and simplifying approximations.\nIn addition to ensuring the analysis is more meaningful, this term guards against\noverfitting, and the subsequent technical and robustness problems this can cause for\na global parameter search. See\n\\cite{Kennedy01_Calibration,Brynjarsdottir:2014aa,Goldstein09_Reify} for extended\ndiscussions\non this point\\footnote{It is worthwhile noting that any analysis that does not\ninclude a model discrepancy is only meaningful given that ``the model $f(x)$ is a\n{\\it precise} match to the real Universe for some input $x$\", and all conclusions\nderived from such an analysis should be written with this conditioning statement\nattached.}.\nThe form of the implausibility comes from the ``best input approach\" which models\nthe link between the \\textsc{galform}\\xspace model evaluated at its best possible input $x^*$ and the\nreal Universe $y$ as $y=f(x^*) + \\epsilon$, where $\\epsilon$ is a random quantity\nrepresenting the model discrepancy with variance $\\sigma^2_{\\epsilon}$, and assumes\nthat the observed data $w$ is measured with uncertain error $e$ with variance\n$\\sigma^2_{e}$, such that $w=y+e$. See\n\\cite{Craig97_Pressure,Vernon2010,Vernon10_CS_rej} for further justifications and\ndiscussions.\n\nMost importantly, a large value of the implausibility $I_i(x,w_i)$ for any output\nimplies that the point $x$ is unlikely to yield an acceptable match between $f(x)$\nand $w$ were we to run the \\textsc{galform}\\xspace model there, hence $x$ is deemed {\\it implausible}\nand can be discarded from further analysis. We therefore impose cutoffs of the form\n$I_i(x,w_i) < c$ to rule out regions of input space, where the choice of $c$ is\nmotivated from Pukelsheim's 3-sigma rule\\footnote{Pukelsheim's 3-sigma rule is the\npowerful, general, but underused\nresult that states for any continuous unimodal distribution, 95\\% of the probability\nmust lie within $\\mu\\pm 3\\sigma$, regardless of its asymmetry or skew.}\n\\citep{threesigma}.\nWe can combine the implausibility measures from several outputs in various ways e.g.\n\\begin{equation}\\label{eq_maximp}\nI_M(x,w) \\;\\;=\\;\\; \\max_{i \\in Q} I_i(x,w_i)\n\\end{equation}\nwhere $Q$ represents the subset of outputs currently considered (often we will only\nemulate a small subset of outputs in early iterations). We may use the second or\nthird maximum implausibility instead for robustness reasons, or use multivariate\nimplausibility\nmeasures to incorporate correlations \\citep{Vernon2010,Vernon10_CS_rej}.\n\nHistory matching proceeds iteratively, discarding implausible regions of the input\nparameter space in waves.\nAt the $k$th wave we define the current set of non-implausible input points as\n$\\mathcal{W}_k$ and the set of outputs that have so far been considered for emulation\nas~$Q_k$. We proceed according to the following algorithm:\n\\begin{itemize}\n\\item[]\n\\vspace{-0.1cm}\n\\begin{enumerate}[1.]\n\\item Design and evaluate a space filling set of wave $k$ runs over the current\nnon-implausible space $\\mathcal{W}_k$. \\label{alg:1}\n\\item Check if there are informative outputs that can now be emulated accurately\n(that were difficult to emulate in previous waves) and add them to $Q_k$, to define\n$Q_{k+1}$.\n\\item Use the wave $k$ runs to construct new, more accurate emulators defined only\nover the region $\\mathcal{W}_k$ for each output in $Q_{k+1}$.\n\\item Recalculate the implausibility measures $I_i(x,w_i)$, $i \\in Q_{k+1}$, over\n$\\mathcal{W}_k$, using the new emulators.\n\\item Impose cutoffs $I_i(x,w_i) < c$ to define a new, smaller\nnon-implausible volume $\\mathcal{W}_{k+1}$ which satisfies $\\mathcal{W} \\subset\n\\mathcal{W}_{k+1} \\subset \\mathcal{W}_k$.\n\\item Unless:\n    \\begin{enumerate}[A)]\n        \\item the emulator variances ${\\rm Var}_{D_i}(f_i(x))$\n         are now small in comparison to the other\n        sources of uncertainty: $\\sigma^2_{\\epsilon_i} +\n\\sigma^2_{e_i}$,\\label{alg:6a}\n        \\item the entire input space has been deemed implausible or\n        \\item computational resources have been exhausted,\n    \\end{enumerate}\n    return to step~\\ref{alg:1}.\n\\item If \\ref{alg:6a} is true, generate a large number of acceptable runs from the\nfinal non-implausible volume $\\mathcal{W}$, using appropriate sampling for the\nscientific purpose.\\label{alg:7}\n\\end{enumerate}\n\\vspace{0.3cm}\n\\end{itemize}\nWe are then free to analyse the structure of the non-implausible volume $\\mathcal{W}$\nand the behaviour of model evaluations from different locations within it. The\nhistory matching approach is powerful for several reasons:\n\\begin{itemize}\n\\item As we progress through the waves and reduce the non-implausible volume, we\nexpect the function $f(x)$ to become smoother, and hence to be more accurately\napproximated by the regression part of the emulator $\\beta_{ij} g_{ij}(x_{A_i})$\n(which is often composed of low order polynomials -- see\nequation~\\ref{eq_emulator}).\n\\item At each new wave we have a higher density of points in a smaller volume,\ntherefore the emulator's Gaussian process term $u_i(x_{A_i})$ will be more effective,\nas it depends mainly on the proximity of $x$ to the nearest runs.\n\\item In later waves the previously strongly dominant active inputs $x_{A_i}$ from\nearly waves will have had their effects curtailed, and hence it will be easier to\nselect additional active inputs, unnoticed before.\n\\item There may be several outputs that are difficult to emulate in early waves\n(often due to their erratic behaviour in scientifically uninteresting parts of the\ninput space) but\nsimple to emulate in later waves, once we have restricted the input space to a much\nsmaller and more physically realistic region.\n\\end{itemize}\nHistory matching can be viewed as the appropriate analysis suitable for model\ninvestigation, model checking and model development.\nShould one wish to perform a fully Bayesian analysis using say MCMC, history\nmatching can be used as a highly effective precursor to such a calculation in order\nto rule out vast regions of input space that would only contain extremely low\nposterior probability.\nHowever such an MCMC analysis would only be warranted assuming one is willing to\nspecify meaningful joint probability distributions over all uncertain quantities\ninvolved, in\ncontrast to only the expectations, variances and covariances required for the Bayes\nLinear history\nmatch.\n\n\n\n\n\\subsection{Application of Emulation and History Matching to \\textsc{galform}\\xspace and the GSMF}\n\n\nWe now apply the above Bayesian emulation and history matching methodology to \\textsc{galform}\\xspace\nand the\nGSMF, and generalise it to the case of multiple available observed data sets.\nWe first identify the \\textsc{galform}\\xspace model outputs $f_i(x)$ that we wish to emulate, and the\ncorresponding observed data $w_i^{(m)}$ to match them to as\n\\begin{equation}\\label{eq_f_and_w}\nf_i(x) = \\log\\phi_{i,\\text{model}} \\quad \\text{and} \\quad w_i^{(m)} =\n\\log\\phi_{i,\\text{obs}_{(m)}}\n\\end{equation}\nwhere\n\\begin{equation}\n \\phi_i = \\left.\\frac{\\dv n}{\\dv \\log M_\\star} \\right|_{M_{\\star,i}, z}\\nonumber\n\\end{equation}\nis the GSMF at the stellar mass bin $M_{\\star,i}$ for redshift $z$. Here $m$ labels\nthe choice of observed data sets we use, represented for output $i$ by\n $w_i^{(m)}$.\n{Following the discussion in \\citet{Bower2010}, we adopt a model\ndiscrepancy of 0.1 dex.  This term summarises the accuracy we expect for the model\ndue to the approximations inherent in the semi-analytic method}.\n{In effect, this means that we will regard models that lie within 5\\% of the\nobserved data-point as a perfectly adequate fit, even if the quoted Poisson\nobservational errors are substantially smaller. This means that if a model has a\nmarginally acceptable implausibility, $\\,I\\sim 3$, it may be 0.3 dex away from the\nobservational data-point.\n}\n\nAs we have multiple sets of observed data for the GSMF which we wish to match to, we\nhave to make an additional decision as to how to combine these within the history\nmatching process. Here we generalise the implausibility measure of\nequation~\\eqref{eq_maximp} by minimizing over the $m$\n data sets:\n\\begin{equation}\\label{eq_maximp2}\nI_M(x,w) \\;\\;=\\;\\; \\max_{i \\in Q} \\{ \\min_m  I_i(x,w_i^{(m)}) \\}\n\\end{equation}\nwith the second and third maximum implausibilities defined similarly.\nThis implies that our history match search will attempt to find all inputs that lead\nto matches to {\\it any} of the observed data sets, judged on an individual bin basis.\nThis is\na simple\nway of incorporating several (possibly conflicting) data sets into the history match\nthat does not involve additional assumptions or further statistical modelling,\nand which is sufficient for our current purposes.\nIt should lead to the identification of all inputs of interest, subsets of which (for\nexample those that match a specific data set, or a combined data set) can be\nsubsequently explored in further detail.\n\nThe emulators used in each wave were constructed following the techniques described\nin \\S\\ref{sec:emconstr}, \\S\\ref{sec:1demul} and specifically the high\ndimensional approaches of \\S\\ref{sec:em_high_dim}.\n\n\\subsection{Observational datasets used}\n\\label{sec:obs}\n\nFor the local Universe GSMF, we use the results of \\citet{Li2009} based on SDSS and\n\\citet{Baldry2012} on the GAMA survey.\n\nFor larger redshifts, we combine the results of \\citet{Tomczak2014} based on the\n\\textsc{zfourge} and \\textsc{candels} surveys, and \\citet{Muzzin2013}, based on the\n\\textsc{UltraVISTA} survey.\nIn these papers, the GSMF is reported for redshift\nintervals\/bins. For simplicity, we adopt the midpoint of each redshift bin as\nthe typical redshift to be compared with the model (e.g. the GSMF obtained for\n$0.5<z<1.0$ will be compared with the model results for $z=0.75$).\n\\green{Both datasets obtain their stellar masses using the \\textsc{fast} code\n\\citep{Kriek2009} to fit the stellar  population synthesis model of\n\\citet{BruzualCharlot2003} to the measured spectral energy distributions of the\ngalaxies, assuming a \\cite{ChabrierIMF} IMF.\n}\n\\green{\nErrors in the determination of galaxy masses at $z>0$ redshifts were assumed to\nfollow the redshift-dependent estimate as \\citet{Behroozi2013}, i.e.\n$\\sigma_M(z) = \\sigma_0 + \\sigma_z z$, with $\\sigma_0=0.07$ and $\\sigma_z=0.04$.\nThese mass errors were accounted for convolving the model GSMF with a Gaussian kernel\n(see \\S\\ref{sec:high_z} for a discussion).\n}\n\n\\begin{table}\n\\begin{center}\n\\caption{\n\\label{tab:cutoffs}Thresholds used for eliminating implausible regions with respect\nto the local Universe GSMF after each wave and the fraction of the initial volume in\nthe non-implausible region.}\n\\begin{tabular}{ccccc}\n\\hline\nWave     & \\mc{3}{c}{Threshold}    &   Fraction of the               \\\\\n& 1$^{\\rm st}$ max. & \\mc{1}{c}{2$^{\\rm nd}$} max.& \\mc{1}{c}{3$^{\\rm rd}$} max.&\ninitial volume \\\\\n\\hline\n1 & -    & \\mc{1}{c}{3.2}  & \\mc{1}{c}{2.5} & 0.2522\\\\\n2 & 4.5  & \\mc{1}{c}{3.0}  & \\mc{1}{c}{2.3} & 0.0494\\\\\n3 & 3.75 & \\mc{1}{c}{2.5}  & \\mc{1}{c}{2.0} & 0.0170\\\\\n4 & 3.5  & \\mc{1}{c}{2.5}  & \\mc{1}{c}{2.0} & 0.0116\\\\\n5 & 3.0  & \\mc{1}{c}{2.25} & \\mc{1}{c}{2.0} & 0.0036\\\\\n6 & 2.4  & \\mc{1}{c}{2.15} & \\mc{1}{c}{1.8} & 0.0010\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Results}\n\n\n\\label{sec:results}\n\nThe parameter space exploration was conducted through successive waves of runs.\nAfter each wave, emulators were generated from its results and used to design\nthe parameter choices for the next wave, discarding a vast, specifically\nimplausible, region of the parameter space. Each wave was designed using a\nLatin Hypercube Sampling of $5000$ points of the non-implausible region of the\nparameter space (full details are given in appendix \\ref{ap:emu_details}).\n\n\\subsection{Matching the local galaxy stellar mass function}\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{.\/hist.pdf}\n \\includegraphics[width=\\columnwidth]{.\/hist_final.pdf}\n \\caption{\n          Histograms of the model implausibilities (with respect to the local\n          Universe only) obtained at each wave.\n          In the top panel, exploratory waves 2-6 are shown.\n          Each new wave reduces the tail of very implausible models.\n          However, the increase in the number of models with $I<2$ occurs only slowly\n          after each wave.\n          In the bottom panel, waves 7a and 7b are also shown (please, note the\n          different range in $I$). Wave 7a was designed specifically to obtain many\n          plausible runs, instead of uniformly covering the non-implausible\nparameter space, and Wave 7b (discussed in \\S\\ref{sec:high_z}) takes into account\n          the constraints by high redshift data.\n          }\n \\label{fig:wall}\n\\end{figure}\n\n\\label{sec:z0}\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{.\/smf_z0.pdf}\n \\caption{\n          Local galaxy stellar mass function.\n          The coloured curves show all the runs with implausibility $I<3.5$, with\n          different shades showing different implausibilities (see colour-bar in the\n          plot).\n          The surrounding light grey curves correspond to the initial set of runs\n          (wave 1).\n          Data-points show the GSMF data obtained by \\citet{Li2009} and\n          \\citet{Baldry2012}.\n          {Note that a 0.1 dex model discrepancy was assumed (see text for\n          details).}\n         }\n \\label{fig:smf_z0}\n\\end{figure}\n\nThere were initially 6 waves of runs, where the implausibilities were computed with\nrespect to the local Universe GSMF data only.\nTable \\ref{tab:cutoffs} shows the implausibility cut-off thresholds applied,\nwhich decreased after each wave as we build more trustworthy emulators.\nTable \\ref{tab:cutoffs} also shows the fraction of the\ninitial volume which corresponds to the region classified as non-implausible after\neach wave.\nTo compute the volumes, the parameters were rescaled following equations\n\\eqref{eq:scale_lin} and \\eqref{eq:scale_log} -- i.e. lengths associated with the\nrange of each parameter were considered equivalent.\nDespite making very conservative choices\nfor the thresholds, there is a strong decrease in the volume after each wave and\nafter wave 6 only $10^{-3}$ of the original volume was classified as\nnon-implausible.\nIn Fig. \\ref{fig:wall} the evolution of the distribution of the\nimplausibilities can be followed: after each wave the number of highly\ndiscrepant models is strongly reduced, but the number of acceptable models,\nwith $I<3$, increases only modestly.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{smf_grid_z0.pdf}\n\\caption{\n         The GSMF of selected models colour-coded by their\n         implausibility, $I$ (see colour bar),\n         calculated \\emph{only} with respect to the GSMF\n         data at the local Universe, from \\citet{Baldry2012} and \\citet{Li2009}.\n         Only runs with $I<3.5$ with are shown.\n         The high redshift observational data\n         shown were obtained from \\citet{Muzzin2013} and \\citet{Tomczak2014}.\n         \\label{fig:smf_grid_z0}\n         The models selected solely by their good match to $z=0$ data produce\n         poor agreement to higher redshift data.\n        }\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{smf_grid_z.pdf}\n\\caption{\\label{fig:smf_grid_high}\n         The GSMF of selected models colour-coded by their\n         implausibility, $I$, calculated \\emph{simultaneously} with respect to\n         the GSMF at\n         $z=0,\\,0.35,\\,0.62,\\,0.75,\\,0.88,\\,1.12,\\,1.25$ and $1.43$.\n         Only runs with $I<3.5$ are shown.\n         The high redshift observational data\n         shown were obtained from \\citet{Muzzin2013} and \\citet{Tomczak2014}\n         \\green{and local Universe data were obtained from \\citet{Baldry2012}\n         and \\citet{Li2009}.}\n         \\textsc{galform}\\xspace models generally lead to a GSMF with a high mass end which is too\n         shallow\n         at $z=0$ and too steep at higher redshifts.\n        }\n\\end{figure*}\n\nAfter the 6$^{\\rm th}$ wave the emulator variance at each point was already\nsmaller or equal to the other uncertainties,\nindicating that no further refinement was possible (condition \\ref{alg:6a}\nin the algorithm described in \\S\\ref{sec:HM}). A final, wave 7a,\nwas then designed, this time using the emulator information to aim for\nthe best possible matches to the GSMF (i.e. step \\ref{alg:7} in the algorithm;\nin contrast to the previous waves where the non-implausible space was uniformly\nsampled to ensure an optimum input for the next wave).\n\nIn Fig. \\ref{fig:smf_z0}, the final results of the history matching are shown,\ntogether with our observational constraints.\n{Error bars in this and other figures show only the quoted\nobservational errors, and do not include the model discrepancy term, so that the\nover all quality of the fits can be judged from figures.\nThe purpose of the model discrepancy is to avoid rejecting models when the\nobservational errors become very small.}\nAll the models in our library with implausibility $I<3.5$ are\nshown {in Fig.~\\ref{fig:smf_z0}}, with the curves\ncolour-coded by the implausibility.\n{The width of the lightest shade, corresponding to $I\\leq 1.0$,\nallows one to visualize the effect of adopted model discrepancy.\n}\nAlso shown are the wave 1 runs given as the grey curves, many of which\nwere far from the observed data.\nThe impact of the history match in terms of the removal of substantial amounts of\nimplausible regions of the parameter space can be seen by comparing the coloured\nregion with the grey curves.\n\nWhile there are models with $I\\sim2.5-3$ which produce an excess in the number of\nsmall\nmass galaxies, the opposite (i.e. $\\phi$ smaller than the observations at the low\nmass end) is very rare. A similar behaviour is also present in the high mass end.\nThus, acceptable ($I\\lesssim 3$) models may display over-abundances of very small\n[$\\log(M_\\star\/\\,\\text{M}_\\odot)\\lesssim 8.5$] or very large\n[$\\log(M_\\star\/\\,\\text{M}_\\odot)\\gtrsim 11.5$] masses, but there are no acceptable models\nwith significant under-abundances in these ranges.\n\nOnce the locus of models with good fits to the local Universe GSMF was found, we\nexamined how well these models performed with respect to high redshift data. In\nFig. \\ref{fig:smf_grid_z0} the GSMF output by the models shown in Fig.\n\\ref{fig:smf_z0} is now compared with higher redshift data.\nOne finds that the models selected only by their ability to\nreproduce the local Universe GSMF data under-represent the abundance of high mass\ngalaxies at higher redshifts while simultaneously generating an excessive number of\ngalaxies of lower masses.\n\nIn the following  section we will examine the parameter space and show that the vast\nmajority of acceptable models have $\\beta_{0,\\text{burst}} > \\beta_{0,\\text{disc}}$ and so lie in a region\nof parameter space not available to the original model.\n\n\n\\subsection{Constraining models with higher redshift data}\n\\label{sec:high_z}\n\n\\begin{figure*}\n\\centering\n\\includegraphics{output_pair.pdf}\n\\caption{\nEach panel compares the output of the GSMF for different mass bins and\/or redshifts.\nWaves 2, 4, 6, 7a, 7b are shown colour-coded as indicated.\nThe observational constraints of previous figures are shown as blue shades.\nThe light grey shade was added to guide the eye, indicating where the GSMF values in\nthe vertical axis are smaller than the horizontal axis, and the dashed\ndiagonal line indicates the case where the GSMF for the two is the same.\nPanels below the diagonal use the same estimates for the errors in the mass\ndetermination as \\citet{Behroozi2013}: $\\sigma_M = 0.07 + 0.04\\,z$ (see text for\ndetails). For comparison, in the panels above the diagonal we double the mass error.\nAfter successive waves there is improvement in the\nagreement with the low redshift data and with the low-mass-end of the high\nredshift data, however, for the high-mass end, there is still tension at higher\nredshifts. The increase in the mass error does not avoid the tension.\nThis tension originates from the data being consistent with small or no evolution\nfor bins of large mass.\n}\\label{fig:output_pair}\n\\end{figure*}\n\nTo investigate if, and to what extent, the present model could reproduce the\nevolution of the GSMF, a new wave of runs was generated from the wave 6 emulator\n(wave 7b), this time computing the implausibilities simultaneously with respect to\nthe GSMF data at higher redshifts, up to $z=1.75$.\n\nAfter just a single additional wave, the emulation technique indicated that no extra\nrefinement was likely: the emulator variances became smaller than the other\nuncertainties, corresponding to step 6A in the algorithm of \u00a73, suggesting that it\nwould be highly unlikely to find a locus of more plausible runs within any sub-volume\nof the parameter space. A new (and final) wave was then designed, to produce runs\nwhich provided a good match to the GSMF at those redshifts (corresponding to step 7).\nThis set of runs was deliberately focused towards the regions of lowest emulator\nimplausibility, where we would now expect the best matches to occur. This is a good\ntechnique for exploring the correlations between parameter sets; however, it is\nimportant to note that the resulting design of runs would not be a suitable basis for\nthe construction of a statistical emulator.\n\nIn Fig. \\ref{fig:smf_grid_high} we show the evolution of the GSMF for all the runs\n(of all waves) with implausibility $I<3.5$ with respect to redshifts up to\n$z=1.12$.\nThe adoption of higher redshift constraints leads to tension with the local Universe\ndata: the least implausible models produce a GSMF with a too shallow high mass end\nat $z=0$ and too steep at any other redshift. In the low mass end, there is an\nexcess of $\\lesssim 10^{10} \\,\\text{M}_\\odot$ galaxies at higher redshift and a small deficit\nof them in the local Universe.\nThis is consistent with behaviour seen in the runs constrained at $z=0$ only.\n{\\green{It should be noted that, despite the tension,\nthe level agreement achieved is still better than what is found in most published models, and is not dissimilar from what is found by \\citet{Henriques2013,Henriques2015}.}}\n\nThis tension becomes clearer when the results for specific mass bins of the GSMF\nare compared. This is shown in panels below the diagonal in\nFig.~\\ref{fig:output_pair}, for two mass bins, $\\log(m\/\\,\\text{M}_\\odot h)=9.5$ and $11.2$,\nredshifts $z=0.0$, $0.35$ and $0.75$. The observational constraints are shown as\nblue bands. By showing the constraints in pairs, we gain insight into the conflicting\npressures imposed on the model.\nInitially, successive waves of runs (shown by colours from red to green\\green{,\nas indicated in the figure}) are increasingly\nfocused towards the point at which the two bands intersect.\nHowever, it becomes increasingly evident that some constraint pairs cannot be matched\nby the model and the successive waves lead to no improvement. For example, the panel\nshowing $\\log(m\/\\,\\text{M}_\\odot h)=11.2$ at $z=0$ and $z=0.35$ has a strong diagonal line above\nwhich\nthe model is never able to cross. The same behaviour is found when comparing the high\nmass end of the GSMF at $z=0$ with other redshifts. For the constraints at\n$\\log(m\/\\,\\text{M}_\\odot h)=9.5$,\nthe outputs of all models are tightly correlated when comparing between redshift.\nComparison\nbetween different mass bins appears to be less constraining.\n\nOne can best interpret Fig.~\\ref{fig:output_pair} by comparing the models to a\nnon-evolving GSMF.  This is shown by the dashed diagonal line in panels that\ncompare the same mass bins. The\ngrey-shaded side of the dashed diagonal line show the case where the number of\ngalaxies decreases with time. It can be seen that the observational data used leads\nto no\nevolution (or even decrease) in the number of $10^{11.2}h^{-1}\\,\\text{M}_\\odot$ galaxies if\n$z=0$ is compared to other datasets. This makes it clear why it is not possible to\nfind models in the exact target region. Since \\textsc{galform}\\xspace is inherently hierarchical, it is\ndifficult\nto conceive of a mechanism which could lead to a significant decrease in the\nabundance of massive galaxies with time.  This would only be possible if\n$10^{11.2}h^{-1}\\,\\text{M}_\\odot$ galaxies\nwere to grow in mass (and so leave the mass bin) faster than lower mass (and more\nabundant) galaxies were able to grow and move into the bin. Clearly, the situation\nnever arises in the \\textsc{galform}\\xspace model and the only way of obtaining points in the grey\nregion for the high\nmass bin panes is due to the distortion caused by errors in the galaxy mass\ndetermination, as we will discuss below.\n\nSystematic errors in the determination of galaxy masses (`mass errors' for short)\narising from the modelling of the star formation history, choice of dust model and\nthe choice of IMF can significantly affect the shape of the GSMF which is inferred\nfrom the observations \\citep{Mitchell2013}.\n\\green{As mentioned in \\S\\ref{sec:obs}, mass errors were accounted for by convolving\nthe model GSMF with a Gaussian kernel.\n}\nThe main effect of such convolution is making the GSMF appear less steep at higher\nredshifts.\nThis raises the question of whether underestimated mass errors could explain the\ndifficulty in simultaneously matching the high mass end of the GSMF at different\nredshifts.\n\n\\bigskip\n\nIn the panels above the diagonal of Fig.~\\ref{fig:output_pair}, we show\nthe consequences of \\green{doubling the mass error -- i.e.}\nconsidering $\\sigma_0=0.14$ and $\\sigma_z=0.08$.\nThis has the effect of loosening the implausibility contours: the\nblue regions are the same as those below\nthe diagonal. The effect of these much increased mass errors is to\nallow models near to the ``no evolution'' region, alleviating the tension by allowing\nthe corrected \\textsc{galform}\\xspace\nresults to get closer to the target region. However, even considering these mass\nerrors, some tension still persists.\n\n\\subsection{Plausible models subspace}\n\\label{sec:subspace}\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{.\/cabral_implaus_report_highz.pdf}\n\\caption{The panels show two dimensional projections of the plausible\n         parameter space. Each circle\n         represents a \\textsc{galform}\\xspace run and is colour coded by its implausibility (as\nindicated by the colour bar); lower implausibility runs are plotted on\n         top facilitating the visualisation of their clustering in the projected\n         space; only runs with $I<3.5$ are shown.\n         In panels above the diagonal, the implausibility is computed with\n        respect to the observed local GSMF only.\n         In panels below the diagonal, the implausibility is computed with\n         respect to the GSMF at redshifts\n         $z=0.0,\\,0.35,\\,0.62,\\,0.75,\\,0.88,\\,1.12,\\,1.25$ and $1.43$.\n         Note that the axes are labelled consistently above and below the diagonal. A\npanel below the diagonal should be rotated and inverted in order to compare it to the\nequivalent panel above the diagonal.\n         This figure summarizes the main constraints imposed by the GSMF and its\n         evolution on the \\textsc{galform}\\xspace parameters.\n        }\n\\label{fig:implaus_space}\n\\end{figure*}\n\nWe examine now what are the main properties of the subspace of plausible models,\nwhich we define as models having implausibility, $I<3.5$, a\nconservative threshold.\n\nWe begin by considering the models that provide a plausible match to the\nGSMF at $z=0$. The distribution of these models are shown above the diagonal\nin Fig.~\\ref{fig:implaus_space}.  In each panel we show the plausible models\nprojected\ninto the two dimensional space of a pair of variables.  The models are coloured by\nimplausibility and the lowest implausibility runs are plotted last to ensure they are\nvisible. This method of plotting also gives\na good impression of the ``optical depth'' of the parameter region in the hidden\nparameters of each panel.\nWe only show the most interesting variables in this plot, the panels for other\nvariable pairs are less informative scatter plots.\n\nThe most constrained parameters are: the disk wind parameters, $\\alpha_\\text{hot}$ and\n$\\beta_{0,\\text{disc}}$, the normalisation of the star formation law, $\\nu_{0,\\text{sf}}$, the AGN feedback\nparameters, $\\alpha_\\text{cool}$, and the disk stability threshold, $f_{\\rm stab}$.  Several\nparameter degeneracies can be picked out in the figure.\nFor example, values of $\\alpha_\\text{hot}$ are strongly\ncorrelated with $\\beta_{0,\\text{disc}}$, with larger $\\beta_{0,\\text{disc}}$ being compensated by a\nsmaller $\\alpha_\\text{hot}$: i.e., the higher mass loading normalisation is compensated by a\nweaker mass dependence so that the level of feedback is similar in low-mass galaxies.\n\n\nOther parameters are more weakly constrained, and it is possible to find plausible\nmodels over most of the range of the parameter considered. The parameter\n$\\alpha_\\text{reheat}$ is a good example. In this case, smaller values of $\\alpha_\\text{reheat}$ can\nbe compensated by reductions in $\\beta_{0,\\text{disc}}$.\nThis makes physical sense. The time-scale on which gas\nis re-incorporated into the halo after ejection depends on $\\alpha_\\text{reheat}^{-1}$\n(equation~\\ref{eq:reheat}), so that increases in the time-scale can be offset by an\noverall lower mass loading of the disk wind \\citep{Mitchell2016}.\n\nOne surprising feature is that the normalization mass loading associated with star\nburst galaxies, $\\beta_{0,\\text{burst}}$, (see \\S\\ref{sec:SN_feedback}) is weakly constrained.\nAlthough the best models (and also the greatest number of models) have $\\beta_{0,\\text{burst}} >\n20$, entirely plausible models can be found with much\nsmaller values. This is presumably because the impact of the large values of\n$\\beta_{0,\\text{burst}}$ can be offset\nby adjusting the values of other parameters. The pairs plot does not, however, reveal\nan obvious\ninteraction with another individual parameter.  In \\S\\ref{sec:pca}, we will use a\nprinciple\ncomponent method to try to isolate simpler interactions between parameter\ncombinations, and we explore the physical interpretation there.\n\nThe panels below the diagonal line show the models that generate plausible fits\nto the GSMF\nover the redshift range $z=0$ to $1.43$. A panel below the diagonal must be rotated\nand inverted in order to compare it to the equivalent panel above the diagonal. As we\nhave already discussed, this is a stringent requirement, and even the best models\nhave $I>2$. The volume of the parameter space within which plausible models can be\nfound is significantly reduced compared to the situation if only the $z=0$\nimplausibility is considered.\nThe plausible range of the parameters $\\alpha_\\text{reheat}$, $\\alpha_\\text{cool}$ and $\\nu_{0,\\text{sf}}$ is\nparticularly affected.\nFor example, the addition of the high redshift GSMF excludes very long gas cycling\ntime-scales (and thus\nsmall values of $\\alpha_\\text{reheat}$).\n\nPlotting the data in this way does not, however, expose any new correlations between\nparameters, or make\nit easy to appreciate the physical differences in the model that result in the very\ndifferent behaviour\nat high redshift that can be seen by comparing Figs. \\ref{fig:smf_grid_z0} and\n\\ref{fig:smf_grid_high}.\nIn order to make it easier to identify these differences, we will analyse the\ndistribution of the plausible models in the Principle Component Analysis (PCA) space.\n This allows us to better identify the critical parameter combinations that are\npicked out by the data. We have already noted that several parameters show\nsignificant (anti-)correlation, and the PCA analysis will identify the\nmost important relations.\n\nOne of the motivations for undertaking a full parameter space exploration is the\npossibility of the existence of multiple disconnected implausibility minima, which\nwould be unlikely to be found in the `traditional' trial-and-error approach to\nchoosing the parameters. Nevertheless, we find that the locus of acceptable \\textsc{galform}\\xspace runs\nis connected and there are no signs of multiple minima or other complex\nshapes. Because of this, the distribution of plausible models is particularly\namenable to the PCA method.\n\n\n\\subsection{Principal component analysis}\n\\label{sec:pca}\n\nIn order to obtain greater insight into the constraints imposed by the GSMF, and in\nparticular the constraints imposed by the higher redshift data, we performed a\nprincipal component analysis (PCA)\non the volume of the input parameter space containing\nruns with $I<3.4$ in all the datasets at\n$z=0,\\,0.35,\\,0.62,\\,0.75,\\,0.88,\\,1.12,\\,1.25$ and\n $1.43$, giving a set of 508 runs in total.\nThe PCA generates a new set of 20 orthogonal variables defined as the\neigenvectors of the\ncovariance matrix formed from the input parameter locations of the 508 runs, ordered\nby size of eigenvalue.\nTherefore the first new variable (Var~1) gives the direction which has the largest\nvariance in the input space,\nwhile the last (Var~20) gives the direction with the smallest variance. Usually, PCA\nis applied to find the directions with\nthe largest variance, but here we are precisely interested in the opposite: we wish\nto learn about those directions\nin input parameter space that have been most constrained by the observed data. This\nanalysis allows the examination of the\nlocation of acceptable runs in the rotated (and translated) PCA space, to identify\npossible hidden features, and the\ntransformation of the (approximately) orthogonal constraints observed in the PCA\nspace back on to the original parameters\nto aid physical interpretation.\nFor example, acceptable model runs all have similar values for Var~20, Var~19 etc,\nand this can be inverted to express the dependencies of the variables on one\nanother. It is important to note that the precise components of the PCA variables\ndepend on their original range (and whether the variables are normalised on to a log\nor linear scale).\nThis can be viewed in a Bayesian sense, in that we are quantifying the increase in\nknowledge about the values of the variables relative to our prior knowledge. It is\nalso important to bear in mind that variables with similar variance are degenerate,\nand that alternative combinations of them will describe the distribution of the data\nsimilarly well, but may have a simpler physical interpretation.\n\n\\begin{figure}\n\\includegraphics{.\/PCA_bars_highz.pdf}\n\\caption{Summary of results of the principal component analysis.\nThe 6 most constrained components of the region with\n$I<3.4$ with respect to $z=0,\\,0.35,\\,0.62,\\,0.75,\\,0.88,\\,1.12,\\,1.25$ and\n$1.43$ (which contains 508 models).\nThe bars show the absolute values of the PCA loads associated with each\nscaled parameter (only parameters with non-negligible loads are shown).\nParameters with larger loads\n($>0.3$)\nare drawn in red\n\\green{and have their names and loads written on the top of the bars.}\nBright (dark)\ncolours show variables with positive (negative) loads.\n}\n\\label{fig:pca}\n\\end{figure}\n\nThe resulting PCA variables (and the centroid of the distribution) are listed in\nAppendix \\ref{ap:fullPCA}.\nThe standard deviation in the directions defined by Var~20 and\nVar~19 is extremely small (less than 0.1 relative to the prior distribution of\n$\\pm1$). Var~18 and Var~17 are also significantly constrained (std. dev. less than\n0.22). The constraints on the other variables are much less significant, Var~14, 15\nand 16 all have std. dev. $\\sim 0.4$. This gives us a quantitative measure of the\ninformation content of the GSMF relative to the freedoms of the model.\n\nThe components of the 6 most constrained variables are shown in Fig.~\\ref{fig:pca}.\nWe begin by considering the strongly constrained components Var~19 and Var~20. The\nvariance of these two components is similar and so we should consider them together.\nAs shown by the colouring of the histogram, Var~19 is dominated by $\\beta_{0,\\text{disc}}$ and\n$\\alpha_\\text{cool}$, with a smaller contribution from $\\alpha_\\text{reheat}$. Qualitatively, this\nsimply confirms that the break of the GSMF is controlled by competition between AGN\nand stellar feedback;\nstronger winds from disks in $200\\,\\text{km}\\,\\text{s}^{-1}$ galaxies (i.e., larger\n        $\\beta_{0,\\text{disc}}$, equation~\\ref{eq:beta}), or a longer re-incorporation\ntime-scale (i.e.,\nsmaller $\\alpha_\\text{reheat}$, equation~\\ref{eq:reheat}), need to be compensated by an\nincrease the halo mass at which AGN become effective (i.e., smaller $\\alpha_\\text{cool}$,\nsince $t_\\text{cool}\/t_\\text{ff}(r_\\text{cool})$ increases with halo mass,\nequation~\\ref{eq:tcool}). As well as providing qualitative insight, this can be\ntranslated\ninto quantitative constraints on the input parameters. To do this we neglect the\ndependence on parameters with small loads ($<0.3$, shown in blue in\nFig.~\\ref{fig:pca}) and assume that they have values close to the centroid\nof the PCA expansion. Using superscripts to denote that this relation applies to\nthe rescaled variables (given by equations \\ref{eq:scale_lin} and\n\\ref{eq:scale_log}),\nthe constraint can then be simplified to:\n\\begin{align}\n\\label{eq:var19}\n|\\text{Var}\\,19|\\, =&\\,\\,\n|\\,-0.669(\\beta_{0,\\text{disc}}^{(s)}+0.464)\n      -0.576(\\alpha_\\text{cool}^{(l)}-0.065) \\nonumber\\\\\n     & +0.356(\\alpha_\\text{reheat}^{(s)}-0.462)\n     \\,|\\,\\, \\lesssim 0.095.\n\\end{align}\n\nVar~20 is mainly composed of $\\alpha_\\text{cool}$ (the AGN feedback parameter), $\\alpha_\\text{hot}$\nand $\\beta_{0,\\text{disc}}$ (the quiescent feedback parameters). Eliminating variables with small\nweight, we arrive at the following inequality:\n\\begin{align}\n|\\text{Var}\\,20|\\, =& \\,\\,|\\,\n     +0.401(\\beta_{0,\\text{disc}}^{(s)}+0.464)\n     +0.583(\\alpha_\\text{hot}^{(s)}-0.673)\\nonumber\\\\\n    &-0.634(\\alpha_\\text{cool}^{(l)}-0.065)\n     \\,|\\,\\, \\lesssim \\,0.071.\n\\end{align}\nPhysicaly, this relation tells us that if we pick the disk feedback parameters\n$\\alpha_\\text{hot}$ and $\\beta_{0,\\text{disc}}$, the AGN feedback must follow from the equality.\nIncreases in  $\\alpha_\\text{hot}$ and\/or $\\beta_{0,\\text{disc}}$ (making supernovae driven feedback) need\nto be compensated by increases $\\alpha_\\text{cool}$ (making AGN feedback effective only in\nhigher mass haloes). Since Var~19\nalready determines $\\alpha_\\text{cool}$, it is more useful to write the constraint as\n(neglecting small weights):\n\\begin{align}\n|\\text{Var}\\,20|\\, \\approx& \\,\\,|\\,\n     +1.137(\\beta_{0,\\text{disc}}^{(s)}+0.464) -0.391(\\alpha_\\text{reheat}^{(s)}-0.462) \\nonumber\\\\\n     &+0.583(\\alpha_\\text{hot}^{(s)}-0.673)\n     \\,|\\,\\, \\lesssim \\,0.175,\n\\label{eq:approx}\n\\end{align}\nwhich expresses the requirement that a given choice of $\\beta_{0,\\text{disc}}$ (and\n$\\alpha_\\text{reheat}$) parameters need to be balanced by a suitable choice of circular\nvelocity dependence of supernova feedback, $\\alpha_\\text{hot}$.\n\nThe next two components, Var~18 and Var~17, have significantly larger variances\n($\\sigma=0.174$ and $0.217$, respectively). Var~17 is almost completely determined\nby $f_{\\rm stab}$, so that successful models require a narrow range of the stability\nparameter, almost independent of the other variables.\n\\begin{equation}\n\\label{eq:var17}\n|\\text{Var}\\,17|\\, =\\,\\,\n|\\,-0.931(f_{\\rm stab}^{(s)}+0.362)\n    \\,\\,\\,|\\,\\,  \\lesssim 0.217\\,.\n\\end{equation}\nVar~18 relates the star formation\nefficiency $\\nu_{0,\\text{sf}}$ to $\\alpha_\\text{hot}$, the halo mass\ndependence of feedback (which in turn relates to the choice of feedback parameters\n$\\beta_{0,\\text{disc}}$ and $\\alpha_\\text{reheat}$, see equation~\\ref{eq:approx}):\n\\begin{align}\n\\label{eq:var18}\n|\\text{Var}\\,18|\\, &= \\,\\,\\nonumber\n|\\,\\,0.613(\\nu_{0,\\text{sf}}^{(s)}+0.456)\\\\\n    &-0.604(\\alpha_\\text{hot}^{(s)}-0.673)\n    \\,\\,\\,|\\,\\, \\lesssim 0.174.\n\\end{align}\nIncreasing the strength of feedback in small galaxies (greater $\\alpha_\\text{hot}$) requires\nthat star formation is made more efficient to compensate\n\\green{(i.e. by increasing star formation at higher mass galaxies, maintaining thus\nthe total amount of stars at low $z$)}.\n\nThe remaining variables are relatively weakly constrained, but have similar variance.\nThey provide addition constraints on the disk and AGN feedback parameters\n($\\alpha_\\text{reheat}$, $\\alpha_\\text{cool}$, $\\alpha_\\text{hot}$ and $\\beta_{0,\\text{disc}}$) and the star formation\nlaw ($\\nu_{0,\\text{sf}}$). Although they are weakly constrained, these relations play an\nimportant role in determining whether models successfully match the higher redshift\nGSMF data as well as the $z=0$ GSMF, as we will show below.\n\n\\subsection{Effect of GSMF constraints in PCA space}\n\\label{sec:pca_comparison}\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{.\/cabral_implaus_report_pca.pdf}\n\\caption{A comparison of runs that provide plausible fits to the $z<1.43$ GSMF\ndatasets (below the diagonal), and those that provide a plausible description of the\n$z=0$ GSMF, but a very implausible match to the high-z data (above the diagonal). We\nshow the comparison  PCA-space, with individual panels showing two dimensional\nprojections. The PCA variables are defined using the set of plausible $z<1.43$ GSMF\ndatasets.  We show the 6 most constrained variables (see text for discussion). Each\ncircle represents a \\textsc{galform}\\xspace run and is colour coded by its implausibility (as indicated\nby the colour bar); lower implausibility runs are plotted on top to facilitate the\nvisualisation of their clustering in the projected spaced. To facilitate comparison\nof the runs above and below the diagonal, we show the full set of runs with plausibly\nfits\nto the $z=0$ GSMF as the underlying grey points.\n        }\n\\label{fig:pca_pairs}\n\\end{figure*}\n\nIn order to better understand why some runs generate a plausible match to the\n$z<1.43$ GSMF (as well as that at $z=0$) while others do not, we select the 5\ncomponents with least variance and rotate the distribution of the full set of runs\nwith plausible $z=0$ into this space. Note that the variables are defined using the\nplausible $z<1.43$ GSMF runs, but we can use the same rotation to examine the\ndistribution of any set of runs. We show projections into pairs of these variables in\nFig.~\\ref{fig:pca_pairs}.  Below the diagonal, we show the runs selected on the basis\nof the full redshift range of GSMF data (as in Fig.~\\ref{fig:implaus_space}). The\ncolouring, and plotting order, of points is the same as in the previous figures.\nAbove the diagonal, we show the set of runs that provide a good match to the $z=0$\nGSMF, but a very implausible match to the full  $z<1.4$ implausibility  ($I>6$). We\nadd the underlying grey points to show the distribution of the runs giving plausible\nfits to the $z=0$ GSMF (regardless of their $z<1.4$ implausibility) in order to make\nit simpler to compare with panels above and below the diagonal.\n\nThe location of the runs in the strongly constrained variables Var~19 and Var~20\nhardly changes.\nThese strong selection rules seem to primarily select runs with a good match to the\n$z=0$ GSMF, and are not particularly important in determining whether a run also\nmatches the higher redshift data or not. Var~15, 16 and 17, however, show systematic\nshifts above and below the diagonal, showing that it is these secondary\nrelationships between the feedback variables and the disk stability parameters that\nare critical in matching the evolution of the mass function. In particular,\nwe recall that Var~17 is almost exclusively dependent on the disk stability\ncriterion: runs which match the $z=0$ GSMF but not the higher redshift data tend to\nhave higher values of Var~17, and thus lower values of $f_{\\rm stab}$ which tends to make\ndisks more unstable at low redshift.\n\\green{Therefore, when larger redshift data is considered, models where\ninstabilities are mostly present at higher redshifts are preferred.\nVar~15 and 16 also show shifts, however, showing that the\nre-incorporation time-scale (ie., $\\alpha_\\text{reheat}$) and the\nstrength of disk feedback also play an important role. In particular, there\nis significant shift in the median value of Var~15 towards smaller values\nwhen higher redshift data is considered,\nwhich implies, simultaneously, an increase in $\\beta_{0,\\text{disc}}$ and a decrease in both\n$\\alpha_\\text{hot}$ and $\\nu_{0,\\text{sf}}$. The combined effect is to reduce the efficiency of star formation in galaxy disks.\n}\n\n\n\n\\subsection{The star formation history of the Universe}\n\nIn this paper we have deliberately focused on the GSMF.  This encoded the star\nformation history of the Universe in the fossil record of the stars that have been\nformed. It is nevertheless of interest to examine the star formation histories of\nthe models that have been selected on this basis.\nFurthermore, it is interesting to separate models in which the mass loading in\nstarbursts, $\\beta_{0,\\text{burst}}$, is comparable to that during quiescent star formation\n($\\beta_{0,\\text{disc}}$).\nFor simplicity, previous versions of \\textsc{galform}\\xspace have assumed that the parameters for the\nnormalization of the mass loading in quiescent discs, $\\beta_{0,\\text{disc}}$, and starbursts,\n$\\beta_{0,\\text{burst}}$,  were equal. By relaxing this assumption in this work, we found in\n\\S\\ref{sec:subspace} that a larger $\\beta_{0,\\text{burst}}$ is favoured.\nWhile it is possible to find\nplausible models for which\n$\\beta_{0,\\text{burst}}\\sim \\beta_{0,\\text{disc}}$, we found that most of the volume\n(and the most plausible runs) of the plausible parameter space has\n$\\beta_{0,\\text{burst}} \\gg \\beta_{0,\\text{disc}}$.\n\\green{Since starbursts are more frequent at earlier times, it is worth noting that\na $\\beta_{0,\\text{burst}} > \\beta_{0,\\text{disc}}$ can lead to stronger supernova feedback at high redshift.\n}\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{SFH_large_betaburst.pdf}\n\\includegraphics[width=\\columnwidth]{SFH_small_betaburst.pdf}\n\\includegraphics[width=\\columnwidth]{dsfrd_dz_burst.pdf}\n\\caption{Panels (a) and (b), show the cosmic star formation history (or\n         comoving star formation rate density, SFRD), of\n         runs with $I<3.5$ with respect to the redshifts\n         $z=0.0,\\,0.35,\\,0.62,\\,0.75,\\,0.88,\\,1.12$ and $1.25$. The\n         colours correspond to their implausibilities as indicated.\n         Observational data from\n         \\citet{Rodighiero2010,Karim2011,Cucciati2012}, and \\citet{Burgarella2013}.\n         Panel (a) shows only models with $\\beta_{0,\\text{burst}}>2\\,\\beta_{0,\\text{disc}}$ while the\n         panel (b) shows the case $\\beta_{0,\\text{burst}} \\leq \\beta_{0,\\text{disc}}$.\n         Panel (c) highlights the slope as a function of the $\\beta_{0,\\text{burst}}\/\\beta_{0,\\text{disc}}$\n         ratio with observational constraints shown as shaded areas (same colours as\n         previous panels).\n         While runs with a larger $\\beta_{0,\\text{burst}}\/\\beta_{0,\\text{disc}}$ {display a}\n         {qualitatively a better fit} to the\n         SFRD, they fail to produce the strong increase in SFRD with redshift\n         between $z=0.5$ and $z=1$, despite providing a better match to the GSMF\n         evolution at the same redshift interval (as it can be seen by the colours).}\n\\label{fig:SFH}\n\\end{figure}\n\nIn Fig.~\\ref{fig:SFH} we show the evolution of the cosmic star formation rate density\n(SFRD) for runs with $\\beta_{0,\\text{burst}} > 2\\,\\beta_{0,\\text{disc}}$  (upper panel) and for\n$\\beta_{0,\\text{burst}} \\leq \\beta_{0,\\text{disc}}$ (middle panel), in both cases selecting only\n``acceptable'' runs,\nwith $I<3.5$ when conditioned on the full range of GSMF data. A selection of\nobservational data are shown as coloured points. Runs with a larger\n$\\beta_{0,\\text{burst}}\/\\beta_{0,\\text{disc}}$ ratio match well the observations for the SFRD at low\nredshifts ($z\\leq 0.5$),\nbut fail to reproduce the steep rise in SFRD with redshift in the interval\n$0.5<z<1.0$.\nWhen runs with $\\beta_{0,\\text{burst}} \\leq \\beta_{0,\\text{disc}}$ are examined, one finds a stronger\nredshift evolution of the SFRD, but the normalization is a factor $\\sim 3$ off.  It\nshould be remembered, however, that the observational data does not measure the star\nformation rate directly,\nbut requires calibration.  This is usually based on \\cite{Kennicutt1998}. However, a\nmore recent study\nby \\cite{Chang2015} has suggested that this calibration needs revision (for Mid-IR\nindicators),\nbringing the observational data into better agreement with the $\\beta_{0,\\text{burst}} \\leq\n\\beta_{0,\\text{disc}}$ models.\nA similar discrepancy was found in the numerical Eagle simulations\n\\green{\\citep{Furlong2015} and other semi-analytic models (\\citealt{Henriques2015},\nsee also \\citealt{Guo2016}).}\nIt is noticeable, however, that these runs \\green{-- Panel~(b) --} are generally in\nless plausible agreement\nwith the mass function data than those in Panel~(a).\n\nWe quantify these differences more clearly in Panel~(c). Here the slope of the star\nformation rate\nrate density is plotted as a function of $\\beta_{0,\\text{burst}}\/\\beta_{0,\\text{disc}}$, with the colour\ncoding indicating\nthe plausibility of the run. The coloured shaded regions indicate the constraints\nimplied by the observational data. The tension between the GSMF and the observed\ndecline in the star formation rate density is now evident. {While} models\nrequire very small ratios {of $\\beta_{0,\\text{burst}}\/\\beta_{0,\\text{disc}}$} to match the {SFRD}\nobservations{,\nthe most plausible models with respect to the GSMF evolution, i.e. those with\n$I<2.5$, all have $1.66<\\beta_{0,\\text{burst}}\/\\beta_{0,\\text{disc}}<2.56$.}\n\n\\section{Summary and Conclusions}\n\\label{sec:summary}\n\nIn this work, using an iterative emulator technique, we explored how the parameter\nspace of \\textsc{galform}\\xspace is constrained by the galaxy stellar mass function.\nAfter 6 waves of emulation, using only the local Universe GSMF data, more\nthan $99.9$ per cent of initial volume of the parameter space was deemed too\nimplausible for further exploration and was eliminated.\nIt was possible to find many parameter choices which provide a good match to\nthe local GSMF. The bi-variate projections of this space are shown\nabove the diagonal in Fig.~\\ref{fig:implaus_space}.  The shape of the GSMF is\nprimarily  controlled by parameters related to star formation and feedback, namely:\n$\\alpha_\\text{cool}$, $\\alpha_\\text{hot}$, $\\beta_{0,\\text{disc}}$, $\\alpha_\\text{reheat}$, $\\nu_{0,\\text{sf}}\\,$ {and}\n$\\,f_{\\rm stab}$.\nConstraints on other parameters are weak.\n\n\nWe then included the requirement that the models also match higher\nredshift data. This parameter space is shown below the diagonal of\nFig.~\\ref{fig:implaus_space}.\nThis proves to be a much more stringent constraint, and the only acceptable runs\nfound had {$I\\gtrsim 2$}.\nThe high mass end of the GSMF produced by this version of \\textsc{galform}\\xspace is\ntypically too shallow for the local universe and becomes too steep at higher\nredshifts.\nThis tension is a consequence of the observational data being consistent\nwith small or no increase in the abundance of high mass galaxies between\n$z>0.35$ and $z=0.0$, compared to the model in which galaxies cannot avoid\ngrowing in mass. This tension would still be present even if mass errors had been\nunderestimated by a factor of 2.\n\nIn order to better understand the dimensionality and most important variables of\nthe parameter space, we performing a PCA of the non-implausible volume of the\nparameter space (constrained using the full range of redshifts,\nFig.~\\ref{fig:pca_pairs}). We show that it is possible to write approximate\nrelations between the parameters, expressing conditions which need to be satisfied\nin order to obtain a model with an acceptable match to the GSMF. Two principal\ncomponents (i.e. 2 directions in the parameter space) contain most of the\ninformation about the basic shape of the GSMF, and these are mainly combinations of\nthe parameters $\\alpha_\\text{cool}$, $\\alpha_\\text{hot}$, $\\beta_{0,\\text{disc}}$, $\\alpha_\\text{reheat}$, i.e. the\nparameters controlling feedback processes.\nThe parameters  $\\nu_{0,\\text{sf}}$, $f_{\\rm stab}$ are also significantly constrained compared to\ntheir initial values.\n\nThe PCA analysis provides a simple way to better understand why some model are able\nto match both the local and high redshift GSMF data (points below the diagonal in\nFig.~\\ref{fig:pca_pairs}),\nwhile other models only match the observational at $z=0$ (points above the diagonal\nin {Fig.~\\ref{fig:pca_pairs}).\nWe show that the primary differences are encoded in Var~15, 16 and (primarily) 17.\nModels which\nmatch the $z=0$ GSMF but not the higher redshift data tend to have higher values of\nVar~17, and thus lower values of $f_{\\rm stab}$ which tends to make disks more unstable at\nlow redshift.\n\n{\nIn this paper, we explored a model in which the we allowed the mass loading in\nstarburst (driven by mergers or disk instabilities) to be different from the mass\nloading in quiescent star formation.\nThe normalization of the quiescent mass, $\\beta_{0,\\text{disc}}$ loading is strongly\nconstrained, while marginally acceptable models can be found for most of\nthe range of values for the burst mass loading, $\\beta_{0,\\text{burst}}$. Nevertheless, this\ndoes not mean that the full range of $\\beta_{0,\\text{burst}}$ is equally plausible: there is a\nmuch larger density of acceptable models ($I\\lesssim 3$) with $20<\\beta_{0,\\text{burst}}<30$\nand the most plausible models, with $I<2.5$, have $1.66<\\beta_{0,\\text{burst}}\/\\beta_{0,\\text{disc}}<2.56$.\n}\n\nWe have deliberately focused the paper on the GSMF. This encoded the star formation\nhistory of the\nUniverse, but we can also compare the models to the observed star formation rates of\ngalaxies. We do this by computing the volume averaged star formation rate density in\nthe model. We find that the star formation history is sensitive to the choice of the\nratio $\\beta_{0,\\text{burst}}\/\\beta_{0,\\text{disc}}$. While models with $\\beta_{0,\\text{burst}}>\\beta_{0,\\text{disc}}$  offer a\nreasonable match to the GSMF evolution,\nthey fail to display sufficiently rapid increase in the cosmic SFRD.\nThese results show the important additional information that can be extracted by\nconfronting the constrained models with additional datasets, but that this needs to\nbe done with care, since it is quite possible that systematic differences may make\nit hard to simultaneously provide a plausible description of all the available data\nif the observational uncertainites are taken at face value. The apparent\ncontradictions inherent in different datasets must be carefully accounted for: as\nthey may point to missing physics in the model. Clearly a future avenue for further\nprogress is to apply the methods we have developed here to a much wider range of\ndatasets.\n\nFinally, we note that the main aim of this paper has been to examine how information\non the formation of galaxies can be extracted from observational dataset. We have\nshown how simple physical results can emerge from the analysis of a highly\ncomplex model. This approach can equally be applied across a wide range of science\ndisciplines where observational data are used to constrain seeming complex numerical\nmodels.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nGiven a complete filtered probability space $(\\Omega,\\F,\\FF,P)$ satisfying the usual hypotheses, let $R$ be an optional process of class $(D)$, and consider the optimal stopping problem\n\\begin{equation}\\label{os}\\tag{OS}\n\\maximize\\quad ER_\\tau\\quad\\ovr\\quad \\tau\\in\\T,\n\\end{equation}\nwhere $\\T$ is the set of stopping times with values in $[0,T]\\cup\\{T+\\}$ and $R$ is defined to be zero on $T+$. We allow $T$ to be $\\infty$ in which case $[0,T]$ is interpreted as the one-point compactification of the positive reals.\n\nWithout further conditions, optimal stopping times need not exist (take any deterministic process $R$ whose supremum is not attained).\nTheorem~II.2 of Bismut and Skalli~\\cite{bs77} establishes the existence for bounded reward processes $R$ such that $R\\ge\\cev R$ and $\\vec R\\le\\pp R$. Here,\n\\[\n\\vec R_t:=\\limsup_{s\\upto t} R_s\\quad\\text{and}\\quad\\cev R_t:=\\limsup_{s\\downto t} R_s,\n\\]\nthe {\\em left-} and {\\em right-upper semicontinuous regularizations} of $R$, respectively. Bismut and Skalli mention on page 301 that, instead of boundedness, it would suffice to assume that $R$ is of class $(D)$.\n\nIn order to extend the above, we study the ``optimal quasi-stopping problem''\n\\begin{equation}\\label{oqs}\\tag{OQS}\n\\maximize\\quad E[R_\\tau+\\vec R_{\\tilde \\tau}]\\quad\\ovr\\quad (\\tau,\\tilde\\tau)\\in\\hat\\T,\n\\end{equation}\nwhere $\\hat\\T$ is the set of {\\em quasi-stopping times} (``split stopping time'' in Dellacherie and Meyer~\\cite{dm82}) defined by\n\\[\n\\hat\\T:=\\{(\\tau,\\tilde\\tau)\\in\\T\\times\\T_p \\mid \\tilde\\tau>0,\\ \\tau\\vee\\tilde\\tau =T+ \\},\n\\]\nwhere $\\T_p$ is the set of predictable times. When $R$ is cadlag, $\\vec R=R_-$, and our formulation of the quasi-optimal stopping coincides with that of Bismut~\\cite{bis79}. Our main result gives the existence of optimal quasi-stopping times when $R\\ge\\cev R$. When $R \\ge \\cev R$ and $\\vec R\\le \\pp R$, we obtain the existence for \\eqref{os} thus extending the existence result of \\cite[Theorem~II.2]{bs77} to possibly unbounded processes $R$ as suggested already on page 301 of \\cite{bs77}. \n\nOur existence proofs are based on functional analytical arguments that avoid the use of Snell envelopes which are used in most analyses of optimal stopping. Our strategy is to first look at a convex relaxation of the problem. This turns out be a linear optimization problem over a compact convex set of random measures whose extremal points can be identified with (quasi-)stopping times. As soon as the objective is upper semicontinuous on this set, Krein-Milman theorem gives the existence of (quasi-)stopping times. Sufficient conditions for upper semicontinuity are obtained as a simple application of the main result of Perkki\\\"o and Trevino~\\cite{pt18a}. The overall approach was suggested already on page 287 of Bismut~\\cite{bis79b} in the case of optimal stopping. We extended the strategy (and provide explicit derivations) to quasi-optimal stopping for a merely right-upper semicontinuous reward process.\n\nThe last section of the paper develops a dual problem and optimality conditions for optimal (quasi-)stopping problems. The dual variables turn out to be martingales that dominate $R$. As a simple consequence, we obtain the duality result of Davis and Karatzas~\\cite{dk94} in a more general setting where the reward process $R$ is merely of class $(D)$.\n\n\n\n\n\\section{Regular processes}\\label{sec:regular}\n\n\nIn this section, the reward process $R$ is assumed to be {\\em regular}, i.e.\\ of class $(D)$ such that the left-continuous version $R_-$ and the predictable projection $\\pp R$ of $R$ are indistinguishable; see e.g.\\ \\cite{bis78} or \\cite[Remark~50.d]{dm82}. Our analysis will be based on the fact that the space of regular processes is a Banach space whose dual can be identified with optional measures of essentially bounded variation; see Theorem~\\ref{thm:reg1} below.\n\n\nThe space $M$ of Radon measures may be identified with the space $X_0$ of left-continuous functions of bounded variation on $\\reals_+$ which are constant on $(T,\\infty]$ and $x_0=0$. Indeed, for every $x\\in X_0$, there exists a unique $Dx\\in M$ such that $x_t=Dx([0,t))$ for all $t\\in\\reals$. Thus $x\\mapsto Dx$ defines a linear isomorphism between $X_0$ and $M$. The value of $x$ for $t>T$ will be denoted by $x_{T+}$. Similarly, the space $\\M^\\infty$ of optional random measures with essentially bounded total variation may be identified with the space $\\N_0^\\infty$ of adapted processes $x$ with $x\\in X_0$ almost surely and $Dx\\in\\M^\\infty$.\n\nLet $C$ the space of continuous functions on $[0,T]$ equipped with the supremum norm and let $L^1(C)$ be the space of (not necessarily adapted) continuous processes $y$ with $E\\|y\\|<\\infty$. The norm $E\\|y\\|$ makes $L^1(C)$ into a Banach space whose dual can be identified with the space $L^\\infty(M)$ of random measures whose pathwise total variation is essentially bounded. The following result is essentially from \\cite{bis78}; see \\cite[Theorem~8]{pp18c} or \\cite[Corollary~16]{pp18b}. It provides the functional analytic setting for analyzing optimal stopping with regular processes.\n\n\n\\begin{theorem}\\label{thm:reg1}\nThe space $\\R^1$ of regular processes equipped with the norm\n\\[\n\\|y\\|_{\\R^1}:=\\sup_{\\tau\\in\\T}E|y_\\tau|\n\\]\nis Banach and its dual can be identified with $\\M^\\infty$ through the bilinear form\n\\[\n\\langle y,u\\rangle =E\\int ydu.\n\\]\nThe optional projection is a continuous surjection of $L^1(C)$ to $\\R^1$ and its adjoint is the embedding of $\\M^\\infty$ to $L^\\infty(M)$. The norm of $\\R^1$ is equivalent to\n\\[\np(y):= \\inf_{z\\in L^1(C)} \\{E\\|z\\| \\mid \\op z = y\\}\n\\]\nwhich has the dual representation\n\\[\np(y)=\\sup\\{\\langle y,u\\rangle\\,|\\,\\esssup(\\|u\\|)\\le 1\\}.\n\\]\n\\end{theorem}\n\nWe first write the optimal stopping problem as\n\\[\n\\maximize\\quad \\langle R, Dx\\rangle\\quad\\ovr\\quad x\\in\\C_e,\n\\]\nwhere \n\\[\n\\C_e:=\\{x\\in\\N_0^\\infty\\,|\\, Dx\\in\\M^\\infty_+,\\ x_t\\in\\{0,1\\}\\}.\n\\]\nThe equation $\\tau(\\omega) = \\inf\\{t\\in\\reals\\mid x_t(\\omega)\\ge 1\\}$ gives a one-to-one correspondence between the elements of $\\T$ and $\\C_e$. Consider also the convex relaxation\n\\begin{equation}\\label{Rcr}\\tag{ROS}\n\\maximize\\quad \\langle R,Dx\\rangle\\quad\\ovr\\quad x\\in\\C,\n\\end{equation}\nwhere\n\\[\n\\C:=\\{x\\in\\N_0^\\infty\\,|\\, Dx\\in\\M^\\infty_+,\\ x_{T+}\\le 1\\}.\n\\]\nClearly, $\\C_e\\subset\\C$ so the optimum value of optimal stopping is dominated by the optimum value of the relaxation. The elements of $\\C$ are {\\em randomized stopping times} in the sense of Baxter and Chacon~\\cite[Section~2]{bc77}. \n\nRecall that $x\\in\\C$ is an {\\em extreme point} of $\\C$ if it cannot be expressed as a convex combination of two points of $\\C$ different from $x$.\n\n\\begin{lemma}\\label{lem:km}\nThe set $\\C$ is convex, $\\sigma(\\N_0^\\infty,\\R^1)$-compact and $\\C_e$ is the set of its extreme points.\n\\end{lemma}\n\n\\begin{proof}\nThe set $\\C$ is a closed convex set of the unit ball that $\\N_0^\\infty$ has as the dual of the Banach space $\\R^1$. The compactness thus follows from Banach-Alaoglu. It is easily shown that the elements of $\\C_e$ are extreme points of $\\C$. On the other hand, if $x\\notin\\C_e$ there exists an $\\bar s\\in(0,1)$ such that the processes\n\\[\nx^1_t:=\\frac{1}{\\bar s}[x_t\\wedge\\bar s]\\quad\\text{and}\\quad x^2_t:=\\frac{1}{1-\\bar s}[(x_t-\\bar s)\\vee 0]\n\\]\nare different elements of $\\C$. Since $x=\\bar s x^1+(1-\\bar s)x^2$, it is not an extreme point of $\\C$.\n\\end{proof}\n\nSince the function $x\\mapsto\\langle R,Dx\\rangle$ is continuous, the compactness of $\\C$ in Lemma~\\ref{lem:km} implies that the maximum in \\eqref{Rcr} is attained. The fact that the maximum is attained at a genuine stopping time follows from the characterization of the extreme points in Lemma~\\ref{lem:km} and the following variant of the Krein-Millman theorem; see e.g.~\\cite[Theorem~25.9]{cho69}.\n\n\\begin{theorem}[Bauer's maximum principle]\\label{thm:bauer}\nIn a locally convex Hausdorff topological vector space, an upper semicontinuous (usc) convex function on a compact convex set $K$ attains its maximum at an extremal point of $K$.\n\\end{theorem}\n\n\nCombining Lemma~\\ref{lem:km} and Theorem~\\ref{thm:bauer} gives the following.\n\n\\begin{theorem}\\label{thm:os}\nOptimal stopping time in \\eqref{os} exists for every $R\\in\\R^1$.\n\\end{theorem}\n\nThe above seems to have been first proved in Bismut and Skalli~\\cite[Theorem~I.3]{bs77}, which says that a stopping time defined in terms of the Snell envelope of the regular process $R$ is optimal. Their proof assumes bounded reward $R$ but they note on page~301 that it actually suffices that $R$ be of class $(D)$. The proof of Bismut and Skalli builds on the (nontrivial) existence of a Snell envelope and further limiting arguments involving sequences of stopping times. In contrast, our proof is based on elementary functional analytic arguments in the Banach space setting of Theorem~\\ref{thm:reg1}, which is of independent interest.\n\nNote that $x$ solves the relaxed optimal stopping problem if and only if $R$ is {\\em normal} to $\\C$ at $x$, i.e.\\ if $R\\in\\partial\\delta_\\C(x)$ or equivalently $x\\in\\partial\\sigma_\\C(R)$, where\n\\[\n\\sigma_\\C(R) = \\sup_{x\\in\\C}\\langle R,Dx\\rangle.\n\\]\nHere, $\\partial$ denotes the {\\em subdifferential} of a function; see e.g.\\ \\cite{roc74}. If $R$ is nonnegative, we have $\\sigma_\\C(R)=\\|R\\|_{\\R^1}$ (by Krein--Milman) and the optimal solutions of the relaxed stopping problem are simply the subgradients of the $\\R^1$-norm at $R$.\n\n\n\n\\section{Cadlag processes}\\label{sec:cadlag}\n\n\n\nThis section extends the previous section to optimal quasi-stopping problems when the reward process $R$ is merely {\\em cadlag and of class $(D)$}. In this case, optimal stopping times need not exist (see the discussion on page 1) but we will prove the existence of a quasi-stopping time by functional analytic arguments analogous to those in Section~\\ref{sec:regular}.\n\nThe Banach space of cadlag functions equipped with the supremum norm will be denoted by $D$. The space of purely discontinuous Borel measures will be denoted by $\\tilde M$. The dual of $D$ can be identified with $M\\times\\tilde M$ through the bilinear form\n\\[\n\\langle y,(u,\\tilde u)\\rangle := \\int ydu + \\int y_-d\\tilde u\n\\]\nand the dual norm is given by \n\\[\n\\sup_{y\\in D}\\left\\{\\left.\\int ydu + \\int y_-d\\tilde u\\,\\right|\\,\\|y\\|\\le 1\\right\\}=\\|u\\|+\\|\\tilde u\\|,\n\\]\nwhere $\\|u\\|$ denotes the total variation norm on $M$. This can be deduced from \\cite[Theorem~1]{pes95} or seen as the deterministic special case of \\cite[Theorem~VII.65]{dm82} combined with \\cite[Remark~VII.4(a)]{dm82}.\n\nThe following result from \\cite{pp18b} provides the functional analytic setting for analyzing quasi-stopping problems with cadlag processes of class $(D)$.\n\n\\begin{theorem}\\label{thm:L1}\nThe space $\\D^1$ of optional cadlag processes of class $(D)$ equipped with the norm\n\\[\n\\|y\\|_{\\D^1}:=\\sup_{\\tau\\in\\T}E|y_\\tau|\n\\]\nis Banach and its dual can be identified with \n\\[\n\\hat\\M^\\infty:=\\{(u,\\tilde u)\\in L^\\infty(M\\times\\tilde M)\\mid u\\text{ is optional},\\, \\tilde u\\text{ is predictable}\\}\n\\]\nthrough the bilinear form\n\\[\n\\langle y,(u,\\tilde u)\\rangle =E\\left[\\int ydu+\\int y_-d\\tilde u\\right].\n\\]\nThe optional projection is a continuous surjection of $L^1(D)$ to $\\D^1$ and its adjoint is the embedding of $\\hat\\M^\\infty$ to $L^\\infty(M\\times \\tilde M)$. The norm of $\\D^1$ is equivalent to\n\\[\n p(y):= \\inf_{z\\in L^1(D)} \\{E\\|z\\| \\mid \\op z = y\\},\n\\]\nwhich has the dual representation\n\\[\np(y)=\\sup\\{\\langle y,(u,\\tilde u)\\rangle\\,|\\,\\esssup(\\|u\\|+\\|\\tilde u\\|)\\le 1\\}.\n\\]\n\\end{theorem}\n\nThe space $M\\times\\tilde M$ may be identified with the space $\\hat X_0$ of (not necessarily left-continuous) functions $x:\\reals_+\\to\\reals$ of bounded variation which are constant on $(T,\\infty]$ and have $x_0=0$. Indeed, every $x\\in\\hat X_0$ can be written uniquely as\n\\[\nx_t = Dx([0,t)) + \\tilde Dx([0,t]),\n\\]\nwhere $\\tilde Dx\\in\\tilde M$ and $Dx\\in M$ are the measures associated with the functions $\\tilde x_t :=\\sum_{s\\le t} (x_s-x_{s-})$ and $x-\\tilde x$, respectively. The linear mapping $x\\mapsto(Dx,\\tilde Dx)$ defines an isomorphism between $\\hat X_0$ and $M\\times\\tilde M$. The value of $x$ for $t>T$ will be denoted by $x_{T+}$. Similarly, the space $\\hat \\M^\\infty$  may be identified with the space $\\hat\\N_0^\\infty$ of predictable processes $x$ with $x\\in \\hat X_0$ almost surely and $(Dx,\\tilde Dx)\\in\\hat\\M^\\infty$.\n\n\nProblem \\eqref{oqs} can be written as\n\\[\n\\maximize\\quad \\langle R, (Dx,\\tilde Dx)\\rangle\\quad\\ovr\\quad x\\in\\hat\\C_e,\n\\]\nwhere \n\\[\n\\hat\\C_e:=\\{x\\in\\hat\\N_0^\\infty\\,|\\, (Dx,\\tilde Dx)\\in\\hat\\M^\\infty_+,\\ x_t\\in\\{0,1\\}\\}.\n\\]\nIndeed, the equations $\\tau(\\omega) = \\inf\\{t\\in\\reals\\mid x_t(\\omega)\\ge 1\\}$ and $\\tilde\\tau(\\omega) = \\inf\\{t\\in\\reals\\mid x_t-x_{t-}(\\omega)\\ge 1\\}$ give a one-to-one correspondence between the elements of $\\hat\\T$ and $\\hat\\C_e$.\n\nConsider also the convex relaxation\n\\begin{equation}\\label{Dcr}\\tag{ROQS}\n\\maximize\\quad \\langle R,(Dx,\\tilde Dx)\\rangle\\quad\\ovr\\quad x\\in\\hat\\C,\n\\end{equation}\nwhere \n\\[\n\\hat\\C:=\\{x\\in\\hat\\N_0^\\infty\\,|\\, (Dx,\\tilde Dx)\\in\\hat\\M^\\infty_+,\\ x_{T+}\\le 1\\}.\n\\]\n\n\\begin{lemma}\\label{lem:cadkm}\nThe set $\\hat\\C$ is convex, $\\sigma(\\hat \\M^\\infty,\\D^1)$-compact and the set of quasi-stopping times $\\hat \\C_e$ is its extreme points. Moreover, the set of stopping times is $\\sigma(\\hat \\M^\\infty,\\D^1)$-dense in $\\hat \\C_e$ and, thus, $\\C$ is $\\sigma(\\hat \\M^\\infty,\\D^1)$-dense in $\\hat\\C$.\n\\end{lemma}\n\n\\begin{proof}\nThe set $\\hat\\C$ is a closed convex set of the unit ball that $\\hat\\N_0^\\infty$ has as the dual of the Banach space $\\D^1$. The compactness thus follows from Banach-Alaoglu. It is easily shown that the elements of $\\hat\\C_e$ are extreme points of $\\hat\\C$.\n\nIf $x\\notin\\hat\\C_e$, there exist $\\bar s\\in(0,1)$ such that\n\\begin{align*}\nx^1_t &:=\\frac{1}{\\bar s }[x_t\\wedge\\bar s],\\quad\\quad x^2_t:=\\frac{1}{1-\\bar s}[(x_t-\\bar s)\\vee 0]\n\\end{align*}\nare distinguishable processes that belong to $\\hat\\C$. Since $x=\\bar s x^1+(1-\\bar s)x^2$, $x$ is not an extremal in $\\hat C$.\n\nTo prove the last claim, let $(\\tau,\\tilde\\tau)$ be a quasi-stopping time and $(\\tau^\\nu)$ an announcing sequence for $\\tilde\\tau$. We then have\n\\[\n\\langle(\\delta_{\\tau\\wedge\\tau^\\nu},0),y\\rangle\\to\\langle(\\delta_\\tau,\\delta_{\\tilde\\tau}),y\\rangle\n\\]\nfor every $y\\in\\D^1$.\n\\end{proof}\n\nJust like in Section~\\ref{sec:regular}, a combination of Lemma~\\ref{lem:cadkm} and Theorem~\\ref{thm:bauer} gives the following existence result which was established in Bismut~\\cite{bis79} using more elaborate techniques based on the existence of Snell envelopes.\n\n\\begin{theorem}\\label{thm:os}\nIf $R\\in\\D^1$, then optimal quasi-stopping time in \\eqref{oqs} exists and the optimal values of \\eqref{oqs}, \\eqref{os} and \\eqref{Dcr} are all equal.\n\\end{theorem}\n\nAs another implication of Lemma~\\ref{lem:cadkm} and Theorem~\\ref{thm:L1}, we recover the following result of Bismut which says that the seminorms in Theorem~\\ref{thm:L1} are not just equivalent but equal\n\n\\begin{theorem}[{\\cite[Theorem~4]{bis78}}]\\label{thm:equiv1}\nFor every $y\\in\\D^1$,\n\\[\n\\|y\\|_{\\D^1}=\\inf_{z\\in L^1(D)}\\{E\\|z\\|_D\\mid \\op z = y\\}.\n\\]\n\\end{theorem} \n\n\\begin{proof}\nThe expression on the right is the seminorm $p$ in Theorem~\\ref{thm:L1} with the dual representation \n\\[\np(y)=p(|y|)=\\sup_{x\\in\\hat\\C}\\langle |y|,(Dx,\\tilde Dx)\\rangle\n\\]\nwhich, by Theorem~\\ref{thm:os}, equals the left side.\n \n\\end{proof}\n\nCombining the above with Theorem~\\ref{thm:reg1} gives a simple proof of the following.\n\n\\begin{theorem}[{\\cite[Theorem~3]{bis78}}]\\label{thm:equivreg}\nFor every $y\\in\\R^1$,\n\\[\n\\|y\\|_{\\R^1}=\\inf_{z\\in L^1(C)}\\{E\\|z\\|_D\\mid \\op z = y\\}.\n\\]\n\\end{theorem} \n\n\\begin{proof}\nBy Jensen's inequality, the left side is less than the right which is the seminorm $p$ in Theorem~\\ref{thm:reg1} with the dual representation\n\\begin{align*}\n  p(y) &= \\sup\\{\\langle y,u\\rangle\\,|\\,\\esssup(\\|u\\|)\\le 1\\}\\\\\n  &\\le \\sup\\{\\langle y,(u,\\tilde u)\\rangle\\,|\\,\\esssup(\\|u\\|+\\|\\tilde u\\|)\\le 1\\}\\\\\n  &=\\sup_{x\\in\\hat\\C}\\langle |y|,(Dx,\\tilde Dx)\\rangle,\n\\end{align*}\nwhich, again by Theorem~\\ref{thm:os}, equals the left side.\n\\end{proof}\n\n\n\\section{Non-cadlag processes}\\label{sec:usc}\n\n\n\nThis section gives a further extension to cases where the reward process is not necessarily cadlag but merely {\\em right-upper semicontinuous} (right-usc) in the sense that $R\\ge\\cev R$. In this case, the objective of the relaxed quasi-optimal stopping problem \\eqref{Dcr} need not be continuous. The following lemma says that it is, nevertheless, upper semicontinuous, so Bauer's maximum principle still applies.\n\n\\begin{lemma}\\label{lem:usc}\nIf $R$ is right-usc and of class $(D)$, then the functional \n\\begin{align*}\n\\hat\\J(u,\\tilde u)=\\begin{cases} \nE\\left[\\int Rdu + \\int \\vec R d\\tilde u\\right]\\quad &\\text{if }(u,\\tilde u)\\in\\hat\\M^\\infty_+\\\\\n-\\infty\\quad&\\text{otherwise}\n\\end{cases}\n\\end{align*}\nis $\\sigma(\\hat\\M^\\infty,\\D^1)$-usc.\n\\end{lemma}\n\n\\begin{proof}\nRecalling that every optional process of class $(D)$ has a majorant in $\\D^1$ (see \\cite[Remark 25, Appendix I]{dm82}), the first example in  \\cite[Section~8]{pt18a} shows, with obvious changes of signs, that $\\hat\\J$ is usc. \n\\end{proof}\n\nCombining Lemma~\\ref{lem:usc} with Theorem~\\ref{thm:bauer} gives the existence of a relaxed quasi-stopping time at an extreme point of $\\C$ which, by Lemma~\\ref{lem:cadkm}, is a quasi-stopping time. We thus obtain the following.\n\n\n\\begin{theorem}\\label{thm:usc}\nIf $R$ is right-usc and of class $(D)$, then \\eqref{oqs} has a solution. \n\\end{theorem}\n\nWe have not been able find the above result in the literature but it can be derived from Theorem~2.39 of El Karoui~\\cite{elk81} on ``divided stopping times'' (temps d'arret divis\\'es). A recent analysis of divided stopping times can be found in Bank and Besslich \\cite{bb18a}. These works extend Bismut's approach on optimal quasi-stopping by dropping the assumption of right-continuity and augmenting quasi-stopping times with a third component that acts on the right limit of the reward process. Much like Bismut's approach, \\cite{elk81,bb18a} build on the existence of a Snell envelope.\n\nTheorem~\\ref{thm:usc} yields the existence of an optimal stopping time when the reward process $R$ is {\\em subregular} in the sense that it is right-usc, of class $(D)$ and $\\vec R\\le\\pp R$.\n\n\n\n\\begin{theorem}\\label{thm:uscreg}\nIf $R$ is subregular, then \\eqref{os} has a solution and its optimum value equals that of \\eqref{oqs}.\n\\end{theorem}\n\n\\begin{proof}\nClearly, the optimum value of \\eqref{oqs} is at least that of \\eqref{os} while for subregular $R$,\n\\[\nE[R_\\tau+\\vec R_{\\tilde\\tau}] \\le E[R_\\tau+\\pp R_{\\tilde\\tau}] = E[R_\\tau+R_{\\tilde\\tau}] = ER_{\\tau\\wedge\\tilde\\tau},\n\\]\nwhere the first equality holds by the definition of predictable projection. The claim now follows from Theorem~\\ref{thm:usc}.\n\\end{proof}\n\n\nThe above seems to have been first established in Bismut and Skalli~\\cite[Section~II]{bs77} for bounded $R$ (again, they mention on page 301 that, instead of boundedness, it would suffice to assume that $R$ is of class $(D)$).\n\nRegularity properties are preserved under compositions with convex functions much like martingale properties. Indeed, if $R$ is regular and $g$ is a real-valued convex function on $\\reals$ then $g(R)$ is subregular as soon as it is of class $(D)$. Indeed, for any $\\tau\\in\\T_p$, conditional Jensen's inequality gives\n\\[\nE[g(\\vec R_\\tau)\\one_{\\tau<+\\infty}]= E[g(\\pp R_\\tau)\\one_{\\tau<+\\infty}] \\le E[g(R_\\tau)\\one_{\\tau<+\\infty}].\n\\]\nSimilarly, if $R$ is subregular and $g$ is a real-valued increasing convex function, then $g(R)$ is subregular as soon as the composition is of class $(D)$.\n\n\n\n\n\n\\section{Duality}\n\n\n\nWe end this paper by giving optimality conditions and a dual problem for the optimal stopping problems. The derivations are based on the conjugate duality framework of~\\cite{roc74} which addresses convex optimization in general locally convex vector spaces.\nThe results below establish the existence of dual solutions without assuming the existence of optimal (quasi-)stopping times. They hold without any path properties as long as the reward process $R$ is of class $(D)$.\n\nWe denote the space of martingales of class $(D)$ by $\\R^1_m$.\n\n\\begin{theorem}\\label{thm:osdual}\nLet $R$ be of class $(D)$. Then the optimum values of \\eqref{oqs} and \\eqref{os} coincide and equal that of\n\\begin{equation}\\label{d}\\tag{DOS}\n\\inf\\{EM_0 \\mid M\\in\\R^1_m,\\ R\\le M\\},\n\\end{equation}\nwhere the infimum is attained.\n\nMoreover, $x\\in\\hat\\C$ is optimal in the convex relaxation of \\eqref{oqs} if and only if there exists $M\\in\\R^1_m$ with $R\\le M$ and\n\\begin{align}\n\\int (M-R)dx+\\int(M_--\\vec R)d\\tilde x &= 0,\\label{eq:oc1}\\\\ \nx_{T+}=1\\quad\\text{or}\\quad M_T&=0\\label{eq:oc2}\n\\end{align}\nalmost surely. Thus, $(\\tau,\\tilde\\tau)\\in\\hat\\T$ is optimal in \\eqref{oqs} if and only if there exists $M\\in\\R^1_m$ with $R\\le M$, $M_\\tau=R_\\tau$, $M_{\\tilde \\tau_-} =\\vec R_{\\tilde\\tau}$ and almost surely either $\\tau+\\tilde\\tau<\\infty+$ or $M_T=0$.\n\nIn particular, $x\\in\\C$ is optimal  in the convex relaxation of \\eqref{os} if and only if there exists $M\\in\\R^1_m$ with $R\\le M$ and\n\\begin{align*}\n\\int (M-R)dx&= 0,\\\\ \nx_{T+}=1\\quad\\text{or}\\quad M_T&=0\n\\end{align*}\nalmost surely. Thus, $\\tau\\in\\T$ is optimal in \\eqref{os} if and only if there exists $M\\in\\R^1_m$ with $R\\le M$, $M_\\tau=R_\\tau$ and almost surely either $\\tau<\\infty+$ or $M_T=0$.\n\\end{theorem}\n\n\\begin{proof}\nBy \\cite[Remark~25, Appendix~I]{dm82}, there are measurable processes $z$ and $\\tilde z$ such that $R =\\op z$, $\\vec R=\\op{\\tilde z}$ and $E[\\sup_tz_t+\\sup_t\\tilde z_t]<\\infty$. The optimum value and optimal solutions of \\eqref{oqs} coincide with those of\n\\begin{align}\\label{eq:ros}\n&\\maximize_{x\\in\\hat\\N^\\infty}\\quad E\\left[\\hat \\J(Dx,\\tilde Dx)   - \\rho(x_{T+}-1)^+\\right],\n\\end{align}\nwhere $\\rho:=\\sup_tz_t+\\sup_t\\tilde z_t+1$ and $\\hat\\J$ is defined as in Lemma~\\ref{lem:usc}. Indeed, if $x$ is feasible in \\eqref{eq:ros} then $\\bar x:=x\\wedge 1$ is feasible in \\eqref{oqs} and since $x-\\bar x$ is an increasing process with $(x-\\bar x)_{T+}=(x_{T+}-1)^+$, we get \n\\begin{align*}\n\\hat\\J(D\\bar x,\\tilde D\\bar x) &= \\hat\\J(Dx,\\tilde Dx) - \\hat\\J(D(x-\\bar x),\\tilde D(x-\\bar x))\\\\\n&\\ge \\hat\\J(Dx,\\tilde Dx) - E\\rho(x_{T+}-1)^+.\n\\end{align*}\n\nProblem~\\eqref{eq:ros} fits the general conjugate duality framework of \\cite{roc74} with $U=L^\\infty$, $Y=L^1$ and\n\\[\nF(x,w) = -\\hat\\J(Dx,\\tilde Dx) + E\\rho(x_{T+}+w-1)^+.\n\\]\nBy \\cite[Theorem~22]{roc74}, $w\\to F(0,w)$ is continuous on $L^\\infty$ in the Mackey topology that it has as the dual of $L^1$. Thus, by \\cite[Theorem~17]{roc74}, the optimum value of \\eqref{eq:ros} coincides with the infimum of the dual objective \n\\[\ng(y) :=-\\inf_{x\\in\\hat\\N^\\infty} L(x,y),\n\\]\nwhere $L(x,y) :=\\inf_{w\\in L^\\infty}\\{F(x,w)-Ewy\\}$, and moreover, the infimum of $g$ is attained. By the interchange rule \\cite[Theorem~14.60]{rw98},\n\\begin{align*}\nL(x,y)&=\n\\begin{cases}\n+\\infty & \\text{if $x\\notin\\hat\\N^\\infty_+$},\\\\\n-\\hat\\J(Dx,\\tilde Dx) + E\\left[\\inf_{u\\in\\reals}\\{\\rho(x_{T+}+u-1)^+ - uy\\}\\right]&\\text{otherwise}\\\\\n\\end{cases}\\\\\n&=\n\\begin{cases}\n+\\infty & \\text{if $x\\notin\\hat\\N^\\infty_+$},\\\\\n-\\hat\\J(Dx,\\tilde Dx)+ E\\left[x_{T+}y-y - \\delta_{[0,\\rho]}(y)\\right]&\\text{otherwise}.\n\\end{cases}\n\\end{align*}\nWe have\n\\[\nE[x_{T+}y] = E[\\int(y\\one)dx+\\int(y\\one)d\\tilde x] = \\langle M,(Dx,\\tilde Dx)\\rangle,\n\\]\nwhere $M=\\op(y\\one)\\in\\R^1_m$. Thus,\n\\[\nL(x,y) = \n\\begin{cases}\n+\\infty & \\text{if $x\\notin\\hat\\N^\\infty_+$},\\\\\n-\\hat\\J(Dx,\\tilde Dx) + \\langle M,(Dx,\\tilde Dx)\\rangle - EM_T & \\text{if $x\\in\\hat\\N^\\infty_+$ and $0\\le M_T\\le\\rho$,}\\\\\n-\\infty & \\text{otherwise}.\n\\end{cases}\n\\]\nThe dual objective can be written as\n\\begin{align*}\ng(y) &= \n\\begin{cases}\nEM_0 & \\text{if $0\\le M_T\\le\\rho$, $M\\ge R$ and $M_-\\ge\\vec R$},\\\\\n+\\infty & \\text{otherwise}.\n\\end{cases}\n\\end{align*}\nSince $M$ is cadlag, $M_-\\ge\\vec R$ holds automatically when $M\\ge R$. In summary, the optimum value of \\eqref{oqs} equals that of \\eqref{d}.\n\nThe dual problem of \\eqref{os} is obtained similarly by defining\n\\[\nF(x,w) = -\\J(Dx) + E\\rho(x_{T+}+w-1)^+.\n\\]\nThe function $w\\to F(0,w)$ is again Mackey-continuous on $L^\\infty$ and one finds that the dual is again \\eqref{d}. Thus, the optimum value of \\eqref{os} equals that of \\eqref{d}.\n\nAs to the optimality conditions, \\cite[Theorem~15]{roc74} says that $x$ is optimal in \\eqref{eq:ros} and $y$ is optimal in the dual if and only i\n\\[\n0\\in\\partial_x L(x,y),\\quad 0\\in\\partial_y[-L](x,y).\n\\]\nThe former means that $x\\in\\hat\\N^\\infty_+$, $M\\ge R$ and \n\\[\n\\int (M-R)dx=0, \\quad \\int (M_--\\vec R)d\\tilde x=0\\quad P\\text{-a.s.}\n\\]\nBy the interchange rule for subdifferentials (\\cite[Theorem~21c]{roc74}), the latter is equivalent to \\eqref{eq:oc2}.\n\\end{proof}\n\n\nNote that for any martingale $M\\in\\R^1_m$,\n\\[\n\\sup_{\\tau\\in\\T}ER_\\tau = \\sup_{\\tau\\in\\T}E(R_\\tau+M_T-M_{\\tau})\\le E\\sup_{t\\in[0,T]}(R_t+M_T-M_t),\n\\]\nwhere the last expression is dominated by $EM_0$ if $R\\le M$. Thus,\n\\begin{align*}\n\\sup_{\\tau\\in\\T}ER_\\tau &\\le \\inf_{M\\in\\R^1_m}E\\sup_{t\\in[0,T]}(R_t+M_T-M_t)\\\\\n&\\le \\inf_{M\\in\\R^1_m}\\{E\\sup_{t\\in[0,T]}(R_t+M_T-M_t)\\,|\\,R\\le M\\}\\\\\n&\\le \\inf_{M\\in\\R^1_m}\\{EM_0\\,|\\,R\\le M\\},\n\\end{align*}\nwhere, by Theorem~\\ref{thm:os}, the last expression equals the first one as soon as $R$ is of class $(D)$. The optimum value of the stopping problem then equals\n\\[\n\\inf_{M\\in\\R^1_m}E\\sup_{t\\in[0,T]}(R_t+M_T-M_t).\n\\]\nThis is the dual problem derived in Davis and Karatzas~\\cite{dk94} and Rogers~\\cite{rog2}. Note also that if $Y$ is the Snell envelope of $R$ (the smallest supermartingale that dominates $R$), then the martingale part $M$ in the Doob--Meyer decomposition $Y=M-A$ is dual optimal. These facts were obtained in \\cite{dk94} and \\cite{rog2} under the assumptions that $\\sup_tR_t$ is integrable.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Supporting information}\n\\subsection{Hydrodynamic transport regime}\nThe hydrodynamic equations for electron drift velocity $\\bf u$, chemical potential $\\mu$ and temperature $T$ can be obtained by supplying the local-equilibrium distribution function\n\\begin{equation}\n    f_0 = [1+e^{(\\epsilon_p - {\\bf p u}_0 - \\mu)\/T}]^{-1}\n\\end{equation}\ninto kinetic equation and integrating it by phase space multiplied by $1$, $\\bf p$ and $\\epsilon_{\\bf p}$. Though a more accurate derivation with explicit account of collisions is possible (see section II), such simplistic derivation may be convenient for analysis of nonlinear effects. The resulting equations can be presented as\n\\begin{gather}\n    \\partial_t n + \\partial_{\\bf r}(n{\\bf u})=0,\\\\\n    \\partial_t(\\rho u_i) + \\partial_{x_j}\\Pi_{ij} = e n \\partial_{x_i}\\varphi,\\\\\n    \\partial_t\\varepsilon + \\partial_{\\bf r}(\\rho v_0^2 {\\bf u}) - e n {\\bf u}{\\partial_{\\bf r}\\varphi} = 0. \n\\end{gather}\n\nAbove, $n$ is the density of electrons, $\\rho$ is the equivalent of mass density, $\\varepsilon$ is the internal energy density, and $\\Pi_{ij}$ is the stress tensor. At given value of chemical potential $\\mu$, all these quantities depend on drift velocity $\\beta = u \/ v_0$. At the same time, they can be expressed via their values in the absence of drift, i.e. at $\\beta = 0$:\n\\begin{gather}\n\\label{Eqs-of-state}\n    n = \\frac{n_{\\beta = 0}}{[1-\\beta^2]^{3\/2}},\\qquad     \\rho = \\frac{\\rho_{\\beta = 0}}{[1-\\beta^2]^{5\/2}},\\\\\n    \\varepsilon = \\varepsilon_{\\beta = 0} \\frac{1 + \\beta^2\/2}{[1-\\beta^2]^{5\/2}},\\\\ \\rho_{\\beta = 0}v_0^2 = \\frac{3}{2}\\varepsilon_{\\beta = 0}\n    \\Pi_{xx} = \\frac{\\varepsilon_{\\beta = 0}}{2}\\frac{1+2\\beta^2}{[1-\\beta^2]^{5\/2}},\\qquad \\Pi_{yy} = \\frac{\\varepsilon_{\\beta = 0}}{2}\\frac{1 - \\beta^2}{[1-\\beta^2]^{5\/2}}\n\\end{gather}\n\nTo avoid dealing with 'relativistic factors' of the type $[1-\\beta^2]^{\\alpha}$, it is convenient to consider density $n$, mass density $\\rho$ and velocity $\\beta$ as hydrodynamic variables. In these variables, the Euler and heat balance equations take on the closed form (assuming one-dimensional motion along $x$-axis)\n\\begin{gather}\n    \\partial_t(\\rho u_x) + \\frac{1}{3}\\partial_{x}[\\rho  (v_0^2 + 2 u^2)] = e n \\partial_{x}\\varphi,\\\\\n   \\frac{2}{3} \\partial_t(\\rho  [v_0^2 + u^2\/2]) + \\partial_{\\bf r}(\\rho v_0^2 {\\bf u}) - e n {\\bf u}{\\partial_{\\bf r}\\varphi} = 0. \n\\end{gather}\nThe equations can now be linearized to find the conductivity of drifting Dirac electrons in the hydrodynamic regime, $n = n_0 + \\delta ne^{i(kx-\\omega t)}$, $u = u_0 + \\delta u e^{i(kx-\\omega t)}$, $\\rho = \\rho_0 + \\delta \\rho e^{i(kx-\\omega t)}$. This results in the system of equations $\\hat M_{\\rm hd}\\delta{\\bf x} = \\delta {\\bf F}_{\\rm hd}$, with the matrix and right-hand side given by\n\\begin{gather}\n\\label{HD-matrix-2}\n \\hat{M}_{\\rm hd}=\\left( \\begin{matrix}\n   -i(\\omega - q u_0) & i q & 0  \\\\\n   0 & -i (\\omega - \\frac{4}{3}qu_0) & - i (\\beta_0\\omega - \\frac{q}{3})  \\\\\n   0 & - i (\\beta_0 \\omega - \\frac{3}{2}q) & -i(\\omega - \\frac{3}{2}qu_0)  \\\\\n\\end{matrix} \\right) \\\\     \n\\label{HD_forces-2}\n  {\\delta {\\bf F}}_{\\rm hd}=\\frac{e\\delta \\varphi }{m v_{0}^{2}}\n      \\left( \n  \\begin{matrix}\n   0 \\\\\n   i q v_0  \\\\\n  \\frac{3}{2}i q u_0  \\\\\n\\end{matrix}\n \\right)  \n\\end{gather}\nthe hydrodynamic mass $m$ si given by\n\\begin{equation}\n    m = \\frac{\\rho_0}{n_0} \\approx \\frac{\\varepsilon_F}{v_0^2 - u_0^2}.\n\\end{equation}\n\nAs a final step, it is possible to solve the system (\\ref{HD-matrix-2}-\\ref{HD_forces-2}) for density $\\delta n$ and velocity $\\delta u$ to find the polarizability and conductivity\n\\begin{gather}\n\\label{HD_polarization}\n    \\Pi = -\\frac{n q^2 (1-\\beta_0^2)}{m \\left[\\omega^2(1-\\frac{\\beta_0^2}{2}) - \\frac{q^2v_0^2}{2}(1-2\\beta^2) - 2 q u_0 \\omega\\right]},\\\\\n    \\sigma = -\\frac{ie^2\\omega}{q^2}\\Pi.\n\\end{gather}\n\nAt zero drift velocity, the polarzation of graphene acquires a simple form common for bulk solids:\n\\begin{equation}\n    \\Pi = -\\frac{n q^2 \/m}{\\omega^2 - v_s^2 q^2},\n\\end{equation}\nwhere $v_s = v_0\/\\sqrt{2}$ is the sound velocity.\n\n\n\\subsection{Polarizability and conductivity for drifting electrons at the hydrodynamic-to-ballistic crossover}\nThe solution of kinetic equation with electron-electron collisions reads\n\\begin{equation}\n\\label{Distribution_funtion}\n    \\delta f = \\frac{i\\nu_{ee}\\delta f_{\\rm hd} - i e\\delta\\varphi {\\bf q} \\partial_{\\bf p}f_0}{\\omega+i\\nu_{ee}-{\\bf qv_p}},\n\\end{equation}\nwhere the perturbed hydrodynamic distribution reads\n\\begin{equation}\n    \\delta f_{hd} = \\delta\\mu \\partial_{\\mu}f_0 + \\delta {\\bf u} \\partial_{\\bf u}f_0 + \\delta T \\partial_{T}f_0,\n\\end{equation}\nand the unperturbed drifting distribution is \n\\begin{equation}\n    f_0 = \\left[1+\\exp\\left\\{ \\frac{\\epsilon_p - {\\bf p u}_0 - \\mu}{T}\\right\\}\\right]^{-1}.\n\\end{equation}\nThe local-equilibrium Fermi energy $\\mu$, drift velocity ${\\bf u}_0$ and temperature $T$ should be determined from the solution of dc transport equations from the known drain and gate voltages. We leave this solution for further work. The variations of local-equilibrium parameters due to ac field $\\delta \\mu$, $\\delta {\\bf u}$ and $\\delta T$ should be obtained by requiring particle, momentum and energy conservation upon e-e collisions:\n\\begin{equation}\n\\label{Conservation_laws}\n    \\sum_{\\bf p}{[\\delta f- \\delta f_{\\rm hd}]} = 0, \\qquad     \n    \\sum_{\\bf p}{{\\bf p}[\\delta f- \\delta f_{\\rm hd}]} = 0, \\qquad\n        \\sum_{\\bf p}{\\epsilon_{\\bf p}[\\delta f- \\delta f_{\\rm hd}]} = 0.\n\\end{equation}\nSubstituting the distribution function (\\ref{Distribution_funtion}) into conservation laws (\\ref{Conservation_laws}), we can obtain a set of generalized hydrodynamic equations for determination of Fermi energy, drift velocity and temperature. In the course of evaluation, one encounters the following integrals\n\\begin{equation}\n\\sum_{\\bf p}{\\frac{p^{m-1} \\cos^n\\theta_{\\bf p} f_0({\\bf p})}{a-\\cos\\theta_{\\bf p}} },\\qquad a=\\frac{\\omega+i\\nu_{ee}}{qv_0}.    \n\\end{equation}\nAnisotropy of $f_0({\\bf p})$ introduces complications that can be handled via the change of momentum\n\\begin{equation}\n    \\tilde p = p(1 - u_0\/v_0 \\cos\\theta_{\\bf p}),\n\\end{equation}\nafter which the integrals over momentum modulus and angle are decoupled\n\\begin{equation}\n\\sum_{\\bf p}{\\frac{p^{m-1} \\cos^n\\theta_{\\bf p} f_0({\\bf p})}{a-\\cos\\theta_{\\bf p}} } = \\frac{1}{(2\\pi)^2}\\int_0^{\\infty}{d\\tilde p \\tilde p^m f_F(\\tilde p)} \\int_0^{2\\pi}{\\frac{\\cos^n\\theta d\\theta}{(a-\\cos\\theta)(1-\\beta \\cos\\theta)^m}}.    \n\\end{equation}\nAbove $f_F$ is already an isotropic Fermi function with the same values of $\\mu$, $T$ and zero drift velocity. We introduce the notation\n\\begin{equation}\n    J_{nm} = \\frac{1}{2\\pi}\\int\\limits_0^{2\\pi}{\\frac{\\cos^n\\theta d\\theta}{(a-\\cos\\theta)(1-\\beta \\cos\\theta)^m}}.\n\\end{equation}\nThe integral is evaluated by setting $z=e^{i\\theta}$ and computing the residues in the poles inside the unit circle $|z|=1$. The remaining integrals over momentum modulus can be expressed via electron density $n$ and energy density $\\varepsilon$ at zero drift velocity $\\beta\\equiv u_0\/v_0$. This results in generalized hydrodynamic system:\n\\begin{gather}\n    \\delta n = \\frac{i\\nu_{ee}}{qv_0}\\delta[n_{\\beta = 0} J_{02}] + i \\frac{e \\delta\\varphi}{qv_0} \\frac{\\partial n_{\\beta = 0}}{\\partial \\mu} [qv_0  J_{12} - {\\bf qu}_0 J_{02} ],\\\\\n    \\delta{{\\bf P}v_0} = \\frac{i \\nu_{ee}}{qv_0} \\delta[\\varepsilon_{\\beta = 0} J_{13}]+ i \\frac{e \\delta\\varphi}{qv_0} 2n_{\\beta = 0} [qv_0  J_{23} - {\\bf qu}_0 J_{13} ],\\\\\n    \\delta{\\varepsilon} = \\frac{i \\nu_{ee}}{qv_0} \\delta[\\varepsilon_{\\beta = 0} J_{03}]+ i \\frac{e \\delta\\varphi}{qv_0} 2n_{\\beta = 0} [qv_0  J_{13} - {\\bf qu}_0 J_{03} ].\n\\end{gather}\n\nFinally, to justigy the form (\\ref{HD-matrix}-\\ref{HD_forces}) of the main text, we express the quantities $n_{\\beta=0}$ and $\\varepsilon_{\\beta=0}$ through $n$ and $\\varepsilon$ at finite drift velocity using the equations of state (\\ref{Eqs-of-state}). We also introduce another set of dimensionless integrals\n\\begin{equation}\n    I_{nm} = (1-\\beta^2)^{m-1\/2}J_{nm},\n\\end{equation}\nand inverse Knudsen number $\\tilde \\gamma_{ee} = (qv_0 \\tau_{ee})^{-1}$. Generally, the right-hand side of generalized hydrodynamic equations has the form \n\\begin{equation}\n  {\\delta {\\bf F}}=-2e\\delta \\varphi \n      \\left( \n  \\begin{matrix}\n   \\frac{I_{12}-{\\beta_0}{I_{02}}}{m_{k,\\beta=0} v_{0}^{2}}  \\\\\n   \\frac{I_{23}-{\\beta_0}{I_{13}}}{m_{hd,\\beta=0} v_{0}^{2}}  \\\\\n  \\frac{3}{2}\\frac{{I}_{13}-{\\beta_0}{I_{03}}}{m_{hd,\\beta=0} v_{0}^{2}}  \\\\\n\\end{matrix}\n \\right)\n\\end{equation}\nwhere the 'kinetic' and 'hydrodynamic' masses have the form\n\\begin{equation}\n    m_{k,\\beta=0} = \\frac{n_{\\beta=0}}{v_0^2 \\partial n_{\\beta=0}\/\\partial\\mu},\\qquad m_{hd,\\beta=0} = \\frac{\\rho_{\\beta=0}}{n_{\\beta=0}}.\n\\end{equation}\nIn the main text, we neglect the difference between these masses which is justified in the degenerate limit $T\/\\epsilon_F \\ll 1$. A small difference between these masses may result in extra plasmon damping due to relaxation of velocity modes by e-e collisions in non-parabolic bands~\\cite{Crossover}.\n\nThe formal solution for polarizability is readily derived from generalized hydrodynamic system~(\\ref{HD-matrix}-\\ref{HD_forces}). Denoting the elements of hydrodynamic matrix as $M_{ij}$ and components of generalized force vector as $\\delta F_i$, we find\n\\begin{equation}\n    \\Pi (q,\\omega) = \\frac{1}{e\\delta\\varphi}\\frac{\\delta F_1}{M_{11}} +\\frac{M_{12}}{M{11}}\\frac{M_{33}\\delta F_2 - M_{23}\\delta F_3}{M_{23}M_{32} - M_{22}M_{33}}\n\\end{equation}\n\n\\subsection{Evaluation of auxiliary spatially dispersive integrals}\nUpon obtaining the generalized hydrodynamic equations, on encounters the following integrals\n\\begin{equation}\n    J_{nm} = \\frac{1}{2\\pi}\\int\\limits_0^{2\\pi}{\\frac{\\cos^n\\theta d\\theta}{(a-\\cos\\theta)(1-\\beta \\cos\\theta)^m}}.\n\\end{equation}\nAt any given integers $n$ and $m$, they are evaluated by passing to the complex variable $z=e^{i\\theta}$. This results in\n\\begin{equation}\n    J_{nm} = \\frac{1}{2\\pi i}\\int\\limits_{|z|=1}{\\frac{(z+z^{-1})^n dz}{2^n z [a - (z+z^{-1})\/2][ 1-\\beta (z+z^{-1})\/2]^m}}.\n\\end{equation}\nThe integrand has poles at the points:\n\\begin{equation}\n    z_0=0,\\qquad z^{(a)}_{\\pm} = a \\pm \\sqrt{a^2-1}, \\qquad z^{(\\beta)}_{\\pm} = \\frac{1\\pm\\sqrt{1-\\beta^2}}{\\beta}.\n\\end{equation}\nAmong these points, $z^{(\\beta)}_{-}$ and $z_0$ lie inside the unit circle $z=1$, $z^{(a)}_{-}$ lies inside the unit circle for $\\rm{Re} a > 0$ and $z^{(a)}_{+}$ -- for $\\rm{Re} a < 0$. These statements are independent of sign of imaginary part of $a$.\n\nEvaluation is completed by computation of integrand residues at these poles. As a result, we arrive at the following expressions:\n\\begin{gather}\n\\label{Integrals}\n    J_{03} = \\frac{1}{2 (a \\beta -1)^3} \\left\\{\\beta \\frac{  \\left(a^2+2\\right) \\beta ^4+\\left(2 a^2-5\\right) \\beta ^2-6 a \\beta +6 }{\\left(1-\\beta\n   ^2\\right)^{5\/2}}+ 2 i \\frac{\\text{sign} \\rm{Im} a }{\\sqrt{1-a^2}}\\right\\},\\\\\n   J_{02} = \\frac{1}{(a\n   \\beta -1)^2} \\left\\{\\frac{\\text{sign} \\rm{Re} a}{\\sqrt{a^2-1}}+ \\beta \\frac{ \\beta  (a+\\beta )-2 }{\\left(1-\\beta ^2\\right)^{3\/2}}\\right\\},\\\\\n   J_{12} = \\frac{1}{{(a \\beta -1)^2}} \\left\\{a \\frac{\\text{sign} \\rm {Re} a}{\\sqrt{a^2-1}}-\\frac{1-a \\beta ^3}{\\left(1-\\beta ^2\\right)^{3\/2}}\\right\\}.\n\\end{gather}\nThe remaining necessary integrals can be obtained with recurrence relations\n\\begin{equation}\n    \\frac{\\partial J_{nm}}{\\partial \\beta} = m J_{n+1,m+1}.\n\\end{equation}\nAs apparent from the forms (\\ref{Integrals}), there is a singularity at $a = \\rightarrow 1$. As the parameter $a$ has finite imaginary part ($a = (\\omega + i\\gamma_{ee})\/qv_0$), the singularity is present only in the ballistic regime (i.e. at $\\omega \\gg \\gamma_{ee}$). Finite strength of e-e collisions softens the singularity. \n\nThe integrals also diverge at $\\beta \\rightarrow 1$, a situation close to that in special relativity. However, this divergence can be re-absorbed into definitions of particle and mass density (\\ref{Eqs-of-state}), so that the resulting generalized hydrodynamic equations are free of divergences.\n\nThere is a spurious singularity at $a \\beta \\rightarrow 1$, however, a closer inspection reveals that it is compensated by the zero value of the numerator. An only special property of the computed dielectric response at $a \\beta \\rightarrow 1$ is the absence of dissipation, as discussed in the main text.\n\n\\subsection{Analysis of instabilities in the double-layer system}\nThe dispersion law for plasmons in a double-layer structure separated by distance $d$ reads\n\\begin{equation}\n\\label{Double-layer}\n   \\epsilon_{2l}(q,\\omega,\\beta)\\equiv (1 + V_0 \\Pi_+) (1 + V_0 \\Pi_-) - V^2_0 \\Pi_- \\Pi_+ e^{-2|q|d} = 0.\n\\end{equation}\nHere $\\Pi_+$ and $\\Pi_-$ are the polarizabilities of individual top and bottom layers, $V_0 = 2\\pi e^2\/\\kappa |q|$ is the Fourier transform of Coulomb interaction and $\\kappa$ is the background dielectric constant. Substitution of hydrodynamic polarizability (\\ref{HD_polarization}) results in biquadratic equation with two eigenmodes\n\\begin{equation}\n\\label{HD-modes-double-layer}\n    \\frac{\\omega^2_{\\pm}}{q^2v_0^2} = \\frac{  2 \\left(\\beta ^4-3 \\beta ^2+2\\right)  s_p^2 + 2 \\beta ^4-3 \\beta ^2+2  \\pm 2 \\sqrt{2 \\beta ^2 \\left(\\beta\n   ^2-1\\right)  \\left(\\left(\\beta ^2-1\\right) + \\left(\\beta ^2-2\\right) s_p^2 \\right)+\\left(\\beta ^4-3 \\beta ^2+2\\right)^2 s_p^4 e^{-2 d\n   q}}}{\\left(\\beta ^2-2\\right)^2},\n\\end{equation}\nhere we have introduced the dimesionless 'plasmon phase velocity' \n\\begin{equation}\n    s_p = \\frac{\\omega_p}{qv_0}, \\qquad \\omega_p = \\sqrt{\\frac{2\\pi n e^2 |q|}{\\kappa m_{hd}}}.\n\\end{equation}\nThe signs $+$ and $-$ in the absove dispersion can be traced back to optical and acoustic modes of the double-layer structure in the absence of drift, respectively. Indeed, at $\\beta = 0$ one obtains\n\\begin{equation}\n    \\omega^2_{\\pm} = \\frac{q^2v_0^2}{2} + \\omega_p^2(1 \\pm e^{-2qd}).\n\\end{equation}\n\nThe instability emerges as the frequency of acoustic mode in (\\ref{HD-modes-double-layer}) passes through zero with increasing the drift velocity. This occurs at\n\\begin{equation}\n\\beta^\\pm_{\\rm th} = \\frac{v_0}{\\sqrt{2}}\\sqrt{\\frac{q^2v_0^2 + 2\\omega^2_p(1 \\pm e^{-qd})}{q^2v_0^2 + \\omega^2_p(1 \\pm e^{-qd})}}. \n\\end{equation}\n\nThe solution of dispersion relation (\\ref{Double-layer}) with polarizability including e-e collisions shows that instability sets on once the acoustic mode frequency crosses zero, i.e. at $\\rm Re \\omega_-  = 0$. A direct verification of this fact is challenging, but the experience of numerical solutions tells that it is the case. The stability diagram in Fig.~\\ref{Fig2} (C) was therefore obtained by numerical solution of $\\epsilon_{2l}(q,0,\\beta)=0$.\n\nWe note that the pattern of instabilities can be much richer if the carrier densities in the two layers are non-equal. In this case, the instability does not necessarily set on once the mode frequency crosses zero. A detailed analysis of these cases will be presented elsewhere.\n\n\\subsection{Diffraction on grating-gated graphene}\nWe consider the diffraction of an electromagnetic wave normally incident on graphene covered by a metal grating. The electric field is polarized along the $x$-axis, i.e. perpendicular to the gratings. The metal grating is assumed to be infinitely thin, its surface conductivity $\\sigma_{\\rm m}$ exceeds the velocity of light, $\\sigma_{\\rm m} \\gg c$, and is set to infinity in the following calculation. The grating-to-graphene distance $d$ is well below the grating period $a$ and with $W = f a$, where $f$ is the filling factor. This ensures efficient coupling of evanescent waves generated by the grating to the surface plasmons. We also assume that the structure is globally gated with a highly conducting substrate, the distance to back gate $z_0$ is the largest length scale in the problem, $z_0 \\gg W \\gg d$. We tune $z_0$ to accommodate nearly quarter of wavelength in the substrate material, $z_0 \\sim \\lambda_0\/4n_{\\rm sub}$, where $\\lambda_0$ is the free-space wavelength.\n\nDue to uniformity of 2DES is the $x$-direction, the diffraction problem can be formulated on surface current in the grating $j_s(x)$, $x \\in [0;W]$, while all the information about 2DES is accommodated in the Green's function of electromagnetic problem. This results in the following integral equation\n\\begin{equation}\n\\label{Diffraction-problem}\n    \\frac{j_s(x)}{\\sigma_{\\rm m}} = \\mathcal{E}_0 + \\int\\limits_{0}^{W}{dx' Z(x-x') j_s(x')},\n\\end{equation}\nwhere $\\mathcal{E}_0$ is the field in the grating plane $z=0$ {\\it in the absence of grating}. It can be presented as\n\\begin{equation}\n\\mathcal{E}_0 = \\mathcal{E}_{\\rm inc}(1 - r_{2d}),\n\\end{equation}\nwhere $r_{2d}$ is the reflection coefficient of bottom-gated 2DES without grating, and $\\mathcal{E}_{\\rm inc}$ is the electric field in the incident wave. \n\nThe impedance kernel $Z(x-x')$ is obtained as follows. First, one finds electric field induced at $z=0$ by $G$-th spatial Fourier harmonic of surface current passing in the grating plane:\n\\begin{equation}\n\\mathcal{E}_{{\\rm ind}{G}} = j_{s,{G}}  Z_{G},\n\\end{equation}\nthe function $Z_{G}$ is easily obtained by plane-wave matching or transfer-matrix methods. Then $Z(x-x')$ is the inverse Fourier transform of $Z_{G}$ with the wave vectors running across the reciprocal wave vectors of the grating, $G_n = 2\\pi n\/a$:\n\\begin{equation}\n\\label{Impedance_kernel}\n    Z(x-x') = \\sum\\limits_{n=-\\infty}^{+\\infty}{Z_{G_n} e^{-iG_n(x-x')}}.\n\\end{equation}\n\nOne should distinguish the cases $n=0$ (corresponding to the normally incident propagating wave) and $|n|\\ge 1$ (corresponding to evanescent waves generated by grating):\n\\begin{gather}\n\\label{Impedance1}\n   Z^{-1}_{|G|\\ge 2\\pi\/a}  = \\frac{i \\omega }{4 \\pi  |G| }\\left\\{ 1 - \\varepsilon_{\\rm sub} + \\frac{2 \\varepsilon_{\\rm sub}  \\left(1+\\frac{2 i \\pi  \\sigma_{2d} |G| }{\\omega \\varepsilon_{\\rm sub} }\\right)}{1+\\frac{2 i \\pi  \\sigma_{2d} |G| \n   \\left(1-e^{-2 d |G| }\\right)}{\\omega \\varepsilon_{\\rm sub} }}  \\right\\},\\\\\n\\label{Impedance2}\n   Z^{-1}_{G=0} = \\frac{\\omega}{4 \\pi  k_1}  \\left\\{\\frac{2 \\varepsilon_{\\rm sub}   \\left(1-\\frac{2 \\pi  k_1 \\sigma}{\\omega  \\varepsilon_{\\rm sub}  }  \\left(-1+e^{2 i  k_1 (z_0 - d)}\\right) \\right)}{\\frac{2 \\pi  k_1 \\sigma}{\\omega  \\varepsilon_{\\rm sub}  }\n    \\left(\\left(-1+e^{-2 i d k_1}\\right) e^{2 i k_1 z_0 }+e^{2 i d\n   k_1}-1\\right)+e^{2 i k_1 z_0}-1}-\\frac{k_1}{k}+\\varepsilon_{\\rm sub} \\right\\};\n\\end{gather}\nhere $k=\\omega\/c$ and $k_1 = \\omega \\sqrt{\\varepsilon_{\\rm sub}}\/c$ are the wave vectors in vacuum and the substrate material.\n\nIt is important to note that impedance kernel (\\ref{Impedance_kernel}) with Fourier components (\\ref{Impedance1}-\\ref{Impedance2}) diverges at large $G$. This is associated with singularities of electric field near the keen edges of thin metal stripe carrying uniform current. The divergence can be cured in two ways. As a first possibility, one can transform Eq.~\\ref{Diffraction-problem} into a second-order differential with respect to $x$ and impose zero boundary conditions on current at the edges \\begin{equation}\n    j_s(x=0) = j_s (x=W) = 0.\n\\end{equation}\nThe Fourier components of modified kernel would have extra two powers of $G$ in the denominator, while the coordinate representation will be non-divergent. \n\nAs a second possibility, one can expand the unknown current over the orthogonal basis functions $\\phi_n(x)$ that already satisfy the zero boundary condition:\n\\begin{equation}\n    j_s(x) = \\sum_n{c_n \\phi_n(x)},\\qquad \\phi(0) = \\phi(W) = 0.\n\\end{equation}\nThe resulting matrix equation\n\\begin{equation}\n    \\left[Z_{nm} - \\frac{\\delta_{nm}}{\\sigma_{\\rm m}}\\right]c_m = (\\mathcal{E}_0)_n\n\\end{equation}\nwould have matrix elements $Z_{nm}$ that quickly converge at large $G$. \n\nWe have chosen the basis functions\n\\begin{equation}\n    \\phi_n(x) = \\sqrt{\\frac{2a}{W}}\\sin\\left(\\frac{\\pi n a}{W}\\right)\n\\end{equation}\nthat are orthogonal with respect to the inner product:\n\\begin{equation}\n    \\int_0^W{\\frac{dx}{a}\\phi_n(x)\\phi_m(x)} = \\delta_{nm}.\n\\end{equation}\nIn this basis, the elements of the impedance matrix are evaluated analytically at each $G$:\n\\begin{equation}\n   (Z_G)_{nm} = 2 \\pi ^2 m n \\frac{W}{a}\\frac{ \\left((-1)^m e^{-i G W} - 1 \\right) \\left((-1)^n e^{i G\n   W} - 1\\right)}{\\left(\\pi ^2 m^2-G^2 W^2\\right) \\left(\\pi ^2 n^2-G^2 W^2\\right)} Z_G.\n\\end{equation}\nExpansion of current density over Chebyshev polynomials $U_n(x)$ multiplied by their respective weight functions $w(x)$ is advantageous in predicting the character of the field at the edges. However, it comes at the cost of numerical approximation to matrix elements $Z_{nm}$.\n\nIn actual calculations, we have truncated the linear system at $n = m = 10$, and evaluated the sums up to $G_{\\max} = 20 (2 \\pi \/a)$. \n\n\n\n\\end{widetext}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\\label{sec:intro}\n\nThe resultant of two polynomials and the discriminant of a polynomial are\nknown since long ago in the mathematical literature\\cite{S1851,G81,BPR03}\nand have already been applied to the analysis of several physical problems.\nSome examples are the determination of singularities in the eigenvalues of\nparameter-dependent matrix eigenvalue problems\\cite{HS91}, level degeneracy\nin a quantum two-spin model\\cite{SM98}, the analysis of the properties of\ntwo-dimensional magnetic traps for laser-cooled atoms\\cite{D02}, the\ndescription of optical polarization singularities\\cite{F04}, the exceptional\npoints (EPs) for the eigenvalues of a modified Lipkin model\\cite{HSG05}, the\nlocation of crossings and avoided crossings between eigenvalues of\nparameter-dependent symmetric matrices\\cite{BR06,B07,BR07a,BR07b} and the\nsolution of two equations with two unknowns that appear in the study of\ngravitational lenses\\cite{TB16}.\n\nThe purpose of this paper is to show that these remarkable mathematical\ntools are not foolproof and, consequently, should be applied with care. In\nsection~\\ref{sec:example} we introduce a simple parameter-dependent\nsymmetric matrix and show that the discriminant of its characteristic\npolynomial vanishes for all values of the model parameter. We reveal the\nreason for the apparent failure of the discriminant and show how to overcome\nthis difficulty in order to obtain the crossings and avoided crossings\nbetween eigenvalues. In section~\\ref{sec:symmetry} we discuss the problem\nfrom the point of view of symmetry and, finally, in section~\\ref\n{sec:conclusions} we summarize the main results and draw conclusions.\n\n\\section{Simple example}\n\n\\label{sec:example}\n\nPresent discussion is based on the simple $6\\times 6$ symmetric matrix\n\\begin{equation}\n\\mathbf{H}(\\lambda )=\\left(\n\\begin{array}{llllll}\n0 & 1 & 0 & 0 & 0 & \\lambda \\\\\n1 & 0 & \\lambda & 0 & 0 & 0 \\\\\n0 & \\lambda & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & \\lambda & 0 \\\\\n0 & 0 & 0 & \\lambda & 0 & 1 \\\\\n\\lambda & 0 & 0 & 0 & 1 & 0\n\\end{array}\n\\right) ,  \\label{eq:H}\n\\end{equation}\nthat depends on a real parameter $\\lambda $. This model was chosen\nsome time ago for the application of perturbation theory to the\nresonance energy given by the H\\\"{u}ckel model for benzene in\nexercise 6.6, page 347, of Szabo and\nOstlund\\cite{SO96}. Its real eigenvalues $E_{j}(\\lambda )$, $j=1,2,\\ldots ,6\n, are roots of the characteristic polynomial\n\\begin{equation}\np(E,\\lambda )=\\left| \\mathbf{H}-E\\mathbf{I}\\right| =E^{6}-3E^{4}\\left(\n\\lambda ^{2}+1\\right) +3E^{2}\\left( \\lambda ^{4}+\\lambda ^{2}+1\\right)\n-\\lambda ^{6}-2\\lambda ^{3}-1,  \\label{eq:charpoly}\n\\end{equation}\nwhere $\\mathbf{I}$ is the $6\\times 6$ identity matrix.\n\nThe values of $\\lambda $ corresponding to level crossings or avoided\ncrossings can be obtained from the discriminant of the characteristic\npolynomial $Disc_{E}\\left( p(E,\\lambda )\\right) $\\cite{BR06,B07,BR07a,BR07b}\n(see Appendix~\\ref{sec:appendix} for definition, notation and properties of\nthe discriminant). According to equation (\\ref{eq:Disc_x(A)}) of the\nAppendix~\\ref{sec:appendix} this discriminant should enable us to obtain the\nvalues of $\\lambda $ for which two (or more) eigenvalues of $\\mathbf{H}$\ncross. However, in the present case this strategy simply produces the\nuseless result $Disc_{E}\\left( p(E,\\lambda )\\right) =0$ for all $\\lambda $.\nIn order to understand the reason for this failure note that the\ncharacteristic polynomial can be factorized as\n\\begin{equation}\np(E,\\lambda )=\\left( E+\\lambda +1\\right) \\left( E-\\lambda -1\\right) \\left(\nE^{2}-\\lambda ^{2}+\\lambda -1\\right) ^{2}.  \\label{eq:p_factorized}\n\\end{equation}\nWe arbitrarily organize the eigenvalues as\n\\begin{equation}\nE_{1}=-E_{6}=-1-\\lambda ,\\;E_{2}=E_{3}=-E_{4}=-E_{5}=-\\sqrt{\\lambda\n^{2}-\\lambda +1}.  \\label{eq:E_j}\n\\end{equation}\nThe discriminant of the characteristic polynomial vanishes because \nE_{2}(\\lambda )=E_{3}(\\lambda )$ and $E_{4}(\\lambda )=E_{5}(\\lambda )$ for\nall $\\lambda $.\n\nWe can obtain the desired crossings or avoided crossings from the\ndiscriminant of the polynomial\n\\begin{eqnarray}\nq(E,\\lambda ) &=&\\left( E+\\lambda +1\\right) \\left( E-\\lambda -1\\right)\n\\left( E^{2}-\\lambda ^{2}+\\lambda -1\\right)   \\nonumber \\\\\n&=&E^{4}-E^{2}\\left( 2\\lambda ^{2}+\\lambda +2\\right) +\\left( \\lambda\n+1\\right) ^{2}\\left( \\lambda ^{2}-\\lambda +1\\right) ,  \\label{eq:q(E,lambda)}\n\\end{eqnarray}\nwhere we have removed the two-fold degeneracy that is independent of \n\\lambda $. It follows from\n\\begin{equation}\nDisc_{E}\\left( q(E,\\lambda )\\right) =1296\\lambda ^{4}\\left( \\lambda\n+1\\right) ^{2}\\left( \\lambda ^{2}-\\lambda +1\\right) ,  \\label{eq:Disc_E(q)}\n\\end{equation}\nthat there are actual level crossings at $\\lambda =-1,0$ and an avoided\ncrossing due to the coalescence of eigenvalues in the complex $\\lambda \n-plane at $\\lambda =\\lambda _{EP}^{\\pm }=\\left( 1\\pm \\sqrt{3}i\\right) \/2$.\nThe multiple level crossing at $\\lambda =0$ comes from the fact that the \n6\\times 6$ matrix exhibits a block diagonal form with three\n$2\\times 2$ sub matrices of the form\n\\begin{equation}\n\\mathbf{H}_{j}=\\left(\n\\begin{array}{ll}\n0 & 1 \\\\\n1 & 0\n\\end{array}\n\\right) ,\\;j=1,2,3,  \\label{eq:H_j_block}\n\\end{equation}\neach one with eigenvalues $E=\\pm 1$. They are representations of\nthe three\nethylene molecules in the H\\\"{u}ckel model discussed by Szabo and Ostlun\n\\cite{SO96}.\n\nFigure~\\ref{Fig:Enes} shows the eigenvalues of the matrix (\\ref{eq:H}) for a\nrange of $\\lambda $ values.We appreciate the crossing between $E_{1}$ and \nE_{6}$ at $\\lambda =-1$, the crossing between the degenerate pair $\\left(\nE_{2},E_{3}\\right) $ and $E_{1}$ at $\\lambda =0$ and the crossing between\nthe degenerate pair $\\left( E_{4},E_{5}\\right) $ and $E_{6}$ also at \n\\lambda =0$. In addition to it, this figure also shows an avoided crossing\nbetween the two degenerate pairs $\\left( E_{2},E_{3}\\right) $ and $\\left(\nE_{4},E_{5}\\right) $. It is well known that only levels of different\nsymmetry cross, while those with the same symmetry exhibit avoided crossings\n(see \\cite{F14} and references therein).\n\nIn passing, we mention that the poor convergence of the perturbation series\nfor the resonance energy of benzene at $\\lambda =1$ mentioned by Szabo and\nOstlund\\cite{SO96} is due to the fact that its radius of convergence is\ndetermined by the pair of branch points at $\\lambda _{EP}^{\\pm }$ and,\nconsequently, given by $\\left| \\lambda _{EP}^{\\pm }\\right| =1$. In other\nwords, $\\lambda =1$ is located on the boundary of the disk of convergence.\n\n\\section{Symmetry and degeneracy}\n\n\\label{sec:symmetry}\n\nIt is well known that degeneracy can be predicted beforehand. Typically,\ndegeneracy is caused by the symmetry of the Hamiltonian operator. In the\npresent case, we expect the existence of orthogonal matrices $\\mathbf{U}_{j}\n, $j=1,2,\\ldots ,N$, that leave the matrix $\\mathbf{H}$ invariant; that is\nto say: $\\mathbf{U}_{j}^{t}\\mathbf{HU}_{j}=\\mathbf{H}$, where $t$ stands for\ntranspose and $\\mathbf{U}_{j}^{t}=\\mathbf{U}_{j}^{-1}$. We can rewrite this\ninvariance expression in terms of commutators: $\\left[ \\mathbf{H},\\mathbf{U\n_{j}\\right] =\\mathbf{HU}_{j}-\\mathbf{U}_{j}\\mathbf{H}=\\mathbf{0}$, where \n\\mathbf{0}$ is the $6\\times 6$ zero matrix.\n\nIn order to construct the orthogonal matrices just mentioned we resort to\nthe graphical representation of the matrix (\\ref{eq:H}) shown in Figure~\\ref\n{Fig:H6}. The hexagon in this figure is regular when $\\lambda =1$ and\nirregular otherwise. Note that any rotation of $2\\pi \/3$ about an axis\nperpendicular to the center of the figure leaves it invariant. From the\neffect of this rotation: $\\left[ c_{1},c_{2},c_{3},c_{4},c_{5},c_{6}\\right]\n\\rightarrow \\left[ c_{5},c_{6},c_{1},c_{2},c_{3},c_{4}\\right] $, we obtain\nthe orthogonal matrix\n\\begin{equation}\n\\mathbf{U}_{1}=\\left(\n\\begin{array}{llllll}\n0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 \\\\\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0\n\\end{array}\n\\right) .  \\label{eq:U_1}\n\\end{equation}\nAnalogously, a rotation of $4\\pi \/3$ about the same axis, $\\left[\nc_{1},c_{2},c_{3},c_{4},c_{5},c_{6}\\right] \\rightarrow \\left[\nc_{3},c_{4},c_{5},c_{6},c_{1},c_{2}\\right] $, leaves the figure invariant\nand is produced by the matrix\n\\begin{equation}\n\\mathbf{U}_{2}=\\left(\n\\begin{array}{llllll}\n0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 1 \\\\\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0\n\\end{array}\n\\right) .  \\label{eq:U_2}\n\\end{equation}\nThe matrices $\\mathbf{U}_{1}$ and $\\mathbf{U}_{2}=\\mathbf{U}_{1}^{2}$ are\nrepresentations of the point-group operations commonly called $C_{3}$ and \nC_{3}^{2}$, respectively\\cite{T64,C90}.\n\nFigure~\\ref{Fig:H6} also shows the existence of three reflection planes\nperpendicular to the plane of the hexagon across the middle of opposite\nsides (commonly called $\\sigma _{v}$\\cite{T64,C90}). The reflection $\\left[\nc_{1},c_{2},c_{3},c_{4},c_{5},c_{6}\\right] \\rightarrow \\left[\nc_{2},c_{1},c_{6},c_{5},c_{4},c_{3}\\right] $ is produced by the matrix\n\\begin{equation}\n\\mathbf{U}_{3}=\\left(\n\\begin{array}{llllll}\n0 & 1 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 \\\\\n0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0\n\\end{array}\n\\right) ,  \\label{eq:U_3}\n\\end{equation}\nwhile $\\left[ c_{1},c_{2},c_{3},c_{4},c_{5},c_{6}\\right] \\rightarrow \\left[\nc_{6},c_{5},c_{4},c_{3},c_{2},c_{1}\\right] $ leads to\n\\begin{equation}\n\\mathbf{U}_{4}=\\left(\n\\begin{array}{llllll}\n0 & 0 & 0 & 0 & 0 & 1 \\\\\n0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0\n\\end{array}\n\\right) ,  \\label{eq:U_4}\n\\end{equation}\nand $\\left[ c_{1},c_{2},c_{3},c_{4},c_{5},c_{6}\\right] \\rightarrow \\left[\nc_{4},c_{3},c_{2},c_{1},c_{6},c_{5}\\right] $ is given by\n\\begin{equation}\n\\mathbf{U}_{5}=\\left(\n\\begin{array}{llllll}\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 \\\\\n0 & 0 & 0 & 0 & 1 & 0\n\\end{array}\n\\right) .  \\label{eq:U_5}\n\\end{equation}\n\nThe set of matrices $G_{6}=\\left\\{ \\mathbf{I},\\mathbf{U}_{1},\\mathbf{U}_{2}\n\\mathbf{U}_{3},\\mathbf{U}_{4},\\mathbf{U}_{5}\\right\\} $ is a group isomorphic\nto $C_{3v}$\\cite{T64,C90}. If instead of the three reflection planes we\nconsider rotation axes on the plane of the figure across the middle of\nopposite the hexagon sides, we appreciate that rotations of $\\pi \/2$ about\nthem also leave the figure invariant (they are commonly called rotation axes\n$C_{2}$\\cite{T64,C90}). In such a case the group $G_{6}$ is isomorphic to \nD_{3}$\\cite{T64,C90}. Both groups exhibit irreducible representations $A_{1}\n, $A_{2}$ and $E$, the latter of dimension $2$ that explains the two-fold\ndegeneracy of the eigenvalues of the matrix (\\ref{eq:H}) for all values of \n\\lambda $. It is clear that we can predict a vanishing discriminant from the\nsymmetry of the problem.\n\nBy means of projection operators\\cite{T64,C90}, constructed\nstraightforwardly from the matrices $\\mathbf{U}_{j}$, we can easily\ndetermine the symmetry of the eigenvectors $\\mathbf{v}_{j}$, $\\mathbf{H\n(\\lambda )\\mathbf{v}_{j}=E_{j}\\mathbf{v}_{j}$, $j=1,2,\\ldots ,6$, of the\nmatrix (\\ref{eq:H}). It is not difficult to verify that $\\mathbf{v}_{1}$ and\n$\\mathbf{v}_{6}$ are bases for the irreducible representations $A_{2}$ and \nA_{1}$, respectively. The pairs of eigenvectors $\\left( \\mathbf{v}_{2}\n\\mathbf{v}_{3}\\right) $ and $\\left( \\mathbf{v}_{4},\\mathbf{v}_{5}\\right) $\nare bases for the two-fold degenerate irreducible representation $E$.\n\nAt $\\lambda =-1$ the eigenvalues $E_{1}$ and $E_{6}$ also become degenerate.\nThis crossing is predicted by the discriminant (\\ref{eq:Disc_E(q)}) as\ndiscussed above and is supposed to lead to an accidental degeneracy because\nit does not appear to be caused by an additional symmetry based on\northogonal matrices like those discussed above.\n\nThe highest symmetry is expected when $\\lambda =1$ because the matrix \n\\mathbf{H}(1)$ is invariant under a rotation of $\\pi \/3$ about an axis\nperpendicular to the center of the \\textit{regular} hexagon in Figure~\\ref\n{Fig:H6}. This axis, commonly called $C_{6}$\\cite{T64,C90} leads to two\nadditional matrix representations of $C_{6}$ and $C_{6}^{3}$ (previously we\nhad $C_{6}^{2}=C_{3}\\rightarrow \\mathbf{U}_{1}$, $C_{6}^{4}=C_{3}^{2\n\\rightarrow \\mathbf{U}_{2}$). In addition to the point-group elements just\ndiscussed we should add reflection planes $\\sigma _{d}$ through opposite\nvertices of the regular hexagon. The resulting group of $12$ elements is\nisomorphic to $C_{6v}$. Alternatively, we may consider three axis $C_{2}$\nthrough opposite vertices in which case the group results to be $D_{6}$\\cite\n{T64,C90}. In both cases the irreducible representations are $A_{1}$, $A_{2}\n, $B_{1}$, $B_{2}$, $E_{1}$ and $E_{2}$ that also predict two-fold\ndegeneracy in agreement with the results in Figure~\\ref{Fig:Enes}.\nWe do not discuss this particular case in detail because it does\nnot add anything relevant to what was said above.\n\n\\section{Conclusions}\n\n\\label{sec:conclusions}\n\nThroughout this paper we have shown that the straightforward application of\nthe discriminant to a quantum-mechanical problem may yield a trivial useless\nresult if there is degeneracy for all values of the model parameter. The\ncause of the failure is the symmetry of the problem which can be determined\nbeforehand by means of suitable tools based on group theory\\cite{T64,C90}.\nMore precisely, one expects to face this difficulty if the symmetry of the\nmodel Hamiltonian does not change with the variation of the model parameter.\nBased on the titles of the papers by Bhattacharya and Raman\\cite{BR06} and\nBhattacharya\\cite{B07} we may say: \\textit{Be careful if you do not look at\nthe spectrum}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\bigskip\n\nWe consider the following problem:\nWhat are possible sizes of Jordan blocks for a pair of commuting nilpotent matrices? \nOr equivalently, for which pairs of nilpotent orbits of matrices (under similarity) there exists a pair of matrices, \none from each orbit, that commute. The answer to the question could be considered as a generalization of Gerstenhaber--Hesselink theorem\non the partial order of nilpotent orbits \\cite{cmc}.\n\nThe structure of the varieties of commuting pairs of matrices and of commuting pairs of nilpotent matrices is not yet well understood. \nIt was proved by Motzkin and Taussky \\cite{mt} (see also Guralnick \\cite{guralnick}), that the variety of pairs of \ncommuting matrices was irreducible. It was Guralnick \\cite{guralnick} who showed that this is no longer the case for \nthe variety of triples of commuting matrices (see also Guralnick and Sethuraman \\cite{gs}, Holbrook and Omladi\\v c \\cite{ho}, \nOmladi\\v c \\cite{omladic}, Han \\cite{han}, \\v Sivic \\cite{sivic}). Recently, it was proved that the variety of \ncommuting pairs of nilpotent matrices was irreducible (Baranovsky \\cite{baranovsky}, Basili \\cite{basili}). \nOur motivation to study the problem is to contribute to better understanding of the structure of this variety\nand which might also help in understanding the (ir)reducibility of the variety of triples of commuting matrices.\n\nWe are also motivated by the problems posed by Binding and Ko\\v sir \\cite{kosiMST} in the multiparameter spectral theory\nand Gustafson \\cite{gustafson} in the module theory over commutative rings. In both problems, certain \npairs of commuting matrices appear. The matrices from the multiparameter spectral theory generate an algebra that \nis a complete intersection, and the matrices from the theory of modules are both functions of another matrix.\n\n\nHere we initiate the study of the problem.\nFirst we list all possible Jordan forms for nilpotent matrices commuting with a single Jordan block.\nIn the Section \\ref{sec:digraphs} we recall the  correspondence between directed graphs and \ngeneric matrices (also known as Gansner--Saks Theorem), which is the main tool to prove our main result, i.e.,\nto compute the maximal index of nilpotency of a nilpotent matrix commuting with a given nilpotent matrix with Jordan canonical form  \n$\\underline{\\mu}$.\nIn the last section we discuss some further examples.\n\n\n\n\n\n\n \\bigskip\n\n \n\n\f\\section{The one Jordan block case and consequences}\\label{sec:general}\n\n\\bigskip\n\nLet us denote by ${\\mathcal N}={\\mathcal N}(n,\\mathbb{F})$ the variety of all $n \\times n$ nilpotent matrices over \na field $\\mathbb{F}$ of characteristic 0 and write ${\\mathcal N}_2=\\{(A,B) \\in {\\mathcal N} \\times {\\mathcal N}; AB=BA\\}$. \n\nWe follow the notations used in Basili \\cite{basili} and write \n${\\mathcal N}_B=\\{A \\in {\\mathcal N}; \\; AB=BA\\}$ for some $B \\in {\\mathcal N}$. Suppose $\\mu_1 \\geq \\mu_2 \\geq \\ldots \\geq \\mu_t > 0$ are\nthe orders of Jordan blocks in the Jordan canonical form for $B$. \nWe call the partition $\\underline{\\mu}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)$ \nthe \\DEF{shape} of the matrix $B$ and denote it by $\\sh{B}$. \nWe also write $\\sh{B}=(m_1^{r_1},m_2^{r_2},\\ldots,m_l^{r_l})$, where $m_1 >m_2>\\ldots > m_l$.\n\n\n\\medskip\n\n  \nLet $\\partition{n}$ denote the set of all partitions of $n \\in \\mathbb{N}$\nand for a subset ${\\cal S} \\subseteq {\\mathcal N}(n,\\mathbb{F})$ write \n$$\\partition{{\\cal S}}=\\{\\underline{\\mu} \\in \\partition{n}; \\; \\underline{\\mu}=\\sh{A} \\text{ for some } A\\in {\\cal S} \\}\\, .$$\nDenote also \n$$\\partition{{\\mathcal N}_2}=\\{(\\sh{A},\\sh{B}); \\; (A,B) \\in {\\mathcal N}_2 \\} \\subseteq \\partition{n} \\times \\partition{n}\\, .$$\nNote that $(\\underline{\\lambda},\\underline{\\mu}) \\in \\partition{{\\mathcal N}_2}$ if and only if\n$(\\underline{\\mu},\\underline{\\lambda}) \\in \\partition{{\\mathcal N}_2}$, i.e. $\\partition{{\\mathcal N}_2}$ is symmetric.\n\n\\medskip\n\nIt is easy to see that for each $t=1,2,\\ldots,n$ there exists a uniquely defined partition \n$\\rpt{n}{t}:=(\\lambda_1,\\lambda_2,\\ldots,\\lambda_t) \\in \\partition{n}$, such that $\\lambda_1-\\lambda_t \\leq 1$.\nIt can be verified that \n$\\rpt{n}{t}=\\left(\\left\\lceil \\frac{n}{t} \\right\\rceil^r,\\left\\lfloor \\frac{n}{t} \\right\\rfloor^{t-r}\\right)$. \nDenote $\\rp{n}=\\{\\rpt{n}{t}; \\; t=1,2,\\ldots,n\\}$.\nFor $\\underline{\\mu}=(\\mu_1,\\mu_2,\\ldots,\\mu_r)$ we define in the same fashion \n$\\rpt{\\underline{\\mu}}{t}=(\\rpt{\\mu_1}{t},\\rpt{\\mu_2}{t},\\ldots,\\rpt{\\mu_r}{t})$.\n\n\\bigskip\n\nTake a matrix $B \\in {\\mathcal N}(n,\\mathbb{F})$ with $\\dim\\ker B=1$ (i.e. $\\sh{B}=(n)$). Then it is well known that any matrix commuting with\n$B$ is a polynomial in $B$. By computing the lengths of the Jordan chains of $B$, we observe that $\\sh{B^k}=\\rpt{n}{k}$ for \n$k=1,2,\\ldots,n$.\t\nThus we have the following.\n\n\\bigskip\n\n\\begin{proposition}\\label{thm:onejorblock}\nFor a matrix $B$ with $\\dim\\ker B=1$ it follows that $\\partition{{\\mathcal N}_B}=\\rp{n}$.\n\\hfill$\\blacksquare$\n\\end{proposition}\n\n\\bigskip\n\n\nFor a sequence $(a_1,a_2,\\ldots,a_k)$, $a_i \\in \\mathbb{N}$, we write\n$\\ord{a_1,a_2,\\ldots,a_k}=(a_{\\pi(1)},a_{\\pi(2)},\\ldots,a_{\\pi(k)})$,  where  \n$a_{\\pi(1)}\\geq a_{\\pi(2)}\\geq \\ldots \\geq a_{\\pi(k)}$ and $\\pi$ is a permutation of $\\{1,2,\\ldots,k\\}$.\n\nFor $\\underline{\\lambda}=(\\lambda_1,\\lambda_2,\\ldots,\\lambda_t)\\in \\partition{n}$ we write\n\\begin{align*}\n \\rp{\\underline{\\lambda}\n             \n             \n              =&\\{\\ord{\\rpt{\\lambda_1}{s_1},\\rpt{\\lambda_2}{s_2},\\ldots,\\rpt{\\lambda_t}{s_t}}; \\; s_i=1,2,\\ldots,i \\; \\}\\, .           \n\\end{align*}\nSo, for example, $(3,3,2) \\in \\rp{(5,3)}$, but $(4,3,1) \\notin \\rp{(5,3)}$.\n\n\n\n\\bigskip\n\n\n\\begin{proposition} \\label{thm:expjor2}\n\n For all $\\underline{\\mu} \\in \\partition{n}$ it follows that\n $\\{(\\underline{\\mu}, \\underline{\\lambda}); \\underline{\\lambda} \\in \\rp{\\underline{\\mu}}\\} \\subseteq \\partition{{\\mathcal N}_2}$.\n\\end{proposition}\n\n\\medskip\n\n\\begin{proof}\n Take an arbitrary $B \\in {\\mathcal N}$ with $\\sh{B}=\\underline{\\mu}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)$ \n and pick $\\underline{\\lambda} \\in \\rp{\\underline{\\mu}}$.\n We want to find a matrix $A \\in {\\mathcal N}_B$ such that $\\sh{A}=\\underline{\\lambda}$.\n \n By definition, $\\underline{\\lambda}$ is of the form $\\underline{\\lambda}=\\ord{\\rpt{\\mu_1}{k_1},\\rpt{\\mu_2}{k_2},\n \\ldots,\\rpt{\\mu_t}{k_t}}$.\n Let $J_{\\mu_i}$ be a $\\mu_i \\times \\mu_i$ Jordan block \n corresponding to eigenvalue 0. Since\n $\\sh{J_{\\mu_i}^{k_i}}=\\rpt{\\mu_i}{k_i} $, it follows that the matrix\n $A=J_{\\mu_1}^{k_1} \\oplus J_{\\mu_2}^{k_2} \\oplus \\ldots \\oplus J_{\\mu_t}^{k_t}$\n has $\\sh{A}=\\underline{\\lambda}$ and clearly $A \\in {\\mathcal N}_B$. \n\\end{proof}\n\n\\bigskip\n\nFor a matrix $A$ denote by $\\nil{A}$ its index of nilpotency, i.e.\n$\\nil{A}=\\min \\{i; \\; A^i=0\\}$. For ${\\cal S} \\subseteq {\\mathcal N}(n,\\mathbb{F})$ write\n$\\nil{{\\cal S}}=\\max \\{i; \\; i=\\nil{A} \\text{ for some } A \\in {\\cal S}\\}$.\n\n\\medskip\n\nThe following Corollaries can be easily obtained from Propositions \\ref{thm:onejorblock} and \\ref{thm:expjor2}.\n\n\\medskip\n\n\n\\begin{corollary} \\label{thm:oneblock}\n If $\\underline{\\lambda} \\in \\partition{n}$, then $\\underline{\\lambda} \\in \\rp{n}$ if and only if \n $((n),\\underline{\\lambda}) \\in \\partition{{\\mathcal N}_2}$.\n\\hfill$\\blacksquare$  \n\\end{corollary}\n\n\\medskip\n\n\\begin{corollary} \\label{thm:rb1}\n For $B \\in {\\mathcal N}$ it follows that $\\sh{B}\\in \\rp{n}$ if and only if $\\nil{{\\mathcal N}_B}=n$. \\hfill \\qedsymbol\n\\end{corollary}\n\n\\medskip\n\n\\begin{example}\nNote that Proposition \\ref{thm:expjor2} gives us a rather large subset of partitions in ${\\mathcal N}_B$.\n\nTake $\\sh{B}=(4,3,2^2,1)$. First we directly observe that \n$(4,3,2^2,1)$, $(4,3,2,1^3)$, $(4,3,1^5)$, $(4,2^3,1^2)$, $(4,2^2,1^4)$, $(4,2,1^6)$, $(4,1^8)$,\n$(3,2^4,1)$, $(3,2^3,1^3)$, $(3,2^2,1^5)$, $(3,2,1^7)$, $(3,1^9)$, $(2^5,1^2)$, $(2^4,1^4)$,\n$(2^3,1^6)$, $(2^2,1^8)$, $(2,1^{10})$, $(1^{12})$ are all in $\\partition{{\\mathcal N}_B}$.\n\nNext, we see that $(4,3,2^2,1)$ is included in $\\rp{\\underline{\\mu}}$ if\n$\\underline{\\mu}$ is $(7,5)$, $(7,4,1)$, $(7,3,2)$, $(7,2^2,1)$, $(5,4,3)$, $(5,4,2,1)$, $(4^2,3,1)$,\n$(4,3^2,2)$ or $(4,3,2^2,1)$. Thus, by symmetry, also all these partitions are in $\\partition{{\\mathcal N}_B}$.\n\nCorollary \\ref{thm:oneblock} shows that $\\partition{{\\mathcal N}_B} \\ne \\partition{12}$.\nThe natural question is whether the list above is the entire $\\partition{{\\mathcal N}_B}$. It can be verified \nwith {\\tt Mathematica} that the answer is negative and that there are more partitions in $\\partition{{\\mathcal N}_B}$. \nHowever, it would take a lot of time to compute $\\partition{{\\mathcal N}_B}$ without computer.\nFrom results in Section \\ref{sec:mainthm} it follows that there exists a partition \n$(\\mu_1,\\mu_2,\\ldots,\\mu_k) \\in \\partition{{\\mathcal N}_B}$ such that $\\mu_1=9$.\n\\hfill$\\square$\n\\end{example} \n\n\n\n\\bigskip\n\n\n\\section{Directed graphs and the Gansner-Saks theory} \\label{sec:digraphs}\n\n\\bigskip\n\n\n\n\nA \\DEF{digraph} is a directed graph (i.e. a graph each of whose edges are directed). \nA \\DEF{path of length} $k$ in a digraph $\\Gamma$ is a sequence of vertices $a_1,a_2,\\ldots,a_k$ such that $(a_i,a_{i+1})$ \nis an edge in $\\Gamma$ for $i=1,2,\\ldots,k-1$. We do allow a path to consist of a single vertex.\nFor a path ${\\mathcal P}$ we denote by $|{\\mathcal P}|$ its length. We call $a_1$ (resp. $a_k$) the \\DEF{initial} (resp. \\DEF{final}) vertex\nof ${\\mathcal P}$.\nA path $a_1,\\ldots,a_k$ with $a_1=a_k$ is called a \\DEF{cycle}. A digraph without cycles is called an \\DEF{acyclic digraph}.\n \nLet $\\Gamma$ be a finite acyclic digraph on $n$ vertices, with the vertices labeled from 1 to $n$.\nA \\DEF{$k$--path} in $\\Gamma$ is a subset of the vertices that can be partitioned into $k$ or fewer disjoint paths. Let \n$\\hat{d}_k=\\hat{d}_k(\\Gamma)$ be the largest cardinality of a $k$--path in $\\Gamma$ and define $\\hat{d}_0=0$ and \n$\\Delta_k=\\Delta_k(\\Gamma)=\\hat{d}_k-\\hat{d}_{k-1}$. Since $\\hat{d}_k \\leq \\hat{d}_{k+1}$, all the $\\Delta_k \\, 's$\nare nonnegative, so we have the infinite sequence $\\Delta=\\Delta(\\Gamma)=(\\Delta_1,\\Delta_2,\\ldots)$ of nonnegative integers.\n\n\\bigskip\n\n\nWe denote by $M(i;j)$ the entry in the $i$-th row and the $j$-th column of the matrix $M$. We say that matrices $M$ and $N$ \nof the same size \\DEF{have the same pattern} if $M(i;j)=0$ if and only if $N(i;j)=0$ for $i,j=1,2,\\ldots,n$.\n\n\nLet $\\overline{\\mathbb{F}}$ be a field that contains $\\mathbb{F}$ and has at least $n^2$ algebraically independent\ntranscendentals over rational numbers $\\mathbb{Q}$. \nA matrix $M \\in {\\mathcal N}(n,\\overline{\\mathbb{F}})$ is called \\DEF{generic} if all its nonzero entries are algebraically independent\ntranscendentals over $\\mathbb{Q}$. \n\n\n\\smallskip\n\nFor a nilpotent matrix $A \\in {\\mathcal N}(n,\\mathbb{F})$ and a generic $M \\in {\\mathcal N}(n,\\overline{\\mathbb{F}})$ of the same pattern it follows that \n$\\nil{A} \\leq \\nil{M}$.\n\n\\bigskip\n\nGiven a finite acyclic digraph $\\Gamma$ with $n$ vertices, we assign to it a generic\n$n \\times n$ matrix $M_\\Gamma=[m_{ij}]$ such that $m_{ij}=0$ if $(i,j)$ is not an edge in $\\Gamma$ \nand such that the rest of the entries of $M_\\Gamma$ are nonzero complex numbers which are independent\ntranscendentals. \nSince $\\Gamma$ is acyclic, $M_\\Gamma$ is nilpotent.\nConversely, given a nilpotent generic matrix, it corresponds, reversing the above assignment,\nto an acyclic digraph on $n$ vertices.\n\n\n\\begin{example}\\label{ex:gs}\nConsider a generic nilpotent matrix   \n$M=\\left[\\begin{array}{cccccccccccc}\n         0&0&a&0&b&0\\\\\n         0&0&0&c&0&d\\\\\n         0&0&0&0&0&0\\\\\n         0&0&0&0&0&0\\\\\n         0&0&0&e&0&f\\\\\n         0&0&0&0&0&0\n     \\end{array}\\right]\\, .$\nIts digraph $\\Gamma_M$ is then equal to the digraph on the Figure \\ref{fig1}. \\pagebreak\n\n       \\begin{figure}[htb]\n       \\begin{center}\n        \\includegraphics[height=2.8cm, width=2.8cm]{fig0.eps}\n       \\end{center}\n       \\caption{}\\label{fig1}\n       \\end{figure}\n\nConversely, every digraph $\\Gamma$ as on the Figure \\ref{fig1}, corresponds to a generic matrix with the\nsame pattern as matrix $M$. \n\\hfill$\\square$\n\\end{example}\n \n \\bigskip\n\nThe following theorem was independently proved by Gansner \\cite{gansner} and Saks \\cite{saks}. \n\n\\bigskip\n \n\\begin{theorem}\\label{thm:gansnersaks}\n Let $M$ be a generic nilpotent matrix. Then \n \\begin{equation}\n  \\Delta(\\Gamma_M)=\\sh{M}\\, . \\tag*{$\\blacksquare$}\n \\end{equation}\n\\end{theorem}\n\n\\bigskip\n\n\\begin{corollary}\\label{eq:longpath}\n The length of the longest path in an acyclic digraph $\\Gamma$ is equal to $\\nil{M_\\Gamma}$.\n\\hfill$\\blacksquare$\n\\end{corollary}\n \\bigskip\n\n\\bigskip\n\n\\begin{example}\n  Take matrix $M$ as in example \\ref{ex:gs} and digraph $\\Gamma$ as on the Figure \\ref{fig1}.  \n  We easily see that $\\hat{d}_1=3$ (a path of length 3 is 1,5,4), $\\hat{d}_2=5$ (a path of length 3 is 1,5,6 and a path of length 2 \n  is 2,4) and $\\hat{d}_3=6$.  By Theorem \\ref{thm:gansnersaks} it follows that $\\sh{M}=(3,2,1)$.\n  \\hfill$\\square$\n\\end{example}\n\n\n \n\n\n\\bigskip\n\n\n\f \\section{The $({\\mathcal N}_B,A)$--digraph and its paths} \\label{sec:preliminaries}\n\n\\bigskip\n\nLet us fix a nilpotent matrix $B \\in {\\mathcal N}(n,\\mathbb{F})$ with \n$\\sh{B}=\\underline{\\mu}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)=(m_1^{r_1},m_2^{r_2},\\ldots,m_l^{r_l})$.\nBy convention we have $m_1 > m_2 > \\ldots > m_l > 0$.\n\n\\bigskip\n\nIn this section we introduce some special digraphs, corresponding to elements in ${\\mathcal N}_B$.\n\n\\bigskip\n\nFor a pair of matrices $(A,B) \\in {\\mathcal N}_2$ denote $\\sh{A,B}=(\\sh{A},\\sh{B})$.\nThen there exists $P \\in GL_n(\\mathbb{F})$ such that $B=P J_B P^{-1}$, where $J_B$ denotes the Jordan\ncanonical form of matrix $B$.\nThus $\\sh{PAP^{-1},J_B}=(\\sh{A},\\sh{B})$ and $(PAP^{-1},J_B)\\in {\\mathcal N}_2$.\nTherefore we can assume that $B$ is already in its upper triangular Jordan canonical form.\n\n\\bigskip\n\nWrite $A=[A_{ij}]$ where $A_{ij} \\in {\\mathcal M}_{\\mu_i \\times \\mu_j}$. It is well known \n(see e.~g.~\\cite[p. 297]{GLR}) that if $AB=BA$, then $A_{ij}$ are all upper triangular Toeplitz matrices, i.e.\n for $1 \\leq j \\leq i \\leq t$ we have\n \\begin{equation}\\label{eq:toeplitz}\n A_{ij}=\n \\left[ \\begin{matrix}\n 0 & \\ldots & 0 & a_{ij}^0 & a_{ij}^1 & \\ldots & a_{ij}^{\\mu_i-1}\\\\\n \\vdots &  & \\ddots & 0 & a_{ij}^0 & \\ddots &  \\vdots \\\\\n \\vdots &  &  & \\ddots & 0 & \\ddots &  a_{ij}^{1} \\\\\n 0 & \\ldots & \\ldots  & \\ldots  & \\ldots  & 0 & a_{ij}^{0}\n \\end{matrix}\n \\right]\n \\;\n \\text{ and }\n \\;\n A_{ji}=\n \\left[ \\begin{matrix}\n a_{ji}^0 & a_{ji}^1 & \\ldots & a_{ji}^{\\mu_i-1}\\\\\n 0 & a_{ji}^0 & \\ddots &  \\vdots \\\\\n \\vdots &  \\ddots & \\ddots &  a_{ji}^{1} \\\\\n \\vdots &   & 0 &  a_{ji}^{0} \\\\\n \\vdots &   &  &  0 \\\\\n \\vdots &   &  &  \\vdots \\\\\n 0 & \\ldots & \\ldots   & 0\n \\end{matrix}\n \\right]\n \\, .\n \\end{equation}\n\nIf $\\mu_{i}=\\mu_j$ then we omit the rows or columns of zeros in $A_{ij}$ or $A_{ji}$ above.\n\n\\bigskip\n\nWe introduce some further notation following Basili \\cite{basili}. For a matrix $B$ with $\\sh{B}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)$ we\ndenote by $r_B$ and $k_i$, $i=1,2,\\ldots,r_B$, the numbers such that \n$k_1=1$, $\\mu_{k_i}-\\mu_{k_{i+1}} \\geq 2$, $\\mu_{k_i}-\\mu_{k_{i+1}-1} \\leq 1$ for $i=1,2,\\ldots,r_B-1$, \n$\\mu_{k_{r_B}}-\\mu_{t} \\leq 1$.  Note that $k_i \\in \\{1,2,\\ldots,t\\}$.\nFor example, if $\\sh{A}=(5^3,3^2,1^3)$ and $\\sh{B}=(5^3,4,3^2,2^4,1^3)$, then $r_A=r_B=3$.\nFurthermore, $\\sh{C}\\in \\rp{n}$ if and only if $r_C=1$.\n\nSet $q_0=0$, $q_l=t$ and for $\\alpha=1,2,\\ldots,l$ let \n$q_\\alpha \\in \\{1,2,\\ldots,t\\}$, be such that $\\mu_i=\\mu_{i+1}$ if $q_{\\alpha-1}+1 \\leq i < q_\\alpha$ and\n$\\mu_{q_\\alpha} \\ne \\mu_{q_\\alpha+1}$. For a block matrix with blocks as in\n\\eqref{eq:toeplitz} we define $\\overline{A}_{\\alpha \\alpha}=[a_{ij}^0]$ where $q_{\\alpha-1}+1 \\leq i,j \\leq q_\\alpha$.\n\n\\medskip\n\n\\begin{lemma}\\label{thm:basiliaa}  \\cite[Proposition 2.3]{basili} \nFor an $n \\times n$ matrix $A$ such that $AB=BA$ it follows that $A \\in {\\mathcal N}_B$ if and only if\n$\\overline{A}_{\\alpha\\alpha}$ are nilpotent for $\\alpha=1,2,\\ldots,l$.\n\nMoreover, if $A \\in {\\mathcal N}_B$ there exists a Jordan basis for $B$ such that $\\overline{A}_{\\alpha\\alpha}$\nare all strictly upper triangular.\n\\hfill{$\\blacksquare$}\n\\end{lemma}\n\n\\bigskip\n\nTherefore, when we study what are possible shapes of pairs from the set ${\\mathcal N}_2$ (or in particular what \nare the indices of nilpotency in ${\\mathcal N}_B$), we may consider only matrices $A \\in {\\mathcal N}_B$\nsuch that we have $a_{ij}^0=0$ for all $1 \\leq j \\leq i \\leq t$ whenever $\\mu_i=\\mu_j$.\nFrom now, we assume that the latter relations hold for $A$.\n\n\\bigskip\n\nLet $\\Gamma$ be a digraph corresponding to a generic matrix $M \\in {\\mathcal N}(n,\\overline{\\mathbb{F}})$ such that $M$ has \nthe same pattern as a matrix $A \\in {\\mathcal N}_B$. We call $\\Gamma$ an \\DEF{$({\\mathcal N}_B,A)$-digraph}.\n\n\\bigskip\n\n\\begin{example} \n Consider a nilpotent matrix $B$ with $\\sh{B}=(4,2)$ and a matrix\n $A=\\left[\\begin{array}{cccccccccccc}\n         0&0&a&0&b&0\\\\\n         0&0&0&a&0&b\\\\\n         0&0&0&0&0&0\\\\\n         0&0&0&0&0&0\\\\\n         0&0&0&c&0&d\\\\\n         0&0&0&0&0&0\n     \\end{array}\\right] \\in {\\mathcal N}_B \\, .$ \n Then the digraph on the Figure \\ref{fig1} is an $({\\mathcal N}_B,A)$-digraph.\n \\hfill$\\square$\n\\end{example}\n\n\n\\bigskip\n\nLet $\\Gamma$ be an $({\\mathcal N}_B,A)$-digraph that corresponds to $A \\in {\\mathcal N}_B$. \nDenote its vertices by $(x,y)$, where $x=1,2,\\ldots,t$ and $y=1,2,\\ldots,\\mu_x$.\nWrite blocks of $A$ as in  \\eqref{eq:toeplitz}. If $a_{ij}^k \\ne 0$ for some $1 \\leq i \\leq j \\leq t$, \nthen $\\Gamma$ contains edges $\\left((i,h),(j,h+k)\\right)$ for $h=1,2,\\ldots,\\mu_j-k$.\nIf $a_{ij}^k \\ne 0$ for some $1 \\leq j \\leq i \\leq t$, \nthen $\\Gamma$ contains edges $\\left((i,h),(j,h+k+\\mu_j-\\mu_i+1)\\right)$ for all $h=1,2,\\ldots,\\mu_i-k$.\n\n\\bigskip\n\nWe say that edges $((i_1,j_1),(i_2,j_2))$ and $((i_3,j_3),(i_4,j_4))$ are \\DEF{parallel} if \n$i_1=i_3$, $i_2=i_4$ and $j_1+j_4=j_2+j_3$. We call two paths ${\\mathcal P}_1$ and ${\\mathcal P}_2$ \\DEF{parallel} if\nthey consist of pairwise parallel edges.\n\n\\bigskip\n\n\n\\begin{example} \\label{ex:43221a}\nConsider the case $\\sh{B}=(4,3,2^2,1) \\in \\partition{12}$. Each $({\\mathcal N}_B,A)$-digraph $\\Gamma$ consists of 12 vertices:\n\n        \\begin{figure}[htb]\n         \\begin{center}\n          \\includegraphics[height=3.6cm, width=6.9cm]{fig43221g}\n         \\end{center}\n         \\end{figure}\n\nDenote by $M$ the generic nilpotent matrix, which corresponds to $\\Gamma$ and has the same pattern as $A \\in {\\mathcal N}_B$.\nSuppose, for example, that there exists an edge $((4,1),(2,2))$ in $\\Gamma$. By the correspondence \nbetween nilpotent matrices and acyclic digraphs it follows that $M_{10,6} \\ne 0$. Since $A$ and \n$M$ have the same pattern, $a_{4,2}^0 \\ne 0$ and thus\n$M_{11,7} \\ne 0$. Therefore, there exists also an edge $((4,2),(2,3))$ in $\\Gamma$, which is parallel to \nan edge $((4,1),(2,2))$.\n\nSimilarly, we can add some other parallel edges to $\\Gamma$ as described above. For example, a possible $({\\mathcal N}_B,A)$-digraph $\\Gamma$ is\n\n        \\begin{figure}[htb]\n         \\begin{center}\n          \\includegraphics[height=4.5cm, width=8.7cm]{fig43221a}\n         \\end{center}\n         \\caption{}\\label{fig3}         \n         \\end{figure}\n\nand the corresponding matrix $A \\in {\\mathcal N}_B$ is equal to\n$$\nA=\\left[\\begin{array}{cccc|ccc|cc|cc|c}\n0 & 0 & a & 0 & b & 0 & c & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & a & 0 & b & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n\\hline\n0 & 0 & d & 0 & 0 & 0 & 0 & e & f & 0 & 0 & 0  \\\\\n0 & 0 & 0 & d & 0 & 0 & 0 & 0 & e & 0 & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & g & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & h & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & h & 0 & 0 & 0 & 0 & 0  \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & i & 0 & 0 & 0  \n\\end{array}\\right]\n$$\nfor some $a,b,c,\\ldots,i \\in \\mathbb{F}$.\n\\hfill$\\square$\n\\end{example}\n\n\\bigskip\n \nSuppose that $r_i,r_{i+1} \\ne 0$ for some $i$.\nThen parts $m_i^{r_i}$, $m_{i+1}^{r_{i+1}}$ in $\\sh{B}$ have to be treated differently if \n$m_i-m_{i+1}=1$ or $m_{i}-m_{i+1} \\geq 2$. We introduce some notation that will help us \nunify the treatments.\n\n\\bigskip\n\nFor $i=1,2,\\ldots,l-1$ such that $r_i \\ne 0$  define \n$$s(i)=\\left\\{\n \\begin{array}{cl}\n  r_i, & \\text{if } m_{i}-m_{i+1}\\geq 2 \\\\\n  r_i+r_{i+1}, & \\text{if } m_{i}-m_{i+1} \\leq 1\n \\end{array}\n \\right.$$\nand $s(l)=r_l$. \n\n\\medskip\n\nWe set $\\mu_0=\\mu_1+1$. Choose any $k$, $1 \\leq k \\leq t$, such that $\\mu_{k-1}> \\mu_k > 1$ \nand let $w$ and $z$ be such that $\\mu_k-\\mu_w \\leq 1$, $\\mu_k-\\mu_{w+1} \\geq 2$ and \n$\\mu_{k}=\\mu_z > \\mu_{z+1}$.\nIf $\\mu_{k-1}> \\mu_k = 1$, then set $w=z=t$.\n\n\\medskip\n\nWe denote by $V_{B,k}$ the set of vertices\n\\begin{align*}\n \\big\\{(x,y); \\;  & x=k,k+1,\\ldots,w, y=1,2,\\ldots,\\mu_{x}\n             \\big\\} \\, .\n\\end{align*}                 \nWe call the path that contains vertices  \\label{Bpath}\n$$V_{1,k}=\\{(x,1)\\; x=1,2,\\ldots,k \\}, $$\n$$ V_{B,k}$$ \nand \n$$ V_{3,k}=\\{(x,\\mu_x)\\; x=1,2,\\ldots,k-1 \\} \\cup \\{(z,\\mu_z)\\}$$ \nthe \\DEF{$B_k$-path} (or \\DEF{$B$-path} for short). We call $s(k)$ the \\DEF{width} of $B_k$-path (or of the set $V_{B,k}$).\n\n\n\\pagebreak\n\n\\begin{example}\\label{ex:43221c}\nConsider again the case $\\sh{B}=(4,3,2^2,1) \\in \\partition{12}$ and an $({\\mathcal N}_B,A)$-digraph $\\Gamma'$\n\n\n        \\begin{figure}[htb]\n         \\begin{center}\n          \\includegraphics[height=4cm, width=6cm]{fig43221z}\n         \\end{center}\n         \\caption{}\\label{fig4}    \n         \\end{figure}\n         \nThen for $k=1,2,3,5$, the set $V_{B,k}$ and $B_k$-path consist of the following vertices\n\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|p{5cm}|p{6cm}|}\n  \\hline\n $k$ & $s(k)$ & $w$ & $z$ & $V_{B,k}$ & vertices of $B_k$-path \\\\\n  \\hline\n  \\hline\n $1$ & $2$ & $2$ & $1$ & $(1,1)$, $(1,2)$, $(1,3)$, $(1,4)$, $(2,1)$, $(2,2)$, $(2,3)$ & $(1,1)$, $(1,2)$, $(1,3)$, $(1,4)$, $(2,1)$, $(2,2)$, $(2,3)$ \\\\   \n  \\hline\n $2$ & $3$ & $4$ & $2$ & $(2,1)$, $(2,2)$, $(2,3)$, $(3,1)$, $(3,2)$, $(4,1)$, $(4,2)$ & $(1,1)$, $(2,1)$, $(2,2)$, $(2,3)$, $(3,1)$, $(3,2)$, $(4,1)$, $(4,2)$, $(1,4)$ \\\\   \n  \\hline\n $3$ & $3$ & $5$ & $4$ & $(3,1)$, $(3,2)$, $(4,1)$, $(4,2)$, $(5,1)$ & $(1,1)$, $(2,1)$, $(3,1)$, $(3,2)$, $(4,1)$, $(4,2)$, $(5,1)$, $(1,4)$, $(2,3)$ \\\\   \n  \\hline\n $5$ & $1$ & $5$ & $5$ & $(5,1)$ & $(1,1)$, $(2,1)$, $(3,1)$, $(4,1)$, $(5,1)$, $(1,4)$, $(2,3)$, $(3,2)$, $(4,2)$\\\\   \n  \\hline\n\\end{tabular}\n\\end{center}\n\\hfill$\\square$\n\\end{example}\n\n\\bigskip\n\nThe $B_k$-path consists of three subpaths ${\\mathcal P}_i$, $i=1,2,3$, such that \n${\\mathcal P}_1 \\cap {\\mathcal P}_2=\\{(k,1)\\}$ and ${\\mathcal P}_2 \\cap {\\mathcal P}_3 = \\{(z,\\mu_z)\\}$.\nThe vertices of ${\\mathcal P}_1$ are from $V_{1,k}$ and its edges are \n$$((i,1),(i+1,1))$$  \nfor $i= 1,2, \\ldots, k-1$. The ${\\mathcal P}_2$ consists of vertices $V_{B,k}$ and\nedges $$((i,j),(i+1,j))$$ \nfor \n$i=k,k+1,\\ldots,w-1$, $j=1,2,\\ldots,\\mu_{k}-1$ or \n$i=k,k+1,\\ldots,z-1$, $j=\\mu_{k}$ and \n$$((w,j),(k,j+1))$$ \nfor $j=1,2,\\ldots,\\mu_{k}-1$.\nThe subpath ${\\mathcal P}_3$ on vertices $V_{3,k}$ consists of edges\n$$((i,\\mu_{i}),(i+1,\\mu_{i+1}))$$ \nif $\\mu_i=\\mu_{i+1}$, $1 \\leq i \\leq k-2$, and\n$$((j,\\mu_{j}),(i,\\mu_{i}))$$ \nif $\\mu_{k} \\leq \\mu_{j+1}< \\mu_j < \\mu_i < \\mu_{i-1}$ and     \nsuch that if $i \\leq h \\leq j$, then it follows that either $\\mu_{h}=\\mu_i \\text{ or } \\mu_h=\\mu_j$.\n\n\\smallskip\n\nFor a $B_k$-path ${\\mathcal P}$ it follows that $|{\\mathcal P}|=2(k-1)+\\mu_{k}+\\mu_{k+1}+\\ldots+\\mu_{w}$.  \n\n\n\\bigskip\n\n\n\\begin{example} \\label{ex:43221b}\nRecall the Examples \\ref{ex:43221a} and \\ref{ex:43221c} where $\\sh{B}=(4,3,2^2,1)$. In the $({\\mathcal N}_B,A)$-digraph $\\Gamma$ from\nFigure \\ref{fig3} there are no $B_k$-paths. However the $({\\mathcal N}_B,A)$-digraph $\\Gamma'$ from Figure \\ref{fig4}\nhas three $B$-paths, namely, $B_1$-path (with width 2 and length 7), $B_2$-path (with width 3 and length 9) and $B_3$-path (also of length 9\nand width 3), as shown below\n\n        \\begin{figure}[htb]\n         \\begin{center}\n          \\includegraphics[height=3.2cm, width=12cm]{fig43221b}\n         \\end{center}\n         \\end{figure}\n\n\n        \\begin{figure}[htb]\n         \\begin{center}\n          \\includegraphics[height=3.2cm, width=4.7cm]{fig43221c}\n         \\end{center}\n         \\end{figure}\n\nNote that in this case $B_3$-path and $B_5$-path coincide.\n\\hfill$\\square$\n\\end{example}\n\n\n\\bigskip\n\n\\begin{example}\nLet $\\sh{B}=(4,3^2)$. By Corollary \\ref{thm:rb1} it follows that $\\nil{{\\mathcal N}_B}=10$. \nThe generic $M$, such that its digraph $\\Gamma_M$ is equal to\n\n        \\begin{figure}[h]\n         \\begin{center}\n          \\includegraphics[height=3.5cm, width=3.5cm]{fig433}\n         \\end{center}\n         \\end{figure}\n\n\\pagebreak\nand $A \\in {\\mathcal N}_B$ such that $a_{12}^0$,  $a_{23}^0$, $a_{31}^0$ are its only nonzero entries,\nhave the same pattern. It is easy to verify that  $\\nil{M}=\\nil{A}=10$. \nThe shown path is the only $B$-path in $\\Gamma_M$.\n\\hfill$\\square$\n\\end{example}\n\n\\bigskip\n\nRecall that $\\nil{A} \\leq \\nil{M}$ for $A \\in {\\mathcal N}_B$ and a generic matrix $M \\in {\\mathcal N}(n,\\overline{\\mathbb{F}})$ with the same pattern. \nBy Corollary \\ref{eq:longpath} it follows that\n$\\nil{{\\mathcal N}_B}$ is less than or equal to the length of the longest path in a $({\\mathcal N}_B,A)$-digraph.\nThe length of the $B_k$-path is equal to \n$2 \\sum_{i=1}^{k-1} r_i + r_k m_k+r_{k+1}m_{k+1}$ if\n$s(k)=r_k+r_{k+1}$ and  $2 \\sum_{i=1}^{k-1} r_i + r_k m_k$ otherwise.\n\n \n\n\\bigskip\n  \n\\section{Maximal index of nilpotency in ${\\mathcal N}_B$} \\label{sec:mainthm}\n\n\\bigskip\n\nHere, we prove the following theorem:\n\n\\bigskip\n\n\\begin{theorem} \\label{thm:main}\n Let $B$ be a nilpotent matrix with $\\sh{B}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)$. Then\n  $$\\nil{{\\mathcal N}_B}= \\max\\limits_{1 \\leq i < t} \\{2i+\\mu_{i+1}+\\mu_{i+2}+\\ldots+\\mu_{i+r}; \\; \\mu_{i+1} - \\mu_{i+r}\\leq 1, \\mu_i \\ne \\mu_{i+1}\\} \n  \\, .$$\n\\end{theorem}\n\n\\bigskip\n\nFirst, we show that the longest path in an $({\\mathcal N}_B,A)$-digraph $\\Gamma$ is actually equal\nto the length of the longest $B$-path in $\\Gamma$.\n\n\n\\bigskip\n\n\\begin{lemma}\\label{thm:path2}\n Suppose that for $x=1,2,\\ldots,t$ vertices $(x,1)$ are contained in a path ${\\mathcal P}$ of an $({\\mathcal N}_B,A)$-digraph $\\Gamma$. \n Then there exists a $B$-path ${\\mathcal P}_B$ in $\\Gamma$ such that $|{\\mathcal P}| \\leq |{\\mathcal P}_B|$.\n\\end{lemma}\n  \n\\medskip\n \n\\begin{proof}\n Let us write $\\sh{B}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)=(m_1^{r_1},m_2^{r_2},\\ldots,m_l^{r_l})$, \n where $m_i=\\mu_1-i+1$ for $i=1,2,\\ldots,l$ and $m_l=\\mu_t$. (Note that $r_i$ can be 0 \n for some $i$.) \n \n Let $\\Gamma$ be an $({\\mathcal N}_B,A)$-digraph and $k$ be the maximal index such that \n $s(k)=\\max\\{s(i); \\; 1 \\leq i \\leq l \\}$. \n By $\\bask{k}$ we denote the $B_k$-path, i.e. the widest $B$-path in $\\Gamma$. \n \n Suppose first that $s(k)=r_k+r_{k+1}$, where $r_{k+1} \\ne 0$. Then\n  $$|\\bask{k}|=2 \\sum_{i=1}^{k-1} r_i + r_k m_k +r_{k+1}m_{k+1}=\n 2 \\sum_{i=1}^{k-1} r_i + (r_k+r_{k+1}) m_{k+1} + r_k \\, .$$\n\n Examine the edge $((i,y_i),(j,y_j))$ of ${\\mathcal P}$, where $\\mu_i \\ne \\mu_j$. \n If $i>j$  (and thus $\\mu_i < \\mu_j$) then  $\\mu_i-y_i > \\mu_j-y_j$.\n If $i<j$ (and thus $\\mu_i > \\mu_j$) then $y_i < y_j$. \n  \n Let ${\\mathcal P}'$ be the subpath of ${\\mathcal P}$ that contains all the vertices $(x,y)$ from ${\\mathcal P}$, where $y > 1$.\n Let $(x_0,y_0)$ be the initial vertex of ${\\mathcal P}'$. Since $((t,1),(x_0,y_0))$, $1 \\leq x_0 \\leq t$, is an edge of ${\\mathcal P}$,\n it follows that $\\mu_t-1 \\geq \\mu_{x_0}-y_0$.\n \n Since $\\bask{k}$ is the widest $B$-path in $\\Gamma$,\n it follows that\n  $|{\\mathcal P}'| \\leq   (\\mu_{x_0}-y_0) (r_{k}+r_{k+1}) + \\sum_{i=1}^{l-1} r_i \\leq  \\sum_{i=1}^{l-1} r_i + (m_l-1) (r_{k}+r_{k+1})$. \n Therefore\n  $$|{\\mathcal P}| \\leq  2 \\sum_{i=1}^{l-1} r_i +r_l +(m_l-1) (r_{k+1}+r_k)\\, .$$\n  \n Since $m_i=\\mu_1-i+1$ for $i=1,2,\\ldots,l$, it follows that $m_i - m_l = l-i$ for $i=1,2,\\ldots,l$. \n In particular, $m_l=m_k-l+k$. Thus\n  $$|{\\mathcal P}| \\leq  2 \\sum_{i=1}^{l-1} r_i +r_l +(m_k-l+k-1) (r_k+r_{k+1})\\, .$$\n \n \n By definition of $k$ it follows that $r_i+r_{i+1} \\leq r_k+r_{k+1}-1$ for all \n $i=k+1,k+2,\\ldots,l-1$.\n Thus \n $r_{k+1}+2r_{k+2}+2r_{k+3}+\\ldots+2r_{l-1}+r_l=\\sum_{i=k+1}^{l-1} (r_i+r_{i+1}) \\leq (l-k-1)(r_k+r_{k+1}-1)$.\n Therefore\n  \\begin{equation*}\n   \\begin{split}\n   |{\\mathcal P}| &\\leq 2 \\sum_{i=1}^{k-1} r_i + r_k + (r_k+r_{k+1}) + (l-k-1)(r_k+r_{k+1}-1)+\\\\\n           & \\qquad \\qquad + (m_k-l+k-1) (r_k+r_{k+1}) =\\\\\n     &= 2 \\sum_{i=1}^{k-1} r_i + r_k + (m_{k}-1)(r_k+r_{k+1})+k+1-l \\leq \\\\\n     &\\leq 2 \\sum_{i=1}^{k-1} r_i + r_k + m_{k+1}(r_k+r_{k+1})\\; = \\;  |\\bask{k}| \\, .\n   \\end{split}\n   \\end{equation*}\n\n \n If $s(k)=r_k$ and $k < l$, then $r_i+r_{i+1} \\leq r_k-1$ for $ i=k+1,k+2,\\ldots,l-1$.\n Thus \n $r_{k+1}+2r_{k+2}+2r_{k+3}+\\ldots+2r_{l-1}+r_l=\\sum_{i=k+1}^{l-1} (r_i+r_{i+1}) \\leq (l-k-1)(r_k-1)$.\n Again, since $\\bask{k}$ is the widest $B$-path in $\\Gamma$, we see similarly as before,\n  \\begin{equation*}\n   \\begin{split}\n    |{\\mathcal P}| & \\leq  2 \\sum_{i=1}^{l-1} r_i + r_l+ (\\mu_t-1) r_k \\leq \\\\\n            & \\leq  2 \\sum_{i=1}^{k-1} r_i +(l-k-1)(r_k-1)+ (\\mu_k-l+k+1) r_k \\leq \\\\\n            & \\leq 2 \\sum_{i=1}^{k-1} r_i + \\mu_k r_k \\; = \\;  |\\bask{k}| \\, ,\n   \\end{split}\n   \\end{equation*}\n which proves the Lemma.\n\n\n\n \\end{proof}\n \n \\bigskip \n \n\n\\begin{theorem}\\label{thm:premain}\n For $A \\in {\\mathcal N}_B$ it follows that\n  $$\\nil{A} \\leq \\max\\limits_{1 \\leq i < t} \n    \\{2i+\\mu_{i+1}+\\mu_{i+2}+\\ldots+\\mu_{i+r}; \\; \\mu_{i+1} - \\mu_{i+r}\\leq 1, \\mu_i \\ne \\mu_{i+1}\\} \\, .$$\n \\end{theorem}\n \n \\medskip\n \n \\begin{proof}\n Write $\\sh{B}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)$.\n We show by induction on $t$ that the longest possible path of $({\\mathcal N}_B,A)$-digraphs is of the\n same length as a $B$-path.\n \n Suppose that the longest path of $({\\mathcal N}_B,A)$-digraphs is included in digraph $\\Gamma$ and\n denote it by ${\\mathcal P}$. \n Note that ${\\mathcal P}$ contains vertices $(1,1)$ and $(1,\\mu_1)$.\n If it does not, we can add an edge from $(1,1)$ to the first vertex of ${\\mathcal P}$ or an edge\n from the last vertex of ${\\mathcal P}$ to $(1,\\mu_1)$ and lenghten it.\n \n \\smallskip\n \n If $t=1$ then the longest path contains all vertices and therefore it is a $B$-path.\n \n Suppose that our claim holds for all partitions with at most $t-1$ parts.\n \n Fix an $x$, $1 \\leq x \\leq t$. If ${\\mathcal P}$ does not contain any of the vertices $(x,y)$ for $y=1,2,\\ldots,\\mu_x$,\n then ${\\mathcal P}$ is a path in a $({\\mathcal N}_{B'},A)$-digraph, where \n $$\\sh{B'}=(\\mu_1,\\mu_2,\\ldots,\\mu_{x-1},\\mu_{x+1},\\mu_{x+2},\\ldots,\\mu_t).$$ By induction, there exists a $B'$-path \n ${\\mathcal P}_{B'}$ such that $|{\\mathcal P}| \\leq |{\\mathcal P}_{B'}|$. If a $B'$-path is not already a $B$-path, \n it can be lengthened to a $B$-path. Therefore there exists a $B$-path ${\\mathcal P}_B$ in $\\Gamma$, such that \n its length is greater than or equal to $|{\\mathcal P}|$. By assumption that ${\\mathcal P}$ is the longest path in $\\Gamma$\n it follows that $|{\\mathcal P}| = |{\\mathcal P}_B|$.\n \n Suppose now that for each $x$ the path ${\\mathcal P}$ contains a vertex $(x,y_x)$ for some $y_x$.\n The basic idea of the proof is to show the following claim: \n \n \\emph{In $\\Gamma$ there exists a path ${\\mathcal P}'$ of the same length as ${\\mathcal P}$ such that its first $t$ vertices are \n $(1,y_1)$, $(2,y_2)$,..., $(t,y_t)$.} \n \n Since ${\\mathcal P}'$ is the longest path in $\\Gamma$ it follows that $y_i=1$ for $i=1,2,\\ldots,t$. \n Otherwise the path that contains vertices $(1,1)$, $(2,1)$,..., $(t,1)$, $(t,2)$,..., $(t,y_t)$ and is after \n $(t,y_t)$ equal to ${\\mathcal P}'$ is longer than ${\\mathcal P}'$.\n \n \\smallskip\n \n Let us prove the claim. Let $y_t > 1$ be the smallest integer such that ${\\mathcal P}$ contains vertex $(t,y_t)$. \n Since ${\\mathcal P}$ is the longest path, $(t,y_t)$ is not the last vertex of ${\\mathcal P}$, and so for $x=1,\\ldots,t$,\n there exists a vertex $(x,y^3_x)$ that ${\\mathcal P}$ visits after $(t,y_t)$, i.e. $y^3_x \\geq y_t$. \n \n Next, suppose that there exists some $k$, where $1 \\leq k < t$ and $y_k^1 < y_k^2 \\leq y_t$, such that ${\\mathcal P}$ \n contains vertices $(k,y_k^1)$ and $(k,y_k^2)$ with the following properties:  there does \n not exist a vertex $(k,y_k^0)$ of ${\\mathcal P}$ such that $1\\leq y_k^0 \\leq y_k^1$ and $y_k^2$ is such that \n there does not exist a vertex between $(k,y_k^2)$ and $(t,y_t)$ in ${\\mathcal P}$ with $k$ as its first coordinate. \n\n\n Denote by ${\\mathcal P}_1$ (resp. ${\\mathcal P}_2)$ the subpath of ${\\mathcal P}$ with its initial vertex $(k,y_k^1)$ \n (resp. $(k,y_k^2)$) and its final vertex $(k,y_k^2)$ (resp. $(k,y_k^3)$) and define \n ${\\mathcal P}_0={\\mathcal P} \\, \\backslash \\, \\{{\\mathcal P}_1, {\\mathcal P}_2\\}$.\n In $\\Gamma$ there exist the path ${\\mathcal P}_1'$ parallel to \n ${\\mathcal P}_2$ starting at the $(k,y_k^1)$ and the path ${\\mathcal P}_2'$ parallel to \n ${\\mathcal P}_1$ ending at the $(k,y_k^3)$. Then, the initial vertex of ${\\mathcal P}'_2$ coincides with the final vertex of ${\\mathcal P}'_1$.\n Thus ${\\mathcal P}_0 \\cup {\\mathcal P}_1'\\cup {\\mathcal P}_2'$ is a path in $\\Gamma$ of the same length as ${\\mathcal P}$ and it contains only \n one vertex with its first coordinate $k$ before $(t,y_t)$.\n \n By repeating this swap for $k=1,2,\\ldots,t$, we obtain a path ${\\mathcal P}'$ such that it does not contain vertices\n $(x,y_x^1)$ and $(x,y_x^2)$, with $y_x^1 < y_x^2 \\leq y_t$ for all $x=1,2,\\ldots, t-1$. \n As argued above, this forces ${\\mathcal P}'$ to be a path in the $({\\mathcal N}_B,A)$-digraph $\\Gamma$ that contains vertices \n $(x,1)$ for all $x=1,2,\\ldots,t$. By Lemma \\ref{thm:path2} there exists a $B$-path $\\bask{k}$ such that \n $|{\\mathcal P}|=|{\\mathcal P}'| \\leq \n |\\bask{k}|=2k+\\mu_{k+1}+\\mu_{k+2}+\\ldots+\\mu_{k+r}$ for some $r$, $\\mu_k-\\mu_{k+r}\\leq 1$.\n \\end{proof}\n\n\\bigskip\n\n To prove Theorem \\ref{thm:main}, it only remains to show that the maximum of Theorem \\ref{thm:premain} is attained. \n\n\\bigskip\n \n\\begin{proof} (of {\\bf Theorem \\ref{thm:main}})\n Let $\\sh{B}=(\\mu_1,\\mu_2,\\ldots,\\mu_t)=(m_1^{r_1},m_2^{r_2},\\ldots,m_l^{r_l})$. We define the $B$-path ${\\mathcal P}$, for which the maximum \n  $$\\max\\limits_{1 \\leq i < t}\\{2i+\\mu_{i+1}+\\mu_{i+2}+\\ldots+\\mu_{i+r}; \\; \\mu_{i+1} - \\mu_{i+r}\\leq 1; \\; \\mu_{i} \\ne \\mu_{i+1}\\}$$\n is attained.\n \n Let $k$ and $s$ be such that \n $\\max\\limits_{1 \\leq i < t}\\{2i+\\mu_{i+1}+\\mu_{i+2}+\\ldots+\\mu_{i+r}; \\; \\mu_{i+1} - \\mu_{i+r}\\leq 1; \\; \\mu_{i} \\ne \\mu_{i+1}\\}=\n 2k+\\mu_{k+1}+\\mu_{k+2}+\\ldots+\\mu_{k+s}$.\n  \n \\smallskip\n \n Denote by  ${\\mathcal P}$ the $B_{(k+1)}$-path in $\\Gamma$. \n It follows that $|{\\mathcal P}|=2k+\\mu_{k+1}+\\mu_{k+2}+\\ldots+\\mu_{k+s}$.  \n The path ${\\mathcal P}$ is such that it can be completed (by drawing parallel edges) \n to an $({\\mathcal N}_B,A)$-digraph. (For examples of such $({\\mathcal N}_B,A)$-digraphs see Example \\ref{ex:2}.) \n By the proof of Theorem \\ref{thm:premain} it follows that ${\\mathcal P}$ is a longest path in $\\Gamma$.\n \n For a generic matrix $M_\\Gamma$, corresponding to $\\Gamma$, it follows that \n $\\nil{M}=|{\\mathcal P}|$. \n Take $A \\in {\\mathcal N}_B$ to be a positive matrix with the same pattern \n as $M$. \n Since $\\ch{\\mathbb{F}}=0$, it follows that $[A^h]_{ij} =0$ if and only if $[M^h]_{ij}=0$ for all $h$, $i$, $j$.\n Therefore $\\nil{A}=\\nil{M}=2k+\\mu_{k+1}+\\mu_{k+2}+\\ldots+\\mu_{k+s}$ which proves the theorem.\n\\end{proof}\n  \n \n\\bigskip\n\n\n\f\\section{Examples} \\label{sec:nilex}\n\n\\bigskip\n\n\\begin{example}\\label{ex:2}\nAs in Examples \\ref{ex:43221a} and \\ref{ex:43221b}, we again consider the case $\\sh{B}=(4,3,2^2,1)$. \nBy Theorem \\ref{thm:main} it \nfollows that $\\nil{{\\mathcal N}_B}=\\max \\{7,9,9\\}=9$. \n\nConsider the following matrices:\n$$\nA_1=\\left[\\begin{array}{cccc|ccc|cc|cc|c}\n0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n\\hline\n0 & d & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0  \\\\\n0 & 0 & d & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0  \\\\\n0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0  \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & e & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & e & 0 & 0 & 0 & 0 & 0  \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \n\\end{array}\\right],$$\n\n$$ A_2=\\left[\\begin{array}{cccc|ccc|cc|cc|c}\n0 & f & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & f & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & f & 0 & 0 & a & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n\\hline\n0 & d & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0  \\\\\n0 & 0 & d & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0  \\\\\n0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0  \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & e & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & e & 0 & 0 & 0 & 0 & 0  \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \n\\end{array}\\right] $$\nand\n$$\nA_3= \\left[\\begin{array}{cccc|ccc|cc|cc|c}\n0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n\\hline\n0 & d & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0  \\\\\n0 & 0 & d & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0  \\\\\n0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0  \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0  \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & e & 0 & 0 & 0 & 0 & 0 & f  \\\\\n0 & 0 & 0 & 0 & 0 & 0 & e & 0 & 0 & 0 & 0 & 0  \\\\\n\\hline\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & g & 0 & 0 & 0 \n\\end{array}\\right],\n$$\n\nwhere $a,b,c,d,e,f,g \\in \\mathbb{F}$ are nonzero.\nThe generic matrices $M_1$, $M_2$ and $M_3$ with the same patterns as $A_1$, $A_2$ and $A_3$\nhave the following digraphs\n \n        \\begin{figure}[htb]\n         \\begin{center}\n          \\includegraphics[height=8.2cm, width=12cm]{fig43221y}\n         \\end{center}\n         \\end{figure}\n\nIt can be easily seen that $\\sh{A_1}=\\sh{M_1}=(9,1^3)$, $\\sh{A_2}=\\sh{M_2}=(9,2,1)$\nand $\\sh{A_3}=\\sh{M_3}=(9,3)$.\n \\hfill$\\square$\n\\end{example}\n\n\\bigskip\n\n\\begin{example}\\label{ex:3}\nNot all $\\underline{\\lambda}=(\\lambda_1,\\lambda_2,\\ldots,\\lambda_k) \\in \\partition{n}$, where $\\lambda_1=\\nil{{\\mathcal N}_B}$,\nare in $\\partition{{\\mathcal N}_B}$. For $\\sh{B}=(6,4)$ it follows that $\\nil{{\\mathcal N}_B}=6$. It can be shown that\n$(6,3,1) \\notin \\partition{{\\mathcal N}_B}$.\n  \\hfill$\\square$\n\\end{example}\n\n \\bigskip\n \n  \n \\begin{example}\n Not all partitions in $\\partition{{\\mathcal N}_B}$ can be obtained from a generic matrix corresponding to\n an $({\\mathcal N}_B,A)$-digraph. \n Let $\\sh{B}=(5,3)$. It can be verified (for example, using {\\tt Mathematica}) that \n \\begin{align*}\n  \\partition{{\\mathcal N}_B}=&\n  \\big\\{(5,3), (5,2,1), (5,1^3), (4^2), (4,2^2), (4,2,1^2), (4,1^4), \\\\\n     & \\; (3^2,2), (3^2,1^2), (3,2^2,1), (3,2,1^3), (3,1^5), \\\\\n     & \\; (2^4),(2^3,1^2), (2^2,1^4), (2,1^6), (1^8)\\big\\}=\\\\\n     = & \\, \\rp{5,3} \\cup \\big\\{(4^2), (4,2^2), (4,2,1^2), (4,1^4), (3^2,1^2), (2^4) \\big\\}\\, .\n \\end{align*}\n One can check that there is no $({\\mathcal N}_B,A)$-digraph $\\Gamma$ with $\\Delta(\\Gamma)$ equal to \n $(4^2)$, $(3^2,1^2)$ or $(2^4)$. However, there are matrices with these shapes in ${\\mathcal N}_B$.\n For example, for\n $$\n A_1=\\left[\\begin{array}{ccccc|ccc}\n 0 & 1 & 0 & 0 & 0 &-1 & 0 & 0   \\\\\n 0 & 0 & 1 & 0 & 0 & 0 &-1 & 0   \\\\\n 0 & 0 & 0 & 1 & 0 & 0 & 0 &-1   \\\\\n 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0   \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0   \\\\\n\\hline\n 0 & 0 & 1 & 0 & 0 & 0 &-1 & 0   \\\\\n 0 & 0 & 0 & 1 & 0 & 0 & 0 &-1   \\\\\n 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \n \\end{array}\\right]$$\n and\n $$A_2=\\left[\\begin{array}{ccccc|ccc}\n 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0   \\\\\n 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0   \\\\\n 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1   \\\\\n 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0   \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0   \\\\\n \\hline\n 0 & 0 &-1 & 1 & 0 & 0 &-1 & 1   \\\\\n 0 & 0 & 0 &-1 & 1 & 0 & 0 &-1   \\\\\n 0 & 0 & 0 & 0 &-1 & 0 & 0 & 0  \n \\end{array}\\right]\\, .\n $$\n \n we have $\\sh{A_1}=(2^4)$, $\\sh{A_2}=(3^2,1^2)$. \n Let $M_i$, $i=1,2$, be generic matrices of the same shape as $A_i$. It can be proved, using\n Gansner-Saks Theorem, that $\\sh{M_1}=\\sh{M_2}=(5,3)$.\n \\hfill$\\square$\n\\end{example}\n\n\\bigskip\n\\bigskip\n\n\n\n\n \n \\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\\subsection{Electron-End-Cap Electromagnetic Calorimeter: EEMC}\nThe electron-end-cap calorimeter will to cover a dynamic energy range of 0.1--18 GeV for electromagnetic showers of the scattered electron based on e+p Pythia simulations at 18x275 GeV$^2$. \nThe choice of technology and detector dimensions are therefore optimized to provide the optimal performance for this expected energy range.\n\nThe EEMC is a high-resolution ECal designed for precision measurements of the energy of scattered electrons and final-state photons in the electron-going region. \nThe requirements for energy resolution in the backward region is driven by inclusive DIS where precise determination of the scattered electron properties is critical to constrain the event kinematics. \nThe EEMC is designed to address the requirements outlined in the EIC Yellow Report.\nIts baseline design is based on an array of approximately 3000 lead tungsten crystals (PbWO$_{4}$) $2\\times 2\\times 20$~cm$^3$ in size, which correspond to approximately 20~$X\/X_0$ longitudinally and a transverse size equal to the PbWO$_{4}$ Moli\\`ere radius. \nThe PbWO$_{4}$ crystal light yield is in the range of 15 to 25 photo-electrons per MeV, providing an excellent energy resolution of $\\sigma_E\/E\\approx2\\%\/\\sqrt{E} \\oplus 1\\%$ \\cite{Inaba:1994jd,Prokoshkin:1995rd} within a very compact design.\\\\\nThe EEEMCAL Consortium is leading the efforts to further develop the EEMC design concept and has summarized their intentions in an Expression Of Interest in 2021.\nThey have begun to organize activities into mechanical design, scintillator, readout, and software\/simulation among the collaborating institutions. \nPre-design activities of the mechanical support structure commenced in 2021 and a document on mechanical design and integration has been prepared \\cite{eeemcal:design}. \nThe concept is based existing detectors that the team has constructed, and in particular the Neutral Particle Spectrometer at Jefferson Lab. \\\\\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/design\/EIC_EEEMCAL_Beampipe_9.png}\n    \\caption{The Electron-End-Cap Calorimeter conceptual design and integration with the beampipe and surrounding detectors as prepared by the EEEMCAL Consortium \\cite{eeemcal:design}. \n    The EEMC consists only of PbWO$_{4}$ crystals and uses the displayed design concept. \n    }\n    \\label{fig:eemcal_config}\n    \\label{fig:eemcal_clearance}\n\\end{figure}\n\\Figure{fig:eemcal_config} shows an overview of the different components of the EEMC as prepared by the EEEMCAL Consortium \\cite{eeemcal:design}. \nIt has four main parts: the detector (PbWO$_{4}$ crystals), the mechanical structure (internal and external), cooling, and electronics (SiPM and cables). \nWith crystal dimensions of $2\\times2\\times20$~cm$^3$, a density of 8.28~g\/cm$^3$, and a mass of 0.6624~kg per crystal the total weight of the EEMC is slightly more than two metric tons. \nThe crystals are aligned and separated using carbon plates of thickness 0.5~mm. \nThe configuration for the first ring of PbWO$_{4}$ crystals depends on the final design of the beam pipe. \nIts minimum diameter will be on the order of 22.5~cm with an additional clearance gap. \nAn additional support and cooling structure with a maximum thickness of 5~mm will be needed to support the crystals directly above the beam pipe.\nFor which detailed calculations will have to be carried out once the final beam pipe design is available.\nThe EEMC is located inside the universal support frame, which also houses the Detection of Internally Reflected Cherenkov light (DIRC) detector \\cite{ecce-paper-det-2022-01}, and covers the pseudorapdity region of $-3.4<\\eta<-1.5$. \nThe main constraints for its acceptance are imposed by the surrounding detector systems and passive materials, as seen in \\Fig{fig:eemcal_config}.\nThe integration of the EEMC into the frame is only possible if the beam pipe is removed, which implies that the flange must be disconnected. \nTo improve the inner diameter of the EEMC and to improve the acceptance up to $-3.7<\\eta<-1.5$, an inner calorimeter is being considered. \nThis option also requires the modification of the overall structure of the EEMC to ensure no significant gaps in scattered electron detection between the electron-end-cap and barrel.\nOverall, the inner diameter of the EEMC will depend on the design of the beam pipe, and in particular the angle between the electron and the hadron tube.\\\\\nCurrently, the EEMC readout is based on silicon photomultipliers (SiPMs) of pixel sizes of 10$\\mu$m or 15$\\mu$m and a photosensitive area of $3\\times3$~mm$^2$. \nThere are two configuration options: 4 SiPM per crystal or 16 SiPM per crystal. \nSince a mechanical structure is required for mounting the PCBs, its width in turn will determine the positioning of the SiPMs.\nAssuming a machined grid with a width of about 5~mm the PCBs can be mounted with small screws. \\\\\nPbWO$_{4}$ crystals are sensitive to temperature changes with a variation of 2\\%\/$^\\circ$C in light output. \nThus, the specification is to keep the crystal temperature stable within $\\pm$0.1 $^\\circ$C.\nTo ensure this stability the additional heat generated by the electronics needs to be removed and the following cooling structures are being considered.\nAs internal cooling structure several machined copper blocks with internal coolant circulation will be used around the beam pipe.\nTo reduce the spatial extend support structures the EEEMC consortium is moreover planning to use cooling plates in between the readout cables which are linked to the support structure surrounding the EEMC with tubes.\nThis system is composed of 12 plates with a 5-8~mm spacing in which water can be circulated. \nThe cooling near the crystals will likely not be enough to meet specification.\nThese challenges could be overcome by outside cooling with standard cooling blocks with airflow in front of the electronics or additional cooling added at the back of the assembly. \nThe main constraint is the space available in the electron end-cap. \\\\\nThe mechanical integration of the EEMC presently envisions that the detector is assembled in its own support structure, mounted on a platform, and then inserted into the universal support frame. \nThe detailed steps and main points of the assembly are described in Ref.~\\cite{eeemcal:design}. \nThe mechanical integration starts when the assembly is complete. \nThe platform is adjusted on rails with an additional support to link the support to the detector. \nThe platform is removed once the EEMC is mounted in the universal support frame. \nClearances of at least 5~mm on all sides between the EEMC and the universal support frame are required to perform maintenance without lifting the detector. \n\\begin{figure*}[ht!]\n    \\centering\n    \\includegraphics[height=0.22\\textheight]{figs\/design\/tower_bemc_unframed.png}\n    \\includegraphics[height=0.22\\textheight]{figs\/design\/becal_eta.png}\n    \\includegraphics[height=0.22\\textheight]{figs\/design\/becal_phi.png}\n    \\caption{Left: Single BEMC tower as implemented in the GEANT4 simulations. \n    Middle: BEMC projective tower layout in $\\eta$. \n    The towers are centered at $z = -10$~cm. \n    Right: BEMC layout layout as a function of $\\varphi$ for one $\\eta$ slice.\n    The towers are tilted in $\\varphi$ by $10^\\circ$ to avoid channeling in the gaps between adjacent towers.\n    }\n    \\label{fig:becal_eta}\n    \\label{fig:becal_phi}\n    \\label{fig:becal_tower}\n\\end{figure*}\n\n\\subsection{Barrel}\nBased on Pythia simulations of e+p collisions at 18x275 GeV$^2$, the expected energy range of particles at mid-rapidity is 0.1--50 GeV in high $Q^2$ events. \nThe ECCE detector therefore requires calorimeters that can cover these expected energies for electromagnetic and hadronic shower reconstruction.\n\\paragraph{Barrel Electromagnetic Calorimeter: BEMC} \\mbox{ }\\\\\nThe barrel electromagnetic calorimeter is designed to cover the central region of the detector ($-1.72 < \\eta <1.31$). \nIts total length along the $z$-axis is 584~cm and the detector is fully contained within solenoid magnet, but positioned at a larger radial position than the DIRC detector.\nThe absolute radial position of the calorimeter is $85<R< 135$~cm from the beampipe, where the inner radius is fixed for all towers but the outer radius varies depending of the position in eta due to the required projective design. \\\\\nThe calorimeter is composed of 8960 towers made out of scintillating glass (SciGlass), which are organized in 128 towers per $\\varphi $ slice and 70 blocks in the $\\eta$ direction.\nThe middle and left panels of \\Fig{fig:becal_eta} show $x$ and $z$ slices of the BEMC geometry as it is modeled in GEANT4. \nThe colors show the different $\\eta $ towers, and the variation in the outer radius can be observed. \nThe interaction point is marked by a red star, from which the left and right directions in the figure correspond to the electron and hadron-going sides, respectively. \nThe detector is designed asymmetric towards the electron and hadron sides, resulting in a length of 389 and 195~cm on the respective sides.\nThe BEMC is designed with offset projectivity in $\\eta $ and $\\varphi$ in order to avoid channeling of particles produced in the collision within the passive material between the towers.\nThus the front face of the towers is tilted such that it is facing the interaction point shifted by $\\Delta z = -10$~cm.\nThis requires that the tower tilt angle depends on its location within the calorimeter.\nAdditionally, the towers have a stronger inclination at higher absolute pseudorapidities, leading to an asymmetric tapered shape of the glass blocks, which increases with $|\\eta|$. \nAll the towers are tilted by $\\Delta\\varphi=10 ^{\\circ}$ to avoid any gaps in $\\varphi$ and further tunneling of particles through inactive detector material. \nA summary of all BEMC detector parameters is given in Table~\\ref{tab:becalpar}.\\\\\n\\begin{table}[!t]\n  \\centering\n  \\begin{tabular}{lr}\n    \\toprule\n    Parameter & Value \\\\\n    \\midrule\n    Inner radius (envelope)   & 85 cm \\\\\n    Outer radius (envelope)    & max. 135 cm ($\\eta-\\text{dependent})$\\\\\n    Length (envelope)           & 584 cm ($-389<z<195$ cm) \\\\\n    Pseudorapidity coverage & $-1.72<\\eta<1.31$\\\\\n    Active material                &  SciGlass \\\\\n    \n    \\# towers in azimuth        & 128 \\\\\n    \\# towers in pseudorapidity & 70 \\\\\n    Tower dimensions \\\\\n    \\quad inner face: & $4\\times4$ cm\\\\\n    \\quad length: & $45.5$ cm\\\\\n    \\quad outer face ($\\eta=0$): & $5\\times5$ cm\\\\\n    \\quad outer face ($|\\eta|>1.1$): & $6.6\\times6.6$ cm\\\\\n   \n    $\\eta$ projectivity point & $z=-10$ cm \\\\\n    $\\varphi$ projectivity tilt & $10^\\circ$ \\\\\n    Sampling fraction & 0.97\\\\\n    Tower depth & $X\/X_0\\approx16.0$ \\\\\n    Moli\\`ere radius & $R_\\mathrm{M}=3.58$ cm\\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{Design parameters for the barrel electromagnetic calorimeter BEMC.}\n  \\label{tab:becalpar}\n\\end{table}\nThe layout for a single tower around $\\eta = 0$ is shown in \\Fig{fig:becal_tower}~(left).\nAll towers currently have an inner size of $4 \\times 4$~cm$^2$ and the same length of 45.5~cm, which corresponds to approximately 16 $X\/X_0$.\nHowever, their outer face dimension varies from $5^2$ to $6.6^2$~cm$^2$ depending on their position in $|\\eta|$.\nIn addition, the considered SciGlass towers have a Moli\\`ere radius of 3.58~cm, which is approximately double the transverse tower size.\nEach tower is composed of a SciGlass core, surrounded by a 1~mm carbon fiber enclosure. \nThe electronics are currently modeled by Kapton, SiO$_{2}$ and carbon fiber layers in the outer part of the blocks. \nThe SciGlass block length is optimized to contain at least $95\\%$ of the energy of a 10 GeV electron, whilst still fitting into the BABAR 1.5T magnet with at most an inner radius of 80~cm and at least 8~cm space for the electronics and support structure.\nThe electron energy mentioned above corresponds to the average scattered electron energy in the BEMC acceptance. \nConstraining the BEMC to not stretch further into the detector allows for more space for other PID and tracking detectors which are necessary for electron, pion, kaon and proton separation. \nIn particular, towards the electron end cap (negative $\\eta$) it could be studied in the future, whether the tower depth could be increased up to 60~cm for higher $|\\eta|$ to decrease the energy leakage for high energetic electrons, which are more probable in this region.\nHere the projective design allows for such an extension at least for parts of the calorimeter. \n\n\\paragraph{Barrel Hadronic Calorimeter: IHCAL \\& OHCAL}\\mbox{ }\\\\\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=0.55\\linewidth]{figs\/design\/OuterHCalSectorWithTiles.png}\n    \\includegraphics[width=0.35\\linewidth]{figs\/design\/OuterHcal-Dimensions-v3-2-crop.pdf}\n    \\caption{Left: Scintillator tiles in a layer of the OHCAL. \n    Right: Transverse cutaway view of an OHCAL module, showing the tilted tapered absorber plates.\n    Light collection and cabling is on the outer radius at the top of the drawing.}\n    \\label{fig:inner_tile_pattern}\n    \\label{fig:hcal-6}\n\\end{figure*}\nThe Outer Hadronic Calorimeter (OHCAL) will be reused from the sPHENIX HCal~\\cite{sPHENIX:2017lqb}, which instruments the large steel-based barrel flux return of the BABAR magnet.\nThe Inner Hadronic Calorimeter (IHCAL), as currently implemented in ECCE, is very similar in design to the sPHENIX inner HCAL in that it instruments the support for the barrel HCal to provide an additional longitudinal segment of hadronic calorimetry. \nThe IHCAL provides useful data for overall calibration of the combined calorimeter system. \\\\\nIn the following, the construction of the scintillating tiles used in the outer and inner HCals is described, followed by a mechanical description of each calorimeter system.\\\\\n\\begin{table}[!t]\n  \\centering\n  \\begin{tabular}{lr}\n    \\toprule\n    Parameter & Value \\\\\n    \\midrule\n    Inner radius (envelope)     & 1820 mm\\\\\n    Outer radius (envelope)     & 2700 mm\\\\\n    Length (envelope)           & 6316 mm\\\\\n    Material                       & 1020 steel \\\\\n    \\# towers in azimuth ($\\Delta \\varphi$)         & 64 \\\\\n    \\# tiles per tower                           & 5 \\\\\n    \\# towers in pseudorapidity ($\\Delta \\eta$)  & 24 \\\\\n    \\# electronic channels (towers)              & $64 \\times 24 = 1536$ \\\\\n    \\# optical devices (SiPMs)                   & 5 $\\times$ 1536 = 7680 \\\\\n    \\# modules (azimuthal slices)                & 32 \\\\\n    \\# towers per module  & $2 \\times 24 = 48$ \\\\\n    Total \\# absorber plates  & $5 \\times 64 = 320$ \\\\\n    Tilt angle (relative to radius)   & 12$^{\\circ}$ \\\\\n    Absorber plate thickness at inner radius  & 10.2 mm\\\\\n    Absorber plate thickness at outer radius  & 14.7 mm\\\\\n    Gap thickness  & 8.5 mm\\\\\n    Scintillator thickness  & 7 mm\\\\\n    Module weight  & 12247 kg\\\\\n    Sampling fraction  & 0.035\\\\\n    Calorimeter depth & 4.0$\\lambda\/\\lambda_0$ \\\\\n    Moli\\`ere radius $R_M$ for $\\pi^\\pm$ & 14.4 cm\\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{Design parameters for the Outer Hadronic Calorimeter (OHCAL).}\n  \\label{tab:ohcalpar}\n\\end{table}\n\\begin{table}[!t]\n    \\centering\n    \\begin{tabular}{lr}\n    \\toprule\n    Parameter & Value \\\\\n    \\midrule\n    Inner radius (envelope)                       & 1350 mm\\\\\n    Outer radius (envelope)                       & 1385 mm\\\\\n    Material                                      & 310 stainless steel \\\\\n    \\# towers in azimuth ($\\Delta \\varphi$)             & 64 \\\\\n    \\# towers per module                             & $2 \\times (12+15) = 56$ \\\\\n    \\# tiles per tower                               & 4 \\\\\n    \\# towers in pseudorapidity ($\\eta > 0$)         & 24 \\\\\n    \\# towers in pseudorapidity ($\\eta < 0$)         & 30 \\\\\n    \\# electronic channels (towers)                  & $64 \\times 27 = 1728$ \\\\\n    \\# optical devices (SiPMs)                       & 4 $\\times$ 1728 = 6912 \\\\\n    Tilt angle (relative to radius)                   & 32 $^{\\circ}$ \\\\\n    Absorber plate thickness                          & 13 mm\\\\\n    Gap thickness                                     & 8.5 mm\\\\\n    Scintillator thickness                            & 7 mm\\\\\n    \\# modules (azimuthal slices)              & 32 \\\\\n    Sampling fraction                                 & 0.059\\\\\n    Calorimeter depth &  0.17$\\lambda\/\\lambda_0$ \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\caption{Design parameters for the Inner Hadronic Calorimeter (instrumented frame) for ECCE.}\n    \\label{tab:ihcalpar}\n\\end{table}\nThe basic calorimeter concept for the IHCAL\/OHCAL is a sampling calorimeter with absorber plates tilted in the radial direction.\nThis design provides more uniform sampling in azimuth and provides information on the longitudinal shower development. \nThe current design uses tapered plates for the OHCAL and non-tapered plates for the IHCAL.\nBased on detailed studies, this design choice lowers the IHCAL machining cost without decreasing its performance.\nExtruded tiles of plastic scintillator with an embedded wavelength shifting fiber are interspersed between the absorber plates and read out at the outer radius with SiPMs.\nThe tilt angle is chosen so that a radial track from the center of the interaction region traverses at least four scintillator tiles. \nEach tile is read out by a single SiPM, and the analog signal from each tile in a tower (five for the OHCAL, four for the IHCAL) are ganged to a single preamplifier channel to form a calorimeter tower. \nTiles are divided in slices of $\\eta$ so that the overall segmentation is $\\Delta \\eta \\times \\Delta \\varphi \\approx 0.1 \\times 0.1$.\\\\\nThe scintillating tiles are similar to the design of the scintillators for the T2K experiment by the INR group (Troitzk, Russia) who designed and built 875~mm long scintillation tiles with a serpentine wavelength shifting fiber readout~\\cite{Izmaylov:2009jq}.\nSimilar extruded scintillator tiles were also developed by the MINOS experiment. \nThe properties of the HCal scintillating tiles and of the WLS fibers are detailed in Ref.~\\cite{sPHENIX:2017lqb}.\nThe Kuraray single clad fiber is chosen due to its flexibility and longevity, which are critical in the geometry with multiple fiber bends.\\\\\nThe OHCAL is north-south symmetric and requires 24 tiles along the $\\eta$ direction, whereas the IHCAL is asymmetric and has 12 towers in the forward direction and 15 towers in the backward direction. \nThe OHCAL design therefore requires 12 different shapes of tiles for each longitudinal segment.\n\\Figure{fig:inner_tile_pattern} shows the tile and embedded fiber pattern for the OHCAL.\\\\\nThe major components of the OHCAL are tapered steel absorber plates and 7680 scintillating tiles which are read out with SiPMs along the outer radius of the detector.\nThe detector consists of 32 modules, which are wedge-shaped sectors containing 2 towers in $\\varphi$ and 24 towers in $\\eta$ equipped with SiPM sensors, preamplifiers, and cables carrying the differential output of the preamplifiers to the digitizer system on the floor and upper platform of the detector.\nEach module comprises nine full-thickness absorber plates and two half-thickness absorber plates, so that as the modules are stacked, adjoining half-thickness absorber plates have the same thickness as the full-thickness absorber plates.\nThe tilt angle is chosen to be 12 degrees relative to the radius, corresponding to the geometry required for a ray from the vertex to cross four scintillator tiles.\nTable~\\ref{tab:ohcalpar} summarizes the major design parameters of the OHCAL, which are illustrated in \\Figure{fig:hcal-6}.\nSince the OHCAL will serve as the flux return of the solenoid, the absorber plates are single, long plates running along the field direction.\nThe IHCAL occupies a radial envelope bounded by a 50~mm clearance inside the solenoid cryostat and the outer radius of the BEMC.\nThe inner radius provides support for the BEMC and the HCal, while the end of the structure carries load to the OHCAL.\\\\\nTable~\\ref{tab:ihcalpar} shows the basic mechanical parameters of the IHCAL reference design.\nThe detector is designed to be built in 32 modules, which are wedge-shaped sectors comprising 8 gaps with 7 full-thickness plates and 2 half-thickness plates (so that as the modules are stacked, adjoining half-thickness plates have the same thickness as the full-thickness plates).\nThe modules contain 2 towers in $\\varphi$ and 27 towers in $\\eta$ equipped with SiPM sensors, preamplifiers, and cables carrying the differential output of the preamplifiers to the digitizer system on the floor and upper platform of the detector.\nThe instrumentation consists of 6912 scintillating tiles and optical devices, 1728 preamplifiers, and cabling.\n\n\\subsection{Hadron-End-Cap}\nWe envision the forward calorimeter system as an integrated ECal and HCal, where the installation units, where appropriate, are constructed in a common casing. \nThese so-called modules consist of an electromagnetic calorimeter segment in the front which is part of the forward EMCal (FEMC) followed by a HCal segment which is part of the longitudinally separated HCal (LFHCAL). \nIn between these segments a read-out section is foreseen for the ECal. \nThe modules of up to 4 different sizes will be installed in half shells surrounding the beam pipe, which are movable on steel trolleys to give access to the inner detectors in the barrel in the hadron going direction. \nEach of these trolley should carry about 150 metric tons of weight.\nThis integrated ECal and HCal design reduces the dead material in the detector acceptance and allows for an easier installation in the experimental hall.\nThis implies that the construction of the modules has to happen in the same location to reduce shipping and assembly costs.\nIn the following, details on the FEMC will be discussed, followed by the design considerations and plans for the longitudinally separated HCal. \\\\\n\\begin{table}[t!]\n  \\centering\n  \\small\n  \\begin{tabular}{lll}\n    \\toprule\n                      parameter &\\textbf{FEMC}  &\\textbf{LFHCAL}\\\\\n        \\midrule\n        inner radius (envelope) & 17 cm  & 17 cm\\\\\n        outer radius (envelope) & 170 cm & 270 cm\\\\\n        $\\eta$ acceptance       & $1.3 < \\eta < 3.5$ & $1.2 < \\eta < 3.5$\\\\\n        tower information \\\\\n        \\hspace{1em} x, y ($R <$\/$> 0.8$ m)     & 1 cm\/ 1.65 cm & 5 cm\\\\\n        \\hspace{1em} z (active depth) & 37.5 cm & 140 cm\\\\\n        \\hspace{1em} z read-out       & 5 cm & 20 cm\\\\\n        \\hspace{1em} $\\#$ scintillor plates   & 66 (0.4 cm each) & 70 (0.4 cm each)\\\\\n        \\hspace{1em} $\\#$ aborber sheets          & 66 (0.16 cm Pb) & 60 (1.6 cm steel)\\\\\n                                    &   & 10 (1.6 cm tungsten)\\\\\n        \\hspace{1em} weight                    & $\\sim 6.4$ kg & $\\sim 30.6$ kg\\\\\n        \\hspace{1em} radiation lengths & 18.5 $X\/X_0$ & - \\\\\n        \\hspace{1em} interaction lengths & 1.0 $\\lambda\/\\lambda_0$ & 6.9 $\\lambda\/\\lambda_0$\\\\\n        Moli\\`ere radius $R_M$ & 5.2 cm (e$^\\pm$ shower) & 21.1 cm ($\\pi^\\pm$ shower)\\\\\n        Sampling fraction $f$ & 0.220 & 0.040\\\\\n        $\\#$ towers (inner\/outer)         & 19,200\/ 34,416 & 9040\\\\\n        $\\#$ read-out channels      & 53,616 & 7 x 9,040 = 63,280\\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{Overview of the calorimeter design properties for the FEMC and the LFHCAL.}\n  \\label{tab:fwdcaloproperties}\n  \\centering\n  \\small\n  \\begin{tabular}{lll}\n    \\toprule   \n        \\textbf{Assembly Module Type} & \\textbf{\\# modules} \\\\\n        \\midrule\n        8 LFHCAL tower modules (8M) & 1091 (total)\\\\\n        \\hspace{1em}    no FEMC towers in front & 538\\\\\n        \\hspace{1em}    200 FEMC towers (inner) & 87\\\\\n        \\hspace{1em}    72 FEMC towers (outer) & 466\\\\\n        4 LFHCAL tower modules (4M)  & 76 (total)\\\\\n        \\hspace{1em}    no FEMC towers in front & 36\\\\\n        \\hspace{1em}    100 FEMC towers (inner) & 16\\\\\n        \\hspace{1em}    36 FEMC towers (outer) & 24\\\\\n        2 LFHCAL tower modules (2M) & 2 (total)\\\\\n        \\hspace{1em}    50 FEMC towers (inner) & 2\\\\\n        1 LFHCAL tower modules (1M) & 4 (total)\\\\\n        \\hspace{1em}    25 FEMC towers (inner) & 4\\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{Number of assembly modules for the full combined FEMC and LFHCAL detector.}\n  \\label{tab:fwdcalomodules}\n\\end{table}\nBoth detector systems need to be able to handle the expected energies of incoming particles up to 150 GeV, based on simulated Pythia events for e+p collisions at 18x275 GeV$^2$. \nDue to the asymmetric collision system, these calorimeters are therefore focused strongly on high energetic particle shower containment while still providing good energy resolution down to lower energies.\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/design\/LFHCAL_FEMC_compilation_propsal_modified.png}  \n    \\caption{Design pictures of the forward calorimeter assembly (left), 8-tower module design (top right) and single scintillator plates of the LFHCAL (bottom middle) and FEMC (bottom right) for an 8M tower module with embedded wavelength shifting fibers.}\n    \\label{fig:FCALdesign}\n\\end{figure*}\n\n\\paragraph{Hadron End-Cap Electromagnetic Calorimeter: FEMC}\\mbox{ }\\\\\n\\label{sec:FEMC}\nThe forward ECal (FEMC) is a Pb-Scintillator shashlik calorimeter. \nIt is placed at a distance of $z=3.07$ m from the interaction point in the hadron-going direction after the tracking and particle identification detectors. \nThe detector is made up of two half disks with a radius of about 1.7m. \nThe calorimeter is based on traditional Pb-Scint-Shashlik calorimeter designs like they have previously been used in ALICE, STAR and PHENIX.\nHowever, it employs more modern techniques for the readout and the scintillation tile separation. \\\\\nIts towers have an active depth of 37.5~cm with additional space for the readout of about 5~cm.\nEach tower consists of 66 layers of alternating 0.16~cm Pb sheets and 0.4~cm scintillator material, as listed in \\Table{tab:fwdcaloproperties}.\nDue to the high occupancy of the detector at large pseudorapities and the collimation of the particles in this area in physical space, the tower size varies depending on the radial position with respect to the beam axis.\nTowers which are close to the beam pipe ($R < 0.8$~m) have an approximate tower size of $1 \\times 1 \\times 37.5$~cm$^3$. \nFor the outer radii this granularity is not necessary and thus the size is increased to $1.65 \\times 1.65 \\times 37.5$~cm$^3$.\nThese numbers are intentionally well below the Moli\\`ere radius of $R_m=5.18$~cm, thus showers will spread transversely over multiple towers.\nIn order to collect the light produced in the scintillator tiles, each scintillator and Pb-plate is pierced by four 0.2~mm diameter wavelength shifting fibers. \nThese fibers are used to collect the light generated in the scintillators across all 66 layers. \nAll four fibers are read out together by a single SiPM. \\\\\nMultiple towers are contained in modules of either $20 \\times 10$~cm$^2$ (8M), $10 \\times 10$~cm$^2$ (4M), $5 \\times 10$~cm$^2$ (2M) or $5 \\times 5$~cm$^2$ (1M) in size. \nThese module sizes match the 8-, 4-, 2- and 1-tower modules of the LFHCAL with which they share a 1.5~mm thin steel enclosure. \nDepending on the radial position, the FEMC packs 72 or 200 read-out towers in an 8M module. \nDue to the integration of the FEMC towers in the LFHCAL modules, the combined ECal and HCal modules are about 2.05 m long. \nA detailed drawing of the 8M inner scintillator tile design for the FEMC can be found in \\Figure{fig:FCALdesign} (bottom right). \nThe full 8M tile is made out of one piece.\nIn order to separate the light produced in different segments of the 8M-tile, the tile surface is subdivided into $1\\times1$~cm$^2$ readout segments by CNC cutting or edging into the scintillator using a laser.\nThese 0.37~mm deep gaps (about 92\\% of the tile thickness) are then refilled with a mixture of epoxy and Titanium-oxide (TiO$_{2}$) in order to reduce the light cross talk among different towers. \nThe 4 fibers per tower are combined in a small light-collecting prism, which is directly attached to the SiPM with an effective photosensitive area of 9-16~mm$^2$ (ie. Hamamatsu S14160-3050HS). \nThese SiPMs are most sensitive around wavelengths of 450~nm, thus the wave length shifting fibers have to be chosen accordingly to peak in a similar region.\\\\\nThe first signal processing happens after the ECal part of the module within the 5~cm space currently assigned for the FEMC read-out, realized using CMS HGCROC chips mounted on custom PCBs \\cite{Thienpont:2020wau}, which can simultaneously process 72 channels. \nThe signals are then transmitted via fiber optic cables to the end of the module for further processing.\\\\\nA first full mechanical design for the joint LFHCAL and FEMC inner 8M module can be seen in \\Figure{fig:FCALdesign}. \nAdditionally, a first full illustration of a half shell is shown. \nThe higher granular 8M and 4M FEMC-LFHCAL modules are indicated in green and red respectively, while the yellow and dark blue towers show the lower granularity 8M and 4M FEMC-LFHCAL modules. \nThe lighter blue and orange modules reflect the modules only containing LFHCAL towers.\\\\\nThe majority of the FEMC is build of 8M modules, supplemented by 4M, 2M and 1M modules as outlined in \\Table{tab:fwdcalomodules} to come closer to the beam pipe and allow for a vertical separation of the two half shells. \nThe entire detector consists of approximately 53600 readout channels and provides a measurement of the energy of photons and electrons created in the collision going in the hadron-going (forward) direction. \n\n\\paragraph{Hadron-End-Cap Hadronic Calorimeter: LFHCAL}\\mbox{ }\\\\\n\\label{sec:LFHCAL}\nThe longitudinally separated forward HCal (LFHCAL) is a Steel-Tungsten-Scintillator calorimeter. \nThe initial idea is based on the PSD calorimeter employed in the forward direction for the NA61\/SHINE experiment \\cite{Guber:109059}, but it has been extensively modified to meet the desired physics performance laid out in the Yellow Report. \nThis longitudinally separated HCal is positioned after the tracking and PID detectors at $z=3.28$m from the center of the detector and is made up of two half disks with a radius of about 2.6m.  \\\\\nThe LFHCAL towers have an active depth of $\\Delta z =1.4$ m with an additional space for the readout of about 20-30~cm depending on their radial position, as summarized in \\Table{tab:fwdcaloproperties}. \nEach tower consists of 70 layers with alternating 1.6~cm absorber and 0.4~cm scintillator material and has transverse dimensions of $5 \\times 5$~cm$^2$.\nFor the first 60 layers the absorber material is steel, while the last 10 layers serve as tail catcher and are thus made of tungsten to maximize the interaction length within the available space.  \\\\\nIn each scintillator, a loop of wavelength shifting fiber is embedded, as can be seen in \\Figure{fig:FCALdesign} (bottom center). \nTen consecutive fibers in a tower are read out together by a single SiPM, leading to 7 samples at different depth per tower. \nThe towers are constructed in units of 8-, 4-, 2- and 1-tower modules to ease the construction and to reduce the dead space between the towers. \nSimilar to the FEMC, the scintillator tiles in the larger modules are made out of one piece and then separated by gaps refilled with epoxy and Titanium oxide to reduce light cross-talk among the different readout towers. \nThe wavelength shifting fibers running on the sides of the towers are grouped early on according to their readout unit and separated by thin plastic pieces over the full length.\nThe corresponding fiber bundles are indicated in \\Figure{fig:FCALdesign} by different colors. \nThey terminate in one common light collector, which is directly attached to a SiPM with an effective photosensitive area of 9-16~mm$^2$ (ie.\\ Hamamatsu S14160-3050HS).\nThese 7 SiPMs per tower are then read out by a common readout design between the FEMC and LFHCAL based on the CMS HGCROC chips.\nAlternatively, a common readout board which could be used for nearly all ECCE calorimeters is being pursued.\nThe entire detector consists of 63280 readout channels grouped in 9040 read-out towers and provides a measurement of the energy of hadronic particles created in the collision in the hadron-going (forward) direction.\\\\\nThe majority of the calorimeter is built of 8-tower modules ($\\sim$1091) which are stacked in the support frame using a lego-like system for alignment and internal stability.\nThe remaining module sizes are necessary to fill the gaps at the edges and around the beam pipe to allow for maximum coverage. \nThe absorber plates in the modules are held to their frame by 4 screws each. To leave space for the read-out fibers, the steel and scintillator plates are not entirely square but equipped with 1.25~mm notches, creating the fiber channels on the sides, as can be seen in \\Figure{fig:FCALdesign} (bottom center) for a scintillator plate. \nIn order to protect the fragile fibers, the notched fiber channels are covered by 0.5~mm thin steel plates after module installation and testing\nFor internal alignment we rely on the usage of 1-2~cm steel pins in the LFHCAL part which are directly anchored to the steel or tungsten absorber plates. \nConsequently, the modules are self-supporting within the outer support frame. \nThe support frame for the half disks is arranged on rails which allows the HCal and ECal to slide out to the sides and gives access to the inner detectors.\nIn addition, the steel in the LFHCAL serves as flux return for the central 1.5T BABAR magnet.\nAs a consequence, a significant force is exerted on the calorimeter, which needs to be compensated for by the frame and internal support structure.  \n\n\\section{Calorimeter Performance}\n\\label{sec:performance}\nThe ECCE electromagnetic and hadronic calorimeters are designed to meet the criteria outlined in the Yellow Report.\nIn the following, the expected performance of the different systems is presented based on standalone and full detector GEANT4 simulations.\n\\begin{figure}[t]\n    \\includegraphics[width=\\linewidth]{figs\/analysis\/clusterizationEIC_paper.pdf}\n    \\caption{Clusterization algorithms visualized on an example energy deposit from two particles ($E_1^\\mathrm{true}=6$ GeV and $E_2^\\mathrm{true}=4$ GeV) in calorimeter towers of arbitrary size. \n    Presented are the AA (Aggregate-All) and the MA (Modified-Aggregation) clusterizers.\n    The found clusters are outlined in color and their reconstructed energy is indicated in the figure. \n    The same seed and aggregation energy thresholds are assumed for both algorithms in this example. }\n    \\label{fig:clusterizersEIC}\n\\end{figure}\n    \n\\subsection{Clusterization}\n    The energy deposit from an electromagnetic or hadronic shower is generally spread over multiple towers.\n    The magnitude of this effect depends on the tower size relative to the Moli\\`ere radius ($R_M$) of the material used and is more prominent for hadronic showers.\n    The Moli\\`ere radius is defined as the radius in which 90\\% of the shower energy is contained, where electron-induced showers are used for the ECals and charged pion-induced showers are used for the HCals.\n    Since $R_M$ is in all cases larger than the individual tower sizes in the different calorimeters, it becomes apparent that the full shower can only be reconstructed when the information of multiple towers is combined.\n    Different reconstruction algorithms can be employed in order to group towers containing energy deposits into so-called clusters, which are the main objects used in physics analyses.\n    The performance of these algorithms mostly depends on the calorimeter occupancy for a given event. \n    While showers from single electromagnetic particles are mostly trivial to reconstruct, a significant challenge is posed by overlapping particle showers, for example in a jet or from high energetic neutral meson decays. \n    In the latter case, the decay photon showers, e.g. from $\\pi^0\\rightarrow\\gamma\\gamma$, can not be separated within the calorimeter granularity above a certain particle energy due to the decay kinematics.\n    Thus, extensive studies were performed to increase the separation power between single and multi-particle showers and to absorb as much of the deposited energy as possible during the so-called clusterization procedure.\n    This procedure always starts with the highest energetic tower in the calorimeter, which is required to contain an energy deposit above a seed energy threshold ($E_\\mathrm{seed}$).\n    \\begin{figure}[t!]\n      \\centering\n     \n      \\includegraphics[width=0.48\\textwidth]{figs\/analysis\/NTowAndMeanClus_Paper.pdf}\n      \\caption{Mean number of clusters per generated particle (a) and average number of towers aggregated within a cluster (b)  as a function of generated particle energy using the MA clusterizer for the different ECCE calorimeters. }\n      \\label{fig:ntowandncluspartFEMC}\n    \\end{figure}\n    \\begin{figure*}[t!]\n      \\centering\n     \n      \\includegraphics[width=\\textwidth]{figs\/analysis\/ClusRecEffEta_E+HCals_PaperPlotAlt.pdf}\\\\\n      \\caption{Cluster reconstruction efficiencies in the EEMC, BEMC and FEMC for electrons (a--c) and in the OHCAL and LFHCAL for charged pions (b and c) reconstructed with the MA-Clusterizer. \n      The efficiencies are calculated according to Equation~\\ref{eq:caloeffi}.}\n      \\label{fig:clusterreceffiEBF}\n    \\label{fig:clusterreceffiHCLF}\n    \\end{figure*}\n    Additional neighboring towers are added to the cluster if their energy exceeds a certain aggregation threshold ($E_\\mathrm{agg}$).\n    The thresholds ($E_\\mathrm{seed}$ and $E_\\mathrm{agg}$) for the different ECals and HCal have been optimized to reduce false seeding from minimum ionizing particles and to suppress noise during aggregation.\n    Their values are tower size and calorimeter type dependent, with approximate values of $E_\\mathrm{seed}^\\mathrm{ECal} = 100$ MeV and  $E_\\mathrm{agg}^\\mathrm{ECal}=$ 5--10 MeV or $E_\\mathrm{seed}^\\mathrm{HCal} =$ 100--500 MeV and $E_\\mathrm{agg}^\\mathrm{HCal}=$ 5--100 MeV for the ECals or HCals, respectively.\n    Two main algorithms have been explored for the cluster reconstruction: the so-called aggregate-all (AA) and modified-aggregation (MA) clusterizers. \n   \n    The AA clusterizer associates all towers sharing a common side with already aggregated towers in the cluster and only stops the aggregation when no further tower above $E_\\mathrm{agg}$ can be found.\n    At this point, the already aggregated towers are removed from the sample and a new seeding starting from the next highest energetic tower is performed.\n    Since this approach can aggregate energy deposits from multiple particles depending on the occupancy, a subsequent splitting of the cluster should be performed based on the number of maxima found in the energy distribution.\n    This cluster splitting procedure is necessary when AA clusters are meant to be used for single particle analyses.\n   \n   \n   \n    The MA clusterizer works similar to the AA clusterizer, however the algorithm stops when a neighboring tower with larger energy than the already aggregated tower is found.\n    It also aggregates towers that share a common corner, thus a $3 \\times 3$ tower window around each already aggregated tower is inspected.\n    This algorithm is preferred for the reconstruction of hadronic showers in high granularity calorimeters, since the energy deposits can fragment over a large amount of towers.\n    \\Figure{fig:clusterizersEIC} shows these algorithms applied to an example energy deposit in a calorimeter, where different clusters are reconstructed based on the various aggregation conditions.\n    The MA clusterizer is also the only clusterizer employed in the LFHCAL cluster reconstruction due to the additional z-segmentation of the calorimeter.\n    For this, the MA clusterizer also allows the inclusion of neighboring towers in z-direction sharing an edge or corner with already aggregated towers.\\\\\n    As can be seen in \\Fig{fig:ntowandncluspartFEMC}, the various ECCE calorimeters show visible differences in the average number of towers they aggregate per MA-based cluster.\n    In addition, the top panel of the same figure shows the mean number of clusters per generated particle, which is approximately one at low energies for the MA clusterizer, but reaches up to two clusters per particle at higher energies for the LFHCAL and FEMC.\\\\    \n    Overall it was found that the MA clusterizer performs slightly better than the AA clusterizer for all calorimeters and especially in events with a higher occupancy in the different detectors.\n    The MA clusterizer is therefore chosen as default for the following detector performance studies.\\\\\n    An important property of any clusterization algorithm and calorimeter is the efficiency with which a cluster can be reconstructed for any given particle.\n    \\Fig{fig:clusterreceffiEBF} shows the reconstruction efficiencies for electrons and charged pions for the different ECals and HCal as a function of generated particle pseudorapidity calculated according to Equation~\\ref{eq:caloeffi}.\n    \\begin{equation}\n    \\epsilon \\cdot a= \\frac{N_{\\mathrm{clus,Ntow}>1} \\text{ in calorimeter}}{N_\\text{MC gen. particles} \\text{ in acceptance}}\n    \\label{eq:caloeffi}\n    \\end{equation}\n    In the calculation, only clusters formed according to the seeding and aggregation thresholds are used and additionally required to be made of more than a single tower.\n    Furthermore, only one cluster per generated particle is considered for the calculation of $\\epsilon \\cdot a$ to avoid counting multiple clusters of a single particle (e.g. due to an induced pre-shower).\n    The latter requirement is necessary to reject secondary low energy clusters or showers that are not contained in the calorimeter (e.g.\\ on the outer and\/or inner edges).\n    The efficiencies show that at low energies, the seed and aggregation thresholds decrease the reconstruction efficiency.\n    Moreover, an edge effect at low and high pseudorapidity (e.g. strong shower leakage) can be observed even at higher energies also reflecting a small remaining effect from acceptance losses in the calorimeters mostly due to deflection of the original particles in the inner detectors. \n    Additionally, the non-uniform $\\varphi$-coverage of the LFHCAL and FEMC is clearly visible, yielding a lower average efficiency for $\\eta > 3$. \n    \\begin{figure*}[!t]\n        \\centering\n       \n        \\includegraphics[width=\\textwidth]{figs\/analysis\/EReso_PaperPlot_ECals.pdf}\\\\ \n        \\includegraphics[width=0.66\\textwidth]{figs\/analysis\/EReso_PaperPlot_HCals.pdf} \n        \\caption{Energy resolution for different particles generated in single particle simulations at fixed energies as measured by the electromagnetic calorimeters EEMC, BEMC, and FEMC (a--c) and the hadronic calorimeters OHCAL and LFHCAL (d and e). }\n         \\label{fig:eresodistributioneemcbecalfemc}\n   \\end{figure*}\n\n\\subsection{Energy resolution}\n  The energy resolution for the ECals and HCals is evaluated based on single particle simulations for photons, electrons, pions and protons generated for $0.2 < E < 30 (50)$ GeV. \n  For these studies the reconstructed energy deposits in the towers are combined into clusters using the MA clusterizer with the aforementioned seed and aggregation energy settings for each calorimeter. \n  The energy scale of the calorimeters is calibrated such that in simulations without material in front of the calorimeter the reconstructed electron energy over the generated energy is approximately unity. \n  Thus, this calibration corrects the ECals and HCal to approximately the same energy scale.\n  No $\\eta$ dependent corrections for the energy response are introduced so far.\n  \\begin{figure}[t!]\n      \\centering\n      \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/BECAL_electron__ExampleBinHighest_1.pdf}\n      \\vspace{-0.2cm}\n      \\caption{Comparison of the energy resolution for electrons generated in single particle simulations at $E= 1$ GeV (top) and $E= 8$ GeV (bottom) as measured by the BEMC (left) and FEMC (right) without additional material in front of the calorimeter and in the full detector setup.}\n      \\label{fig:materialresopeak}\n  \\end{figure}\n  \\begin{figure}[t!]\n      \\centering\n      \\includegraphics[width=0.45\\textwidth]{figs\/analysis\/Resolution_electron_Fitted_wFWHM.pdf}\\\\\n      \\includegraphics[width=0.45\\textwidth]{figs\/analysis\/Resolution_pion_Fitted_wTB}\n      \\caption{Energy resolution for electrons (charged pions) generated in single particle simulations with energies between 0.2 and 20 (50) GeV as measured by the different ECals (top) and the different HCals (bottom) in the central acceptance of the corresponding detectors.\n      The shaded bands show the requirements as extracted from the yellow report for the different calorimeters. \n      The data points and fits indicated as $\\sigma_g\/E$ are based on the Gaussian width of the resolution peaks, while $\\sigma_F\/E$ is based on the FWHM.\n      The energy resolution based on a test beam for the OHCAL is shown for comparison \\cite{sPHENIX:2017lqb}.}\n      \\label{fig:eresocalos}\n  \\end{figure}\n  \\Figure{fig:eresodistributioneemcbecalfemc} shows the energy response $E^\\mathrm{rec}\/E^\\mathrm{MC}$ for the various particle species and in each calorimeter.\n  By construction, the electron and photon response in the ECals peaks around unity with a strong excess that is accompanied by a visible tail towards lower values. \n  This tail is a result of multiple effects.\n  First, the clusterization in the calorimeter is not perfect (see clusterization chapter) and thus not all energy of an incoming particle is reconstructed.\n  In addition, for these studies only the highest energetic cluster in each event is selected, which combined with the clusterizer performance leads to a smearing to lower $E^\\mathrm{rec}\/E$ values.\n  Further smearing comes from bremsstrahlung losses of the electrons in the magnetic field as well as from material interaction of photons that could lead to photon conversions, as seen in \\Fig{fig:materialresopeak}.\n  The figure shows a comparison of the energy response for the BEMC with and without the remaining ECCE detector material in front, highlighting an increasing tail at lower $E^\\mathrm{rec}\/E$ due to the additional material.\n  In the following studies, contributions from photon conversions are not rejected and thus are still contained in the photon sample.\n  The left side tails of the resolution peaks can also arise through particles hitting the support material in between the towers.\n  The reconstructed energy loss from hitting and subsequently channeling in the passive support structures is a major factor to be considered for the calorimeter design.\n  Initial studies have shown that already a 2 mm carbon fiber support structure between the EEMC towers is enough to significantly deteriorate the energy resolution.\n  As such, the supports were optimized to the current design of 0.5 mm carbon sheets, which greatly improves the energy resolution.\n  Further improvements are possible with carbon support grids holding multiple crystals that are further separated by a thin foil.\n  Similar support material considerations are to be made for the BEMC, where the current design employs 2 mm carbon fiber sheets.\n  Charged hadrons deposit in the majority of cases only a minimum ionizing signal in the ECals, which is visible as a strong peak at low $E^\\mathrm{rec}\/E$ values.\n  However, there is also a non-negligible amount of charged hadrons that deposit 40\\% or more of their energy in the ECals, which can negatively impact the HCal energy resolution.\n  For the HCals, the charged pions and protons peak around unity, whereas remaining shower leakage from electron showers out of the ECals is mostly negligible.\n  \\Figure{fig:eresodistributioneemcbecalfemc} also highlights a shifted peak for protons compared to pions in the HCals which can be explained by a loss of visible energy for baryons.\n  In future studies, this effect could be counteracted for the LFHCAL by shower depth analyses and subsequent application of a correction factor for the loss of visible energy.\\\\\n  In order to determine the energy resolutions of the different calorimeters, the $E^\\mathrm{rec}\/E$ distributions are fitted with crystalball functions in order to determine the peak width.\n  This width can either be taken from the Gaussian component or from the full width at half maximum (FWHM).\n  The slightly larger values of the latter are a reflection of the asymmetric $E^\\mathrm{rec}\/E$ distribution as described above.\n  Based on the fit values, \\Fig{fig:eresocalos} shows the energy resolution for electrons in their generated energy range in the ECals and for charged pions in the HCals.\\\\\n  All ECal resolutions, based either on the Gaussian sigma ($\\sigma_\\mathrm{g}$) or the FWHM ($\\sigma_\\mathrm{F}$), are well within the limits imposed by the YR and even exceed the requirements in the case of the BEMC by a significant amount.\n  Thus, despite the smeared $E^\\mathrm{rec}\/E$ peaks from the full ECCE detector simulation, the resolution is still within the imposed limits.\n  In addition, a minimal pseudorapidity dependence for all calorimeters is observed, but none of the $\\eta$-regions fail to deliver the required YR performance.\\\\    \n  For the HCals, I\/OHCAL and LFHCAL, a similar behavior is observed, where the resolutions are found to be better than the YR requirements with $\\sigma\/E=(31-34\\%)\/\\sqrt{E}\\oplus(17-18\\%)$ and $\\sigma\/E=(33-44\\%)\/\\sqrt{E}\\oplus(1.4\\%)$ by about 1--20\\% and constant 8\\%, respectively.\n  This also holds true for both tested particle species ($\\pi^\\pm$ and protons) and in each $\\eta$ region individually.\n  In addition, the HCal resolution is compared to the sPHENIX test beam data and shows a better resolution in the presented simulations \\cite{sPHENIX:2017lqb}, which can be explained by an imperfect simulation setup of the details of the calorimeter response.\n  \n\\subsection{Position resolution}\n\\label{sec:PositionResolution}\n  A significant fraction of physics observables either directly or indirectly require a good position resolution of the reconstructed clusters in the calorimeters. \n  For example, the jet reconstruction clusters objects which are reconstructed in a given radial cone and thus position inaccuracies especially in difficult pseudorapdity regions can deteriorate the physics performance. \n  Moreover, charged particle association or neutral cluster determination via track matching (see next section) depends on the cluster position resolution as much as on the tracking resolution.\n  \\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/analysis\/PositionResolution_Paper.pdf} \n    \\caption{ Position resolution in $\\eta$ (top) and $\\varphi$ (bottom) for electrons or charged pions generated in single particle simulations with energies between 0.2 and 40 GeV as measured by the different calorimeters in the central acceptance of the corresponding destectors without a magnetic field. }\n    \\label{fig:posreso}\n  \\end{figure}\n\n  \\begin{figure*}[t!]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/analysis\/TMEffEta_E+HCal_PaperPlotAlt.pdf}\n    \\caption{Track matching efficiencies for electrons reconstructed with the MA-Clusterizer in the EEMC (a), the BEMC (b) and the FEMC (c) and for charged pions reconstructed in the OHCAL (b) and the LFHCAL (c). The efficiencies are relative to the number of reconstructed tracks according to \\Equation{eq:tmeffitrk}. }\n    \\label{fig:tmeffiEBF}\n  \\end{figure*}\n  To determine the pure position resolution of the clusterization algorithm and intrinsic calorimeter granularity single particle simulations without a magnetic field have been used.\n  This setup allows to separate between the intrinsic position resolution in the respective calorimeters and effects arrising from a larger inclination angle at the calorimeter surface as well as inaccuracies in the particle propagation through the material due to the 1.4T magnetic field.\n  For the track-to-cluster matching under realisitic conditions within a magnetic field the $\\eta$ and $\\varphi$ coordinates for charged particles are calculated by propating the tracks through the detector material to approximately half the depth of each calorimeter.\n  The median cluster depth is however $\\eta$ dependent for non projective calorimeters.\n  Consequently, the mean shift in the $\\eta$-position has to be corrected for the forward and backward calorimeters based on the zero-field data.\n  \\Figure{fig:posreso} presents the width of the difference of the generated particle $\\eta$($\\varphi$) and the reconstructed cluster position in $\\eta$($\\varphi$) in the different calorimeters.\n  For all electro-magnetic calorimeters an excellent resolution of about $0.01-0.015$ in pseudorapidity is observed which only degrades slightly towards lower energies.\n  The $\\varphi$-resolution for highly energetic particles is similarly good with $\\Delta \\varphi = 0.02$ (corresponding to $1.15$ degrees). \n  It is mainly determined by the size of the single towers in $\\Delta \\varphi$ of the respective calorimeter and the width of the electro magnetic shower.\n  Due to the larger tower sizes and wider spread of hadronic showers without a very well defined core the $\\eta$ and $\\varphi$ resolutions of the hadronic calorimeters are slightly worse in both dimensions. \n  The resolutions for the LFHCAL could be further improved in the future by taking into account the correct depth of the shower as well, which so far has not been considered in the position calculation.\n\\subsection{Track-Cluster matching}\n  The position resolution described in the previous chapter is a necessary ingredient for performance studies of the cluster-to-track matching.\n  This matching is needed for particle identification studies, like electron selection via charged pion rejection or cluster neutralization for photon analyses.\n  Moreover, the track matching procedure is a crucial ingredient for particle flow-based jet measurements.\\\\\n  The track matching efficiency can be calculated as the number of track-matched clusters relative to the number of reconstructed tracks in the full calorimeter acceptance via\n  \\begin{equation}\n    \\epsilon_\\mathrm{TM} = N_\\mathrm{clus}^{\\text{matched}} \/ N_\\mathrm{tracks}^{\\text{in acc.}},\n    \\label{eq:tmeffitrk}\n  \\end{equation}\n  or relative to the number of reconstructed clusters via\n  \\begin{equation}\n    \\kappa_\\mathrm{TM} = N_\\mathrm{clus}^{\\text{matched}} \/ N_\\mathrm{clus}.\n  \\end{equation}\n  \\Figure{fig:tmeffiEBF} shows the track matching efficiencies ($\\epsilon_\\mathrm{TM}$) for the different calorimeters for single particle simulations of either electrons or charged pions in the full ECCE GEANT4 detector setup.\n  For a majority of the ECal acceptance, an excellent efficiency of $\\epsilon_\\mathrm{TM}>95\\%$ is observed.\n  Expected deviations towards lower particle momenta are observed, where the track to cluster association breaks down due to a cut-off in the particle cluster reconstruction imposed by the minimum seeding and aggregation thresholds.\n  An additional pseudorapdity dependence for the track matching efficiencies is expected due to the previously observed cluster position resolution, which deteriorates for certain $\\eta$ regions as wellas the reducted cluster finding efficiency due to remaining acceptance effects.\n  Moreover, the particles might be stronger deflected in the $\\eta$-regions with higher amounts of material due to support structures.\\\\\n  Further insights into the track matching efficiency are given by \\Fig{fig:tmeffitrkclus}, where the track matching efficiencies $\\kappa_\\mathrm{TM}$ are shown for all calorimeters in their nominal acceptance.\n  The comparison of $\\epsilon_\\mathrm{TM}$ and $\\kappa_\\mathrm{TM}$ highlights that the track matching efficiency depends equally on the cluster finding efficiency and the track finding efficiency.\n  This can clearly be seen in the electron matching efficiency for the ECals, where $\\epsilon_\\mathrm{TM}$ is nearly unity when calculated according to \\Equation{eq:tmeffitrk}, meaning that if a track is found, it is nearly always matched to a cluster.\n  On the other hand, $\\kappa_\\mathrm{TM}$ shows a reduced efficiency, meaning that for a large portion of clusters no track is found for matching, especially in the forward region.\n  For the HCals, the performance is generally worse as particles can pre-shower in the ECals, resulting in clusters with distorted positions on the HCals, thus not for all tracks a matching cluster is found.\n\n  \\begin{figure}[!ht]\n    \\centering\n   \n    \\includegraphics[width=0.49\\textwidth]{figs\/analysis\/ClusterTMEfficiency_KappaOnly_Paper.pdf}\n    \\caption{Track matching efficiencies calculated relative to the number of reconstructed clusters ($\\kappa_\\mathrm{TM}$). Single particle simulations of electrons in the detector acceptance of the ECals and of charged pions in the acceptance of the HCals are used to obtain the efficiencies.}\n    \\label{fig:tmeffitrkclus}\n  \\end{figure}\n\\subsection{Particle Identification}\n  The information provided by the ECals and HCal can help distinguish between particle species and thus provide highly efficient particle identification, which is crucial for a variety of physics analyses.\n  This section therefore focuses first on the PID capabilities of the ECals and subsequently the additional benefits from the HCals.\n \n  The electromagnetic calorimeters (EEMC, BEMC and FEMC) are most commonly used to identify electromagnetic showers coming from a single particle.\n  They can differentiate between photons and their background from merged $\\pi^0$ decay photons.\n  If tracking information is used in addition, the calorimeters can be used to provide a strong separation power between electrons and charged hadrons like $\\pi^\\pm$, kaons or protons.\n  In the following, the different PID approaches are briefly explained and the expected performance is shown based on full detector GEANT4 simulations.\n  \\begin{figure}[!t] \n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/Pi0_InvMassBin_AllCalos.pdf}\n    \\caption{Invariant mass $M_{\\gamma\\gamma}$ distribution for generated $\\pi^0$ mesons in the energy range from 2.5 to 3.0 GeV for EEMC, BEMC, and FEMC including a composite Gaussian fit function that includes a left-sided exponential tail component.}\n    \\label{fig:invmasspi0sep}\n  \\end{figure}\n  \\begin{figure}[!t] \n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/Pi0_MassAndWidth.pdf}\n    \\caption{Invariant mass $M_{\\gamma\\gamma}$ peak width (top) and position (bottom) obtained from a composite Gaussian fit including a left-sided exponential tail.}\n    \\label{fig:invmassfittedvalues}\n  \\end{figure}\n\n  \\begin{figure*}[!t] \n    \\centering\n    \\includegraphics[width=\\textwidth]{figs\/analysis\/EoP_ExampleBin_PaperPlot.pdf}\n    \\caption{$E\/p$ distribution for electrons (blue) and charged pions (red) in the EEMC (right) and the BEMC (left). The $E\/p$ distribution is shown for two different approaches where $E\/p$ is either calculated using the generated (true) particle momentum or the reconstructed tracking based (rec.) momentum. For both distributions, the full ECCE detector has been simulated using its GEANT4 implementation.}\n    \\label{fig:eopdistribution}\n  \\end{figure*}\n\n  \\begin{figure}[!ht] \n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/Pi0_MergingFraction.pdf}\n    \\caption{Fraction of neutral pions for which the showers from their decay photons are merged into a single cluster and can not be reconstructed using an invariant-mass-based approach for the different ECals.\n    }\n    \\label{fig:pi0merging}\n  \\end{figure}\n\n\\subsubsection{Single photon and neutral pion separation}\n  A significant background for photon analyses originates from $\\pi^0$ meson decay photons, which end up in the same reconstructed cluster due to their close proximity.\n  In general, when the decay photons can still be reconstructed separately, their calculated invariant mass ($M_{\\gamma\\gamma}=\\sqrt{2E_{\\gamma_1}E_{\\gamma_2}(1-\\cos{\\theta_{12}})}$) can be used to veto decay photon clusters if the mass falls in a certain window around the nominal $\\pi^0$ mass.\n  Example invariant mass distributions for the ECals are shown in \\Fig{fig:invmasspi0sep} for a selected energy range of the two-photon meson candidates including a composite Gaussian fit with a left-sided exponential tail.\n  The BEMC invariant mass distribution is significantly wider than that of the EEMC or FEMC, as can also be seen in \\Fig{fig:invmassfittedvalues}, where the peak width (obtained from the width of the Gaussian fit component) is shown as a function of $\\pi^0$ energy.\n  The broadening of the peak width with increasing energy and the cutoff of the BEMC data at $E\\approx12$ GeV is further elaborated in the following.\n  Above a certain energy, the decay kinematics of the $\\pi^0$ together with the granularity and resolution of the calorimeter no longer allow to reconstruct separate decay photons due to a merging of their showers.\n  Thus, the separation power between meson decay photons and single photons decreases and the reconstructed meson mass starts to deviate from the nominal meson mass as seen in \\Fig{fig:invmassfittedvalues}.\\\\\n  The energy dependence of this cluster merging effect is shown in \\Fig{fig:pi0merging} for the three ECals.\n  The close proximity of the BEMC to the interaction point together with its $4 \\times 4$~cm tower size results in a large fraction of merged decay photon clusters already at 5 GeV around $\\eta = 0$.\n  For $|\\eta| > 0.9$ within the BEMC the merging starts to set in at around $10$~GeV due to the larger distance of the calorimeter surface from the interaction point and thus a larger average distance of the two decay photons on the calorimeter surface.\n  In contrast, the higher granularity and larger distance from the IP of the EEMC and FEMC, respectively, results in a much later onset of the cluster merging. \n  For the FEMC this effect becomes only significant above 25 GeV, while the EEMC experiences this effect already above 15 GeV.\n\n\\subsubsection{Electron PID via charged pion rejection}\nSeveral observables of EIC physics require a clean electron sample \\cite{AbdulKhalek:2021gbh}.\nOne of the largest backgrounds for electrons stems from charged pions ($\\pi^\\pm$), which can be distinguished on a statistical basis from electrons with a high efficiency using ECal information.\nThe so-called pion rejection factor is a handle on how strong this $e^\\pm$--$\\pi^\\pm$ separation is for a given calorimeter.\nIt can be calculated by simulating the response for single electron and separate single pion events.\nThe quantity $E\/p$, meaning the reconstructed cluster energy relative to the incident particle momentum exhibits only slightly overlapping distributions for both particles.\nThis is shown in \\Fig{fig:eopdistribution} where electrons (blue) show a strong enhancement around $E\/p\\approx1$, while charged pions (red) are smeared towards lower $E\/p$ values for all three ECCE ECals.\nIn realistic events, e.g. based on the Pythia event generator, one expects significantly more hadrons relative to electrons and thus the hadronic tail overlap is expected to be stronger than shown in \\Fig{fig:eopdistribution}.\nThis effect is further enhanced by the presence of jets which result in shower overlaps in the calorimeters.\nThe track momentum in the following is determined using the full ECCE tracking capabilities \\cite{ecce-paper-det-2022-03}.\n\nDue to the small overlap of the $E\/p$ distribution for different particle species, a minimum $E\/p$ cut can be employed to reject the majority of charged pions in the sample while retaining a high efficiency electron sample.\nIn previous studies, a minimum cut of $\\Delta=1.6\\,\\sigma_E\/E$ has been determined to result in a high electron efficiency of $\\varepsilon_e\\approx 95\\%$.\nHowever, the asymmetric electron resolution distributions of the calorimeters within the full ECCE integration, as shown in \\Fig{fig:eresodistributioneemcbecalfemc} lead to a significantly reduced electron efficiency when applying a 1.6$\\sigma$-based cut.\nEspecially for the BEMC where a strong tail in the energy resolution distribution is visible, the cut results in an electron efficiency of $\\varepsilon_e\\approx70\\%$, while for the other ECals values of about 90--95\\% are observed.\nThus, an additional $E\/p$ cut value has been determined that allows for $\\varepsilon_e= 95\\%$, which is indicated as $\\varepsilon_{95\\%}$ in the following.\nThis cut value corresponds to approximately $2\\sigma$ for the EEMC, $6\\sigma$ for the BEMC, and $3\\sigma$ for the FEMC, highlighting the difference in the energy resolution peak asymmetry for the various ECCE ECals.\n\\begin{figure}[!t] \n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/EoP_pionrejection_AllCalo_95comp.pdf}\n    \\caption{Pion rejection factor for the different ECals with $E\/p>1-1.6\\,\\sigma_e\/E$ or based on a $\\varepsilon_e\\approx 95\\%$ cut.\n    }\n    \\label{fig:pionrejection}\n\\end{figure}\nApplying these cuts on the single pion event simulations results in the rejection factors shown in \\Fig{fig:pionrejection}.\nValues up to $6\\times10^4$ are reached for the EEMC using the $1.6\\,\\sigma$-based cut, while for the other calorimeters $\\pi^\\pm$ rejection factors ranging from 20 to more than $10^3$ are reached.\nFor the EEMC the pion rejection capabilities are so striking that an accurate pion rejection factor is hard to determine with the currently available single particle production statistics and the reported values should be interpreted as lower limits.\nA significant reduction of about an order of magnitude in the $\\pi^\\pm$ rejection is observed for the $\\varepsilon_e=95\\%$ based cut for the FEMC and BEMC, which therefore stands in no reasonable relation to the efficiency loss observed for the other $E\/p$ cut values.\nThis loss mainly arrises from the significant tails observed for these two calorimeters in their current configuration.\n\n\\begin{figure}[!t] \n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/M02_ExampleBin_EEMC.pdf}    \n    \\caption{The shower shape distribution (\\ensuremath{\\sigma_\\mathrm{long}^{2}}) for electrons (blue) and charged pions (red) in the EEMC using the full ECCE detector simulations at a fixed generated energy of $E=1.3$ GeV. \n             The gray vertical line indicates the \\ensuremath{\\sigma_\\mathrm{long}^{2}}\\ cut value below which 90\\% of the electrons would be kept.}\n    \\label{fig:SSdistribution}\n\\end{figure}\n\n\\subsubsection{Hadron PID}\nBesides using an $E\/p$ cut to differentiate between electrons and hadrons the shape of the shower and thus the cluster can be used.\nThe distribution of energy within a cluster is referred to as ``shower shape'', which is described using a parametrization of the shower surface ellipse axes~\\cite{Alessandro:2006yt, Abeysekara:2010ze}. \nThe shower surface is defined by the intersection of the cone containing the shower with the front plane of the calorimeter. \nThe energy distribution along the $\\eta$ and $\\varphi$ directions is represented by a covariance matrix with terms ${\\sigma_{\\varphi\\varphi}}$, ${\\sigma_{\\eta\\eta}}$ and ${\\sigma_{\\varphi\\eta}}$, which are calculated using logarithmic energy weights $w_i$.\nThe tower dependent weights are expressed as:\n\\begin{equation}\n\\label{eq:w_i}\nw_i = {\\rm Maximum}(0,w_0+\\ln(E_{i}\/E_\\mathrm{cluster}))\n\\end{equation}\nand\n\\begin{equation}\nw_\\mathrm{tot} = \\sum_i w_i,\\\\\n\\end{equation}\nwhere $w_0$~=~4.5 for the EEMC \\cite{Awes:1992yp},  which excludes towers with energy smaller than 1.1\\% of the cluster energy.\nFor the BEMC and FEMC  $w_0$~=~4.0 and $w_0$~=~3.5 are used, respectively, in order to compensate for the different Moliere radii and tower size.\nThe covariance matrix terms can then be calculated as follows\n\\begin{equation}\n\\sigma^{2}_{\\alpha\\beta} = \\sum_i \\frac{w_i\\alpha_i\\beta_i}{w_\\mathrm{tot}}-\\sum_i \\frac{w_i\\alpha_i}{w_\\mathrm{tot}}\\sum_i \\frac{w_i\\beta_i}{w_\\mathrm{tot}}\\,,\n\\label{eq:ss_centroid}\n\\end{equation}\nwhere $\\alpha_{i}$ and $\\beta_{i}$ are the tower indices in the  $\\eta$ or $\\varphi$ direction. \nSimilarly, also the average cluster position in the $\\eta$ and $\\varphi$ direction in the calorimeter plane is determined using the tower positions weighted logarithmically by their deposited energy \\cite{Awes:1992yp}.\\\\  \nThe shower shape parameters \\ensuremath{\\sigma_\\mathrm{long}^{2}}\\ (long axis) and \\ensuremath{\\sigma_\\mathrm{short}^{2}}\\ (short axis) are defined as the eigenvalues of the covariance matrix, and are calculated as\n\\begin{eqnarray}\n\\ensuremath{\\sigma_\\mathrm{long}^{2}} = 0.5(\\sigma^{2}_{\\varphi\\varphi}+\\sigma^{2}_{\\eta\\eta})+\\sqrt{0.25(\\sigma^{2}_{\\varphi\\varphi}-\\sigma^{2}_{\\eta\\eta})^2+\\sigma^{2}_{\\eta\\varphi}}, \\label{eq:ss1}\\\\\n\\ensuremath{\\sigma_\\mathrm{short}^{2}} = 0.5(\\sigma^{2}_{\\varphi\\varphi}+\\sigma^{2}_{\\eta\\eta})-\\sqrt{0.25(\\sigma^{2}_{\\varphi\\varphi}-\\sigma^{2}_{\\eta\\eta})^2+\\sigma^{2}_{\\eta\\varphi}} \\label{eq:ss2},\n\\end{eqnarray}\nPrevious experiments have determined that the short axis \\ensuremath{\\sigma_\\mathrm{short}^{2}}\\ carries significantly less discriminative power compared to \\ensuremath{\\sigma_\\mathrm{long}^{2}}\\ and thus only the long axis is considered in the following.\n\n\\begin{figure}[!htb] \n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/RejectPi_AllCalo_true.pdf}\n    \\includegraphics[width=0.4\\textwidth]{figs\/analysis\/EoP_pionrejection_AllCalo_withPID.pdf}\n    \\caption{Left: Fraction ($R_\\pi$) of cluster originating from charged pions, which can be rejected due to the chosen shower shape cut as a function of the incident pion energy. \n             Right: Pion rejection factor for the different ECals with $E\/p>1-1.6\\,\\sigma_e\/E$ (w$\/$o PID) or $E\/p>1-1.6\\,\\sigma_e\/E$ and the additional \\ensuremath{\\sigma_\\mathrm{long}^{2}}\\ selection (w$\/$ PID) applied as a function of the true track momentum.}\n    \\label{fig:RejectionPID}\n\\end{figure}\nUsing these parameters symmetric electromagnetic showers with a small spread originating either from photons or electrons can be distinguish from non-symmetric showers caused by hadronic interactions.\nThe shower shape of charged particles can also be elongated by the angle of incidence.  calorimeters.\nFurthermore, the merging of showers from electromagnetic processes, i.e. ${\\rm e^{+}e^{-}}$ pairs from conversions within a close distance to the calorimeter or photons from neutral meson decays with high transverse momenta, also lead to asymmetric shower shapes.\\\\\nAn example distribution of the shower shape parameter \\ensuremath{\\sigma_\\mathrm{long}^{2}}\\ for electrons (blue) and pions (red) as seen by the EEMC can be found in \\Fig{fig:SSdistribution}.\nAs can be seen, the energy deposits from an electrons at the same incident energy are significantly more collimated than those of charged pions.\nConsequently, electron clusters have predominantly lower shower shape values. \nAs these distributions strongly change as a function of the incident energy a \\ensuremath{\\sigma_\\mathrm{long}^{2}}\\ cut value function is calculated that preserves 90\\% of the electrons. \nUsing these cut values based on the shower shape alone up to 90\\% of the pions can be rejected in the EEMC as shown in  \\Fig{fig:RejectionPID}~(top).\nBy simultaneously using the aforementioned $E\/p$ and \\ensuremath{\\sigma_\\mathrm{long}^{2}}\\ cuts, the pion rejection quoted in \\Fig{fig:pionrejection} is improved by at least a factor two in most momentum bins as seen in \\Fig{fig:RejectionPID}~(bottom).\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nWe report the design and performance of the calorimeter systems for the ECCE detector~\\cite{ecce-paper-det-2022-01}.\nHomogeneous and sampling calorimeter technologies are employed in the different pseudorapidity regions (backwards, central, and forward) aiming to achieve the overall performance requirements outlined in the EIC Yellow Report~(YR)~\\cite{AbdulKhalek:2021gbh} cost effectively and with consideration of technical and schedule risks.\nThe main physics program of the EIC imposes strong detector performance requirements on the calorimeter systems.\nWhile single inclusive DIS, jets and heavy quark reconstruction require an excellent energy resolution for the electromagnetic and hadronic calorimeters, further requirements for $\\pi$ \/ $e$ separation at the $3\\sigma$ level are imposed, for example, by spin asymmetry measurements, TMD evolution, and $XYZ$ spectroscopy.\nIn order to probe the requested kinematic regions for such processes, a large acceptance in pseudorapidity for the calorimeters is required with special focus on continuous coverage from the backward region to the forward region.\nThe key performances of the ECCE calorimeter systems are reported and put in context to their impact on physics analyses.\nThis includes the reconstruction performance, expected energy and position resolution, as well as particle identification via matching to charged particle tracks obtained from the ECCE tracking systems \\cite{ecce-paper-det-2022-03}. \n\n\\section{Acknowledgements}\n\\label{acknowledgements}\n\nWe acknowledge support from the Office of Nuclear Physics in the Office of Science in the Department of Energy, the National Science Foundation, and the Los Alamos National Laboratory Laboratory Directed Research and Development (LDRD) 20200022DR.\n\nWe thank (list of individuals who are not coauthors) for their useful discussions and comments.\n\n\\section{Summary}\n\\label{summary}\n\nIn summary, the ECCE calorimeter systems have been designed to support the full scope of the EIC physics program as presented in the EIC white paper~\\cite{Accardi:2012qut} and in the 2018 report by the National Academies of Science (NAS)~\\cite{NAP25171}. These systems can be built within the budget envelope set out by the EIC project while simultaneously managing cost and schedule risks.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nCollisions of asymmetric nuclear systems, like proton-nucleus or deuteron-nucleus \ncollisions, attract significant experimental and theoretical attention at present.  \nAt the Relativistic Heavy Ion Collider (RHIC), Run 8 included deuteron-gold\ncollisions at 200 $A$GeV with a luminosity increase of about an order of magnitude\ncompared to the deuteron-gold run of 2003. One of the physics goals was to provide \na high-statistics ``cold'' nuclear matter data set to establish a definitive baseline \nfor ``hot'' nuclear matter, created in gold-gold collisions. The benchmark role of the\n``the deuteron-gold control experiment'' for e.g. energy-loss studies has often been \nemphasized~\\cite{Gyulassy:2004jj,Hemmick:2004jc}.\nProton-nucleus and deuteron-nucleus reactions were \nalso used to study the Cronin-effect~\\cite{Cron75,Antr79}.  \n\nLight-on-heavy nucleus-nucleus collisions offer unique information about the \nunderlying dynamics, not available in symmetric proton-proton or \nnucleus-nucleus collision systems. The light-on-heavy asymmetry\nmanifests itself in an asymmetric (pesudo-)rapidity distribution of \ncharged particles with respect to\nzero rapidity (or pseudorapidity) as measured by BRAHMS~\\cite{Arsene:2004cn} and \nPHOBOS~\\cite{Back:2004mr}. The asymmetry of the yields can be quantified by \nintroducing the ratio of pseudorapidity densities at a given negative \npseudorapidity relative to that at\nthe positive pseudorapidity of the same magnitude. This backward\/forward ratio\n(forward being the original direction of motion of the light partner) is referred to\nas pseudorapidity asymmetry. The STAR Collaboration published pseudorapidity \nasymmetries in 200 $A$GeV $dAu$ collisions for several identified hadron species \nand total charged hadrons in the pseudorapidity intervals $|\\eta| \\le 0.5$ and \n$0.5 \\le |\\eta| \\le 1.0$~\\cite{Abelev:2006pp}. Asymmetries with the\nbackward\/forward ratio above unity for transverse momenta up to $\\approx 5$ GeV\/c\nare observed for charged pion, proton+anti-proton, and total charged hadron production \nin both rapidity regions. Because of the importance of asymmetric collisions, we \nanticipate that proton-lead or deuteron-lead data will be \ncollected at the Large Hadron Collider (LHC). \n\nThe theoretical relevance of asymmetric collision systems (in particular colliding\nthe lightest nuclei like protons or deuterons on a heavy partner) in \ntesting parameterizations for nuclear shadowing and models for initial multiple \nscattering was recognized prior to the availability of 200 $A$GeV RHIC deuteron-gold \ndata~\\cite{Vitev:2003xu,Wang:2003vy,Barnafoldi:2004kh}. One can take advantage\nof the fact that\npositive (forward) rapidities correspond to large parton momentum fractions from\nthe light partner and small momentum fractions from the heavy nucleus\n(and vice versa for negative rapidities), and that there are more collisions \nsuffered traversing the heavy partner. Wang made predictions for pseudorapidity \nasymmetries on this basis~\\cite{Wang:2003vy}. A subsequent calculation focused on the \nlowest transverse momenta where pQCD may be applicable~\\cite{Barnafoldi:2005rb}.\nMore recently, two versions of a pQCD-based model with nuclear modifications were \nused to address deuteron-gold collisions and results were compared to available\ndata~\\cite{Adeluyi:2008qk,Barnafoldi:2008rb}. The related subject of\nforward-backward rapidity correlations in asymmetric systems is treated in\n~\\cite{Brogueira:2007ub} using a derivative of string percolation models, and \nin \\cite{Armesto:2006bv} under the framework of color glass condensate.  \n\nIn the present study we investigate the roles of nuclear shadowing and multiple \nscattering in the generation of rapidity asymmetry \nin the backward\/forward yields\nat intermediate and high transverse-momenta. Here we use the HIJING\nshadowing parameterization~\\cite{Li:2001xa} and the recently released \nEskola--Paukkunen--Salgado (EPS08) nuclear parton distribution functions\n(nPDFs)~\\cite{Eskola:2008ca}. While the former has been applied widely, \nthe latter was not available at the time of the\nearlier studies mentioned above. We calculate the pseudorapidity asymmetry in three \nrepresentative asymmetric light-on-heavy systems: $pBe$ at $30.7$ GeV (Fermilab), \n$dAu$ at $200$ $A$GeV (RHIC), and $dPb$ at $8.8$ $A$TeV (LHC). \nWe concentrate on neutral pion production and compare results to experimental data \nfrom the E706 experiment~\\cite{Apanasevich:2002wt},\nPHENIX~\\cite{PHENIXdAu}, and STAR~\\cite{Abelev:2006pp}.\n\nThe paper is organized as follows: in Sec.~\\ref{col} we review the basic \nformalism of the pQCD-improved parton model with intrinsic transverse momentum and\nmultiple scattering as applied to proton-nucleus ($pA$) and deuteron-nucleus ($dA$) \ncollisions. This section also includes the definition of the pseudorapidity asymmetry \nand a discussion of the roles of nuclear shadowing and multiple scattering in asymmetric \ncollisions. We present the results of our calculation in Sec.~\\ref{res} and conclude \nin Sec.~\\ref{concl}.\n\n\\section{Calculational framework}\n\\label{col}\n\n\\subsection{Description of the model}\n\\label{partmd}\nThe invariant cross section for the production of \nfinal hadron $h$ from the collision of nucleus $A$ and nucleus $B$ \n($A+B \\!\\to\\! h+X$), can be written, in collinearly factorized pQCD, as\n\\begin{eqnarray}\\label{eq:pdA}\nE_h\\f{\\dd ^3 \\sigma_{AB}^{h}}{\\dd^3 p} =\n\\sum_{\\!\\!abcd}\\!\\int\\!\\! \\dd^2b\\ \\dd^2r\\ t_A(b) t_B(|\\vec{b}-\\vec{r}|)\\dd{x_a} \\dd{x_b}\n\\nonumber \\\\ \n\n\\dd{z_c} \\,\nf_{\\!a\/A}(x_a,Q^2)\\ f_{\\!b\/B}(x_b,Q^2)\\\n\\nonumber \\\\ \n\\f{\\dd\\sigma(ab\\!\\to\\!cd)}{\\dd\\hat{t}}\\,\n\\frac{D_{h\/c}({z_c},\\!Q_f^2)}{\\pi z_c^2}\n\\hat{s} \\, \\delta(\\hat{s}\\!+\\!\\hat{t}\\!+\\!\\hat{u})   \\,\\,  ,\n\\end{eqnarray}\nwhere $x_a$ and $x_b$ are parton momentum fractions in $A$ and $B$, respectively, \nand $z_c$ is the fraction of the parton momentum carried by the final-state \nhadron~$h$.\nThe factorization and fragmentation scales are $Q$ and $Q_f$, respectively. \nAs usual, $\\hat{s}$, $\\hat{t}$, and $\\hat{u}$ refer to the partonic Mandelstam variables, \nthe massless parton approximation is used, and\n\\begin{equation} \\label{eq:tf}\nt_{A}(\\!{\\vec s}) = \\int\\!\\! \\dd z \\rho_{A}(\\!{\\vec s},z)\n\\end{equation}\nis the Glauber thickness function of nucleus $A$, with the nuclear density distribution,\n$\\rho_{A}({\\vec s},z)$ subject to the normalization condition\n\\begin{equation}\n\\int\\!\\! \\dd^2s \\ \\dd z \\rho_{A}({\\vec s},z) = A \\,\\, .\n\\end{equation}\nThe quantity $\\dd\\sigma(ab\\!\\to\\!cd)\/\\dd\\hat{t}$ in eq.~(\\ref{eq:pdA}) represents the \nperturbatively calculable partonic cross section, and $D_{h\/c}(z_c,\\!Q_f^2)$ stands \nfor the fragmentation function of parton $c$ to produce hadron $h$, evaluated at momentum \nfraction $z_c$ and fragmentation scale $Q_f$. \n\nThe collinear parton distribution functions (PDFs) can be generalized to include a transverse\nmomentum degree of freedom, ${\\vec k}_T$, as required by the uncertainty principle. This\ncan be formally implemented in terms of unintegrated PDFs~\\cite{Collins:2007ph,Czech:2005vy}. \nTo avoid some of the complications associated with using unintegrated\nPDFs, it is expedient to parameterize phenomenologically the ${\\vec\n  k}_T$ dependence of the parton distributions. In this light   \na phenomenological model for proton-proton ($pp$), proton-nucleus ($pA$), \nand nucleus-nucleus ($AB$) collisions incorporating parton transverse \nmomentum in the collinear pQCD formalism was developed in\nRef.~\\cite{Zhang:2001ce}. In order to make the present study\nreasonably self-contained, we include details relevant to the\npresent treatment.\n\nIn this model, the invariant cross section can be written as\n\\begin{eqnarray}\\label{eq:pdA_kt}\nE_h\\f{\\dd ^3 \\sigma_{AB}^{h}}{\\dd^3 p} =\n\\sum_{\\!\\!abcd}\\!\\int\\!\\! \\dd^2b\\ \\dd^2r\\ t_A(b) t_B(|\\vec{b}-\\vec{r}|)\\dd{x_a} \\dd{x_b}\n\\nonumber \\\\ \n \\dd{\\vec k}_{Ta} \\, \\dd{\\vec k}_{Tb} \\, \\dd{z_c} \\,\nf_{\\!a\/A}(x_a,\\!{\\vec k}_{Ta},Q^2)\\ f_{\\!b\/B}(x_b,\\!{\\vec k}_{Tb},Q^2)\\\n\\nonumber \\\\ \n\\f{\\dd\\sigma(ab\\!\\to\\!cd)}{\\dd\\hat{t}}\\,\n\\frac{D_{h\/c}({z_c},\\!Q_f^2)}{\\pi z_c^2}\n\\hat{s} \\, \\delta(\\hat{s}\\!+\\!\\hat{t}\\!+\\!\\hat{u})   \\,\\,  ,\n\\end{eqnarray}\nwhere the $k_T$-broadened parton distribution in the nucleon is\nwritten, in a simple product approximation, as\n\\begin{equation}\nf_{a\/N}(x,{\\vec k}_{T},Q^2) \\longrightarrow  \\ g({\\vec k}_T) \\cdot  f_{a\/N}(x,Q^2)  \\,\\, ,\n\\end{equation} \nwith  $f_{a\/N}(x,Q^2)$ denoting the standard collinear PDF in the nucleon.\nThe transverse momentum distribution is taken to be\na Gaussian,\n\\begin{equation}\\label{eq:kT}\ng({\\vec k}_T)  =  \\frac{\\exp(-k_T^2\/\\langle k_T^2 \\rangle_{pp})} {\\pi \\langle k_T^2 \\rangle_{pp}} \\,\\, ,\n\\end{equation}\nwhere $\\langle k_T^2 \\rangle_{pp}$ is the two-dimensional width of the transverse-momentum\ndistribution in the proton.\nBased on the then-available pion and unidentified hadron production data, an estimate \nfor the model width of the transverse-momentum distribution of partons in the proton\n($\\langle k_T^2 \\rangle_{pp}$) \nwas presented in Ref.~\\cite{Zhang:2001ce}.\nA recent summary of the energy dependence of the model parameter $\\langle k_T^2 \\rangle$, \nrelated to the average transverse momentum of the created pair by\n\\begin{equation}\\label{eq:pT_kT}\n\\langle k_T^2 \\rangle_{pp} = \\frac{\\langle p_T \\rangle^{2}_{pair}}{\\pi}  \\,\\, ,\n\\end{equation}\ncan be found in Ref.~\\cite{Barnafoldi:2007uw}. The value of $\\langle k_T^2 \\rangle_{pp}$\nincreases logarithmically with $\\sqrt{s}$, and the data are well described by the function\n\\begin{equation}\\label{eq:pT_rs}\n\\langle p_T \\rangle_{pair} = (1.74 \\pm 0.12) \\centerdot \\log_{10}(\\sqrt s) +\n(1.23 \\pm 0.2)  \\,\\, .\n\\end{equation}\nUsing eq.~(\\ref{eq:pT_rs}) one can estimate \n$\\langle k_T^2 \\rangle$ at required cms energies.\nIt should be noted that eq.~(\\ref{eq:pT_rs}) is a fit to the data at\nlower energies. Since no data exists at LHC energies, we\nextrapolate $\\langle k_T^2 \\rangle$. We observe that: \n(a) Since the dependence is logarithmic, the energy step \nwe make from RHIC to LHC is comparable to the one from SPS to RHIC.\nWe thus do not a priori expect a radical change in the trend.\n(b) The $k_T$ effects have been shown to be appreciable at low $p_T$, and since \nour predictions at LHC cover a wide $p_T$ range up to hundreds of GeV\/c in\n$p_T$, potential uncertainties arising from the extrapolation of Eq. 8 will \naffect a small fraction of this range at the low $p_T$ end.\n(c) The present study, at LHC energies, is exploratory. Results of pp\ncollisions at the LHC will be helpful in determining the magnitude of\nthe presently uncertain $\\langle k_T^2 \\rangle_{pp}$.\n\nIt is easy to appreciate the physical necessity of the presence of a \ntransverse-momentum degree of freedom in proton-proton and, therefore, nuclear \ncollisions. However, the handling of the effect of the nuclear environment \non transverse momenta is one of the specific features of the given description. \nMost shadowing parameterizations include at least some of the effects\nof multiple scattering in the nuclear medium, while the HIJING parameterization\n(as we discuss further in Sec.~\\ref{shad_multiscatt}) needs to be augmented\nwith modeling nuclear multiscattering. For this purpose, in $pA$ collisions\nwe use a broadening of the width of the transverse momentum distribution \n(\\ref{eq:kT}) according to \n\\begin{equation}\\label{eq:broad}\n\\langle k_T^2\\rangle_{pA} = \\langle k_T^2\\rangle_{pp} + C\\ h_{pA}(b) \\,\\, ,\n\\end{equation}\nwhere $\\langle k_T^2\\rangle_{pp} $ is the width already present in proton-proton \ncollisions, $h_{pA}(b)$ is the number of effective nucleon-nucleon ($NN$) collisions as a \nfunction of nucleon impact parameter $b$, and $C$ is the average increase in width \nper $NN$ collision. In nucleus-nucleus ($AB$) collisions, the $p_T$ distributions of\nboth nuclei are subject to the broadening represented by eq.~(\\ref{eq:broad}). \n\nThe function $h_{pA}(b)$ can be written in terms of the number of collisions suffered \nby the incoming proton in the target nucleus, $\\nu_{A}(b) =\n\\sigma_{NN}t_{A}(b)$, where $\\sigma_{NN}$ is the inelastic nucleon-nucleon \ncross sections. It was found in \nRef.~\\cite{Zhang:2001ce} that only a limited number of collisions is effective in broadening\nthe transverse momentum distribution. This model scenario was referred to as\n``saturation'', and an optimal description was found with\nthe effective number of nucleon-nucleon collisions maximized at $4$, \nand the average width increase per $NN$ collision, $C$, set to $0.35$  GeV$^2\/c^2$.\nWe do not change the values of these parameters in the present study. The resulting\ntransverse momentum broadening may appear too large relative to what is observed \nin Drell-Yan data at FNAL. However, we focus on meson production in the present\napplication, where the PHENIX experiment extracts intrinsic transverse momenta\nin $pp$ collisions at RHIC energies that are similar in magnitude to the ones\ndiscussed here\\cite{Adler:2006sc}.\nFurther details about this aspect of the model\ncan be found in Ref.~\\cite{Zhang:2001ce}.\n\nThe intrinsic transverse momentum $k_T$ is treated \nphenomenologically in Ref.~\\cite{Zhang:2001ce}  and in the \npresent study. While next-to-leading-order (NLO) calculations provide\na more accurate description of the parton-level cross section, they continue\nto rely on the factorization theorem and represent the non-perturbative\ninformation in terms of PDFs and fragmentation functions. Since these\nfunctions are fitted to the same data as in LO, the expected change is a shift\nin responsibilty between the perturbative and non-perturbative sectors in\ndescribing the data used to define the non-perturbative ingredients. In \naplications to a larger set of data this will not eliminate the need for the \nphenomenological use of a transverse momentum distribution, in particular\nconsidering the fact that even NLO may be far from a full perturbative\nexpansion. In addition, above we argue for inclusion of a transverse momentum \ndegree of freedom on fundamental physical grounds as basic as the uncertatinty\nprinciple. The simple Gaussian representation of this physics provides a \nphenomenologically useful additional parameter (the width). This holds true\nboth at the LO and NLO levels. Going to NLO may change the range of transverse\nmomenta where intrinsic $k_T$ is important (see e.g. \nRefs.~\\cite{MRTS:1995,CTEQ2:1995,Vogelsang:2007}).\nNo attempt is made in the present work to discuss the $x$,  \nflavor (quarks and gluons), and rapidity dependence of $k_T$.  We have \nlimited ourselves here to a simple effective description. This seems adequate for now, \nsince $k_T$ effects are appreciable only at relatively low $p_T$, and a \nmajor focus of this work is asymmetry at high $p_T$ (except at very forward \nrapidities where $p_T$-s are low due to phase space constraints). The intrinsic \ntransverse momentum, $\\langle k_T^2\\rangle_{pp}$, enhances hadron production \nyields at low $p_T$ in both negative and positive pseudorapidity regions, thus tending \nto cancel out in the asymmetry ratio. Multiscattering, on the other hand, does impact \nappreciably on the asymmetry even at low $p_T$.\nOverall, calculations within this framework \nhave proven their value in the interpretation of hadron-production \ndata~\\cite{Zhang:2001ce,Levai:2003at}. (For a comparison of the leading-order \n$k_T$-factorized approach and next-to-leading order collinear approach see \nRef.~\\cite{Szczurek:2007bt}.)\n\nThe collinear nPDFs $f_{a\/A}(x,Q^2)$ are \nexpressible as convolutions of nucleonic parton distribution functions (PDFs)\n$f_{a\/N}(x,Q^2)$ and a shadowing function ${\\cal S}_{a\/A}(x,Q^2)$ which encodes the \nnuclear modifications of parton distributions.  We use the MRST2001\nPDFs~\\cite{Martin:2001es} for the nucleon \nparton distributions, and for the shadowing function we employ both \nthe EPS08 shadowing routine~\\cite{Eskola:2008ca} and HIJING~\\cite{Li:2001xa}.\n(Other nPDFs, like FGS~\\cite{Frankfurt:2003zd}, HKN~\\cite{Shad_HKN}, \nand the earlier EKS~\\cite{Eskola:1998df}, are used elsewhere\nto calculate pseudorapidity asymmetries~\\cite{Adeluyi:2008qk}.) \nFor the final hadron fragmentation we utilize the fragmentation functions\nin the AKK set~\\cite{Albino:2005me}. The factorization scale is tied to the \nparton transverse momentum via $Q = (2\/3) p_T\/z_c$, while the\nfragmentation scale varies with the transverse momentum of the outgoing hadron,\naccording to $Q_f = (2\/3) p_T$, following the best-fit results obtained in\nRef.~\\cite{Levai:2006yd}. This scale fixing is not modified from earlier applications,\nand is used consistently throughout the present calculations.\nTo protect against divergences in the partonic cross sections, a cutoff \nregulator mass is necessary. We have tested the sensitivity of our\ncalculations at different cms energies by varying the regulator mass \nbetween $0.5-2.0$ GeV. For $p_T>1.5$ GeV\/c our results show little\nsensitivity to variation of regulator mass. We present results for \n$p_T>2.25$ GeV\/c here, and thus regulator mass effects are expected \nto be minimal.\nWe obtain the density distribution of the deuteron from the Hulthen \nwave function~\\cite{Hulthen1957} (as in Ref.~\\cite{Kharzeev:2002ei}), while a \nWoods-Saxon density distribution is used for gold and lead with parameters from\nRef.~\\cite{DeJager:1974dg}.\n\n\\subsection{Forward and backward nuclear modifications}\n\\label{rapasym}\n\nThe nuclear modification factor is designed to compare, as a ratio, spectra \nof particles produced in nuclear collisions to a hypothetical scenario \nin which the nuclear collision is assumed to be a superposition of the \nappropriate number of nucleon-nucleon collisions. The ratio can be \ndefined as a function of $p_T$\nfor any produced hadron species $h$ at any pseudorapidity $\\eta$:\n\\begin{equation}\nR^h_{AB}(p_T, \\eta) = \\frac{1}{\\langle N_{bin}\\rangle} \\cdot\n\\frac{E_h \\dd^3\\sigma_{AB}^{h}\/\\dd^3 p |_{\\eta}}\n{E_h \\dd^3\\sigma_{pp}^{h}\/\\dd^3 p |_{\\eta}}   \\,\\, ,\n\\label{rdau}\n\\end{equation}\nwhere $\\langle N_{bin} \\rangle$ is the average number of binary collisions in \nthe various impact-parameter bins.\nNuclear effects manifest themselves in $R^h_{AB}(p_T, \\eta)$ values greater or \nsmaller than unity, representing enhancement or suppression, respectively, \nrelative to the $NN$ reference.\n\nIn asymmetric collisions, hadron production at forward rapidities\nmay be different from what is obtained at backward \nrapidities. It is thus of interest to study ratios of particle\nyields between a given pseudorapidity value and its negative in these\ncollisions. The pseudorapidity asymmetry $Y_{Asym}(p_T)$ is defined \nfor a hadron species $h$ as \n\\begin{equation}\nY^h_{Asym}(p_T) = \\left. E_h\\f{\\dd ^3 \\sigma_{AB}^{h}}{\\dd^3 p} \\right|_{-\\eta} \n \\left\/ \n\\left. E_h\\f{\\dd ^3 \\sigma_{AB}^{h}}{\\dd^3 p} \\right|_{\\eta} \\right. \\,\\, .\n\\label{yasym}\n\\end{equation}\n\nLet us consider the (double) ratio of the  \nforward and backward nuclear modification factors in $dAu$\ncollisions for species $h$:\n\\begin{eqnarray}\nR^h_{\\eta}(p_T) = \\frac{R^h_{dAu}(p_T,-\\eta)}{R^h_{dAu}(p_T,\\eta)}= \\nonumber \\\\  \n\\frac{E_h \\dd^3\\sigma_{dAu}^{h}\/\\dd^3 p |_{-\\eta}}\n{E_h \\dd^3\\sigma_{pp}^{h}\/\\dd^3 p |_{-\\eta}} \\left\/  \n\\frac{E_h \\dd^3\\sigma_{dAu}^{h}\/\\dd^3 p |_{\\eta}}\n{E_h \\dd^3\\sigma_{pp}^{h}\/\\dd^3 p |_{\\eta}} \\right. \\,\\, . \n\\label{y-r:eq}\n\\end{eqnarray}\nAs discussed in Ref.~\\cite{Barnafoldi:2008rb}, since the $pp$ \nrapidity distribution is symmetric around $y=0$,\nif the same backward and forward (pseudo)rapidity ranges are taken\nin both directions (i.e. $ |\\eta_{min}| \\leq |\\eta| \\leq |\\eta_{max}| $),  \nthen the $pp$ yields cancel in eq.~(\\ref{y-r:eq}) and one obtains\nthat the ratio defined in (\\ref{y-r:eq}) is identical to \nthe pseudorapidity asymmetry (\\ref{yasym}):  \n\\begin{equation}\nY^h_{Asym}(p_T)= R^h_{\\eta}(p_T)=\\frac{R^h_{dAu}(p_T,-\\eta)}{R^h_{dAu}(p_T,\\eta)} \\,\\, .\n\\label{y-r2:eq}\n\\end{equation}\n\nEq. (\\ref{y-r2:eq}) is useful from the experimantal point of view to handle the \nsystematic errors of data and extract the proper pseudorapidity asymmetry. It also \nprovides a connection between measured rapidity asymmetry and the nuclear \nmodification factors. \n\n\n\n\\subsection{Nuclear shadowing and multiple scattering}\n\\label{shad_multiscatt}\n\nWe expect that particle production in $pA$ ($dA$) collisions will have \ndifferent yields in the forward and backward directions. This is because the\nrespective partons have different momentum fractions (shadowing differences)\nand because the forward-going parton has to traverse a large amount of matter.\nWe found in our earlier studies that the HIJING shadowing parameterization\nis particularly useful for the study of multiple scattering, as it requires\nan explicit treatment of $k_T$ broadening (see eq.~(\\ref{eq:broad})) for a \nsuccessful description of data~\\cite{Zhang:2001ce}. In this Section we therefore \nstudy the interplay of shadowing and multiple scattering in more detail \nusing the HIJING parameterization. \n\n\\begin{figure}[!htb]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=0]{lin_etadist_hij_pt_3.eps}\n\\end{center}\n\\caption[...]{(Color Online) Illustration of calculated yields at $p_T=3.0$ GeV\/c in\nminimum bias neutral pion production from $dAu$ collisions at 200 $A$GeV.  \nThe solid line represents HIJING shadowing with intrinsic $k_T$ in the\nproton, the dashed line is HIJING with intrinsic $k_T$ and multiple \nscattering, while the dot-dashed curve is obtained by turning off\nany transverse momentum.}\n\\label{fig:multiscatt}\n\\end{figure}\n\nIn Fig.~\\ref{fig:multiscatt} we display the \nneutral pion yield as a function of pseudo-rapidity in $dAu$ collisions \nat a fixed $p_T$, namely at $ 3.0$ GeV\/c. This illustrates how the \nnuclear effects we examine modify the asymmetry of the yields. \nThe figure shows the calculated distribution using the HIJING shadowing\nparameterization without any intrinsic $k_T$ (dot-dashed curve), \nusing HIJING shadowing and including the intrinsic $k_T$ in the nucleon (solid), \nand HIJING shadowing with intrinsic $k_T$ and multiple scattering (dashed).\nThe parameters of eq.~(\\ref{eq:broad}) are unchanged from our previous \nstudies at midrapidity~\\cite{Levai:2003at,Levai:2006yd}, and \nwe have chosen a transverse momentum value comparable to $\\langle k_T \\rangle_{pp}$,\nwhere the various effects are clearly displayed.  \n\nWe checked that when shadowing and all nuclear effects are turned off, \nthe distribution is symmetric around midrapidity, as expected. \nWith shadowing only, we find an asymmetry in the distribution: the yield \nat a fixed negative pseudorapidty, say $\\eta=-2$ ($Au$ side or ``backward'') \nis higher than at the corresponding positive pseudorapidity ($\\eta=2$, $d$ side,  \nor ``forward''). Thus the pseudorapidity asymmetry, $Y_{Asym}$, is greater \nthan unity in this case. The inclusion of the intrinsic transverse momentum \nin the proton significantly increases the yields on both the $Au$ side and the\n$d$ side at this transverse momentum, as it makes larger $p_T$-s accessible.\nWe tested that adding intrinsic transverse momentum in the proton in the \nabsence of shadowing does not destroy the forward\/backward symmetry. When\nintrinsic $k_T$ in the proton is added to the calculation using shadowing,\nbut without the multiple scattering contribution, the sense of the asymmetry \ngiven by shadowing (backward\/forward $\\geq$ 1) is preserved. However, when \nmultiple scattering is included, the yield shows a stronger increase in \nthe forward direction. This is understandable in the present picture, since \nforward-going products originate in the partons of the deuteron, and have\nto traverse a large amount of nuclear matter resulting in strong multiple \nscattering, while the backward products suffer no or little multiple \nscattering. This has the effect of reversing the asymmetry: the yield on the \n$d$-side is now greater than that on the $Au$-side. The calculated pseudorapidity \nasymmetry, $Y_{Asym}$, turns out to be less than unity in this case.\nAn alternative way to summarize the situation is to say that shadowing suppresses\nthe yield more on the $d$ side (forward) relative to a symmetric collision,\nwhile the multiple scattering contribution is understandably large in the \nforward direction.\n \nWe have carried out similar studies at higher $p_T$ values. When the intrinsic\ntransverse momenta are small relative to the $p_T$ of the final hadron, $k_T$\neffects become naturally smaller. Shadowing effects also become smaller as\nthe antishadowing region of the HIJING parameterization is approached. Thus,\nat $p_T \\gtrsim 15$ GeV\/c the influence of multiple scattering and intrinsic\n$k_T$ become negligible. These phenomena are most important at intermediate $p_T$ \nvalues, 2 GeV\/c $\\lesssim p_T \\lesssim$ 8 GeV\/c at RHIC. It is interesting to note\nthat a similar transverse-momentum region is sensitive to nuclear effects \nat lower (e.g. CERN SPS) energies, due to the $\\sim \\log(\\sqrt{s})$ scaling of the \nCronin peak~\\cite{Barnafoldi:2007uw,e706,Zielinski}. \n\n\\section{Results}\n\\label{res}\n\nTo judge the success of the model in reproducing spectra,  \nhere we first present calculated spectra for neutral pion production \nat midrapidity and non-zero rapidities. This will help select the model \nchoices providing the best agreement with the experimental information. \nThe selected model variants are then used to calculate pseudorapidity \nasymmetries.\n \n\\subsection{Midrapidity spectra}\n\\label{centrap}\n\nFigure~\\ref{fig:dAuspect1} displays midrapidity RHIC $\\pi^0$ spectra from \n$dAu$ collisions at $|\\eta| < 0.35$, and for reference, the spectra \nfrom $pp$ collisions at the same energy ($\\sqrt{s}= 200 A$GeV). The results are \ncompared to data from the PHENIX collaboration~\\cite{PHENIXdAu}. In the top left\npanel we show calculated $dAu$ spectra using EPS08 nPDFs with and without intrinsic $k_T$ \nin the proton, but without any multiple scattering contribution. The top right depicts \nthe $dAu$ spectra using HIJING plus $k_T$ with and without multiple scattering.\nThe bottom panels contain data\/theory (i.e. data\/model) ratios. \nAs can be seen clearly in the bottom left $dAu$ data\/theory panel, including \nintrinsic $k_T$ in the proton increases the calculated yield mostly at low $p_T$.\nWe consider the EPS08 description with intrinsic $k_T$ in the proton\nsatisfactory in the 4 GeV\/c \n$\\lesssim p_T \\lesssim$ 10 GeV\/c interval. Comparing the two bottom \ndata\/theory panels it is easy to see that the HIJING parameterization gives\na very similar accuracy for $dAu$ when multiple scattering is included. This is \nalso very close to the data\/theory for $pp$ collisions given by HIJING. \n\n\\begin{figure}[!h]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=270]{dAus1.ps}\n\\end{center}\n\\caption[...]{(Color Online) Top: spectra for $dAu$ $\\pi^0$ production at \n$|\\eta| < 0.35$. The left panel shows EPS08 with (dashed) and without (solid) \nintrinsic $k_T$. The right panel is HIJING plus $k_T$ with (solid) and without \n(dashed) multiple scattering. Filled triangles denote the PHENIX data~\\cite{PHENIXdAu}. \nThe $pp$ spectra from the same experiment and calculated with (dashed) and without \n(solid) intrinsic $k_T$ are also included. Bottom: corresponding data\/theory\nratios for (left) $dAu$ with and without $k_T$  and (right) $pp$ with $k_T$\nand HIJING plus $k_T$ with multiple scattering.}\n\\label{fig:dAuspect1}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=270]{pBes1.ps}\n\\end{center}\n\\caption[...]{(Color Online) Spectra for $p+Be \\rightarrow \\pi{^0}+X$ at $|\\eta| < 0.2$.\nThe left panel depicts EPS08 calculations with (dashed) and without (solid) intrinsic $k_T$. \nThe right panel is HIJING plus $k_T$ with (solid) and without (dashed)\nmultiple scattering. Filled triangles denote the E706\ndata~\\cite{Apanasevich:2002wt}. Data and calculations for proton-proton (pp)\ncollisions are at $530$ GeV\/c.}\n\\label{fig:pBespect1}\n\\end{figure}\n\nFig.~\\ref{fig:pBespect1} shows the spectra for neutral pion \nproduction from $pBe$ collisions at $|\\eta| < 0.2$. The top left \npanel shows the spectra using EPS08 nPDFs with and without intrinsic\n$k_T$. The top right depicts the spectra using HIJING plus $k_T$ \nwith and without multiple scattering. The agreement with the E706 \ndata from Fermilab is quite good as can be seen from the lower panels\nwhich display the data per theory ratio. The HIJING \nparameterization with and without multiple scattering gives almost \nidentical data\/theory ratios, i.e. multiple scattering has only\na small effect on spectra from $pBe$ \ncollisions. This is reasonable in view of the fact that in \nthe light $Be$ nucleus there are few \nscattering centers. The effect of intrinsic \n$k_T$ is to increase the calculated yield, leading to data\/theory ratios\nless than unity for all relevant $p_T$. \n\nWe conclude from Fig.s~\\ref{fig:dAuspect1} and \\ref{fig:pBespect1} that\nit is necessary to include the intrinsic transverse \nmomentum of partons in the proton in our model to obtain a satisfactory\ndescription of available data. The EPS08 and HIJING parameterizations differ\nto the extent that while HIJING calls for the inclusion of the broadening of\nthe intrinsic transverse momentum distribution via multiple scattering, \nEPS08 appears to incorporate this physics in their nPDFs. We thus \nconcentrate on three model variants (EPS08 with proton intrinsic $k_T$ and \nHIJING with intrinsic $k_T$ and with\/without multiscattering) in the remainder\nof this study.\n\n\\subsection{Spectra at non-zero rapidities}\n\\label{noncentrap}\n\nWe now consider spectra at non-zero pseudorapidities. \nWe display results from EPS08 with proton \nintrinsic $k_T$ and HIJING with intrinsic $k_T$ and multiscattering.\nFig.~\\ref{fig:pBespect2} shows the spectra and corresponding data\/theory ratio \nfor $p+Be \\rightarrow \\pi{^0}+X$ at $-0.7 < \\eta < -0.2$ (``backward'', left panel) \nand $0.2 < \\eta < 0.7$ (``forward'', right panel) using EPS08 nPDFs with $k_T$ \nand HIJING plus $k_T$ with multiscattering. The two sets (EPS08 and HIJING) give\nvery similar data\/theory ratios for both pseudorapidity intervals. There is\nreasonable agreement with the E706 data~\\cite{Apanasevich:2002wt} as is apparent \nfrom the quality of the data\/theory ratios.\n\\begin{figure}[!h]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=270]{pBes2.ps}\n\\end{center}\n\\caption[...]{(Color Online) Spectra for $p+Be \\rightarrow \\pi{^0}+X$ at \n$-0.7 < \\eta < -0.2$ (left panel) and $0.2 < \\eta < 0.7$ (right panel). \nThe solid lines represent the EPS08 nPDFs with intrinsic $k_T$, while \nthe dashed line is obtained from HIJING \nplus $k_T$ with the inclusion of multiscattering. Stars denote \nthe E706 data~\\cite{Apanasevich:2002wt}.}\n\\label{fig:pBespect2}\n\\end{figure}\n\nThe $dAu$ spectra are displayed in Fig.~\\ref{fig:dAuspect2} for both \nEPS08 and HIJING at $-1.0 < \\eta < -0.5$ (left panel) and $0.5 < \\eta < 1.0$\n(right panel). Note that the experimental data from the STAR \ncollaboration~\\cite{Abelev:2006pp} are not separated into negative and\npositive pseudorapidities, but rather averaged over both \nintervals. Therefore, the data points in Fig.~\\ref{fig:dAuspect2} only\nserve to guide the eye; the relevance of the Figure is to highlight the \ndifference between the EPS08 and HIJING results visible in the bottom panels.\nIt can be seen that the EPS08 results do not differ\nmuch in the forward and backward directions, while HIJING gives significantly larger\ndata\/theory ratios forward. Thus, EPS08 and HIJING \ndiffer appreciably at forward pseudorapidities, $0.5 < \\eta < 1.0$.\nThis is not unexpected, because multiple scattering influences\nthe $d$-side (forward) more than the $Au$-side (backward).\nTo be able to draw stronger conclusions, separated forward and backward\ndata will be necessary, hopefully forthcoming from the high-statistics Run 8.\n\n\\begin{figure}[!h]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=270]{dAus2.ps}\n\\end{center}\n\\caption[...]{(Color Online) Spectra for $d+Au \\rightarrow \\pi{^0}+X$ at \n$-1.0 < \\eta < -0.5$ (left panel) and $0.5 < \\eta < 1.0$ (right panel). \nThe solid line represents the\nEPS08 with intrinsic $k_T$, while the dashed line is obtained from HIJING \nplus $k_T$ with the inclusion of multiscattering. Stars denote STAR data\naveraged over $0.5 < |\\eta| < 1.0$}\n\\label{fig:dAuspect2}\n\\end{figure}\n\n\\subsection{Pseudorapidity asymmetry}\n\\label{etasmy}\n\n\\subsubsection{Asymmetry in $pBe$ collisions at $30.7$ GeV}\n\\label{asympBe}\n\nThe pseudorapidity asymmetry for the rapidity interval\n$0.2 < |\\eta| < 0.7$ is shown in the upper panel of Fig.~\\ref{fig:asyfnal} \ncompared with the E706 data~\\cite{Apanasevich:2002wt}.\nThe solid line represents EPS08 with proton intrinsic $k_T$, the dot-dashed line \ndisplays HIJING with intrinsic $k_T$, and the dashed is HIJING plus \nintrinsic $k_T$ and multiple scattering. Both EPS08 with $k_T$ and HIJING \nwithout multiple scattering give very small asymmetries and the data are\nalso consistent with $Y_{Asym}=1$ for low $p_T$. The HIJING parameterization\nwith multiple scattering yields somewhat lower values at all transverse momenta.\nThis is in line with the effect of multiple scattering moderately increasing \nthe yield on the $p$-side relative to that of the $Be$-side. In view of the rather \nlarge error bars, all three sets are in reasonable agreement with the data. \n\\begin{figure}[!h]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=270]{asympBe.ps}\n\\end{center}\n\\caption[...]{(Color Online) Pseudorapidity asymmetry,\n$Y_{Asym}$ for $p+Be \\rightarrow \\pi{^0}+X$ at $0.2 < |\\eta| < 0.7$ (top)\nand  $1.0 < |\\eta| < 1.5$ (bottom). The solid line represents the\nEPS08 nPDFs, while the dashed line is obtained from HIJING with \nthe inclusion of multiscattering. The dot-dashed line \ncorresponds to HIJING without multiscattering, and filled triangles denote \nthe E706 data~\\cite{Apanasevich:2002wt}.}\n\\label{fig:asyfnal}\n\\end{figure}\nThe lower panel is our prediction for the interval $1.0 < |\\eta| < 1.5$. \nHere, the calculated effects are larger, but have a similar structure to\nwhat is seen at lower $\\eta$. This trend is similar to what will be seen at \nother energies. \n\n\n\\subsubsection{Asymmetry in $dAu$ collisions at $200$ $A$GeV}\n\\label{asymdAu}\n\nFigure~\\ref{fig:asyrhic} shows the pseudorapidity asymmetry for $\\pi^0$ \nproduction from $dAu$ collisions at RHIC, for different pseudorapidity intervals. \nThe two uppermost panels are our results for the asymmetry at $|\\eta| < 0.5$ \nand $0.5 < |\\eta| < 1.0$ compared with the STAR data~\\cite{Abelev:2006pp}. For\n$p_T > 4.0$ GeV\/c, the agreement with data is quite good for all three sets.\nAt lower $p_T$, multiple scattering increases the calculated yield mostly in the \nforward direction as discussed in Sec.~\\ref{shad_multiscatt}, leading to \nasymmetries below unity. At very high $p_T$ we observe a divergence in \nthe model predictions.\n\\begin{figure}[!h]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=270]{asymdAu1.ps}\n\\end{center}\n\\caption[...]{(Color Online) Pseudorapidity asymmetry,\n$Y_{Asym}$ for $d+Au \\rightarrow \\pi{^0}+X$ at different \npseudorapidity intervals. The solid line represents the\nEPS08 nPDFs, while the dashed line is obtained using HIJING shadowing with \nthe inclusion of multiscattering. The dot-dashed line \ncorresponds to HIJING without multiscattering, and filled triangles denote \nthe STAR data~\\cite{Abelev:2006pp}.}\n\\label{fig:asyrhic}\n\\end{figure}\nThe lower four panels are our predictions for the asymmetry as pseudorapidity \nincreases. The first three of these correspond to the BRAHMS\npseudorapidity intervals~\\cite{Arsene:2004ux}. The general trend is that the\nasymmetry becomes larger as $\\eta$ increases. This mainly arises\nfrom the strong shadowing in the larger nucleus at lower $x$ values. Also, \nsince increasing $\\eta$ leads to decreasing accessible $p_T$ \ndue to phase space constraints, the effects of multiple scattering become \nmore pronounced. In fact, for the largest $\\eta$ considered, \n$3.7 < |\\eta| < 4.3$, there is a marked difference between HIJING with \nand without multiple scattering for all $p_T$ considered. \n\nIt is pertinent at this point to make some remarks about\npseudorapidity asymmetry in the BRAHMS pseudorapidity intervals.\nThe BRAHMS Collaboration~\\cite{Arsene:2004ux} has observed a progressive \nsuppression of both minimum bias nuclear modification $R_{dAu}$ and \ncentral-to-peripheral ratios, $R_{CP}$, with increasing\n$\\eta$. The present study is limited to minimum bias pseudorapidity\nasymmetry, and both shadowing parameterizations (EPS08 and HIJING) adequately \ndescribe the existing experimental data at very forward rapidities. \nThe EPS08 parameterization incorporates RHIC data at large rapidities, i.e. at\nlow $x$, and thus reproduces the data. Therefore, at least in the \nminimum bias case, shadowing seems sufficient for a good description of \nthe suppression observed at low $x$. \nThe situation is different for the geometry-dependent $R_{CP}$, \nwhere ($b$-independent) shadowing plus conjectured \nimpact parameter dependencies~\\cite{Adeluyi:2008qk,Vogt:2004hf} \nare clearly inadequate in describing the observed suppression.\n\n\\subsubsection{Asymmetry in $dPb$ collisions at $8.8$ $A$TeV}\n\\label{asymdPb}\n\nLet us now turn to our predictions for the pseudorapidity asymmetry in a potential future \n$dPb$ collision at LHC energy of $8.8$ $A$TeV. \nThe calculated results are displayed in \nFig.~\\ref{fig:asylhc}, where the upper panel is for the interval $|\\eta| < 0.9$ \nand the lower panel is for $2.4 < |\\eta| < 4.0$. These \nintervals correspond to acceptance in the central detector and in the muon \narm, respectively, of the ALICE experiment~\\cite{Alessandro:2006yt}. All three \nsets predict minimal asymmetry of the order of a few percent for the interval \n$|\\eta| < 0.9$. \n\\begin{figure}[!h]\n\\begin{center} \n\\includegraphics[width=8.5cm, height=8.5cm, angle=270]{asymdPb1.ps}\n\\end{center}\n\\caption[...]{(Color Online) Predicted pseudorapidity asymmetry,\n$Y_{Asym}$ for $d+Pb \\rightarrow \\pi{^0}+X$ at $\\sqrt{s} = 8.8$ $A$TeV\nfor $|\\eta| < 0.9$ and $2.4 < |\\eta| < 4.0$. The solid line represents the\nEPS08 nPDFs, while the dashed line is obtained from HIJING with \nthe inclusion of multiscattering. The dot-dashed line \ncorresponds to HIJING without multiscattering.}\n\\label{fig:asylhc}\n\\end{figure}\nAs we move to higher $\\eta$, the predicted asymmetry becomes more significant. \nAs can be seen in the lower panel of Fig.~\\ref{fig:asylhc}, both EPS08 and \nHIJING predict substantial asymmetry up to $\\sim 10$ GeV\/c, and both variants\nof HIJING asymmetries remaining significant up to $\\sim 100$ GeV\/c, in \ncontrast to EPS08. \n\nAt the present level, neither model variant gives agreement with all aspects of \nthe data: in an earlier calculation we have found that shadowing \nparameterizations which do not need to be augmented by a multiple scattering \nprescription~\\cite{Eskola:1998df,Shad_HKN,Frankfurt:2003zd} have difficulty \ndescribing central-to-peripheral ratios at forward rapidity~\\cite{Adeluyi:2008qk}. \nWe have checked that this also holds for the EPS08 nPDFs. On the other hand,\nthe HIJING parameterization with multiscattering yields pseudorapidity \nasymmetries below unity at low transverse momenta. This deficiency may be cured \nby allowing an $\\eta$-dependent multiscattering~\\cite{gergely:2008}. \n\\section{Conclusion}\n\\label{concl}\n\nHere we give a concise summary of the results of our\ncalculations. We have demonstrated the usefulness of asymmetric \n(light-on-heavy) nuclear collisions at relativistic energies. As illustrated\nin Sec.~\\ref{shad_multiscatt}, the physical differences between forward- and \nbackward-going produced particles arise from the different ranges of $x$ \nsampled (different shadowing) and different amount of multiple scattering.\nThis leads to observable pseudorapidity asymmetries\nat some collision energies and transverse momenta.\nWe have considered the effects of nuclear shadowing \nand multiple scattering on pseudorapidity asymmetry for three asymmetric \nsystems: $pBe$, $dAu$, and $dPb$ in a wide energy range.\n\nTo calibrate and fine-tune our model we first examined spectra of\nproduced neutral pions. We found that there are two avenues in the model for \nthe reasonable description of these data: (i) HIJING shadowing, intrinsic\n$k_T$, plus multiscattering, or (ii) some other nPDFs (we use EPS08 here) \nand intrinsic $k_T$ (but no additional multiscattering). We then calculated \npseudorapidity asymmetries with these prescriptions.\nOverall, the calculated asymmetries are in reasonable agreement with available \nexperimental data. Intrinsic transverse momentum in the nucleon is\nseen to be important at low $p_T$. Multiple scattering \nincreases the yield in the ``forward'' or positive pseudorapidity region,\nthus leading to a tendency for asymmetries less than unity at low $p_T$\nin the scheme explicitely relying on multiple scattering,\nat variance with the data. An LHC measurement at a high pseudorapidity \nand high $p_T$ (where $k_T$ effects no longer make a difference) may be \nable to distinguish between strong shadowing (as in the HIJING prescription) \nand an nPDF with a relatively weaker gluon suppression (like e.g. EPS08).\n\nA major constraint in assessing pseudorapidity asymmetries is the limited \navailability of data for direct comparison with theoretical calculations. \nMore data in asymmetric light-on-heavy collisions separated with respect\nto positive and negative pseudorapidities are needed\nto judge calculated pseudorapidity asymmetries. At RHIC, it is expected\nthat the high-statistic $dAu$ Run 8 is going to provide such a large data set.\n\n\n\\section{Acknowledgments}\n\\label{ack}\n\nThis work was supported in part by Hungarian OTKA PD73596,\nT047050, NK62044, and IN71374, by the U.S. Department of Energy under\ngrant U.S. DOE DE-FG02-86ER40251, and jointly by the U.S. and Hungary under\nMTA-NSF-OTKA OISE-0435701.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nAn important issue in general relativity is the nature of singularities. While it is widely believed that the strong cosmic censorship conjecture holds, which can be loosely stated as the expectation that timelike singularities do not form by evolution from generic spatially compact or asymptotically flat initial data sets \\cite{Penrose1979}, the issue is wide open. From this perspective it is of interest to consider timelike singularities and therefore the ways in which cosmic censorship could be violated.\n\nThere are numerous exact solutions with timelike singularities (e.g. \\cite{Griffiths2009}). Such solutions are typically obtained in searches of solutions with symmetries. This leads naturally to the question, whether there exist solutions with timelike singularities and without symmetries. We prove in this work that this is indeed the case: We construct an analog to the class of the non-chaotic solutions without symmetries and with controlled asymptotics of \\cite{Chrusciel2015,Klinger2015}, by changing the time parameter $\\tau$ from a timelike to a spacelike coordinate. As the behavior of the Hamiltonian differs from the spacelike case only by sign changes which do not affect the analysis in the analytic case, we obtain a family of solutions with the same free functions and asymptotics (in terms of the now spacelike $\u03c4$ coordinate) but with a timelike instead of spacelike singularity.\n\nThe construction of the solutions is based on the cosmological billiard formalism using the  Iwasawa decomposition of the metric. This method was introduced by Damour, Henneaux, and Nicolai in \\cite{Damour2003} to give a heuristic argument for the chaotic picture of spacelike singularities provided by the  BKL conjecture, and later by Damour and DeBuyl in \\cite{Damour2008} to provide a precise statement of the conjecture.\n\nWe also show that the change of the time parameter $\\tau$ from a timelike to a spacelike coordinate, i.e. considering timelike instead of spacelike singularities, switches the signs of some of the terms (walls) in the Hamiltonian considered. These changes violate the property of the spacelike case that the coefficients of the dominant wall terms are positive, thus rendering the arguments of Damour et al. in \\cite{Damour2003} inapplicable. The affected terms become attractive rather than repulsive, allowing subdominant walls lying behind the dominant ones to become relevant. This does not affect the class of solutions we construct here, as these are non-generic and use an ansatz that suppresses the wall terms asymptotically.\n\n\\section{Derivation of Hamiltonian for spacelike ``time''-variable}\nWe follow the derivation of the Hamiltonian formalism by Wald \\cite[Appendix E.2]{Wald1984}. The spacetime metric is denoted by $\\bar{g}_{\u03b1\u03b2}$ while the induced lorentzian metric on the timelike hypersurfaces of constant $\u03c4$ is denoted by $g_{ij}$.  We choose a zero shift gauge, i.e. the metric takes the form\n\\begin{equation}\n\\mathrm{d}s^2=N^2\\mathrm{d}\u03c4^2+g_{ij}\\mathrm{d}x^i\\mathrm{d}x^j\\,.\n\\end{equation}\n\nAs the hypersurfaces of constant $\u03c4$ are timelike, their normal vector is spacelike. This means that the Gauss equation takes the form\n\\begin{equation}\nR\\indices{_{abc}^d}=g_a{}^f g_b{}^g g_c{}^k g^d{}_j\\bar{R}\\indices{_{fgk}^j}+K_{ac}K\\indices{_b^d}-K_{bc}K\\indices{_a^d}\n\\end{equation}\nwhere $R\\indices{_{abc}^d}$ and $\\bar{R}\\indices{_{abc}^d}$ are the Riemann curvature tensors of the induced and full metric respectively and $K_{ab}$ is the second fundamental form of the hypersurface. Compared to the case of spacelike hypersurfaces the signs of the $KK$ terms are interchanged.\n\nUsing $g_{\u03b1\u03b2}=\\bar{g}_{\u03b1\u03b2}-n_\u03b1 n_\u03b2$ with $n^\u03b1$ the unit normal vector of the hypersurface ($n_\u03b1n^\u03b1=1$) gives\n\\begin{equation}\n\\bar{R}_{\u03b1\u03b2\u03b3\u03b4}g^{\u03b1\u03b3}g^{\u03b2\u03b4}=-2\\bar{G}_{\u03b1\u03b2}n^\u03b1n^\u03b2\\,.\n\\end{equation}\n\nThis leads to a change of sign in the constraint equation:\n\\begin{equation}\\label{const}\n0=\\bar{G}_{\u03bc\u03bd}n^\u03bc n^\u03bd = -\\frac{1}{2}R-(K\\indices{^\\mu_\\mu})^2+K_{\u03bc\u03bd}K^{\u03bc\u03bd}\\,.\n\\end{equation}\n\nContracting the Einstein tensor twice with the normal vector $n^a$ gives an expression for the scalar curvature:\n\\begin{equation}\\label{scurv}\n\\bar{R}=-2n^\u03b1n^\u03b2(\\bar{G}_{\u03b1\u03b2}-\\bar{R}_{\u03b1\u03b2})\\,.\n\\end{equation}\n\nThe definition of the Riemann tensor gives for the last term\n\\begin{equation}\\label{riemnn}\n\\begin{split}\n\\bar{R}_{\u03b1\u03b2}n^\u03b1n^\u03b2=&\\bar{R}_{\u03b1\u03b3\u03b2}{}^\u03b3n^\u03b1n^\u03b2=-n^\u03b1(\\nabla_\u03b1\\nabla_\u03b3-\\nabla_\u03b3\\nabla_\u03b1)n^\u03b3\\\\\n=&(\\nabla_\u03b1 n^\u03b1)(\\nabla_\u03b3 n^\u03b3)-(\\nabla_\u03b3 n^\u03b1)(\\nabla_\u03b1 n^\u03b3)\\\\\n&-\\nabla_\u03b1(n^\u03b1\\nabla_\u03b3 n^\u03b3)+\\nabla_\u03b3(n^\u03b1\\nabla_\u03b1n^\u03b3)\\\\\n=&(K^\u03b1_\u03b1)^2-K_{\u03b1\u03b3}K^{\u03b1\u03b3}-\\nabla_\u03b1(n^\u03b1\\nabla_\u03b3 n^\u03b3)+\\nabla_\u03b3(n^\u03b1\\nabla_\u03b1n^\u03b3)\n\\end{split}\n\\end{equation}\nwhere the last two terms are divergences, which will be discarded in the Lagrangian.\n\nUsing \\eqref{const}, \\eqref{scurv}, \\eqref{riemnn} and $\\sqrt{-\\bar{g}}=N\\sqrt{-g}$ to express the Einstein-Hilbert action gives\n\\begin{equation}\n\\mathcal{L}=\\sqrt{-\\bar{g}}\\bar{R}=-\\sqrt{-g}N\\left(R-K_{ab}K^{ab}+(K^a{}_a)^2\\right)\\,.\n\\end{equation}\nThe momenta canonically conjugate to the components $g_{ij}$ are given by\n\\begin{equation}\n\u03c0^{ij}=\\frac{\u2202\\mathcal{L}}{\u2202\\dot{g}_{ij}}=\\sqrt{-g}(K^{ij}-K^k{}_k g^{ij})=N^{-1}\\sqrt{-g}\\frac{1}{2}(\\dot{g}_{kl}g^{ki}g^{lj}-\\dot{g}_{kl} g^{kl}g^{ij})\\,,\n\\end{equation}\nunchanged from the standard case.\n\nThe Hamitonian, expressed in terms of the canonical coordinates $g_{ab}$ and momenta $\u03c0^{ab}$ is finally\n\\begin{equation}\\label{hamiltonian}\n\\mathcal{H}=\u03c0^{ab}\\dot{g}_{ab}-\\mathcal{L}=(-g)^{-1\/2}N\\left(\u03c0^{ab}\u03c0_{ab}-\\frac{1}{2}(\u03c0^a{}_a)^2\\right)+RN\\sqrt{-g}\n\\end{equation}\ni.e. the standard one with the sign of the curvature term changed.\n\n\\section{Iwasawa variable Hamiltonian}\nHere we will describe the changes to the derivation of the Iwasawa variable Hamiltonian, as given in Appendix A of \\cite{Klinger2015}.\n\nSince the level sets of $\\tau$ are timelike, we need to decide which frame vector is the timelike one. As the Iwasawa ansatz breaks the symmetry between the frame vectors, different choices will lead to different dynamical systems. We will use an index  $J\\in\\{1,2,3\\}$ to distinguish between those cases: $x^J$ will denote the timelike coordinate.\n\nThe Lorentzian metric $g_{ij}$ on the $\u03c4=\\text{const}$ hypersurfaces is split in Iwasawa variables as\n\\begin{equation}\ng_{ij}=\\sum_a m^J_a e^{-2\u03b2^a}\\N{^a_i}\\N{^a_j}\n\\end{equation}\nwhere $m^J_a=1-2\u03b4_{Ja}$, i.e. $-1$ for $a=J$ and $1$ otherwise. \n\nWe set the lapse function $N$ equal to $\\sqrt{-g}$ where $g$ is the determinant of the metric $g_{ij}$. The (timelike) singularity will be approached as $\u03c4\\to \\infty$.\n\nThe conjugate momenta $\u03c0_a$to the $\u03b2^a$ and $P\\indices{^i_a}$ to the $\\N{^a_i}$ are given by\n\\begin{equation}\n\u03c0_a=\\frac{\u2202\\mathcal L}{\u2202\\dot{\u03b2}^a}=\\frac{\u2202\\mathcal L}{\u2202\\dot{g}_{ij}}\\frac{\u2202\\dot{g}_{ij}}{\u2202\\dot{\u03b2^a}}=-2\u03c0^{ij}m^J_ae^{-2\u03b2^a}\\N{^a_i}\\N{^a_j}\n\\end{equation}\nand\n\\begin{equation}\nP\\indices{^i_a}=2m^J_a \u03c0^{ij}\\N{^a_j}e^{-2\u03b2^a}\\,,\n\\end{equation}\ni.e. the same as in the spacelike case except for the additional factor $m^J_a$.\n\nThe non-curvature terms of the Hamiltonian \\eqref{hamiltonian}, with $N=\\sqrt{-g}$ inserted, are\n\\begin{equation}\\label{2015X30.2}\n\u03c0^{ab}\u03c0_{ab}-\\frac{1}{2}(\u03c0^a{}_a)^2\\,.\n\\end{equation}\n\nThe first term can be split into\n\\begin{equation}\\label{2015X30.1}\n\\frac{1}{4}\\sum_a \u03c0_a^2+\\frac{1}{2}\\sum_{a<b}m^J_a m^J_b e^{-2(\u03b2^b-\u03b2^a)}\\left(P\\indices{^j_a}\\N{^b_j}\\right)^2\\,.\n\\end{equation}\nThe first term of \\eqref{2015X30.1} together with the second term of \\eqref{2015X30.2} give the kinetic term $G^{ab}\u03c0_a\u03c0_b$ of the Hamiltonian, unchanged from the spacelike case. The second term of \\eqref{2015X30.1} is the symmetry wall term, with the addition of $m^J_a$ and $m^J_b$. These cause a sign change for two of the symmetry walls.\n\nThe Iwasawa form of the curvature term in the Hamiltonian \\eqref{hamiltonian}, which gives the gravitational potential walls, is calculated in the Iwasawa frame, where the metric takes the form\n\\begin{equation}\n\u03b3_{ab}=\u03b4_{ab}m^J_ae^{-2\u03b2^a}=\u03b4_{ab}m^J_a A_a^2\n\\end{equation}\nwith $A_a^2:=\\exp(-2\u03b2^a)$ i.e. with an additional factor $m^J_a$ compared to the spacelike case.\n\nThis is the only change in the derivation of the curvature term, as the Cartan formulas remain unchanged. The terms corresponding to the dominant gravitational walls are\n\\begin{equation}\n\\frac{1}{4}\\hspace{-0.9em}\\sum_{a\\neq b\\neq c \\neq a}\\hspace{-0.9em}(C\\indices{^a_b_c})^2\\frac{A_a^2}{A_c^2A_b^2}\\frac{m^J_a}{m^J_b m^J_c}\\,.\n\\end{equation}\nIn $3+1$ dimensions exactly one of $a, b, c$ is equal to $J$, which adds an additional minus in front of this term. This cancels the change of sign in the Hamiltonian.\n\nIn addition there are sign changes in the subdominant gravitational terms but as they have indeterminate sign even for the spacelike case this does not affect the analysis.\n\nIn conclusion, for $3+1$ dimensions, the prefactors of two of the symmetry walls change sign. In the case $J=1$ this involves the $\u03b2^2-\u03b2^1$ and $\u03b2^3-\u03b2^1$ walls, for $J=2$ the $\u03b2^3-\u03b2^2$ and $\u03b2^2-\u03b2^1$ walls and for $J=3$ the $\u03b2^3-\u03b2^2$ and $\u03b2^3-\u03b2^1$ walls. In all cases the sign of at least one dominant wall term changes.\n \n The potential (i.e. the Hamiltonian without the terms containing $\u03c0^a$) for the spacelike case and the 3 choices of $J$ is sketched in figures \\ref{fig:t} to \\ref{fig:x3}.\n \n \\begin{figure}[p]\n\n    \\centering\n    \\subfloat[$\u03c4$ timelike, $\u03b2^3=3$ slice]{\n        \\includegraphics[width=0.45\\textwidth]{t_b1_b2.pdf}\n\t} \n    \\subfloat[$\u03c4$ timelike, $\u03b2^1=1$ slice]{\n        \\includegraphics[width=0.45\\textwidth]{t_b2_b3.pdf}\n    }\n    \\caption{Sketch of the potential for the case of timelike $\u03c4$ (i.e. with spacelike singularity). Only the exponential terms are plotted, the coefficients are set to $1$. The potential increases from dark blue through orange to light yellow. Black lines mark the dominant walls, grey lines the subdominant walls and the allowed region (i.e. the ``billiard table'') is hatched.}\n\t\\label{fig:t}\n\\end{figure}\n\\begin{figure}[p]\n\n\t\\centering\n    \\subfloat[$x^1$ timelike, $\u03b2^3=3$ slice]{\n        \\includegraphics[width=0.45\\textwidth]{x1_b1_b2.pdf}\n\t}\n    \\subfloat[$x^1$ timelike, $\u03b2^1=1$ slice]{\n        \\includegraphics[width=0.45\\textwidth]{x1_b2_b3.pdf}\n    }\n    \\caption{$J=1$, i.e. $x^1$ timelike case. The walls with negative coefficients (set to $-1$ in the plots) are marked by red lines. The hatched region is the same as in figure \\ref{fig:t} but no longer corresponds to an allowed region, as the potential approaches $-\\infty$ outside it.}\n    \\label{fig:x1}\n\\end{figure}\n\\begin{figure}[p]\n\t\\centering\n    \\subfloat[$x^2$ timelike, $\u03b2^3=3$ slice]{\n        \\includegraphics[width=0.45\\textwidth]{x2_b1_b2.pdf}\n\t}\n    \\subfloat[$x^2$ timelike, $\u03b2^1=2$ slice]{\n        \\includegraphics[width=0.45\\textwidth]{x2_b2_b3.pdf}\n    }\n    \\caption{$J=2$ i.e. $x^2$ timelike case.}\n    \\label{fig:x2}\n\\end{figure}\n\\begin{figure}[p]\n\n\t\\centering\n    \\subfloat[$x^3$ timelike, $\u03b2^3=3$ slice]{\n        \\includegraphics[width=0.45\\textwidth]{x3_b1_b2.pdf}\n\t} \n    \\subfloat[$x^3$ timelike, $\u03b2^1=2$ slice]{\n        \\includegraphics[width=0.45\\textwidth]{x3_b2_b3.pdf}\n    }\n    \\caption{$J=3$ i.e. $x^3$ timelike case.}\n    \\label{fig:x3}\n\\end{figure}\n\\begin{figure}[htbp]\n\t\\centering\n    \\subfloat[$\u03c4$ timelike (spacelike singularity)]{\n        \\includegraphics[width=0.44\\textwidth]{hyp_t.pdf}\n\t} \n    \\subfloat[$J=1$, i.e. $x^1$ timelike]{\n        \\includegraphics[width=0.44\\textwidth]{hyp_x1.pdf}\n    }\\\\\n    \\subfloat[$J=2$, i.e. $x^2$ timelike]{\n        \\includegraphics[width=0.44\\textwidth]{hyp_x2.pdf}\n\t} \n    \\subfloat[$J=3$, i.e. $x^3$ timelike]{\n        \\includegraphics[width=0.44\\textwidth]{hyp_x3.pdf}\n    }\n    \\caption{Sketch of the potentials in the hyperbolic space of the $\u03b3^a$, projected onto the Poincar\\'{e} disk. As before the allowed region for the case of spacelike singularity is hatched, thick black lines mark the dominant walls, thick gray lines the subdominant walls and red lines the walls with negative coefficients.}\n    \\label{fig:hyp}\n\\end{figure}\n\n\n\\section{Consequences for cosmological billiard}\nThe arguments for the asymptotic billiard picture depend on the positive sign of the prefactors of the dominant wall terms: In hyperbolic coordinates $\u03c1$ and $\u03b3^a$, such that $G_{ab}\u03b3^a\u03b3^b=-1$ ($G_{ab}$ is the constant matrix $G_{ab}=-\\sum_{c\\neq d}\u03b4^c_a\u03b4^d_b$) and $\u03b2^a=\u03c1 \u03b3^a$ (such a splitting is possible, assuming the solution is close to a Kasner state asymptotically \\cite{Damour2003}) the Hamiltonian takes the form\n\\begin{equation}\nH=\\frac{1}{4}(-\u03c1^2\u03c0_\u03c1^2+\u03c0_\u03b3^2)+\u03c1^2\\sum_Ac_Ae^{-2\u03c1 \u03c9_A(\u03b3)}\\,.\n\\end{equation}\nIf the prefactors $c_A$ for the dominant walls are positive this approaches the ``sharp wall Hamiltonian''\n\\begin{equation}\nH=\\frac{1}{4}(-\u03c1^2\u03c0_\u03c1^2+\u03c0_\u03b3^2)+\\sum_{A'}\u0398_\\infty(-\u03c9_{A'}(\u03b3))\n\\end{equation}\nwhere the sum over $A'$ only covers the dominant walls and\n\\begin{equation}\n\u0398_\\infty(x)=\n\\begin{cases}\n  0&x<0\\,,\\\\\n  \\infty&x>0\\,.\n\\end{cases}\n\\end{equation}\nIf, however, some of the prefactors are negative the corresponding terms are potential wells instead of walls. In the timelike case in 3+1 dimensions this affects at least one of the dominant symmetry walls.\n\nFigure \\ref{fig:hyp} shows the potentials in the hyperbolic space of the $\u03b3^a$, projected onto the Poincar\\'{e} disk.\n\n\\section{Consequences for solutions constructed in  \\cite{Chrusciel2015,Klinger2015}}\nThe class of solutions constructed in \\cite{Chrusciel2015,Klinger2015} for the case of timelike $\u03c4$ also exists for spacelike $\u03c4$. The sign changes in the Hamiltonian have no effect on the arguments concerning the evolution equations in the context of the analytic Fuchs theorem, as the decay of the exponential terms is unchanged.\n\nAn additional factor $m^J_b$ appears in the term $\\tilde{\u03c0}^b{}_a$, which enters  in the Iwasawa variable momentum constraint:\n\\begin{equation}\\label{pi_iwa}\n\t\\tilde{\u03c0}\\indices{^b_a}=\\frac{1}{2}\\begin{cases}\n\t-\u03c0_b&\\text{for }a=b\\,,\\\\\n\t\\N{^b_i}P\\indices{^i_a}&\\text{for }b>a\\,,\\\\\n\tm^J_be^{-2(\u03b2^a-\u03b2^b)}\\N{^a_i}P\\indices{^i_b}&\\text{for }a>b\\,.\n\t\\end{cases}\n\\end{equation}\nAs the factor $m^J_b$ is only present in the asymptotically decaying case $a>b$, which is discarded in the asymptotic constraints, this leaves the conditions on the free functions unchanged.\n\nSimilarly there are sign changes in the derivation of the evolution equations for the constraints, given in Appendix D of \\cite{Klinger2015}, which cancel out in the final equations.\n\nAs in the case of a spacelike singularity, the presence of a cosmological constant does not affect the result (see Appendix F of \\cite{Klinger2015}).\n\nThis leads to the following theorem, in close analogy with the results of \\cite{Chrusciel2015,Klinger2015}:\n\n\\begin{theorem}\nFor any choice of $J\\in\\{1,2,3\\}$ and analytic functions $\u03b2_\\circ^2$, $\u03b2_\\circ^3$ and $P\\indices{_\\circ^2_1}$\ndepending on coordinates $x^i\\,,i\\in\\{1,2,3\\}$,\nand for any two analytic functions, $p_\\circ^2$ and $p_\\circ^3$ depending on $x^i$, which satisfy the inequalities\n %\n\\bel{13VI15.1}\n0<p_\\circ^2<(\\sqrt{2}-1)p_\\circ^3\\,.\n\\end{equation}\nwe obtain a solution of the vacuum Einstein equations with arbitrary cosmological constant given by the metric\n\\begin{equation}\\label{10II15.1}\ng=e^{-2\\sum_{a=1}^3  \u03b2^a}\\mathrm{d} \u03c4^2+\\sum_{a=1}^3m^J_a  e^{-2\u03b2^a}\\N{^a_i}\\N{^a_j}\\mathrm{d}x^i\\mathrm{d}x^j\\,.\n\\end{equation}\nHere $m^J_a=1-2\u03b4_{Ja}$, $\u03b2^a$ and $\\N{^a_i}$, $i,a\\in\\{1,2,3\\}$,  depend on all coordinates $\u03c4$, $x^i$ and behave asymptotically as\n\\begin{equation}\\label{10II15.2}\n\u03b2^a=\u03b2_\\circ^a+\u03c4p_\\circ^a+O(e^{-\u03c4\u03bd})\\quad\\text{and}\\quad\n\\N{^a_i}=:\u03b4^a_i+ \\N{_s^a_i}=\u03b4^a_i+O(e^{-\u03c4\u03bd})\n \\,,\n\\end{equation}\nwhere $\u03bd$ is a positive constant, the $\u03b2_\\circ^a$'s and  $p_\\circ^a$'s depend only upon $x^i$ and $\\N{_s^a_i}=0$ for $a\\ge i$ with the non-vanishing terms given by\n\\begin{align}\\label{16VI15.4}\n    \\N{_s^1_2}\n     &=\n     -\\frac{\\p{_\\circ^2_1}e^{-2(\u03b2_\\circ^2-\\beta_\\circ^1)}}{2(p_\\circ^2-p_\\circ^1)}\n      e^{-\u03c4(2p_\\circ^2-2p_\\circ^1 )}+O(e^{-\u03c4(2p_\\circ^2-2p_\\circ^1 +\u03bd)})\n     \\,,\n\\\\\n \\label{16VI15.5}\n    \\N{_s^2_3}\n     & =\n       -\\frac{\\p{_\\circ^3_2}e^{-2(\u03b2_\\circ^3-\u03b2_\\circ^2)}}{2(p_\\circ^3-p_\\circ^2)}\n         e^{-\u03c4(2p_\\circ^3-2p_\\circ^2 )}\n      +O(e^{-\u03c4(2p_\\circ^3-2p_\\circ^2 +\u03bd)})\n      \\,,\n\\\\\n    \\N{_s^1_3}\n     &=\n      e^{-2(\u03b2_\\circ^3-\u03b2_\\circ^1)}\\bigg(\\p{_\\circ^3_1}\n        -\\frac{\\p{_\\circ^2_1}\\p{_\\circ^3_2}}{2p_\\circ^3-2p_\\circ^2}\\bigg)\\frac{1}{2p_\\circ^3\n        -2p_\\circ^1}\n       e^{-\u03c4(2p_\\circ^3-2p_\\circ^1 )}\n     \\nonumber\n\\\\\n     &\\hspace{1em}\n      +O(e^{-\u03c4(2p_\\circ^3-2p_\\circ^1 +\u03bd)})\n     \\,,\n\\label{16VI15.6}\n\\end{align}\nwhere the functions $\\{P\\indices{_\\circ^i_a}\\}_{1\\le a < i\\le 3}$ depend only on $x^i$.\n\n\nThe remaining functions $p_\\circ^1$, $\u03b2^1_{\\circ}$,  $\\p{_\\circ^3_2 }$ and $\\p{_\\circ^3_1 }$ are then determined from the asymptotic constraint equations:\n\\begin{align}\n \\label{final_my_p_cond}\n   \n    p_\\circ^1 &=-\\frac{p_\\circ^2 p_\\circ^3}{p_\\circ^2+p_\\circ^3}\\,,\n\\\\\n\u03b2^1_{\\circ,3}\n &=-(p_\\circ^2+p_\\circ^3)^{-1}(p_{\\circ,3}^2+p_{\\circ,3}^1+\u03b2_{\\circ,3}^2(p_\\circ^1\n +p_\\circ^3)+\u03b2^3_{\\circ,3}(p_\\circ^1+p_\\circ^2))\\,,\n \\\\\n \\p{_\\circ^3_2_{,3}}&=2\\left(G_{2c}p_{\\circ,2}^c+\u03b2_{\\circ,2}^d p_\\circ^f G_{df}\\right)\\,,\n\\\\\n \\label{28X14.1}\n\\p{_\\circ^3_1_{,3}}&=-\\p{_\\circ^2_1_{,2}}+2\\left(G_{1c}p_{\\circ,1}^c+\u03b2^d_{\\circ,1}p_\\circ^f G_{df}\\right)\n \\,.\n\\end{align}\nHere $G_{ab}$ denotes the constant matrix $G_{ab}=-\\sum_{c\\neq d}\u03b4^c_a\u03b4^d_b$.\n\nFinally the Kretschmann scalar behaves as\n\\[\n R_{\u03b1\u03b2\u03b3\u03b4}R^{\u03b1\u03b2\u03b3\u03b4}=\\left(\\frac{16e^{4(\u03b2_\\circ^1+\u03b2_\\circ^2+\u03b2_\\circ^3)}\\big(p_\\circ^2 p_\\circ^3\\big)^2  }{(p_\\circ^2+p_\\circ^3)^2}\n       \\big((p_\\circ^2)^2+p_\\circ^2p_\\circ^3+(p_\\circ^3)^2\\big)+O(e^{-\\nu\\tau})\n \\right)e^{\\tau 4(p_\\circ^1+p_\\circ^2+p_\\circ^3)}\n \\,,\n\\]\nand therefore, since $p_\\circ^2p_\\circ^3>0$, the curvature diverges as $\u03c4\\to \\infty$. Along curves $\u03b3(\u03c4)=(\u03c4,\u03b3^i(\u03c4))$, $\u03c4\\in [\u03c4_0,\\infty)$, fulfilling $|\u03b3'^i(\u03c4)|=O(e^{(p_\\circ^i(\u03b3^j(\u03c4))-\u03b5)\u03c4})$ for some $\u03b5>0$ and for $i=1,2,3$, the curvature diverges in finite proper time \/ length. \n\\end{theorem}\n\n\\section{Conclusion}\nWe have constructed a large class of vacuum spacetimes containing a timelike singularity. The solutions asymptotically approach a timelike Kasner metric at each point $(x^i)$, which can be interpreted as the field of an infinitely extended thin rod, with positive mass for $J\\neq1$ and negative mass for $J=1$  \\cite{Israel1977}. As the Kasner exponents now depend upon the coordinates $x^i$ the solutions might represent the field of more complicated, non-symmetric and non-static, one-dimensional sources.\n\nWe have also noted that the cosmological billiards arguments of Damour, Henneaux, and Nicolai \\cite{Damour2003} are not directly applicable to this case, because of the transformation of asymptotically infinite potential walls into infinite wells. \nOne should keep in mind the results of Parnovsky~\\cite{Parnovsky1980,Parnovsky1999}, who applied the original procedure used by BKL to the timelike case, and concluded that the heuristic construction of chaotic singularities remains applicable. It would be of interest to resolve this apparent contradiction.\n\nIn \\cite{Shaghoulian2016} the authors argue, using a model Bianchi IX spacetime, that the change of sign of some of the wall terms is an artifact of the Iwasawa decomposition and that the affected walls vanish in a different gauge. It is not clear to us whether their arguments apply to the general inhomogeneous case.\n\n\\bigskip\n\n\\noindent{\\sc Acknowledgements:} Supported in part by a uni:docs grant of the University of Vienna.\nWe are grateful to W.~Piechocki for drawing our attention to~\\cite{Parnovsky1980,Parnovsky1999} and to P. Chru\\'{s}ciel for helpful comments and discussions.\n\\printbibliography\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe {\\it fast} (asymptotically superlinear or arbitrarily Q-linear) local convergence rate is perhaps computationally the most crucial property of the augmented Lagrangian method.\nThis topic has long been discussed ever since the birth of the augmented Lagrangian method in \\cite{powell}, and proceeded thereafter in \\cite{rockafellar}, \\cite{tre73}, \\cite{ber76}, \\cite{conn}, \\cite{ito}, \\cite{cont93}, \\cite{zhang}, \\cite{sun07}, \\cite{chenjota} and \\cite{cui}, for various problem settings.\nGenerally speaking, it has been observed that if the augmented Lagrangian method achieves a Q-linear convergence rate, the rate should be inversely proportional to the penalty parameter in the algorithm, i.e., the corresponding modulus should tend to zero as the penalty parameter increases to infinity.\nAs a result, both theoretically and practically, large (but not too large to influence the numerical stability) penalty parameters are desirable in implementing the augmented Lagrangian method.\nIn fact, the fast local convergence of the augmented has been exploited in many efficient solvers for large-scale convex optimization problems, e.g., \\cite{zhaoxy}, \\cite{yanglq}, \\cite{chench}, \\cite{lixd}, \\cite{lixd2}, \\cite{lixd3}, \\cite{zhangyj} and \\cite{liu2019}, to name a few.\n\n\nThe main objective of this paper is to investigate whether such a fast local Q-linear convergence is still preserved for the inexact augmented Lagrangian method with a practical relative error criterion proposed by \\cite{eckstein13}.\nThis criterion is of some practical interests since it is checkable whenever the gradient (or the subgradient) of the augmented Lagrangian function is computable.\nMoreover, this criterion is relative in the sense that only {\\it one} tolerance parameter, instead of a summable sequence of nonnegative real numbers, such as that in \\cite{rockafellar}, is used to control the error.\nAs was reported in \\cite{eckstein13}, this approximation criterion improves the augmented Lagrangian method and therefore has significant practical value, but the corresponding algorithm can not be interpreted, at least currently, as an application of the proximal point algorithm (PPA) in \\cite{rockafellar76} or its inexact variants. \nAs a result, the convergence analysis in \\cite{eckstein13} is not based on the convergence properties of the PPAs, which is\nvery different from the convergence analysis for the augmented Lagrangian method in \\cite{rockafellar76}, where the convergence properties, including the fast Q-linear convergence rate, are heavily dependent on those of PPAs.\nConsequently, the convergence properties of the augmented Lagrangian method with the relative error criterion in \\cite{eckstein13} can not be obtained by directly applying some known results.\nTherefore, it is of great interest to know if the algorithm in \\cite{eckstein13} also admits a {\\em fast} (local) convergence and if the modulus is inversely proportional to the penalty parameter.\n\nThe classic augmented Lagrangian method, also known as the method of multipliers, was initiated by \\cite{hestenes69} and \\cite{powell} for solving equality constrained nonlinear programming problems.\nFor general nonlinear programming problems, one may refer to the monograph of \\cite{berbook} and the references therein, where the details for various aspects of the augmented Lagrangian methods were systematically provided.\nIn the context of convex programming, as was illustrated in \\cite{rockafellar}, the classic augmented Lagrangian method can be viewed as a PPA applied to the dual of the given problem.\nMoreover, in the seminal papers \\cite{rockafellar} and \\cite{rockafellar76}, the author has studied both the convergence and the {\\em fast} local convergence rate of the augmented Lagrangian method, even if in the subproblems are approximately solved with a few summable sequences of nonnegative real numbers controlling the errors.\nIn fact, the convergence rate in \\cite{rockafellar} for the augmented Lagrangian method is inherited from the convergence rate of the inexact PPA established in \\cite{rockafellar76}, where the errors are also controlled by a summable sequence of nonnegative real numbers.\n\nAlong a different line, inexact PPAs with relative error criteria, in which summable sequences of real numbers are no longer necessary, also have been well studied in \\cite{solodov99,solodov992,solodov00}.\nMoreover, the corresponding local Q-linear convergence rate was also available in \\cite{solodov992}.\nThere are many concrete instances of applications of these inexact PPAs, but direct applications of them to augmented Lagrangian methods would bring conditions that are not practically verifiable.\nFortunately, in \\cite{eckstein13}, the authors developed a practical relative error criterion for approximately minimizing the subproblems in the augmented Lagrangian method.\nThis criterion is precisely implementable in the sense that all the information needed for checking it can be obtained directly at every candidate approximate solution to the subproblems, instead of using certain values such as the infimum of the objective functions to the subproblems in \\cite{rockafellar}, which are generally not available\\footnote{It wa until recently that practical surrogates of such criteria has been developed for a class of convex composite conic programming problems in \\cite{cui}.}.\nIn \\cite{eckstein13}, an informative discussion on the design of this relative error criterion for the augmented Lagrangian method was provided, and the comparisons of this criterion to those of \\cite{solodov99,solodov992,solodov00} for inexact PPAs are also elaborated.\n\n\nTo be consistent with, and even more general\\footnote{In \\cite{eckstein13}, ${\\mathcal X}\\equiv{\\mathds R}^n$, and the objective function $f$ is assumed to be either a continuous differentiable convex function, or the sum of a continuous differentiable convex function and the indicator function of a closed convex box in ${\\mathds R}^n$. However, the algorithm in \\cite{eckstein13} and all the results established in \\cite{eckstein13} and \\cite{alves} are valid for problem \\eqref{prob}, with the corresponding gradients and normal cones being replaced by subgradients.} than, that of \\cite{eckstein13}, the problem setting of this paper is as follows\n\\(\n\\label{prob}\n\\begin{array}{cl}\n\\displaystyle\\min_{x\\in{\\mathcal X}} &f(x)\\\\[2mm]\n\\mbox{s.t.}& h(x)=0, \\\\[2mm]\n&g(x) \\le 0,\n\\end{array}\n\\)\nwhere ${\\mathcal X}$ is a finite dimensional real Hilbert space endowed with an inner product $\\langle\\cdot,\\cdot\\rangle$ and $\\|\\cdot\\|$ is the corresponding norm,\n$f:{\\mathcal X}\\to(-\\infty,+\\infty]$ is a closed proper convex function, $h:{\\mathcal X}\\to{\\mathds R}^{m_1}$ is an affine mapping,\nand $g:{\\mathcal X}\\to{\\mathds R}^{m_2}$ is a nonlinear mapping, i.e., $g(x)=(g_1(x);\\ldots; g_{m_2}(x))$, with each $g_i:{\\mathcal X}\\to(-\\infty,\\infty)$, $i=1,\\ldots,m_2$ being a continuously differentiable convex function.\n\nIn fact, the original convergence results and the corresponding convergence analysis are not sufficient for analyzing the convergence rate of the algorithm in \\cite{eckstein13}. However, fortunately, the convergence analysis has been further improved in \\cite{alves}, where a novel technique for treating F\\'ejer monotone sequence in product spaces was developed.\nThis partially paves the way for studying the convergence rate of the algorithm in \\cite{eckstein13}.\n\n\nMotivated by the above expositions and the fact that allowing the subproblems being solved approximately would further contribute to the efficiency of the augmented Lagrangian method,\nwe are interested in further exploring the convergence properties of the augmented Lagrangian method with the relative error criterion developed in \\cite{eckstein13} for solving problem \\eqref{prob}.\nIn this paper, we will show that, under a mild local error bound condition, this algorithm also admits a {\\em fast} Q-linear local convergence in the sense that the convergence rate of the dual sequence is Q-linear and the modulus of this rate tends to zero if the penalty parameter increases to infinity.\nBesides, we show that such a local error bound condition is also sufficient to guarantee the convergence of the distance from the primal sequence, generated by the algorithm, to the optimal solution set of problem \\eqref{prob}.\nHere, we should emphasize that neither \\cite{eckstein13} nor \\cite{alves} has established the convergence of the primal sequence.\n\nThis remaining parts of this paper are organized as follows.\nIn Sect. \\ref{sec:pre}, we provide the notation and preliminaries that will be used throughout this paper. In Sect. \\ref{sec:algo}, we present the inexact augmented Lagrangian method for solving problem \\eqref{prob} with the practical relative error criterion developed in \\cite{eckstein13}. Also, we will summarize the corresponding convergence properties from \\cite{eckstein13} and \\cite{alves}.\nHere, we also present some useful equalities and inequalities for further use.\nIn Sect. \\ref{sec:conv}, we introduce a local error bound condition and establish the fast local convergence of the algorithm presented in Sect. \\ref{sec:algo}, as well as the global convergence of the primal sequence.\nWe conclude this paper in Sect. \\ref{sec:conclusion}.\n\n\\subsection*{Notation}\nLet ${\\mathcal H}$ be a finite dimensional real Hilbert space endowed with the inner product $\\langle\\cdot,\\cdot\\rangle$.\nWe use $\\|\\cdot\\|$ to denote the norm induced by this inner product.\nFor any given $z\\in{\\mathcal H}$, we use ${\\mathds B}_\\epsilon(z)$ to denote the closed ball centring at $z$ with the radius $\\epsilon\\ge 0$, i.e.,\n$$\n{\\mathds B}_\\epsilon(z):=\\{z'\\in{\\mathcal H}\\mid \\|z-z'\\|\\le \\epsilon\\}.\n$$\nLet $C\\subset{\\mathcal H}$ be a nonempty closed convex set. The projection of a vector $z\\in{\\mathcal H}$ onto the set $C$ is defined by\n\\[\n\\Pi_C(z):=\\argmin_{z'\\in C}\\left\\{\\frac{1}{2}\\|z'-z\\|^2\\right\\},\n\\]\nwhile the distance from $z\\in{\\mathcal H}$ to the set $C$ is defined by\n\\[\n\\dist (z,C):=\\|z-\\Pi_C(z)\\|.\n\\]\n\nLet $\\theta:{\\mathcal H}\\to(-\\infty,+\\infty]$ be an arbitrary closed proper convex function, we use $\\dom\\, \\theta$ to denote its effective domain, i.e.,\n$$\n\\dom\\, \\theta:=\\{z\\in{\\mathcal H}\\mid \\theta(z)<+\\infty\\}$$\nand $\\partial\\theta$ to denote its subdifferential mapping, i.e.,\n$$\n\\partial \\theta(z):=\\{\n\\gamma \\in{\\mathcal H} \\mid \\theta(z')-\\theta(z)\\ge\\langle \\gamma, z'-z\\rangle,\\ \\forall z'\\in{\\mathcal H}\n\\},\\quad\\forall z\\in{\\mathcal H}.\n$$\nMoreover, if $\\theta$ is continuously differentiable at $z\\in{\\mathcal H}$, one has $\\partial\\theta(z)=\\{\\nabla\\theta(z)\\}$, where $\\nabla\\theta(z)$ is the gradient of $\\theta$ at $z$.\n\nIf ${\\mathcal H}\\equiv{\\mathds R}^l$, i.e., the $l$-dimensional real vector space, we use the dot product as the inner product, i.e.,  $\\langle z,z'\\rangle:=z^Tz'$, so that $\\|\\cdot\\|$ is the conventional $\\ell_2$-norm. In this case we define\n$$\\max\\{z,z'\\}:=\\{\\bar z\\mid \\bar z_i=\\max\\{z_i,z'_i\\}, i=1\\ldots,l\\},\\quad\\forall z,z'\\in{\\mathcal H}, \n$$ i.e., the component-wise maximum between $z$ and $z'$. The component-wise minimum is also defined in the same fashion. \n\n\n\\section{Preliminaries}\n\\label{sec:pre}\nGenerally, the notation and definitions used in this paper are the same as those used in \\cite{eckstein13} and \\cite{alves}. In fact, they are consistent with those in \\cite{rocbook,rocbookcg}.\n\nThe Lagrangian function ${\\mathcal L}:{\\mathcal X}\\times{\\mathds R}^{m_1}\\times{\\mathds R}^{m_2}\\to[-\\infty,\\infty]$ of problem \\eqref{prob} is defined by\n\\[\n{\\mathcal L}(x;\\lambda,\\mu):=\\left\\{\n\\begin{array}{ll}\nf(x)+\\langle \\lambda, h(x)\\rangle+\\langle \\mu, g(x)\\rangle,\\quad &\\mbox{if }\\mu\\ge 0,\\\\[2mm]\n-\\infty, & \\mbox{otherwise}.\n\\end{array}\n\\right.\n\\]\nThen, the dual of problem \\eqref{prob} is given by\n\\(\n\\label{dual}\n\\max_{\\lambda\\in{\\mathds R}^{m_1},\\mu\\in{\\mathds R}^{m_2}}\\left\\{ d(\\lambda,\\mu):=\\inf_{x\\in{\\mathcal X}}{\\mathcal L}(x;\\lambda,\\mu)\\right\\},\n\\)\nwhere $d(\\cdot)$ is called the dual objective function.\nHence, we call $x\\in{\\mathcal X}$ as the primal variable and call $(\\lambda,\\mu)\\in{\\mathds R}^{m_1+m_2}$ as the dual variable.\nNote that the Lagrangian function ${\\mathcal L}$ is convex in $x$ and concave in $(\\lambda,\\mu)$.\nMoreover, for this Lagrangian function,\nthe subdifferential mapping\\footnote{For concave-convex functions, c.f. \\cite[p. 374]{rockafellar}.}\n$\\partial{\\mathcal L}:{\\mathcal X}\\times{\\mathds R}^{m_1}\\times{\\mathds R}^{m_2}\\to{\\mathcal X}\\times{\\mathds R}^{m_1}\\times{\\mathds R}^{m_2}$ is defined as follows\n\\begin{equation*}\n\\begin{array}{lll}\n&(y;u,v)\\in\\partial {\\mathcal L}(x,\\lambda,\\mu)\n\\\\[2mm]\n&\\Leftrightarrow\n\\left\\{\n\\begin{array}{ll}\n{\\mathcal L}(x';\\lambda,\\mu)\\ge {\\mathcal L}(x,\\lambda,\\mu)+\\langle y, x'-x\\rangle, &\\forall x'\\in{\\mathcal X},\n\\\\[2mm]\n{\\mathcal L}(x;\\lambda',\\mu')\\le {\\mathcal L}(x,\\lambda,\\mu)-\\langle u,\\lambda'-\\lambda\\rangle-\\langle v,\\mu'-\\mu\\rangle,\\quad &\\forall (\\lambda',\\mu')\\in{\\mathds R}^{m_1+m_2}.\n\\end{array}\n\\right.\n\\end{array}\n\\end{equation*}\nBased on the above definition, it is easy to verify that the subdifferential mapping\n$\\partial{\\mathcal L}$ is a maximal monotone operator.\nMoreover, if $(x^*,\\lambda^*,\\mu^*)\\in{\\mathcal X}\\times{\\mathds R}^{m_1}\\times{\\mathds R}^{m_2}$ satisfies\n$0\\in\\partial{\\mathcal L}(x^*,\\lambda^*,\\mu^*)$, it holds that $x^*$ is an optimal solution to problem \\eqref{prob} and $(\\lambda^*,\\mu^*)$ is an optimal solution to problem \\eqref{dual}.\nIn this case, $(x^*,\\lambda^*,\\mu^*)$ is called as a saddle point of the Lagrangian function ${\\mathcal L}$, and it holds that $f(x^*)=d(\\lambda^*,\\mu^*)$.\n\nSince $g$ and $h$ are continuously differentiable, we define\n$$\n\\nabla h(x):=\\big(\n\\nabla h_1(x),\\cdots,\n\\nabla h_{m_1}(x)\n\\big)\n\\quad\n\\mbox{and}\n\\quad\n\\nabla g(x):=\\big(\n\\nabla g_1(x),\\cdots,\\nabla g_{m_2}(x)\n\\big).\n$$\nThen, the Karush-Kuhn-Tucker (KKT) system of problem \\eqref{prob} is given by\n\\(\n\\label{kkt}\\left\\{\n\\begin{array}{l}\n0\\in\\partial_x {\\mathcal L}(x,\\lambda,\\mu):=\\partial f(x)+\\nabla h(x) \\lambda+ \\nabla g(x) \\mu,\\\\[2mm]\nh(x)=0,\\\\[2mm]\ng(x)\\le 0,\\ \\mu\\ge0,\\ \\langle\\mu,g(x)\\rangle=0.\n\\end{array}\n\\right.\n\\)\nIf the solution set to the KKT system \\eqref{kkt} is nonempty, from \\cite[Theorem 30.4 \\& Corollary 30.5.1]{rocbook} one knows that a vector\n$(x^*, \\lambda^*,\\mu^*)\\in{\\mathcal X}\\times{\\mathds R}^{m_1}\\times{\\mathds R}^{m_2}$ is a solution to the KKT system \\eqref{kkt} if and only if $x^*$ is an optimal solution to problem \\eqref{prob} and $(\\lambda^*, \\mu^*)$ is an optimal solution to problem \\eqref{dual}.\nMoreover, the solution set to the KKT system \\eqref{kkt} can be written as $X^*\\times P^*$ with $X^*$ being the solution set to problem \\eqref{prob} and $P^*$ being the solution set to problem \\eqref{dual}.\nIn this case, the optimal values of problem \\eqref{prob} and problem \\eqref{dual}\nare equal, and the solution set to the KKT system \\eqref{kkt} is exactly the set of saddle points to the Lagrangian function ${\\mathcal L}$.\n\n\n\n\n\\section{An Inexact Augmented Larangian Method}\n\n\n\n\n\n\n\n\n\\label{sec:algo}\nIn this section, we present the inexact augmented Lagrangian method with the practical relative error criterion of \\cite{eckstein13}, and summarize its convergence properties from \\cite{eckstein13} and \\cite{alves}. Some equalities and inequalities, which are useful to study its convergence rate, are also prepared in this section.\n\nWe dfine the closed convex cone\n$${\\mathcal K}={\\mathds R}^{m_2}_+:=\\{v\\in{\\mathds R}^{m_2}:v\\ge 0\\}.\n$$\nLet $c>0$ be a given real number. \nThe augmented Lagrangian function of problem \\eqref{prob} is defined (with $c$ being the penalty parameter) by\n\\[\n\\begin{array}{l}\n\\displaystyle\n{\\mathcal L}_c(x,\\lambda,\\mu):\n\\displaystyle =f(x)+\\langle\\lambda, h(x)\\rangle+\\frac{c}{2}\\|h(x)\\|^2+\\frac{1}{2c}\\big(\\|\\Pi_{\\cal K}(\\mu+cg(x))\\|^2-\\|\\mu\\|^2\\big)\\\\[4mm]\n\\displaystyle =f(x)+\\sum_{i=1}^{m_1}\\big(\\lambda_ih_i(x)+\\frac{c}{2}(h_i(x))^2\\big)\n+\\frac{1}{2c}\\sum_{i=1}^{m_2}\\Big(\\big(\\max\\{0,\\mu_i+cg_i(x)\\}\\big)^2-\\mu_i^2\\Big),\\quad\n\\\\[4mm]\n\\hfill \\forall x\\in{\\mathcal X}, \\lambda\\in{\\mathds R}^{m_1}, \\mu\\in{\\mathds R}^{m_2}.\n\\end{array}\n\\]\n\nWhen applied to solving problem \\eqref{prob}, the augmented Lagrangian method with the practical relative error criterion proposed in \\cite[(11)-(15)]{eckstein13} can be described as the following Algorithm.\n\n\n\n\\begin{center}\n\\fbox{\\begin{minipage}{.97\\textwidth}\n\\begin{algo}\n\\label{alg:alm}\n{\\bf An inexact augmented Lagrangian method with a practical relative error criterion for solving problem \\eqref{prob}.}\n\\end{algo}\nLet $\\sigma\\in[0,1)$ and let $\\{c_k\\}$ be a sequence of positive real numbers such that $\\inf_{k\\ge 1}\\{c_k\\}>0$.\nChoose $\\lambda^0\\in{\\mathds R}^{m_1}$, $\\mu^0\\in{\\mathds R}^{m_2}_+$ and $w^0\\in{\\mathcal X}$. For $k=1, 2, \\ldots$,\n\\begin{enumerate}\n\\item[\\bf 1.] find $x^{k}\\in{\\mathcal X}$ and $y^k\\in{\\mathcal X}$ such that\n\\(\n\\label{conditionsub}\ny^{k}\\in\\partial_x {\\mathcal L}_{c_k}(x^{k},\\lambda^{k-1},\\mu^{k-1}),\n\\)\nand\n\\begin{equation}\n\\label{condition}\n\\frac{2}{c_k}\n\\Big|\\langle w^{k-1}-x^{k},y^{k}\\rangle\\Big|\n+\n\\|y^k\\|^2\\le \\sigma\n\\left(\n\\|h(x^k)\\|^2\n+\\left\\|\\min\\left\\{\\frac{1}{c_k}\\mu^{k-1},-g(x^k)\\right\\}\\right\\|^2\n\\right);\n\\end{equation}\n\\item[\\bf 2.] set\n\\begin{eqnarray*}\n&&\\lambda^k:=\\lambda^{k-1}+c_kh(x^k),\n\\\\[2mm]\n&&\\mu^k:=\\max\\{0,\\mu^{k-1}+c_kg(x^k)\\},\n\\\\[2mm]\n&&w^k:=w^{k-1}-c_k y^k.\n\\end{eqnarray*}\n\\end{enumerate}\n\\end{minipage}}\n\\end{center}\n\n\\begin{remark}\nIn the above algorithm, each $x^k$ is an approximate solution to the corresponding subproblem with $y^k$ being a subgradient of the corresponding objective function at $x^k$, i.e., $x^k$ approximately solves the problem\n$$\n\\min_{x}{\\mathcal L}_{c_k}(x; \\lambda^{k-1},\\mu^{k-1}).\n$$\nIf this subproblem admits a solution, say $\\tilde x^k$, one can let $y^k=0$ so that \\eqref{condition} is satisfied.\nTherefore, one can always find the sequence $\\{(x^k,y^k)\\}$ via Algorithm \\ref{alg:alm} if the subproblems are well-defined.\n\n\n\n\n\\end{remark}\n\nFor the convenience of further discussions, we denote $m:=m_1+m_2$ and define the sequences $\\{p^k\\}$ and $\\{u^k\\}$ in ${\\mathds R}^m$ via\n\\(\n\\label{def:pu}\np^k:=(\\lambda^k,\\mu^k)\n\\quad\\mbox{and}\\quad\nu^k:=\\frac{1}{c_k}(p^{k-1}-p^k).\n\\)\nThe following theorem is an immediate extension of \\cite[Proposition 1]{eckstein13}.\n\\begin{theorem}\n\\label{prop:main}\nAssume that the solution set to the KKT system \\eqref{kkt} of problem \\eqref{prob} is non-empty.\nSuppose that the infinite sequences generated by Algorithm \\ref{alg:alm} are well-defined. Then,\n\\begin{enumerate}\n\\item[(a)] The sequences $\\{p^k\\}$ and $\\{w^k\\}$ are bounded;\n\\item[(b)] $u^k\\to 0$ and $y^k\\to0$ as $k\\to\\infty$;\n\\item[(c)] Any accumulation point of the sequence $\\{x^k\\}$ is a solutions to problem \\eqref{prob}, and any accumulation point of $\\{p^k\\}$ is a solution to the dual problem \\eqref{dual}.\n\n\n\n\\end{enumerate}\n\\end{theorem}\n\n\nAs can be observed from Theorem \\ref{prop:main}, the convergence results in \\cite{eckstein13} show that the sequence $\\{p^k\\}$ is bounded and each accumulation point of this sequence is a solution to the dual problem \\eqref{dual}.\nTo ensure the full convergence of the sequence $\\{p^k\\}$, in \\cite[Proposition 2]{eckstein13}, the authors introduced the additional criterion to \\eqref{condition} that\n$$\nc_k\\|y^k\\|\\le\\xi\\|p^{k-1}-p^k\\|^2,\n$$\nwhere $\\xi\\ge 0$ is a given constant. As was commented in \\cite{eckstein13}, despite $\\xi$ can be arbitrarily large, this additional criterion seems stringent. Fortunately, in \\cite{alves},\nthe authors showed that it is not necessary to use this extra criterion to guarantee the convergence of the sequence $\\{p^k\\}$,\nthanks to their insightful investigation of F\\'ejer monotone sequences in product spaces.\nThe following theorem directly comes from \\cite[Proposition 2]{alves}.\n\\begin{theorem}\n\\label{theo:main}\nSuppose that all the assumptions and conditions in Proposition \\ref{prop:main} hold.\nThen, the whole sequences $\\{p^k\\}$ converges to a solution to problem \\eqref{dual}.\n\\end{theorem}\n\n\nAs can be seen from Proposition \\ref{prop:main} and Theorem \\ref{theo:main}, the global convergence of Algorithm \\ref{alg:alm} has been well established in the sense that the sequence of the generated dual variables is globally convergent,\nwhich is quite similar to the global convergence properties of the (inexact) augmented Lagrangian method established in \\cite{rockafellar}.\nIn fact, Proposition \\ref{prop:main}, together with Theorem \\ref{theo:main}, lays the foundation for further investigating the convergence rate of Algorithm \\ref{alg:alm} in the next section.\n\nFinally, we should emphasize that, referring to the primal sequence $\\{x^k\\}$, the best possible result in both \\cite{eckstein13} and \\cite{alves} is that any accumulation point of this sequence is a solutions to problem \\eqref{prob}.\n\n\n\n\n\n\\section{Convergence Rate Analysis}\n\\label{sec:conv}\nThis section establishes the local convergence rate of Algorithm \\ref{alg:alm} for solving problem \\eqref{prob}, under a mild error bound condition.\nWe first discuss this error bound condition and then given the main result of this paper.\n\n\\subsection{An error bound condition}\nLet $T:{\\mathcal H}\\to{\\mathcal H}$ be a maximal monotone mapping with $T^{-1}$ being its inverse mapping\\footnote{One may refer to \\cite[Chapter 12]{va} for more information about (maximal) monotone mappings.}.\nFor solving the general inclusion problem of finding $z\\in{\\mathcal H}$ such that\n\\(\n\\label{inclu}\n0\\in T(z),\n\\)\nthe (inexact) PPA in \\cite{rockafellar76} takes the following iteration scheme\n\\(\n\\label{algppa}\nz^{k}\\approx (I+c_k T)^{-1}(z^{k-1}),\\quad k=1,2,\\ldots,\n\\)\nwhere $c_k>0$, and $z^0\\in{\\mathcal H}$ is the given initial point.\nIn \\cite{rockafellar76},\nthe convergence rate of this (inexact) PPA has been analyzed under a local error bound condition in which the solution to \\eqref{inclu} should be a singleton. In \\cite{luque}, the author extended the convergence rate analysis of \\cite{rockafellar76} for the PPA to the case that the solution set of \\eqref{inclu} is not necessarily a singleton.\nThe following definition introduces an error bound condition, which has been used, in the name of a growth condition of maximal monotone operators, in \\cite{luque} for the convergence rate analysis of PPA.\n\n\n\\begin{definition}[\\cite{robinson}]\n\nThe mapping $T^{-1}$ is called locally upper Lipschitz continuous at the origin if $T^{-1}(0)$ is nonempty and there exist constants $\\epsilon>0$ and $\\kappa>0$ such that\n\\(\n\\label{growth}\n\\dist(z,T^{-1}(0))\\le\\kappa\\|\\beta\\|,\\quad\n\\forall \\beta\\in{\\mathds B}_{\\epsilon}(0)\\ \\ \\mbox{and}\\ \\ \\forall z\\in T^{-1}(\\beta).\n\\)\n\\end{definition}\n\nThe augmented Lagrangian method and its inexact version in \\cite{rockafellar} can be explained as an application of the PPA to a certain maximal monotone operator\\footnote{In particular, for problem \\eqref{prob} considered in this paper, this maximal monotone operator is given by $\\partial d$  with $d$ being defined in \\eqref{dual}. Here the subdifferential mapping is in the concave sense, c.f. \\cite[pp. 307--308]{rocbook}.}.\nTherefore, the convergence rate analysis in \\cite{rockafellar} for the augmented Lagrangian method is heavily dependent on the convergence rate analysis in \\cite{rockafellar76} for the PPA.\nHowever, Algorithm \\ref{alg:alm} can not be explained as a certain variant of the PPA, at least currently.\nTherefore, its convergence rate should be studied via a totally different approach.\nIn this paper, to analyze the convergence rate of Algorithm \\ref{alg:alm}, we make the following assumption, which is an error bound condition on problem \\eqref{prob}.\n\n\n\n\n\n\n\\begin{assumption}\n\\label{ass:growth}\nThe mapping $(\\partial{\\mathcal L})^{-1}$ is locally upper Lipschitz continuous at the origin.\n\\end{assumption}\nBefore analyzing the local Q-linear convergence rate of Algorithm \\ref{alg:alm}, we make the following comment to Assumption \\ref{ass:growth}.\n\\begin{remark}\nAssumption \\ref{ass:growth} is not a very restrictive condition. On the one hand, it contains the case that $(\\partial{\\mathcal L})^{-1}$ is locally Lipschitz continuous at $0$, which was used extensively in {\\rm \\cite{rockafellar,rockafellar76}} for deriving the Q-linear convergence rate for PPA and augmented Lagrangian methods.\nNote that even the stronger condition than Assumption \\ref{ass:growth} that $(\\partial{\\mathcal L})^{-1}$ is locally Lipschitz continuous at $0$ can be satisfied, at least heuristically, for ``most'' convex optimization problems in the form of \\eqref{prob} with $f$ being the sum of a twice continuously differentiable convex function plus an indicator function of a closed convex set and $g$ and $h$ being twice continuously differentiable \\cite[p. 105, Remark]{rockafellar}.\nOne may also refer to {\\rm\\cite[Proposition 4]{rockafellar}} and the discussions after this proposition for more information. In addition, we mention that Assumption \\ref{ass:growth} is more general in the sense that $(\\partial{\\mathcal L})^{-1}(0)$ is not necessarily to be a singleton.\n\\end{remark}\n\n\n\n\n\n\\subsection{The convergence rate}\nNow we present the main result of this paper.\n\\begin{theorem}\n\\label{thm:main}\nSuppose that the solution set $X^*\\times P^*$ to the KKT system \\eqref{kkt} of problem \\eqref{prob} is nonempty and the infinite sequences generated by Algorithm \\ref{alg:alm} are well-defined. Suppose that Assumption \\ref{ass:growth} holds (with the parameters $\\kappa>0$, $\\epsilon>0$ such that \\eqref{growth} holds for $T=\\partial{\\mathcal L}$).\nThen, the following statements hold:\n\\begin{enumerate}\n\\item[(a)] for any sufficiently large $k$, it holds that\n\\(\n\\label{result1}\n\\dist(x^k,X^*)\\le\n\\frac{\\kappa(1+\\sqrt{\\sigma})}{c_k}\\|p^{k-1}-p^k\\|,\n\\)\nso that\n\\(\n\\label{pconv}\n\\lim_{k\\to\\infty}\\dist(x^k,X^*)=0;\n\\)\n\\item[(b)]\nif,\nadditionally, one has that\n\\(\n\\label{cond:c}\n\\liminf_{k\\to\\infty}c_k>2\\kappa(\\sigma+\\sqrt{\\sigma}),\n\\)\nthen, for any sufficiently large $k$\n\\(\n\\label{resmain}\n\\dist(p^{k},P^*)\\le \\rho_k \\dist(p^{k-1},P^*)\n\\)\nwith\n$$\n\\rho_k:\n=\\frac{\\kappa\\sqrt{1+\\sigma}}\n{\\sqrt{c_k^2-2\\kappa(\\sigma+\\sqrt{\\sigma}){c_k}+\\kappa^2(1+\\sigma)}}\n$$\nand\n\\(\n\\label{limrho}\n\\limsup_{k\\to\\infty}\\{\\rho_k\\}\n<1.\n\\)\nThis means that the local rate of convergence for Algorithm \\ref{alg:alm} is Q-linear and the modulus of the convergence rate tends to zero if the sequence $\\{c_k\\}$ increases to infinity.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n{\\it (a)}\nAccording to \\eqref{def:pu} and \\cite[eq. (24)]{eckstein13}, one has that\n$$\n\\begin{array}{l}\n\\big(y^k,\\frac{1}{c_k}(p^{k-1}-p^k)\\big)\n=\n(y^k,u^k)\\in\\partial {\\mathcal L}(x^k,p^k).\n\\end{array}\n$$\nFrom \\cite[eq. (38)]{eckstein13} we know that, for any $p^*\\in P^*$, it holds that\n\\(\n\\label{eq:p}\n\\|p^k-p^*\\|=\\|p^{k-1}-p^*\\|^2-2c_k\\langle u^k,p^k-p^*\\rangle-\\|p^{k-1}-p^k\\|^2.\n\\)\nSince $(0,0)\\in\\partial{\\mathcal L}(x^*,p^*)$ and $\\partial{\\mathcal L}$ is maximally monotone, it holds that for any $x^*\\in X^*$ and $p^*\\in P^*$,\n\\(\n\\label{ineq:px}\n\\langle y^k, x^k-x^*\\rangle+\\langle u^k, p^k-p^*\\rangle\\ge 0.\n\\)\nDefine the sequences $\\{\\bar x^k\\}$ in ${\\mathcal X}$ and $\\{\\bar p^k\\}$ in ${\\mathds R}^{m}$ via\n\\[\n\\bar x^k:=\\Pi_{X^*}(x^k)\n\\quad\\mbox{and}\\quad\n\\bar p^k:=\\Pi_{P^*}(p^k).\n\\]\nCombining \\eqref{eq:p} and \\eqref{ineq:px} together implies that, for any $p^*\\in P^*$,\n\\(\n\\label{ineq:pin1}\n\\begin{array}{ll}\n&\\|p^{k-1}-p^*\\|^2\n-\\|p^{k}-p^*\\|^2\\\\[2mm]\n&=\\|p^{k}-p^{k-1}\\|^2+2c_k\n\\langle u^k,p^k-p^*\\rangle\\\\[2mm]\n&\\ge\\|p^{k}-p^{k-1}\\|^2-2c_k\\langle x^k-\\bar x^k,y^k\\rangle\n\\\\[2mm]\n&\\ge\\|p^{k}-p^{k-1}\\|^2-2c_k\\|x^k-\\bar x^k\\|\\|y^k\\|.\n\\end{array}\n\\)\nSince Assumption \\ref{ass:growth} holds with $\\kappa>0$ and $\\epsilon>0$,\nfrom \\eqref{growth} we know that\n$$\n\\dist\\Big((x,p),(X^*,P^*)\\Big)\\le\\kappa\\big\\|(y,u)\\big\\|,\\quad\n\\forall (y,u)\\in\\partial {\\mathcal L}(x,p) \\mbox{ with }\n\\big\\|(y,u)\\big\\|\\le\\epsilon.\n$$\nFrom Proposition \\ref{prop:main}(b) and \\eqref{ineq:yp1} we know that $\\|(y^k,u^k)\\|\\le\\epsilon$ if $k$ is sufficiently large. Then, there exists a positive integer $k_0$ such that for any $k\\ge k_0$,\n\\(\n\\label{growthbound}\n\\dist\\left((x^{k},p^{k}),(X^*,P^*)\\right)\n\\le\n\\kappa\\left\\|(y^k,u^k)\\right\\|.\n\\)\nThis, together with \\eqref{def:pu}, implies that\n\\[\n\\|x^k-\\bar x^k\\|^2\n\\le\n\\kappa^2\\|y^k\\|^2+\\frac{\\kappa^2}{c_k^2}\\|p^{k-1}-p^k\\|^2.\n\\]\nHence, it holds that, for any $k\\ge k_0$,\n\\(\n\\label{ineq:qq}\n\\|x^k-\\bar x^k\\|\n\\le\n\\kappa\\|y^k\\|+\\frac{\\kappa}{c_k}\\|p^{k-1}-p^k\\|.\n\\)\nNote that \\eqref{condition} can be equivalently written as\n\\(\n\\label{ineq:yp1}\n{2}{c_k}\n\\big|\\langle w^{k-1}-x^{k},y^{k}\\rangle\\big|\n+\nc_k^2\\|y^k\\|^2\\le \\sigma\\|p^{k-1}-p^k\\|^2,\n\\)\nwhich implies that\n\\(\n\\label{ineq:yp2}\n\\|y^k\\|\\le \\frac{\\sqrt{\\sigma}}{c_k}\\|p^{k-1}-p^k\\|.\n\\)\nAs a result, by combining \\eqref{ineq:qq} and \\eqref{ineq:yp2} together one gets\n\\[\n\\|x^k-\\bar x^k\\|\n\\le\n\\frac{\\kappa(1+\\sqrt{\\sigma})}{c_k}\\|p^{k-1}-p^k\\|,\n\\]\nso that \\eqref{result1} is established. Moreover, since $\\inf_{k\\ge 0}\\{c_k\\}>0$,\nthe above inequality, together with \\eqref{def:pu} and Proposition \\ref{prop:main}(b), implies \\eqref{pconv}.\n\n\\medskip\n{\\it (b)}\nOne can get from \\eqref{ineq:qq} that\n\\(\n\\label{ineq:xp1}\n\\|x^k-\\bar x^k\\|\\|y^k\\|\\le\n\\kappa\\|y^k\\|^2+\\frac{\\kappa}{c_k}\\|y^k\\|\\|p^{k-1}-p^k\\|.\n\\)\nBy taking \\eqref{ineq:xp1} into \\eqref{ineq:pin1}, one has that\n\\(\n\\label{ineq:main1}\n\\begin{array}{ll}\n&\\|p^{k-1}-p^*\\|^2\n-\\|p^{k}- p^*\\|^2\n\\\\[2mm]\n&\\displaystyle\n\\ge\\|p^{k}-p^{k-1}\\|^2-2c_k\\left(\\kappa\\|y^k\\|^2+\\frac{\\kappa}{c_k}\\|y^k\\|\\|p^{k-1}-p^k\\|\\right)\n\\\\[4mm]\n&\\displaystyle\n=\\|p^{k}-p^{k-1}\\|^2-2c_k\n\\kappa\\|y^k\\|^2-2\\kappa\\|y^k\\|\\|p^{k-1}-p^k\\|.\n\\end{array}\n\\)\nThen, by combining \\eqref{ineq:yp2} and \\eqref{ineq:main1} together we can get that for any $p^*\\in P^*$,\n\\(\n\\label{ineq:pp2}\n\\begin{array}{llll}\n&\\displaystyle\n\\|p^{k-1}-p^*\\|^2\n-\\|p^{k}-p^*\\|^2\n\\\\[3mm]\n&\\displaystyle\n\\ge\\|p^{k}-p^{k-1}\\|^2-2\n\\kappa\\frac{\\sigma}{c_k}\\|p^{k-1}-p^k\\|^2-2\\kappa\\frac{\\sqrt{\\sigma}}{c_k}\\|p^{k-1}-p^k\\|^2\n\\\\[3mm]\n&\\displaystyle\n=\\left(\n1-\\frac{2\\kappa(\\sigma+\\sqrt{\\sigma})}{c_k}\\right)\n\\|p^{k}-p^{k-1}\\|^2.\n\\end{array}\n\\)\nOn the other hand, as \\eqref{growthbound} holds for $k\\ge k_0$, from \\eqref{ineq:yp2} one has that for any $k\\ge k_0$,\n\\(\n\\label{ineq:pp3}\n\\begin{array}{lllll}\n\\|p^{k}-\\bar p^k\\|^2\n&\\displaystyle\n\\le\n\\kappa^2\\|y^k\\|^2+\\frac{\\kappa^2}{c_k^2}\\|p^{k-1}-p^k\\|^2\n\\\\[3mm]\n&\\displaystyle\\le\n\\frac{\\kappa^2\\sigma}{c^2_k}\\|p^{k-1}-p^k\\|^2\n+\\frac{\\kappa^2}{c_k^2}\\|p^{k-1}-p^k\\|^2\n\\\\[3mm]\n&\\displaystyle\n=\\frac{(1+\\sigma)\\kappa^2}{c_k^2}\\|p^{k-1}-p^k\\|^2.\n\\end{array}\n\\)\nBy combining \\eqref{ineq:pp2} and \\eqref{ineq:pp3} together, one has that for any $k\\ge k_0$,\n\\(\n\\label{ineq:final}\n\\begin{array}{lll}\n&\\displaystyle\n\\|p^{k-1}-\\bar p^{k-1}\\|^2\n-\\|p^{k}-\\bar p^{k}\\|^2\n\\\\[3mm]\n&\\displaystyle\n\\ge\\|p^{k-1}-\\bar p^{k-1}\\|^2\n-\\|p^{k}-\\bar p^{k-1}\\|^2\n\\\\[3mm]\n&\\displaystyle\n\\ge\\frac{c_k^2-{2\\kappa(\\sigma+\\sqrt{\\sigma})}{c_k}}\n{\\kappa^2(1+\\sigma)}\n\\|p^{k}-\\bar p^k\\|^2.\n\\end{array}\n\\)\nAccording to \\eqref{cond:c}, it is easy to see that there exists a certain positive integer $k_0'$ such that for all $k\\ge k'_0$, it holds that $c_k>2\\kappa(\\sigma+\\sqrt{\\sigma})$,\nso that\n$$\n\\frac{c_k^2-{2\\kappa(\\sigma+\\sqrt{\\sigma})}{c_k}}\n{\\kappa^2(1+\\sigma)}>0.\n$$\nHence, from \\eqref{ineq:final} one can get that for any $k\\ge\\max\\{k_0,k'_0\\}$,\n\\[\n\\frac{\\|p^{k}-\\bar p^k\\|^2}{\\|p^{k-1}-\\bar p^{k-1}\\|^2}\n\\le\n\\frac{{\\kappa^2(1+\\sigma)}}{{\\kappa^2(1+\\sigma)}\n+{c_k^2}-{2\\kappa(\\sigma+\\sqrt{\\sigma})}{c_k}}.\n\\]\nThis proves \\eqref{resmain}.\nFinally, denote\n\\(\n\\delta:=\\liminf_{k\\to\\infty}\\{c_k\\}-2\\kappa(\\sigma+\\sqrt{\\sigma})>0.\n\\)\nSince $c_k>2\\kappa(\\sigma+\\sqrt{\\sigma})$ for $k>k_0'$, one has in this case that\n$$\nc_k^2-2\\kappa(\\sigma+\\sqrt{\\sigma}){c_k}>\\delta c_k>2\\delta\\kappa(\\sigma+\\sqrt{\\sigma}),\n$$\nso that\n$$\n\\rho_k=\n\\frac{\\kappa\\sqrt{1+\\sigma}}\n{\\sqrt{c_k^2-2\\kappa(\\sigma+\\sqrt{\\sigma}){c_k}+\\kappa^2(1+\\sigma)}}\n<\n\\frac{\\kappa\\sqrt{1+\\sigma}}\n{\\sqrt{2\\delta\\kappa(\\sigma+\\sqrt{\\sigma})+\\kappa^2(1+\\sigma)}}<1.\n$$\nThis proves \\eqref{limrho} and hence completes the proof of the theorem.\n\\end{proof}\n\nBefore concluding this paper, we should make some comments on the results established above.\n\\begin{remark}\nApparently,\nThe result on the rate of convergence established in Theorem \\ref{thm:main} depends on the condition \\eqref{cond:c}.\nThis can partially be explained by the fact that due to the subproblems of Algorithm \\ref{alg:alm} are solved inexactly, one can not always guarantee the sequence $\\{p^k\\}$ is F\\'ejer monotone to $P^*$ even if $k$ is sufficiently large.\nHowever, this is not something ``bad'' since\n\\eqref{cond:c} is very likely to be satisfied.\nThe reason is that, if $c_k$ increases to $+\\infty$, \\eqref{cond:c} must be satisfied when $k$ is sufficiently large and $\\rho_k$ converges to $0$, no matter what the values of $\\kappa$ and $\\sigma$ are.\nAs a result, large (but not too large to cause the numerical issues when solving the subproblems) penalty parameters are preferred when implementing Algorithm \\ref{alg:alm} as they can both bring better modulus of the convergence rate and enhance the likelihood that \\eqref{cond:c} holds.\n\\end{remark}\n\n\n\n\\begin{remark}\nIt was established in {\\rm\\cite{eckstein13}} that any accumulation point of the primal sequence $\\{x^k\\}$ is a solution to problem \\eqref{prob}, but this sequence is not guaranteed to be bounded.\nAs a result, the convergence properties of this sequence has not been explored.\nMoreover,\neven the sequence $\\{w^k\\}$ is bounded, it is just a sequence of auxiliary variables so that the importance of its convergence properties is not apparent.\nHence, the convergence result of the sequence $\\{x^k\\}$ established in Theorem \\ref{thm:main} (a), to some extent, makes real progress.\n\\end{remark}\n\n\n\n\\begin{remark}\nRecently, in \\cite[Proposition 1]{cui}, the authors have improved the convergence rate analysis of the PPA \\eqref{algppa} for solving \\eqref{inclu} with the error tolerance criteria in \\cite{rockafellar76} by relaxing the upper Lipschitz continuous of $T^{-1}$ at the origin to\nthe condition that $T^{-1}$ is calm\\footnote{\nSuch a calmness assumption is weaker than the upper Lipschitz continuous property of $T^{-1}$, and one may refer to \\cite[Section 2.1]{cui} for details.} at the origin for $z^{\\infty}$ with a certain positive modulus, where $z^{\\infty}$ is the limit of the sequence ${z^k}$ generated by the PPA (which always exists due to the global convergence property of the PPA).\nAs can be seen from Proposition \\ref{prop:main} and Theorem \\ref{theo:main}, the primal sequence generated by Algorithm \\ref{alg:alm} is not guaranteed to be convergent in general. Hence, it is impracticable to impose similar calmness assumptions on $(\\partial {\\mathcal L})^{-1}$ instead of using Assumption \\ref{ass:growth}.\n\n\n\\end{remark}\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusion}\nIn this paper, we have analyzed the fast Q-linear convergence rate of the inexact augmented Lagrangian method with a practical relative error criterion, under a mild local error bound condition.\nThe results established in this paper imply that if the penalty parameter increases to infinity, the modulus of the convergence rate would tend to zero, which constitutes an asymptotically superlinear convergence rate.\nBesides, without introducing any additional condition, the distance from the primal sequence to the solution set of the primal problem is guaranteed to vanish.\nThe results here can serve as the guideline for choosing the penalty parameter when implementing the inexact augmented Lagrangian method of \\cite{eckstein13}.\n\n\n\n\n\n\n\\section*{Acknowledgments}\nThe authors would like to thank Prof. Defeng Sun at the Hong Kong Polytechnic University for bringing the paper of \\cite{eckstein13} to their attentions.\n \n\n\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecent developments in technology make it possible to perform for \nthe first time large-area sensitive blind surveys at 21-cm to search\nfor extragalactic H{\\sc i}. This opens many scientific opportunities. Most \nimportantly, the selection effects through the 21-cm window will be quite different \nfrom the well known optical selection effects which may presently bias \nour understanding of galaxy populations (e.g. Impey \\& Bothun 1997, \nDisney 1998). For instance, low surface brightness (LSB) galaxies are \ndimmer than the sky and nearby gas-rich dwarf galaxies show a morphology\nconfusingly similar to background giant galaxies which make both galaxy\ntypes very difficult to  identify optically.  But they may well become\n``visible'' in a 21-cm survey if  they    have  a significant amount \nof H{\\sc i}. LSB galaxies appear to be common - with approximately equal numbers of \ngalaxies, of a given size, in each one-magnitude bin of blue central \nsurface brightness $\\mu_{B,0}$     \nmeasured between $21.5<\\mu_{B,0}<26.5$ (Turner \net al.~1993; McGaugh 1996). Considering the relative narrowness of \nthe bins at fainter magnitudes this is a remarkable result. The \ncosmological significance of LSB galaxies remains controversial\nbecause of uncertain  optical selection effects at faint luminosities \n(see Impey \\& Bothun 1997, Disney 1998 \n   and references therein). Likewise there is \nserious disagreement about the relative number of faint dwarf \ngalaxies. Magnitude-limited surveys tend to find a flat faint-end slope \n(Schechter, 1976, \n   parameter $\\alpha \\sim 1$) to the optical luminosity function \n(e.g. Loveday et al.~1992, Lin et al.~1996). This would suggest that the \ntotal light contribution from dwarfs in the universe is insignificant, \nwhereas surveys within clusters of galaxies (e.g.  \n   de Propris et al. 1995;  Driver \\& Phillipps 1996;  \nTrentham 1997;  Phillipps, \n Parker, Schwartzenberg \\& Jones 1998) find steep faint end slopes with $\\alpha>2$, suggesting a  \ndominant  role for dwarfs in the determination of the light content of \nthe universe. Dwarf galaxies may also contain appreciable amounts of \ndark matter (e.g. Freeman 1997), implying that they may add even more to the \noverall mass in the universe than to the light. Finally, any extragalactic survey \nin which every object is automatically tagged with its velocity-distance \noffers rare opportunities for statistical studies. \n\nBlind 21-cm surveys have a long history but their ambitions \nhave been severely curtailed by technological limitations. \nSingle dish radio telescopes large enough to reach high sensitivity \nhave had beam areas too small to survey large areas of sky, while \ninterferometric surveys are still confined to relatively narrow \nvelocity windows by the computer power required to carry out \nthe necessary cross-correlations between every interferometer \npair. Nevertheless,  in recent times \n  blind surveys have been carried out in \nthe galaxy clusters Hydra by McMahon  et al.~(1993) and Hercules\nby Dickey (1997); in voids by Krumm \\& Bosch (1984) and \nWeinberg et al. (1991);  behind the Galactic plane by Kerr \\& \nHenning (1987); and in the field by Henning (1995). \n Latterly, narrow zenith-strip scans have been done at Arecibo  \n  (see later)  by \nSpitzak \\& Schneider (1998) and Zwaan et al.~(1997).\n\nThe most recent 21-cm survey is our  H{\\sc i} Parkes All-Sky Survey \n(H{\\sc i}PASS ) which started early in 1997 and is expected to be \ncomplete within $\\sim$3 years. This project takes advantage of the \nnew multibeam instrument at the 64m radio telescope at \nParkes (Australia) equipped with 13 separate beams (26 receivers) and \nenough correlator power in each beam to survey 1024 velocity channels \nsimultaneously. The scientific aims of the project include a  \n study  of  the Galactic H{\\sc i} and  of the galaxy content of the local \nuniverse. For this purpose     we scan  \n  the complete southern sky in the \nvelocity range $-1200$ km\\,s$^{-1}$  $<v_{\\odot}<12,700$ km\\,s$^{-1}$ . \n\nFor the purpose of testing the performance of H{\\sc i}PASS   the present \nlimited study was carried out using  the \n   first set of H{\\sc i}PASS    data completed\nto full sensitivity. The selected area covers a substantial fraction of the nearby \nCentaurus A group of galaxies. The main reasons for this choice are twofold:  \n   (1) \nthe Cen\\,A group is among the closest galaxy groups to the Local Group\nat $\\sim 3.5$\\,Mpc (Hui et al.~1993; Sandage et al.~1994) and \ncontains the richest population of   different morphological galaxy types (de \nVaucouleurs 1979), including the peculiar giant elliptical NGC 5128, the \nlarge disk galaxies M 83 and NGC 4945, the S0 galaxy NGC 5102, and the active \nstar forming galaxy NGC 5253. All main group members appear disturbed in \ntheir morphology. It has been suggested (Graham 1979; van Gorkom et al.~1990) \nthat the recent accretion of a population of gas-rich dwarfs could be the \nreason.  The group lies in the Supergalactic Plane with its southern \nextension running down to low galactic latitudes making optical studies \nchallenging because of high extinction. (2) The group has recently been the \n subject of a careful optical survey (C\\^ot\\'e et al.~1997)  enabling \n comparison between contemporary optical \n searches for gas-rich type (irregular) \n galaxies and   our ``blind'' H{\\sc i} \n  survey.   \n\n\n Most \n   earlier  H{\\sc i} studies of groups of galaxies have been done based on  \n  previous  \noptical identifications of candidate members.  In the M~81 group,  \nvan Driel et al.\\ (1998) and Huchtmeier \\& Skillman (1998) have independently \ndone 21~cm searches for optically selected dwarf candidates.  Neither group \ndetected any new members. \n Huchtmeier, Karachentsev \\& Karachentseva (1997) followed up on  \npotential dwarfs found on the Second Palomar Sky Survey (POSS II), and found \n1 new candidate of the IC~342\/Maffei group, and three new dwarf members each \nin the NGC~6946 and NGC~672\/IC~1727 groups.   \nMaia \\&  Willmer (1998) have studied the neutral \nhydrogen content of a number of ``loose groups'', but they also used  \npointed observations of optical candidates. \n\n\n   Several H{\\sc i} searches of  \ncompact groups of galaxies have been done (Oosterloo \\& Iovino 1997; Williams \n\\& van Gorkom 1997; Williams, McMahon \\& \n  van Gorkom 1991), however these groups are usually \nfar enough away that both the spatial resolution and the sensitivity preclude \ndetecting dwarf galaxies.   However, a \n   few nearby groups have been surveyed \n blind at 21~cm, and they include \nM~81, CVnI and  NGC~1023, in which Lo \\&  Sargent (1979) found 4 \n previously uncatalogued members, and a part of  the \n Sculptor   group, studied by \n Haynes \\& Roberts (1979) who found  uncatalogued H{\\sc i} \nobjects thought to be  group members.   \n \n\n\nThe present paper is organised as follows: \\S 2 describes the 21-cm survey;  \nthe observations with results are given in \\S 3 and \\S 4. The optical\nfollow-up on the newly detected galaxies and their optical properties are \npresented in \\S 5 and \\S 6. In \\S 7 we derive the H{\\sc i} Mass Function\nfor the Cen\\,A group. Finally, we discuss detection limits and future projects \nin \\S 8.   \n\n\\section{Survey}\n\n   The Cen\\,A group  covers approximately 1000 square degrees of sky down to and perhaps \n through  the galactic plane. Since it is also projected  against \n the Supergalactic  plane, the area is rich in galaxies and therefore \n well suited to test the performance of H{\\sc i}PASS .  \n\n  The comparison optical search for dwarf irregulars in the area by \nC\\^ot\\'e et \nal.~(1997; hereafter C97)  covered 900 square degrees \nbetween   \n  $12^{h}30^m< \\alpha <15^{h}00^m$ \n   and $-50^{\\circ}< \\delta <-20^{\\circ}$ whereas we have  surveyed a \n smaller  area of 600 square degrees, chosen because it fits \n into the H{\\sc i}PASS   survey grid. C97 found 18 H{\\sc i} rich dIrr's \n(2 new)  of which 14 were in our search-area.  Adding \n AM 1321-304  found by  Matthews \\& Gallagher (1996) and the six well known \n bright group members NGC 5128, M 83, NGC 4945, NGC 5102, \n NGC 5253 and ESO 270-G017  means there were in all 21 \n suspected group members in our 600 square degree \n search area prior to our survey, with another 7 \n outside (see Figure 1). Our search was  confined to \n the velocity range 200 $< v_{\\odot} <$ 1000  km\\,s$^{-1}$ , cut off at \n the lower end because of the possibility of \n confusion with High Velocity Clouds which are numerous in \n the Cen\\,A group direction (Wakker, 1991). One \n group member - CEN 5 ($v_{\\odot}$=160  km\\,s$^{-1}$ ) \n with a velocity lower than 200  km\\,s$^{-1}$  has been previously claimed  (C97) and there remains \n the possibility  that other group members in our data  remain excluded by the 200  km\\,s$^{-1}$  \n restriction. \n\n\n  If we exclude CEN 5 from the statistics  then  there were in our search \n area, prior to this survey, 18 suspected group members with H{\\sc i} \ndetections and two with H$\\alpha$ (NGC 5206 and \n ESO 272-G025).  \n\n\n\n\n\\section{ Observations and Data Reduction}\nThe 21-cm observations for this pilot study were carried out in February and \nMarch 1997 as part of the  H{\\sc i}PASS  survey which is ongoing  at the 64m \nsingle dish Parkes Radio Telescope. This data set comprises the first \ncompleted area observed to full sensitivity with H{\\sc i}PASS . The instrument is a \nmultibeam system consisting of 13 beams, each with 2 orthogonal polarisations. The \nscanning rate across the sky is 1 degree\/minute at constant right ascension. \nThe data from the 26 correlators are stored every 5 seconds. Each correlator has \n1024 channels tuned to a velocity range between $-1200$  km\\,s$^{-1}$  to $12,700$  km\\,s$^{-1}$ , \n corresponding  to a channel separation of $13.2$ km\\,s$^{-1}$ . The \n   system temperature, $T_{\\rm sys}$, is $\\sim 21$ K and the \n  scanning pattern is \norganised so \n   that every point in the sky passes through several beams during a scan, \nand through several different scans separated by days or weeks of time depending on \nthe observing schedule. This allows for the rejection of time-dependent interference. \nThe total integration time per point on the sky is 450 seconds. The 5$\\sigma$ sensitivity of the \nH{\\sc i}PASS  survey is measured at 70 mJy per channel per beam. For more details \nabout the multibeam system we refer to Staveley-Smith et al.~(1996).  \nThe observed spectra were smoothed on-line by applying a 25\\% Tukey filter, to  \nreduce ``ringing'' caused by the strong Galactic signal entering through spectral \nside lobes. This process accounts for a loss in velocity resolution of 37\\%. \nConsequently the actual velocity resolution is $18.2$ km\\,s$^{-1}$ , or $35$ km\\,s$^{-1}$  after \nHanning smoothing. A description of the on-line data reduction at the telescope \nwhich involves bandpass correction can be found in Barnes et al.~(1998). \n Three-dimensional data cubes ($\\alpha$, $\\delta$, $v_{\\odot}$)\nwere produced by putting individual spectra onto a pixel grid of \n$4^{\\prime} \\times 4^{\\prime}$. Due to the lack of an automatic source finder \nat this early stage of the H{\\sc i}PASS  survey, these cubes were visually searched \nfor H{\\sc i} sources by moving through the velocity range $0<v_{\\odot}<1000$  km\\,s$^{-1}$ ,   \nwhich covers the velocity distribution of the Cen\\,A group,  and corresponds \n   to \n1000 km\\,s$^{-1}$ \/35 km\\,s$^{-1}$ =28 independent velocity resolution elements at a given \nspatial pixel on the sky. Given that there are 10,000  \n  such  pixels in this survey, we searched \nthrough 280,000 independent ``cubicles''  (i.e. 3-dimensional pixels). \n\n  The 1$\\sigma$ sensitivity of \nthe H{\\sc i}PASS  survey  was   measured at  \n   14 mJy\/channel. Assuming  noise  normally distributed  about \n  zero, we expect \n statistically to find $\\sim$ 0.1  spurious positive 5-sigma detections in \nthe data. However, 27 objects with velocities $v_{\\odot}>200$ km\\,s$^{-1}$     \nwere found at this detection threshold (plus CEN 5). \n Among others we recovered  main Cen\\,A group \nmembers such as NGC 4945, NGC 5102, M 83, NGC 5237, and NGC 5253. NGC 5128 remained undetected \nbecause  the H{\\sc i} emission was swamped by  \n  strong continuum emission from that galaxy.  \n   \n\nMany more H{\\sc i} sources have been identified with velocities $v_\\odot < 200$ km\\,s$^{-1}$ .\nThese are most likely not galaxies but galactic high velocity clouds,  as \nthere is a clear overlap in velocity space between nearby galaxies and the system of \nHVCs of our Galaxy towards  the Cen\\,A group direction (Wakker 1991). \nBecause of the number dominance of high velocity clouds,  and because only one Cen\\,A \ngroup galaxy is known below $200$ km\\,s$^{-1}$  to date (CEN 5, see Table 1),  \n   we \nrestricted our study and optical follow-up program (see \\S 5) \n   to   detections with \n$v_\\odot>200$  km\\,s$^{-1}$ .   \n\n\n\\section{H{\\sc i} properties}\n\nWe determined the H{\\sc i} fluxes of our  detections by making zero-th \nmoment maps. A MIRIAD (Sault et al.~1995) gauss fitting routine yielded an estimate \nof the object's coordinates  $(\\alpha,\\delta)$ (cols.~[2] and [3] in Table \n 1).   \nPositional accuracy should equal the half-power beamwidth ($14^{\\prime}.8$) divided by \nthe signal-to-noise ratio, which corresponds to $< 3^{\\prime}$ (3.2 kpc at the \ndistance of Cen\\,A) for all sources with a 5$\\sigma$ detection or better. A \nspectrum was extracted from the data cube at the sky position from which the \nheliocentric radial velocity, $v_{\\odot}$ (col.~[4]), was measured at the mid-point \nat 50\\% of the peak of emission, and the total flux was determined (col.~[5]). \nFurthermore, we measured the velocity widths $\\Delta V_{50}$ (cols.~[6]) and \n$\\Delta V_{20}$ (cols.~[7]) at 50\\% and 20\\% of the peak, respectively. The \nH{\\sc i} mass (col.~[8]) was computed by employing the standard formula: \n\n$$\n\\frac {M_{\\rm H{\\sc i}}}{M_{\\odot}} = 2.356 \\times 10^{5} (D[{\\rm Mpc}])^{2} \n\\int{S_{\\nu}dV} \n$$\n\n\\noindent\nwhere D is 3.5 Mpc, our assumed distance to the \n   Cen\\,A   group. \n\n\nThe examination of the Hanning smoothed \n spectra (Figure 2) reveal that none of the detections are \nmarginal. All sources are found to have signals above 3.9 Jy km\\,s$^{-1}$ . \n  The coordinates of the H{\\sc i} detections were cross-correlated with \nthe NASA Extragalactic Database and compared with UK Schmidt films to find optical \ncounterparts. Each source could be associated unambiguously with  a galaxy \n(col.~[1] of Table 1 \n ) which is located within the positional \n   uncertainty \nof 3$^{\\prime}$ radius. In some cases more accurate position informations were  \n  acquired with the Compact Array of the Australia Telescope (see Staveley-Smith et al.~1999)  where   the identifications were \n always confirmed.  \n\nAmong the 28 detection we found ten new Cen\\,A group members. \n  Five of these  galaxies were \npreviously catalogued but not known to be associated with  the group, \n while the  other five\nwere previously uncatalogued galaxies and are identified  according to the H{\\sc i}PASS  \nsurvey nomenclature (HIPASS 1321-31 has recently been detected as \n 131820.5-311605 by  Karachentseva \\& Karachentsev  1998). \n\nThe most gas-rich new group member is ESO 174-G?001 with a H{\\sc i} mass of \n$M_{H{\\sc i}}=2.1\\times 10^{8}$ M$_{\\odot}$,  \n   which is larger than the amount of \nH{\\sc i} observed for instance in NGC5253 or NGC5408. It is a low surface brightness \ngalaxy close to the Galactic plane ($b=8^{\\circ}$) which may explain why its identification as a \ngalaxy was previously uncertain (Lauberts 1982). CEN 5 was previously optically \nidentified and classified as a group member candidate by C97, and so \n we include the spectrum for completeness, although it is excluded from the \n statistics by its low velocity ($<$ 200  km\\,s$^{-1}$ ).   \n  Looking at the spectrum (Fig. 2), we see two peaks. However, \n the   higher velocity   signal is from the   nearby  \n   galaxy  NGC 4945,   the complete profile of \n   which  can   be seen    in Figure 2.  \n\n\n\nFor completeness reasons we also list in Table 1\n  the H{\\sc i} limits \nNGC 5206, ESO 272-G025, and NGC 5128 which remained undetected in this study: NGC 5206\nis an early-type galaxy and ESO 272-G025 was classified as peculiar (Lauberts, 1982).   These  two galaxies  were also not detected in  H{\\sc i} emission \n  by C97, to a more sensitive detection limit.  \nSince the Multibeam failed to detect the powerful radio galaxy NGC 5128, \n the listed H{\\sc i} mass for this galaxy \nis taken   from de Vaucouleurs et al. (1991),  derived from absorbing gas. \n\nIn Figure 3 and Figure 4 we compared the total fluxes and velocity-widths \n($\\Delta$V$_{20}$) with the corresponding data measured by C97. \n   In   general we found\ngood agreement,  showing H{\\sc i}PASS is achieving \n the expected sensitivity,   although for three galaxies we measured   larger integrated fluxes, suggesting that there may be a 21-cm signal \n outside C97's chosen search areas.  \n\n\n\\section{Optical properties}\n\nWe attempted to get CCD images in the optical photometric bands $B$ and $R$ for \nthose galaxies without reliable photometry in the literature. The optical data\nwas acquired as part of an extensive optical follow-up program of H{\\sc i}PASS \ngalaxies. This program employs three telescopes: (i) the 40-inch telescope at \nthe Siding Spring Observatory, (ii) the 1m Swope Telescope at Las Campanas Observatory,\nand (iii) the Curtis Schmidt telescope at CTIO. \n We used $2048^{2}$ CCDs with a $\\sim$ 24$^{\\prime}$ field of \nview and  a pixel size of $0.7^{\\prime\\prime}$, except for the \n Siding Spring  observations, where \n a 800$^{2}$  CCD was used with a $\\sim$ 8$^{\\prime}$ field of \nview and the same pixel size.  For each galaxy a series\n   of 6 exposures of 300 \nseconds each were taken.   The optical data were  \n de-biased and flattened \n with the standard STARLINK CCDPACK (Draper 1993) \n reduction techniques,   then calibrated using \n  Landolt (1992) standards.   \n The seeing was typically \n  1$^{\\prime\\prime}$.  \nIn the case of \\hipass1337-39, we had to rely on the image from the \nDigitised Sky Survey, DSS\\footnotemark  (a   similar   method   to that  \n    employed by   Spitzak \\& \n Schneider, 1998)    to calibrate   an  image for which  we  \n  had no  calibrated   CCD \n  data.  \n\\footnotetext{The Digitized Sky Surveys  \n  were produced at the Space Telescope Science Institute under U.S.  \nGovernment grant NAG W-2166.}\n\n\nFigure 5 shows $B$-band images of the newly identified Cen\\,A group galaxies \nto illustrate the variety in optical morphology. Previously catalogued \ngalaxies (single *) \nare high surface brightness objects and thus likely to be visually identified as \nbackground galaxies. The uncatalogued galaxies  \n  (**),   on the other hand, are faint, low \nsurface brightness galaxies with small angular size and very hard to see.  \n\nIn Table 2 \n  we list photometric parameters of the 10 new group members \n(col.[1]) drawn  from various sources - either   The Third Reference \n Catalogue of Bright Galaxies, RC3 (de Vaucouleurs et al. 1991), the \nDSS, or from our own CCD data (if available). The coordinates (cols.~[2] \nand [3]) are the optical positions either quoted in the literature or measured by us at the \ncentroid of the  optical  image. When CCD data were available, \nthe galaxy image was cleaned by removing superimposed foreground stars. Afterwards, \nthe total apparent magnitude $m_B$ (col.~[4]) was determined by  circular aperture \nphotometry and fitting a growth curve.  \nOtherwise the total apparent $B$ magnitude was \n  taken from the literature. Where \nwe could fit an exponential profile $\\mu(r)=\\mu_0^{exp}+ 1.086(r\/r_0) $ to \nour own optical data we also quote the extrapolated central surface brightness \n$\\mu_{0}$ (col.~[6]) and the exponential scale length $r_0$\n   (cols.~[7] and [8]). \n   In some cases the galaxy was too faint, or too obscured by  stars, \n for a reliable profile fit to be made. \n\n\n\\section{Results} \nTo summarise our results, a shallow   integration  H{\\sc i} survey of the inner 600 square degrees of \nthe Cen\\,A group and the velocity range $200  <v_{\\odot}<1000$ km\\,s$^{-1}$    \n  revealed 28 H{\\sc i} sources \nwhich all have been \n   identified as Cen\\,A group galaxies. All detections have integrated \nflux larger than 3.9 Jy km\\,s$^{-1}$ .  \n Eighteen   objects coincide with previously known group members.  A  \n  further   \n 10 are \n  suggested as new group members.  Of these 10,  5 coincide with previously catalogued \ngalaxies and 5 coincide with previously uncatalogued galaxies. Three main group \nmembers were not detected at our survey sensitivity of 10$^{7}$ M$_{\\odot}$ of H{\\sc i}, assuming an H{\\sc i} velocity width of   \n   50 km\\,s$^{-1}$ , but one (NGC 5128) is \n known as an  H{\\sc i} source  from absorption \n studies.    All new galaxies  detected in this study have optical \n  counterparts.  \n \n\nTable 3 \n   lists various intrinsic parameters for the new group members. \nThe optical radius $R_{opt}$ (col.~[2]) has been estimated by eye on the DSS images.  \nThe model-dependent total $B$ magnitudes,  m$_{T}$ (col.~[4]),  have been calculated for \neach galaxy using the parameters of the exponential profile, $\\mu_{0}^{exp}$ and $r_0$ \n(cols.~[6] and [7], Table~\\ref{photomone}),    using \n\n\n$$\nm_{T}  =  \\mu_{0}^{exp} - 2.5 log(2 \\pi r_0^2)\n$$\n\n\n  However, this  depends on an extrapolation to infinity, and so \n the model magnitudes should be \ncompared with the m$_{B}$ values in Table 2\n   obtained from aperture \nphotometry. The galactic extinction $A_B$ for each galaxy, given in col.(4), \n has been obtained from the dust maps of Schlegel et al.~(1998), wherein dust is \nmapped from the energy it emits in the far infra-red as seen in the high  spatial \nresolution observations made with IRAS and DIRBE. In the case of the highly obscured \nESO174-G?001 we have checked that estimate by measuring the colours of neighbouring background \nelliptical galaxies in our frames, and find excellent agreement ($\\pm$ 0.1 magnitudes). \n\nAssuming a mean distance modulus of $(m-M)=27.8$ for the Cen\\,A group \n  (Saha et al. 1995),  and \ncorrecting for the extinction, we converted the absolute magnitude M$_{B}$ (col.~[5]) \nand determined M$_{\\rm H{\\sc i}}$\/L$_{\\rm B}$ (col.~[6]).  The mass M$_{\\star}$, quoted in \ncol.~(7) is arrived at by simply assuming a value of the stellar mass-to-light ratio \nM$_{\\star}$\/L$_{\\rm B}$ = 1  for such gas-rich dwarf galaxies - a result suggested \nby previous authors (e.g. Staveley-Smith et al.~1990 and de Blok, McGaugh \\&  \nvan der Hulst, 1996). Finally, \nthe ratio between indicative dynamical mass  M$_{\\rm i}$ $\\equiv$ R$_{opt}$($\\Delta$V$_{50,corr})^{2}$\/G \n(Roberts 1978) and photometric mass M$_{\\star}$ is given in col.~(8).   \n   $R_{opt}$   is the quoted \noptical radius   and $\\Delta V_{50}$ is the \nFWHM, corrected for  inclination if the minor and major \n axes of the galaxy could be reliably determined.  \n\n\nAlthough M$_{\\rm i}$  may be a crude estimator of the true dynamical mass, it should provide \na safe lower limit because H{\\sc i} radii are usually  larger than the optical radii. For \ninstance, Bosma (1978) suggests H{\\sc i} disks extend out to 1.5 $\\times$ the optical Holmberg \nradius, while for gas-rich galaxies, the ratio of H{\\sc i} to optical diameter can be much \ngreater (e.g. van Zee et al.~1995).  \n\nThe newly found group members fall in two main categories. Of the five previously \ncatalogued galaxies, ESO 321-G014 (Figure 5)  and ESO 269-G058 \n  (Figure 5)  are not in the C97 area: ESO 321-G014  \nis obviously a dwarf irregular, while ESO 269-G058  is a highly irregular object with obvious \nH{\\sc ii} regions and strong dust lanes. For the latter galaxy,  an incorrect redshift (1853  km\\,s$^{-1}$ ) is \npublished in the literature (de Vaucouleurs et al.~1991). \n Our velocity \nfor IC 4247 (418  km\\,s$^{-1}$ ), Figure 5,  is rather  different \n  from  the value of 274  km\\,s$^{-1}$    published by da Costa et \nal.~(1987), and now places it firmly in the Cen~A group. \n\nThe five previously uncatalogued galaxies exhibit a quite different morphology. They are faint \n($\\sim$ 17$^{m}$),  low surface brightness, inconspicuous objects picked up in a blind survey \nsolely because they are anomalously rich in H{\\sc i}. The survey wouldn't \npick up objects as faint as m$_{B}$=17 unless they had M$_{\\rm H{\\sc i}}$\/L$_{\\rm B}$ \nratios of 4 or more. Indeed, if we  look at their  \n  ratios of   M$_{\\rm H{\\sc i}}$\/L$_{\\rm B}$   \n{\\bf uncorrected} for Galactic extinction we find they have values ranging  \nbetween 4 and 10, with a median of 5.5. Higher values  are \nknown in the literature (e.g. Lequeux \\& Viallefond 1980; Bergvall \\& \nJ\\\"ors\\\"ater 1988; Brinks \\& Klein 1988; van Zee, Haynes \\& \n Giovanelli, 1995)  but are very rare. If such objects have a \nstellar mass-to-light ratio M$_{\\star}$\/L$_{\\rm B}$~=~1,  then they contain significantly \nmore mass in the form of gas than of stars, and it will be interesting to investigate what \ntheir evolutionary status may be. \n\n\n\n\\section {The H{\\sc i} Mass Function}\n\n\n   There is considerable controversy  over the faint-end slope of the  galaxy \n Luminosity Function (see \\S 1) and thus added interest in the low-mass \n end of the  H{\\sc i} Mass Function (H{\\sc i}MF).  \n  The   H{\\sc i}MF   has been recently investigated  by three \n different  groups (Spitzak \\& Schneider 1998; Zwaan et al.~1997; and Kilborn, Staveley-Smith \\& Webster  1998).  The first two groups  used \n deep pencil beam samples observed with Arecibo, whereas Kilborn \nused H{\\sc i}PASS  objects to study a flux-limited sample. Each of these  surveys  contain  between 50 \n and 100 objects  but,  because they are flux-limited, \n they  contain  \n  very few low mass objects, and are therefore  not suited to \n determining  the low mass slope. Our survey, on the \n   other hand, being principally distance-limited  is much better suited \n to determining  that low-mass \n slope, though it lacks the higher masses needed to determine \n $M^{\\star}_{H{\\sc i}}$ well in the usual Schechter function:\n\n\\begin{displaymath}\n\\Phi(M_{\\rm H{\\sc i}}\/M^{\\star}_{\\rm H{\\sc i}}) = \n \\Phi_{\\star} \n(M_{\\rm H{\\sc i}}\/M^{\\star}_{\\rm H{\\sc i}})^{-\\alpha} e^{-(M_{\\rm H{\\sc i}}\/M^{\\star}_{\\rm H{\\sc i}})}  \n\\end{displaymath}\n\n\n\n   In Figure 6  we plot the H{\\sc i} mass function   from the 27 detections \n  of our survey  which have accurate  H{\\sc i} masses. \n \n\n  In order   not to over-interpret  the data  we do not    attempt \n  to find a fit  for \n  $M^{\\star}_{H{\\sc i}}$, only to find   the faint-end slope $\\alpha$. To \n do this we make a weighted linear least-squares fit of \n log $\\rho$ against log $M_{H{\\sc i}}$ (where $\\rho$ is the number of sources per log$_{10}$ $M_{HI}$ bin) for sources with $M_{H{\\sc i}} < 10^9 \n M_\\odot$  (Figure 6)  and find : \n\n log $\\rho$  = (-0.30 $\\pm$ 0.15) log $M_{H{\\sc i}}$ + (3.21 $\\pm$ 1.20)\n\n \\noindent Hence the Schechter  faint-end slope is \n\n $\\alpha$ = 1.30 $\\pm$ 0.15\n\n \\noindent where the   formal error on the slope  (see Bevington, 1969)  \n   arises from assuming  purely Poisson  statistics \n for  the numbers in each $\\rho$-bin in \n Figure 6.    Since the most likely systematic error  is the loss \n from our lowest mass bin of sources  which did not make the first cut, and \n therefore  were not re-measured  more accurately, we \n can ask how $\\alpha$  would change if we added \n 50\\% more sources to the 12 already in \n that bin. In fact $\\alpha$ rises \n to only 1.45 $\\pm$ 0.15.  Similarly the formal error on the slope allows \n$\\alpha = 1.45$ at 1 standard deviation.\n Our data therefore does not rule out a faint-end slope of 1.5 but we\n can exclude a slope as steep as 1.7 at the 95\\% level.   Table 4 \n contains a comparison  of  our Cen~A results  with those from the three other \n H{\\sc i}MF surveys mentioned above.  \n\n\\section{Discussion}\n\nIt is encouraging that the present comparatively shallow blind H{\\sc i} survey  already   increases    \nthe number of known galaxies of the Cen\\,A group by 50\\%, i.e. from 20 to 30 in our search area. \nIt is in particular interesting that this happened after the group was \n  so  carefully studied \nin the optical (C97). \n\n\nA glance at the spectra  (Figure 2) confirms that \nnone of our objects are marginal detections.   All have \nsignals above 3.9 Jy km\\,s$^{-1}$ , whereas the formal 5$\\sigma$ noise limit for \n  optimally smoothed  sources  that are   \n50  km\\,s$^{-1}$  broad is closer to 1.5 Jy km\\,s$^{-1}$ . The dispersion in \nthe velocity of the less significant peaks in our data map     \nmimic the velocity distribution of the Cen\\,A group   galaxies, \nsuggesting that many of them will also turn out to be real members \nof the group.  \n\nWe find only one low surface brightness galaxy with an \n H{\\sc i} mass of more than \n   10$^{8}$ M$_{\\odot}$ - ESO174-G?001  \nwhich contains 2.8  $\\times$ 10$^{8}$ M$_{\\odot}$ of H{\\sc i}. It is 3 \narc minutes across  and its measured  central  surface brightness \n is  24.5 $\\mu_B$, which is indeed low. However, it is the \ngalaxy closest to the Galactic plane ($b=8^{\\circ}$) and turns out to \nhave 2.1 magnitudes of  \n  foreground  blue extinction. When this is allowed for, the central \nsurface brightness rises to 22.4 $\\mu_B$ - which is not very low; \nM$_{\\rm H{\\sc i}}$\/L$_{\\rm B}$ drops to 0.8 and the absolute magnitude,  \nM$_{B}=-15.6$.  \n   ESO174-G?001  was missed by previous studies simply because \n of the large extinction towards it.   Despite its comparatively \nlarge size, ESO174-G?001 is still much smaller than the instrument beam.  If there \nwere a giant LSB galaxy in the group, i.e. one with a diameter larger than \nthe projected beam (15 kpc), we could only have detected it, on \nsignal to noise grounds, if its projected mass were more than 10$^{7}$ M$_{\\odot}$ \nof H{\\sc i}  per  beam; thus it would need to  have a column density, \nN$_{H{\\sc i}}$, of at least 10$^{19}$  atoms cm$^{-2}$. In reality, our \nautomatic baseline fitting procedure would at present almost certainly \n``remove'' the signal from the data. Irrespective of telescope size, \n for sources that fill the beam  \nthere is a lowest column density \n\n\\begin{equation}\n N_{\\rm H{\\sc i}} > 10^{18} \\cdot T_{sys}{\\sqrt{\\frac{\\Delta V [kms^{-1}]}{t_{obs} [s]}}}\\hbox{cm}^{-2}\n\\end{equation}\n\n\n\\noindent  \ndetectable by any radio telescope, set by receiver noise, where \n$\\Delta$V  is the projected linewidth in  km\\,s$^{-1}$  (e.g. Disney \\& Banks \n1997). \nExtreme objects, close to this column density limit, are best found when they \nare matched to the beam. For example, an  L$_{\\star}$ galaxy ten times the \nradius of our own and containing the same amount of H{\\sc i} would \nhave an N$_{H{\\sc i}} < 10^{19}$ atoms cm$^{-2}$ and would be \nfound only in integrations significantly longer than H{\\sc i}PASS , and \nmost probably out near 6000 km\\,s$^{-1}$ .  So the results from the present\nH{\\sc i} survey do not rule out the existence of extended LSB galaxies \nin the Cen\\,A group.  The present study \n is being followed by a much deeper H{\\sc i} survey in  the same region, \nusing the same instrument to address this issue. \n\nAlthough our survey has increased the number of known Cen\\,A group \nmembers by 50\\%, the new members add only 6\\% to the total \nH{\\sc i}, and 4\\% to its light. The H{\\sc i} mass function has \na slope at the low mass end of  1.33 $\\pm$ 0.15, which is unlikely to be\nsteeper than 1.5 due to either statistical or systematic effects.\n\n The five newly \ncatalogued dwarfs have indicative mass to stellar mass ratios \n(col.[8], Table~3) between 40 and 80  (mean 63  $\\pm$ 16). \nSuch ratios probably underestimate the dynamical masses because they \nassume the H{\\sc i} radii (R$_{H{\\sc i}}$) are equal to the optical \nradii (R$_{opt}$). However, H{\\sc i} maps of similar objects by other authors \ngenerally show that R$_{H{\\sc i}} >$ R$_{opt}$  - the most \nstriking example being the VLA observations of H{\\sc i}1225+01, \nChengalur, Giovanelli \\& Haynes 1995  (for comparison, the five \n  new group members that are \n  previously catalogued \n  have a mean ratio of 14). The high ratios add to the \naccumulating evidence (e.g. Kormendy 1985) that H{\\sc i}-rich \ndwarfs have their dynamics controlled by dark matter.\n\nThis survey of the Cen\\,A group was aimed chiefly at validating \nthe performance of the wider H{\\sc i}PASS  survey of the southern sky - \nwhich will be over 90\\% complete by the time this goes to press. It \nconfirms that performance matches expectation, and that large area \nblind H{\\sc i} surveys will open a new window onto the extragalactic \nsky. As to the Cen\\,A group itself, we expect to find further new group\nmembers when we run the data through the automatic galaxy-finding \nalgorithm now under development. We have also completed a much deeper \nsurvey (t$_{obs}$=12 $\\times$ H{\\sc i}PASS ) of a limited region \n(8$^{\\circ}$ x 4$^{\\circ}$)  in the area, to  investigate the \nperformance and promise of long multibeam integrations (Banks 1998). \n\n\\acknowledgements\n\nThe development of the Parkes Multibeam Receiver was supported by \nthe Australia Telescope National Facility, the Universities of \nMelbourne, Western Sydney, Sydney, Cardiff and  Manchester \n and by  Mount \nStromlo Observatory,  PPARC and the ARC. The receiver and hardware \nwere constructed by the staffs of the CSIRO Division of Telecommunications \nand Industrial Physics with assistance from Jodrell Bank. The Multibeam \nsoftware was developed by the AIPS++ team which included staff and students \nof the ATNF and University of Melbourne. \nThe authors are very grateful to \n an anonymous referee  whose comments  have led to \n significant  changes in the \n presentation of the results.  GDB acknowledges the \n support of a PPARC postgraduate research award.  \n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDirichlet p-branes are p+1-dimensional hypersurfaces on which\nsuperstrings can begin and end (see~\\cite{Pol96,wati} for a review).\nThe low energy dynamics of an ensemble of N parallel Dp-branes can be\ndescribed by the U(N) supersymmetric gauge theory obtained by\ndimensional reduction of ten dimensional supersymmetric Yang-Mills\ntheory to the p+1-dimensional world-volume of the brane.\\cite{witt2}\nThe Yang-Mills theory gives an accurate perturbative representation of\nthe Dp-brane dynamics when the separations between the branes is\nlarge.\\cite{gab,kp,dkps} It represents a truncation of the full string\nspectrum to the lowest energy modes.  The full string theoretical\ninteractions between a pair of Dp-branes is computed by considering\nthe annulus diagram shown in fig.~\\ref{f:annulus}.  The short distance\nasymptotics of this diagram are dominated by the open string sector\nwhose lowest modes are the fields of ten dimensional supersymmetric\nYang-Mills theory.  On the other hand, long distance asymptotics are\nmost conveniently described by the dual description of this diagram as\na closed string exchange and the relevant field theoretical modes are\nthose of ten dimensional supergravity.  That these are also\nrepresented by the dimensionally reduced super Yang-Mills theory is a\nresult of supersymmetry and the fact that, for fixed Dp-brane\npositions, the ground state is a BPS state.  At zero temperature,\nbecause of supersymmetry, the interaction potential between a pair of\nstatic D0-branes vanishes independently of their separation.  Their\neffective action has been computed in an expansion in their\nvelocities, divided by powers of the separation and is known to\nbe~\\cite{b,dkps}\n\\begin{equation}\nS_{\\rm eff}(T=0)= \\int dt \\left( \n\\frac{1}{2g_s\\sqrt{\\alpha'}} \\sum_{\\alpha=1}^2 ( {\\dot {\\vec q}}\\,{}^\\alpha \n) ^2 - \n\\frac{15}{16}\\left( \\alpha'\\right)^3\n\\frac{\\vert {\\dot{\\vec q}}\\,{}^1-{\\dot{\\vec q}}\\,{}^2\n\\vert^4}{\\vert {\\vec q}\\,{}^1-{\\vec q}\\,{}^2 \\vert^7}+\\ldots \\right) \n\\label{T=0}\n\\end{equation}\nThis result agrees with the effective potential for the interaction \nof D0-branes in ten dimensional supergravity. Note that, for weak string \ncoupling, the D0-brane is very heavy.\n\n\\begin{figure}\n\\vspace{-0.3in} \n\\hspace{1.5in}\n\\psfig{figure=cylinder.eps,height=1.5in} \n\\vspace{-0.3in} \n\\caption[x]{\\footnotesize Annulus diagram for D0-brane interactions.\nThe bold lines represent world-lines of the D0-branes, separated\nby the distance $L$, which go along the periodic temporal direction. \nThey bound the string world-sheet.} \n\\label{f:annulus} \n\\end{figure} \n\nIn this paper, we shall consider the description of D0-brane\ninteractions in type IIA superstring theory using matrices.  Even at\nvery low temperatures, non-BPS states are important to the leading\ntemperature dependence.\nWe perform 1-loop computation of the effective interaction between\nstatic D0-branes in the matrix theory at finite temperature and\ncompare with the known superstring computations.  We show that the\nresults of the two computations are similar in the low temperature\nlimit but an extra integration over \n the temporal component of the gauge\nfield, is present in the matrix\ntheory. At finite temperature, because the Euclidean \ntime is compact, the  temporal gauge field  can not be removed  \nby a gauge transformation.  {\\it This integration\nis needed in order to describe correctly thermodynamics of $D0$-branes\nboth in the matrix and superstring theories}.\n\nThe paper is organized as follows.  In section~2 we discuss the\nformulation of the matrix theory at finite temperature. In section~3\nwe perform one loop computation of the effective interaction between\nstatic D0-branes at finite temperature and show that it is \nattractive, and short-ranged. In section~4 we compare this result with\nthe superstring computations and discuss the conditions under which\nthe two computations agree. Section~5 is devoted to the discussion of\nour results and, in particular, the origin of the divergence of the\nclassical thermal partition function of D0-branes which is cured by\nquantum statistics.\n                              \n\n\n\\section{Matrix theory at finite temperature}\n\nWe shall consider the matrix theory description~\\cite{bfss} of the effective\ndynamics of D0-branes in a type-IIA superstring theory which is\nderived by the reduction of ten dimensional supersymmertic \nYang-Mills theory which has the action\n\\begin{equation}\nS_{\\rm YM}[A,\\theta]=\\frac{1}{g_{YM}^2}\\int d\\tau {\\rm TR}\\left(\n\\frac{1}{4}F_{\\mu\\nu}^2+\\frac{i}{2}\\theta\\gamma_\\mu D_\\mu\\theta\\right)\n\\end{equation}\nto zero spatial dimension: $A_\\mu=A_\\mu(\\tau)$, $\\theta=\\theta(\\tau)$.\n\nThe thermal partition function of this theory is given by\n\\begin{equation}\nZ_{\\rm YM}=\\int[dA(\\tau)][d\\theta(\\tau)]\\exp\\left( -S_{\\rm YM}[A,\\theta]\\right)\n\\label{thermal}\n\\end{equation}\nwhere $S_{\\rm YM}$ is the Euclidean action and the time coordinate is\nperiodic. The bosonic and fermionic coordinates have periodic and\nanti-periodic boundary conditions,\n\\begin{eqnarray}\nA_\\mu(\\tau+\\beta)= A_\\mu(\\tau),   \\\\\n\\theta(\\tau+\\beta)=-\\theta(\\tau),\\\\\n\\beta=1\/k_BT ,\n\\label{b.c}\n\\end{eqnarray}\nwhere $T$ is the temperature and $k_B$ is Boltzmann's constant.  Gauge\nfixing will be necessary and will involve introducing ghost fields\nwhich will have periodic boundary conditions.\n\nThe representation (\\ref{thermal}) of the  thermal partition \nfunction can be derived in the standard way starting from the known\nHamiltonian of the matrix theory~\\cite{bfss} and representing\nthe  thermal partition function \n\\begin{equation}\nZ_{\\rm YM}= {\\rm tr}\\, e^{-\\beta H}  \n\\end{equation}\nvia the path integral. The trace is calculated here over all states\n obeying Gauss's law which is taken care by the integration\nover $A_0$ in (\\ref{thermal}). This representation of the  matrix theory\nat finite temperature have been already discussed \\cite{OZ98,MOP98,Sat98},\nbut the temperature induced interaction between $D0$-branes described\nbelow was never identified. \n\nIn matrix theory, the diagonal components of the gauge fields,\n$\\vec{a}^\\alpha\\equiv \\vec{A}^{\\alpha\\alpha}$, are interpreted as the\nposition coordinates of the $\\alpha$-th D0-brane and they should be\ntreated as collective variables. Static configurations play a special\nrole since they satisfy classical equations of motion \nwith the periodic boundary conditions and dominate\nthe path integral as $g^2_{\\rm YM}\\rightarrow 0$. Notice that there\nare no such static zero modes for fermionic components since they would not\nsatisfy the antiperiodic boundary conditions.\n\\footnote{This is a difference between our computation at finite temperature \nand computations of the Witten index for the matrix theory where\nfermions obey periodic boundary conditions.}\nThis is an important\ndifference from the  zero temperature case and a manifestation of the fact\nthat supersymmetry is explicitly broken at non-zero temperature.\n\nIn the following, we will construct\nan effective action for these coordinates by integrating the\noff-diagonal gauge fields, the fermionic variables and the ghosts,\n\\begin{equation}\nS_{\\rm eff}[ \\vec a^\\alpha]\\equiv -\\ln \\int[d a^\\alpha_0]\n\\prod_{\\alpha\\neq\\beta}[dA^{\\alpha\\beta}_\\mu][d\\theta][d{\\rm ghost}]\\exp\\left(\n-S_{\\rm YM}-S_{\\rm gf}-S_{\\rm gh}\\right).\n\\end{equation}\nGenerally, this integration can only be done in the a simultaneous\nloop expansion and expansion in the number of derivatives of the\ncoordinates $\\vec a^\\alpha$.  Such an expansion is accurate in the\nlimit where $\\left| \\vec a^\\alpha-\\vec a^\\beta\\right|$ are large for\neach pair of D0-branes and where the velocities are small.  Since\nthese variables are periodic in Euclidean time, small velocities are\nonly possible at low temperatures.  The remaining dynamical problem\nthen defines the statistical mechanics of a gas of D0-branes,\n\\begin{equation}\nZ_{\\rm YM}=\\int\\prod_{\\tau,\\alpha}\n[d \\vec a^\\alpha(\\tau)]\\exp\\left( -S_{\\rm eff}[\\vec a^\\alpha]\n\\right).\n\\label{effec}\n\\end{equation}\nWe expect that the zero temperature limit of $S_{\\rm eff}$ reduces to\n(\\ref{T=0}).  \n\nWe shall find several subtleties with this formulation.  If the\neffective D0-brane action is to reproduce the results of a string\ntheoretical computation, the integration over $a_0^\\alpha$ must be\nperformed in both cases.   \n\nThe effective action is a symmetric functional of the position\nvariables $\\vec a^\\alpha(\\tau)$.  Only the configuration of these\ncoordinates needs to be periodic.  Therefore the individual position\nshould be periodic up to a permutation. The variables in the path\nintegral (\\ref{effec}) should therefore be periodic up to a\npermutation and the integral should be summed over the permutations.\n\n\\section{One loop computation}\n\nWe will compute the effective action $S_{\\rm eff}$ in a simultaneous \nexpansion in the number of loops and in powers of time derivatives \nof the D0-brane positions.\n\nWe decompose the gauge field into diagonal and off-diagonal parts,\n\\begin{equation}\nA_\\mu^{\\alpha\\beta}=a_\\mu^{\\alpha}\\delta^{\\alpha\\beta}+g_{\\rm YM}\n\\bar A^{\\alpha\\beta}_\\mu\n\\end{equation}\nwhere $\\bar A_{\\mu}^{\\alpha\\alpha}=0$ so that the curvature is\n\\begin{equation}\nF_{\\mu\\nu}^{\\alpha\\beta}=\\delta^{\\alpha\\beta}f^\\alpha_{\\mu\\nu}+\ng_{\\rm YM}D_\\mu^{\\alpha\\beta}\n\\bar A^{\\alpha\\beta}_\\nu-g_{\\rm YM}D_\\nu^{\\alpha\\beta}\n\\bar A_\\mu^{\\alpha\\beta}- ig^2_{\\rm YM}\n\\left[ \\bar A_\\mu, \\bar A_\\nu \\right]^{\\alpha\\beta}\n\\end{equation}\nwhere\n\\begin{equation}\nf_{\\mu\\nu}^\\alpha= \\partial_\\mu a^\\alpha_\\nu-\\partial_\\nu a^\\alpha_\\mu\n\\end{equation}\nand\n\\begin{equation}\nD_\\mu^{\\alpha\\beta}=\\partial_\\mu-i\\left(a^\\alpha_\\mu-a^\\beta_\\mu\\right).\n\\end{equation}\nIn the Yang-Mills term in the action, we keep all orders of the\ndiagonal parts of the gauge field and expand up to second order in the\noff-diagonal components,\n\\begin{equation}\n\\frac{1}{g_{\\rm YM}^2}{\\rm TR}\\left( F_{\\mu\\nu}^2 \n\\right)=\\sum_\\alpha \\frac{1}{g_{\\rm YM}^2}\\left(f^\\alpha_{\\mu\\nu}\\right)^2\n+2\\sum_{\\alpha\\beta}\\bar A_{\\mu}^{\\beta\\alpha}\\left(\n\\delta_{\\mu\\nu}{\\buildrel \\leftarrow\\over D}_\\lambda^{\\beta\\alpha}\n\\vec D_\\lambda^{\\alpha\\beta}-{\\buildrel \\leftarrow\\over\nD}_\\mu^{\\beta\\alpha}\\vec D_\\nu^{\\alpha\\beta}+2i\\left(\nf^\\alpha_{\\mu\\nu}- f^\\beta_{\\mu\\nu} \\right)\\right)\\bar\nA_\\nu^{\\alpha\\beta}+\\ldots\n\\end{equation}\nWe will fix the gauge\n\\begin{equation}\nD_\\mu^{\\alpha\\beta}\\bar A_\\mu^{\\alpha\\beta}=0.\n\\end{equation} \nThis entails adding the Fadeev-Popov ghost term to the action\n\\begin{equation}\nS_{\\rm gh}=\\int\\sum_{\\alpha\\beta} \\left\\{\\bar c^{\\alpha\\beta}\\left( \n-D^{\\alpha\\beta}_\\mu\\right)^2c^{\\beta\\alpha}+i\ng_{\\rm YM}\\bar c^{\\beta\\alpha}D_\\mu^{\\alpha\\beta}\n\\left[\\bar A_\\mu,c\\right]\\right\\}\n\\end{equation}\nThere is a residual gauge invariance under the abelian transformation,\n\\begin{eqnarray}\n\\bar A^{\\alpha\\beta}_\\mu\\rightarrow \\bar A^{\\alpha\\beta}_\\mu \ne^{i(\\chi^\\alpha -\\chi^\\beta)},\n\\nonumber \\\\\na_\\mu^\\alpha\\rightarrow a_\\mu^\\alpha+\\partial_\\mu\\chi^\\alpha.\n\\end{eqnarray}\nWe shall use this gauge freedom to set the additional condition\n\\begin{equation}\n\\partial_0 a^{\\alpha}_0=0\n\\end{equation}\nand to fix the constant%\n\\footnote{These constants are related to the eigenvalues of the\nholonomy \n$$\n{\\rm P} \\exp{\\left(i \\int_0^\\beta d\\tau A_0(\\tau)\\right)}= \n\\Omega^\\dagger\\;{\\rm diag}\\; \\left(\ne^{i\\beta a_0^1},\\ldots,e^{i\\beta a_0^{\\rm N}}\\right) \\Omega\n$$\nknown as the Polyakov loop winding along the compact Euclidean time.\nIt can not be made trivial by the gauge transformation if $T\\neq 0$.}\n\\begin{equation}\n-\\pi\/\\beta < a_0^\\alpha\\leq\\pi\/\\beta.\n\\end{equation}\n\nThe ghost for this gauge fixing condition decouples.\n\nKeeping terms up to quadratic order in $\\bar A, c,\\bar c, \\theta$, the\naction is\n\\begin{eqnarray}\nS=\\int\\left\\{\\frac{1}{4g_{\\rm YM}^2}\\sum_\\alpha \\left( \nf^\\alpha_{\\mu\\nu}\\right)^2 +\\frac{1}{2}\n\\sum_{\\alpha\\beta}\\bar A_{\\mu}^{\\beta\\alpha}\n\\left( -\\delta_{\\mu\\nu} D_\\lambda^{\\beta\\alpha} \nD_\\lambda^{\\alpha\\beta}+D_\\mu^{\\beta\\alpha}\nD_\\nu^{\\alpha\\beta}+2i\\left( f^\\alpha_{\\mu\\nu}- f^\\beta_{\\mu\\nu}\n\\right)\\right)\\bar A_\\nu^{\\alpha\\beta}\n\\right.\\nonumber\\\\\n\\left.\n+\\sum_{\\alpha\\beta}\\bar\nc^{\\beta\\alpha}(D_\\mu^{\\alpha\\beta})^2c^{\\alpha\\beta}\n+\\frac{i}{2}\\theta^{\\beta\\alpha}\\gamma_\\mu D_\\mu^{\\alpha\\beta}\n\\theta^{\\alpha\\beta}\n\\right\\}.\n\\end{eqnarray}\nThe effective action obtained by integrating over $\\bar A,\\bar\nc,c,\\theta$ is\n\\begin{eqnarray}\nS_{\\rm eff}=\\int\\sum_\\alpha\\frac{1}{4g_{\\rm YM}^2}(f^\\alpha_{\\mu\\nu})^2\n+\\sum_{\\alpha\\neq\\beta}\\left\\{ \\frac{1}{2}{\\rm TR}\\ln\\left( -\\delta_{\\mu\\nu}(D^{\\alpha\\beta}_\\mu)^2 +2i(f_{\\mu\\nu}^\\alpha-f_{\\mu\\nu}^\\beta)\\right)\n\\right. \\nonumber \\\\  \\left.\n-{\\rm TR}\\ln\\left( -(D_\\mu^{\\alpha\\beta})^2\\right)-\\frac{1}{2}{\\rm TR}\\ln\n\\left( i\\gamma_\\mu D^{\\alpha\\beta}_\\mu\\right) \\right\\}.\n\\label{seff}\n\\end{eqnarray}\n\n\\subsection{Leading order in time derivatives}\n\nWe will evaluate the determinants on the right-hand-side of\n(\\ref{seff}) in an expansion in powers of the derivatives of $\\vec\na(\\tau)$.  The leading order term can be found by setting $\\vec a={\\rm\nconst.}$. In this case, $f^\\alpha_{\\mu\\nu}=0$ and \n(here we retain the tree-level term with time derivatives)\n\\begin{equation}\nS_{\\rm eff}=8\\sum_{\\alpha <\\beta}\\left\\{{\\rm\nTR}_B\\ln\\left( -(D^{\\alpha\\beta}_\\mu)^2 \\right) -{\\rm TR}_F\\ln\\left(\n-(D^{\\alpha\\beta}_\\mu)^2 \\right)\\right\\}\n\\end{equation}\nwhere the subscript $B$ denotes contributions from the gauge fields\nand ghosts, whereas $F$ denotes those from the adjoint fermions.  The\ndeterminants should be evaluated with periodic boundary conditions for\nbosons and anti-periodic boundary conditions for fermions.  (Note\nthat, because of supersymmetry, if both bosons and fermions had\nidentical boundary conditions the determinants would cancel.  This\nwould give the well-known result that the lowest energy state is a BPS\nstate whose energy does not depend on the relative separation of the\nD0-branes.)  \nThe boundary conditions are taken into account by introducing Matsubara \nfrequencies, so that\n\\begin{equation}\ne^{-S_{\\rm eff}}=e^{-S_{0}} \\beta^{\\rm N}\n\\int \\frac{da^\\alpha_0}{2\\pi} \\prod_{\\alpha<\\beta}\n\\prod_{n=-\\infty}^\\infty\n\\left( \\frac{ \\left(\\frac{2\\pi n}{\\beta}\n+\\frac{\\pi}{\\beta}+a_0^\\alpha-a_0^\\beta\\right)^2+\\vert\\vec \na_\\alpha-\\vec a_\\beta\\vert^2}{\\left(  \n\\frac{2\\pi n}{\\beta}+a_0^\\alpha-a_0^\\beta\\right)^2 +\\vert\\vec \na_\\alpha-\\vec a_\\beta\\vert^2}\n\\right)^8\n\\end{equation}\nUsing the formula\n\\begin{equation}\n\\prod_{n=-\\infty}^\\infty\n\\left( \\frac{2\\pi n}{\\beta}+\\omega\\right)=\\sin\\left( \\frac{\\beta\\omega}\n{2}\\right)\n\\label{21}\n\\end{equation}\nwe obtain the result\n\\begin{equation}\ne^{-S_{\\rm eff}}=e^{-S_{0}} \\beta^{\\rm N}\n\\int \\frac{da^\\alpha_0}{2\\pi} \\prod_{\\alpha<\\beta}\n\\left(  \\frac{ \\cosh \\beta\\vert\\vec a^\\alpha-\\vec a^\\beta\\vert+\n\\cos \\beta \\left(a_0^\\alpha- a_0^\\beta\\right)}\n{\\cosh\\beta\\vert\\vec a^\\alpha-\\vec a^\\beta\\vert - \n\\cos\\beta \\left( a_0^\\alpha-a_0^\\beta\\right) }\\right)^8\n\\label{23}\n\\end{equation}\nIn order to find the effective action for $\\vec a^\\alpha$, \nit is now necessary to \nintegrate the temporal gauge fields $a_0^\\alpha$ over the domain \n$(-\\pi\/\\beta,\\pi\/\\beta]$.  This integration implements the projection \nonto the gauge invariant eigenstates of the matrix theory Hamiltonian.\n\nIn the case where there is a single pair of D0-branes, N=2, the integration\nover $a_0^\\alpha$ in (\\ref{23}) can be done explicitly to obtain\nthe effective action\n\\begin{equation}\nS_{\\rm eff}=\\int_0^\\beta d\\tau\\left\\{\n\\sum_1^2\\frac{(\\dot{\\vec a}{}^\\alpha)^2}{2g_{\\rm YM}^2} \n -\\frac{1}{\\beta}\\ln\\left(\n\\frac{ P(z) }{  (1-z^2)^{15} }\n\\right)\\right\\}\n\\label{effac}\n\\end{equation}\nwhere \n\\begin{eqnarray}\nP(z)= 1+241z^2+12649z^4+254009z^6+2434901z^8+12456773z^{10}\n+36119181z^{12}\\nonumber \\\\*+61178589z^{14}+6191459z^{16}\n+36109171z^{18}\n+12462779z^{20}+2432171z^{22}\\nonumber\\\\* \n+254919z^{24}+12439z^{26}+271z^{28}\n-z^{30},\n\\end{eqnarray}\n$z=\\exp(-\\beta\\vert {\\vec a}^1-{\\vec a}^2\\vert)$ and we have\nincluded the tree level term, which gives the non-relativistic kinetic\nenergies of the D0-branes.\n\nThe effective action has the\nlow temperature expansion\n\\begin{eqnarray}\nS_{\\rm eff}=\\int_0^\\beta d\\tau\\left( \\frac{1}{2 g_{\\rm YM}^2}\\sum_\\alpha \n\\left(\\dot{\\vec a}^\\alpha\\right)^2-\\frac{1}{\\beta}\\left( 256 e^{-2\\beta\\vert \n\\vec a^1-\\vec a^2\\vert}-16384 e^{-4\\beta\\vert \\vec a^1-a^2\n\\vert} \\right.\\right. \\nonumber \\\\*   \\left.\\left.\n+\\frac{5614336}{3}e^{-6\\beta\\vert \\vec a^1-\\vec a^2\\vert}+\n\\ldots\\right)\\right)\n\\label{finalSeff}\n\\end{eqnarray}\nWe shall compare in the following section this result with\nthe superstring computation of the effective interaction between D0-branes. \n\n\\section{String theoretical interactions}\n\nThe effective interactions of D0-branes \nin superstring theory is given by computing the annulus diagram \nshown in fig.~\\ref{f:annulus}. \nThis was done in ref.~\\cite{green} (and in~\\cite{VM96} for Dp-branes).  \nThe result of summing over physical (GSO projected) superstring states gives \nthe free energy\n\\begin{equation}\nF[L,\\beta,\\nu]=\\frac{8}{\\sqrt{\\pi \\alpha'}}\\int_0^\\infty \\frac{dl}{l^{3\/2}}\\,\ne^{ -L^2l\/4\\pi^2\\alpha'}\n\\Theta_2 \\left(\\nu \\left \\vert \\frac{i\\beta^2}{\\pi \\alpha' l} \\right. \\right)\n\\prod_{n=1}^\\infty\\left( \\frac{1+e^{-nl}}{1-e^{-nl}}\\right)^8\n\\label{GreenF}\n\\end{equation}\nwhere\n\\begin{equation} \n\\Theta_2 \\left(\\nu \\left \\vert iz \\right. \\right)\n=\\sum_{q=-\\infty}^\\infty\ne^{ -\\pi z(2q+1)^2\/4+i\\pi(2q+1)\\nu},\n\\end{equation}\n$L$ is the brane separation \nand $\\nu$ is a parameter\nwhich weights the winding numbers of strings around the periodic time \ndirection. An extra factor of 2 accounting for the exchange of the\ntwo ends of the superstring~\\cite{Pol96} ending on each of the two \nD0-branes is inserted in~(\\ref{GreenF}). \n\nThe product in the integrand represents the sum over string states, \nwith requisite degeneracies,\n\\begin{equation}\n8\\prod_{n=1}^\\infty\\left( \\frac{1+e^{-nl}}{1-e^{-nl}}\\right)^8\n=\\sum_{N=0}^{\\infty} d_N e^{-Nl}\n\\label{degeneracy}\n\\end{equation}\nwhere $d_N$ is the degeneracy of the \neither superstring state at level $N$.  For \nthe lowest few levels, $d_0=8$ and $E_0=L\/2\\pi\\alpha'$.\n\nInserting (\\ref{degeneracy}) in (\\ref{GreenF}) and integrating over $l$,\nthe free energy has the form\n\\begin{equation}\nF(\\beta,L,\\nu)=\\frac{2}{\\beta}\\sum_{N=0}^\\infty d_N~\n\\ln\\left| \\frac{ 1-e^{-\\beta E_N+i\\pi\\nu}}{ 1+e^{-\\beta E_N+i\\pi\\nu} }\n\\right|\n\\label{Fsuperstring}\n\\end{equation}\nwhere the string energies are given by the formula\n\\begin{equation}\nE_N=\\frac{L}{2\\pi\\alpha'}\\sqrt{ 1+\\frac{4\\pi^2\\alpha'N}{L^2}}.\n\\label{Esuperstring}\n\\end{equation}\nThis results in the partition function\n\\begin{equation}\nZ_{\\rm str}(\\beta,L,\\nu)\\equiv e^{-\\beta F}= \\prod_{N=0}^\\infty \\left|\n\\frac{1+e^{-\\beta E_N+i\\pi \\nu}}{1-e^{-\\beta E_N+i\\pi \\nu}}\n\\right|^{2d_N}.\n\\label{Zsuperstring}\n\\end{equation}\nThe physical meaning of the last formula is obvious:\nthe partition function equals the ratio of the Fermi and\nBose distributions with the power being twice the degeneracy of states\nand $i\\nu$ playing the role of a chemical potential. The factor of 2\nin the exponent $2d_N$ in (\\ref{Zsuperstring}) \nis due to the interchange of the superstring ends as is already\nmentioned. It will provide the agreement with the matrix theory \ncomputation.\n\nIn order to compare with the Yang-Mills computation, we should first re-scale \nthe coordinates so that the mass of the D0-brane appears in the kinetic term\nas in (\\ref{T=0}).  \nThe mass is given by the formula\n\\begin{equation}\nM=\\frac{1}{g_s\\sqrt{\\alpha'}}\n\\end{equation}\nand the Yang-Mills coupling $g_{\\rm YM}$ is related to the string coupling \n$g_s$ by the equation\n\\begin{equation}\ng_{\\rm YM}^2=\\frac{g_s}{4\\pi^2(\\alpha')^{3\/2}}.\n\\end{equation}\nThe physical coordinate of the $\\alpha$-th D0-brane is identified with\n\\begin{equation}\n\\vec q^\\alpha= 2\\pi\\alpha'\\vec a^\\alpha .\n\\label{qvsa}\n\\end{equation}\nTaking N=2 in (\\ref{23}) and identifying\n$L=2\\pi\\alpha' \\vert \\vec a^1-\\vec a^2 \\vert$, we see that the \nintegrand in (\\ref{23}) coincides with (\\ref{Zsuperstring})\ntruncated to the massless modes ($N=0$) provided \n$\\nu=\\beta(a_0^1-a_0^2)$.\n\nIt is clear that the integral over $a_0^\\alpha$ is responsible for \nthe ``mismatch'' of  the effective \nactions between the string theory computation \nand matrix theory computation of the free energy. \nIn the string theory done in the spirit of ref.~\\cite{green}, \nthe parameter $\\nu$ appears in the same place as \n$\\beta(a_0^1-a_0^2)\/\\pi$ but is not integrated. It is associated\nwith the interaction of the ends of the open string with an Abelian\ngauge field background ${A}_{\\mu}(\\tau,\\vec x)$: \n\\begin{equation}\nS_{\\rm int}=\\int dx^\\mu {A}_\\mu.\n\\end{equation}\nIf the ends are separated by the distance $L$, e.g.\\ along the\nfirst spatial axis, then\n\\begin{equation}\n\\nu=\\int_0^\\beta d\\tau \\left( {A}_0(\\tau,0,\\ldots)-\n{A}_0(\\tau,L,\\ldots)\\right)\n\\end{equation}\nsince $\\dot x_\\mu (\\tau)=(1,\\vec 0)$ on the boundaries.\nThe matrix theory automatically takes into account the\nintegration over the background field while in the string theory\ncalculation of ref.~\\cite{green} the background field is fixed.\nThis is just a reflection of the fact that matrix theory is \nan effective low-energy theory of $D$-branes, while the \nolder string theory did not treat the boundaries as dynamical \nobjects. However, it is interesting to notice how close some \nof the earlier string papers came to such a description simply \nby the requirement of consistency \\cite{green1}. Further, \nin the context of matrix theory it is natural to take the \nexponential of the free energy (\\ref{Fsuperstring}) as in\neq.\\ (\\ref{Zsuperstring}), and only integrete over $\\nu$\nafterwards, a procedure not entirely obvious in \na string theory where the boundaries are not dynamical objects.\nThis will be discussed further in the next section.\n\nAn exact coincidence between the matrix theory and superstring results\nis possible only when the higher stringy modes are suppressed.\nUsually, the truncation of the string spectrum to get\nYang-Mills theory is valid for small $\\alpha'$, that is when we are interested\nin temperatures which are much smaller than $1\/\\sqrt{\\alpha'}$.  In fact, the \ncondition in our case  \nis a little different than this once the length $L$ appears as a parameter in \nthe spectrum (\\ref{Esuperstring}).  \nThen, the spectrum can be truncated at the first level only when\n\\begin{equation}\n\\frac{1}{\\beta}\\equiv k_BT\\ll \n\\sqrt{ \\left(  \\frac{L}{2\\pi\\alpha'}\\right)^2+\\frac{1}{\\alpha'} }-\\frac{L}\n{2\\pi\\alpha'}\n\\label{trunk}\n\\end{equation}\nwhich is the energy gap between the first two levels.\nIf the temperature is small, this condition is always satisfied unless\nthe length $L$ is not too large. In other words the truncation of the\nspectrum to the lightest modes is valid for $\\beta \\gg L$\n(or $TL\\ll 1$).\n\nIt is also interesting to discuss what happens in the opposite\nlimit $L \\gg \\beta$ where the interaction between D0-branes\nis mediated by the lightest closed string modes.\nThe superstring free energy can be evaluated in this limit by the\nstandard modular transformation which relates the annulus diagram\nfor an open string with a cylinder diagram for a closed string.\nIntroducing the new integration variable $s=2\\pi^2\/l$, we \nrewrite (\\ref{GreenF}) as~\\cite{green}\n\\begin{equation}\n F[L,\\beta,\\nu]= \\frac{8 \\pi^4}{\\sqrt{2 \\pi \\alpha'}} \n\\int_0^\\infty \\frac{ds}{s^{9\/2}}\\;\ne^{s}\\,e^{-L^2\/2s\\alpha'}\n\\Theta_2 \\left(\\nu \\left \\vert \\frac{i\\beta^2 s}{2\\pi^3 \\alpha' } \n\\right. \\right)\n\\prod_{n=1}^\\infty\\left( \\frac{1-e^{-(2n+1)s}}{1-e^{-2ns}}\\right)^8.\n\\label{viathetas}\n\\end{equation}\nIn the limit where the brane separation is large the integration over $s$ is \nconcentrated in the region of large $s\\sim L^{2}$.\nSubstituting the large-$z$ asymptotics\n\\begin{equation}\n\\Theta_2 \\left(\\nu \\left \\vert i z  \\right. \\right)\n\\rightarrow\n2\\cos{(\\pi\\nu)}e^{-\\pi z\/4\n\\end{equation}\nand evaluating the saddle-point integral, we get\n\\begin{equation}\n F[L,\\beta,\\nu] \\propto \\cos{(\\pi\\nu)} \n\\frac{\\left(\\beta^2-8\\pi^2\\alpha'\\right)^{3\/2}}{L^4}\ne^{-L\\sqrt{\\beta^2-8\\pi^2\\alpha'}\/2\\pi \\alpha'}.\n\\end{equation}\n\nExponentiating and integrating over $\\nu$, we have\n\\begin{equation}\n\\int_{-1}^{1} {d\\nu} Z_{\\rm str}(\\beta,L,\\nu) \n\\propto \\frac{\\beta^2 \\left(\\beta^2-8\\pi^2\\alpha'\\right)^{3}}{L^8}\ne^{-L\\sqrt{\\beta^2-8\\pi^2\\alpha'}\/\\pi \\alpha'}.\n\\end{equation}\nTaking into account (\\ref{qvsa}) the exponent at the low temperatures is\nthe same as in (\\ref{finalSeff}) but the pre-exponential differs.\nThe dependence of the pre-exponential on $L$ in the superstring case \nemerges because the splitting between energy states in\n(\\ref{Esuperstring}) is of order $1\/L$ and the truncation condition\n(\\ref{trunk}) is no longer satisfied when $L$ is large.\nHigher stringy modes are then not separated \nby a gap and the continuum spectrum \nresults in the $L$-dependence of the preexponential.\nAs usual, the limits of $L\\rightarrow\\infty$ and $T\\rightarrow 0$ are not \ninterchangeable in the superstring theory.\n\n\n\\section{Discussion}\n\n\nOur main results concern D0-brane dynamics at finite temperatures. \nWe have computed the 1-loop effective action for the interaction\nof static D0-branes in the matrix theory at finite temperature and\ncompared it with the analogous superstring computation.\nWe have seen that an extra integration over the eigenvalues\nof the holonomy along the compactified Euclidean time is present\nin the matrix theory. The two computations agrees in the \nlow temperature limit provided the superstring thermal partition\nfunction is integrated over the Abelian gauge fields $a_0$'s living on\nD0-branes. \n\nThe integration over $a_0$'s is of course natural in the context \nof the Yang-Mills theory, where it expresses that only \ngauge-invariant states should contribute to ${\\rm TR}\\,e^{-\\beta H}$.\nBut it is also natural from the point\nof view of the D0-brane physics. It can be seen as follows. \nSuppose we make a T-duality transformation, which interchanges the Neumann\nand Dirichlet boundary conditions,\nalong the compactified Euclidean time direction. Then $a_0$'s become\nthe coordinates of D-instantons on the dual circle. The integration\nover $a_0$'s becomes now the integration over the positions of D-instantons.\nThe partition function should involve such an integration over\nthe collective coordinates and since they are collective coordinates\nthe integration appears in front of the exponential of the effective \naction, not in the action itself. Viewed in terms of D0-branes and \nopen strings, we have a gas of D0-branes with open strings between \nthem. The individual strings might have a winding number $q$ (more \nprecisely $2q+1$ in the case of superstrings), describing the \nwinding around the finite-temperature space-time cylinder. The energy \nof such states are $\\propto \\beta q\/2\\pi \\alpha'$. However, the \n$q$'s satisfy $\\sum q=0$ as a result of the integration over $a_0$.\nPhysically this constraint is most easily understood  by going to \nthe closed string channel where we have closed string boundary state \non the dual circle with radius $\\tilde{\\beta} = 4\\pi^2 \\alpha'\/\\beta$\nlocalized at the point $(\\nu \\tilde \\beta, \\vec q ) $. Passing to the momentum\nrepresentation, we write\n\\begin{equation}\n\\Big\\vert B,\\vec q, \\nu \\Big\\rangle =\n\\sum_{q=-\\infty}^\\infty e^{-2i\\pi\\nu q} \\, \\Big\\vert B,\\vec q, \np_0=2\\pi q \/\\tilde{\\beta} \\Big\\rangle .\n\\end{equation}\nHere the temporal momentum is quantized as \n$p_0=2\\pi q \/\\tilde{\\beta}=q\\beta\/2\\pi\\alpha'$, which \nlead to the same energy as the above mentioned open string states.\nIn this representation $\\sum q=0$ simply expresses momentum conservation\nin the thermal direction.\n \n\nThe effective static potential between two D0-branes emerges because \nsupersymmetry\nis broken by finite temperature. This effect of breaking supersymmetry\nis somewhat analogous to the velocity effects at zero temperature\nwhere the matrix theory and superstring computations agree to\nthe leading order of the velocity expansion~\\cite{dkps}.\nWe have thus shown that the leading term in a low temperature expansion \nis correctly reproduced by the matrix theory.\nThe discrepancy between the matrix theory and superstring computations, \nwhich we have observed in the limit of large distances $L T\\gg 1$,\ndoes not contradict this statement since temperature the limits of large \ndistances and small temperatures are not interchangeable.\n\nAn interesting feature of the effective static potential between\nD0-branes is that it is logarithmic and attractive at short distances.\nIn the matrix theory, the singularity of the computed 1-loop potential\noccurs when the distance between the D0-branes vanishes and the \nSU(N) symmetry which is broken by finite distances is restored.\nThe integration over the off-diagonal components can no longer be treated\nin the 1-loop approximation! \nIn the superstring theory, the singularity is exactly the same as in the\nmatrix theory since it is determined only by the massless bosonic modes\n(the NS sector in the superstring theory). Its origin is {\\em not}\\\/ due \nto the presence of massless photon states in the spectrum.\nPutting $\\nu_1=\\nu_2$ in the above D-instanton picture on the dual\ntemporal circle, we see that the mass of the lowest states,\nassociated with the winding numbers $2q+1=\\pm 1$ is \n$\\tilde \\beta{} \/2\\pi\\alpha'$. The divergence at $L=0$ emerges, in\nthis picture, after summing over all the open string states since\nno single state has such a divergence. It shows up only\nat finite temperature where the winding number $q$ exists. \n\n\n\n\nIt is important to notice that the computed partition functions take\ninto account only thermal fluctuations of superstring stretched\nbetween D0-branes but not the fluctuations of D0-branes themselves.  \nTo calculate the thermal partition function of D0-branes,\na further path integration over their periodic trajectories $\\vec a(\\tau)$ \nis to be performed as in (\\ref{effec}). One might think that \nclassical statistics is applicable to this problem since the D0-branes\nare very heavy as $g^2_{\\rm YM}\\rightarrow0$ so that one could restrict\nhimself by the static approximation. \nThis is however not the case\ndue to the singularity of the effective static potential at small distances.\nThe integral over the D0-brane positions $\\vec{a}$'s is divergent\nwhen the two positions coincide.\n\nHowever, this singularity is only in the classical partition function.  \nThe path integral over the periodic trajectories $\\vec a(\\tau)$ \nthat we actually have to do can not diverge since \nthe 2-body quantum mechanical problem has a well-defined spectrum. \n The path integral can then be evaluated as\n$\\sum_n \\exp(-\\beta E_n)$\nwhere $E_n$ are in the spectrum of the operator\n$ H=P^2\/M+V_{\\rm eff}$. There certainly should not be the bound state energy  \neigenvalue at negative infinity for this\nquantum mechanical problem which implies the convergence of the\npath integral. These issues which are related to thermodynamics of\nD0-branes will be considered in a separate publication. \n\nLet us finally discuss when the 1-loop appoximation\nthat we have done is applicable.\nThe loop expansion in Yang-Mills theory computation \nis valid only in the limit where\n$$\ng_{\\rm YM}^2\/\\vert\\vec a\\vert^3\\sim\ng_s \\left(\\frac{ \\sqrt{\\alpha'}}{L}\\right)^3\n$$ \nis small.  This is due to the fact that the distance $L$ plays the role of a \nHiggs mass which cuts off the infrared divergences of the loop expansion in \nthe 0+1~-dimensional gauge theory.\nThus, the perturbative Yang-Mills theory computation is good when \n\\begin{equation}\ng_s^{1\/3}\\sqrt{\\alpha'} \\ll L.\n\\end{equation}   \nThis can be satisfied if either the string coupling is small or if the \nD0-brane \nseparation is large compared with the string length scale. In the latter case,\nthe truncation of the spectrum to the lightest modes is still valid when\n\\begin{equation}\nk_BT\\ll \\frac{1}{L} \\ll\\frac{1}{g_s^{1\/3}\\sqrt{\\alpha'}}\n\\end{equation} \nNote that the first inequality which is independent of both the string \nscale and the string coupling is the one already discussed in\nthe previous section. In this case, the temperature must be less than \nthe inverse distance between D0-branes.   \nIn the case where the string coupling $g_s$ is small, the criterion for \nvalidity of truncation of  the spectrum becomes\n\\begin{equation}\nk_BT<1\/\\sqrt{\\alpha'}\n\\end{equation}\nthat is the usual one.\n\n\n\\section*{Acknowlegments}\n\nWe are gratefull to P.~Di~Vecchia, G.\\ Ferretti, A.~Gorsky, M.~Green, \nM.~Krogh,  \nN.~Nekrasov, N.~Ohta, P.~Olesen, L.~Thorlacius and K.~Zarembo \nfor very useful discussions. \nJ.A. and Y.M. acknowledge the support \nfrom MaPhySto financed by the Danish National Research Foundation\nand from INTAS under the grant 96--0524.  \nThe work by Y.M. is supported in part by the grants\nCRDF 96--RP1--253 and RFFI 97--02--17927.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDistributed algorithms are an important part of solving many applied optimization problems \\cite{shalev2014understanding,mcdonald2010distributed,mcmahan2017communication}. They help to parallelize the computation process and make it faster. In this paper, we focus on the distributed methods for the saddle point problem:\n\\begin{equation}\n    \\label{SPP}\n    \\min_{x \\in \\mathcal{X}} \\max_{y \\in \\mathcal{Y}} f(x,y) := \\frac{1}{M} \\sum\\limits_{m=1}^{M} f_m(x,y).\n\\end{equation}\nIn this formulation of the problem, the original function $f$ is divided into $M$ parts, each of part $f_m$ is stored on its own local device. Therefore, only the device with the number $m$ knows information about $f_m$. Accordingly, in order to obtain complete information about the function $f$, it is necessary to establish a communication process between devices. This process can be organized in two ways: centralized and decentralized. In a centralized approach, communication takes place via a central server, i.e. all devices can send some information about their local $f_m$ function to the central server, the server collects information from the devices and does some additional calculations, and then can send new information or request to the devices. Then the process continues. With this approach, one can easily write centralized gradient descent for distributed sum minimization: $\\min_{x} g(x) := \\frac{1}{M} \\sum_{m=1}^{M} g_m(x)$. All devices compute local gradients in the same current point and then send these gradients to the server, in turn, the server averages the gradients and makes a gradient descent step, thereby obtaining a new current point, which it sends to the devices. Centralized methods for \\eqref{SPP} are discussed in detail, for example, in \\cite{beznosikov2020local}.  However, centralized approach has several problems, e.g. synchronization drawback or high requirements to the server. Possible approach to deal with these drawbacks is to use decentralized architecture \\cite{bertsekas1989parallel}.  In this case, there is no longer any server, and the devices are connected into a certain communication network and workers are able to communicate only with their neighbors and communications are simultaneous. The most popular and frequently used communication methods are the gossip protocol \\cite{kempe2003gossip,boyd2006randomized,nedic2009distributed} and accelerated gossip protocol \\cite{scaman2017optimal,ye2020multi}. In the gossip protocol, nodes iteratively exchange data with their immediate neighbors using a communication matrix and in this way the information diffuses over the network. The rate of convergence depends on the ratio $\\chi$ of maximal and minimal non-zero eigenvalues of the communication matrix, which is typically proportional to diameter of the graph squared. Accelerated consensus can be achieved i.e. by Chebyshev acceleration \\cite{scaman2017optimal} and improves the dependence from $\\chi$ to $\\sqrt\\chi$, which is optimal. However, the non-accelerated variant is more robust, e.g. it can be applied to time-varying (wireless) communication networks.\n\n\n\n\n\\subsection{Our contribution}\n\nIn particular, our contribution can be briefly described as follows\n\n\\textit{Lower bounds.} We present lower bounds for decentralized smooth strongly-convex-strongly-concave and convex-concave saddle-point problems on the time-varying networks. The lower bounds are derived under the assumption that the network is always a connected graph.\n\n\\textit{Optimal algorithm.} The paper constructs an optimal algorithm that meets the lower bounds. The analysis of the algorithm is carried out for smooth strongly-convex-strongly-concave and convex-concave saddle-point problems\n\nSee our results in the column \"time-varying\" of Table 1.\n\n\\subsection{Related works} \n\nOur work is one of the first dedicated to decentralized saddle problems over time-varying networks. Among other works, we can highlight the following paper \\cite{beznosikov2021decentralized}.\nThis work looks at a more general time-varying setting and suggests a new method. The upper bounds for their method are worse than for our method. We also mention papers on related topics:\n\n\\textit{Decentralized saddle point problems.}\n\nThe next work is devoted to centralized and decentralized distributed saddle problems \\cite{beznosikov2020local}. It carries out lower bounds and optimal algorithms in the case when the communication network is constant (non-time-varying). See Table 1 for comparison our results for time-varying topology and results from \\cite{beznosikov2020local} for constant network.\nAlso note the following works devoted to decentralized min-max problems:  in deterministic case \\cite{liu2019decentralizedprox,9304470,rogozin2021decentralized}, in stochastic case \\cite{liu2019decentralized}.\n\n\\textit{Minimization on time-varying networks.}\n\nDecentralized methods are built upon combining iterations of classical first-order methods with communication steps. In the case of time-varying networks, a non-accelerated communication procedure is employed. Paper \\cite{nedic2009distributed} can be named as an initial work on decentralized sub-gradient methods, and \\cite{nedic2017achieving} proposed DIGing -- the first first-order minimization algorithm with linear convergence over time-varying networks. After that, PANDA, which is a dual method capable of working over time-varying graphs, was proposed in \\cite{maros2018}. Analysis of DIGing and PANDA assumes that the underlying network is B-connected, that is, the union of B consequent networks is connected, while the network is allowed to be disconnected at some steps. Considering the time-varying graphs which stay connected at each iteration, decentralized Nesterov method \\cite{rogozin2019optimal} has an accelerated rate under the condition that graph changes happen rarely enough, ADOM \\cite{kovalev2021adom} and ADOM+ \\cite{kovalev2021lower} are first-order optimization methods which achieve lower complexity bounds \\cite{kovalev2021lower}. APM-C \\cite{rogozin2019projected}, Acc-GT \\cite{li2021accelerated} are accelerated methods over time-varying graphs, as well. The mentioned results are devoted to minimization algorithms and can be generalized to saddle-point problems. In this paper we generalize lower bounds of \\cite{kovalev2021lower} to min-max problems and obtain an algorithm which reaches them up to a logarithmic factor.\n\n\\renewcommand{\\arraystretch}{1.5}\n\\renewcommand{\\tabcolsep}{10pt} \n\\begin{table}[h!]\n\\vspace{-0.3cm}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\hline\n\\multicolumn{1}{c}{} & \\multicolumn{1}{c}{\\textbf{time-varying network}} & \\multicolumn{1}{c}{\\textbf{constant network \\cite{beznosikov2020local}}} \\\\ \\hline\n\\multicolumn{1}{c}{}   & \\multicolumn{2}{c}{{\\tt lower}} \\\\ \\hline\n{\\tt sc} & {$\\Omega\\left( R_0^2 \\exp\\left( - \\frac{\\mu K}{256L \\chi} \\right)\\right)$} & {$\\Omega\\left( R_0^2 \\exp\\left( - \\frac{\\mu K}{128L \\sqrt{\\chi}} \\right)\\right)$} \\\\ \\hline\n{\\tt c} & {$\\Omega\\left(\\frac{L D^2 \\chi}{K}\\right)$} & {$\\Omega\\left(\\frac{L D^2 \\sqrt{\\chi}}{K}\\right)$} \\\\ \\hline\n\\multicolumn{1}{c}{}   & \\multicolumn{2}{c}{{\\tt upper}} \\\\ \\hline\n{\\tt sc} & {$\\mathcal{y}{\\overline{\\lambda} O}\\left( R_0^2 \\exp\\left( - \\frac{\\mu K}{8L \\chi} \\right)\\right)$} & {$\\mathcal{y}{\\overline{\\lambda} O}\\left( R_0^2 \\exp\\left( - \\frac{\\mu K}{8L \\sqrt{\\chi}} \\right)\\right)$} \\\\ \\hline\n{\\tt c} & {$\\mathcal{y}{\\overline{\\lambda} O} \\left(\\frac{L D^2 \\chi}{K}\\right)$} & {$\\mathcal{y}{\\overline{\\lambda} O}\\left(\\frac{L D^2 \\sqrt{\\chi}}{K} \\right)$} \\\\ \\hline\n\\end{tabular}\n\\vspace{0.4cm}\n\\caption{Lower and upper bounds for distributed smooth stochastic strongly-convex--strongly-concave ({\\tt sc}) or convex-concave ({\\tt c}) saddle-point problems in centralized and decentralized cases. Notation:\n$L$ -- smothness constant of $f$, $\\mu$ -- strongly-convex-strongly-concave constant, $R^2_0 = \\|x_0 - x^* \\|^2_2 + \\|y_0 - y^* \\|^2_2$, $D$ -- diameter of optimization set, $\\chi$ -- condition number of communication graph (in time-varying case maximum of all graphs), $K$ -- number of communication rounds. In the case of upper bounds in the convex-concave case, the convergence is in terms of the \"saddle-point residual\", in the rest -- in terms of the (squared) distance to the solution.}\n\\end{center}\n\\label{table1}\n\\end{table}\n\n\\section{Preliminaries}\n\nWe use $\\langle z,u \\rangle := \\sum_{i=1}^d z_i u_i$ to denote standard inner product of $z,u\\in\\mathbb R^d$. It induces $\\ell_2$-norm in $\\mathbb R^d$ in the following way $\\|z\\| := \\sqrt{\\langle z, z \\rangle}$. We also introduce the following notation $\\text{proj}_{\\mathcal{Z}}(z) = \\min_{u \\in \\mathcal{Z}}\\| u - z\\|$ -- the Euclidean projection onto $\\mathcal{Z}$.\n\nWe work with the problem \\eqref{SPP}, where the sets $\\mathcal{X} \\subseteq \\mathbb{R}^{n_x}$ and $\\mathcal{Y} \\subseteq \\mathbb{R}^{n_y}$ are convex sets. Additionally, we introduce the set $\\mathcal{Z} = \\mathcal{X} \\times \\mathcal{Y}$, $z = (x,y)$ and the operator $F$:\n\\begin{equation}\n\\label{opSP}\n    F_m(z) = F_m(x,y) = \\begin{pmatrix}\n\\nabla_x f_m(x,y)\\\\\n-\\nabla_y f_m(x,y)\n\\end{pmatrix}.\n\\end{equation}\nThis notation is needed for shortness. \n\n\\textbf{Problem setting.} Next, we introduce the following assumptions:\n\n\\textit{Assumption 1(g).}\n $f(x,y)$ is $L$ - smooth, if for all $z_1, z_2 \\in \\mathcal{Z}$\n    \\begin{eqnarray}\\label{as1g}\n    \\|F(z_1) - F(z_2)\\| \\leq L\\|z_1-z_2\\|.\\end{eqnarray}\n    \n\\textit{Assumption 1(l).}\n For all $m$, $f_m(x,y)$ is Lipschitz continuous with constant $L_{\\max}$, it holds that for all $z_1, z_2 \\in \\mathcal{Z}$\n    \\begin{eqnarray}\\label{as1l}\n    \\|F_m(z_1) - F_m(z_2)\\| \\leq L_{\\max}\\|z_1-z_2\\|.\\end{eqnarray}\n    \n\\textit{Assumption 2(s).}\n$f(x,y)$ is strongly-convex-strongly-concave with constant $\\mu$, if for all $z_1, z_2 \\in \\mathcal{Z}$\n    \\begin{eqnarray} \\label{as2g}\\langle F(z_1) - F(z_2), z_1 - z_2 \\rangle \\geq \\mu\\|z_1-z_2\\|^2.\\end{eqnarray}\n    \n\\textit{Assumption 2.}\n$f(x,y)$ is convex-concave, if $f(x,y)$ is strongly-convex-strongly-concave with $0$.\n\n\\textit{Assumption 3.} $\\mathcal{Z}$ -- compact bounded, i.e. for all $z, z'\\in \\mathcal{Z}$\n    \\begin{eqnarray} \\label{as5}\n    \\| z - z'\\| \\leq D.\n    \\end{eqnarray}\nAll assumptions are standard in the literature.\n\n\\textbf{Network setting.} In each moment of time (iteration) $t$, the communication network  is modeled  as a connected, undirected graph graph $\\mathcal{G}(t) \\triangleq (\\mathcal{V}, \\mathcal{E}(t))$, where $\\mathcal{V} := \\{1,\\ldots, M\\}$ denotes  the vertex    set--the set of devices (does not change in time) and  $\\mathcal{E}(t) := \\{(i,j) \\, |\\, i,j \\in \\mathcal{V} \\}$ represents the set of edges--the communication links at the moment $t$;  $(i,j) \\in \\mathcal{E} (t)$ iff there exists a communication link between devices $i$ and $j$ in moment $t$.  \n\nAs mentioned earlier, the gossip protocol is the most popular communication procedures in decentralized\nsetting. This approach uses a certain matrix $W$. Local vectors during communications are \"weighted\" by multiplication of a vector with $W$. The convergence of decentralized algorithms is determined by the properties of this matrix. Therefore, we introduce it:\n\n\\textbf{Assumption 4.} We call a matrix $W(t)$ a gossip matrix at the moment $t$ if it satisfies the following conditions: 1) $W(t)$ is an $M \\times M$ symmetric, 2) $W(t)$ is positive semi-definite, 3) the kernel of $W(t)$ is the set of constant vectors, 4) $W(t)$ is defined on the edges of the network at the moment $t$: $W_{ij}(t) \\neq 0$ only if $i=j$ or $(i,j) \\in \\mathcal{E}(t)$.\n\nLet $\\lambda_1(W(t)) \\geq \\ldots \\geq \\lambda_M(W(t)) = 0$ the spectrum of $W(t)$, and let define condition number $\\chi = \\max_t \\chi(W(t)) = \\frac{\\lambda_1(W(t))}{\\lambda_{M-1}(W(t))}$. Note that in practice we will use not the matrix $W(t)$, but $y}{\\overline{\\lambda} W(t) = I - \\frac{W(t)}{\\lambda_1(W(t))}$. It is these matrices that are used in consensus algorithms \\cite{boyd2006randomized}. To describe the convergence, we introduce $\\rho = \\max_t \\lambda_2(y}{\\overline{\\lambda} W(t)) = \\max_t \\left[1 - \\frac{\\lambda_{M-1}(W(t))}{\\lambda_1(W(t))}\\right] = \\max_t \\left[1 - \\frac{1}{\\chi(W(t))}\\right] = 1 - \\frac{1}{\\max_t \\chi(W(t))} = 1 - \\frac{1}{\\chi}$.\n\n\\section{Main part}\n\nWe divide our contribution into two main parts, first we discuss lower bounds for decentralized saddle point problems over time-varying graphs. In the second part, we present an algorithm that achieves the lower bounds (up to logarithmic factors and numerical constants).\n\n\\subsection{Lower bounds}\n\nBefore presenting lower bounds, we must restrict the class of algorithms for which our lower bounds are valid. For this we introduce the following back-box procedure.\n\n\\begin{definition} \\label{proc}\n    Each device $m$ has its own local memories $\\mathcal{M}^x_{m}$ and $\\mathcal{M}^y_{m}$ for the $x$- and $y$-variables, respectively--with  initialization $\\mathcal{M}_{m}^x = \\mathcal{M}_{m}^y= \\{0\\}$.    $\\mathcal{M}_{m}^x$ and $\\mathcal{M}_{m}^x$ are updated as follows:\\\\\n   \n    $\\bullet$ \\textbf{Local computation:} Each device $m$ computes and adds to its $\\mathcal{M}^x_{m}$ and $\\mathcal{M}^y_{m}$ a finite number of points $x,y$, each satisfying \n    \\begin{equation}\\begin{aligned}\\label{eq:oracle-opt-step}\n        x \\in \\text{span} \\big\\{x'~,~\\nabla_x f_m(x'',y'')\\big\\},\\quad \n        y \\in \\text{span} \\big\\{y'~,~\\nabla_y f_m(x'',y'')\\big\\}, \n    \\end{aligned}\\end{equation}\n    for given $x', x'' \\in \\mathcal{M}^x_{m}$ and  $y', y'' \\in \\mathcal{M}^y_{m}$.\n\n    $\\bullet$ \\textbf{Communication:} Based upon communication round among neighbouring nodes at the moment $t$,   $\\mathcal{M}^x_{m}$ and $\\mathcal{M}^y_{m}$ are updated according to\n    \\begin{equation}\\label{eq:oracle-comm}\n        \\mathcal{M}^x_{m} := \\text{span}\\left\\{\\bigcup_{(i,m) \\in \\mathcal{E}(t)} \\mathcal{M}^x_{i} \\right\\}, \\quad \n        \\mathcal{M}^{y}_{m} := \\text{span}\\left\\{\\bigcup_{(i,m) \\in \\mathcal{E}(t)} \\mathcal{M}^y_{i} \\right\\}.\n    \\end{equation}\n\n    $\\bullet$ \\textbf{Output:} \n    The final global output at the current moment of time is calculated as: \n    \\begin{align*}\n        x \\in \\text{span}\\left\\{\\bigcup_{m=1}^M \\mathcal{M}^x_{m} \\right\\},~~y \\in \\text{span}\\left\\{\\bigcup_{m=1}^M \\mathcal{M}^y_{m} \\right\\}.\n    \\end{align*}\n\\end{definition} \n\nThis definition includes all algorithms capable of making local gradient updates, as well as exchanging information with neighbors.\nNotice that the proposed oracle builds on   \\cite{scaman2017optimal}  for minimization problems over networks. \n\n\\begin{theorem}\\label{th-LB-distributed}\nFor any $L$, $\\mu$ and $\\chi \\leq 1$, there exist a  saddle point problem in the form (\\ref{SPP}) with $\\mathcal{Z}=\\mathcal{R}^{2d}$(where $d$ is sufficiently\nlarge), and  local functions $f_m$ being  $L$-smooth, $\\mu$-strongly-convex-strongly-concave, and a gossip matrices $W(t)$ over the connected (at each moment) graph  $\\mathcal{G} (t)$,  satisfying Assumption 4 and  with $\\chi$, such that any decentralized algorithm  satisfying Definition~\\ref{proc} and using the gossip matrices $W(t)$  produces    the following estimate on the global output $z=(x,y)$ after $K$ communication rounds\n\\begin{equation*}\n    \\|z^{K} - z^*\\|^2 = \\Omega\\left(\\exp\\left( - \\frac{256\\mu}{L - \\mu} \\cdot \\frac{K}{\\chi} \\right) \\| y^*\\|^2\\right).\n\\end{equation*}\n\\end{theorem}\n\n\\begin{corollary}\nIn the setting of Theorem~\\ref{th-LB-distributed}, the number of communication rounds required to obtain a $\\varepsilon$-solution is lower bounded by\\vspace{-0.2cm}\n\\begin{equation*}\n    \\Omega\\left( \\chi \\frac{L}{\\mu} \\cdot  \\log \\left(\\frac{\\| y^*\\|^2}{\\varepsilon}\\right)\\right).\n\\end{equation*}\nAdditionally, we can get a lower bound for the number of local calculations on each of the devices:\n\\begin{equation*}\n    \\Omega\\left( \\frac{L}{\\mu} \\cdot  \\log \\left(\\frac{\\| y^*\\|^2}{\\varepsilon}\\right)\\right).\n\\end{equation*}\n\\end{corollary}\n\nAlso we want to find lower bounds for the case of  (non strongly) convex-concave problems, one can use  regularization and consider the following objective function \n\\begin{align*}\n    f(x,y) + \\frac{\\varepsilon}{4D^2} \\cdot \\|x - x^0 \\|^2 - \\frac{\\varepsilon}{4D^2}\\cdot \\|y - y^0 \\|^2,\n\\end{align*}\nwhich is   strongly-convex-strongly-concave with constant  $\\mu = \\frac{\\varepsilon}{4D^2}$, where $\\varepsilon$ is a precision of the solution and $D$ is the  diameter of the  sets $\\mathcal{X}$ and $\\mathcal{Y}$. The resulting new SPP problem is solved to $\\varepsilon\/2$-precision in order to guarantee an accuracy $\\varepsilon$ in computing the solution of  the original problem. Therefore, we can easily deduce the lower bounds for convex-concave case\n\\begin{equation*}\n\\label{lower_cc}\n    \\Omega\\left( \\chi \\frac{L D^2}{\\varepsilon} \\right)~\\text{communic. rounds}\\quad \\text{and}\\quad  \\Omega\\left(\\frac{L D^2}{\\varepsilon}\\right) ~\\text{local computations}.\n\\end{equation*}\nSee Table 1 to compare with lower bounds for constant networks. The full proof of Theorem 1 one can find in the full version of our paper [for reviewers: here will be the link to the full version in arxiv.org, but we did not publish the paper before review].\n\n\\subsection{Optimal algorithm}\n\nIn this part, we present an Algorithm that achieves lower bounds (up to logarithmic terms). Our Algorithm uses an auxiliary procedure for communication. This is a classic procedure - Gossip Algorithm.\n\n\\begin{algorithm} [th]\n\t\\caption{Gossip Algorithm ({\\tt Gossip})}\n\t\\label{alg3}\n\t\\begin{algorithmic}\n\\State\n\\noindent {\\bf Parameters:} Vectors $z_1, ..., z_M$, communic. rounds $H$.\n\\State \\noindent {\\bf Initialization:}\nConstruct matrix $\\textbf{z}$ with rows $z^T_1, ..., z^T_M$.\n\\State Choose $\\textbf{z}^0 = \\textbf{z}$.\n\\For {$h=0,1, 2, \\ldots, H$ } \n\\State $\\textbf{z}^{h+1} = y}{\\overline{\\lambda} W (h) \\cdot  \\textbf{z}^{h}$\n\\EndFor\n\\State\n\\noindent {\\bf Output:} rows $z_1, ..., z_M$ of  $\\textbf{z}^{H+1}$ .\n\t\\end{algorithmic}\n\\end{algorithm}\n\nThe essence of the {\\tt Gossip} is very simple. Initially, there are vectors $z_1$ and $z_M$, which are stored on their devices. Our goal is to get a vector close to the $\\bar z = \\frac{1}{M} \\sum_{m=1}^M z_m$ vector on all devices. At each iteration, each device exchange local vectors with its neighbors, and then modify its local vector by averaging local vector and vectors of neighbors with weights from the matrix $W (h)$.\n\nWe are now ready to present our main algorithm. It is based on the classical method for smooth saddle point problems - Extra Step Method (Mirror Prox) \\cite{Nemirovski2004,juditsky2008solving}. With the right choice of $H$, we can achieve averaging of all vectors with good accuracy. In particular, we can assume that $z^k_1 \\approx \\ldots \\approx z^k_M$. For more details about the choice of $H$ and a detailed analysis of the algorithm (taking into account that in the general $z^k_1 \\neq \\ldots \\neq z^k_M$), see in the full version of the paper [for reviewers: here will be the link to the full version in arxiv.org, but we did not publish the paper before review].\n\n\n\\begin{algorithm} [th]\n\t\\caption{Time-Varying Decentralized Extra Step Method ({\\tt TVDESM})}\n\t\\label{alg2}\n\t\\begin{algorithmic}\n\\State\n\\noindent {\\bf Parameters:}  Stepsize $\\gamma \\leq \\frac{1}{4L}$, number of {\\tt Gossip} steps $H$.\\\\\n\\noindent {\\bf Initialization:} Choose  $(x^0,y^0)=z^0\\in \\mathcal{Z}$, $z^0_m = z^0$.\n\\For {$k=0,1, 2, \\ldots, $ } \n \\State Each machine $m$ compute  \\hspace{0.1cm} $\\hat z_m^{k+1\/2} = z_m^{k} - \\gamma \\cdot F_m(z^k_m)$\n \\State  Communication: \\hspace{0.1cm} $y}{\\overline{\\lambda} z^{k+1\/2}_1, \\ldots,  y}{\\overline{\\lambda} z^{k+1\/2}_M$ ={\\tt Gossip}$(\\hat z^{k+1\/2}_1, \\ldots, \\hat z^{k+1\/2}_M, H)$\n\\State Each machine $m$ compute $z^{k+1\/2}_m = \\text{proj}_{\\mathcal{Z}}(y}{\\overline{\\lambda} z^{k+1\/2}_m)$,\n\\State Each machine $m$ compute \\hspace{0.1cm} $\\hat z_m^{k+1} = z_m^{k} - \\gamma \\cdot F_m(z^{k+1\/2}_m)$\n\\State Communication: \\hspace{0.1cm} $y}{\\overline{\\lambda} z^{k+1}_1, \\ldots,  y}{\\overline{\\lambda} z^{k+1}_M$ ={\\tt Gossip}$(\\hat z^{k+1}_1, \\ldots, \\hat z^{k+1}_M, H)$\n\\State Each machine $m$ compute \\hspace{0.1cm} $z^{k+1}_m = \\text{proj}_{\\mathcal{Z}}(y}{\\overline{\\lambda} z^{k+1}_m)$\n\\EndFor\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{theorem} \nLet $\\{ z_m^k\\}^K_{k \\geq 0}$ denote the iterates of Algorithm~\\ref{alg2} for solving problem \\eqref{SPP} after $K$ communication rounds. Let Assumptions 1(g,l) and 4 be satisfied. Then, if $\\gamma \\leq \\frac{1}{4L}$, we have the following estimates in\n\n$\\bullet$ $\\mu$-strongly-convex--strongly-concave case (Assumption 2(s)):\n\\begin{eqnarray*}\n \\mathbb E[\\| \\bar z^{k+1}\\!-\\!z^* \\|^2]\\!=\\!\\mathcal{y}{\\overline{\\lambda} O}\\left( \\| z^{0} - z^* \\|^2 \\exp\\left( -\\frac{\\mu K}{8L\\sqrt{\\chi}} \\right) \\right) ,\n \\end{eqnarray*}\n\n$\\bullet$ convex--concave case (Assumption 2 and 3):\n\\begin{equation*}\n    \\mathbb E[\\text{gap}(\\bar z^{k+1}_{avg})] = \\mathcal{y}{\\overline{\\lambda} O}\\left(\\frac{L \\Omega_z^2 \\chi}{K} \\right),\n\\end{equation*}\n\nwhere $\\bar z^{t} = \\frac{1}{M} \\sum\\limits_{m=1}^M z_m^{t}$, $\\bar z^{k+1}_{avg} = \\frac{1}{M(k+1)} \\sum\\limits_{t=0}^k \\sum\\limits_{m=1}^M z_m^{t+1\/2}$ and $$\\text{gap}(z) = \\max_{y'\\in \\mathcal{Y}} f(x, y') -  \\min_{x'\\in \\mathcal{X}} f(x', y).$$\n\\end{theorem}\n\n\\begin{corollary}\nIn the setting of Theorem~2, the number of communication rounds required for Algorithm 2 to obtain a $\\varepsilon$-solution is upper bounded by\\vspace{-0.2cm}\n\\begin{equation*}\n    \\mathcal{y}{\\overline{\\lambda} O}\\left( \\chi \\frac{L}{\\mu} \\cdot  \\log \\left(\\frac{\\| z^0 - z^*\\|^2}{\\varepsilon}\\right)\\right)\n\\end{equation*}\nin $\\mu$-strongly-convex--strongly-concave case and \n\\begin{equation*}\n\\mathcal{y}{\\overline{\\lambda} O}\\left( \\chi \\frac{L D^2}{\\varepsilon} \\right)\n\\end{equation*}\nin convex-concave case.\nAdditionally, one can obtain upper bounds for the number of local calculations on each of the devices:\n\\begin{equation*}\n    \\mathcal{O}\\left( \\frac{L}{\\mu} \\cdot  \\log \\left(\\frac{\\| z^0 - z^*\\|^2}{\\varepsilon}\\right)\\right)\n\\end{equation*}\nin $\\mu$-strongly-convex--strongly-concave case and \n\\begin{equation*}\n\\mathcal{O}\\left(\\frac{L D^2}{\\varepsilon} \\right)\n\\end{equation*}\nin convex-concave case.\n\\end{corollary}\n\n\\section{Conclusion}\n\nIn conclusion, we briefly summarize the contributions of this paper and discuss the directions for future work. Our findings consist of two parts: lower bounds and optimal (up to a logarithmic factor) algorithms.\n\nFirst, we derived the lower bounds for the classes of convex-concave and strongly-convex-strongly-concave min-max problems over time-varying graphs. The graph is assumed to be connected at each communication round. However, we studied only one class of time-varying networks. Other classes are connected to different assumptions on the network structure. In particular, in B-connected networks \\cite{nedic2017achieving} the graph can be disconnected at some times, but the union of any B consequent graphs must be connected. Yet another possible assumption is the randomly changing graph with a contraction property of $W$ in expectation \\cite{koloskova2020unified}. Developing lower bounds for min-max problems for these two classes is an open question in decentralized optimization.\n\nSecond, we proposed a near-optimal algorithm with a gossip subroutine resulting in squared logarithmic factor. Developing an algorithm without an additional logarithmic factor would close the gap in theory and result in a more practical algorithm with less parameters to fine-tune. Possible directions for developing such an algorithm are generalizations of dual-based approaches for minimization \\cite{kovalev2021adom,maros2018} and gradient-tracking \\cite{nedic2017achieving,maros2018}.\n\nFinally, the comparison of our algorithm to existing works requires additional numerical experiments, which is left for future work.\n\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe are concerned in this paper with the varieties $W^r_d(C)$ of special linear series on a general curve of fixed genus $g$ and gonality $k$ over an algebraically closed field $K$ with mild restrictions on the characteristic. See Situation \\ref{sit_main} for the necessary hypotheses.\n\nFor a general curve $C$ of genus $g$, we have $k = \\lfloor \\frac{g+3}{2} \\rfloor$ and the Brill-Noether theorem \\cite{gh} says that $\\dim W^r_d(C)$ is equal to the Brill-Noether number\n$$\n\\rho_g(d,r) = g - (r+1)(g-d+r),\n$$\n\nunless $\\rho_g(d,r) < 0$, in which case $W^r_d(C) = \\emptyset$.\n\nOur objective is to compute $\\dim W^r_d(C)$ for non-generic values of $k$. We propose a modification $\\overline{\\rho}_{g,k}(d,r)$ of the Brill-Noether number, incorporating the value of $k$, and prove the following results. By convention, a negative estimate on $\\dim W^r_d(C)$ means that $W^r_d(C)$ is empty.\n\n\\begin{thm} \\label{thm_main} In Situation \\ref{sit_main},\n$\\dim W^r_d(C) \\leq \\overline{\\rho}_{g,k}(d,r)$.\n\\end{thm}\nIn characteristic $0$, we may combine Theorem \\ref{thm_main} with a lower bound from results of Coppens and Martens \\cite{cm02}, to obtain the following sharp results.\n\\begin{thm} \\label{thm_eq}\nIn Situation \\ref{sit_main}, if $\\chr K=0$ and either $k \\leq 5$ or $k \\geq \\frac15 g + 2$, then $\\dim W^r_d(C) = \\overline{\\rho}_{g,k}(d,r)$.\n\\end{thm}\n\n\\begin{thm} \\label{thm_howgeneral}\nIn Situation \\ref{sit_main}, if $\\chr K = 0$ and  $\\rho_g(d,r) \\geq 0$, then $\\dim W^r_d(C) = \\rho_g(d,r)$ if and only if $r=0$, $g-d+r = 1$, or $g - k \\leq d - 2r$.\n\\end{thm}\n\nAll three theorems use the following notation and hypotheses.\n\n\\begin{sit} \\label{sit_main}\nFix nonnegative integers $g,k,r,d$ such that $2 \\leq k \\leq~\\frac{g+3}{2}$ and $g-d+r > 0$. Let $K$ be an algebraically closed field, and $C$ be a general $k$-gonal curve of genus $g$ over $K$. Also assume that \n\\begin{itemize}\n\\item If $k$ is odd, then $\\chr K \\neq 2$,\n\\item If $k=4$ or $10$, then $\\chr K \\neq 3$, and\n\\item If $k=6$, then $\\chr K \\neq 5$.\n\\end{itemize}\n\\end{sit}\n\nThe hypothesis $g-d+r > 0$ is harmless, since $g-d+r \\leq 0$ would imply automatically that $W^r_d(C) = \\Pic^d(C)$. The peculiar restrictions on the characteristic of $K$ arise in our proof when we must construct a \\emph{tame} morphism of metrized complexes with certain properties. The characteristic $0$ assumption in Theorems \\ref{thm_eq} and \\ref{thm_howgeneral} is included because these make use of constructions from \\cite{cm99} and \\cite{cm02}, which assume this hypothesis.\n\n\n\n\\subsection{The estimate $\\overline{\\rho}_{g,k}(d,r)$} The estimate $\\overline{\\rho}_{g,k}(d,r)$ we refer to above is the following.\n\n\\begin{defn} \\label{def_ovr}\nLet $r'$ denote the minimum of $r$ and $g-d+r-1$. Define\n$$\n\\overline{\\rho}_{g,k}(d,r) = \\max_{\\ell \\in \\{0,1,2,\\cdots,r'\\}} \\left( \\rho_g(d,r-\\ell) - \\ell k \\right).\n$$\n\\end{defn}\n\n\\begin{rem} \\label{rem_whatsell}\nThe expression being maximized in this definition is a quadratic function in $\\ell$, which takes its maximum (among all real numbers $\\ell$) at $\\ell_0 = \\frac12(g-d+2r+1-k)$. It is therefore straightforward to write a closed-form expression for $\\overline{\\rho}_{g,k}(d,r)$; see Remark \\ref{rem_threecases} and the equation preceding it.\n\\end{rem}\n\nNote that, by Riemann-Roch, $W^r_d(C) \\cong W^{g-d+r-1}_{2g-2-d}(C)$. This explains the apparently strange appearance of $r'$ in Definition \\ref{def_ovr}: it ensures that $\\overline{\\rho}_{g,k}(d,r)$ is invariant under this duality.\n\nWe will relate $\\overline{\\rho}_{g,k}(d,r)$ to $\\dim W^r_d(C)$ by specializing to a metric graph $\\Gamma_{g,k,\\ell}$ very similar to the metric graphs used in the tropical proof of the Brill-Noether Theorem \\cite{cdpr}, but chosen with \\textit{special} edge lengths. Making use of our description in \\cite{pfl} of the Brill-Noether loci of such chains, we will see (Corollary \\ref{cor_tropdim}) that $\\dim W^r_d(\\Gamma_{g,k,\\ell})$ is equal to $\\overline{\\rho}_{g,k}(d,r)$. The tropical lifting results of \\cite{abbr}, combined with semicontinuity results for tropicalization, give Theorem \\ref{thm_main}.\n\n\\subsection{The lower bound $\\underline{\\rho}_{g,k}(d,r)$} \nTo deduce Theorems \\ref{thm_eq} and \\ref{thm_howgeneral} from Theorem \\ref{thm_main}, we require a lower bound on $\\dim W^r_d(C)$.\n\nCoppens and Martens provide constructions in \\cite{cm99} and \\cite{cm02} that establish lower bounds on $\\dim W^r_d(C)$ in Situation \\ref{sit_main}, with an additional characteristic $0$ hypothesis. Their results imply the existence of irreducible components of $W^r_d(C)$ of dimension $\\rho_g(d,r-\\ell) - \\ell k$ for certain values of $\\ell$ in $\\{0,1,\\cdots,r'\\}$. From their results, one can deduce the following lower bound on $\\dim W^r_d(C)$, which coincides with the ``optimistic guess'' at the end of \\cite{cm99} (they observe in \\cite[Remark on p. 39]{cm02} that this optimistic guess is too small in general, though it is correct for $k \\leq 5$).\n\n\\begin{defn}\nLet $r' = \\min(r,g-d+r-1)$. If $r' \\geq 1$, let\n$$\n\\underline{\\rho}_{g,k}(d,r) = \\max_{\\ell \\in \\{0,1,r'-1,r'\\}} \\left( \\rho_g(d,r-\\ell) - \\ell k \\right).\n$$\nIf $r' = 0$, let $\\underline{\\rho}_{g,k}(d,r) = \\rho_g(d,r)$ (i.e. only allow $\\ell = 0$).\n\\end{defn}\n\n\\begin{thm}[\\cite{cm02}] \\label{thm_cm}\nIn Situation \\ref{sit_main}, if $\\chr K = 0$ then \n$\\dim W^r_d(C) \\geq \\underline{\\rho}_{g,k}(d,r)$.\n\\end{thm}\n\nCoppens and Martens do not state Theorem \\ref{thm_cm} in the form that we have stated it here, but Theorem \\ref{thm_cm} can be rapidly deduced from the main theorem in \\cite{cm02}; the argument is provided in Section \\ref{sec_proofs}.\n\n\\begin{rem} \\label{rem_ab}\nWe will see in our analysis that it is useful to consider instead the quantites $g - \\overline{\\rho}_{g,k}(d,r)$ $g - \\underline{\\rho}_{g,k}(d,r)$ (which are bounds on the \\textit{codimension} of $W^r_d(C)$ in $\\Pic^d(C)$). For fixed $k$, both of these quantities are functions of the following two variables.\n\\begin{eqnarray*}\na &=& r+1\\\\\nb &=& g-d+r\n\\end{eqnarray*}\nObserve that, for \\emph{any} smooth curve $C$ of genus $g$, $W^r_d(C) \\neq \\Pic^d(C)$ if and only if both $a$ and $b$ are positive. It is useful to use these variables, rather than $r$ and $d$, for visualization purposes, such as in Figures \\ref{fig_g20} and \\ref{fig_disagree}; by the previous remark, the region of interest is the first quadrant. The maximizing value $\\ell_0$ mentioned in Remark \\ref{rem_whatsell} is easier to understand in these variables as well: it is $\\ell_0 = \\frac12( a+b-k)$.\n\\end{rem}\n\n\\begin{eg}\nConsider the case $g=20$ in characteristic $0$. Since $\\frac15 g + 2 = 6$, Theorem \\ref{thm_eq} shows that $\\dim W^r_d(C) = \\overline{\\rho}_{20,k}(d,r)$ in all cases. We can visualize how the existence of specific types of linear series varies with the gonality by plotting, for each possible gonality $k$, all points $(r+1,20-d+r)$ such that $W^r_d(C) \\neq \\emptyset$ for a general $k$-gonal curve of genus $20$. These plots are shown in Figure \\ref{fig_g20}. We show only the points with both coordinates positive, since all other points correspond to values of $r,d$ for which $W^r_d(C) = \\Pic^d(C)$.\n\n\\begin{figure}\n\\begin{tabular}{ccccc}\n\\includegraphics{boxplot20-2.png}&\n\\includegraphics{boxplot20-3.png}&\n\\includegraphics{boxplot20-4.png}&\n\\includegraphics{boxplot20-5.png}&\n\\includegraphics{boxplot20-6.png}\\\\\n$k=2$ & $k=3$ & $k=4$ & $k=5$ & $k=6$\\\\\n\\includegraphics{boxplot20-7.png}&\n\\includegraphics{boxplot20-8.png}&\n\\includegraphics{boxplot20-9.png}&\n\\includegraphics{boxplot20-10.png}&\n\\includegraphics{boxplot20-11.png}\\\\\n$k=7$ & $k=8$ & $k=9$ & $k=10$ & $k=11$\n\\end{tabular}\n\\caption{\nThese plots show, for genus $20$ and every possible gonality $k$, the points $(g-d+r,r+1)$ such that $W^r_d(C)$ is nonempty for a general $k$-gonal curve $C$ in characteristic $0$.\n} \\label{fig_g20}\n\\end{figure}\n\\end{eg}\n\nWe consider the form of the results proved in \\cite{cm02}, together with the analysis of the tropical version of the problem, as evidence for the following conjecture. Note in particular that we believe that the hypotheses on the characteristic of the field should be superfluous, although they are needed for technical reasons in our arguments.\n\n\\begin{conj}\nIf $g,k,r,d$ are nonnegative integers with $g-d+r > 0$ and $2 \\leq k \\leq \\frac{g+3}{2}$, and $C$ is a general $k$-gonal curve of genus $g$ over an algebraically closed field, then $\\dim W^r_d(C) = \\overline{\\rho}_{g,k}(d,r)$.\n\\end{conj}\n\nBeyond the dimension itself, one can ask about the dimensions of \\textit{all} irreducible components of $W^r_d(C)$. The form of the upper bound, and the results of \\cite{cm02}, suggest the following question.\n\n\\begin{qu}\nDoes every irreducible component of $W^r_d(C)$ have dimension equal to $\\rho_g(d,r-\\ell) - \\ell k$ for some integer $\\ell$?\n\\end{qu}\n\n\\subsection{The gap between $\\underline{\\rho}_{g,k}(d,r)$ and $\\overline{\\rho}_{g,k}(d,r)$} \\label{ss_gap}\n\nThe definitions of $\\overline{\\rho}_{g,k}(d,r)$ and $\\underline{\\rho}_{g,k}(d,r)$, along with Remark \\ref{rem_whatsell}, reveal that there are four cases where we can immediately conclude that $\\underline{\\rho}_{g,k}(d,r) = \\overline{\\rho}_{g,k}(d,r)$ and compute $\\dim W^r_d(C)$ (for curves over characteristic $0$ fields). These are the cases where the maximum in Definition \\ref{def_ovr} occurs at $\\ell = 0,1,r'-1,r'$, respectively (note that these cases may overlap in situations where the maximum occurs for two integers $\\ell$, or when $r' \\leq 2$). Here $r' = \\min(r,g-d+r-1)$, as in Definition \\ref{def_ovr}.\n\nThe issue of comparing $\\underline{\\rho}_{g,k}(d,r)$ to $\\overline{\\rho}_{g,k}(d,r)$ therefore is confined to the situation where the maximum in Definition \\ref{def_ovr} occurs for a value $\\ell$ such that $2 \\leq \\ell \\leq r'-2$. In the variables $(a,b) = (r+1,g-d+r)$, this is equivalent to the inequalities $2 \\leq \\frac{a+b-k}{2} \\leq \\min(a-3,b-3)$; equivalently $a+b \\geq 4+k$ and $|a-b| \\leq k-6$ (where we assume $a,b > 0$, as is implied by Situation \\ref{sit_main}). See Figure \\ref{fig_disagree}.\n\n\\begin{figure}\n\\begin{tikzpicture}[scale=0.2]\n\\draw[<->,thick] (0,20) -- (0,0) -- (20,0);\n\\draw[<->,fill=lightgray] (17,20) -- (5,8) -- (8,5) -- (20,17);\n\\end{tikzpicture}\n\\caption{The region in which $\\underline{\\rho}_{g,k}(d,r) < \\overline{\\rho}_{g,k}(d,r)$, where the coordinates are $a = r+1$ and $b = g-d+r$. It is bounded by the lines $a+b = 4+k$ and $a-b = \\pm (k-6)$, and empty for $k \\leq 5$.} \\label{fig_disagree}\n\\end{figure}\n\n\nOur proof of Theorem \\ref{thm_eq} proceeds by showing that as long as $k \\geq \\frac15 g + 2$ or $k \\leq 5$, \\textit{all} $d,r$ corresponding to points in this region give $\\overline{\\rho}_{g,k}(d,r) < 0$. The following example shows that even for smaller $k$, the ambiguous cases are nonetheless quite sparse, even in the worst case.\n\n\\begin{eg}\nConsider the case $g=1000$. For each possible value of the gonality $k$, we can enumerate all possible pairs of positive integers $r,d$ with $d \\leq g-1$ and $\\overline{\\rho}_{g,k}(d,r) \\geq 0$ (we will restrict to $d \\leq g-1$, since the other cases are symmetric via Riemann-Roch), and count the proportion of these for which $\\underline{\\rho}_{g,k}(d,r) < \\overline{\\rho}_{g,k}(d,r)$. The largest such proportion is $0.042$, for gonality $k=40$; in that case there are $13123$ possible pairs where $\\overline{\\rho}_{g,k}(d,r) \\geq 0$, of which $552$ satisfy $\\overline{\\rho}_{g,k}(d,r) < \\underline{\\rho}_{g,k}(d,r)$. Of those $552$ cases, there are only $69$ cases where $\\underline{\\rho}_{g,k}(d,r) < 0$. In all other cases there is no ambiguity about whether $W^r_d(C)$ is nonempty, but only about its exact dimension.\n\\end{eg}\n\n\\subsection{Outline of the paper}\n\nIn Section \\ref{sec_lifting}, we define the metric graphs $\\Gamma_{g,k,\\ell}$ where we formulate the tropical analog we will study, and prove that these graphs can be lifted to algebraic curves using results of \\cite{abbr}. We also cite the necessary statements of our \\cite{pfl} that translate the computation of $\\dim W^r_d(\\Gamma_{g,k,\\ell})$ into the analysis of certain \\textit{displacement tableaux}. This analysis is performed in Section \\ref{sec_disp}, which is purely combinatorial. In section \\ref{sec_proofs} we deduce the theorems stated in the introduction.\n\n\\begin{rem}\nAn alternative to the tropical approach that we use in Section \\ref{sec_lifting} would be to instead consider limit linear series on a chain of elliptic curves, where the two attachment points on each elliptic curve differ by $k$-torsion. Such an approach would reduce to the same analysis of displacement tableaux.\n\\end{rem}\n\n\\section{Lifting metric graphs} \\label{sec_lifting}\n\nIn this section, we construct the metric graphs which will be the basis of our argument, and prove the necessary lifting results to algebraic curves.\n\n\\begin{defn}\nLet $\\Gamma_{g,k,\\ell}$ denote the metric graph formed by connecting $g$ cycles in a chain as in Figure \\ref{fig_chain}, with edge lengths as follows.\n\\begin{itemize}\n\\item The length of the top (clockwise) edge from $w_i$ to $w_{i+1}$ is $\\ell$.\n\\item The length of the bottom (counterclockwise) edge from $w_i$ to $w_{i+1}$ is $k-\\ell$.\n\\end{itemize}\n\\end{defn}\n\n\\begin{figure}\n\\begin{tikzpicture}[scale=2]\n\\draw (0,0) arc (105:75:1.931);\n\\draw (1,0) arc (0:-180:0.5);\n\\draw (1,0) arc (105:75:1.931);\n\\draw (2,0) arc (0:-180:0.5);\n\\draw (2,0) arc (105:75:1.931);\n\\draw (3,0) arc (0:-180:0.5);\n\\draw (3.5,0) node {$\\cdots$};\n\\draw (4,0) arc (105:75:1.931);\n\\draw (5,0) arc (0:-180:0.5);\n\n\\foreach \\x in {0,1,2,4} {\n\\draw (\\x+0.5,0.05) node[above] {$\\ell$};\n\\draw (\\x+0.5,-0.5) node[below] {$k - \\ell$};\n\\draw (\\x+0.5,-2) node[below] {$\\ell(k-\\ell)$};\n}\n\n\\draw[->,ultra thick] (2.5,-1) -- (2.5,-1.5);\n\\draw (2.5,-1.25) node[right] {degree $k$};\n\\draw (0,-2) -- (3.25,-2);\n\\draw[fill] (0,-2) circle[radius=0.05] node[above] {$u_0$};\n\\draw[fill] (1,-2) circle[radius=0.05] node[above] {$u_1$};\n\\draw[fill] (2,-2) circle[radius=0.05] node[above] {$u_2$};\n\\draw[fill] (3,-2) circle[radius=0.05] node[above] {$u_3$};\n\\draw (3.5,-2) node {$\\cdots$};\n\\draw (3.75,-2) -- (5,-2);\n\\draw[fill] (4,-2) circle[radius=0.05] node[above] {$u_{g-1}$};\n\\draw[fill] (5,-2) circle[radius=0.05] node[above] {$u_g$};\n\n\\draw[fill] (0,0) circle[radius=0.05] node[above] {$w_0$};\n\\draw[fill] (1,0) circle[radius=0.05] node[above] {$w_1$};\n\\draw[fill] (2,0) circle[radius=0.05] node[above] {$w_2$};\n\\draw[fill] (3,0) circle[radius=0.05] node[above] {$w_3$};\n\\draw[fill] (4,0) circle[radius=0.05] node[above] {$w_{g-1}$};\n\\draw[fill] (5,0) circle[radius=0.05] node[above] {$w_g$};\n\\end{tikzpicture}\n\\caption{The chain of cycles $\\Gamma_{g,k,\\ell}$ and the harmonic morphism to an interval.} \\label{fig_chain}\n\\end{figure}\n\n\\begin{defn}\nLet $k,\\ell$ be positive integers with $\\ell < k$, and let $p$ be either a prime number or $0$. Call the triple $(p,k,\\ell)$ \\emph{admissible} if $\\gcd(\\ell,k) = \\gcd(\\ell,p) = \\gcd(k-\\ell,p) = 1$.\n\\end{defn}\n\nThis notion of admissibility underlies the restrictions on the characteristic of the field in Situation \\ref{sit_main}.\n\n\\begin{lemma} \\label{lem_lexists}\nLet $p,k$ be integers, with $k \\geq 2$ and either $p=0$ or $p$ prime. There exists an integer $\\ell \\in \\{1,2,\\cdots,k-1\\}$ such that $(p,k,\\ell)$ is admissible if and only if \n$$\n(p,k) \\not\\in \\{(2,k):\\ k \\mbox{ odd}\\} \\cup \\{(3,4),(3,10),(5,6)\\}.\n$$\n\\end{lemma}\n\\begin{proof}\nAssume that $(p,k)$ are chosen, not belonging to any of these four cases stated in the lemma. Let $\\ell$ be an integer, chosen as follows.\n\\begin{enumerate}\n\\item If $k \\not\\equiv 1 \\pmod{p}$, let $\\ell = 1$ (this case includes the case $p=0$).\n\\item If $k \\equiv 1 \\pmod{p}$ and $k$ is odd, let $\\ell = 2$.\n\\item If $k \\equiv 1 \\pmod{p}$ and $k$ is even, choose $\\ell$ according to the following grid. Observe that our assumptions imply, in this case, that $p$ is an odd prime.\n\\begin{center}\n{\n\\setlength\\extrarowheight{5pt}\n$\\begin{array}{|l |l l l |}\\hline\n& p=3 & p=5 & p > 5 \\\\\\hline\nk \\equiv 0 \\pmod{4} & \\ell = \\frac12 k - 3 & \\ell = \\frac12 k - 1 & \\ell = \\frac12 k - 1\\\\\nk \\equiv 2 \\pmod{4} & \\ell = \\frac12 k - 6 & \\ell = \\frac12 k - 4 & \\ell = \\frac12 k - 2\\\\\\hline\n\\end{array}$\n}\n\\end{center}\n\\end{enumerate}\nIn cases (1) and (2), it follows immediately that $(p,k,\\ell)$ is admissible (note in case (2) that our assumption has ruled out $p=2$). In case $3$, we must verify that $\\ell \\not\\equiv 0,1 \\pmod{p}$, $\\gcd(\\ell,k) = 1$, and $\\ell \\geq 1$. This is a routine verification in each of the six sub-cases.\n\nConversely, it is straightforward to check that no such $\\ell$ exists in the four cases mentioned in the statement.\n\\end{proof}\n\n\\begin{lemma} \\label{lem_lift}\nLet $K$ be a complete algebraically closed non-Archimedean field of characteristic $p$ (possibly $p=0$), with nontrivial valuation and residue field of the same characteristic. Let $\\ell$ be a positive integer such that $(p,k,\\ell)$ is admissible. There exists a smooth projective curve $C$ over $K$ with the following properties:\n\\begin{enumerate}\n\\item The genus of $C$ is $g$;\n\\item There is a degree $k$ map $C \\rightarrow \\textbf{P}^1$;\n\\item The minimal skeleton of $C$ is isometric to $\\Gamma_{g,k,\\ell}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe apply the results of \\cite{abbr} on lifting harmonic morphisms of metrized complexes to morphisms of algebraic curves.\n\nLet $I$ denote a metric graph consisting of vertices $u_0,u_1,\\cdots,u_g$ joined (in that order) into a path, with all edge lengths equal to $\\ell(k-\\ell)$ (see Figure \\ref{fig_chain}).\n\nDefine a piecewise-linear map $\\pi: \\Gamma_{g,k,\\ell} \\rightarrow I$ by $\\pi(w_i) = u_i$, with $\\pi$ linear between these vertices. This map of metric graphs has integer slopes, with expansion factor $k-\\ell$ on the top (clockwise) edges $w_i w_{i+1}$, and expansion factor $\\ell$ on the bottom (counterclockwise) edges $w_i w_{i+1}$. Since $(p,k,\\ell)$ is admissible, all expansion factors are positive integers, none divisible by the characteristic of $K$, such that at any vertex of $\\Gamma_{g,k,\\ell}$, the expansion factors along any outward rays mapping to the leftward ray in $I$ add up to $k$ (and likewise for outward rays mapping to the rightward ray in $I$). Therefore $\\pi$ is a \\textit{finite harmonic} map of metric graphs, of degree $k$ \\cite[Definitions 2.4,\\ 2.6]{abbr}.\n\nThis harmonic map of metric graphs gives rise to harmonic morphism of \\textit{metrized complexes} \\cite[Definition 2.19]{abbr} as follows: to each vertex $v$ in either $\\Gamma_{g,k,\\ell}$ or $I$ we associate a curve $C_v$ isomorphic to $\\textbf{P}^1$ over the residue field of $K$; for each vertex $w_i$ of $\\Gamma_{g,k,\\ell}$ we associate a degree $k$ morphism $C_{w_i} \\rightarrow C_{u_i}$ defined by a rational function with divisor $(k-\\ell) \\cdot A + \\ell \\cdot B - (k-\\ell) \\cdot C - \\ell \\cdot D$, where $A,B,C,D$ denote any four rational points of $C_{w_i}$. We identify the two points $A,B$ of $C_{w_i}$ with the two tangent directions from $w_i$ that point to the left (if they exist), and we identify with $C,D$ the tangent directions that points to the right (if they exist). We identify the rational point $(0)$ of $C_{u_i}$ with the leftward tangent direction in $I$, and the rational point $(\\infty)$ with the rightward tangent direction.\n\nThe morphism $\\pi'$ is \\textit{tame} \\cite[Definition 2.21]{abbr}, since none of the expansion factors of $\\pi$ are divisible by the characteristic of the residue field of $K$. Therefore \\cite[Proposition 7.15]{abbr} implies the existence of a degree $k$ morphism $C \\rightarrow \\textbf{P}^1$ of smooth projective curves over $K$ specializing to the morphism $\\pi'$ of metrized complexes. In particular, the minimal skeleton of $C$ is isometric to $\\Gamma_{g,k,\\ell}$, and $C$ has a degree $k$ map to $\\textbf{P}^1$, as desired.\n\\end{proof}\n\nLemma \\ref{lem_lift} makes it possible to deduce results about algebraic curves from the tropical Brill-Noether theory of the graph $\\Gamma_{g,k,\\ell}$, i.e. of the loci $W^r_d(\\Gamma_{g,k,\\ell})$, which can be defined in terms of the Baker-Norine rank of divisors on metric graphs (see \\cite{gk} or \\cite{mz}). Our \\cite{pfl} develops the tools necessary for this analysis, applicable to any chain of cycles with any edge lengths. For convenience, we summarize the needed facts in the context necessary for this paper.\n\nThe structure of the tropical Brill-Noether loci $W^r_d(\\Gamma_{g,k,\\ell})$ depends on the \\textit{torsion profile} of the metric graph $\\Gamma_{g,k,\\ell}$, which is defined as the tuple $m_2, m_3, \\cdots, m_{g-1}$, where $m_i$ is the generator of the arithmetic progression $\\{m \\in \\textbf{Z}: m \\cdot w_{i-1} \\sim m \\cdot w_i \\}$ \\cite[Definition 1.9]{pfl} (in \\cite{pfl}, the torsion profile also includes a final number $m_g$, but this number is irrelevant in the context of this paper). For the metric graph $\\Gamma_{g,k,\\ell}$, where $\\ell$ is chosen relatively prime to $k$ (which is part of the definition of $(p,k,\\ell)$ being admissible), all of the numbers $m_i$ are equal to $k$. We therefore make the following definition, which is a simplified version of \\cite[Definition 2.1]{pfl} for the case where all of the $m_i$ equal the same number $k$.\n\n\\begin{defn} \\label{def_disp}\nLet $a,b$ be positive integers, and let $\\lambda$ be the set $\\{1,2,\\cdots,b\\} \\times \\{1,2,\\cdots,a\\}$. A \\emph{$k$-uniform displacement tableau} on $\\lambda$ is a function $t: \\lambda \\rightarrow \\textbf{Z}_{>0}$ subject to the following two conditions.\n\\begin{enumerate}\n\\item $t(x+1,y) > t(x,y)$ and $t(x,y+1) > t(x,y)$ whenever both sides of the inequality are defined.\n\\item If $t(x,y) = t(x',y')$, then $x-y \\equiv x'-y' \\pmod{k}$.\n\\end{enumerate}\nThe elements of $\\lambda$ will be called ``boxes'' and the integer $t(x,y)$ will be called the ``label'' of the box $(x,y)$.\n\\end{defn}\n\n\\begin{eg}\nThe following is a $3$-uniform displacement tableau, with $a = b = 3$. The labels $3$ and $5$ are repeated, which is permitted since the occurrences are in boxes of lattice distance $k=3$ from each other.\n$$\n\\young(567,345,123)\n$$\n\\end{eg}\n\n\\begin{lemma} \\label{l_td}\nFix integers $r,d$, and let \n$$\n\\lambda = \\{1,2,\\cdots,g-d+r\\} \\times \\{1,2,\\cdots,r+1\\}.$$\nSuppose that $\\ell$ is relatively prime to $k$. The locus $W^r_d(\\Gamma_{g,k,\\ell})$ is nonempty if and only if there exists a $k$-uniform displacement tableau on $\\lambda$ with at most $g$ distinct labels. If it is nonempty, then its dimension is equal to $g - | t(\\lambda) |$, where $t$ has the fewest possible number of distinct labels in such a tableau.\n\\end{lemma}\n\n\\begin{proof}\nIf such a tableau exists, then we may assume without loss of generality that the labels used are all chosen from $\\{1,2,\\cdots, g\\}$, by applying an order-preserving function to the labels used. The result is now \\cite[Corollary 3.7]{pfl}, applied to the case where $\\lambda$ is rectangular and all the torsion orders are equal to $k$.\n\\end{proof}\n\n\\begin{cor} \\label{cor_punchline}\nLet $C$ be a curve of the form guaranteed to exist by Lemma \\ref{lem_lift}. Then the dimension of $W^r_d(C)$ is bounded above by the maximum number of omitted labels from $\\{1,2,\\cdots,g\\}$ in a $k$-uniform displacement tableau on $\\{1,2,\\cdots,g-d+r\\} \\times \\{ 1,2,\\cdots, r+1 \\}$.\n\\end{cor}\n\n\\begin{proof}\nThis follows from \\cite[Proposition 5.1]{pfl} applied to the case of a rectangular partition, which states that the dimensions of the Brill-Noether loci of $C$ are bounded above by the dimensions of the Brill-Noether loci of $\\Gamma_{g,k,\\ell}$.\n\\end{proof}\n\nWe have therefore reduced our task to the analysis of $k$-uniform displacement tableaux, which we undertake in the following section.\n\n\\section{Uniform displacement tableaux} \\label{sec_disp}\n\nOur purpose in this section is to compute the minimum number of symbols needed to form a $k$-uniform displacement tableau on a rectangular partition. The answer is given by the following function.\n\n\\begin{defn} \\label{def_cd}\nLet $a,b,k$ be positive integers with $k \\geq 2$. Let $\\cd(a,b,k)$ denote the fewest number of distinct symbols used in a $k$-uniform displacement tableau on an $a \\times b$ rectangular partition. Define $\\delta(a,b,k)$ by\n$$\n\\delta(a,b,k) = \\min_{\\ell \\in \\{0,1,\\cdots,\\min(a,b)-1\\}} \\left( (a-\\ell)(b-\\ell) + k \\ell \\right).\n$$\n\\end{defn}\n\nThe name $\\cd$ is chosen since this quantity will be used to bound the codimensions of Brill-Noether loci. Indeed, note that by definition, $$\\overline{\\rho}_{g,k}(d,r) = g - \\delta(r+1,g-d+r,k).$$\n\nObserve that $\\cd(a,b,k) = \\cd(b,a,k)$ and $\\delta(a,b,k) = \\delta(b,a,k)$. Therefore it suffices to consider the case $a \\leq b$, which will simplify some statements.\n\n\\begin{rem} \\label{rem_threecases}\nThe function being minimized in the definition of $\\delta(a,b,k)$ is a quadratic function in $\\ell$, which achieves its minimum (for real values of $\\ell$) at $\\ell = \\frac12 (a+b-k)$. If $\\ell$ is constrained to integer values, the minimum is attained by both the floor or the ceiling of $\\frac12 (a+b-k)$. An elementary calculation shows that if $a \\leq b$, then\n$$\n\\delta(a,b,k) = \\begin{cases}\n(k-1)(a-1) + b & \\mbox{if $k \\leq b-a+3$},\\\\\nab - \\left\\lfloor \\left( \\frac{a+b-k}{2} \\right)^2 \\right\\rfloor & \\mbox{if $b-a+1 \\leq k \\leq a+b+1$},\\\\\nab & \\mbox{if $k \\geq a+b-1.$}\n\\end{cases}\n$$\nThe overlap of the hypotheses for these cases reflects the fact that there are two minimizing values of $\\ell$ when $a+b-k$ is odd. If $a \\geq b$, we obtain similar formulas by symmetry.\n\\end{rem}\n\n\\begin{prop} \\label{p_cd}\nFor any positive integers $a,b,k$ with $k \\geq 2$, $$\\cd(a,b,k) = \\delta(a,b,k).$$\n\\end{prop}\n\nWe prove Proposition \\ref{p_cd} by proving inequalities in both directions. In fact, only the first inequality is needed for this paper's main results, but we present the second inequality to emphasize that our result is the strongest possible using this method.\n\n\\begin{lemma} \\label{l_cdlower}\nFor any positive integers $a,b,k$ with $k \\geq 2$, $$\\cd(a,b,k) \\geq \\delta(a,b,k).$$\n\\end{lemma}\n\n\\begin{proof} \nWe assume without loss of generality that $a \\leq b$.\n\nLet $\\lambda$ denote the rectangular partition $\\{1,2,\\cdots,b\\} \\times \\{1,2,\\cdots,a\\}$, and let $t: \\lambda \\rightarrow \\textbf{Z}_{>0}$ be any $k$-uniform displacement tableau on $\\lambda$. \nWe consider three cases separately (corresponding to the three cases of Remark \\ref{rem_threecases}). In each case, we will give a subset $S$ of the boxes of $\\lambda$, with $\\delta(a,b,k)$ elements, such that each box must have a different value in $t$. This will imply the statement of the lemma.\n\nThroughout, we will say that a box $(x',y')$ \\emph{dominates} another box $(x,y)$ if $x' \\geq x$ and $y' \\geq y$. If so, then $(x',y')$ and $(x,y)$ cannot have the same label.\n\n\n\\textit{Case 1:} $k \\geq a+b-1$. For any box $(x,y) \\in \\lambda$, $1-a \\leq x-y \\leq b-1$. Therefore, if $(x,y),(x',y')$ are two boxes with $x-y \\equiv x'-y' \\pmod{k}$, then $x-y = x'-y'$ and one box dominates the other. Therefore we take $S$ to be all of $\\lambda$; all boxes must have distinct labels, so $\\cd(a,b,k) = \\delta(a,b,k) = ab$.\n\n\\textit{Case 2:} $k \\leq b-a+2$. Define $S \\subseteq \\lambda$ to be the disjoint union $S_1 \\cup S_2$, where\n\\begin{eqnarray*}\nS_1 &=& \\left\\{ (x,y):\\ 1 \\leq y \\leq a-1 \\textrm{ and } y \\leq x \\leq y+k-1 \\right\\} \\\\\nS_2 &=&  \\left\\{ (x,a):\\ a \\leq x \\leq b \\right\\}.\n\\end{eqnarray*}\nAn example is shown in Figure \\ref{fig_s}. Any box $(x,y)$ in $S_1$ satisfies $0 \\leq x-y \\leq k-1$, hence no two boxes in $S_1$ can have the same label, by the same reasoning as in case $1$. Any two boxes in $S_2$ must have different labels since one dominates the other. Finally, if $(x,y) \\in S_1$ and $(x',a) \\in S_2$ satisfy $x-y \\equiv x'-a \\pmod{k}$, then $(x',a)$ either dominates or is equal to $(x-y+a, a)$ (since $x' \\geq a$, $x' \\equiv x-y + a \\pmod{k}$, and $0 \\leq x-y \\leq k-1$), which dominates $(x,y)$. Therefore no box of $S_1$ can have the same label as a box of $S_2$. It follows that the boxes of $S$ all have distinct labels in $t$. The size of $S$ is $k(a-1) + b-a+1 = (k-1)(a-1) + b = \\delta(a,b,k)$, so $\\cd(a,b,k) \\geq \\delta(a,b,k)$.\n\n\\begin{figure}\n\\begin{tabular}{cc}\n\\begin{tikzpicture}[scale=0.5]\n\\draw (0,0) rectangle (8,3);\n\\draw[fill=lightgray] (0,0) -- (4,0) -- (4,1) -- (5,1) -- (5,2) -- (8,2) -- (8,3) -- (2,3) -- (2,2) -- (1,2) -- (1,1) -- (0,1) -- cycle;\n\\foreach \\x in {1,2,...,7} {\\draw (\\x,0) -- (\\x,3);}\n\\foreach \\y in {1,2} {\\draw (0,\\y) -- (8,\\y);}\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=0.5]\n\\draw (0,0) rectangle (6,5);\n\\draw[fill=lightgray] (0,0) -- (3,0) -- (3,1) -- (4,1) -- (4,2) -- (5,2) -- (5,3) -- (6,3) -- (6,5) -- (3,5) -- (3,4) -- (2,4) -- (2,3) -- (1,3) -- (1,2) -- (0,2) -- cycle;\n\\foreach \\x in {1,2,...,5} {\\draw (\\x,0) -- (\\x,5);}\n\\foreach \\y in {1,2,3,4} {\\draw (0,\\y) -- (6,\\y);}\n\\end{tikzpicture}\n\\\\\n$a=3,\\ b=8,\\ k=4$ (case 2)\n& \n$a=5,\\ b=6,\\ k=4$ (case 3)\n\\end{tabular}\n\\caption{Examples of the regions $S$ is cases $2$ and $3$ of the proof of Lemma \\ref{l_cdlower}.} \\label{fig_s}\n\\end{figure}\n\n\\textit{Case 3:} $b-a+3 \\leq k \\leq a+b-2$. Define\n$$\nS = \\left\\{ (x,y) \\in \\lambda:\\ - \\left\\lceil \\frac{k-1-(b-a)}{2} \\right\\rceil \\leq x-y \\leq \\left\\lfloor \\frac{k-1+(b-a)}{2} \\right\\rfloor \\right\\}.\n$$\n\nSince $(b-a) \\leq k-3$, the lower bound on $x-y$ in this definition is indeed negative. An example is shown in Figure \\ref{fig_s}.\n\nObserve that the difference between the upper bound and the lower bound for $x-y$ in the definition of $S$ is exactly $k-1$. Therefore any two boxes in $S$ with values of $x-y$ that are congruent modulo $k$ must in fact have equal values of $x-y$, hence one dominates the other. It follows that $\\cd(a,b,k) \\geq |S|$. It remains to calculate the size of the set $S$.\n\nThe complement of $S$ can be written as the union of two disjoint parts.\n\\begin{eqnarray*}\nT_1 &=& \\left\\{ (x,y) \\in \\lambda:\\ y \\geq x + 1 + \\left\\lceil \\frac{k-1-(b-a)}{2} \\right\\rceil \\right\\}\\\\\nT_2 &=& \\left\\{ (x,y) \\in \\lambda:\\ x \\geq y + 1 + \\left\\lfloor \\frac{k-1+(b-a)}{2} \\right\\rfloor \\right\\}\n\\end{eqnarray*}\n\nBoth $T_1$ and $T_2$ are triangular regions in the upper left and lower right corners of $\\lambda$. By determining the side lengths of these two regions, it follows that\n\n\\begin{eqnarray*}\n|T_1| &=& \\binom{ \\lfloor (a+b-k+1)\/2 \\rfloor }{2},\\\\\n|T_2| &=& \\binom{ \\lceil (a+b -k+1)\/2 \\rceil }{2}.\n\\end{eqnarray*}\n\nLet $z = \\frac{a+b-k+1}{2}$ and $\\varepsilon = z - \\lfloor z \\rfloor$. Then\n\n\\begin{eqnarray*}\n|T_1| + |T_2| &=& \\frac12 (z - \\varepsilon)(z - 1 - \\varepsilon) + \\frac12 (z + \\varepsilon)(z-1+\\varepsilon)\\\\\n&=& z(z-1) + \\varepsilon^2\\\\\n&=& \\left(z-\\frac12\\right)^2 + \\varepsilon^2 - \\frac14\\\\\n&=& \\left \\lfloor \\left( \\frac{a+b-k}{2} \\right)^2 \\right\\rfloor\n\\end{eqnarray*}\n\nTherefore $|S| = ab - \\left \\lfloor \\left( \\frac{a+b-k}{2} \\right)^2 \\right\\rfloor = \\delta(a,b,k)$ (by the formula in Remark \\ref{rem_threecases}). Since all the boxes of $S$ must have distinct labels, it follows that $\\cd(a,b,k) \\geq \\delta(a,b,k)$.\n\\end{proof}\n\n\\begin{lemma} \\label{l_cdupper}\nFor any positive integers $a,b,k$ with $k \\geq 2$,\n$$\n\\cd(a,b,k) \\leq \\delta(a,b,k).\n$$\n\\end{lemma}\n\\begin{proof}\nWe assume without loss of generality that $a \\leq b$. The case $a=1$ can be checked directly (both $\\cd(1,b,k)$ and $\\delta(1,b,k)$ are equal to $b$), so we also assume that $a \\geq 2$.\n\nWe give an explicit construction for $k$-uniform displacement tableaux with exactly $\\delta(a,b,k)$ symbols.\n\n\\textit{Case 1:} $k \\geq a+b-1$. In this case, $\\delta(a,b,k) = ab$, so we can take $t$ to be any tableau with all distinct symbols.\n\n\\textit{Case 2:} $k \\leq a+b-2$. Define $\\ell = \\min \\left( a-1,\\ \\left\\lceil \\frac{a+b-k}{2} \\right\\rceil \\right)$; this value is chosen so that $\\delta(a,b,k) = (a-\\ell)(b-\\ell) + k \\ell$. Note that our assumptions imply that $\\ell \\geq 1$. We will construct a tableau $t$ on $\\lambda$ with exactly $(a-\\ell)(b-\\ell) + k \\ell$ distinct symbols, which will establish that $\\cd(a,b,k) \\leq \\delta(a,b,k)$.\n\n\\textit{Definition of $t$}. Informally, $t$ will be defined by first dividing the partition $\\lambda$ into strips of width $1$ and height $a-\\ell$ (with the top row of strips truncated to a smaller height), and filling each strip with consecutive numbers from bottom to top. In each row of strips, we fill the strips in order, from left to right. In order to minimize the distinct symbols used, we begin filling each row of strips before the previous row is complete: the first strip of one row is filled simultaneously with the $(k+\\ell-a+1)$th strip of the row below.\n\nFormally, define first a function $t: \\lambda \\rightarrow \\textbf{Z}_{>0}$ as follows.\n\n\\begin{eqnarray*}\nt(x,y) &=& (a-\\ell) \\cdot \\left[ (x-1) + q (k+\\ell-a) \\right] + r,\\\\\n\\mbox{where } q &=& \\left\\lfloor \\frac{y-1}{a-\\ell} \\right\\rfloor\\\\\n\\mbox{and } r &=& y - q \\cdot (a-\\ell).\n\\end{eqnarray*}\n\nAn example is shown in Figure \\ref{fig_construction}.\n\n\\begin{figure}\n\\begin{tikzpicture}[scale=0.7]\n\\draw (0,0) -- (7,0) -- (7,7) -- (0,7) -- (0,0);\n\\draw (1,0) -- (1,7);\n\\draw (2,0) -- (2,7);\n\\draw (3,0) -- (3,7);\n\\draw (4,0) -- (4,7);\n\\draw (5,0) -- (5,7);\n\\draw (6,0) -- (6,7);\n\\draw (0,3) -- (7,3);\n\\draw (0,6) -- (7,6);\n\\draw[fill=lightgray] (0,6) rectangle (1,7);\n\\draw[fill=lightgray] (3,3) rectangle (4,6);\n\\draw[fill=lightgray] (6,0) rectangle (7,3);\n\\draw (0.500000,0.500000) node {$1$};\n\\draw (0.500000,1.500000) node {$2$};\n\\draw (0.500000,2.500000) node {$3$};\n\\draw (0.500000,3.500000) node {$10$};\n\\draw (0.500000,4.500000) node {$11$};\n\\draw (0.500000,5.500000) node {$12$};\n\\draw (0.500000,6.500000) node {$19$};\n\\draw (1.500000,0.500000) node {$4$};\n\\draw (1.500000,1.500000) node {$5$};\n\\draw (1.500000,2.500000) node {$6$};\n\\draw (1.500000,3.500000) node {$13$};\n\\draw (1.500000,4.500000) node {$14$};\n\\draw (1.500000,5.500000) node {$15$};\n\\draw (1.500000,6.500000) node {$22$};\n\\draw (2.500000,0.500000) node {$7$};\n\\draw (2.500000,1.500000) node {$8$};\n\\draw (2.500000,2.500000) node {$9$};\n\\draw (2.500000,3.500000) node {$16$};\n\\draw (2.500000,4.500000) node {$17$};\n\\draw (2.500000,5.500000) node {$18$};\n\\draw (2.500000,6.500000) node {$25$};\n\\draw (3.500000,0.500000) node {$10$};\n\\draw (3.500000,1.500000) node {$11$};\n\\draw (3.500000,2.500000) node {$12$};\n\\draw (3.500000,3.500000) node {$19$};\n\\draw (3.500000,4.500000) node {$20$};\n\\draw (3.500000,5.500000) node {$21$};\n\\draw (3.500000,6.500000) node {$28$};\n\\draw (4.500000,0.500000) node {$13$};\n\\draw (4.500000,1.500000) node {$14$};\n\\draw (4.500000,2.500000) node {$15$};\n\\draw (4.500000,3.500000) node {$22$};\n\\draw (4.500000,4.500000) node {$23$};\n\\draw (4.500000,5.500000) node {$24$};\n\\draw (4.500000,6.500000) node {$31$};\n\\draw (5.500000,0.500000) node {$16$};\n\\draw (5.500000,1.500000) node {$17$};\n\\draw (5.500000,2.500000) node {$18$};\n\\draw (5.500000,3.500000) node {$25$};\n\\draw (5.500000,4.500000) node {$26$};\n\\draw (5.500000,5.500000) node {$27$};\n\\draw (5.500000,6.500000) node {$34$};\n\\draw (6.500000,0.500000) node {$19$};\n\\draw (6.500000,1.500000) node {$20$};\n\\draw (6.500000,2.500000) node {$21$};\n\\draw (6.500000,3.500000) node {$28$};\n\\draw (6.500000,4.500000) node {$29$};\n\\draw (6.500000,5.500000) node {$30$};\n\\draw (6.500000,6.500000) node {$37$};\n\\end{tikzpicture}\n\\caption{The construction in the proof of lemma \\ref{l_cdupper}, in the case $a=7,\\ b=7,\\ k=6$. The tableau is constructed by filling $3 \\times 1$ strips in sequence. Although the labels of the tableau go up to $37$, there are four values that do not occur ($32,33,35,36$), so there are exactly $33$ distinct labels.} \\label{fig_construction}\n\\end{figure}\n\n\\textit{Validity of $t$}. We first verify that this function does in fact give a $k$-uniform displacement tableau. Clearly $t(x+1,y) > t(x,y)$. The value $t(x,y+1)-t(x,y)$ is either $1$ (if $y \\not \\equiv 0 \\pmod{a-\\ell}$), or $(a-\\ell)(k+\\ell-a) - (a-\\ell-1)$ otherwise. The latter quantity is equal to $(a - \\ell)( k + \\ell - a -1) + 1$, which is positive since $k + \\ell -a - 1 \\geq 0$ and $\\ell \\leq a-1$ (these follow from our choice of the number $\\ell$). Hence $t(x,y)$ is strictly increasing in rows and columns; it remains to show that any two boxes $(x,y)$ with the same label in $t$ have values of $x-y$ that are congruent modulo $k$.\n\nSince $1 \\leq r \\leq a - \\ell$, two boxes satisfy $t(x,y) = t(x',y')$ if and only if $r = r'$ and $(x-1) + q(k+\\ell-a) = (x'-1) + q'(k+\\ell-a)$ (where $q',r'$ are obtained from $y'$ in the same manner as $q,r$ are obtained from $y$). The latter equation implies that \n$$\nx - q (a - \\ell) \\equiv x' - q'(a-\\ell) \\pmod{k},\n$$\nand subtracting $r = r'$ from both sides gives $x-y \\equiv x' - y' \\pmod{k}$. So $t$ is indeed a $k$-uniform displacement tableau.\n\n\\textit{Number of symbols in $t$.} We now enumerate the number of distinct symbols that occur in $t$ on $\\lambda$. The primary difficulty in this computation is that, as we have constructed it, the symbols of $t$ are not consecutive; some symbols are skipped.\n\nEvery value $t(x,y)$ for $(x,y) \\in \\lambda$ has the form $(a - \\ell) B + r$, where $1 \\leq r \\leq a - \\ell$; clearly two such values are equal if and only if they arise from equal values of $B$ and $r$, so it suffices to enumerate the number of distinct pairs $(B,r)$ which occur.\n\nThe number $B$ is equal to $(x-1) + q(k + \\ell - a)$. The number $q$ depends on $y$ alone; its possible values are the integers $0$ through $\\left\\lfloor \\frac{a-1}{a - \\ell} \\right\\rfloor$, inclusive. Note that the largest possible value of $q$ is at least $1$, since our assumptions imply that $\\ell \\geq 1$. For each possible value of $q$ in $\\{0,1,\\cdots, \\left\\lfloor \\frac{a-1}{a - \\ell} \\right\\rfloor - 1\\}$, the possible values of $y$ giving this value of $q$ give all possible values of $r$ (from $1$ to $a - \\ell$). Since $x-1$ takes all values from $0$ to $b-1$ inclusive, this shows that all pairs $(B,r)$ satisfying the following inequalities will occur.\n\n\\begin{eqnarray*}\n0 &\\leq B \\leq& b-1 + \\left( \\left\\lfloor \\frac{a-1}{a - \\ell} \\right\\rfloor - 1 \\right) (k + \\ell -a)\\\\\n1 &\\leq r \\leq& a - \\ell\n\\end{eqnarray*}\n\nThis accounts for \n\\begin{equation} \\label{monster1}\n(a - \\ell) \\left( b + \\left( \\left\\lfloor \\frac{a-1}{a - \\ell} \\right\\rfloor - 1 \\right) (k + \\ell -a) \\right) \n\\end{equation}\ndistinct values of $t(x,y)$.\n\nAll of the remaining values of $B$ must occur for the largest possible value of $q$, namely $q = \\left\\lfloor \\frac{a-1}{a-\\ell} \\right\\rfloor$. For this value of $q$, the possible values of $r$ range from $1$ to $a - \\left\\lfloor \\frac{a-1}{a-\\ell} \\right\\rfloor (a - \\ell)$, inclusive. For this value of $q$, the value of $B$ is distinct from those listed above if and only if $x$ is taken to be one of the $(k+\\ell-a)$ largest values from the range $\\{1,2,\\cdots,b\\}$ (here we use the fact that $k+\\ell-a \\leq b$, which follows from the assumption $a+b-k \\geq 2$ and the choice of $\\ell$). Therefore the number of values of $t(x,y)$ not accounted for is precisely\n\n\\begin{equation} \\label{monster2}\n\\left( a - \\left\\lfloor \\frac{a-1}{a-\\ell} \\right\\rfloor (a - \\ell) \\right) (k + \\ell - a).\n\\end{equation}\n\nAdding the quantities (\\ref{monster1}) and (\\ref{monster2}) and rearranging terms gives the quantity $(a-\\ell)(b-\\ell) + k \\ell$, which is therefore the number of distinct symbols in $t$. It follows that $\\cd(a,b,k) \\leq (a-\\ell)(b-\\ell) + k \\ell$, as desired.\n\\end{proof}\n\nLemmas \\ref{l_cdlower} and \\ref{l_cdupper} together prove Proposition \\ref{p_cd}.\n\n\\begin{cor} \\label{cor_tropdim}\nIf $k$ and $\\ell$ are relatively prime, then the dimension of $W^r_d(\\Gamma_{g,k,\\ell})$ is equal to $\\overline{\\rho}_{g,k}(d,r)$.\n\\end{cor}\n\\begin{proof}\nBy Riemann-Roch, we may assume without loss of generality that $d \\leq g-1$. Then $\\dim W^r_d(\\Gamma_{g,k,\\ell}) = g - \\cd(r+1,g-d+r,k)$ by Lemma \\ref{l_td}, which is $g- \\delta(r+1,g-d+r,k)$ by Proposition \\ref{p_cd}. By definition, this is equal to $\\overline{\\rho}_{g,k}(d,r)$.\n\\end{proof}\n\n\\section{Proofs of main theorems} \\label{sec_proofs}\n\nWe can now combine our results from the previous two sections to prove the theorems from the introduction. We assume that Situation \\ref{sit_main} holds throughout, i.e. integers $g,k,r,d$, a field $K$, and a general $k$-gonal curve $C$ are fixed, with the characteristic of $K$ meeting our mild restrictions. Before each proof we briefly recall the theorem statement.\n\nOur main result is Theorem \\ref{thm_main}, which asserts that in Situation \\ref{sit_main}, $\\overline{\\rho}_{g,k}(d,r)$ is an upper bound for $\\dim W^r_d(C)$.\n\n\\begin{proof}[Proof of theorem \\ref{thm_main}]\nWe first prove that the result holds when $K$ is a complete algebraically closed non-Archimedean field with nontrivial valuation. By Lemma \\ref{lem_lexists}, there exists an integer $\\ell$ such that $(\\chr K, k, \\ell)$ is admissible. By Lemma \\ref{lem_lift} and Corollary \\ref{cor_punchline}, there exists a smooth projective curve $C'$ over $K$ such that for all $r,d$,\n$$\n\\dim W^r_d(C') \\leq g - \\cd(g-d+r,r+1,k),\n$$\nwhere $\\cd$ is the function from Definition \\ref{def_cd}. Proposition \\ref{p_cd}, together with the fact that $\\overline{\\rho}_{g,k}(d,r) = g - \\delta(r+1,g-d+r,k)$, gives the inequality $\\dim W^r_d(C') \\leq \\overline{\\rho}_{g,k}(d,r)$.\n\nIn particular, letting $r=1$, these bounds show that $W^1_d(C')$ is empty for all $d < k$ (the inequalities $d \\leq k-1$ and $k \\leq \\frac{g+3}{2}$ imply that $\\overline{\\rho}_{g,k}(d,1) = d-k \\leq -1$). So the gonality of $C'$ is at least $k$. By construction, $C'$ does have a degree $k$ line bundle with two linearly independent sections, so the gonality of $C'$ is exactly $k$. By the irreducibility of the $k$-gonal locus and upper semicontinuity, it follows that these upper bounds on dimension hold on a dense open subset of the set of $k$-gonal curves over the field $K$. Hence $\\dim W^r_d(C) \\leq \\overline{\\rho}_{g,k}(d,r)$ for a \\textit{general} $k$-gonal curve $C$ of genus $g$ over $K$.\n\n\\textit{Extension to arbitrary algebraically closed fields.} So far we have deduced the theorem only for a specific type of field. To obtain the result for all fields, observe that the coarse moduli space of curves of genus $g$ is a scheme of finite type over $\\Spec \\textbf{Z}$, and the locus of $k$-gonal curves for which the theorem holds is a locally closed subscheme whose image in $\\Spec \\textbf{Z}$ includes the generic point and $\\Spec \\textbf{F}_p$ for all $p$ not excluded by Situation \\ref{sit_main}. Therefore it has points in every algebraically closed field of characteristic not excluded by Situation \\ref{sit_main}.\n\\end{proof}\n\nTheorem \\ref{thm_cm} states that, under a characteristic $0$ hypothesis, $\\underline{\\rho}_{g,k}(d,r)$ is a lower bound for $\\dim W^r_d(C)$; it follows from the main theorem of \\cite{cm02} by an elementary argument. We give this elementary argument below for completeness. First, we restate the main theorem of \\cite{cm02} in the notation of this paper.\\\\\n\n\\noindent\\textbf{Main theorem of \\cite{cm02}}. \\textit{\nLet $C$ be a general $k$-gonal curve of genus $g$ over the complex numbers, and let $r,d$ be positive integers. Let $\\ell \\in \\{0,1,\\cdots,r\\}$ be an integer satisfying the following hypotheses.\n\\begin{enumerate}\n\\item $\\ell \\geq r- k$,\n\\item $r+1 - \\ell$ divides either $r$ or $r+1$, and\n\\item $\\rho_g(d,r-\\ell) - \\ell k \\geq \\max(0,\\rho_g(d,r))$.\n\\end{enumerate}\nThen $W^r_d(C)$ has an irreducible component of dimension $\\rho_g(d,r-\\ell) - \\ell k$.\n}\n\n\\begin{proof}[Proof of Theorem \\ref{thm_cm}]\nNote first of all that to prove this results for all algebraically closed fields of characteristic $0$, it suffices to verify the result for the field $K = \\textbf{C}$ of complex numbers, by standard arguments (e.g. as in the last paragraph of the proof of Theorem \\ref{thm_main}). We use the notation of Situation \\ref{sit_main}. By Riemann-Roch, we may assume without loss of generality that $d \\leq g-1$, so that $r' = r$ in the notation of Definition \\ref{def_ovr}. Assume also that $\\underline{\\rho}_{g,k}(d,r) \\geq 0$ and $\\underline{\\rho}_{g,k}(d,r) > \\rho_g(d,r)$, since otherwise the desired result is either vacuous or follows from $\\dim W^r_d(C) \\geq \\rho_g(d,r)$. Note that this implies $r > 0$.\n\nLet $\\ell$ be whichever integer $\\ell \\in \\{0,1,r-1,r\\}$ is closest to $\\ell_0 = \\frac{g-d+2r - k + 1}{2}$, where we choose the largest value in case of a tie. In this case $\\rho_g(d,r-\\ell) - \\ell k = \\underline{\\rho}_{g,k}(d,r)$. So it suffices to verify hypotheses (1), (2), and (3) in the theorem statement above.\n\nHypothesis (3) follows from our assumptions. Hypothesis (2) holds for all $\\ell \\in \\{0,1,r-1,r\\}$ for elementary reasons.  It remains to verify hypothesis (1). If $\\ell \\in \\{r-1,r\\}$ then hypothesis (1) follows from $k \\geq 2$. We are assuming that $\\underline{\\rho}_{g,k}(d,r) > \\rho_g(d,r)$, hence $\\ell \\neq 0$; it remains to consider the case $\\ell = 1$. We may assume that $r \\geq 3$, otherwise either $r=0$, $\\ell = 1 = r-1$ or $\\ell = 1 = r$. Since $\\ell = 1$ must be closer to $\\ell_0$ than $r-1$ (which is strictly greater than $1$), it follows that $\\ell_0 < \\frac12 r$. From the definition of $\\ell_0$, this implies that $r < k + d - g - 1$. We are assuming that $d < g$, hence $r < k - 1$. So certainly $1 \\geq r - k$ in this case, and hypothesis (1) holds.\n\nTherefore all three hypotheses hold, and the main theorem of \\cite{cm02} implies that $W^r_d(C)$ has a component of dimension $\\underline{\\rho}_{g,k}(d,r)$.\n\\end{proof}\n\nTheorem \\ref{thm_eq} asserts that, in characteristic $0$, when $k \\leq 5$ or $k \\geq \\frac15g+2$, the dimension of $W^r_d(C)$ is exactly $\\overline{\\rho}_{g,k}(d,r)$ (unless this number is negative, in which case $W^r_d(C)$ is empty). The proof is elementary, given Theorems \\ref{thm_main} and \\ref{thm_cm}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm_eq}]\nWe must show that if $k \\leq 5$ or $k \\geq \\frac15 g + 2$, then either $\\underline{\\rho}_{g,k}(d,r) = \\overline{\\rho}_{g,k}(d,r)$ or $\\overline{\\rho}_{g,k}(d,r) < 0$. We assume without loss of generality that $d \\leq g-1$.\n\nSuppose that $\\underline{\\rho}_{g,k}(d,r) \\neq \\overline{\\rho}_{g,k}(d,r)$. By the discussion in \\ref{ss_gap},\n$$\n2 \\leq \\frac{g-d+2r-k+1}{2} \\leq r-2.\n$$\n\n\nDefine two auxiliary variables, as follows.\n\\begin{eqnarray*}\n\\ell_0 &=& \\frac{g-d+2r-k+1}{2}\\\\\n\\delta &=& \\frac{g-d-1}{2}\n\\end{eqnarray*}\n\nWe deduce from the inequalities above that $\\ell_0 \\geq 2$ and $\\delta \\leq \\frac{k}{2} - 3$.  An elementary (but tedious) rearrangement of terms shows that, for any integer $\\ell$,\n$$\n\\rho_g(d,r-\\ell) -k \\ell = g - (\\ell - \\ell_0)^2 - \\left( \\frac{k}{2} \\right)^2 - k \\ell_0 + \\delta^2.\n$$\nThe maximum value occurs at $\\ell = \\lfloor \\ell_0 \\rfloor$, hence\n\\begin{eqnarray*}\n\\overline{\\rho}_{g,k}(d,r) &=& \\rho_g(d,r-\\lfloor \\ell_0 \\rfloor) - k \\lfloor \\ell_0 \\rfloor\\\\\n&\\leq& g - (\\lfloor \\ell_0 \\rfloor - \\ell_0)^2 - \\left( \\frac{k}{2} \\right)^2 - k \\ell_0 + \\delta^2\\\\\n&\\leq& g - 0 - \\left( \\frac{k}{2} \\right)^2 - k \\cdot 2 + \\left( \\frac{k}{2} - 3 \\right)^2\\\\\n&\\leq& g - 5k + 9.\n\\end{eqnarray*}\nIn the case $k > \\frac15 g + 2$, it follows that $\\overline{\\rho}_{g,k}(d,r) \\leq -1$. In the case $k \\leq 5$, the inequality $\\delta \\leq \\frac{k}{2} -3$ would imply that $\\delta < 0$, which contradicts the assumption that $d \\leq g-1$. Therefore we conclude that in either case, $\\underline{\\rho}_{g,k}(d,r) = \\overline{\\rho}_{g,k}(d,r)$ unless $\\overline{\\rho}_{g,k}(d,r) < 0$. The result now follows from Theorems \\ref{thm_main} and \\ref{thm_cm}.\n\\end{proof}\n\nTheorem \\ref{thm_howgeneral} characterizes, in characteristic $0$, those values of $d,r$ for which a general $k$-gonal curve of genus $g$ satisfies $\\dim W^r_d(C) = \\rho_g(d,r)$ (i.e. the same as for a general curve of genus $g$). This occurs if and only if $r=0,\\ g-d+r = 1,$ or $g-k \\leq d-2r$. \n\n\\begin{proof}[Proof of Theorem \\ref{thm_howgeneral}]\nThe $r=0$ case (and dually, the $g-d+r = 1$ case) follow from the fact that the $d$th symmetric product of $C$ surjects onto $W^0_d(C)$, so assume that $r \\geq 1$ and $g-d+r \\geq 2$. Let $\\ell_0$ be as in Section \\ref{ss_gap}. Then $\\ell_0 \\leq \\frac12$ is equivalent to $g-k \\leq d-2r$; the discussion in Section \\ref{ss_gap} shows that in this case $\\dim W^r_d(C) = \\rho_g(d,r)$. For the converse, suppose that $g-k > d - 2r$; equivalently, $\\ell_0 \\geq 1$. Then $\\rho_g(d,r-1) - k > \\rho_g(d,r)$, hence $\\underline{\\rho}_{g,k}(d,r) > \\rho_g(d,r)$. By Theorem \\ref{thm_cm}, $\\dim W^r_d(C) > \\rho_g(d,r)$, as desired.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nFlash memory is a non-volatile technology that is both electrically\nprogrammable and electrically erasable. It incorporates a set of cells\nmaintained at a set of levels of charge to encode information.\nWhile raising the charge level of a cell is an easy operation,\nreducing the charge level requires the erasure of the whole block to which the cell belongs.\nFor this reason charge is injected into the cell over several iterations.\nSuch programming is slow and can cause errors since cells may be injected with extra unwanted charge.\nOther common errors in flash memory cells are due to charge leakage and reading disturbance that\nmay cause charge to move from one cell to its adjacent cells.\nIn order to overcome these problems, the novel framework\nof \\emph{rank modulation} was introduced in~\\cite{JMSB}.\nIn this setup the information is carried by the relative ranking of the\ncells' charge levels and not by the absolute values of the charge levels.\nThis allows for more efficient programming of cells, and coding by the ranking of the cells' levels\nis more robust to charge leakage than coding by their actual values.\nIn this model codes are subsets of~$S_n$,\nthe set of all permutations on $n$ elements,\nand the codewords are members of $S_n$,\nwhere each permutation corresponds to a ranking of $n$ cells' levels from the highest one to the lowest.\nFor example, the charge levels $(c_1,c_2,c_3,c_4)=(5,1,3,4)$ are represented by the codeword $[1,4,3,2]$\nsince the first cell has the highest level, the forth cell has the next highest level and so on.\n\nTo detect and\/or correct errors caused by injection of extra charge or due\nto charge leakage we will use an appropriate\ndistance measure.\nSeveral metrics on permutations are used for this purpose. In this\npaper we will consider only the Kendall's $\\tau$-metric~\\cite{JSB10,KeGi90}.\nThe Kendall's $\\tau$-distance between two permutation $\\pi_1$ and $\\pi_2$ in $S_n$\nis the minimum adjacent transpositions required to obtained $\\pi_2$ from $\\pi_1$, where adjacent transposition is an exchange of two distinct adjacent elements.\nFor example, the Kendall's $\\tau$-distance between $\\pi_1=[2,1,4,3]$ and $\\pi_2=[2,4,3,1]$ is $2$ as\n$[2,1,4,3]\\to [2,4,1,3] \\to [2,4,3,1]$.\nTwo permutations in this metric are at distance\none if they differ in exactly one pair of adjacent elements.\nDistance one between these two permutations represent an exchange of two\ncells, which are adjacent in the permutation, due to a small change in\ntheir charge level which change their order.\n\nGray codes are very important in the context of rank modulation\nas was explained in~\\cite{JMSB}. They are used in many other applications,\ne.g.~\\cite{EtPa96,SaWi95}. An excellent survey on Gray codes is given in~\\cite{Sav97}.\nThe usage of Gray codes for\nrank modulation was also discussed in~\\cite{GLSB11,GLSB13,JMSB,YeSc12}.\nThe permutations of $S_n$ in the rank modulation scheme\nrepresent \"new\" logical levels of the flash memory. The codewords in\nthe Gray code provide the order of these levels which should be implemented\nin various algorithms with the rank modulation scheme.\nUsually, a Gray code is just a simple cycle in a graph, in which the edges are defined between\nvertices with distance one in a given metric. Two adjacent\nvertices in the graph represent on one hand two elements\nwhose distance is one by the given metric; and on\nthe other hand a move from a vertex to a vertex implied\nby an operation defined by the metric. A snake-in-the-box code\nis a Gray code in which two elements in the code are not\nadjacent in the graph, unless they are consecutive in the\ncode. Such a Gray code can detect a single error in a codeword.\nSnake-in-the-box codes were mainly discussed in the context of\nthe Hamming scheme, e.g.~\\cite{AbKa88}.\n\nIn the rank modulation scheme the Gray code is defined slightly\ndifferent since the operation is not defined by a metric.\nThe permutation is defined by the\norder of the charge levels, from the highest one to the lowest one.\nFrom a given ranking of the charge levels, which defines a permutation,\nthe next ranking is obtained by raising\nthe charge level of one of the cells to be the highest level.\nThis operation, called \"push-to-the-top\", is used in the rank modulation scheme.\nFor example, the charge levels $(c_1,c_2,c_3,c_4)=(5,1,3,4)$ are represented by the codeword $[1,4,3,2]$,\nand by applying push-to-the-top operation on the second cell which has the lowest charge level,\nwe have, for example, the charge levels $(c_1,c_2,c_3,c_4)=(5,6,3,4)$ which are represented by the codeword $[2,1,4,3]$.\nHence, the permutation $\\pi_2$ can follow the permutation $\\pi_1$\nif $\\pi_2$ is obtained from $\\pi_1$ by applying a push-to-the-top operation on $\\pi_1$.\nTherefore, the related graph is directed with an outgoing edge from the vertex\nwhich represents $\\pi_1$ into the vertex which represents $\\pi_2$.\nOn the other hand, one possible metric for the scheme is\nthe Kendall's $\\tau$-metric.\nA Gray code\n(and a snake-in-the-box code as a special case)\nrelated to the rank modulation scheme is a directed simple cycle in the graph.\nIn a snake-in-the-box code, related to this scheme, there is another requirement that\nthe Kendall's $\\tau$-distance between any two codewords\nis at least two, including consecutive codewords.\nFor example,\n$C=([1,2,3,4],\\ [4,1,2,3],\\ [2,4,1,3],\\ [3,2,4,1],\\ [4,3,2,1], \\newline [1,4,3,2],\\ [3,1,4,2],\\ [2,3,1,4])$\nis a snake-in-the-box code in $S_4$\nobtained by applying a push-to-the-top operation on the lowest cell at each time.\nThe Kendall's $\\tau$-distance between any two permutations in $C$ is at least~$2$.\n\nOne of the most important problems in the research on\nsnake-in-the-box codes is to construct the\nlargest possible code for the given graph. In a snake-in-the-box\ncode for the rank modulation scheme\nwe would like to find such a code with the largest number of permutations.\nIn a recent paper by Yehezkeally and Schwartz~\\cite{YeSc12}, the authors\nconstructed a snake-in-the-box code of length\n${M_{2n+1}=(2n+1)(2n-1) M_{2n-1}}$ for permutations of $S_{2n+1}$,\nfrom a snake of length $M_{2n-1}$ for permutations of~$S_{2n-1}$.\nWe will improve on this result by constructing a snake of length\n$M_{2n+1}=((2n+1)2n-1) M_{2n-1}$ for permutations of $S_{2n+1}$,\nfrom a snake of length $M_{2n-1}$ for permutations of~$S_{2n-1}$.\nThus, we have\n$\\lim\\limits_{n\\to \\infty} \\frac{M_{2n+1}}{S_{2n+1}}\\approx 0.4338$,\nimproving on the previous known ratio of\n$\\lim\\limits_{n\\to \\infty} \\frac{1}{\\sqrt{\\pi n}}$ \\cite{YeSc12}.\nFor these constructions of snake-in-the-box codes we\nneed an initial snake-in-the-box code and the largest one known\nto start both constructions is a snake of length 57 for\npermutations of $S_5$. We also propose a direct construction\nto form a snake of length\n$\\frac{(2n+1)!}{2} -2n+1$ for permutations\nof $S_{2n+1}$. The direct construction was applied successfully for\n$S_7$ and $S_9$. This implies better initial condition for the\nrecursive constructions, and the ratio $\\lim\\limits_{n\\to \\infty} \\frac{M_{2n+1}}{S_{2n+1}}\\approx 0.4743$.\n\nThe rest of this paper is organized as follows.\nIn Section~\\ref{sec:pre} we will define the basic concepts\nof Gray codes in the rank modulation\nscheme, the push-to-the-top operation,\nand the Kendall's $\\tau$-metric required in this paper.\nIn Section~\\ref{sec:main} we present the main ideas\nand a framework for constructions of snake-in-the-box codes.\nIn Section~\\ref{sec:recursive} we present a recursive construction\nbased on the given framework. This construction is used to obtain\nsnake-in-the-box codes longer than the ones known before.\nIn~Section~\\ref{sec:direct}, based on the framework,\nwe present an idea for a\ndirect construction based on necklaces.\nThe construction is used to obtain snake-in-the-box codes\nof length $\\frac{(2n+1)!}{2} -2n+1$ in $S_{2n+1}$,\nwhich we believe are optimal. The construction was\napplied successfully on $S_7$ and on $S_9$,\nand we conjecture that it can be applied\non $S_n$ for any odd $n > 6$.\nConclusions and problems for future research are\npresented in Section~\\ref{sec:conclude}.\n\n\\section{Preliminaries}\n\\label{sec:pre}\n\nIn this section we will repeat some notations defined and mentioned\nin~\\cite{YeSc12}, and we also present some other definitions.\n\nLet $[n]\\triangleq\\{1, 2, \\ldots , n\\}$\nand let $\\pi= [a_1, a_2, \\ldots , a_n]$ be a permutation\nover $[n]$, i.e., a permutation in $S_n$, such that for each $i\\in [n]$ we have that $\\pi(i)=a_i$.\n\nGiven a set $\\cS$ and a subset of transformations\n$T\\subseteq \\{f|f:\\cS\\to \\cS\\}$, a \\emph{Gray code}\nover $\\cS$ of size $M$, using transitions from\n$T$, is a sequence $C=(c_0, c_1, \\ldots , c_{M-1})$ of\n$M$ distinct elements from $\\cS$, called \\emph{codewords}, such that for each\n$j\\in[M-1]$ there exists a $t\\in T$ for which $c_j=t(c_{j-1})$.\nThe Gray code is called \\emph{complete} if $M=|\\cS|$,\nand \\emph{cyclic} if there exists $t\\in T$ such that $c_0=t(c_{M-1})$.\nThroughout this paper we will consider only cyclic Gray codes.\n\n\nIn the context of rank modulation for flash memories,\n$\\cS=S_n$ and the set of transformations $T$ comprises of push-to-the-top\noperations. We denote by $t_i$ the \\emph{push-to-the-top}\noperation on index $i$, $2 \\leq i \\leq n$, defined by\n\\begin{align*}\nt_i(&[a_1,\\ldots,a_{i-1},a_i,a_{i+1},\\ldots,a_n])=\\\\\n&[a_i,a_1,\\ldots,a_{i-1},a_{i+1},\\ldots,a_n].\n\\end{align*}\nand a \\emph{p-transition} will be an abbreviated notation for a push-to-the-top operation.\n\nA sequence of p-transitions will be called a \\emph{transitions sequence}.\nA permutation $\\pi_0$ and a transitions sequence\n$t_1 , t_2, \\ldots t_\\ell$ define a sequence of permutations\n$\\pi_0 , \\pi_1 , \\pi_2 ,\\ldots , \\pi_{\\ell-1} ,\\pi_\\ell$,\nwhere $\\pi_i = t_i (\\pi_{i-1})$, for each $i$, $1 \\leq i \\leq \\ell$. This\nsequence is a cyclic Gray code, if $\\pi_\\ell = \\pi_0$\nand for each $0 \\leq i < j < \\ell$, $\\pi_i \\neq \\pi_j$.\nIn the sequel the word cyclic will be omitted\n\nGiven a permutation $\\pi=[a_1,a_2,\\ldots,a_n] \\in S_n$,\nan \\emph{adjacent transposition} is an exchange of two distinct adjacent elements\n$a_i,a_{i+1}$, in $\\pi$, for some $1\\leq i\\leq n-1$.\nThe result of such an adjacent transposition is the permutation\n$[a_1,\\ldots,a_{i-1},a_{i+1},a_i,a_{i+2},\\ldots,a_n]$.\nThe \\emph{Kendall's} $\\tau$-\\emph{distance}~\\cite{KeGi90} between two permutations\n$\\pi_1,\\pi_2 \\in S_n$ denoted by $d_K (\\pi_1,\\pi_2)$\nis the minimum number of adjacent transpositions\nrequired to obtain the permutation $\\pi_2$ from the permutation $\\pi_1$.\nA \\emph{snake-in-the-box code} is a Gray code in which for each two\npermutations $\\pi_1$ and $\\pi_2$ in the code we have\n$d_K(\\pi_1,\\pi_2) \\geq 2$.\nHence, a snake-in-the-box code is a Gray code capable of detecting\none Kendall's $\\tau$-error. We will call such a snake-in-the-box\ncode a $\\mK$\\emph{-snake}.\nWe further denote by $(n,M,\\mK)$\\emph{-snake} a $\\mK$-snake of size $M$\nwith permutations from $S_n$. A $\\mK$-snake can be\nrepresented in two different equivalent ways:\n\\begin{itemize}\n{\\setlength\\itemindent{-9pt}\n\\item the sequence of codewords (permutations),\n\\item the transitions sequence along with the first permutation.}\n\\end{itemize}\n\nLet $\\cT$ be a transitions sequence and let $\\pi$ be a permutation in $S_n$.\nIf a $\\mK$-snake is obtained by applying $\\cT$ on $\\pi$\nthen a $\\mK$-snake will be obtained by using any other\npermutation from $S_n$ instead of $\\pi$. This is a simple observation\nfrom the fact that $t( \\pi_2 ( \\pi_1 )) = \\pi_2 (t(\\pi_1))$,\nwhere $t$ is a p-transition and $\\pi_2 (\\pi_1)$ refers to applying the\npermutation $\\pi_2 \\in S_n$ on the permutation $\\pi_1 \\in S_n$.\nIn other words applying $\\cT$ on a different permutation\njust permute the symbols, by a fixed given permutation,\nin all the resulting permutations when $\\cT$ is applied on $\\pi$. Therefore,\nsuch a transitions sequence~$\\cT$ will be called an \\emph{S-skeleton}.\n\nFor a transitions sequence ${\\sigma=t_{k_1}, t_{k_2}, \\ldots t_{k_{\\ell}}}$\nand a permutation $\\pi \\in S_n$,\nwe denote by $\\sigma\\left(\\pi\\right)$, the permutation obtained by applying\nthe sequence of p-transitions in $\\sigma$ on $\\pi$,\ni.e., $t_{k_1}$ is applied on $\\pi$, $t_{k_2}$ is applied on $t_{k_1} (\\pi)$, and so on.\nIn other words, $\\sigma\\left(\\pi\\right)= (t_{k_1}\\circ t_{k_2}\\circ \\ldots \\circ t_{k_{\\ell}})(\\pi)=\nt_{k_{\\ell}}\\left(t_{k_{l-1}}\\left(\\ldots t_{k_2}\\left(t_{k_1}\\left(\\pi\\right)\\right)\\right)\\right)$.\nLet $\\sigma_1,\\sigma_2$ be two transitions sequences.\nWe say that $\\sigma_1$ and  $\\sigma_2$ are \\emph{matching sequences},\nand denote it by $\\sigma_1 \\leftrightsquigarrow \\sigma_2$,\nif for each ${\\pi\\in S_n}$ we have $\\sigma_1(\\pi)=\\sigma_2(\\pi)$.\n\n\nIn~\\cite{YeSc12} it was proved that a Gray code with permutations from $S_n$ using only\np-transitions on odd indices is a $\\mK$-snake. By starting with an even permutation and using only\np-transitions on odd indices we get a sequence\nof even permutations, i.e., a subset of the $A_n$, the alternating group of order $n$.\nThis observation saves us the need to check whether a Gray code is\nin fact a $\\mK$-snake, at the cost of restricting the\npermutations in the $\\mK$-snake to the set of even permutations.\nHowever, the following assertions were also proved in~\\cite{YeSc12}.\n\\begin{itemize}\n\\item If $C$ is an $(n,M,\\mK)$-snake then $M\\leq \\frac{{|S_n|}}{2}$.\n\\item If $C$ is an $(n,M,\\mK)$-snake which contains a p-transition\non an even index then ${M\\leq{\\frac{{|S_n|}}{2}-\\frac{1}{n-1}\\binom{\\lfloor n\/2 \\rfloor -1}{2}}}$.\n\\end{itemize}\nThis motivates not to use p-transitions on even indices.\nSince we will use only p-transitions on odd indices,\nwe will describe our constructions only for even permutations with odd length.\n\n\n\n\\section{Framework for Constructions of $\\mK$-Snakes}\n\\label{sec:main}\n\nIn this section we present a framework for constructing $\\mK$-snakes in $S_{2n+1}$.\nOur snakes will contain only even permutations. We start by partitioning\nthe set of even permutations of $S_{2n+1}$ into classes. Next, we describe\nhow to merge $\\mK$-snakes of different classes into one $\\mK$-snake.\nWe conclude this section by describing how to combine most of these classes\nby using a hypergraph whose vertices represent the classes and whose\nedges represent the classes that can be merge together in one step.\n\n\nWe present two constructions for a $(2n+1,M_{2n+1},\\mK)$-snake,\n$C_{2n+1}$, one recursive and one direct. In this section\nwe present the framework for these constructions.\nFirst, the permutations of $A_{2n+1}$,\nthe set of even permutations from~$S_{2n+1}$, are partitioned into classes,\nwhere each class induces one $\\mK$-snake which contains permutations only from the class.\nAll these snakes have the same S-skeleton.\nLet $L_{2n+1}$ be the set of all the classes.\n\nThe construction of $C_{2n+1}$ from the $\\mK$-snakes of $L_{2n+1}$\nproceeds by a sequence of joins, where at each step\nwe have a main $\\mK$-snake, and\ntwo $\\mK$-snakes from the remaining $\\mK$-snakes\nof $L_{2n+1}$ are joined to the current main $\\mK$-snake.\nA join is performed by replacing one transition\nin the main $\\mK$-snake with a matching sequence.\n\nIn order to join the $\\mK$-snakes we need the following lemmas,\nfor which the first can be easily verified. In the sequel, let\n${\\sigma^{k}\\triangleq\\underbrace{\\sigma\\circ \\sigma \\circ \\ldots \\circ \\sigma}_{k \\, times}}$,\ni.e., performing the transitions sequence $\\sigma$, $k$ times.\n\\begin{lemma}\n\\label{lem:helpLem1}\nIf $\\alpha,\\beta \\in S_n$ then $\\beta{=}t_{i}(\\alpha)$ if and only if $\\alpha{=}t_i^{i-1}(\\beta)$.\n\\end{lemma}\n\\begin{lemma}\n\\label{lem:mainLem}\nIf ${i \\in [n-2]}$ then $t_i \\leftrightsquigarrow t_{i+2}\\circ (t_i^{i-1} \\circ t_{i+2})^{2}$.\n\\end{lemma}\n\\begin{proof}\nLet ${\\alpha=[a_1, a_2, \\ldots ,a_i, a_{i+1}, a_{i+2}, \\ldots , a_n]}$ be a permutation over [n].\n\n\\begin{align*}\n\\begin{tabular}{lllllllll}\n$t_{i+2}(\\alpha)$  & \\\\\n\\hspace*{\\parindent} $=[a_{i+2}, a_1, \\ldots , a_i, a_{i+1}, a_{i+3},$  &  $\\ldots,a_n]$,\\tabularnewline\n$t_i^{i-1}(t_{i+2}(\\alpha))$ & \\\\\n\\hspace*{\\parindent} $=[a_1, a_2, \\ldots , a_{i-1}, a_{i+2}, a_i, a_{i+1}, a_{i+3},$  &  $\\ldots,a_n]$,\\tabularnewline\n$t_{i+2}(t_i^{i-1}(t_{i+2}(\\alpha)))$  & \\\\\n\\hspace*{\\parindent} $=[a_{i+1}, a_1, a_2, \\ldots , a_{i-1},  a_{i+2}, a_i, a_{i+3},$  &  $\\ldots,a_n]$,\\tabularnewline\n$t_i^{i-1}(t_{i+2}(t_i^{i-1}(t_{i+2}(\\alpha))))$  & \\\\\n\\hspace*{\\parindent} $=[a_1, a_2, \\ldots ,a_{i-1}, a_{i+1}, a_{i+2}, a_i, a_{i+3},$  &  $\\ldots,a_n]$,\\tabularnewline\n\\intertext{and hence we have,}\n$t_{i+2}(t_i^{i-1}(t_{i+2}(t_i^{i-1}(t_{i+2}(\\alpha)))))$ & \\\\\n\\hspace*{\\parindent} $=[a_i, a_1, \\ldots , a_{i-1}, a_{i+1}, a_{i+2},$  &  $\\ldots,a_n]$ \\tabularnewline\n\\hspace*{\\parindent} $=t_i(\\alpha)$.\n\\end{tabular}\n\\end{align*}\n\\end{proof}\n\\vspace{-4pt}\n\\begin{cor}\n\\label{cor:mainLem}\nIf $\\pi \\in S_{2n+1}$ then $t_{2n-1}(\\pi)=t_{2n+1}\\left(t_{2n-1}^{2n-2}\\left(t_{2n+1}\\left(t_{2n-1}^{2n-2}\\left(t_{2n+1}(\\pi)\\right)\\right)\\right)\\right)$.\n\\end{cor}\n\nLemma~\\ref{lem:mainLem} can be generalized as follows\n(the following lemma is given\nfor completeness, but it will not be used in the sequel, and hence\nits proof is omitted).\n\n\\begin{lemma}\n\\label{lem:mainLemExp}\nIf ${i,j\\in [n]}$ and ${|i-j|{=}k}$, then ${t_i \\leftrightsquigarrow t_j\\circ (t_i^{i-1} \\circ t_j)^{k}}$.\n\\end{lemma}\n\n\nThe partition of $A_{2n+1}$ into the set of classes $L_{2n+1}$\nshould satisfy the following properties:\n\\begin{itemize}\n\\item[(P1)] The last two ordered elements of two permutations in same class are equal.\n\\item[(P2)] Any two permutations which differ only by a cyclic shift of the first $2n-1$ elements, belong to the same class.\n\\end{itemize}\n\n\\begin{cor}\nLet $\\pi$ be a permutation in $A_{2n+1}$.\n\\begin{itemize}\n\\item $\\pi$ and $t_{2n+1}(\\pi)$ belong to different classes in $L_{2n+1}$.\n\\item $\\pi$ and $t_{2n-1}(\\pi)$ belong to the same class in $L_{2n+1}$.\n\\end{itemize}\n\\end{cor}\n\nWe continue now with the description of the method to join the\n$\\mK$-snakes of $L_{2n+1}$ into $C_{2n+1}$.\nIn the rest of the paper, $A_{2n+1}$ is partitioned into classes according to the last two ordered elements in the permutations.\nLet $[x,y]$ denote the class of $A_{2n+1}$ in which the last ordered pair in the permutations is $(x,y)$.\nLet $\\cT$ be the S-skeleton of the $\\mK$-snakes in $L_{2n+1}$.\nLet $C_{\\cT}^{\\pi}$ be a $\\mK$-snake for which\n$\\cT$ is its transitions sequence, and $\\pi$ is its first permutation.\nIf $\\pi$ belongs to the class $[x,y]$,\nwe say that $C_{\\cT}^{\\pi}$ represents the class $[x,y]$.\nNote that all the permutations in $C_{\\cT}^{\\pi}$ belong to the same class.\n\nThe transitions sequence $\\cT$\nshould satisfy the following properties (these properties are needed in order to make the required joins of cycles):\n\\begin{itemize}\n\\item[(P3)] $t_{2n-1}$ is the last transition in $\\cT$.\n\\item[(P4)] Given a permutation $\\pi=[a_1,\\ldots, a_{2n},a_{2n+1}]$,\nfor each $x\\in [2n+1]\\setminus \\{a_{2n},a_{2n+1}\\}$\nthere exists a permutation $\\pi' \\in C_{\\cT}^{\\pi}$\nwhose last ordered three elements are $(x,a_{2n},a_{2n+1})$.\n\\end{itemize}\n\n\\begin{cor}\n\\label{cor:mergeCyclesCondition}\nFor each class $[x,y]$, a permutation $\\pi \\in [x,y]$, and $z\\in [2n+1]\\setminus \\{x,y\\}$,\nthere exists a permutation $\\pi' \\in C_{\\cT}^{\\pi}$ whose last ordered three elements are $(z,x,y)$,\nfollowed by the permutation $t_{2n-1}(\\pi')$.\n\\end{cor}\n\n\\begin{lemma}\n\\label{prop:mergeClasses1}\nLet $C$ be a $\\mK$-snake which doesn't contain any permutation from the classes $[y,z]$ or $[z,x]$, let\n$\\pi=[a_1, a_2, \\ldots, a_{2n-2},z, \\mathbf{x,y}]$\nbe a permutation in $C$ followed by $t_{2n-1}$, and let\n$\\sigma$ be a transitions sequence such that $\\cT = \\sigma \\circ t_{2n-1}$.\nThen replacing this $t_{2n-1}$ transition in $C$, with\n\\begin{equation*}\nt_{2n+1}\\circ \\sigma \\circ t_{2n+1}\\circ \\sigma \\circ t_{2n+1} ,\n\\end{equation*}\njoins two $\\mK$-snakes representing the classes ${[y, z]}$ and ${[z, x]}$ into $C$ (after $\\pi$).\n\\end{lemma}\n\\begin{proof}\nObserve that by Lemma \\ref{lem:helpLem1} we have~${\\sigma \\leftrightsquigarrow t_{2n-1}^{2n-2}}$.\nThus, we have\n\\begin{align*}\n\\pi=&\n    \\left.\\begin{array}{l}\n    \t\t\\left[ a_1, a_2, \\ldots,  a_{2n-2}, z, \\mathbf{{x},y} \\right]\n    \\\\ \\end{array}\\right. \\\\\n    &\n    \\left.\\begin{array}{l}\n    \t\t\\downarrow {t_{2n+1}}\n    \\\\ \\end{array}\\right. \\\\\n    &\n    \\left.\\begin{array}{l}\n    \t\t\\left[ y, a_1, a_2, \\ldots, a_{2n-2}, \\mathbf{z, x} \\right]   \\\\\n    \t\t\\downarrow {\\sigma \\leftrightsquigarrow t_{2n-1}^{2n-2}} \\\\\n    \t\t\\left[ a_1, a_2, \\ldots, a_{2n-2}, y, \\mathbf{z, x} \\right]\n    \\\\ \\end{array}\\right\\}\n    \\left.\\begin{array}{l}\n    \\mK-snake\\\\\n    for\\ \\left[z,x\\right] \\\\\n    \\end{array}\\right. \\\\\n    &\n    \\left.\\begin{array}{l}\n    \t\t\\downarrow {t_{2n+1}}\n    \\\\ \\end{array}\\right. \\\\\n    &\n    \\left.\\begin{array}{l}\n    \t\t\\left[ x, a_1, a_2, \\ldots, a_{2n-2}, \\mathbf{y, z}\\right] \\\\\n    \t\t\\downarrow {\\sigma \\leftrightsquigarrow t_{2n-1}^{2n-2}} \\\\\n    \t\t\\left[ a_1, a_2, \\ldots, a_{2n-2}, x,  \\mathbf{y, z}\\right]\n    \\\\ \\end{array} \\right\\}\n    \\left.\\begin{array}{l}\n    \\mK-snake\\\\\n    for\\ \\left[y, z\\right] \\\\\n    \\end{array}\\right. \\\\\n    &\n    \\left.\\begin{array}{l}\n    \t\t\\downarrow {t_{2n+1}}\n    \\\\ \\end{array}\\right. \\hphantom{\\left[ a_1, a_2, \\ldots, x,\\mathbf{y, z} \\right]}\n     \\left.\\begin{array}{l}\n        return\\ to\\ the\\ \\\\\n        \\mK-snake\\ C  \\\\\n        \\end{array}\\right. \\\\\nt_{2n-1}(\\pi)=&\n    \\left.\\begin{array}{l}\n    \\left[ z, a_1, a_2, \\ldots, a_{2n-2}, \\mathbf{x, y}\\right]\n    \\end{array}\\right.\n\\end{align*}\n\\end{proof}\nThe next step is to present an order for merging all the $\\mK$-snakes of $L_{2n+1}$,\nexcept one, into $C_{2n+1}$.\nThis step will be performed by\ntranslating the merging problem into a 3-graph problem.\nWe start with a sequence of definitions taken from~\\cite{hypergraph}.\n\\begin{definition}\n\\label{def:3-graph}\nA 3-graph (also called a 3-uniform hypergraph) $H=(V,E)$ is a hypergraph where $V$ is a set of vertices\nand $E\\subseteq \\binom{V}{3}$. A~hyperedge of $H$ will be called triple.\\\\\nA path in $H$ is an alternating sequence of $\\ell+1$ distinct\nvertices and $\\ell$ distinct triples:\n$v_0, e_1, v_1, \\ldots ,v_{\\ell-1}, e_{\\ell}, v_{\\ell}$,\nwith the property that $\\forall i\\in [\\ell]: v_{i-1},v_{i} \\in e_{i}$.\\\\\nA cycle is a closed path, i.e. $v_0 = v_{\\ell}$.\\\\\nA sub-3-graph contains a subset $E'\\subseteq E$ and the subset\n$V'\\subseteq V$ which contains all the vertices in $E'$.\\\\\nA tree $T$ in $H$ is a connected sub-3-graph of $H$ with no cycles.\n\\end{definition}\n\n\\pagebreak\nLet $H_{2n+1}=(V_{2n+1},E_{2n+1})$ be a 3-graph defined as follows:\n\\begin{align*}\nV_{2n+1}&=\\{[x,y]~:~x,y \\in [2n+1], x\\ne y\\},\\\\\nE_{2n+1}&=\\{\\{[x,y],[y,z],[z,x]\\}~:\\\\\n&\\hphantom{=\\{\\}} x,y,z\\in [2n+1], x\\ne y, x\\ne z, y\\ne z\\}.\n\\end{align*}\nWe denote a hyperedge $\\{[x,y],[y,z],[z,x]\\}$, where $x<y$ and $x<z$, by the triple $\\langle x,y,z \\rangle$.\n\nThe vertices in $H_{2n+1}$ correspond to the classes in the set $L_{2n+1}$.\nEach $e\\in E_{2n+1}$ contains three vertices,\nwhich correspond to three classes.\nThese three classes can be represented\nby three $\\mK$-snakes, generated from the S-skeleton,\nwhich can be merged together\nby Corollary~\\ref{cor:mergeCyclesCondition} and Lemma~\\ref{prop:mergeClasses1}.\nNote that for any two edges $e_1 , e_2$ in $H_{2n+1}$\neither $e_1 \\cap e_2 = \\varnothing$ or $| e_1 \\cap e_2 | =1$.\nLet $T_{2n+1}=(V_{T_{2n+1}},E_{T_{2n+1}})$ be a tree in $H_{2n+1}$.\nWe join $|V_{T_{2n+1}}|$ $\\mK$-snakes which represent $|V_{T_{2n+1}}|$ classes of $L_{2n+1}$\nto form the $\\mK$-snake $C_{2n+1}$,\nby Corollary~\\ref{cor:mergeCyclesCondition} and Lemma~\\ref{prop:mergeClasses1}.\nThe hyperedges which represent the joins which are performed are determined by $T_{2n+1}$,\nbut these joins are not unique,\nand hence they can yield different final $\\mK$-snakes.\nThe order in which the hyperedges are selected for these joins is also not unique,\nbut this order doesn't affect the final $\\mK$-snakes.\nThe size of the $\\mK$-snake $C_{2n+1}$ depends on the number of vertices\nin the tree $T_{2n+1}$.\nA tree in a 3-graph\ncontains an odd number of vertices \\cite{hypergraph}.\nSince in $H_{2n+1}$ there are $(2n+1)(2n)$ vertices\nit follows that there is no tree in $H_{2n+1}$ which contains\nall the vertices of $V_{2n+1}$. This motivates the following definition.\n\\begin{definition}\nA nearly spanning tree in a 3-graph $H=(V,E)$\nis a tree in $H$\nwhich contains all the vertices of $V$ except one.\n\\end{definition}\n\nNow, let $T_{2n+1}$ be a nearly spanning tree in $H_{2n+1}$.\n\n\\begin{example}\n\\label{Exm:T5}\nOne choice for $T_{5}$ is given below.\\\\\nThe edges in the tree $T_{5}$ are:\n\n\\begin{tabular}{lll}\n\\label{treeFor5}\n\\hspace*{-\\parindent}\n\\hspace*{-\\parindent}\n$\\langle 1,2,5 \\rangle,$  &  $\\langle 1,2,4 \\rangle,$  &  $\\langle 1,2,3 \\rangle,$ \\tabularnewline\n\\hspace*{-\\parindent}\n\\hspace*{-\\parindent}\n$\\langle 1,4,5 \\rangle,\\langle 2,5,4 \\rangle,$  &  $\\langle 1,3,4 \\rangle,\\langle 2,4,3 \\rangle,$  &  $\\langle 1,5,3 \\rangle, \\langle 2,3,5 \\rangle.$\n\\end{tabular}\nThe order of merging $\\mK$-snakes\nfrom these classes obtained by this choice of $T_5$\ncan be chosen as follows.\n\\begin{enumerate}[(1)]\n{\\setlength\\itemindent{10pt}\n\\item vertex $\\hphantom{ii}[1,2]$;\n\\item vertices $[3,1]$, $[2,3]$, (through the edge $\\langle 1,2,3 \\rangle$);\n\\item vertices $[4,1]$, $[2,4]$, (through the edge $\\langle 1,2,4 \\rangle$);\n\\item vertices $[5,1]$, $[2,5]$, (through the edge $\\langle 1,2,5 \\rangle$);\n\\item vertices $[5,3]$, $[1,5]$, (through the edge $\\langle 1,5,3 \\rangle$);\n\\item vertices $[5,2]$, $[3,5]$, (through the edge $\\langle 2,3,5 \\rangle$);\n\\item vertices $[3,4]$, $[1,3]$, (through the edge $\\langle 1,3,4 \\rangle$);\n\\item vertices $[3,2]$, $[4,3]$, (through the edge $\\langle 2,4,3 \\rangle$);\n\\item vertices $[4,5]$, $[1,4]$, (through the edge $\\langle 1,4,5 \\rangle$);\n\\item vertices $[4,2]$, $[5,4]$, (through the edge $\\langle 2,5,4 \\rangle$).}\n\\end{enumerate}\n\nUsing the S-skeleton $\\cT=t_3,t_3,t_3$ of the $(3,3,\\mK)$-snake,\nthe snake-in-the-box code which is obtained\nby $T_{5}$ is a $(5,57,\\mK)$-snake presented in Figure~\\ref{fig:snake5}.\nThere is no $(5,M,\\mK)$-snake for which $M>57$ \\cite{YeSc12}.\nThe S-skeleton of this code is $\\sigma ^ 3$, where\n\\begin{align*}\n\\sigma=\t& t_5,t_5,t_3,t_3,t_5,t_3,t_3,t_5,t_3,\\\\\n\t\t& t_5,t_5,t_3,t_3,t_5,t_3,t_3,t_5,t_3,t_5\n\\end{align*}\n\\begin{figure*}[htbp]\n\\begin{center}\n\\small\n\\setlength{\\tabcolsep}{1.5pt}\n\\begin{tabular}{*{56}{c|}c}\n 3 & 2 & 1 & 3 & 2 & 5 & 3 & 2 & 4 & 3 & 1 & 5 & 3 & 1 & 2 & 3 & 1 & 4 & 3 & 5 &\n 2 & 1 & 5 & 2 & 4 & 5 & 2 & 3 & 5 & 1 & 4 & 5 & 1 & 2 & 5 & 1 & 3 & 5 & 4 & 2 &\n 1 & 4 & 2 & 3 & 4 & 2 & 5 & 4 & 1 & 3 & 4 & 1 & 2 & 4 & 1 & 5 & 4 \\\\\n 4 & 3 & 2 & 1 & 3 & 2 & 5 & 3 & 2 & 4 & 3 & 1 & 5 & 3 & 1 & 2 & 3 & 1 & 4 & 3 &\n 5 & 2 & 1 & 5 & 2 & 4 & 5 & 2 & 3 & 5 & 1 & 4 & 5 & 1 & 2 & 5 & 1 & 3 & 5 & 4 &\n 2 & 1 & 4 & 2 & 3 & 4 & 2 & 5 & 4 & 1 & 3 & 4 & 1 & 2 & 4 & 1 & 5 \\\\\n 5 & 4 & 3 & 2 & 1 & 3 & 2 & 5 & 3 & 2 & 4 & 3 & 1 & 5 & 3 & 1 & 2 & 3 & 1 & 4 &\n 3 & 5 & 2 & 1 & 5 & 2 & 4 & 5 & 2 & 3 & 5 & 1 & 4 & 5 & 1 & 2 & 5 & 1 & 3 & 5 &\n 4 & 2 & 1 & 4 & 2 & 3 & 4 & 2 & 5 & 4 & 1 & 3 & 4 & 1 & 2 & 4 & 1 \\\\\n 1 & 5 & 4 & 4 & 4 & 1 & 1 & 1 & 5 & 5 & 2 & 4 & 4 & 4 & 5 & 5 & 5 & 2 & 2 & 1 &\n 4 & 3 & 3 & 3 & 1 & 1 & 1 & 4 & 4 & 2 & 3 & 3 & 3 & 4 & 4 & 4 & 2 & 2 & 1 & 3 &\n 5 & 5 & 5 & 1 & 1 & 1 & 3 & 3 & 2 & 5 & 5 & 5 & 3 & 3 & 3 & 2 & 2 \\\\\n 2 & 1 & 5 & 5 & 5 & 4 & 4 & 4 & 1 & 1 & 5 & 2 & 2 & 2 & 4 & 4 & 4 & 5 & 5 & 2 &\n 1 & 4 & 4 & 4 & 3 & 3 & 3 & 1 & 1 & 4 & 2 & 2 & 2 & 3 & 3 & 3 & 4 & 4 & 2 & 1 &\n 3 & 3 & 3 & 5 & 5 & 5 & 1 & 1 & 3 & 2 & 2 & 2 & 5 & 5 & 5 & 3 & 3 \\\\\n\\end{tabular}\n\\end{center}\n\\caption{A $(5,57,\\mK)$-snake obtained by $T_5$} \\label{fig:snake5}\n\\end{figure*}\n\\end{example}\n\n\\begin{theorem}\n\\label{thm:treeCompose}\nIf $n\\geq 2$, then there exists a nearly spanning tree $T_{2n+1}$ in $H_{2n+1}$ which doesn't include the vertex $[2,1]$.\n\\end{theorem}\n\\begin{proof}\nWe present a recursive construction for such a nearly spanning tree.\nWe start with the nearly spanning tree given in Example~\\ref{Exm:T5}.\nNote that $T_5$ doesn't include the vertex $[2,1]$.\nAssume that there exists a nearly spanning tree,~$T_{2n-1}$,\nin $H_{2n-1}$, which doesn't include the vertex $[2,1]$.\nNote that $H_{2n-1}$ is a sub-graph of~$H_{2n+1}$\nand therefore~$T_{2n-1}$ is a tree in $H_{2n+1}$.\nThe vertices of $H_{2n+1}$ which are not spanned by $T_{2n-1}$ are\n\\begin{itemize}\n\\item $[x,2n], [2n,x], [x,2n+1], [2n+1,x]$ for each $x\\in [2n-1]$,\n\\item $[2n,2n+1], [2n+1,2n]$,\n\\item $[2,1]$.\n\\end{itemize}\nThe nearly spanning tree $T_{2n+1}$ is constructed from $T_{2n-1}$ as follows.\nFor each $x$, $2\\leq x \\leq 2n-2$, the edges\n${\\langle x,x+1,2n \\rangle}$ and $\\langle x,x+1,2n+1 \\rangle$\nare joined to $T_{2n+1}$;\nalso the edges\n$\\langle 1,2,2n \\rangle$,\n$\\langle 1,2n,2n-1 \\rangle$,\n$\\langle 1,2n+1,2n-1 \\rangle$,\n$\\langle 1,2n,2n+1 \\rangle$, and\n$\\langle 2,2n+1,2n \\rangle$ are joined to $T_{2n+1}$.\nIt is easy to verify that all the vertices of $H_{2n+1}$\nwhich are not spanned by $T_{2n-1}$ (except for $[2,1]$)\nare contained in the list of the edges which are joined to $T_{2n-1}$.\nWhen an edge is joined to the tree it has one vertex which is already in the\ntree and two vertices which are not on the tree. Hence, connectivity is\npreserved and no cycle is formed.\nHence, it is easy to verify that by joining these edges to $T_{2n-1}$\nwe form a nearly spanning tree in $H_{2n+1}$.\n\\end{proof}\n\n\\begin{example}\n\\label{Exm:T7}\nBy using Theorem~\\ref{thm:treeCompose} and the\nnearly spanning tree $T_5$ of Example~\\ref{Exm:T5}\nwe obtain the spanning tree $T_7$ depicted in Figure~\\ref{fig:T7}.\nThe dashed boxes edges and the double lines nodes are added to $T_5$ in order to form $T_7$.\n\\pagestyle{empty}\n\\tikzstyle{level 1}=[sibling angle=90]\n\\tikzstyle{level 2}=[sibling angle=80]\n\\tikzstyle{level 3}=[sibling angle=50]\n\\tikzstyle{level 4}=[sibling angle=50]\n\\tikzstyle{every node}=[draw]\n\\tikzstyle{edge from parent}=[draw]\n\\begin{figure*}[htbp]\n\\centering\n\\begin{tikzpicture}[grow cyclic,very thick,level distance=10mm,\n                    cap=round, scale=0.9]\n                    \\tikzset{VertexStyle\/.style = {shape          = circle,\n                                                     draw,\n                                                     text           = black,\n                                                     inner sep      = 1pt,\n                                                     outer sep      = 0pt,\n                                                     minimum size   = 8 pt,\n                                                     scale=0.9}}\n                    \\tikzset{Vertex7Style\/.style = {shape          = circle,\n                                                     draw,\n                                                     thick,\n                                                     double,\n                                                     text           = black,\n                                                     inner sep      = 1pt,\n                                                     outer sep      = 1pt,\n                                                     minimum size   = 8 pt,\n                                                     scale=0.9}}\n                      \\tikzset{EdgeStyle\/.style   = {draw, thin, scale=0.9}}\n                      \\tikzset{LabelStyle\/.style =   {text = black, draw=none, scale=0.9}}\n                      \\tikzset{Label7Style\/.style =   {text = black,outer sep = 1pt,dashed,thin, scale=0.9}}\n\\node[VertexStyle] {12}\n\tchild\n\t{\n\t\tnode[Label7Style] {126}\n\t\tchild{\n\t\t\t\tnode[Vertex7Style] {26}\n\t\t\t\t}\n\t\tchild{\n\t\t\t\tnode[Vertex7Style] {61}\n\t\t\t\t}\n\t\t}\n\tchild{\n\t\tnode[LabelStyle] {124}\n\t\tchild{\n\t\t\t\tnode[VertexStyle] {24}\n\t\t\t\tchild{\n\t\t\t\t\t\tnode[LabelStyle] {243}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[VertexStyle] {43}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[VertexStyle] {32}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t}\n\t\t\t\t}\n\t\tchild{\n\t\t\t\tnode[VertexStyle] {41}\n\t\t\t\tchild{\n\t\t\t\t\t\tnode[LabelStyle] {134}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[VertexStyle] {13}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[VertexStyle] {34}\n\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\tnode[Label7Style] {346}\n\t\t\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\t\t\tnode[Vertex7Style] {46}\n\t\t\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\t\t\tnode[Vertex7Style] {63}\n\t\t\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\tnode[Label7Style] {347}\n\t\t\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\t\t\tnode[Vertex7Style] {47}\n\t\t\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\t\t\tnode[Vertex7Style] {73}\n\t\t\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t}\n\t\t\t\tchild [missing]\n\t\t\t\tchild [missing]\n\t\t\t}\n\t\t}\n\tchild{\n\t\tnode[LabelStyle] {123}\n\t\tchild{\n\t\t\t\tnode[VertexStyle] {23}\n\t\t\t\tchild{\n\t\t\t\t\t\tnode[LabelStyle] {235}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[VertexStyle] {35}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[VertexStyle] {52}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t}\n\t\t\t\tchild\n\t\t\t\t{\n\t\t\t\t\t\tnode[Label7Style] {236}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[Vertex7Style] {36}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[Vertex7Style] {62}\n\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tnode[Label7Style] {276}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tnode[Vertex7Style] {27}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tnode[Vertex7Style] {76}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t}\n\t\t\t\tchild\n\t\t\t\t{\n\t\t\t\t\t\tnode[Label7Style] {237}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[Vertex7Style] {37}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[Vertex7Style] {72}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t}\n\t\t\t\t}\n\t\tchild{\n\t\t\t\tnode[VertexStyle] {31}\n\t\t\t\tchild{\n\t\t\t\t\t\tnode[LabelStyle] {153}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[VertexStyle] {15}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[VertexStyle] {53}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t}\n\t\t\t}\n\t\t}\n\tchild{\n\t\tnode[LabelStyle] {125}\n\t\tchild{\n\t\t\t\tnode[VertexStyle] {25}\n\t\t\t\tchild{\n\t\t\t\t\t\tnode[LabelStyle] {254}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[VertexStyle] {54}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[VertexStyle] {42}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t}\n\t\t\t\t}\n\t\tchild{\n\t\t\t\tnode[VertexStyle] {51}\n\t\t\t\tchild{\n\t\t\t\t\t\tnode[LabelStyle] {145}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[VertexStyle] {14}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[VertexStyle] {45}\n\t\t\t\t\t\t\t\tchild\n\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\tnode[Label7Style] {456}\n\t\t\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\t\t\tnode[Vertex7Style] {56}\n\t\t\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\t\t\tnode[Vertex7Style] {64}\n\t\t\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\tchild\n\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\tnode[Label7Style] {457}\n\t\t\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\t\t\tnode[Vertex7Style] {57}\n\t\t\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\t\t\tnode[Vertex7Style] {74}\n\t\t\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\tchild [missing]\n\t\t\t\t\t\t}\n\t\t\t\tchild\n\t\t\t\t{\n\t\t\t\t\t\tnode[Label7Style] {165}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[Vertex7Style] {16}\n\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\tnode[Label7Style] {167}\n\t\t\t\t\t\t\t\t\t\tchild [missing]\n\t\t\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\t\tnode[Vertex7Style] {67}\n\t\t\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\t\t\t\tnode[Vertex7Style] {71}\n\t\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[Vertex7Style] {65}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t}\n\t\t\t\tchild\n\t\t\t\t{\n\t\t\t\t\t\tnode[Label7Style] {175}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[Vertex7Style] {17}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\tchild{\n\t\t\t\t\t\t\t\tnode[Vertex7Style] {75}\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t}\n\t\t\t}\n\t\t}\n\t;\n\\end{tikzpicture}\n\\vspace{-15pt}\n\\caption{The nearly spanning tree $T_7$ constructed from $T_5$} \\label{fig:T7}\n\\end{figure*}\n\\end{example}\n\n\\section{A Recursive Construction}\n\\label{sec:recursive}\n\nIn this section we present the recursive construction for a\n$(2n+1,M_{2n+1},\\mK)$-snake from a\n$(2n-1,M_{2n-1},\\mK)$-snake. The construction is based on the\nnearly spanning tree $T_{2n+1}$ presented in the previous section.\nEach of its vertices represent a class in which a $\\mK$-snake based\non the $(2n-1,M_{2n-1},\\mK)$-snake is generated. Those $\\mK$-snakes\nare merged together into one $(2n+1,M_{2n+1},\\mK)$-snake using the\nframework presented in the previous section. We conclude this\nsection with an analysing the length of the generated\n$\\mK$-snake compared the total number of permutations in $S_{2n+1}$.\n\nWe generate a $(2n+1,M_{2n+1},\\mK)$-snake, $C_{2n+1}$,\nwhose transitions sequence is\n${t_{k_1} , t_{k_2} ,\\ldots , t_{k_{M_{2n+1}}}}$.\n$C_{2n+1}$ has the following properties:\n\\begin{enumerate}\n\\item[(Q1)] $k_j$ is odd for all ${j\\in[M_{2n+1}]}$.\n\\item[(Q2)] $k_{M_{2n+1}}=2n+1$.\n\\item[(Q3)] For each $z\\in [2n+1]$ there exists a permutation $\\pi \\in C_{2n+1}$ such that $\\pi(2n+1)=z$.\n\\end{enumerate}\nThe starting point of the recursive construction is $2n+1=3$.\nThe transitions sequence for $2n+1=3$ is $t_3,t_3,t_3$,\nand the complete $(3,3,\\mK)$-snake is $C_3\\eqdef \\{[1, 2, 3], [3, 1, 2], [2, 3, 1]\\}$.\nClearly (Q1), (Q2), and (Q3) hold for this transitions sequence and $C_3$.\n\nNow, assume that there exists a $(2n-1,M_{2n-1},\\mK)$-snake, $C_{2n-1}$,\nwhich satisfies properties (Q1), (Q2), (Q3), and let\n$\\cT_{2n-1}=t_{k_1} , t_{k_2} , \\ldots , t_{k_{M_{2n-1}}}$\nbe its S-skeleton, i.e., $\\cT_{2n-1}$ is the transitions sequence of $C_{2n-1}$.\nNote that (Q1), (Q2), and (Q3) depend on the transitions sequence $\\cT_{2n-1}$\nand are independent of the first permutation of $C_{2n-1}$.\nWe construct a ${(2n+1,M_{2n+1},\\mK)}$-snake, $C_{2n+1}$,\nwhere $M_{2n+1}=((2n+1)(2n)-1)M_{2n-1}$, which also satisfies (Q1), (Q2), and (Q3).\n\nFirst, all the permutations of $A_{2n+1}$ are\npartitioned into ${(2n+1)(2n)}$ classes according\nto the last ordered two elements in the permutations.\nThis implies that (P1) and (P2) are satisfied.\nIn addition, (P3) and (P4) for $\\cT_{2n-1}$ are immediately\nimplied by (Q2) and (Q3) for  $C_{2n-1}$, respectively.\nHence $\\cT_{2n-1}$ can be used\nas the S-skeleton for the $\\mK$-snakes in $L_{2n+1}$.\nNow, we merge the $\\mK$-snakes of the classes in $L_{2n+1}$ (except $[2,1]$),\nby using Lemma~\\ref{prop:mergeClasses1} and the nearly spanning tree $T_{2n+1}$ of\nTheorem~\\ref{thm:treeCompose}.\nWe have to show that (Q1), (Q2), and (Q3) are satisfied for $C_{2n+1}$. (Q1) is readily verified.\nClearly, $t_{2n+1}$ was used to obtain $C_{2n+1}$ (see Lemma~\\ref{prop:mergeClasses1}),\nand therefore we can always define $\\cT_{2n+1}$\nin such a way that its last transition is $t_{2n+1}$, and hence (Q2) is satisfied.\nFor each $z\\in [2n+1]$ there exists a class $[x,z]$\nwhose $\\mK$-snake is joined into $C_{2n+1}$,\nand therefore (Q3) is satisfied.\nThus, we have\n\\begin{theorem}\nGiven a $(2n-1,M_{2n-1},\\mK)$-snake which satisfies (Q1), (Q2), and (Q3),\nwe can obtain a $(2n+1,M_{2n+1},\\mK)$-snake,\nwhere $M_{2n+1}=((2n+1)(2n)-1)M_{2n-1}$,\nwhich also satisfies (Q1), (Q2), and (Q3).\n\\end{theorem}\n\nFollowing~\\cite{YeSc12},\nwe define $D_{2n+1}=\\frac{M_{2n+1}}{(2n+1)!}$ as the ratio between\nthe number of permutations in the given ${(2n+1, M_{2n+1},\\mK)}$-snake\nand the size of $S_{2n+1}$.\nRecall that if $C$ is an $(2n+1,M,\\mK)$-snake then $M\\leq \\frac{{|S_{2n+1}|}}{2}$,\nand we conjecture that optimal size is ${M=(2n+1)!-2n+1}$.\nThus, it is desirable to obtain a value $D_{2n+1}$ close to half as much as possible.\nIn our recursive construction $M_{2n+1}=((2n+1)(2n)-1)M_{2n-1}$.\nThus, we have\n\\begin{align*}\nD_3&=\\frac{1}{2},\\\\\n\\prod_{n=2}^{\\infty}\\frac{D_{2n+1}}{D_{2n-1}}&=\n\\frac{12\\sqrt{\\pi}}{5(1+\\sqrt{5})\\Gamma(\\frac{1}{4}(5-\\sqrt{5}))\\Gamma(\\frac{1}{4}(1+\\sqrt{5}))},\n\\end{align*}\nwhich implies that\n\\begin{align*}\n\\lim\\limits_{n\\to \\infty} D_{2n+1}&= \\frac{1}{2}\\cdot \\frac{12\\sqrt{\\pi}}{5(1+\\sqrt{5})\\Gamma(\\frac{1}{4}(5-\\sqrt{5}))\\Gamma(\\frac{1}{4}(1+\\sqrt{5}))}\\\\\n&\\approx 0.4338.\n\\end{align*}\nThis computation can be done by any mathematical tool, e.g., WolframAlpha.\nThis improves on the construction described in \\cite{YeSc12}, which yields $M_{2n+1}=(2n+1)(2n-1)M_{2n-1}$ and $\\lim\\limits_{n\\to \\infty} D_{2n+1}=\\lim\\limits_{n\\to \\infty} \\frac{1}{\\sqrt{\\pi n}}$.\n\n\n\\section{A Direct Construction based on Necklaces}\n\\label{sec:direct}\n\nIn this section we describe a direct construction to\nform a $(2n+1,M_{2n+1},\\mK)$-snake. First, we describe\na method to partition the classes which were used before\ninto subclasses that are similar to necklaces. Next, we show how\nsubclasses from different classes are merged into\ndisjoint chains. Finally, we present a hypergraph and a graph\nin which we have to search for certain trees to form our desired $\\mK$-snake which\nwe believe is of maximum length. Such $\\mK$-snakes were found in $S_7$ and $S_9$.\n\n\nWe present a direct construction for a $(2n+1,M_{2n+1},\\mK)$-snake, $C_{2n+1}$.\nThe goal is to obtain $M_{2n+1}=\\frac{(2n+1)!}{2}-(2n-1)$,\nand hence ${\\frac{D_{2n+1}}{D_{2n-1}}\\geq 1-\\frac{1}{(2n)!}}$.\nWe believe that there is always a $(2n+1,M_{2n+1},\\mK)$-snake with\n$M_{2n+1}=\\frac{(2n+1)!}{2}-(2n-1)$ and there is no such $\\mK$-snake\nwith more codewords.\nWe are making a slight change in the framework discussed in Section~\\ref{sec:main}.\nFirst, all the permutations of $A_{2n+1}$ are\npartitioned into ${(2n+1)(2n)}$ classes according to the last ordered two elements.\nWe denote by $[x,y]$ the class of all even permutations\nin which the last ordered pair in the permutation is $(x,y)$.\nEach class is further partitioned into subclasses\naccording to the cyclic order of the first $2n-1$ elements in the permutations,\ni.e., in each class $[x,y]$, the $\\frac{(2n-1)!}{2}$ permutations\nare partitioned into $\\frac{(2n-2)!}{2}$ disjoint subclasses.\nThis implies that (P1) and (P2) are satisfied for both classes and subclasses.\nLet's denote each one of the subclasses by $[\\alpha]-[x,y]$ where $\\alpha$\nis the cyclic order of the first $2n-1$ elements in the permutations of the subclass.\nLet $\\alpha_1, \\alpha_2$ be two permutations over $[2n+1]\\setminus \\{x,y\\}$.\nIf $\\alpha_1$ and $\\alpha_2$ have the same cyclic order, we denote it by $\\alpha_1 \\simeq \\alpha_2$,\notherwise $\\alpha_1 \\not\\simeq \\alpha_2$.\nNote that if $\\alpha_1 \\simeq \\alpha_2$ then $[\\alpha_1]-[x,y]=[\\alpha_2]-[x,y]$.\nFor example $[1,2,3]-[4,5]$ represents the subclass\nwith the permutations $[1,2,3,4,5]$, $[3,1,2,4,5]$, and $[2,3,1,4,5]$.\n\nLet $L_{2n+1}$ be the set of all classes,\nand let $\\cT=t_{2n-1}^{2n-1}$ be the S-skeleton of the $\\mK$-snakes in $L_{2n+1}$.\nNote that a $\\mK$-snake generated by $\\cT$\nspans exactly all the permutations in one subclass.\nHence (P3) and (P4) are immediately implied for both classes and subclasses.\nSuch a $\\mK$-snake will be called a \\emph{necklace}.\nThe slight change in the framework is that instead of one $\\mK$-snake,\neach class contains $\\frac{(2n-2)!}{2}$ $\\mK$-snakes,\nall of them have the same S-skeleton.\n\nThe necklaces (subclasses) $[\\alpha]-[x,y]$ are similar to\nnecklaces on $2n-1$ elements. Joining the necklaces into\none large $\\mK$-snake might be similar to the join of cycles from\nthe pure cycling register of order $2n-1$, PCR$_{2n-1}$,\ninto one cycle, which is also known as a de Bruijn sequence~\\cite{EtLe84,Fred82}.\nThere are two main differences between the two types of necklaces.\nThe first one is that in de Bruijn sequences\nthe necklaces do not represent permutations, but words\nof a given length over some finite alphabet. The second is\nthat there is rather a simple mechanism to join all the\nnecklaces into a de Bruijn sequence.\nWe would like to have such a mechanism to join\nas many as possible necklaces from all the classes into one $\\mK$-snake.\n\nLet $T_{2n+1}$ be the nearly spanning tree constructed by Theorem~\\ref{thm:treeCompose}.\nBy repeated application of Lemma~\\ref{prop:mergeClasses1}\naccording to the hyperedges of $T_{2n+1}$ starting from a necklace in the class $[1,2]$\nwe obtain a $\\mK$-snake which contains exactly one necklace\nfrom each class $[x,y]\\ne [2,1]$.\nSuch a $\\mK$-snake will be called a \\emph{chain}.\nIf the chain contains the necklace $[\\alpha]-[1,2]$, we will denote it by $c[\\alpha]$.\nFor two permutations $\\alpha_1$ and $\\alpha_2$ over $[2n+1]\\setminus \\{1,2\\}$\nsuch that $\\alpha_1 \\simeq \\alpha_2$ we have $c[\\alpha_1]=c[\\alpha_2]$.\nNote that there is a unique way to merge the three necklaces which correspond to a hyperedge of $T_{2n+1}$,\nand hence there is no ambiguity in $c[\\alpha]$ (even so the order of the joins is not unique),\nNote also that the transitions sequence of two distinct chains is usually different.\nThe number of permutations in a chain is $((2n+1)(2n)-1)(2n-1)$.\nThe following lemma is an immediate consequence of Lemma~\\ref{prop:mergeClasses1}.\n\n\\begin{lemma}\n\\label{lem:key1}\nLet $[x,y]$, $[y,z]$, and $[z,x]$ be three classes,\nand let $\\alpha$ be a permutation of $[2n+1]\\setminus \\{x,y,z\\}$.\nThe necklaces $[\\alpha,z]-[x,y]$, $[\\alpha,y]-[z,x]$,\nand $[\\alpha,x]-[y,z]$ can be merged together, where\n$\\alpha,z$ is the sequence formed by concatenation of $\\alpha$\nand $z$.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lem:key2}\nLet $[x,y]$, $[y,z]$, and $[z,x]$ be three classes.\nAll the subclasses in these classes can be partitioned into disjoint sets,\nwhere each set contains exactly one necklace\nfrom each of the above three classes.\nThe necklaces of each set can be merged together into one $\\mK$-snake.\n\\end{lemma}\n\\begin{proof}\nFor each permutation $\\alpha$ over $[2n+1]\\setminus \\{x,y,z\\}$,\nthe necklaces $[\\alpha,z]-[x,y]$, $[\\alpha,y]-[z,x]$,\nand $[\\alpha,x]-[y,z]$ can be merged by Lemma \\ref{lem:key1}.\nThus, all the subclasses in these classes can be partitioned into disjoint sets.\n\\end{proof}\n\n\\begin{cor}\n\\label{cor:disjointChains}\nThe permutations of all the classes except for $[2,1]$\ncan be partitioned into disjoint chains.\n\\end{cor}\n\nBy Corollary~\\ref{cor:disjointChains} we construct $\\frac{(2n-2)!}{2}$\ndisjoint chains which span $A_{2n+1}$, except for all the even permutations of the class $[2,1]$.\nRecall that we have the same number, $\\frac{(2n-2)!}{2}$, of $[2,1]$-necklaces, which\nspan all the permutations of the class~$[2,1]$.\nNow, we need a method to merge all these chains and necklaces, except\nfor one necklace from the class~$[2,1]$, into one $\\mK$-snake $C_{2n+1}$.\nNote that for $2n+1=5$ we have only one chain. Thus, this chain is the final $\\mK$-snake $C_5$.\nThis $\\mK$-snake is exactly the same $\\mK$-snake as the one generated\nby the recursive construction in Section~\\ref{sec:recursive}.\n\n\\begin{lemma}\n\\label{lem:chainConnection}\nLet $x$ be an integer such that $3\\leq x\\leq 2n+1$,\nlet $\\alpha$ be a permutation of $[2n+1]\\setminus \\{x,2,1\\}$,\nand assume that the permutations $[\\alpha,1,x,2]$ and $[\\alpha,2,1,x]$ are contained in two distinct chains.\nWe can merge these two chains via the necklace $[\\alpha,x]-[2,1]$.\n\\end{lemma}\n\\begin{proof}\nLet $c_1$ be the chain which contains the permutation $\\pi_1=[\\alpha,1,x,2]$,\n$c_2$ be the chain which contains the permutation $\\pi_2=[\\alpha,2,1,x]$, and\n$\\eta$ be the necklace which contains the permutation $\\pi_3=[\\alpha,x,2,1]$.\nNote that all the chains contains only the p-transitions $t_{2n+1}$ and $t_{2n-1}$.\nThe permutation $t_{2n+1}(\\pi_1)$ appears in $c_2$,\nthe permutation $t_{2n+1}(\\pi_2)$ appears in $\\eta$,\nand the permutation $t_{2n+1}(\\pi_3)$ appears in $c_1$.\nTherefore, $\\pi_1$, $\\pi_2$, and $\\pi_3$ are followed\nby $t_{2n-1}$ in $c_1$, $c_2$, and $\\eta$, respectively.\nLet $\\sigma_i$, $i\\in \\{1,2\\}$, be a transitions sequence such that $\\sigma_i, t_{2n-1}$ is the transitions sequence of $c_i$,\nand therefore $t_{2n-1}(\\sigma_i(\\pi_i))=\\pi_i$.\nBy Lemma \\ref{lem:helpLem1} we have $\\sigma_1 \\leftrightsquigarrow t_{2n-1}^{2n-2} \\leftrightsquigarrow \\sigma_2$.\nSimilarly to Lemma~\\ref{prop:mergeClasses1},\nby replacing the transition $t_{2n-1}$ which follows $\\pi_3$ in $\\eta$,\nwith $t_{2n+1} \\circ \\sigma_{1} \\circ t_{2n+1} \\circ \\sigma_2 \\circ t_{2n+1}$,\nwe merge $c_1$, $c_2$ and $\\eta$ into a $\\mK$-snake.\nThus, we have\n\\begin{align*}\n\\pi_3=&[a_1, a_2, \\ldots,  a_{2n-2},x,2,1] \\\\\n&\\downarrow {t_{2n+1}}  \\\\\n&[ 1, a_1, a_2, \\ldots, a_{2n-2}, x, 2] \\\\\n&\\downarrow {\\sigma_1 \\leftrightsquigarrow t_{2n-1}^{2n-2}} \\hphantom{aaaaaaaaaaaa}  the\\ chain\\ c_1 \\\\\n\\pi_1=&[ a_1, a_2, \\ldots, a_{2n-2}, 1,x,2] \\\\\n&\\downarrow {t_{2n+1}}  \\\\\n&[ 2, a_1, a_2, \\ldots, a_{2n-2},1,x] \\\\\n&\\downarrow {\\sigma_2 \\leftrightsquigarrow t_{2n-1}^{2n-2}} \\hphantom{aaaaaaaaaaaa}  the\\ chain\\ c_2 \\\\\n\\pi_2=&[ a_1, a_2, \\ldots, a_{2n-2}, 2, 1, x] \\\\\n&\\downarrow {t_{2n+1}} \\hphantom{aaaaaaa} return\\ to\\ the\\ necklace\\ \\eta \\\\\nt_{2n-1}(\\pi_3) =&[x, a_1, a_2, \\ldots, a_{2n-2}, 2,1]\n\\end{align*}\n\\end{proof}\n\nFor each $x$, $3\\leq x\\leq 2n+1$, and for each permutation $\\alpha$ of $[2n+1]\\setminus\\{x,1,2\\}$,\nthe merging of two distinct chains which contain the permutations\n$[\\alpha,1,x,2]$ and $[\\alpha,2,1,x]$ via the necklace $[\\alpha,x]-[2,1]$\nas described in Lemma~\\ref{lem:chainConnection}, will be denoted by $M[x]$-connection.\nNote that if $x\\in\\{3,4,5\\}$ then the permutations $[\\alpha,1,x,2]$ and $[\\alpha,2,1,x]$\nare contained in the same chain.\nThus, there are no $M[3]$-connections, $M[4]$-connections, or $M[5]$-connections.\n\nLemma~\\ref{lem:chainConnection} suggests a method\nto join all the chains and all the $[2,1]$-necklaces except one into\na $\\mK$-snake of length $\\frac{(2n+1)!}{2}-(2n-1)$.\nThis should be implemented by $\\frac{(2n-2)!}{2}-1$\niterations of the merging suggested by Lemma~\\ref{lem:chainConnection}.\nThe current merging problem is also translated into a $3-graph$ problem (see Definition~\\ref{def:3-graph}).\nLet $\\hat{H}_{2n+1}=(\\hat{V}_{2n+1},\\hat{E}_{2n+1})$ be a 3-graph defined as follows.\n\\begin{align*}\n\\hat{V}_{2n+1}&=\\{c[\\alpha]~:~\\alpha \\text{ is a permutation of } [2n+1]\\setminus \\{1,2\\}\\}\\\\\n& \\cup \\{[\\beta]-[2,1]~:\\\\\n& \\hphantom{\\cup\\{\\}} \\beta \\text{ is a permutation of } [2n+1]\\setminus \\{1,2\\}\\} \\\\\n\\hat{E}_{2n+1}&=\\{\\{c[\\alpha_1], c[\\alpha_2], [\\beta]-[2,1]\\}~:\\\\\n& \\hphantom{=\\{\\}} c[\\alpha_1] \\text{ and } c[\\alpha_2] \\text{ can be merged together} \\\\\n& \\hphantom{=\\{\\}} \\text{via } [\\beta]-[2,1] \\text{ by Lemma \\ref{lem:chainConnection}}\\}.\n\\end{align*}\nThe vertices in $\\hat{V}_{2n+1}$ are of two types, chains and $[2,1]$-necklaces.\nEach $e\\in \\hat{E}_{2n+1}$ contains three vertices, two chains and one necklace,\nwhich can be merged together by Lemma \\ref{lem:chainConnection}.\nTherefore, the edge will be signed by $M[x]$ as described before.\nNote that $\\hat{E}_{2n+1}$ might contains parallel edges with different signs.\n\nLet $\\hat{T}_{2n+1}=(V_{\\hat{T}_{2n+1}},E_{\\hat{T}_{2n+1}})$ be a nearly spanning tree in $\\hat{H}_{2n+1}$.\nNote that such a nearly spanning tree must contain all the vertices in $\\hat{V}_{2n+1}$\nexcept for one $[2,1]$-necklace.\nIf such a nearly spanning tree exists then\nby Lemma~\\ref{lem:chainConnection},\nwe can merge all the chains via $[2,1]$-necklaces\nto form the $\\mK$-snake $C_{2n+1}$.\nThis $\\mK$-snake contains all the permutations\nof $A_{2n+1}$ except for $2n-1$ permutations\nwhich form one $[2,1]$-necklace.\n\nThe joins which are performed are determined by the edges of $\\hat{T}_{2n+1}$.\nNote that there is a unique way to merge the three vertices\nwhich correspond to a hyperedge of $\\hat{T}_{2n+1}$ signed by $M[x]$.\nHence, by using the given spanning trees $T_{2n+1}$ and $\\hat{T}_{2n+1}$,\nthere is no ambiguity in $C_{2n+1}$ (even so the orders of the joins are not unique).\nHowever, different nearly spanning trees\ncan yield different final $\\mK$-snakes.\nNote that the $\\mK$-snake $C_{2n+1}$ generated by this construction\nhas only $t_{2n+1}$ and $t_{2n-1}$ p-transitions, where usually $t_{2n-1}$ is used.\nThe p-transition $t_{2n-1}$ is the only transition in the $\\mK$-snake of the subclasses.\nOn average $3$ out of $4n-2$ sequential p-transitions of $C_{2n+1}$ are the p-transition $t_{2n+1}$.\nA similar property exists when a de Bruijn sequence is generated\nfrom the necklaces of pure cycling register of order~$n$~\\cite{EtLe84,Fred82}.\n\nFinding a nearly spanning tree $\\hat{T}_{2n+1}$ is an open question.\nBut, we found such trees for $n=3$ and $n=4$.\nWe believe that a similar construction to the\none which follows in the sequel for $n=3$ and $n=4$, exists for all $n>4$.\n\n\\begin{conjecture}\n\\label{conj:consB}\nFor each $n\\geq 2$, there exists\na $(2n+1,M_{2n+1},\\mK)$-snake,\nwhere $M_{2n+1}=\\frac{(2n+1)!}{2}-(2n-1)$\nin which there are only $t_{2n-1}$ and $t_{2n+1}$ p-transitions.\n\\end{conjecture}\n\n\n\\begin{example}\n\\label{ex:consB_7}\nFor $n=3$, a $(7,2515,\\mK)$-snake is constructed by using the tree $T_7$ of Example \\ref{Exm:T7},\nand the tree $\\hat{T}_7$ defined below.\n$\\hat{T}_7$ contains $12$ chains, where each chain contains $41$ necklaces.\nIt also contains $11$ $[2,1]$-necklaces and $11$ hyperedges.\nDenote an edge in $\\hat{H}_{7}$ by $(\\{ c_i, c_j, \\eta_k \\}, x)$\nwhere $M[x]$ is the sign of the edge.\n$\\hat{T}_{7}$ is defined as follows.\n\\begin{tabular}{ll}\n\\underline{The chains in $\\hat{T}_{7}$:} & \\tabularnewline\n$c_1\\ =[3, 4, 5, 6, 7]-[1, 2]$, &\n$c_2\\ =[3, 4, 6, 7, 5]-[1, 2]$, \\tabularnewline\n$c_3\\ =[3, 4, 7, 5, 6]-[1, 2]$, &\n$c_4\\ =[3, 5, 4, 7, 6]-[1, 2]$, \\tabularnewline\n$c_5\\ =[3, 5, 6, 4, 7]-[1, 2]$, &\n$c_6\\ =[3, 5, 7, 6, 4]-[1, 2]$, \\tabularnewline\n$c_7\\ =[3, 6, 4, 5, 7]-[1, 2]$, &\n$c_8\\ =[3, 6, 5, 7, 4]-[1, 2]$, \\tabularnewline\n$c_9\\ =[3, 6, 7, 4, 5]-[1, 2]$, &\n$c_{10}=[3, 7, 4, 6, 5]-[1, 2]$,\\tabularnewline\n$c_{11}=[3, 7, 5, 4, 6]-[1, 2]$,&\n$c_{12}=[3, 7, 6, 5, 4]-[1, 2]$.\n\\end{tabular}\n\\begin{tabular}{ll}\n\\underline{The necklaces in $\\hat{T}_{7}$:} & \\tabularnewline\n$\\eta_1\\ =[3, 4, 5, 7, 6]-[2, 1]$, &\n$\\eta_2\\ =[3, 4, 6, 5, 7]-[2, 1]$, \\tabularnewline\n$\\eta_3\\ =[3, 4, 7, 6, 5]-[2, 1]$, &\n$\\eta_4\\ =[3, 5, 4, 6, 7]-[2, 1]$, \\tabularnewline\n$\\eta_5\\ =[3, 5, 6, 7, 4]-[2, 1]$, &\n$\\eta_6\\ =[3, 5, 7, 4, 6]-[2, 1]$, \\tabularnewline\n$\\eta_7\\ =[3, 6, 4, 7, 5]-[2, 1]$, &\n$\\eta_8\\ =[3, 6, 5, 4, 7]-[2, 1]$, \\tabularnewline\n$\\eta_9\\ =[3, 6, 7, 5, 4]-[2, 1]$, &\n$\\eta_{10}=[3, 7, 4, 5, 6]-[2, 1]$, \\tabularnewline\n$\\eta_{11}=[3, 7, 5, 6, 4]-[2, 1]$.\n\\end{tabular}\n\\begin{tabular}{ll}\n\\underline{The edges in $\\hat{T}_{7}$:} & \\tabularnewline\n$e_1\\ =(\\{ c_{11}, c_6\\hphantom{\\scriptsize{i}}, \\eta_9\\hphantom{\\scriptsize{i}} \\}, 6)$, &\n$e_2\\ =(\\{ c_6\\hphantom{\\scriptsize{i}}, c_1\\hphantom{\\scriptsize{i}}, \\eta_2\\hphantom{\\scriptsize{i}} \\}, 6)$, \\tabularnewline\n$e_3\\ =(\\{ c_2\\hphantom{\\scriptsize{i}}, c_{12}, \\eta_{11} \\}, 6)$, &\n$e_4\\ =(\\{ c_{12}, c_7\\hphantom{\\scriptsize{i}}, \\eta_4\\hphantom{\\scriptsize{i}} \\}, 6)$, \\tabularnewline\n$e_5\\ =(\\{ c_5\\hphantom{\\scriptsize{i}}, c_3\\hphantom{\\scriptsize{j}}, \\eta_3\\hphantom{\\scriptsize{i}} \\}, 6)$, &\n$e_6\\ =(\\{ c_3\\hphantom{\\scriptsize{i}}, c_4\\hphantom{\\scriptsize{i}}, \\eta_7\\hphantom{\\scriptsize{i}} \\}, 6)$, \\tabularnewline\n$e_7\\ =(\\{ c_9\\hphantom{\\scriptsize{i}}, c_{10}, \\eta_{10} \\}, 6)$, &\n$e_8\\ =(\\{ c_{10}, c_8\\hphantom{\\scriptsize{i}}, \\eta_5\\hphantom{\\scriptsize{i}} \\}, 6)$, \\tabularnewline\n$e_9\\ =(\\{ c_{12}, c_9\\hphantom{\\scriptsize{i}}, \\eta_8\\hphantom{\\scriptsize{i}} \\}, 7)$, &\n$e_{10}=(\\{ c_9\\hphantom{\\scriptsize{i}}, c_3\\hphantom{\\scriptsize{i}}, \\eta_1\\hphantom{\\scriptsize{i}} \\}, 7)$, \\tabularnewline\n$e_{11}=(\\{ c_2\\hphantom{\\scriptsize{i}}, c_{11}, \\eta_6\\hphantom{\\scriptsize{i}} \\}, 7)$.\n\\end{tabular}\n\n\n$\\hat{H}_{7}$ contains another $[2,1]$-necklace, $\\eta_{12}=[3, 7, 6, 4, 5]-[2, 1]$,\nand the following additional edges:\n\\begin{align*}\n\\begin{tabular}{ll}\n$e_{12}=(\\{ c_1\\hphantom{\\scriptsize{'}}, c_{11}, \\eta_{12} \\}, 6),$\n&\n$e_{13}=(\\{ c_7\\hphantom{\\scriptsize{i}}, c_2\\hphantom{\\scriptsize{i}}, \\eta_1\\hphantom{\\scriptsize{i}} \\}, 6)$, \\tabularnewline\n$e_{14}=(\\{ c_4\\hphantom{\\scriptsize{i}}, c_5\\hphantom{\\scriptsize{1}}, \\eta_8\\hphantom{\\textbf{\\scriptsize{i}}} \\}, 6)$,  &\n$e_{15}=(\\{ c_8\\hphantom{\\scriptsize{i}}, c_9\\hphantom{\\scriptsize{i}}, \\eta_6\\hphantom{\\scriptsize{i}} \\}, 6)$, \\tabularnewline\n$e_{16}=(\\{ c_{10}, c_2\\hphantom{\\scriptsize{i}}, \\eta_2\\hphantom{\\scriptsize{i}} \\}, 7)$,  &\n$e_{17}=(\\{ c_8\\hphantom{\\scriptsize{i}}, c_1\\hphantom{\\scriptsize{i}}, \\eta_3\\hphantom{\\scriptsize{i}} \\}, 7)$, \\tabularnewline\n$e_{18}=(\\{ c_{11}, c_{10}, \\eta_4\\hphantom{\\scriptsize{'}} \\}, 7)$,  &\n$e_{19}=(\\{ c_3\\hphantom{\\scriptsize{i}}, c_{12}, \\eta_5\\hphantom{\\scriptsize{i}} \\}, 7)$,  \\tabularnewline\n$e_{20}=(\\{ c_6\\hphantom{\\scriptsize{i}}, c_7\\hphantom{\\scriptsize{i}}, \\eta_7\\hphantom{\\scriptsize{i}} \\}, 7)$, &\n$e_{21}=(\\{ c_4\\hphantom{\\scriptsize{i}}, c_8\\hphantom{\\scriptsize{i}}, \\eta_9\\hphantom{\\scriptsize{i}} \\}, 7)$, \\tabularnewline\n$e_{22}=(\\{ c_1\\hphantom{\\scriptsize{i}}, c_4\\hphantom{\\scriptsize{i}}, \\eta_{10} \\}, 7)$,  &\n$e_{23}=(\\{ c_5\\hphantom{\\scriptsize{i}}, c_6\\hphantom{\\scriptsize{i}}, \\eta_{11} \\}, 7)$, \\tabularnewline\n$e_{24}=(\\{ c_7\\hphantom{\\scriptsize{i}}, c_5\\hphantom{\\scriptsize{i}}, \\eta_{12} \\}, 7)$.\n\\end{tabular}\n\\end{align*}\nAn additional different illustration of $\\hat{H}_{7}$ is presented in the sequel (see Example~\\ref{ex:g_7}).\n\\end{example}\n\nFor each $n\\geq 3$,\nlet $\\cG_{2n+1}=(\\cV_{2n+1},\\cE_{2n+1})$ be a multi-graph (with parallel edges) with labels and signs on the edges.\nThe vertices of $\\cV_{2n+1}$ represent the $\\frac{(2n-2)!}{2}$ chains\nand hence $|\\cV_{2n+1}|=\\frac{(2n-2)!}{2}$.\nThere is an edge signed with $M[x]$, where $6\\leq x\\leq 2n+1$,\nbetween the vertex (chain) $c_1$ and vertex (chain) $c_2$,\nif $c_1$ contains a permutation $[\\alpha,2,1,x]$\nand $c_2$ contains the permutation $[\\alpha,1,x,2]$, where $c_1\\ne c_2$.\nThe label on this edge is the necklace $[\\alpha,x]-[2,1]$.\nNote that the label on the edge is a necklace\nwhich can merge together the chains of its corresponding endpoints by $M[x]$-connection.\nNote also that the pair $\\alpha, x$ might not be unique\nand hence the graph might have parallel edges.\nA tree in $\\cG_{2n+1}$ which doesn't have two edges with the same label,\nwill be called \\emph{a chain tree}.\nThe following Lemma can be easily verified.\n\\begin{lemma}\nThere exists a nearly spanning tree in $\\hat{H}_{2n+1}$\nif and only if there exists a chain tree in $\\cG_{2n+1}$.\n\\end{lemma}\n\nHenceforth, $T_{2n+1}$ will be the nearly spanning tree constructed in Theorem \\ref{thm:treeCompose},\nand the chains are constructed via~$T_{2n+1}$.\n\\begin{definition}\nLet $\\cG_1=(\\cV_1, \\cE_1)$ and $\\cG_2=(\\cV_2,\\cE_2)$ be two multi-graphs with labels and signs on the edges,\nwhere the set of the labels of $\\cG_i$ denoted by $\\cL_i$, $i\\in \\{1,2\\}$.\nWe say that\n$\\cG_1$ is isomorphic to $\\cG_2$ if there exist two bijective functions $f: \\cV_1\\to \\cV_2$ and $g: \\cL_1 \\to \\cL_2$,\nwith the following property:\n$(u,v)\\in \\cE_1$ with the label $\\eta$ and sign $M[x]$, if and only if\n$(f(u),f(u)) \\in \\cE_2$ with the label $g(\\eta)$ and sign $M[x]$.\n\\end{definition}\n\\begin{definition}\nFor each $n\\geq 4$, a sub-graph of $\\cG_{2n+1}$ which is isomorphic to $\\cG_{2n-1}$\nis called a component of $\\cG_{2n+1}$,\nand denoted by $A=(\\cV_A,\\cL_A)$\nwhere $\\cV_A$ consists of the vertices (chains) of the component,\n$\\cL_A$ consists of the labels ($[2,1]$-necklaces) on the edges in the component.\nNote that $|\\cV_A|=|\\cL_A|$, i.e., the numbers of the distinct labels is equal to the number of the vertices.\n\\end{definition}\n\\begin{definition}\nTwo components, $A=(\\cV_A,\\cL_A)$ and $B=(\\cV_B,\\cL_B)$, in $\\cG_{2n+1}$ are called disjoint\nif $\\cV_A\\cap \\cV_B =\\varnothing$ and $\\cL_A\\cap \\cL_B =\\varnothing$, i.e., there is no a common vertex (chain) or a common label ($[2,1]$-necklace) in $A$ and $B$.\n\\end{definition}\n\\begin{lemma}\n\\label{lem:2n+1structure}\nFor each $n\\geq 4$, $\\cG_{2n+1}$ consists of $(2n-3)(2n-2)$ disjoint copies\nof isomorphic graphs to $\\cG_{2n-1}$, called components.\nThe edges between the vertices of two distinct components are signed\nonly with $M[2n]$ and $M[2n+1]$.\n\\end{lemma}\n\\begin{proof}\nThe $M[x]$-connections are\ndeduced by the tree $T_{2n+1}$, which was used for the construction of the chains.\nIn particular, the path between the vertices $[1,x]$ and $[x,2]$\nin $T_{2n+1}$ determines the $M[x]$-connections in~$\\cG_{2n+1}$.\nBy Theorem~\\ref{thm:treeCompose}, $T_{2n-1}$ is a sub-graph of $T_{2n+1}$.\nTherefore, for each $x$, $x\\geq 3$,\nthe path between the vertices $[1,x]$ and $[x,2]$ in $T_{2n+1}$ is equal to\nthe path between the vertices $[1,x]$ and $[x,2]$ in $T_{2k+1}$ for each ${x\\leq 2k+1\\leq 2n+1}$.\nThe number of the vertices (chains) in $\\cG_{2n+1}$ is equal to $\\frac{(2n-2)!}{2}$,\nand each component contains $\\frac{(2n-4)!}{2}$ vertices.\nThus, $\\cG_{2n+1}$ consists of $(2n-3)(2n-2)$ disjoint copies\nof isomorphic graphs to $\\cG_{2n-1}$ connected by edges signed only with $M[2n]$ and $M[2n+1]$.\n\\end{proof}\n\nFor each $n\\geq 4$, let $\\hat{\\cG}_{2n+1}=(\\hat{\\cV}_{2n+1},\\hat{\\cE}_{2n+1})$ be the \\emph{component graph} of\n$\\cG_{2n+1}$.\nThe vertices of $\\hat{\\cV}_{2n+1}$ represent the components of $\\cG_{2n+1}$,\nThere is an edge signed with $M[x]$, $x\\in \\{2n,2n+1\\}$,\nbetween the vertices (components) $A$ and $B$,\nif the chain that contains the permutation $[\\alpha,2,1,x]$ is contained in $A$,\nand the chain that contains the permutation $[\\alpha,1,x,2]$ is contained in $B$.\nThe label on this edge is the necklace $[\\alpha,x]-[2,1]$.\nWe define $\\hat{\\cG}_{7}$ to be $\\cG_{7}$, i.e., each component of $\\hat{\\cG}_{7}$ consists of exactly one chain\n(and also one distinct $[2,1]$-necklace in order to follow\nthe properties of $\\hat{\\cG}_{2n+1}$).\n\\begin{definition}\nA components spanning tree, $\\hat{T}_{2n+1}$\nis a spanning tree in $\\hat{\\cG}_{2n+1}$,\nwhere in the set of the labels of the tree's edges,\nthere are no two labels from the same component, i.e.,\neach label in the set of the labels of the tree's edges belongs to a different component.\n\\end{definition}\n\\begin{example}\n\\label{ex:g_7}\n$\\hat{\\cG}_{7}$ is depicted in Figure~\\ref{fig:hatG7}, where the vertices numbers and the edges labels\ncorresponds to the chains and the necklaces in Example \\ref{ex:consB_7}, respectively.\nThe vertical edges are signed with $M[6]$,\nwhile the horizontal edges are signed with $M[7]$.\nThe double lines edges correspond to the edges of $\\hat{T}_7$.\n\\begin{figure}[H]\n\\centering\n\\vspace{-10pt}\n\\begin{tikzpicture} [-,scale=0.6,node distance=3cm,\n  thick,main node\/.style={font={\\normalsize}, circle,draw,minimum size=25 pt,scale=0.6}\n  ,edge_node\/.style={font={\\normalsize},scale=0.6}\n  ,edgeTree_node\/.style={font={\\large},scale=0.6}]\n\n  \\node[main node] (1) {1};\n  \\node[main node] (11) [below of=1] {11};\n  \\node[main node] (6) [below of=11] {6};\n\n  \\node[main node] (2) [right of=1] {2};\n  \\node[main node] (12) [below of=2] {12};\n  \\node[main node] (7) [below of=12] {7};\n\n  \\node[main node] (3) [right of=2] {3};\n  \\node[main node] (4) [below of=3] {4};\n  \\node[main node] (5) [below of=4] {5};\n\n  \\node[main node] (8) [right of=3] {8};\n  \\node[main node] (9) [below of=8] {9};\n  \\node[main node] (10) [below of=9] {10};\n\n  \\path[]\n    (1) edge node[edge_node] [left, xshift=.1cm] {12} (11)\n\n    (7) edge [bend left] node[edge_node] [left,xshift=.1cm] {1} (2)\n\n    (4) edge node[edge_node] [left,xshift=.1cm] {8} (5)\n\n    (8) edge node[edge_node] [right,xshift=-.1cm] {6} (9);\n\n  \\path[]\n \n   \n    (11) edge[double, thin] node[edgeTree_node] [left,xshift=.1cm] {9} (6)\n    (6) edge [double, thin,bend left] node[edgeTree_node] [left,xshift=.1cm] {2} (1)\n\n    (2) edge[double,thin] node[edgeTree_node] [left,xshift=.1cm, yshift=-.6cm] {11} (12)\n    (12) edge[double,thin] node[edgeTree_node] [left,xshift=.1cm] {4} (7)\n\n    (5) edge [double,thin,bend left] node[edgeTree_node] [left,xshift=.1cm] {3} (3)\n    (3) edge[double,thin] node[edgeTree_node] [left,xshift=.1cm, yshift=-.1cm] {7} (4)\n\n    (9) edge[double,thin] node[edgeTree_node] [right,xshift=-.1cm] {10} (10)\n    (10) edge [double,thin,bend left] node[edgeTree_node] [right,xshift=-.1cm] {5} (8)\n\n   \n    (2) edge[double,thin] node[edge_node] [above] {6} (11)\n\n    (12) edge[double,thin,bend right] node[edge_node] [above, xshift=.7cm] {8} (9)\n    (9) edge[double,thin] node[edge_node] [above, xshift=-0.8cm, yshift=0.8cm] {1} (3);\n\n  \\path[]\n    (6) edge node[edge_node] [above,yshift=-.1cm] {7} (7)\n    (7) edge node[edge_node] [above,yshift=-.1cm] {12} (5)\n    (5) edge [bend left] node[edge_node] [above,yshift=-.1cm] {11} (6)\n\n    (1) edge node[edge_node] [above, xshift=-2cm, yshift=1cm] {10} (4)\n    (4) edge node[edge_node] [right, xshift=0.5cm, yshift=0.9cm] {9} (8)\n    (8) edge[bend right=20] node[edge_node] [above] {3} (1)\n\n    (11) edge node[edge_node] [above] {4} (10)\n    (10) edge [bend right] node[edge_node] [above,xshift=-.4cm,yshift=.4cm] {2} (2)\n\n    (3) edge node[edge_node] [above] {5} (12);\n\n`\\end{tikzpicture}\n\\vspace{-5pt}\n\\caption{The graph $\\hat{\\cG}_7$ and its component spanning tree $\\hat{T}_7$} \\label{fig:hatG7}\n\\end{figure}\n\\end{example}\n\n\\begin{conjecture}\n\\label{conj:main}\nFor each component $A$ in $\\hat{\\cG}_{2n+1}$, $n\\geq 3$,\nand for each label $\\eta$ of $A$,\nthere exists a components spanning tree,\nwhere there is no edge in the tree with the label $\\eta$.\n\\end{conjecture}\nConjecture \\ref{conj:main} implies Conjecture \\ref{conj:consB}, i.e.,\n\\begin{theorem}\nIf Conjecture \\ref{conj:main} is true then for each $n\\geq 2$, there exists\na $(2n+1,M_{2n+1},\\mK)$-snake,\nwhere $M_{2n+1}=\\frac{(2n+1)!}{2}-(2n-1)$\nin which there are only $t_{2n-1}$ and $t_{2n+1}$ p-transitions.\n\\end{theorem}\nConjecture \\ref{conj:main} was verified by computer search for $n=3$ and $n=4$.\nBy using Conjecture \\ref{conj:main} recursively,\nfor each $n\\geq 3$, and for each necklace $\\eta$ in class $[2,1]$,\nwe can construct a chain tree $T$ in $\\cG_{2n+1}$,\nwhich doesn't include $\\eta$ as a label on an edge in $T$.\n\n\\begin{cor}\nThere exist a $(7,2515,\\mK)$-snake and a~$(9,181433,\\mK)$-snake,\nand hence $\\lim\\limits_{n\\to \\infty} \\frac{M_{2n+1}}{S_{2n+1}}\\approx 0.4743$.\n\\end{cor}\n\nNote that the ratio $\\lim\\limits_{n\\to \\infty} \\frac{M_{2n+1}}{S_{2n+1}}$ would be improved,\nif there exists a $(2m+1,\\frac{(2m+1)!}{2}-(2m-1),\\mK)$-snake for some $m>4$.\n\n\\begin{conjecture}\nThe $(2n-3)(2n-2)$ components in $\\hat{\\cG}_{2n+1}$\ncan be arranged in a $(2n-3)\\times (2n-2)$ grid.\nThe edges which are sign by $M[2n]$ define $2n-2$ cycles of length $2n-3$.\nEach cycle contains the vertices of exactly one column, and is called an $M[2n]$-cycle.\nThe edges which are sign with $M[2n+1]$ are between two components in different columns,\nand they also define $2n-2$ cycles of length $2n-3$.\nSuch a cycle will be called an $M[2n+1]$-cycle.\nEach multi-edge between two components has $\\frac{(2n-4)!}{2}$ parallel edges (the number of chains in the component).\nParallel edges have the same sign $x$, $x\\in \\{2n,2n+1\\}$,\nbut different labels (i.e., $M[x]$-connection, but with different $[2,1]$-necklaces).\n\\end{conjecture}\n\\begin{example}\nAn illustration for the structure of $\\hat{\\cG}_{2n+1}$ for $n=3$ is presented in Example~\\ref{ex:g_7},\nand for $n=4$ is depicted in Figure~\\ref{fig:hatG9}.\nIn $\\hat{\\cG}_9$ there are $30$ components,\nwhere each component is isomorphic to $\\hat{\\cG}_7$\n(thus, it contains $12$ chains and $12$ $[2,1]$-necklaces).\n\\begin{figure}[H]\n\\vspace{-5pt}\n\\begin{tikzpicture} [-, scale=0.5, node distance=1.5 cm,\n  thick,main node\/.style={font={\\normalsize},circle,draw,scale=0.5,minimum size=20 pt}\n  ,edge node\/.style={font={\\normalsize},scale=0.5}]\n\n  \\node[main node] (6) {1};\t\n  \\node[main node] (7) [below of=6]{2};\t\n  \\node[main node] (8) [below of=7]{3};\t\n  \\node[main node] (9) [below of=8]{4};\t\n  \\node[main node] (10) [below of=9]{5};\t\n\n  \\node[main node] (1)[right= 1.15cm of 6] {6};\t\t\t\t\n  \\node[main node] (2) [below of=1]{7};\t \n  \\node[main node] (3) [below of=2]{8};\t\t\n  \\node[main node] (4) [below of=3]{9};\t\t\n  \\node[main node] (5) [below of=4]{10};\t\t\n\n  \\node[main node] (26) [right= 1.15cm of 1]{11};\t\n  \\node[main node] (27) [below of=26]{12};\t\n  \\node[main node] (28) [below of=27]{13};\t\n  \\node[main node] (29) [below of=28]{14};\t\n  \\node[main node] (30) [below of=29]{15};\t\n\n  \\node[main node] (11) [right= 1.15cm of 26]{16};\t\n  \\node[main node] (12) [below of=11]{17};\t\n  \\node[main node] (13) [below of=12]{18};\t\n  \\node[main node] (14) [below of=13]{19};\t\n  \\node[main node] (15) [below of=14]{20};\t\n\n  \\node[main node] (16) [right= 1.15cm of 11]{21};\t\n  \\node[main node] (17) [below of=16]{22};\t\n  \\node[main node] (18) [below of=17]{23};\t\n  \\node[main node] (19) [below of=18]{24};\t\n  \\node[main node] (20) [below of=19]{25};\t\n\n  \\node[main node] (21) [right= 1.15cm of 16]{26};\t\n  \\node[main node] (22) [below of=21]{27};\t\n  \\node[main node] (23) [below of=22]{28};\t\n  \\node[main node] (24) [below of=23]{29};\t\n  \\node[main node] (25) [below of=24]{30};\t\n\n\n  \\path[thin]\n    (1) edge (2)\n    (2) edge (3)\n    (3) edge (4)\n    (4) edge (5)\n    (5) edge [bend left=20] (1)\n\n\n    (6) edge (7)\n    (7) edge (8)\n    (8) edge (9)\n    (9) edge (10)\n    (10) edge [bend left=20] (6)\n\n\n    (11) edge (12)\n    (12) edge (13)\n    (13) edge (14)\n    (14) edge (15)\n    (15) edge [bend left=20] (11)\n\n    (16) edge (17)\n    (17) edge (18)\n    (18) edge (19)\n    (19) edge (20)\n    (20) edge [bend left=20] (16)\n\n    (21) edge (22)\n    (22) edge (23)\n    (23) edge (24)\n    (24) edge (25)\n    (25) edge [bend left=20] (21)\n\n\n    (26) edge (27)\n    (27) edge (28)\n    (28) edge (29)\n    (29) edge (30)\n    (30) edge [bend left=20] (26);\n\n\n\\path[thin]\n    (25) edge (28)\n    (28) edge[bend left=16] (10)\n    (10) edge (3)\n    (3)  edge (12)\n    (12) edge (25)\n\n    (15) edge[bend right=16] (23)\n    (23) edge[bend left=12, out=60, in=200] (5)\n    (5)  edge (9)\n    (9)  edge[bend left=16] (19)\n    (19) edge (15)\n\n    (1)  edge (26)\n    (26) edge (11)\n    (11) edge (16)\n    (16) edge (21)\n    (21) edge[bend right=16] (1)\n\n    (7)  edge (2)\n    (2)  edge (29)\n    (29) edge[bend left=16] (17)\n    (17) edge (13)\n    (13) edge (7)\n\n    (22) edge[bend right=3, out=10, in=200] (8)\n    (8)  edge (14)\n    (14) edge (30)\n    (30) edge[bend right=16] (18)\n    (18) edge (22)\n\n    (24) edge[out=125, in=305] (6)\n    (6)  edge (27)\n    (27) edge (4)\n    (4)  edge (20)\n    (20) edge (24);\n\\end{tikzpicture}\n\\vspace{-30pt}\n\\caption{The graph $\\hat{\\cG}_9$} \\label{fig:hatG9}\n\\end{figure}\n\\end{example}\n\n\n\\section{Conclusions and Future Research}\n\\label{sec:conclude}\n\nGray codes for permutations using the operation push-to-the-top\nand the Kendall's $\\tau$-metric were discussed.\nWe have presented a framework for constructing snake-in-the-box codes\nfor $S_n$. The framework for the construction yield a recursive construction\nwith large snakes. A direct construction to obtain snakes which might\nbe optimal in length was also presented. Several questions arise from our discussion\nand they are considered for current and future research.\n\\begin{enumerate}\n\\item Complete the direct construction for snakes of length\n$\\frac{(2n+1)!}{2} - 2n+1$ in $S_{2n+1}$.\n\n\\item Can a snake in $S_{2n+1}$ have size larger than\n$\\frac{(2n+1)!}{2} - 2n+1$?\n\n\\item Prove or disprove that the length of the longest snake in $S_{2n}$\nis not longer than the length of the longest snake in~$S_{2n-1}$.\n\n\\item Examine the questions in this paper for the $\\ell_\\infty$ metric.\n\\end{enumerate}\n\n\\begin{center}\n{\\bf Acknowledgment}\n\\end{center}\nThe authors would like to thank the anonymous reviewers for their careful reading of the paper.\nEspecially, one of the reviewers pointed out on the good ratio $\\lim\\limits_{n\\to \\infty} \\frac{M_{2n+1}}{S_{2n+1}}\\approx 0.4338$ compared to one in \\cite{YeSc12}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe quantum Rabi model (QRM) is widely recognized as the simplest and most fundamental model describing quantum light-matter interactions, that is, the interaction between a two-level system and a bosonic field mode. Indeed, it is considered a milestone in the long history of quantum physics \\cite{HR2008, KG2004}. In \\cite{bcbs2016} we can find a recent collection of introductory, survey and original articles from both experimental and theoretical viewpoints not limited to light-matter interaction but also in diverse fields of research. For a notable achievement in recent experimental studies, we refer the reader to \\cite{YS2018}.\n\nThe Hamiltonian \\(H_{\\text{\\upshape Rabi}}\\) of the QRM is precisely given by\n\\[\n  H_{\\text{\\upshape Rabi}} := \\omega a^{\\dagger}a + \\Delta \\sigma_z + g (a + a^{\\dagger}) \\sigma_x .\n\\]\nHere, \\(a^{\\dagger}\\) and \\(a\\) are the creation and annihilation operators of the single bosonic mode (\\([a,a^{\\dagger}]=1 \\)), $\\sigma_x, \\sigma_z$ are the Pauli matrices (sometimes written as \\(\\sigma_1\\) and \\(\\sigma_3\\), but since there is no risk of confusion with the variable \\(x\\) to appear below in the heat kernel, we use the usual notations), $2\\Delta$ is the energy difference between the two levels, $g$ denotes the coupling strength between the two-level system and the bosonic mode with frequency $\\omega$ (subsequently, we set $\\omega=1$ without loss of generality). The Hamiltonian \\(H_{\\text{\\upshape Rabi}}\\) of the QRM has a $\\Z_2$-symmetry that gives the parity decomposition\n\\[\n  H_{\\text{Rabi}} = H_{+} \\oplus H_{-}.\n\\]\nWe note that the integrability of the QRM was established in \\cite{B2011PRL} using the $\\Z_2$-symmetry. \n\nThe aim of the present paper is to provide a explicit formula for the propagator of the Schr\\\"odinger equation for the\nQRM and the same for the two systems defined by the parity decomposition using the explicit analytical formula of\nthe corresponding heat kernels \\(K_{\\text{Rabi}}(x,y,t)\\) and \\(K_\\pm(x,y,t, \\Delta)\\) obtained\nin \\cite{RW2020hk}. The explicit formula may be used for precise computation of time evolution of quantum states without the\nnumerical drawbacks appearing in prior methods (see e.g. \\cite{WKB2012}).\n\nWe recall that the propagator is the integral kernel associated to the Schr\\\"odinger equation\ncorresponding to $H_{\\text{\\upshape Rabi}}$. Precisely, it is a (two-by-two matrix valued) function $U_{\\text{Rabi}}(x,y,t)$ satisfying\n\\begin{align*}\n& i \\frac{\\partial}{\\partial t} U_{\\text{Rabi}}(x,y,t) = -H_{\\text{\\upshape Rabi}} U(x,y,t) \\quad \\text{for all}\\quad t>0, \\\\ \n& \\lim_{t \\to 0}U_{\\text{Rabi}}(x,y,t)=\\delta_x(y) {\\bf I}_2 \\quad \\text{for}\\quad x,y \\in \\R.\n\\end{align*}\nIn other words, we have the expression as a time evolution operator \n\\begin{align*}\nU_{\\text{Rabi}}(x,y,t)=\\langle x\\,|\\,e^{-itH_{\\text{Rabi}}} \\,|\\,y\\rangle (= K_{\\text{Rabi}}(x,y,it)) \\quad \\text{for} \\quad t>0,\n\\end{align*}\nwhenever the Wick rotation, the analytic continuation of the heat kernel $K_{\\text{Rabi}}(x,y,t)$ from real to imaginary time,\nexists (possibly with singularities). \n\nLet us mention works related to the computation of the propagator or heat kernel of QRM.\nIn \\cite{ZZ1988} it has been given an approximated formula for the propagator using path-integral techniques. For the Spin-Boson model, and the QRM as a special case, the Feynman-Kac formula for the heat kernel was obtained in \\cite{HH2012,HHL2012} via a Poisson point process and a Euclidean field. However, for the study of longtime behavior of the system, the use of numerical computations or approximations is inevitable (see e.g. \\cite{LF2011, CPMP}). Moreover, recently in \\cite{DSGKSN2017}, it was shown that even approximation forms which are obtained by a perturbative approach provides can provide a significant insight. In fact, it was shown in \\cite{DSGKSN2017} that perturbative diagrammatic approach provides a direct visualization of virtual and physical photons in the physical process using the Jaynes-Cummings (the RWA of the QRM)) propagator.\n\nOur second theme is the study of the spectral (functional) determinant, that is, the zeta regularized product of the spectrum for $H_{\\pm}$. The spectral determinant of an operator is a function that has zeros at the eigenvalues, that is, it is the generalization of the characteristic polynomial for a finite matrix, It describes various important topological and\/or number theoretical invariants (see e.g. \\cite{RS1974, QHS1993TAMS}). In \\cite{KRW2017}, a significant relation was found between Braak's $G$-function, the power series of transcendental function used to prove the integrability of the QRM \\cite{B2011PRL}, and the spectral zeta function of the QRM \\cite{Sugi2016} (the Mellin transform of the partition function). Actually, the $G$-function is (up to a non-vanishing function) equal to the spectral determinant\nof \\(H_{\\text{\\upshape Rabi}}\\), that is, the zeta-regularized product associated to the spectral zeta function of the QRM.\nThis result is significant because, on the one hand, the $G$-function is defined through the solutions of system of ordinary differential equations (which is equivalent with the confluent Heun ODE picture of the QRM) that assures the existence of the entire solutions (see Appendix \\ref{sec:bargm-space-confl}) and, on the other hand, the spectral determinant arises from the linear term of the Taylor expansion of the spectral zeta function at the origin. The identification of $G$-function and spectral determinant suggests a deeper relation between the exact solvability of a quantum interaction system and the meromorphic continuation of its spectral zeta function.\n\nWe extend this result for the case of the Hamiltonians $H_{\\pm}$ of the parity decomposition of the QRM. In order to accomplish this, we show the meromorphic continuation of the spectral zeta function of the Hamiltonians $H_{\\pm}$. \nIn addition, by using the explicit formula of the partition function $Z_{\\text{Rabi}}(t)$ of the QRM\n  we obtain a contour integral representation of spectral zeta function $\\zeta_{\\text{QRM}}(s; \\tau)$\n  (Theorem \\ref{IntRep_SZF}). We leave the proof of the integral representation to Appendix \\ref{sec:proofmero}. There, we also discuss certain interesting properties of the spectral zeta function that are the consequence of the contour integral expression.\n\nThe common tool behind the two results in this paper, that is, the propagator formula and the identification of the spectral determinant with the $G$-function, is the meromorphic continuation to the complex plane of the heat kernel and partition function allowed by precise estimates obtained from the analytical formulas obtained in \\cite{RW2020hk}.\n\n\\section{Revisited: the discrete path integral for the heat kernel of QRM} \\label{sec:limit}\n\nIn this section we recall the formulas for the heat kernel and partition function of\nthe QRM obtained in \\cite{RW2020hk}. \nAs an introduction for the reader and to complement the discussion of the aforementioned paper, we give a brief overview of the method of computation for the heat kernel which we refer here as the method of discrete paths. We also provide a new interpretation on the resulting expression of the heat kernel using representation theory.\n\n\\subsubsection{Trotter-Kato's product formula}\n\nThe Hamiltonian of the QRM is given by\n\\[\n  H_{\\text{\\upshape Rabi}} := a^{\\dagger}a + \\Delta \\sigma_z + g (a + a^{\\dagger}) \\sigma_x,\n\\]\nand it is easy to see that it can be written as\n\\begin{align*}\n  H_{\\text{\\upshape Rabi}} = b^{\\dagger} b - g^2 + \\Delta \\sigma_z,\n\\end{align*}\nwhere $b = a +  g \\sigma_x$ and $b^{\\dagger} = a^\\dag +  g \\sigma_x $ are the annihilation and creation operators of\na non-commutative version of the quantum harmonic oscillator (i.e. $[b^{\\dagger},b] = \\bm{I}_2$).\n\nBy the Trotter-Kato product formula (see e.g. \\cite{Calin2011}), the heat semigroup is given by\n\\[\n  e^{- t H_{\\text{\\upshape Rabi}}} = e^{- t (b^{\\dagger}b -g^2 + \\Delta \\sigma_z)} = \\lim_{N\\to \\infty} (e^{-t (b^{\\dagger}b -g^2)\/N} e^{-t(\\Delta \\sigma_z)\/N})^N,\n\\]\nwith convergence in the strong operator topology, and the heat kernel $K_{\\text{Rabi}}(x,y,t)$ is obtained\nfrom this formula. In \\S\\ref{sec:Propagator}, the propagator $U_{\\text{Rabi}}(x,y,t)$ is obtained from the Heat kernel by\nthe Wick rotation $t \\to i t$ from the heat kernel $K_{\\text{Rabi}}(x,y,t)$. \n\nThe first step for the computation of the heat kernel is to obtain the explicit form of the integral kernel $K^{(N)}(x,y,t)$ of\n$(e^{-t (b^{\\dagger}b -g^2)\/N} e^{-t(\\Delta \\sigma_z)\/N})^N$. \nSince the kernel $K^{(N)}(x,y,t)$ is a two-by-two matrix-valued function, we can write it\nin terms of scalar and a non-commutative (matrix) parts in a reasonable way. More precisely, we write \n\\begin{align}\n  \\label{eq:sumGI}\n  K^{(N)}(x,y,t) = \\sum_{\\mathbf{s} \\in \\btZ{N}} G_N(u,\\Delta,\\mathbf{s}) I_N(x,y,u, \\, \\mathbf{s}),\n\\end{align}\nwhere $G_N(u,\\Delta,\\mathbf{s})$ is a matrix-valued function and the scalars $I_N(x,y,u, \\, \\mathbf{s})$ correspond to the evaluation of multivariate Gaussian integrals. The matrices appearing in the Hamiltonian $H_{\\text{\\upshape Rabi}}$ give rise to the finite group \\(\\Z_2^N\\)  structure of equation \\eqref{eq:sumGI}.\nAn important observation is that the $\\Z_2^{N}$-group structure can be interpreted in terms of finite paths as in Figure \\ref{fig:paths}, we refer to Section 4.2 of \\cite{RW2020hk} for more details. \n\nBy taking the limit, we see that the components of the heat kernel are, up to Mehler's type factor inherited by the\nquantum harmonic oscillator, given by\n\\begin{align}\n  \\label{eq:limit0}\n  & \\lim_{N \\to \\infty}\\left( \\frac{1-u^{\\frac{2\\Delta}N}}{2 u^{\\frac{\\Delta}N}} \\right) \\sum_{k \\geq 3}^N   \\left( \\frac{1+u^{\\frac{2\\Delta}N}}{2 u^{\\frac{\\Delta}N}} \\right)^{N-k}  J_\\eta^{(k,N)}(x,y,u^{\\frac{1}N},g)  \\sum_{ \\mathbf{s} \\in \\Z_2^{k-3}} g^{(\\alpha,\\eta)}_{k-1}(u^{\\frac1N},\\mathbf{s}) R^{(\\alpha,\\eta)}_\\eta(u^{\\frac{1}N},\\mathbf{s}),\n\\end{align}\nwith $\\alpha,\\eta \\in \\{0,1\\}$ and where we omitted some trivial terms for clarity (see equation (33) in \\cite{RW2020hk}). This form\nof the heat kernel suggests a possible evaluation as a type of Riemann integral. However, the presence of multiple changes of\nsigns depending on $\\mathbf{s} \\in \\Z_2^{k-3}$ and $k \\geq 3$ in the term $g^{(\\alpha,\\eta)}_{k-1}(u^{\\frac1N},\\mathbf{s}) R^{(\\alpha,\\eta)}_\\eta(u^{\\frac{1}N},\\mathbf{s})$ make such an approach unfeasible.\n\n\\subsubsection{Fourier transform on $\\Z_2^n  \\, (n\\geq0)$}\n\nAt this point the method described in \\cite{RW2020hk} greatly differs from the usual evaluation by path integrals.\nLet us denote by $|\\rho|$ the norm (or length) in $\\Z_2^{N}$, that is,\n$ |\\rho| = \\sum \\rho_i $ for $\\rho=(\\rho_1,\\rho_2,\\cdots,\\rho_{N})\\in \\Z_2^{N}$.\nBy using Fourier analysis, and more concretely Parseval's identity, on the group algebra \\(\\C[\\Z_2^{k-3}]\\), we see that\n\\[\n  \\sum_{ \\mathbf{s} \\in \\Z_2^{k-3}} g^{(\\alpha,\\eta)}_{k-1}(u^{\\frac1N},\\mathbf{s}) R^{(\\alpha,\\eta)}_\\eta(u^{\\frac{1}N},\\mathbf{s}) = \\sum_{ \\rho \\in \\Z_2^{k-3}}  \\hat{g}^{(\\alpha,\\eta)}_{k-1}(u^{\\frac1N},\\rho) \\hat{R}^{(\\alpha,\\eta)}_\\eta(u^{\\frac{1}N},\\rho).\n\\]\nThen, by fixing $|\\rho|=\\lambda \\in \\Z_{\\geq0}$, we observe that in the right-hand side of the above equation the function $\\hat{R}^{(\\alpha,\\eta)}_\\eta(u^{\\frac{1}N},\\rho)$ is given as the exponential of certain $q$-polynomials and that these $q$-polynomials do not depend on $k$. This process resembles the separation of a function defined on $\\R^{n}$ into  radial and non-radial parts. We remark, however, that the function $\\hat{R}^{(\\alpha,\\eta)}_\\eta(u^{\\frac{1}N},\\rho)$ is not radial, it is only determined (as a $q$-polynomial) by $|\\rho|$. In fact, it turns out that it actually determined by the $\\mathfrak{S}_\\infty$ orbit of $\\rho$ for certain action on $\\Z_2^{\\infty}$.\n\n\\subsubsection{$\\mathfrak{S}_\\infty$-action on $\\Z_2^{\\infty}$ (Geometric)}\n\\label{sec:mathfr-acti-z_2infty}\n\nThe infinite sum in the limit appearing in the heat kernel is identified with the inductive limit\n\\[\n  \\Z_2^{\\infty} =  \\varinjlim_{n} \\Z_2^{n},\n\\]\nwhere, for $ i \\leq j$, the injective homomorphisms $f_{i j} : \\Z_2^i \\to \\Z_2^j$ are given by\n\\[\n  f_{i j}(\\rho) = (\\rho_1,\\rho_2,\\cdots,\\rho_i,0,\\cdots,0) \\in \\Z_2^{j}\n\\]\nfor $\\rho = (\\rho_1,\\rho_2,\\cdots,\\rho_i) \\in \\Z_{2}^{i}$, that is, the natural group embeddings. As usual, we consider the inductive limit $\\Z_2^{\\infty}$ to\nbe equipped with the discrete topology.\n\nLet us also consider the infinite symmetric group $\\mathfrak{S}_\\infty$ obtained by the inductive limit of the finite symmetric groups $\\mathfrak{S}_n$ where the inductive homomorphisms are also given by the natural group embeddings. Then $\\mathfrak{S}_\\infty$ acts naturally on $\\Z_2^{\\infty}$ and the orbits are given by\n\\[\n  \\mathcal{O}_\\lambda := \\left\\{ \\sigma \\in \\Z_2^{\\infty} \\, : \\, |\\sigma|= \\lambda  \\right\\}\n\\]\nfor $\\lambda\\geq0$ and where the function $|\\cdot|$ is induced by the norms on each group $\\Z_2^{n}\\, (n\\geq0)$. Equivalently, we have $ \\mathcal{O}_{\\lambda} = \\mathfrak{S}_\\infty . \\sigma$, for $\\sigma \\in \\Z_2^{\\infty} $ with $|\\sigma|=\\lambda$ for $\\lambda \\in \\Z_{\\geq 0}$.\n\nIn other words, $|\\cdot|$ is an orbit invariant for the action and we get an orbit decomposition of $\\Z_2^{\\infty}$ by\n\\begin{equation*}\n  \\Z_2^{\\infty} = \\bigsqcup_{n=0}^{\\infty} \\mathfrak{S}_{\\infty} . [\\underbrace{1,1,\\cdots,1}_{n}] \\xleftrightarrow{ \\text{labeled by $|\\cdot|$}} \\Z_{\\geq 0},\n\\end{equation*}\nwhere \n\\[\n  [\\underbrace{1,1,\\cdots,1}_{n}] := (\\underbrace{1,1,\\cdots,1}_{n},0,0,\\cdots) \\in \\Z_2^{\\infty}\n\\]\nis the image of $(1,1,\\cdots,1) \\in \\Z_2^{n}$ in $\\Z_2^{\\infty}$. In Figure \\ref{fig:paths} we give an example of an element in $\\Z_2^{\\infty}$ and its corresponding canonical coset representative. We note that for the orbit $\\mathcal{O}_\\lambda$, the value $\\lambda$ of the invariant coincides with the length of the canonical element $(1,1,\\cdots,1)\\in \\Z_2^\\lambda$ described above.\n\n\\begin{figure}[!ht]\n  \\centering\n  \\begin{tikzpicture}[domain=0:4]\n\n   \n\n   \n    \\draw[step=1,very thin,color=gray] (0.6,2) grid (9,3);\n\n   \n    \\draw[thick] (0.6,2) -- (1,2) ;\n    \\draw[thick] (1,2) -- (9.1,2) \n    node[pos=1,below] {$9$}\n    node[pos=0,below]{1} node[pos=0.125,below]{2} node[pos=0.25,below]{3}\n    node[pos=0.375,below]{4} node[pos=0.5,below]{5} node[pos=0.625,below]{6}\n    node[pos=0.750,below]{7} node[pos=0.875,below]{8};\n\n   \n    \n    \\draw[thick,dashed] (9.1,3) -- (11,3) ;\n    \\draw[thick,dashed] (9.1,2) -- (11,2) ;\n\n   \n    \\draw[thick] (0.6,1.8) node[below] {$0$} -- (0.6,3.2) node[above] {$1$};\n\n    \\draw[color=blue,very thick]\n    (0.6,3) -- (1,3) to[out=0,in=180] (2,2) to[out=0,in=180] (3,3) to[out=0,in=180]\n    (4,3) to[out=0,in=180] (5,3) to[out=0,in=180] (6,2) to[out=0,in=180] (7,3)\n    to[out=0,in=180] (8,3) to[out=0,in=180] (9,2) --  (11,2);\n\n   \n\n    \\node at (1.5,0.5) {$\\equiv$};\n\n   \n    \n   \n    \\draw[step=1,very thin,color=gray] (2.6,0) grid (11,1);\n\n   \n    \\draw[thick] (2.6,0) -- (3,0) ;\n    \\draw[thick] (3,0) -- (11.1,0) \n    node[pos=1,below] {$9$}\n    node[pos=0,below]{1} node[pos=0.125,below]{2} node[pos=0.25,below]{3}\n    node[pos=0.375,below]{4} node[pos=0.5,below]{5} node[pos=0.625,below]{6}\n    node[pos=0.750,below]{7} node[pos=0.875,below]{8};\n\n   \n    \n    \\draw[thick,dashed] (11.1,1) -- (13,1) ;\n    \\draw[thick,dashed] (11.1,0) -- (13,0) ;\n\n   \n    \\draw[thick] (2.6,-0.2) node[below] {$0$} -- (2.6,1.2) node[above] {$1$};\n\n    \\draw[color=orange,very thick]\n    (2.6,1) -- (3,1) to[out=0,in=180] (4,1) to[out=0,in=180] (5,1) to[out=0,in=180]\n    (6,1) to[out=0,in=180] (7,1) to[out=0,in=180] (8,1) to[out=0,in=180] (9,0)\n    to[out=0,in=180] (10,0) to[out=0,in=180] (11,0) --  (13,0);\n\n   \n\n    \\node at (14.2,0.5) {$\\pmod{\\mathfrak{S}_{\\infty}}$};\n    \n  \\end{tikzpicture}\n  \\caption{A path in $\\Z_2^{9} \\subset \\Z_2^{\\infty} $ (above) and the corresponding canonical  $\\mathfrak{S}_{\\infty}$-orbit representative in $\\Z_2^{\\infty}$ (below) of the orbit $\\mathcal{O}_6$.}\n  \\label{fig:paths}\n\\end{figure}\n\nThus, using this idea, we rearrange the limit for the main body of heat kernel in the following way\n\\begin{equation}\n  \\label{eq:limit1}\n  \\sum_{\\lambda=0}^{\\infty} \\lim_{N \\to \\infty} \\left( \\frac{1-u^{\\frac{2\\Delta}N}}{2 u^{\\frac{\\Delta}N}} \\right) \\sum_{k \\geq \\lambda}^N   \\left( \\frac{1+u^{\\frac{2\\Delta}N}}{2 u^{\\frac{\\Delta}N}} \\right)^{N-k}  h_\\eta^{(k,N)}(x,y,u^{\\frac{1}N},g) \\int_{\\mathcal{O}_\\lambda^{k}}  f^{(\\alpha,\\eta)}_{k-1}(u^{\\frac{1}N},\\mu) d\\mu_{\\lambda},\n\\end{equation}\nfor certain functions $h_\\eta^{(k,N)}(x,y,u^{\\frac{1}N},g)$ and $f^{(\\alpha,\\eta)}_{k-1}(u^{\\frac{1}N},\\mu) d\\mu_{\\lambda}$ and where the innermost integral is the orbital integral of the action and $\\mathcal{O}_\\lambda^{k} := \\mathcal{O}_\\lambda \\cap \\Z_2^k$ (here, $\\Z_2^k$ is regarded as a subgroup of $\\Z_2^{\\infty}$).\nLet us describe the orbital integral. First, notice that when we fix $|\\rho| = \\lambda$, the elements of $\\mathcal{O}_\\lambda^{k}$ are determined by the position of the ones, in other words there is a bijection\n\\[\n  \\left\\{ \\rho \\in \\Z_2^k \\, : \\, |\\rho|= \\lambda \\right\\} \\longleftrightarrow \\left\\{ (j_1,j_2,\\cdots,j_\\lambda) \\in \\Z_{\\geq1}^{\\lambda} \\,;\\, j_1 <j_2<\\cdots<j_\\lambda \\leq k \\right\\}.\n\\]\nFor instance, if we write $\\tfrac{j}N$ to indicate the presence of a $1$ in the $j$-th position of a given $\\rho \\in \\Z_2^N$, the elements of $\\mathcal{O}_\\lambda^{N}$ are given by\n\\[\n  \\left(\\tfrac{1}N, \\tfrac{2}N, \\cdots, \\tfrac{\\lambda}{N}\\right), \\left(\\tfrac{1}N, \\tfrac{3}N,\\tfrac{4}{N} \\cdots, \\tfrac{\\lambda+1}{N}\\right), \\left(\\tfrac{1}N, \\tfrac{2}N,\\tfrac{4}{N} \\cdots, \\tfrac{\\lambda+1}{N}\\right),\\cdots,\\left(\\tfrac{N-\\lambda}N, \\tfrac{N-\\lambda+1}N,\\cdots, \\tfrac{N}{N}\\right) \n\\]\nfor $\\lambda << N$. The orbit integral is then realized by a sum\n\\[\n  \\int_{\\mathcal{O}_\\lambda^{k}}  f^{(\\alpha,\\eta)}_{k-1}(u^{\\frac{1}N},\\mu) d\\mu_{\\lambda} = \\sum_{1\\le j_1 < j_2 < \\cdots < j_\\lambda\\leq k} f^{(\\alpha,\\eta)}_{k-1}(u^{\\frac{1}N},\\bm{j}), \n\\]\nand moreover, the sum in the right hand side is identified, up to certain terms of bounded order,  by an integral\non the $\\lambda$-simplex by the usual procedure using the Riemann-Stieljes integration\n\\begin{equation}\n  \\label{eq:simplexint}\n  \\sum_{1\\le j_1 < j_2 < \\cdots < j_\\lambda\\leq k} f^{(\\alpha,\\eta)}_{k-1}(u^{\\frac{1}N},\\bm{j}) = \\int_{0}^k \\int_0^{z_\\lambda} \\cdots \\int_0^{z_{2}} f^{(\\alpha,\\eta)}_{k-1}(u^{\\frac{1}N},\\bm{z})  d \\bm{z} + O(k^{\\lambda-1}).\n\\end{equation}\n Finally, by using an appropriate change of variables, we are able to evaluate the limit in equation\n\\eqref{eq:limit1} as a Riemann sum, thus obtaining the explicit formula for the heat kernel.\nThe lower order terms in \\eqref{eq:simplexint} vanish at the\nlimit appearing in the heat kernel. \n\nTo summarize, in place of Feynman or Feynman-Kac path integrals, in this method we consider only the ``discrete paths'' in the inductive limit  $\\Z_2^{\\infty}$. In fact, the heat kernel $K_{\\text{Rabi}}(x,y,t)$ can be expressed as the summation over the orbit of the infinite\nsymmetric group $\\mathfrak{S}_{\\infty}$ on $\\Z_2^{\\infty}$ by the decomposition described above.\nWe would like to emphasize the role of the harmonic analysis on the inductive family of groups $\\{\\Z_2^n\\}_{n\\geq0}$. Concretely, it allows us to transform the (infinitely many) changes of sign in the limit \\eqref{eq:limit0} into an expression that is evaluated into a series of the form\n\\[\n  \\sum_{\\lambda\\geq 0} (t \\Delta)^{2 \\lambda} \\int_{ ( \\mathfrak{S}_{\\infty}.[1,1,\\cdots,1] )\/ \\mathfrak{S}_\\lambda  } \\exp\\left(\\text{hyperbolic functions of $t, \\bm{\\mu}_\\lambda$} \\right) d \\bm{\\mu}_{\\lambda},\n\\]\nwhere the orbital integral over $(\\mathfrak{S}_{\\infty} . [1,1,\\cdots,1]) \/ \\mathfrak{S}_\\lambda $\nis realized as an integral over the $\\lambda$-th simplex.\n\n\\subsubsection{$\\mathfrak{S}_\\infty$-module decomposition for $\\Z_2^{\\infty}$ (Algebraic)}\n\nIn the foregoing discussion, we  described geometrically the discrete path integral based\non the group $\\Z_2^{\\infty}$. However, since the computation is effectively done in the dual space, that is,\nthe space of its Fourier image, which is also identified by $\\Z_2^{\\infty}$, it would be natural to understand\nthis discrete path integral in the framework of the representation theory of the infinite symmetric group\n$\\mathfrak{S}_\\infty$ (see e.g. \\cite{Ol2001, TV2007}).\n\nActually, the representative $[1,1,\\ldots,1]$ in $\\Z_2^{\\infty}$ can be regarded as the partition $\\Pi_{\\lambda}=(\\lambda, \\infty)$ of $\\N$.\nThen the corresponding Young subgroup $\\mathfrak{S}_{\\Pi_{\\lambda}}$ of $\\mathfrak{S}_\\infty$ is given by\n$\\mathfrak{S}_{\\Pi_{\\lambda}}= \\tilde{\\mathfrak{S}}_\\lambda \\times \\tilde{\\mathfrak{S}}_\\infty $, where $\\tilde{\\mathfrak{S}}_\\lambda$ is isomorphic to\n$\\mathfrak{S}_\\lambda$ in $\\mathfrak{S}_\\infty$, that is, its tail is infinite identity permutation. Also $\\tilde{\\mathfrak{S}}_\\infty $\nis isomorphic to $\\mathfrak{S}_\\infty $ by the first $\\lambda$ component is the trivial permutation. Notice that the pair of Young diagrams\n\\[\n  \\bigg(\\, {\\underbrace{\\ydiagram{2}\\cdots\\ydiagram{2}}_{\\lambda} \\; , \\; \\underbrace{\\ydiagram{5}\\cdots}_{\\infty}}\\bigg)\n\\]\ncorrespond to the trivial representation of $\\mathfrak{S}_{\\Pi_{\\lambda}}$.\nThen the representation $\\mathrm{Ind}_{\\mathfrak{S}_{\\Pi_{\\lambda}} }^{\\mathfrak{S}_\\infty }1$ induced from the identity representation\nof $\\mathfrak{S}_{\\Pi_{\\lambda}}$ is irreducible (the most simple example of the type I representation in \\cite{TV2007}) and as\na $\\mathfrak{S}_\\infty$-module\n\\[\n  \\mathfrak{S}_\\infty \\, {}^\\curvearrowright \\, \\Z_2^{\\infty}=\\oplus_{\\lambda=0}^\\infty \\mathrm{Ind}_{\\mathfrak{S}_{\\Pi_{\\lambda}} }^{\\mathfrak{S}_\\infty }1 .  \n\\]\nThis decomposition gives another interpretation of the expression for the heat kernel. Therefore, one of the ways to explore the relation between the discrete path integral and the Feynman path integral is to study the asymptotic combinatorics as developed in e.g. \\cite{Hora2001}.\n\n\\begin{rem}\n  In the setting of discrete paths, we expect the existence of an equivalence relation $\\sim$ of (continuous) paths such that\n\\begin{equation}\n  \\label{eq:equiv}\n  \\{\\text{paths}\\}\/\\sim \\, = \\, \\Z_2^{\\infty},\n\\end{equation}\nin a way that the path integral can be identified with the ``discrete path integral''. The existence of the equivalence relation $\\sim$ is highly non-trivial since, as it is well-known, the set of paths $\\{\\text{paths}\\}$ is uncountable and, in general, there is no known  way to associate a reasonable measure to the path integral, while the set of discrete (countably many) paths $\\Z_2^{\\infty}$ has a well behaved topology induced from the point measure in each group $\\Z_2^{n}$. Thus, further understanding of the conjectural relation $\\sim$ may be of importance for the mathematical formalization of the path integral for a certain class of quantum systems.  In addition, the derivation of the discrete path integral, that is, the explicit formula of the heat kernel, from the concrete Feynman-Kac formula for the QRM (obtained for the QRM in \\cite{HH2012}) may also be an interesting problem. In Figure \\ref{fig:paths2} we give a schematic picture of this discussion. \n\n\\begin{figure}[!ht]\n  \\centering\n  \\begin{tikzpicture}[domain=0:4]\n\n   \n\n    \\node at (0,2.5) { {\\small\\begin{tabular}{c}  Trotter-Kato \\\\ product formula \\end{tabular}}};\n\n    \\node at (7,5) { {\\small Path integral}};\n    \\node at (7,3.9) {{\\Large $\\bigcup$}};\n    \n    \\node at (7,2.6) { {\\small\\begin{tabular}{c}  Feynman-Kac \\\\ path integral \\end{tabular}}};\n\n    \\node at (7,0) { {\\small\\begin{tabular}{c}  Discrete \\\\ path integral \\end{tabular}}};\n\n    \\node at (5.5,1.1) {\\begin{tabular}{c} ${\\scriptstyle \\Z_2^{\\infty}}$ {\\tiny harmonic}  \\\\ {\\tiny analysis } \\end{tabular}};\n    \n   \n\n    \\draw[->,dashed,thick] (1.8,2.5) --  (5.4,5);\ng    \n   \n\n    \\draw[->,thick] (1.8,2.5) -- (5.4,0);\n   \n\n    \\draw[->, dashed,thick,double] (7,1.9) --  (7,0.7);\n\n    \\draw[->] (7.2,1.5) -- (11,0.9);\n    \n    \n   \n\n    \\draw[->,dashed,thick] (8.3,5) to[out=0,in=0] (8.5,0);\n    \n    \\node at (11.5,2.5) { {\\small$ \\{\\text{paths}\\}\/\\sim = \\Z_2^{\\infty}  \\; \\bm{?}$}};\n\n   \n\n    \\node[fill=green!20,draw] at (1.4,0.5) { {\\tiny \\begin{tabular}{c}  Gaussian integrations \\\\ + \\\\ Fourier analysis on $\\Z_2^{n} (n\\geq 0)$ \\\\ (dual space picture) \\end{tabular}}};\n    \n    \\node[fill=green!20,draw] at (12.1,0.5) { {\\tiny \\begin{tabular}{c}  $\\mathfrak{S}_{\\infty}$-orbital integrals \\\\ with probability measure \\\\ (asymptotic combinatorics) \\end{tabular}}};\n    \n  \\end{tikzpicture}\n  \\caption{Schematic picture of computation of the heat kernel for the QRM}\n  \\label{fig:paths2}\n\\end{figure}\n\nWe remark that since the $\\Z_2^{\\infty}$-group structure is inherent to the Hamiltonian $H_{\\text{\\upshape Rabi}}$, the conjectural equivalence \\eqref{eq:equiv} may be determined only up to the given quantum system. It is difficult to expect a unique relation of the type \\eqref{eq:equiv} for path integrals in general.\n\\end{rem}\n\n\\begin{rem}\n  The $\\Z_2^{\\infty}$ group structure appearing in the computation of the heat kernel is unrelated to \n  the $\\Z_2$-parity of the QRM. In fact, it is not difficult to note that a similar $\\Z_2^{\\infty}$ structure also\n  appears in the case of the asymmetric quantum Rabi model\n  \\[\n    H_{\\text{\\upshape Rabi}}^{\\varepsilon} := a^{\\dagger}a + \\Delta \\sigma_z + g (a + a^{\\dagger}) \\sigma_x + \\varepsilon \\sigma_x,\n  \\]\n  even thought a $\\Z_2$-parity is known only for the case $\\varepsilon = 0$ (QRM case).\n\\end{rem}\n\n\n\n\\subsection{Explicit formulas for heat kernel and partition function}\n\\label{sec:expl-form-heat}\n\n\n\\subsubsection{Heat kernel}\n\\label{sec:heat-kernel}\n\nIn the expressions for the heat kernel and the partition function, the\nintegral over the $\\lambda$-th simplex for \\(\\lambda = 0 \\) is used with the meaning\n\\[\n  \\idotsint\\limits_{0\\leq \\mu_1 \\leq \\cdots \\leq \\mu_\\lambda \\leq 1} f(x) d \\bm{\\mu_0} = f(x),\n\\]\nfor any function \\(f(x)\\).\n\n\\begin{thm}[Thm 4.2 of \\cite{RW2020hk}]\n  The heat kernel $K_{\\text{Rabi}}(x,y,t)$ of the QRM is given by the uniformly convergent series\n  \\begin{align*}    \n    &K_{\\text{Rabi}}  (x,y,t) =  K_0(x,y,g,t) \\Bigg[ \\sum_{\\lambda=0}^{\\infty} (t\\Delta)^{\\lambda} e^{-2g^2 (\\coth(\\tfrac{t}2))^{(-1)^\\lambda}}\n    \\\\\n    &\\qquad\\times \\idotsint\\limits_{0\\leq \\mu_1 \\leq \\cdots \\leq \\mu_\\lambda \\leq 1} e^{4g^2 \\frac{\\cosh(t(1-\\mu_\\lambda))}{\\sinh(t)}(\\frac{1+(-1)^\\lambda}{2}) + \\xi_{\\lambda}(\\bm{\\mu_{\\lambda}},t)}  \n          \\begin{bmatrix}\n            (-1)^{\\lambda} \\cosh  &  (-1)^{\\lambda+1} \\sinh  \\\\\n            -\\sinh &  \\cosh\n          \\end{bmatrix}\n                     \\left( \\theta_{\\lambda}(x,y,\\bm{\\mu_{\\lambda}},t) \\right) d \\bm{\\mu_{\\lambda}} \\Bigg],\n  \\end{align*}\n  with \\(\\bm{\\mu_0} := 0\\) and \\(\\bm{\\mu_{\\lambda}}= (\\mu_1,\\mu_2,\\cdots,\\mu_\\lambda)\\) and \\(d \\bm{\\mu_{\\lambda}} = d \\mu_1 d \\mu_2 \\cdots d \\mu_{\\lambda} \\)\n  for \\(\\lambda \\geq 1\\).\n\n  Here, $K_0(x,y,g,t)$ is given by\n  \\begin{align*}\n    K_0(x,y,g,t)\n    & := \\frac{e^{t(g^2+\\tfrac12)}}{\\sqrt{2\\pi \\sinh(t)}} \\exp\\left( -\\frac{(x^2 + y^2) \\cosh(t) - 2x y}{2\\sinh(t)} \\right)\\\\\n  \\end{align*}\n  and the functions \\(\\theta_{\\lambda}(x,y, \\bm{\\mu_{\\lambda}},t)\\) and $\\xi_\\lambda(\\bm{\\mu_{\\lambda}},t)$ are given by\n  \\begin{align*}\n    \\theta_{\\lambda}(x,y, \\bm{\\mu_{\\lambda}},t) &:= \\frac{2\\sqrt{2} g}{\\sinh(t)}\\left( x \\cosh(t) - y \\right) \\left( \\frac{1-(-1)^{\\lambda}}{2} \\right) - \\sqrt{2} g (x-y) \\coth(\\tfrac{t}2) \\\\\n    & \\quad +   \\frac{2\\sqrt{2} g (-1)^{\\lambda} }{\\sinh(t)}  \\sum_{\\gamma=0}^{\\lambda} (-1)^{\\gamma} \\Big[ x  \\cosh(t(1 -   \\mu_{\\gamma})) -  y  \\cosh(t \\mu_{\\gamma})  \\Big] \\nonumber \\\\\n    \\xi_\\lambda(\\bm{\\mu_{\\lambda}},t) &:=  -\\frac{8g^2 }{\\sinh(t)} \\left(\\sinh(\\tfrac12t(1-\\mu_\\lambda))\\right)^2 (-1)^{\\lambda}  \\sum_{\\gamma=0}^{\\lambda} (-1)^{\\gamma} \\cosh( t \\mu_{\\gamma}) \\nonumber  \\\\\n   &\\qquad  - \\frac{4 g^2  }{\\sinh(t)} \\sum_{\\substack{0\\leq\\alpha<\\beta\\leq \\lambda-1\\\\ \\beta - \\alpha \\equiv 1 \\pmod{2}  }}  \\left( \\cosh(t(\\mu_{\\beta+1}-1)-\\cosh(t(\\mu_{\\beta}-1)) \\right) \\nonumber \\\\\n   &\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\times ( \\cosh(t  \\mu_{\\alpha}) - \\cosh(t \\mu_{\\alpha+1})), \\nonumber \n  \\end{align*}\nwhere we use the convention \\( \\mu_0 = 0 \\) whenever it appears in the formulas above.\n\\end{thm}\n\nAs mentioned in the Introduction, the Hamiltonian of the QRM has a parity decomposition \\(H_{\\text{\\upshape Rabi}} = H_{+}\\oplus H_{-} \\).\nThe formula for the heat kernel $K_{\\pm}$ of $H_{\\pm}$ can be obtained directly from the analytical formula of heat kernel\nof the QRM, this is the method used in \\cite{RW2020hk}. A direct computation of the heat kernel for the Hamiltonians $H_{\\pm}$ is also possible by using the method described in \\S\\ref{sec:limit} but appears to be more complicated. A brief overview of the decomposition in the Bargmann space and $G$-functions for the QRM can be found in Appendix  \\ref{sec:bargm-space-confl}.\n\n\\begin{thm}[Thm 4.4 of \\cite{RW2020hk}]\n  The heat kernel $K_{\\pm}(x,y,t, \\Delta)$ of $H_\\pm= H_{\\text{\\upshape Rabi}} |_{\\mathcal{H}_\\pm}$ is given by \n  \\begin{align*}\n    K_{\\pm}(x,y,t, \\Delta)\n    = & K_0(x,y,g,t)\\sum_{\\lambda=0}^{\\infty} (t\\Delta)^{2\\lambda} \\Phi^-_{2\\lambda}(x,y,t) \\mp \\tilde{K}_0(x,-y,g,t) \\sum_{\\lambda=0}^{\\infty}\n        (t\\Delta)^{2\\lambda+1} \\Phi^+_{2\\lambda+1}(x,-y,t),\n  \\end{align*}\n  where for $\\lambda\\geq1$, the function $\\Phi^\\pm_{\\lambda}(x,y,t)$ is given by\n  \\begin{align*}\n    \\Phi^\\pm_{\\lambda}(x,y,t) := e^{-2g^2 (\\coth(\\tfrac{t}2))^{(-1)^\\lambda}} \\idotsint\\limits_{0\\leq \\mu_1 \\leq \\cdots \\leq \\mu_{\\lambda} \\leq 1}  e^{4g^2 \\frac{\\cosh(t(1-\\mu_\\lambda))}{\\sinh(t)}(\\frac{1+(-1)^\\lambda}{2}) +  \\xi_{\\lambda}(\\bm{\\mu_{\\lambda}},t)\\pm \\theta_{n}(x,y, \\bm{\\mu_{\\lambda}},t)} d \\bm{\\mu_{n}}\n  \\end{align*}\n  and  \n  \\begin{align*}\n    \\Phi^\\pm_0(x,y,t) := e^{-2g^2\\tanh\\big(\\frac{t}2\\big) \\pm\\sqrt2 g(x+y)\\tanh\\big(\\frac{t}2\\big)}.\n  \\end{align*}\n\\end{thm}\n\nIt is not difficult to verify that in fact, we have\n\\[\n  K_{\\text{Rabi}}(x,y,t, \\Delta)= K_{+}(x,y,t, \\Delta)\\oplus  K_{-}(x,y,t, \\Delta).\n\\]\n\n\\subsubsection{Partition function}\n\\label{sec:partition-function}\n\nThe partition function \\( Z_{\\text{Rabi}}(\\beta)\\) for the QRM is obtained by direct computation from the formula of the heat kernel $K_{\\text{Rabi}}(x,y,t)$ by the identity\n\\begin{equation*}\n  Z_{\\text{Rabi}}(\\beta):=  \\int_{-\\infty}^\\infty  \\tr K_{\\text{Rabi}}(x,x,\\beta) dx.\n\\end{equation*} \nThe partition functions \\(Z_{\\rm{Rabi}}^{\\pm}(\\beta) \\) of the Hamiltonians $H_{\\pm}$ are obtained in an analogous way.\n\n\\begin{cor}[Cor. 4.3 of \\cite{RW2020hk}] \\label{cor:Partition_function}\n  The partition function \\( Z_{\\text{Rabi}}(\\beta)\\) of the QRM is given by\n    \\begin{align*}\n    Z_{\\text{Rabi}}(\\beta) &= \\frac{e^{\\beta(g^2+1)}}{\\sinh(\\beta)} \\Bigg[ 1 + e^{-2g^2 \\coth(\\frac{\\beta}2)} \\sum_{\\lambda=1}^{\\infty} (\\beta \\Delta)^{2\\lambda} \\idotsint\\limits_{0\\leq \\mu_1 \\leq \\cdots \\leq \\mu_{2 \\lambda} \\leq 1} e^{ 4g^2\\frac{\\cosh(\\beta(1-\\mu_{2\\lambda}))}{\\sinh(\\beta)} +  \\xi_{2 \\lambda}(\\bm{\\mu_{2\\lambda}},\\beta) +\\psi^-_{2 \\lambda}(\\bm{\\mu_{2 \\lambda}},\\beta)} d \\bm{\\mu_{2 \\lambda}}  \\Bigg],\n  \\end{align*}\n  where the function $\\psi_\\lambda^{-}(\\bm{\\mu_{\\lambda}},t)$ is given by\n  \\begin{equation*}\n  \\psi_\\lambda^{-}(\\bm{\\mu_{\\lambda}},t) :=  \\frac{4 g^2 }{\\sinh(t)}\\left[ \\sum_{\\gamma=0}^{\\lambda} (-1)^{\\gamma}  \\sinh(t\\left(\\tfrac12 - \\mu_{\\gamma}\\right)  \\right]^2.\n\\end{equation*}\nfor  \\(\\lambda \\geq 1\\) and \\(\\bm{\\mu_{\\lambda}} = (\\mu_1,\\mu_2,\\cdots,\\mu_\\lambda) \\) and where \\( \\mu_0 = 0 \\). \n\\end{cor}\n\n\\begin{cor}[Cor. 4.4 of \\cite{RW2020hk}]\n  The partition function \\(Z_{\\rm{Rabi}}^{\\pm}(\\beta) \\) for the Hamiltonian \\(H_{\\pm}\\) is given by\n  \\begin{align*}\n    &Z_{\\text{Rabi}}^{\\pm}(\\beta) = \\frac{ e^{\\beta(g^2+1)}}{2\\sinh(\\beta)} \\Bigg[ 1 + e^{-2g^2 \\coth(\\frac{\\beta}2)}  \\sum_{\\lambda =1}^{\\infty} (\\beta\\Delta)^{2 \\lambda} \\idotsint\\limits_{0\\leq \\mu_1 \\leq \\cdots \\leq \\mu_{2 \\lambda} \\leq 1} e^{4g^2\\frac{\\cosh(\\beta(1-\\mu_{2\\lambda}))}{\\sinh(t)} +   \\xi_{2\\lambda}(\\bm{\\mu_{2 \\lambda}},\\beta) +\\psi^-_{2 \\lambda} (\\bm{\\mu_{2\\lambda}},\\beta)} d \\bm{\\mu_{2\\lambda}}  \\Bigg] \\\\\n     &\\qquad \\qquad \\mp \\frac{ e^{\\beta(g^2 +1)}}{2\\cosh(\\beta)} e^{ - 2g^2 \\tanh(\\frac{\\beta}2)}  \\sum_{\\lambda = 0}^{\\infty} (\\beta \\Delta)^{2\\lambda+1} \\idotsint\\limits_{0\\leq \\mu_1 \\leq \\cdots \\leq \\mu_{2 \\lambda+1} \\leq 1} e^{\\xi_{2\\lambda+1}(\\bm{\\mu_{2 \\lambda+1}},\\beta) +\\psi^+_{2 \\lambda+1} (\\bm{\\mu_{2\\lambda+1}},\\beta)} d \\bm{\\mu_{2\\lambda+1}},\n  \\end{align*}\n  where the function $\\psi_\\lambda^{-}(\\bm{\\mu_{\\lambda}},t)$ is as in Corollary \\ref{cor:Partition_function} and\n \\begin{equation*}\n  \\psi_\\lambda^{+}(\\bm{\\mu_{\\lambda}},t) :=  \\frac{4 g^2 }{\\sinh(t)}\\left[ \\sum_{\\gamma=0}^{\\lambda} (-1)^{\\gamma}  \\cosh(t\\left(\\tfrac12 - \\mu_{\\gamma}\\right)  \\right]^2.\n\\end{equation*}\n\\end{cor}\n\n\\section{Propagator of the QRM}\n\\label{sec:Propagator}\n\nThe propagator is the integral kernel associated to the solution of the Schr\\\"odinger equation corresponding to $H_{\\text{\\upshape Rabi}}$.\nPrecisely, it is a (two-by-two matrix valued) function $U_{\\text{Rabi}}(x,y,t)$ satisfying\n$i \\frac{\\partial}{\\partial t} U_{\\text{Rabi}}(x,y,t)= -H_{\\text{\\upshape Rabi}} U_{\\text{Rabi}}(x,y,t)$ for all $t>0$ and $\\lim_{t \\to 0}U_{\\text{Rabi}}(x,y,t)=\\delta_x(y) \\bf{I}_2$ for $x,y \\in \\R$.\n\nClearly, we may obtain $U_{\\text{Rabi}}(x,y,t)$ from an explicit expression for the propagator from the heat kernel\n$K_{\\text{Rabi}}$ by a change of variable $t \\to i t$ (known as  the Wick rotation).\nIn this section we formalize this idea by extending the domain $K_{\\text{Rabi}}$ to the complex plane as\na function with respect to the variable $t$.\n\nFirst, a simple technical lemma is needed to establish the holomorphicity of \\(K_{\\text{Rabi}}(x,y,t)\\).\nThe lemma is also used later to give the meromorphic continuation of the spectral zeta function\nof the QRM and the Hamiltonian of each parity.\n\n\\begin{lem}\n  \\label{lem:bound}\n  Suppose \\(\\lambda \\in \\Z_{\\geq 1}\\) and let\n  \\[\n    \\mathcal{R}^* = \\{ z \\in \\C \\, | \\, z \\neq n \\pi i \\, , n \\in \\Z\\}\n  \\]\n  Then, for \\( t \\in \\mathcal{R}^* \\) there are real valued functions \\( C_1(x,y,t), C_2(t), C_3(t) \\geq 0 \\)\n  bounded in compact subsets of \\(\\mathcal{R}^*\\), such that\n  \\begin{align*}\n    \\left| \\theta_{\\lambda}(\\bm{\\mu_\\lambda},x,y,t) \\right| &\\leq \\left|\\frac{\\sqrt{2} g }{1-e^{-2 t}} \\right| C_1(x,y,t) \\\\\n    \\left| \\psi_{\\lambda}^{\\pm}(\\bm{\\mu_\\lambda},t) \\right| &\\leq \\left|\\frac{2 g^2 }{1-e^{-2 t}} \\right| C_2(t) \\\\\n    \\left\\vert \\xi_{\\lambda}(\\bm{\\mu_{\\lambda}},t) \\right\\vert  &\\le \\left|\\frac{2 g^2 }{1-e^{-2 t}} \\right| C_3(t) \\lambda \n  \\end{align*}\n  uniformly for \\(0 \\leq  \\mu_1 \\leq \\mu_2 \\leq \\cdots \\leq \\mu_\\lambda \\leq 1\\).\n\\end{lem}\n\n\\begin{proof}\n  Let \\( t = a+ b i \\in \\mathcal{R}^*\\). For the proof it\n    is convenient to use the exponential form of the hyperbolic functions appearing in the functions $\\theta_{\\lambda}(\\bm{\\mu_{\\lambda}},x,y,t)$,$\\psi_{\\lambda}^{\\pm}(\\bm{\\mu_{\\lambda}},t)$  and $\\xi_{\\lambda}(\\bm{\\mu_{\\lambda}},t)$.\n  Lets consider first the case of $\\theta_{\\lambda}(\\bm{\\mu_\\lambda},x,y,t)$.  Clearly, we have\n  \\begin{align*}\n    &\\left|\\frac{2\\sqrt{2} g e^{-t}}{1-e^{-2t}}\\left( x (e^{t}+e^{- t}) - 2 y \\right) \\left( \\frac{1-(-1)^{\\lambda}}{2} \\right) - \\sqrt{2} g (x-y) \\frac{1+e^{-t}}{1-e^{-t}}\\right| \\\\\n    & \\qquad \\qquad \\leq \\frac{\\sqrt{2}|g| e^{-a}}{|1-e^{-2t}|} \\left(2 (|x|(e^a + e^{-a}) + 2|y|) + (|x|+|y|)(1+e^{-a})^2 \\right) = \\frac{\\sqrt{2} g c_1(x,y,t)}{|1-e^{-2t}|},\n  \\end{align*}\n  Next, we notice that, for \\( \\lambda \\equiv 1 \\pmod{2} \\), we have\n  \\begin{align*}\n    &s(x,t,y) := \\sum_{\\gamma=0}^{\\lambda} (-1)^{\\gamma} \\Big[ x  (e^{t(1 -   \\mu_{\\gamma}) } + e^{ t( \\mu_{\\gamma} - 1)})  -  y  (e^{- t \\mu_{\\gamma} }+ e^{ t \\mu_{\\gamma}})  \\Big] \\\\\n    &\\qquad = - t \\sum_{\\gamma = 0}^{\\frac{2\\lambda-1}{2}} \\left( x \\int_{\\mu_{2\\gamma}}^{\\mu_{2\\gamma+1}} (e^{t(1-x)} + e^{t(x-1)})d x - y \\int_{\\mu_{2\\gamma}}^{\\mu_{2\\gamma+1}} (e^{t x} + e^{-t x}) d x  \\right).\n  \\end{align*}\n  Next, we have\n  \\[\n    |s(x,t,y)| \\leq  |t| \\left(|x| \\int_{0}^{1} (e^{a(1-x)} + e^{a(x-1)})d x + |y| \\int_{0}^{1} (e^{a x)} + e^{-a x)})d x  \\right),\n  \\]\n  giving\n  \\[\n    |s(x,y,t)| \\leq 2\\frac{|t|}{|a|}(|x|+|y|)\\left(e^a + e^{-a}\\right).\n  \\]\n  If  \\( \\lambda \\equiv 0 \\pmod{2} \\), we apply the estimate above to the first \\(\\lambda\\) terms of the sum resulting in\n  \\[\n    |s(x,y,t)| \\leq 2\\frac{|t|}{|a|}(|x|+|y|)\\left(2e^{|a|} + e^{-|a|} + 1 \\right) \n  \\]\n  and we set \\(c_2(x,y,t)\\) as the right hand side of the inequality. Setting $C_1(x,y,t) = c_1(x,y,t) + \\sqrt{2}g e^{-a} c_2(x,y,t)$ gives the desired result. The case of \\(\\psi_{\\lambda}^{\\pm}(\\bm{\\mu_\\lambda},t) \\) and the first sum in\n  $\\xi_{\\lambda}(\\bm{\\mu_{\\lambda}},t)$ are dealt in the same way.\n  \n  For the second sum in $\\xi_{\\lambda}(\\bm{\\mu_{\\lambda}},t)$, we fix \\(0 \\leq n < \\lambda-1\\) and consider the sum\n   \\begin{align*}\n     S_{n}(t) &= \\sum_{\\substack{n <\\beta\\leq \\lambda-1 \\\\ \\beta - n \\equiv 1 \\pmod{2}} } \\left( (e^{- t \\mu_{\\beta+1}} +  e^{ t \\mu_{\\beta+1}-2t} )-(e^{- t \\mu_{\\beta}} +  e^{ t \\mu_{\\beta}-2t}) \\right) ( (e^{ t \\mu_{n}} + e^{- t \\mu_{n}}) - (e^{ t \\mu_{n+1}} + e^{- t \\mu_{n+1}}))\\\\\n     &=  ( (e^{ t \\mu_{n}} + e^{- t \\mu_{n}}) - (e^{ t \\mu_{n+1}} + e^{- t \\mu_{n+1}})) \\sum_{\\substack{n<\\beta\\leq \\lambda-1 \\\\ \\beta - n \\equiv 1 \\pmod{2}} } \\left( (e^{- t \\mu_{\\beta+1}} +  e^{ t \\mu_{\\beta+1}-2t} )-(e^{- t \\mu_{\\beta}} +  e^{ t \\mu_{\\beta}-2t}) \\right).\n   \\end{align*}\n   Transforming the sums into definite integrals as in the case above we see that \\(S_{n}(t) \\) is equal to\n   \\begin{gather*}\n     - t^2 \\left( \\int_{ \\mu_{n}}^{ \\mu_{n+1}} (e^{-t x} + e^{t x} ) d x \\right) \\sum_{\\substack{n<\\beta\\leq \\lambda-1 \\\\ \\beta - \\alpha \\equiv 1 \\pmod{2}} }\n     \\left( \\int_{ \\mu_{\\beta}}^{ \\mu_{\\beta+1} } e^{- t x} d x + e^{-2 t}\\int_{ \\mu_{\\beta}}^{ \\mu_{\\beta+1} } e^{t x} d x  \\right).\n   \\end{gather*}\n   It follows that\n   \\begin{align*}\n     |S_{n}(t)| &\\leq |t|^2 \\left(\\int_{0}^{1} (e^{- a x} + e^{a x} ) d x \\right) \\left( \\int_{0}^{1 } e^{- a x} d x + e^{-2 a}\\int_{0}^{1 } e^{a x } d x \\right) \\\\\n     &\\leq \\left|\\frac{t}{a} \\right|^2 \\left( e^a - e^{-a} \\right) \\left(1 - e^{-2 a} \\right),\n   \\end{align*}\n   with a limit interpretation for \\(a = 0 \\).\n   It follows that\n   \\[\n     \\left\\vert \\xi_{\\lambda}(\\bm{\\mu_{\\lambda}},t) \\right| \\leq \\left|\\frac{2 g^2 }{1-e^{-2 t}} \\right| \\left|\\sum_{n=0}^{\\lambda-2} S_n(t) \\right|\n     \\leq \\left|\\frac{2 g^2 }{1-e^{-2 t}} \\right| c_3(t) \\lambda,\n   \\]\n   completing the proof.\n\\end{proof}\n\nFor the purpose of giving an explicit formula for the propagator it is only necessary to extend the function $K_{\\text{Rabi}}  (x,y,t)$ to the line $i \\R$. However, for completeness we give the holomorphic extension to a larger region in the complex plane. We remark that the region is chosen according to the principal branch of logarithm (equivalently, branch of square root).\n\n\\begin{prop} \\label{prop:MeromExtK}\n  For fixed $x,y \\in \\R$, the series defining any of the entries of the heat kernel $K_{\\text{Rabi}}  (x,y,t)$ is\n  uniformly convergent in compacts in the complement in the complex plane of the region\n  $\\bigcup_{n \\in \\Z} \\{ t = a + i  \\pi n \\in \\C  : \\, a\\leq0  \\}$. In particular, $K_{\\text{Rabi}}  (x,y,t)$ is (entrywise)\n  holomorphic in said region.\n\\end{prop}\n\n\\begin{proof}\n  Denote by \\(\\mathcal{D}\\) the region given by the complement of $\\bigcup_{n \\in \\Z} \\{ t = a + i  \\pi n \\in \\C  : \\, a\\leq0  \\}$\n  and consider a compact \\(K \\subset \\mathcal{D}\\). \n  Since \\(\\mathcal{D}\\) does not contain any zero of \\(1-e^{-2t}\\), we have\n  \\[\n    |K_0(x,y,g,t)| \\leq c_0\n  \\]\n  for some constant $c_0 \\ge 0$. Similarly, we set $c_1 = \\max_{t\\in K}(|t|)$ and\n  \\[\n    c_2 = \\max_{t \\in K}\\left(\\left|-2g^2\\coth(\\tfrac{t}2) + 4g^2 (1+e^{|t|})\/(\\sinh(t))\\right|,\\left|2g^2 \\tanh(\\tfrac{t}2)\\right| \\right),\n  \\]\n  then we have\n  \\begin{align*}\n    & \\left|\\sum_{\\lambda=0}^{\\infty} (t\\Delta)^{\\lambda} \\idotsint\\limits_{0\\leq \\mu_1 \\leq \\cdots \\leq \\mu_\\lambda \\leq 1} e^{-2g^2 (\\coth(\\tfrac{t}2))^{(-1)^\\lambda}+ 4g^2 \\frac{\\cosh(t(1-\\mu_\\lambda))}{\\sinh(t)}(\\frac{1+(-1)^\\lambda}{2}) + \\xi_{\\lambda}(\\bm{\\mu_{\\lambda}},t) \\pm \\theta_{\\lambda}(x,y,\\bm{\\mu_{\\lambda}},t)} d \\bm{\\mu_{\\lambda}}\\right| \\\\\n    & \\quad \\leq  \\sum_{\\lambda=0}^{\\infty} ( c_1 \\Delta)^{\\lambda} \\idotsint\\limits_{0\\leq \\mu_1 \\leq \\cdots \\leq \\mu_\\lambda \\leq 1} e^{c_2 + |\\xi_{\\lambda}(\\bm{\\mu_{\\lambda}},t)| +|\\theta_{\\lambda}(x,y,\\bm{\\mu_{\\lambda}},t)| } d \\bm{\\mu_{\\lambda}} \\leq  e^{c_2+ c_3(x,y)} \\sum_{\\lambda=0}^{\\infty} ( c_1 \\Delta)^{\\lambda}  e^{ \\lambda c_4 } \\idotsint\\limits_{0\\leq \\mu_1 \\leq \\cdots \\leq \\mu_\\lambda \\leq 1} d \\bm{\\mu_{\\lambda}} \\\\\n    & \\quad \\leq e^{c_2+ c_3(x,y)} \\sum_{\\lambda=0}^{\\infty} \\frac{( c_1 \\Delta e^{c_4})^{\\lambda}  e^{ \\lambda c_4 }}{k!} = e^{c_2+ c_3(x,y)+ c_1\\Delta e^{c_4}},\n  \\end{align*}\n  uniformly in $K$, where the constants $c_3$ and $c_4$ are given by Lemma \\ref{lem:bound}. Therefore, we see\n  that the series defining any entry of $K_{\\text{Rabi}}(x,y,t)$ is bounded uniformly in $K$ by\n  \\[\n    c_0 e^{c_2+ c_3(x,y)+ c_1\\Delta e^{c_4}},\n  \\]\n  and the result follows from Weierstrass convergence theorem since $K$ is an arbitrary compact in $\\mathcal{D}$.\n\\end{proof}\n\nWith these preparations, we are ready to give the formula for the propagator of the QRM.\n\n\\begin{thm}\n  The integral kernel $U_{\\text{Rabi}}(x,y,t)$ of $e^{-i t H_{\\text{\\upshape Rabi}}}$ (the propagator of QRM) is given by\n  \\(K_{\\text{Rabi}}(x,y,i t) \\). Concretely, $U_{\\text{Rabi}}(x,y,t)$ is given by \n  \\begin{align*}    \n    &U_{\\text{Rabi}}  (x,y,t) =  U_0(x,y,g,t) \\Bigg[ \\sum_{\\lambda=0}^{\\infty} (i t\\Delta)^{\\lambda} e^{-  2  g^2 (-i\\cot(\\tfrac{t}2))^{(-1)^\\lambda}}\n    \\\\\n    &\\qquad \\times \\idotsint\\limits_{0\\leq \\mu_1 \\leq \\cdots \\leq \\mu_\\lambda \\leq 1}  e^{-4 i g^2 \\frac{\\cos(t(1-\\mu_\\lambda))}{\\sin(t)}(\\frac{1+(-1)^\\lambda}{2}) + \\bar{\\xi}_{\\lambda}(\\bm{\\mu_{\\lambda}},t)}  \n          \\begin{bmatrix}\n            (-1)^{\\lambda} \\cosh  &  (-1)^{\\lambda+1} \\sinh  \\\\\n            -\\sinh &  \\cosh\n          \\end{bmatrix}\n             \\left( \\bar{\\theta}_{\\lambda}(x,y,\\bm{\\mu_{\\lambda}},t) \\right) d \\bm{\\mu_{\\lambda}} \\Bigg]\n  \\end{align*}\n  with \\(\\bm{\\mu_0} := 0\\) and \\(\\bm{\\mu_{\\lambda}}= (\\mu_1,\\mu_2,\\cdots,\\mu_\\lambda)\\) and \\(d \\bm{\\mu_{\\lambda}} = d \\mu_1 d \\mu_2 \\cdots d \\mu_{\\lambda} \\)\n  for \\(\\lambda \\geq 1\\).\n  Here, \n  \\begin{align*}\n    U_0(x,y,g,t)\n    & := \\frac{e^{i t(g^2+\\tfrac12)}}{\\sqrt{2 i \\pi \\sin(t)}} \\exp\\left( - \\frac{(x^2 + y^2) \\cos(t) - 2 x y}{2i \\sin(t)}  \\right)\n  \\end{align*}\n  and the functions \\(\\bar{\\theta}_{\\lambda}(x,y, \\bm{\\mu_{\\lambda}},t)\\) and $\\bar{\\xi}_\\lambda(\\bm{\\mu_{\\lambda}},t)$ are given by\n  \\begin{align*}\n    \\bar{\\theta}_{\\lambda}(x,y, \\bm{\\mu_{\\lambda}},t) &:= \\frac{2\\sqrt{2} g}{i \\sin(t)}\\left( x \\cos(t) - y \\right) \\left( \\frac{1-(-1)^{\\lambda}}{2} \\right)  + i \\sqrt{2}g  (x-y)  \\cot(\\tfrac{t}2) \\\\\n     & \\quad +   \\frac{2\\sqrt{2} g (-1)^{\\lambda} }{i \\sin(t)}  \\sum_{\\gamma=0}^{\\lambda} (-1)^{\\gamma} \\Big[ x  \\cos(t(1 -   \\mu_{\\gamma})) -  y  \\cos(t \\mu_{\\gamma})  \\Big] \\nonumber \\\\\n  \\bar{\\xi}_\\lambda(\\bm{\\mu_{\\lambda}},t) &:=  \\frac{8g^2 }{i\\sin(t)} \\left(\\sin(\\tfrac12t(1-\\mu_\\lambda))\\right)^2 (-1)^{\\lambda}  \\sum_{\\gamma=0}^{\\lambda} (-1)^{\\gamma} \\cos( t \\mu_{\\gamma})  \\\\\n                &  - \\frac{4 g^2  }{i\\sin(t)} \\sum_{\\substack{0\\leq\\alpha<\\beta\\leq \\lambda-1\\\\ \\beta - \\alpha \\equiv 1 \\pmod{2}  }}  \\left( \\cos(t(\\mu_{\\beta+1}-1)-\\cos(t(\\mu_{\\beta}-1)) \\right)  ( \\cos(t  \\mu_{\\alpha}) - \\cos(t \\mu_{\\alpha+1})), \\nonumber \n  \\end{align*}\n  In the formulas above, the square root is taken according to the principal branch.\n\\end{thm}\n\nWe note that \\(U_{\\text{Rabi}}(x,y,t) \\) can be written completely in terms of circular functions, in contrast with the case of \\(K_{\\text{Rabi}}(x,y, t) \\) which is given in terms of hyperbolic functions.\n\n\\begin{thm}\n  The propagator $U_{\\pm}(x,y,t, \\Delta)$ of $H_\\pm= H_{\\text{\\upshape Rabi}} |_{\\mathcal{H}_\\pm}$ is given by \n  \\begin{align*}\n    U_{\\pm}(x,y,t, \\Delta)\n    = & U_0(x,y,g,t)\\sum_{\\lambda=0}^{\\infty} (i t\\Delta)^{2\\lambda} \\bar{\\Phi}^-_{2\\lambda}(x,y,t) \\mp U_0(x,-y,g,t) \\sum_{\\lambda=0}^{\\infty}\n        (i t\\Delta)^{2\\lambda+1} \\bar{\\Phi}^+_{2\\lambda+1}(x,-y,t),\n  \\end{align*}\n  where for $\\lambda\\geq1$, the function $\\bar{\\Phi}^\\pm_{\\lambda}(x,y,t)$ is given by\n  \\begin{align*}\n    \\bar{\\Phi}^\\pm_{\\lambda}(x,y,t) := e^{-2  g^2 (-i\\cot(\\tfrac{t}2))^{(-1)^\\lambda}} \\idotsint\\limits_{0\\leq \\mu_1 \\leq \\cdots \\leq \\mu_{\\lambda} \\leq 1}  e^{- 4 i g^2 \\frac{\\cos(t(1-\\mu_\\lambda))}{\\sin(t)}(\\frac{1+(-1)^\\lambda}{2}) +  \\bar{\\xi}_{\\lambda}(\\bm{\\mu_{\\lambda}},t)\\pm \\bar{\\theta}_{n}(x,y, \\bm{\\mu_{\\lambda}},t)} d \\bm{\\mu_{n}}\n  \\end{align*}\n  and  \n  \\begin{align*}\n    \\bar{\\Phi}^\\pm_0(x,y,t) := e^{-2 ig^2\\tan\\big(\\frac{t}2\\big) \\pm\\sqrt2 i g(x+y)\\tan\\big(\\frac{t}2\\big)}.\n  \\end{align*}\n\\end{thm}\n\n\n\\section{Spectral determinant for parity Hamiltonians and Braak's $G$-functions}\n\\label{sec:spectral-determinant}\n\nThe $G$-function for the QRM (and the respective ones for the parity Hamiltonians) was\noriginally defined by Braak \\cite{B2011PRL} to establish the exact solvability of the QRM. In \\cite{KRW2017}\n(see also \\cite{Sugi2016}) a significant relation was found between the Braak $G$-function and the spectral zeta\nfunction of the QRM (the Mellin transform of the partition function). Actually, the $G$-function is\n(up to a non-vanishing function) equal to the spectral determinant of \\(H_{\\text{\\upshape Rabi}}\\), that is, the zeta-regularized\nproduct associated to the spectral zeta function of the QRM. In this section, we extend this result for the case\nthe Hamiltonians \\(H_{\\pm}\\) of each of the parities.\n\nLet us start by recalling the definitions of the spectral zeta functions and the spectral determinant\nspecialized to the case of the QRM and the Hamiltonians of each parity.\nLet\n\\[\n  \\lambda^{\\pm}_1 < \\lambda^{\\pm}_2 \\leq \\lambda^{\\pm}_3 \\leq \\ldots \\leq \\lambda^{\\pm}_n \\leq \\ldots (\\nearrow \\infty)\n\\]\nbe the eigenvalues of \\(H_{\\pm}\\), then the (Hurwitz-type) spectral zeta function \\(\\zeta^{\\pm}_{\\text{QRM}}(s; \\tau)\\) is given by the Dirichlet series\n\\[\n  \\zeta^{\\pm}_{\\text{QRM}}(s; \\tau):= \\sum_{j=1}^\\infty (\\lambda^{\\pm}_j +\\tau)^{-s}.\n\\]\nSimilarly, the spectral zeta function \\(\\zeta_{\\text{QRM}}(s; \\tau)\\) of the QRM is given by\n\\[\n  \\zeta_{\\text{QRM}}(s; \\tau) = \\zeta^{+}_{\\text{QRM}}(s; \\tau)+  \\zeta^{-}_{\\text{QRM}}(s; \\tau).\n\\]\nIn both cases, it is easily verified (cf. \\cite{Sugi2016}) that the zeta functions above are absolutely convergent\nfor \\(\\Re(s)>1 \\) for \\( \\tau \\in \\C - \\Spec({H_{\\text{\\upshape Rabi}}})\\). \n\nFix the log-branch by $-\\pi\\leq \\arg(\\tau- \\lambda_i)<\\pi$. For a sequence $\\mathcal{A} =\\{a_i\\}_{i\\geq 1}, \\, a_i \\in \\C$, by\ndefining the associated zeta function\n\\[\n  \\zeta_{\\mathcal{A}}(s) = \\sum_{n=1}^{\\infty} a_n^{-s},\n\\]\nassumed to be holomorphic at $s=0$, the zeta regularized product (cf. \\cite{QHS1993TAMS}) associated to $\\mathcal{A}$ is given by\n\\begin{equation*}\n  \\regprod_{i=0}^\\infty a_i:= \\exp\\left(-\\frac{d}{ds}\\zeta_{\\mathcal{A}}^{\\pm}(s)\\big|_{s=0}\\right).\n\\end{equation*}\nBy introducing an auxiliary parameter, the zeta regularized product is also one of the ways to define a\nfunction with prescribed zeros.\n\nIf the sequence $\\mathcal{A}$ correspond to the eigenvalues of an operator (e.g. a Hamiltonian), the zeta\nregularized product is a generalization of the characteristic polynomials of finite matrices. \nFor the case of the QRM, the zeta regularized product  associated to \\(\\zeta_{\\text{QRM}}^{\\pm}(s; \\tau)\\) is defined by\n\\begin{equation*\n  \\regprod_{i=0}^\\infty (\\tau-\\lambda^{\\pm}_i):= \\exp\\left(-\\frac{d}{ds}\\zeta_{\\text{QRM}}^{\\pm}(s; \\tau)\\big|_{s=0}\\right),\n\\end{equation*}\nwhere the product is over the eigenvalues \\(\\lambda_i^{\\pm}\\) in the spectrum of \\(H_{\\pm}\\). Now we define the spectral determinant of the Hamiltonians \\(H_{\\pm} \\) as\n\\begin{equation*}\n  \\det (\\tau-H_{\\pm}):= \\regprod_{i=0}^\\infty (\\tau-\\lambda^{\\pm}_i).\n\\end{equation*}\nThe spectral determinant, as a function of $\\tau$ vanishes exactly at the eigenvalues of $H_{\\pm}$.\n\nIn \\cite{RW2017} the authors proved that the zeta regularized product of \\(\\zeta_{\\text{Rabi}}(s;\\tau)\\), equivalently the spectral determinant of  \\( H_{\\text{\\upshape Rabi}}\\), is given (up to a non-vanishing entire function) by the complete $G$-function (called generalized $G$-function in \\cite{RW2017} ) given by\n\\[\n  \\mathcal{G}(x;g,\\Delta) = G_{+}(x;g,\\Delta) G_{-}(x;g,\\Delta) \\Gamma(-x)^{-2},\n\\]\nwhere \\(G_{\\pm}(x;g,\\Delta)\\) are the parity $G$-functions defined in \\cite{B2011PRL,B2011PRL-OnlineSupplement} (see also Appendix \\ref{sec:bargm-space-confl} and cf. \\cite{LB2015JPA}).\n\n\nTo extend the result to the parity Hamiltonians we need some preparations. First, we need to show that the spectral\nzeta function $\\zeta^{\\pm}_{\\text{QRM}}(s; \\tau)$ is holomorphic around $s = 0$. In \\cite{Sugi2016}, it was shown, without using an explicit formula for the heat kernel, that $\\zeta_{\\text{QRM}}(s; \\tau)$ extends meromorphically to the complex plane with a simple pole at \\(s= 1\\) (cf. \\cite{IW2005a,IW2005b} for a reference to the method for the case of NCHO).\n\nUsing the Mellin transform expression of $\\zeta_{\\text{QRM}}(s)$ by the partition function $Z_{\\text{Rabi}}(t)$ we can give another proof for the meromorphic continuation, similar to one of Riemann's original proofs for the zeta function. In addition, by the same method we obtain the analytic continuation of the parity zeta function \\(\\zeta_{\\text{QRM}}^{\\pm}(s;\\tau)\\). The details are given in Appendix \\ref{sec:proofmero}.\n\n\\begin{thm} \\label{IntRep_SZF}We have\n\\begin{align}\\label{ContourSZF}\n  \\zeta_{\\text{QRM}}(s;\\tau)= -\\frac{\\Gamma(1-s)}{2\\pi i}\\int_\\infty^{(0+)} \\frac{(-w)^{s-1} \\Omega(w)e^{-\\tau w}}{1-e^{-w}}dw.\n\\end{align}\nHere the contour integral is given by the path which starts at $\\infty$ on the real axis, encircles the origin (with a radius smaller than $2\\pi$) in the positive direction and returns to the starting point and it is assumed $|\\arg(-w)|\\leq \\pi$. This gives a meromorphic continuation of $\\zeta_{\\text{QRM}}(s;\\tau)$  to the whole plane where the only singularity is a simple pole with residue $2$ at $s=1$.\n\\end{thm}\n\n\\begin{cor\n  With the notation of Theorem \\ref{IntRep_SZF}, we have\n  \\begin{align*}\n    \\zeta_{\\text{QRM}}^{\\pm}(s;\\tau)= -\\frac{\\Gamma(1-s)}{4\\pi i}\\int_\\infty^{(0+)}\\left( \\frac{(-w)^{s-1} \\Omega(w)e^{-\\tau w}}{1-e^{-w}} \\mp \\frac{(-w)^{s-1} \\Omega_{\\text{odd}}(w)e^{-\\tau w}}{1+e^{-w}}  \\right)dw.\n  \\end{align*}\n  This gives a meromorphic continuation of $\\zeta^{\\pm}_{\\text{QRM}}(s;\\tau)$ to the whole plane where the only singularity is a simple pole with residue $1$ at $s=1$. \\qed\n\\end{cor}\n\nAs an important consequence of the meromorphic continuation of $\\zeta_{\\text{QRM}}^{\\pm}(s;\\tau)$ we obtain the Weyl law for the distribution of the eigenvalues of the parity Hamiltonians \\(H_{\\pm}\\) in the usual way (cf. \\cite{IW2005a,Sugi2016}).\n\nLet us define the spectral counting functions\n\\begin{align*}\n  N_{\\text{Rabi}}(T) &= \\# \\{\\lambda \\in \\Spec(H_{\\text{\\upshape Rabi}}) \\, | \\, \\lambda \\le T \\}, \\\\\n  N_{\\pm}(T) &= \\# \\{\\lambda \\in \\Spec(H_{\\pm}) \\, | \\, \\lambda \\le T \\}\n\\end{align*}\nfor \\(T > 0 \\).\n\n\\begin{cor}\n  We have\n  \\[\n    N_{\\pm}(T) \\sim \\frac12 N_{\\text{Rabi}}(T)  \\sim T,\n  \\]\n  as \\(T \\to \\infty \\). \\qed\n\\end{cor}\n\nThe Weyl law shows that the positive and negative parity eigenstates are equally distributed  and supports also the original Braak conjecture concerning the number of eigenvalues (in each parity) in the consecutive intervals (see e.g. \\cite{B2011PRL,KRW2017}). We note that\nequality of the distribution between the parities also follows from the relation $G_-(x, g, \\Delta)=G_+(x,g,-\\Delta)$ between $G_\\pm$-functions \\cite{B2011PRL} and the properties of the constraint functions\/polynomials \\cite{KRW2017}.\n\nNext, we compute the residue at the poles for the $G$-functions \\(G_{\\pm}(x;g,\\Delta)\\).\n\n\\begin{lem}\\label{lem:respole}\n  The residue of the $G$-function \\(G_{\\pm}(x;g,\\Delta)\\) at the (simple) pole at \\(x = N  \\in \\Z_{\\geq 0}\\) is given by\n  \\[\n    \\Res_{x = N} G_{\\pm}(x;g,\\Delta) = \\frac{\\Delta^2 g^N}{2(N+1)} K_N(N;g,\\Delta) G_{\\pm}^{(N)}(g,\\Delta). \n  \\]\n\\end{lem}\n\n\\begin{proof}\n  The result follows directly by computation and comparison with the definition of \\(K_N(N;g,\\Delta)\\) and \\(G_{\\pm}^{(N)}(g,\\Delta)\\) (see Appendix \\ref{sec:bargm-space-confl} and the proof of Proposition 6.8 of \\cite{KRW2017}).\n\\end{proof}\n\nNext, we show that the zeros of the complete $G$-function $\\mathcal{G}_{\\pm}$ for each parity defined in the following captures the complete spectrum of $H_{\\pm}$.  \n  \\[\n    \\mathcal{G}_{\\pm}(x;g,\\Delta) := G_{\\pm}(x;g,\\Delta) \\Gamma(-x)^{-1}. \n  \\]\n\n\\begin{thm}\\label{thm:eigenparity}\n  There is a one-to-one correspondence between eigenvalues \\(\\lambda\\) in \\(\\Spec{H_{\\pm}}\\) and zeros \\(x = \\lambda + g^2 \\) of the generalized\n  \\(G\\)-function \\(\\mathcal{G}_{\\pm}(x;g,\\Delta)\\).\n\\end{thm}\n\n\\begin{proof}\n  Let \\(\\lambda \\in \\R\\) be a regular eigenvalue of \\(H_{\\pm}\\), then by the definition \\( x = \\lambda + g^2 \\)  is a zero of \\(G_{\\pm}(x;g,\\Delta)\\).\n  Now, suppose \\(\\lambda= N - g^2\\) is an exceptional eigenvalue of \\(H_{\\pm}\\), then by Lemma \\ref{lem:respole}, we see that at \\(x = \\lambda -g^2 = N \\) the\n  function \\(G_{\\pm}(x;g,\\Delta)\\) has a finite value, and then  \\(\\mathcal{G}_{\\pm}(x;g,\\Delta)\\) vanishes by the zero of \\(\\Gamma(-x)^{-1}\\).\n  Conversely, let \\(x \\in \\R\\) be a zero of \\(\\mathcal{G}_{\\pm}(x;g,\\Delta) \\). If \\(x \\notin \\Z_{\\geq0} \\) then \\(x \\) is a zero of \\(G_{\\pm}(x;g,\\Delta)\\)  and  \\(\\lambda = x - g^2 \\) is a regular eigenvalue of \\(H_{\\pm}\\). If \\(x = N \\in \\Z_{\\geq 0}\\), then, since the zero of \\(\\Gamma(-x)^{-1}\\)\n  at \\(x=N \\) is canceled by the pole of \\(G_{\\pm}(x;g,\\Delta)\\), \\(x = N \\) must be a zero of the residue of \\(G_{\\pm}(x;g,\\Delta) \\) at \\(x=N \\),\n  in other words, the tuple \\((g,\\Delta) \\) must be a zero of \\(K_N(N;g,\\Delta)\\) or \\(G_{\\pm}^{(N)}(g,\\Delta)\\) and thus \\(\\lambda = N - g^2 \\) is an exceptional eigenvalue\n  (the Juddian or non-Juddian exceptional, respectively).\n\\end{proof}\n\n  \\begin{rem}\n    The spectrum of the QRM can be captured by irreducible representations of $\\mathfrak{sl}_2$ (cf. \\cite{KRW2017,W2017JPA}).\n    For instance, the Juddian (resp. non-Juddian \\cite{MPS2014}) exceptional solutions are obtained from the irreducible finite dimensional (resp. lowest weight) representations. The existence of these exceptional eigenvalues inherited from the quantum harmonic oscillator (or as its ruins) which are described by the oscillator representation of $\\mathfrak{sl}_2$  is the reason for the presence of the gamma factor in \\(\\mathcal{G}_{\\pm}(x;g,\\Delta)\\).\n\\end{rem}\n\nThe meaning of Theorem \\ref{thm:eigenparity} is that the complete $G$-function \\(\\mathcal{G}_{\\pm}(x;g,\\Delta)\\) vanishes\nexactly at the eigenvalues of \\(H_{\\pm}\\). Immediatly it follows that it is equal, up to non-vanishing constant,\nto the spectral determinant of the parity Hamiltonian.\n\n\\begin{cor} \\label{cor:specdet}\n  There exists a non-vanishing entire function \\(c_{\\pm}(\\tau; g, \\Delta)\\) such that\n  \\[\n    \\det (\\tau-H_{\\pm}) = c_{\\pm}(\\tau; g, \\Delta) \\,  \\mathcal{G}_{\\pm}(\\tau;g,\\Delta). \\qed\n  \\]\n\\end{cor}\n\nWe conclude by making a remark on Corollary \\ref{cor:specdet}. As mentioned before, in \\cite{B2011PRL}\nBraak proved the integrability of the QRM by defining the $G$-function of the parity Hamiltonians \\(H_{\\pm}\\).\nHere, in Corollary \\ref{cor:specdet} above, we see that the $G$-function is, up to a non-vanishing constant,\nequal to the spectral determinant of \\(H_{\\pm}\\), in other words, the zeta regularized product of the spectral zeta function \\(\\zeta_{\\text{QRM}}^{\\pm}(s;\\tau)\\).\nThe zeta regularized product of a zeta function \\(\\zeta(s)\\) is defined when the \\(\\zeta(s)\\) function is holomorphic in a neighborhood around \\(s = 0 \\)\n(in case \\(\\zeta(s)\\) has a pole at \\(s = 0 \\), a modified zeta regularized product may be used, cf. \\cite{KW2004}).\nIt would be interesting to investigate the relationship between the integrability (or exact solvability) of  the Hamiltonian \\(H\\) of a\nquantum interaction model, that is, the existence of entire solutions of the corresponding Fuchsian ODE (Bargmann model), and the existence of a zeta regularized\nproduct for its corresponding spectral zeta function \\(\\zeta_{H}(s;t) \\), or equivalently, the meromorphic continuation of the spectral zeta function to a region containing \\(s=0\\). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nConvolutional neural networks (CNNs) have shown promising results on supervised learning tasks. However, the performance of a learned model always degrades severely when dealing with data from the other domains. Considering that constantly annotating massive samples from new domains is expensive and impractical, unsupervised domain adaptation (UDA), therefore, has emerged as a new learning framework to address this problem \\cite{csurka2017domain}. UDA aims to utilize full-labeled samples in source domain to annotate the completely-unlabeled target domain samples. Thanks to deep CNNs, recent advances in UDA show satisfactory performance in several computer vision tasks \\cite{hoffman2018cycada}. Among them, most methods bridge the source and target domain by learning domain-invariant features. These dominant methods can be further divided into two categories: (1) Learning domain-invariant features by minimizing the discrepancy between feature distributions \\cite{long2015learning,sun2016deep,long2017deep,zellinger2017central,chen2019towards}. (2) Encouraging domain confusion by a domain adversarial objectives whereby a discriminator (domain classifier) is trained to distinguish between the source and target representations. \\cite{ganin2016domain,tzeng2017adversarial,shu2018dirt,hoffman2018cycada}.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{fig0.pdf}\n\\caption{The metrics of using higher-order moment tensor for domain alignment. 300 points in $\\mathbb{R}^2$ and the level set of the moment tensor with different order. As observed, using higher-order moment tenser captures the shape of the cloud of samples more accurately.}\n\\label{fig0}\n\\end{figure}\n\nFrom the perspective of moment matching, most of the existing discrepancy-based methods in UDA are based on Maximum Mean Discrepancy (MMD) \\cite{long2017deep} or Correlation Alignment (CORAL) \\cite{sun2016deep}, which are designed to match the first-order (Mean) and second-order (Covariance) statistics of different distributions. However, for the real world applications (such as image recognition), the deep features are always a complex, non-Gaussian distribution \\cite{jia2011heavy,xu2016blind}, which can not be completely characterized by its first-order or second-order statistics. Therefore, aligning the second-order or lower statistics only guarantees coarse-grained alignment of two distributions. To address this limitation, we propose to perform domain alignment by matching the higher-order moment tensor (mainly refer to third-order and fourth-order moment tensor), which contain more discriminative information and can better represent the feature distribution \\cite{pauwels2016sorting,gou2017mom}. Inspired by \\cite{pauwels2016sorting}, Fig.\\ref{fig0} illustrates the metrics of higher-order moment tensor, where we plot a cloud of points (consists of three different Gaussians) and the level sets of moment tensor with different order.  As observed, the higher-order moment tensor characterizes the distribution more accurately.\n\nOur contribution can be concluded as: (1) We propose a Higher-order Moment Matching (HoMM) method to minimize the domain discrepancy, which is expected to perform fine-grained domain alignment. The HoMM integrates the MMD and CORAL into a unified framework and generalizes the first-order and second-order moment matching to higher-order moment tensor matching. Without bells and whistles, the third- and fourth-order moment matching outperform all existing discrepancy-based methods by a large margin. The source code of the HoMM is released. (2) Due to lack of labels in the target domain, we propose to learn discriminative clusters in the target domain by assigning the pseudo-labels for the reliable target samples, which also improves the transfer performance.\n\n\\section{Related Work}\n\\textbf{Learning Domain-Invariant Features} To minimize the domain discrepancy and learn domain-invariant features, various distribution discrepancy metrics have been introduced. The representative ones include Maximal Mean Discrepancy (MMD) \\cite{gretton2012kernel,tzeng2014deep,long2015learning,long2017deep},  Correlation Alignment \\cite{sun2016deep,morerio2018,chen2019joint,chen2019selective} and Wasserstein distance \\cite{lee2019sliced,chendeep}. MMD was first introduced for the two-sample tests problem \\cite{gretton2012kernel}, and is currently the most widely used metric to measure the distance between two feature distributions. Specifically, Long \\etal proposed DAN \\cite {long2015learning} and JAN \\cite{long2017deep} which perform domain matching via multi-kernel MMD or a joint MMD criteria in multiple domain-specific layers across domains. Sun \\etal proposed the correlation alignment (CORAL) \\cite{sun2016return,sun2016deep} to align the second order statistics of the source and target distributions. Some recent work also extended the CORAL into reproducing kernel Hilbert spaces (RKHS) \\cite{zhang2018aligning} or deployed alignment along geodesics by considering the log-Euclidean distance \\cite{morerio2018}. Interestingly, \\cite{li2017demystifying} theoretically demonstrated that matching the second order statistics is equivalent to minimizing MMD with the second order polynomial kernel. Besides, the approach most relevant to our proposal is the Central Moment Discrepancy (CMD) \\cite{zellinger2017central}, which matches the higher order central moments of probability distributions by means of order-wise moment differences. Both CMD and our HoMM propose to match the higher-order statistics for domain alignment. The CMD matches the higher-order central moment while our HoMM matches the higher-order cumulant tensor. Another fruitful line of work tries to learn the domain-invariant features through adversarial training \\cite{ganin2016domain,tzeng2017adversarial}. These efforts encourage domain confusion by a domain adversarial objective whereby a discriminator (domain classifier) is trained to distinguish between the source and target representations. Also, recent work performing pixel-level adaptation by image-to-image transformation \\cite{murez2018image,hoffman2018cycada} has achieved satisfactory performance and obtained much attention. In this work, we propose a higher-order moment tensor matching approach to minimize the domain discrepancy, which shows great superiority over existing discrepancy-based methods.\n\n\n\\noindent\\textbf{Higher-order Statistics} The statistics higher than first-order has been success fully used in many classical and deep learning methods \\cite{de2007fourth,koniusz2016higher,pauwels2016sorting,li2017second,gou2017mom}. Especially in the field of fine-grained image\/video recognition \\cite{lin2015bilinear,cui2017kernel}, second-order statistics such as Covariance and Gaussian descriptors, have demonstrated better performance than descriptors exploiting zeroth- or first-order statistics \\cite{li2017second,lin2015bilinear,wang2017g2denet}. However, using second-order or lower statistical information might not be enough when the feature distribution is non-Gaussian \\cite{gou2017mom}. Therefore, the higher-order (greater than two) statistics have been explored in many signal processing problems \\cite{mansour1995fourth,jakubowski2002higher,de2007fourth,gou2017mom}. In the field of Blind Source Separation (BSS) \\cite{de2007fourth,mansour1995fourth}, for example, the fourth-order statistics are widely used to identify different signals from mixtures. Gou \\etal utilizes the third-order statistics for person ReID \\cite{gou2017mom}, Xu \\etal exploits the third-order cumulant for blind image quality assessment \\cite{xu2016blind}. In \\cite{jakubowski2002higher,koniusz2016higher}, the authors exploit higher-order statistics for image recognition and detection. Matching the second order statistics can not always ensure two distributions inseparable, just as using the second order statistics can not identifies different signals from underdetermined mixtures \\cite{de2007fourth}. That's why we explore higher-order moment tensor for domain alignment.\n\n\\noindent\\textbf{Discriminative Clustering} Discriminative clustering is a critical task in the unsupervised and semi-supervised learning paradigms \\cite{grandvalet2005semi,lee2013pseudo,xie2016unsupervised,caron2018deep}. Due to the paucity of labels in the target domain, how to obtain the discriminative representations in the target domain is of great importance for the UDA tasks. Therefore, a large body of work pays attention to learn the discriminative clusters in the target domain via entropy minimization \\cite{grandvalet2005semi,morerio2018,shu2018dirt}, pseudo label \\cite{lee2013pseudo,saito2017asymmetric,xie2018learning} or distance-based metrics \\cite{chen2019joint,kang2019contrastive}. Specifically, Saito \\etal \\cite{saito2017asymmetric} assign pseudo-labels to the reliable unlabeled samples to learn discriminative representations for the target domain. Shu \\etal \\cite{shu2018dirt} consider the cluster assumption and minimize the conditional entropy to ensure the decision boundaries not cross high-density data regions. MCD \\cite{saito2018maximum} also considers to align distributions of source and target by utilizing the task-specific decision boundaries. Besides, JDDA \\cite{chen2019joint} and CAN \\cite{kang2019contrastive} propose to model the intra-class domain discrepancy and the inter-class domain discrepancy to learn more discriminative features.\n\n\n\\section{Method}\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{fig1.pdf}\n\\caption{Two-stream CNNs with shared parameters are adopted for unsupervised deep domain adaptation. The first stream operates the source data and the second stream operates the target data. The last FC layer (the input of the output layer) is used as the adapted layer.}\n\\label{fig1}\n\\end{figure}\nIn this work, we consider the unsupervised domain adaptation problem. Let $\\mathcal{D}_s=\\{\\bm{x}_s^i,y_s^i\\}_{i=1}^{n_s}$ denotes the source domain with $n_s$ labeled samples and $\\mathcal{D}_t=\\{\\bm{x}_t^i\\}_{i=1}^{n_t}$ denotes the target domain with $n_t$ unlabeled samples. Given $\\mathcal{D}_s\\cup\\mathcal{D}_t$, we aim to train a cross-domain CNN classifier $f_{\\bm\\theta}(\\bm{x})$ which can minimize the target risks $\\epsilon_t=\\mathbb{E}_{\\bm{x}\\in\\mathcal{D}_t}[f_{\\bm\\theta}(\\bm{x})\\neq \\bm{y}_t]$. Here $f_{\\bm\\theta}(\\bm{x})$ denotes the outputs of the deep neural networks, $\\bm\\theta$ denotes the model parameter to be learned. Following \\cite{long2017deep,chen2019joint}, we adopt the two-stream CNNs architecture for unsupervised deep domain adaptation.  As shown in Fig. \\ref{fig1}, the two streams share the same parameters (tied weights), operating the source and target domain samples respectively. And we perform the domain alignment in the last full-connected (FC) layer \\cite{sun2016deep,chen2019joint}. According to the theory proposed by Ben-David \\etal \\cite{ben2010theory}, a basic domain adaptation model should, at least, involve the source domain loss and the domain discrepancy loss, i.e.,\n\\begin{equation}\\label{eq1}\n\\mathcal{L}(\\bm\\theta|\\mathbf{X}_s,\\mathbf{Y}_s,\\mathbf{X}_t)=\\mathcal{L}_s+\\lambda_{d}\\mathcal{L}_{d}\n\\end{equation}\n\\begin{equation}\\label{eq2}\n\\mathcal{L}_s=\\dfrac{1}{n_s}\\sum\\limits_{i=1}^{n_s}J(f_{\\bm\\theta}(\\bm{x}_i^s), \\bm{y}_i^s)\n\\end{equation}\nwhere $\\mathcal{L}_s$ represents the classification loss in the source domain, $J(\\cdot,\\cdot)$ represents the cross-entropy loss function. $\\mathcal{L}_{d}$ represents the domain discrepancy loss and $\\lambda_{d}$ is the trade-off parameter. As aforementioned, most of existing discrepancy-based methods are designed to minimize distance of the second-order or lower statistics between different domains. In this work, we propose a higher-order moment matching method, which matches the higher-order statistics of different domains.\n\\subsection{Higher-order Moment Matching}\nTo perform fine-grained domain alignment, we propose a higher-order moment matching as\n\\begin{equation}\\label{eq3}\n\\mathcal{L}_d=\\frac{1}{L^p}\\Vert\\frac{1}{n_s}\\sum\\limits_{i=1}^{n_s}\\phi_{\\bm\\theta}(\\bm{x}_s^i)^{\\otimes p}-\\frac{1}{n_t}\\sum\\limits_{i=1}^{n_t}\\phi_{\\bm\\theta}(\\bm{x}_t^i)^{\\otimes p}\\Vert_F^2\n\\end{equation}\nwhere $n_s=n_t=b$ ($b$ is the batch size) during the training process. $\\phi_{\\bm\\theta}(\\bm{x})$ denotes the activation outputs of the adapted layer. As illustrated in Fig. \\ref{fig1}, $\\bm{h}^i=\\phi_{\\bm\\theta}(\\bm{x}^i)=[\\bm{h}^i(1),\\bm{h}^i(2),\\cdots,\\bm{h}^i(L)]\\in\\mathbb{R}^L$ denotes the activation outputs of the $i$-th sample, $L$ is the number of hidden neurons in the adapted layer. Here, $\\bm{u}^{\\otimes p}$ denotes the $p$-level tensor power of the vector $\\bm{u}\\in\\mathbb{R}^c$. That is\n\\begin{equation}\\label{eq4}\n\\bm{u}^{\\otimes p}=\\underbrace{\\bm{u}\\otimes\\bm{u}\\cdots\\otimes\\bm{u}}_{\\text{p times}}\\in\\mathbb{R}^{c^p}\n\\end{equation}\nwhere $\\otimes$ denotes the outer product (or tensor product). We have $\\bm{u}^{\\otimes 0}=1$, $\\bm{u}^{\\otimes 1}=\\bm{u}$ and $\\bm{u}^{\\otimes 2}=\\bm{u}\\otimes\\bm{u}$. The 2-level tensor product $\\bm{u}^{\\otimes 2}\\in\\mathbb{R}^{c^2}$ defined as\n\\begin{equation}\\label{eq4}\n\\bm{u}^{\\otimes 2}=\\bm{u}^T\\bm{u}=\\left[\\begin{array}{cccc}\nu_1u_1 & u_1u_2 & \\cdots & u_1u_c\\\\\nu_2u_1 & u_2u_2 & \\cdots & u_2u_c\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\nu_cu_1 & u_cu_2 & \\cdots & u_cu_c\n\\end{array}\\right]\n\\end{equation}\nwhen $p\\geq 3$, $\\mathbf{T}=\\bm{u}^{\\otimes p}$ is a $p$-level tensor with $\\mathbf{T}[i,j,\\cdots,k]=u_iu_j\\cdots u_k$.\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{fig2.pdf}\n\\caption{An illustration of first-order, second-order and third-order moments in the source domain. HoMM matches the higher-order ($p\\geq3$) moment across different domains.}\n\\label{fig2}\n\\end{figure}\n\n\n\\textbf{Instantiations}\nAccording to Eq. \\eqref{eq3}, when $p=1$, the first-order moment matching is equivalent to the linear MMD \\cite{tzeng2014deep}, which is expressed as\n\\begin{equation}\\label{eq6}\n\\mathcal{L}_d=\\frac{1}{L}\\Vert\\frac{1}{b}\\sum\\limits_{i=1}^{b}\\bm{h}_s^i-\\frac{1}{b}\\sum\\limits_{i=1}^{b}\\bm{h}_t^i\\Vert_F^2\n\\end{equation}\nWhen $p=2$, the second-order HoMM is formulated as,\n\\begin{equation}\\label{eq7}\n\\begin{split}\n\\mathcal{L}_d=&\\frac{1}{L^2}\\Vert\\frac{1}{b}\\sum\\limits_{i=1}^{b}\\bm{h}_s^i{^T}\\bm{h}_s^i-\\frac{1}{b}\\sum\\limits_{i=1}^{b}\\bm{h}_t^i{^T}\\bm{h}_t^i\\Vert_F^2\\\\\n=&\\frac{1}{b^2L^2}\\Vert\\bm{G}(\\bm{h}_s)-\\bm{G}(\\bm{h}_t)\\Vert_F^2\n\\end{split}\n\\end{equation}\nwhere $\\bm{G}(\\bm{h})=\\bm{H}^T\\bm{H}\\in\\mathbb{R}^{L\\times L}$ is the Gram matrix, $\\bm{H}=[\\bm{h}^1;\\bm{h}^2;\\cdots,\\bm{h}^b]\\in\\mathbb{R}^{b\\times L}$, $b$ is the batch size. Therefore, the second-order HoMM is equivalent to the Gram matrix matching, which is also widely used for cross-domain matching in neural style transfer \\cite{gatys2016image,li2017demystifying} and knowledge distillation \\cite{yim2017gift}. Li \\etal \\cite{li2017demystifying} theoretically demonstrate that matching the Gram matrix of feature maps is equivalent to minimize the MMD with the second order polynomial kernel. Besides, when the activation outputs are normalized by subtracting the mean value, the centralized Gram matrix turns into the Covariance matrix. In this respect, the second-order HoMM is also equivalent to CORAL, which matches the Covariance matrix for domain matching \\cite{sun2016deep}.\n\nAs illustrated in Fig. \\ref{fig2}, in addition to the first-order moment matching (e.g. MMD) and the second-order moment matching (e.g. CORAL and Gram matrix matching), our proposed HoMM can also perform higher-order moment tensor matching when $p\\geq 3$. Since higher-order statistics can characterize the non-Gaussian distributions better, applying higher-order moment matching is expected to perform fine-grained domain alignment. However, the space complexity of calculating the higher-order tensor $\\bm{u}^{\\otimes p}$ ($p\\geq3$) reaches $\\mathcal{O}(L^p)$, which makes the higher-order moment matching infeasible in many real-world applications. Adding bottleneck layers to shrink the length of adaptive layer does not even solve the problem. When $L=128$, for example, the dimension of a third-order tensor still reaches $\\mathcal{O}(10^6)$, and the dimension of a fourth-order tensor reaches $\\mathcal{O}(10^8)$, which is absolutely computational-unfriendly. To address this problem, we propose two practical techniques to perform the compact tensor matching.\n\n\\textbf{Group Moment Matching.} As the space complexity grows exponentially with the number of neurons $L$, one practical approach is to divide the hidden neurons in the adapted layer into $n_g$ groups, with each group $\\lfloor L\/n_g\\rfloor$ neurons. Then we can calculate and match the high-level tensor in each group respectively. That is,\n\\begin{equation}\\label{eq8}\n\\mathcal{L}_d=\\frac{1}{b^2\\lfloor L\/n_g\\rfloor^p}\\sum\\limits_{k=1}^{n_g}\\Vert\\sum\\limits_{i=1}^{b}\\bm{h}_{s,k}^i{^{\\otimes p}}-\\sum\\limits_{i=1}^{b}\\bm{h}_{t,k}^i{^{\\otimes p}}\\Vert_F^2\n\\end{equation}\nwhere $\\bm{h}_{:,k}^i\\in\\mathbb{R}^{\\lfloor L\/n_g\\rfloor}$ is the activation outputs of $k$-th group. In this way, the space complexity can be reduced from $\\mathcal{O}(L^p)$ to $\\mathcal{O}(n_g\\cdot\\lfloor L\/n_g\\rfloor^p)$. In practice, $\\lfloor L\/n_g\\rfloor\\geq25$ need to be satisfied to ensure satisfactory performance.\n\n\\textbf{Random Sampling Matching.} The group moment matching can work well when $p=3$ and $p=4$, but it tends to fail when $p\\geq5$. Therefore, we also propose a random sampling matching strategy which is able to perform arbitrary-order moment matching. Instead of calculating and matching two high-dimensional tensors, we randomly select $N$ values in the high-level tensor, and only calculate and align these $N$ values in the source and target domains. In this respect, the $p$-order moment matching with random sampling strategy can be formulated as,\n\\begin{equation}\\label{eq9}\n\\small\n\\mathcal{L}_d=\\frac{1}{b^2N}\\sum\\limits_{k=1}^{N}[\\sum\\limits_{i=1}^{b}\\prod_{j=rnd[k,1]}^{rnd[k,p]}\\bm{h}_s^i(j)-\\sum\\limits_{i=1}^{b}\\prod_{j=rnd[k,1]}^{rnd[k,p]}\\bm{h}_t^i(j)]^2\n\\end{equation}\nwhere $rnd\\in\\mathbb{R}^{N\\times p}$ denotes the randomly generated position index matrix, $rnd[k,j]\\in\\{1,2,3,\\cdots,L\\}$. Therefore, $\\prod_{j=rnd[k,1]}^{rnd[k,p]}\\bm{h}_s^i(j)$ denotes a randomly sampled value in the $p$-level tensor $\\bm{h}_{s}^i{^{\\otimes p}}$. With the random sampling strategy, we can perform arbitrarily-order moment matching, and the space complexity can be reduced from $\\mathcal{O}(L^p)$ to $\\mathcal{O}(N)$. In practice, the model can achieve very competitive results even $N=1000$.\n\\subsection{Higher-order Moment Matching in RKHS}\nSimilar to the KMMD \\cite{long2017deep}, we generalize the higher-order moment matching into reproducing kernel Hilbert spaces (RKHS). That is,\n\\begin{equation}\\label{eq10}\n\\begin{split}\n\\mathcal{L}_d=\\frac{1}{L^p}\\Vert\\frac{1}{b}\\sum\\limits_{i=1}^{b}\\psi(\\bm{h}_s^i{^{\\otimes p}})-\\frac{1}{b}\\sum\\limits_{i=1}^{b}\\psi(\\bm{h}_t^i{^{\\otimes p}})\\Vert_F^2\n\\end{split}\n\\end{equation}\nwhere $\\psi(\\bm{h}_s^i{^{\\otimes p}})$ denotes the feature representation of $i$-th source sample in RKHS. According to the proposed random sampling strategy, $\\bm{h}_s^i{^{\\otimes p}}$ and $\\bm{h}_t^i{^{\\otimes p}}$ can be approximated by two $N$-dimensional vectors $\\bm{h}_{sp}^i\\in\\mathbb{R}^N$ and $\\bm{h}_{tp}^i\\in\\mathbb{R}^N$, where $\\bm{h}_{sp}^i(k)=\\prod_{j=rnd[k,1]}^{rnd[k,p]}\\bm{h}_s^i(j)$, $k=1,\\cdots,N$. In this respect, the domain matching loss can be formulated as\n\\begin{equation}\\label{eq11}\n\\begin{split}\n\\mathcal{L}_d&=\\frac{1}{b^2}\\sum\\limits_{i=1}^{b}\\sum\\limits_{j=1}^{b}k(\\bm{h}_{sp}^i,\\bm{h}_{sp}^j)-\\frac{2}{b^2}\\sum\\limits_{i=1}^{b}\\sum\\limits_{j=1}^{b}k(\\bm{h}_{sp}^i,\\bm{h}_{tp}^j) \\\\\n&+\\frac{1}{b^2}\\sum\\limits_{i=1}^{b}\\sum\\limits_{j=1}^{b}k(\\bm{h}_{tp}^i,\\bm{h}_{tp}^j)\n\\end{split}\n\\end{equation}\nwhere $k(\\bm{x},\\bm{y})=\\exp({-\\gamma\\Vert \\bm{x}-\\bm{y}\\Vert_2})$ is the RBF kernel function. Particularly, when $p=1$, the kernelized HoMM (KHoMM) is equivalent to the KMMD.\n\n\\subsection{Discriminative Clustering}\nWhen the target domain features are well aligned with the source domain features, the unsupervised domain adaptation turns into the semi-supervised classification problem, where the discriminative clustering in the unlabeled data is always encouraged \\cite{grandvalet2005semi,xie2016unsupervised}. There have been a lot of work trying to learn the discriminative clusters in the target domain \\cite{shu2018dirt,morerio2018}, most of which minimize the conditional entropy to ensure the decision boundaries do not cross high-density data regions,\n\\begin{equation}\\label{eq12}\n\\mathcal{L}_{ent}=\\frac{1}{n_t}\\sum_{i=1}^{n_t}\\sum_{j=1}^{c}-p_j\\log p_j\n\\end{equation}\nwhere $c$ is the number of classes, $p_j$ is the softmax output of $j$-th node in the output layer. We find that the entropy regularization works well when the target domain has high test accuracy, but it helps little or even downgrades the accuracy when the test accuracy is unsatisfactory. The reason can be drawn that the classifier may be misled as a result of entropy regularization enforcing over-confident probability on some misclassified samples. Instead of clustering in the output layer by minimizing the conditional entropy, we propose to cluster in the shared feature space. First, we pick up highly confident predicted target samples whose predicted probabilities are greater than a given threshold $\\eta$, and assign pseudo-labels to these reliable samples. Then, we penalize the distance of each pseudo-labeled sample to its class center. The discriminative clustering loss can be given as\n\\begin{equation}\\label{eq13}\n\\mathcal{L}_{dc}=\\frac{1}{n_t}\\sum_{i=1}^{n_t}\\Vert\\bm{h}_t^i-\\bm{c}_{\\hat{y}_t^i}\\Vert_2^2\n\\end{equation}\nwhere $\\hat{y}_t^i$ is the assigned pseudo-label of $\\bm{x}_t^i$, $\\bm{c}_{\\hat{y}_t^i}\\in \\mathbb{R}^L$ denotes its estimated class center. As we perform update based on mini-batch, the centers can not be accurately estimated by a small size of samples. Therefore, we update the class center in each iteration via moving average method. That is,\n\\begin{equation}\\label{Eq14}\n\\mathbf{c}_j^{t+1}=\\alpha\\mathbf{c}_j^t+(1-\\alpha)\\Delta\\mathbf{c}_j^t\n\\end{equation}\n\\begin{equation}\\label{Eq15}\n\\Delta\\mathbf{c}_j=\\dfrac{\\sum_{i=1}^b\\delta(\\hat{y}_i=j)\\bm{h}_t^i}{1+\\sum_{i=1}^b\\delta(\\hat{y}_t^i=j)}\n\\end{equation}\nwhere $\\alpha$ is the learning rate of the center. $\\bm{c}_j^t$ is the class center of $j$-th class in $t$-th iteration. $\\delta(\\hat{y}_t^i=j)=1$ if $\\bm{x}_t^i$ belongs to $j$-th class, otherwise it should be 0.\n\n\n\\subsection{Full Objective Function}\nBased on the aforementioned analysis, to enable effective unsupervised domain adaptation, we propose a holistic approach with an integration of (1) source domain loss minimization, (2) domain alignment with the higher-order moment matching and (3) discriminative clustering in the target domain. The full objective function is as follows,\n\\begin{equation}\\label{eq16}\n\\mathcal{L}=\\mathcal{L}_{s}+\\lambda_d\\mathcal{L}_{d}+\\lambda_{dc}\\mathcal{L}_{dc}\n\\end{equation}\nwhere $\\mathcal{L}_s$ is the classification loss in the source domain, $\\mathcal{L}_d$ is the domain discrepancy loss measured by the higher-order moment matching, and $\\mathcal{L}_{dc}$ denotes the discriminative clustering loss. Note that in order to obtain reliable pseudo-labels for discriminative clustering, we set $\\lambda_{dc}=0$ during the initial iterations, and enable the clustering loss $\\mathcal{L}_{dc}$ after the total loss tends to be stable.\n\n\\section{Experiments}\n\\subsection{Setup}\n\\noindent\\textbf{Dataset.}\nWe conduct experiments on three public visual adaptation datasets: digits recognition dataset, Office-31 dataset, and Office-Home dataset. The digits recognition dataset includes four widely used benchmarks: MNIST, USPS, Street View House Numbers (SVHN), and SYN (synthetic digits dataset). We evaluate our proposal across three typical transfer tasks, including: \\textbf{SVHN}$\\rightarrow$\\textbf{MNIST}, \\textbf{USPS}$\\rightarrow$\\textbf{MNIST} and \\textbf{SYN}$\\rightarrow$\\textbf{MNIST}. The details of this dataset can be seen in \\cite{chen2019joint}. Office-31 is another commonly used dataset for real-world domain adaptation scenario, which contains 31 categories acquired from the office environment in three distinct image domains: \\textbf{A}mazon (product images download from amazon.com), \\textbf{W}ebcam (low-resolution images taken by a webcam) and \\textbf{D}slr (high-resolution images taken by a digital SLR camera). The office-31 dataset contains 4110 images in total, with 2817 images in \\textbf{A} domain, 795 images in \\textbf{W} domain and 498 images in \\textbf{D} domain. We evaluate our method on all the six transfer tasks as \\cite{long2017deep}. The Office-Home dataset \\cite{venkateswara2017deep} is a more challenging dataset for domain adaptation, which consists of images from four different domains: Artistic images (\\textbf{A}), Clip Art images (\\textbf{C}), Product images (\\textbf{P}) and Real-world images (\\textbf{R}). The dataset contains around 15500 images in total from 65 object categories in office and home scenes.\n\n\n\\noindent\\textbf{Baseline Methods.}\nWe compare our proposal with the following methods, which are most related to our work:\nDeep Domain Confusion (\\textbf{DDC}) \\cite{tzeng2014deep}, Deep Adaptation Network (\\textbf{DAN}) \\cite{long2015learning}, Deep Correlation Alignment (\\textbf{CORAL}) \\cite{sun2016deep},  Domain-adversarial Neural Network (\\textbf{DANN}) \\cite{ganin2016domain}, Adversarial Discriminative Domain Adaptation (\\textbf{ADDA}) \\cite{tzeng2017adversarial}, Joint Adaptation Network (\\textbf{JAN}) \\cite{long2017deep}, Central Moment Discrepancy (\\textbf{CMD}) \\cite{zellinger2017central}  Cycle-consistent Adversarial Domain Adaptation (\\textbf{CyCADA}) \\cite{hoffman2018cycada}, Joint Discriminative feature Learning and Domain Adaptation (\\textbf{JDDA}) \\cite{chen2019joint}. Specifically, DDC, DAN, JAN, CORAL and CMD are representative moment matching based methods, while DANN, ADDA and CyCADA are representative adversarial training based methods.\n\n\n\n\\begin{table}[ht]\n\\centering\n\\caption{ Test accuracy (\\%) on digits recognition dataset for unsupervised domain adaptation based on modified LeNet}\\label{tab:aStrangeTable\n\\label{tab1}\n\\small\n\\begin{tabular}{ccccc}\n\\toprule\nMethod&  SN$\\rightarrow$MT&US$\\rightarrow$MT&SYN$\\rightarrow$MT&Avg\\\\\n\\midrule\nSource Only &67.3$\\pm$0.3&$66.4\\pm$0.4&89.7$\\pm$0.2&74.5\\\\\nDDC &71.9$\\pm$0.4&75.8$\\pm$0.3&89.9$\\pm$0.2&79.2\\\\\nDAN &79.5$\\pm$0.3&$89.8\\pm$0.2&75.2$\\pm$0.1&81.5\\\\\nDANN &70.6$\\pm$0.2&76.6$\\pm$0.3&90.2$\\pm$0.2&79.1\\\\\nCMD &86.5$\\pm$0.3&86.3$\\pm$0.4&96.1$\\pm$0.2&89.6\\\\\nADDA &72.3$\\pm$0.2&92.1$\\pm$0.2&96.3$\\pm$0.4&86.9\\\\\nCORAL &89.5$\\pm$0.2&96.5$\\pm$0.3&96.5$\\pm$0.2&94.2\\\\\nCyCADA &92.8$\\pm$0.1&97.4$\\pm$0.3&97.5$\\pm$0.1&95.9\\\\\nJDDA &94.2 $\\pm$0.1&96.7$\\pm$0.1&97.7$\\pm$0.0&96.2\\\\\n\\midrule\n\\textbf{HoMM}(p=3) &96.5$\\pm$0.2&97.8$\\pm$0.0&97.6$\\pm$0.1&97.3\\\\\n\\textbf{HoMM}(p=4) &95.7$\\pm$0.2&97.6$\\pm$0.0&97.6$\\pm$0.0&96.9\\\\\n\\textbf{KHoMM}(p=3) &97.2$\\pm$0.1&97.9$\\pm$0.1&98.2$\\pm$0.1&97.8\\\\\n\\textbf{Full} &\\textbf{98.8}$\\pm$0.1&\\textbf{99.0}$\\pm$0.1&\\textbf{99.0}$\\pm$0.0&\\textbf{98.9}\\\\\n\\midrule\n\\textbf{KHoMM+$\\mathcal{L}_{ent}$} &\\textbf{99.0}$\\pm$0.0&\\textbf{99.1}$\\pm$0.1&\\textbf{99.2}$\\pm$0.0&\\textbf{99.1}\\\\\n\\bottomrule\n\\end{tabular}\n\\footnotesize \\small We denote SVHN, MNIST, USPS as SN, MT and US.\n\\end{table}\n\n\n\n\n\n\n\\begin{table*}[ht]\n\\centering\n\\caption{Test accuracy (\\%) on Office-31 dataset for unsupervised domain adaptation based on ResNet-50}\\label{tab:aStrangeTable\n\\label{tab2}\n\\begin{tabular}{cccccccc}\n\\toprule\nMethod& A$\\rightarrow$W&D$\\rightarrow$W&W$\\rightarrow$D&A$\\rightarrow$D&D$\\rightarrow$A&W$\\rightarrow$A&Avg\\\\\n\\midrule\nSource Only &73.1$\\pm$0.2&93.2$\\pm$0.2&$98.8\\pm$0.1&72.6$\\pm$0.2&55.8$\\pm$0.1&56.4$\\pm$0.3&75.0\\\\\nDDC \\cite{tzeng2014deep}&74.4$\\pm$0.3&94.0$\\pm$0.1&98.2$\\pm$0.1&74.6$\\pm$0.4&56.4$\\pm$0.1&56.9$\\pm$0.1&75.8\\\\\nDAN \\cite{long2015learning}&78.3$\\pm$0.3&$95.2\\pm$0.2&$99.0\\pm$0.1&75.2$\\pm$0.2&58.9$\\pm$0.2&64.2$\\pm$0.3&78.5\\\\\nDANN \\cite{ganin2016domain}&73.6$\\pm$0.3&94.5$\\pm$0.1&99.5$\\pm$0.1&74.4$\\pm$0.5&57.2$\\pm$0.1&60.8$\\pm$0.2&76.7\\\\\nCORAL \\cite{sun2016deep}&79.3$\\pm$0.3&94.3$\\pm$0.2&99.4$\\pm$0.2&74.8$\\pm$0.1&56.4$\\pm$0.2&63.4$\\pm$0.2&78.0\\\\\nJAN \\cite{long2017deep}&85.4$\\pm$0.3&97.4$\\pm$0.2&99.8$\\pm$0.2&84.7$\\pm$0.4&68.6$\\pm$0.3&70.0$\\pm$0.4&84.3\\\\\nCMD \\cite{zellinger2017central}&76.9$\\pm$0.4&94.6$\\pm$0.3&99.2$\\pm$0.2&75.4$\\pm$0.4&56.8$\\pm$0.1&61.9$\\pm$0.2&77.5\\\\\nCyCADA \\cite{hoffman2018cycada}&82.2$\\pm$0.3&94.6$\\pm$0.2&99.7$\\pm$0.1&78.7$\\pm$0.1&60.5$\\pm$0.2&67.8$\\pm$0.2&80.6\\\\\nJDDA\\cite{chen2019joint}&82.6$\\pm$0.4&95.2$\\pm$0.2&99.7$\\pm$0.0&79.8$\\pm$0.1&57.4$\\pm$0.0&66.7$\\pm$0.2&80.2\\\\\n\\midrule\n\\textbf{HoMM}(p=3) &87.6$\\pm$0.2&96.3$\\pm$0.1&99.8$\\pm$0.0&83.9$\\pm$0.2&66.5$\\pm$0.1&68.5$\\pm$0.3&83.7\\\\\n\\textbf{HoMM}(p=4) &89.8$\\pm$0.3&97.1$\\pm$0.1&100.0$\\pm$0.0&86.6$\\pm$0.1&69.6$\\pm$0.3&69.7$\\pm$0.3&85.5\\\\\n\\textbf{KHoMM}(p=4) &90.5$\\pm$0.2&98.3$\\pm$0.1&100.0$\\pm$0.0&87.7$\\pm$0.2&70.4$\\pm$0.2&70.3$\\pm$0.2&86.2\\\\\n\\textbf{Full} &\\textbf{91.7}$\\pm$0.3&\\textbf{98.8}$\\pm$0.0&\\textbf{100.0}$\\pm$0.0&\\textbf{89.1}$\\pm$0.3&\\textbf{71.2}$\\pm$0.2&\\textbf{70.6}$\\pm$0.3&\\textbf{86.9}\\\\\n\\midrule\n\\textbf{KHoMM+$\\mathcal{L}_{ent}$} &90.8$\\pm$0.1&\\textbf{99.3}$\\pm$0.1&\\textbf{100.0}$\\pm$0.0&87.9$\\pm$0.2&69.3$\\pm$0.3&69.5$\\pm$0.4& 86.1\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table*}\n\n\n\n\n\n\n\n\n\n\\begin{table}[ht]\n\\centering\n\\caption{Test accuracy (\\%) on Office-Home dataset for unsupervised domain adaptation based on ResNet-50}\\label{tab:aStrangeTable\n\\label{tab3}\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{ccccccc}\n\\toprule\nMethod& A$\\rightarrow$P&A$\\rightarrow$R&C$\\rightarrow$R&P$\\rightarrow$R&R$\\rightarrow$P\\\\\n\\midrule\nSource Only  &50.0 &58.0 &46.2 &60.4 &59.5 \\\\\nDDC          &54.9 &61.3 &50.5 &64.1 &65.9 \\\\\nDAN          &57.0 &67.9 &60.4 &67.7 &74.3 \\\\\nDANN         &59.3 &70.1 &60.9 &68.5 &76.8 \\\\\nCORAL        &58.6 &65.4 &59.8 &68.3 &74.7 \\\\\nJAN          &61.2 &68.9 &61.0 &70.3 &76.8 \\\\\n\\midrule\n\\textbf{HoMM}(p=3)         &60.7 &68.3 &61.4 &69.2 &76.7 \\\\\n\\textbf{HoMM}(p=4)         &63.5 &70.2 &64.6 &72.6 &79.3 \\\\\n\\textbf{KHoMM}(p=4)        &63.9 &70.5 &65.3 &73.3 &79.8 \\\\\n\\textbf{Full}    &\\textbf{64.7} &\\textbf{71.8} &\\textbf{66.1} &\\textbf{74.5} &\\textbf{81.2} \\\\\n\\midrule\n\\textbf{KHoMM+$\\mathcal{L}_{ent}$}    &64.2 &70.1 &65.5 &73.2 &80.1 \\\\\n\\bottomrule\n\\end{tabular}}\n\\end{table}\n\n\n\\noindent\\textbf{Implementation Details.}\nIn our experiments on digits recognition dataset, we utilize the modified LeNet whereby a bottleneck layer with $90$ hidden neurons is added before the output layer. Since the image size is different across different domains, we resize all the images to $32\\times32$ and convert the RGB images to grayscale. For the experiments on Office-31, we use ResNet-50 pretrained on ImageNet as our backbone networks. And we add a bottleneck layer with 180 hidden nodes before the output layer for domain matching. It is worth noting that the \\textbf{relu} activation function can not be applied to the adapted layer, as relu activation function will make most of the values in the high-level tensor $\\bm{h}_{s}^i{^{\\otimes p}}$ to be zero, which will make our HoMM fail. Therefore, we adopt \\textbf{tanh} activation function in the adapted layer. Due to the small samples size of Office-31 and Office-Home datasets, we only update the weights of the full-connected layers (fc) as well as the final block (scale5\/block3), and fix other parameters pretrained on ImageNet. Follow the standard protocol of \\cite{long2017deep}, we use all the labeled source domain samples and all the unlabeled target domain samples for training. All the comparison methods are based on the same CNN architecture for a fair comparison. For DDC, DAN, CORAL and CMD, we embed the official implementation code into our model and carefully select the trade-off parameters to get the best results. When training with ADDA, our adversarial discriminator consists of 3 fully connected layers: two layers with 500 hidden units followed by the final discriminator output. For other compared methods, we report the results in the original paper directly.\n\n\\noindent\\textbf{Parameters.} Our model is trained with Adam Optimizer based on Tensorflow. Regarding the optimal hyper-parameters, they are determined by applying multiple experiments using grid search strategy. The optimal hyper-parameters may be distinct across different transfer tasks. Specifically, the trade-off parameters are selected from $\\lambda_{d}=\\{1,10,10^2,\\cdots,10^8\\}$, $\\lambda_{dc}\\in\\{0.01,0.03,0.1,0.3,1.0\\}$. For the digits recognition tasks, the hyper-parameter $\\lambda_d$ is set to $10^4$ for third-order HoMM and set to $10^7$ for fourth-order HoMM \\footnote{Note that the trade-off on the fourth-order HoMM is much larger than the third-order HoMM. This is because most deep features of the digits are very small, higher-order moments calculating the cumulative multiplication between different features become very close to zeros. Therefore, on digits dataset, the higher the order, the larger the trade-off should be.}. For the experiments on Office-31 and Office-Home, $\\lambda_d$ is set to $300$ for the third-order HoMM and set to $3000$ for the fourth-order HoMM.  Besides, the hyper-parameter $\\gamma$ in RBF kernel is set to 1e-4 across the experiments, the learning rate of the centers is set to $\\alpha=0.5$ for digits dataset and set to $\\alpha=0.3$ for Office-31 and Office-Home dataset. The threshold $\\eta$ of the predicted probability is chosen from $\\{0.6,0.65,0.7,0.75,0.8,0.85,0.9,0.95\\}$, and the best results are reported. The parameter sensitivity can be seen in Fig. \\ref{fig4}.\n\n\n\\subsection{Experimental results}\n\\textbf{Digits Dataset} For the experiments on digits recognition dataset, we set the batch size as 128 for each domain and set the learning rate as 1e-4 throughout the experiments. Table \\ref{tab1} shows the adaptation performance on three typical transfer tasks based on the modified LeNet. As can be seen, our proposed HoMM yields notable improvement over the comparison methods on all of the transfer tasks. In particular, our method improves the adaption performance significantly in the hard transfer tasks SVHN$\\rightarrow$MNIST. Without bells and whistles, the proposed third-order KHoMM achieve 97.2\\% accuracy, improving the second-order moment matching (CORAL) by +8\\%. Besides, the results also indicate that the third-order HoMM outperforms the fourth-order HoMM and slightly underperforms the KHoMM.\n\n\\noindent\\textbf{Office-31} Table \\ref{tab2} lists the test accuracies on Office-31 dataset. We set the batchsize as 70 for each domain. The learning rate of the fc layer parameters is set as 3e-4 and the learning rate of the conv layer (scale5\/block3) parameters is set as 3e-5. As we can see, the fourth-order HoMM outperforms the third-order HoMM and achieves the best results among all the moment-matching based methods. Besides, it is worth noting that the fourth-order HoMM outperforms the second-order statistics matching (CORAL) by more than 10\\% on several representative transfer tasks A$\\rightarrow$W, A$\\rightarrow$D and D$\\rightarrow$A, which demonstrates the merits of our proposed higher-order moment matching.\n\n\\noindent\\textbf{Office-Home} Table 3 gives the results on the challenged Office-Home dataset. The parameter settings are the same as in Office-31. We only evaluate our method on 5 out of 12 representative transfer tasks due to the space limitation. As we can see, on all the five transfer tasks, the HoMM outperforms the DAN, CORAL, DANN by a large margin and also outperforms the JAN by 3\\%-5\\%. Note that the experimental results of the compared methods are reported from \\cite{wang2019transferable} directly.\n\n\nThe results in Table \\ref{tab1}, Table \\ref{tab2} and Table \\ref{tab3} reveal several interesting observations: (1) All the domain adaptation methods outperform the source only model by a large margin, which demonstrates that minimizing the domain discrepancy contributes to learning more transferable representations. (2) Our proposed HoMM significantly outperforms the discrepancy-based methods (DDC, CORAL, CMD), and the adversarial training based methods (DANN, ADDA and CyCADA), which reveals the advantages of matching the higher-order statistics for domain adaptation. (3) The JAN performs slightly better than the third-order HoMM on several transfer tasks, but it's always not as good as the fourth-order HoMM in spite of aligning the joint distributions of multiple domain-specific layers across domains. The performance of our HoMM will be improved as well if we utilize such a strategy. (4) The kernelized HoMM (KHoMM) consistently outperforms the plain HoMM, but the improvement seems limited. We believe the reason is that, the higher-order statistics are originally the high-dimensional features, which conceals the advantages of embedding the features into RKHS. (5) In all transfer tasks, the performance increases consistently by employing the discriminative clustering in target domain. In contrast, entropy regularization improves the transfer performance when the test accuracy is high, but it helps little or even downgrades the performance when the test accuracy is not that confident.\n\n\\begin{figure*}[!ht]\n    \\centering\n  \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{source-eps-converted-to.pdf}\n    \\caption{Source Only}\n    \\label{2D1}\n  \\end{subfigure}\n   \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{KMMD_1-eps-converted-to.pdf}\n    \\caption{KMMD}\n    \\label{2D2}\n  \\end{subfigure}\n    \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{CORAL_1-eps-converted-to.pdf}\n    \\caption{CORAL}\n    \\label{2D3}\n  \\end{subfigure}\n    \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{HoMM3_1-eps-converted-to.pdf}\n    \\caption{HoMM(p=3)}\n    \\label{2D4}\n  \\end{subfigure}\n    \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{Full_1-eps-converted-to.pdf}\n    \\caption{Full Loss}\n    \\label{2D5}\n  \\end{subfigure}\n\n\n    \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{source_1-eps-converted-to.pdf}\n    \\caption{Source Only}\n    \\label{2D21}\n  \\end{subfigure}\n   \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{KMMD-eps-converted-to.pdf}\n    \\caption{KMMD}\n    \\label{2D22}\n  \\end{subfigure}\n    \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{CORAL-eps-converted-to.pdf}\n    \\caption{CORAL}\n    \\label{2D23}\n  \\end{subfigure}\n    \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{HoMM3-eps-converted-to.pdf}\n    \\caption{HoMM(p=3)}\n    \\label{2D24}\n  \\end{subfigure}\n    \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{Full-eps-converted-to.pdf}\n    \\caption{Full Loss}\n    \\label{2D25}\n  \\end{subfigure}\n\\caption{ 2D visualization of the deep features generated by different model on SVHN$\\rightarrow$MNIST. The first row illustrates the t-SNE embedding of deep features\nwhich are marked by category information, each color represents a category. The second row illustrates the t-SNE embedding of deep features which are marked by domain information, red and blue points represent the samples drawn from the source and target domains.}\n\\label{fig3}\n\\end{figure*}\n\n\n\\begin{figure*}[!ht]\n    \\centering\n  \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{sen_d-eps-converted-to.pdf}\n    \\caption{$\\lambda_d$}\n    \\label{2D11}\n  \\end{subfigure}\n   \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{sen_dc-eps-converted-to.pdf}\n    \\caption{$\\lambda_{dc}$}\n    \\label{2D22}\n  \\end{subfigure}\n    \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{sen_N-eps-converted-to.pdf}\n    \\caption{$N$}\n    \\label{2D33}\n  \\end{subfigure}\n    \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{sen_delta-eps-converted-to.pdf}\n    \\caption{$\\eta$}\n    \\label{2D55}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.19\\textwidth}\n    \\includegraphics[width=\\textwidth]{Convergence_1-eps-converted-to.pdf}\n    \\caption{Convergence}\n    \\label{2D55}\n  \\end{subfigure}\n\\caption{ Analysis of parameter sensitivity (a)-(d) and convergence analysis (e). The dash line in (b) and (d) indicate the performance of HoMM without the clustering loss $\\mathcal{L}_{dc}$}\n\\label{fig4}\n\\end{figure*}\n\n\n\\subsection{Analysis}\n\\textbf{Feature Visualization}\nWe utilize t-SNE to visualize the deep features on the tasks SVHN$\\rightarrow$MNIST by ResNet-50, KMMD, CORAL, HoMM(p=3) and the Full Loss model. As shown in Fig. \\ref{fig3}, the feature distributions of the source only model in (a) suggests that the domain shift between SVNH and MNIST is significant, which demonstrates the necessity of performing domain adaptation. Besides, the global distributions of the source and target samples are well aligned with the KMMD (b) and CORAL (c), but there are still many samples being misclassified. With our proposed HoMM, the source and target samples are aligned better and categories are discriminated better as well.\n\n\\noindent\\textbf{First\/Second-order versus Higher-order}\nSince our proposed HoMM can perform arbitrary-order moment matching, we compare the performance of different order moment matching on three typical transfer tasks. As shown in table \\ref{tab4}, the order is chosen from $p\\in\\{1,2,3,4,5,6,10\\}$. The results show that the third-order and fourth-order moment matching significantly outperform the other order moment matching. When $p\\leq3$, the higher the order, the higher the accuracy. When $p\\geq4$, the accuracy will decrease as the order increases. Besides, the fifth-order moment matching also achieves very competitive results. Regarding why the fifth-order and above perform worse than the fourth-order, one reason we believe is that the fifth-order and above moments can't be accurately estimated due to the small sample size problem \\cite{raudys1991small}.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Test accuracy (\\%) comparison of different-order moment matching on three transfer tasks}\n\\label{tab4}\n\\small\n\\begin{tabular}{cccccccc}\n\\toprule\norder &1 &2 &3 &4 &5 &6 &10\\\\\n\\midrule\nSN$\\rightarrow$MT &71.9 &89.5 &\\textbf{96.5} &95.7 &94.8 &91.5 &58.6 \\\\\nA$\\rightarrow$W &74.4 &79.3 &87.6 &\\textbf{89.8} &86.6 &85.3 &80.2 \\\\\nA$\\rightarrow$P &54.9 &58.6 &60.7 &\\textbf{63.5} &60.9 &58.2 &57.3\\\\\n\\bottomrule\n\\end{tabular}\n\\footnotesize \\small We denote SVHN and MNIST as SN and MT respectively.\n\\end{table}\n\n\n\n\n\\noindent\\textbf{Parameter Sensitivity and Convergence}\nWe conduct empirical parameter sensitivity on SVHN$\\rightarrow$MNIST and A$\\rightarrow$W in Fig. \\ref{fig4}(a)-(d). The evaluated parameters include two trade-off parameters $\\lambda_c$, $\\lambda_{dc}$, the number of selected values in Random Sampling Matching $N$, and the threshold $\\eta$ of the predicted probability. As we can see, our model is quite sensitive to the change of $\\lambda_{dc}$ and the bellshaped curve illustrates the regularization effect of $\\lambda_d$ and $\\lambda_{dc}$. The convergence performance is provided in Fig. \\ref{fig4}(e), which shows that our proposal converges fastest compared with other methods. It is worth noting that, the test error of the Full Loss model has a obvious mutation at the $2.0\\times10^4$ iteration where we enable the clustering loss $\\mathcal{L}_{dc}$, which also demonstrates the effectiveness of the proposed discriminative clustering loss.\n\n\n\n\\section{Conclusion}\nMinimizing statistic distance between source and target distributions is an important line of work for domain adaptation. Unlike previous methods that utilize the second-order or lower statistics for domain alignment, this paper exploits the higher-order statistics for domain alignment. Specifically, a higher-order moment matching method has been presented, which integrates the MMD and CORAL into a unified framework and generalizes the existing first-order and second-order moment matching to arbitrary-order moment matching. We experimentally demonstrate that the third-order and fourth-order moment matching significantly outperform the existing moment matching methods. Besides, we also extend the HoMM into RKHS and learn the discriminative clusters in the target domain, which further improves the adaptation performance. The proposed HoMM can be easily integrated into other domain adaptation model, and it is also expected to benefit the knowledge distillation and image style transfer.\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:Intro}\nIndustrial drying accounts for approximately 12\\% of the total end-use energy used in manufacturing, corresponding to 1.2 quads annually \\cite{DOEBarrier}. The US Department of Energy estimates a reduction of around 40\\% (0.5 quads\/year) of this amount by using more efficient process controls and new drying technologies, which lower the operating costs by \\$8 B\/year \\cite{DOEReview}. In addition, the drying process significantly affects the \\textit{quality} of food products. For instance, prolonged exposure to excessive heat negatively impacts products' physical and nutritional properties \\cite{NutritionalP}.\n\nIn recent times, more efficient drying technologies have been proposed in the literature. \nDielectrophoresis (DEP) technology \\cite{DEP}\\cite{DEP2}, ultrasound drying (US) \\cite{US}\\cite{US2}, slot jet reattachment nozzle (SJR) \\cite{SJR}\\cite{SJR2}, and infrared (IR) drying \\cite{IR} are examples of these new technologies which have helped improve product quality and energy efficiency. Industrial drying units typically employ one of the possible technologies out of all the available options to achieve the end goal of the drying process. However, in most cases, these technologies perform with different efficiencies in different settings. Hence, depending on the operating conditions, some of them are more favorable to use than others. For instance, contact-based ultrasound technology is more effective in the initial phase of the process, where the moisture content of the food sample is relatively high. In contrast, pure hot air drying consumes less energy and is more effective when the moisture content is low. Thus, if these two processes are conjoined, we can take advantage of both technologies to compensate for the other's inefficiencies. Therefore, knowing (a) the sequence in which these different drying techniques should be put together, and (b) the operating parameters of each of them, helps us make use of their capabilities. To put it in another way, dividing the drying process into sub-processes that differ in drying method and operating conditions paves the way to alleviate their individual limitation.\n\n\\captionsetup[figure]{font=small,labelfont=small}\n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=0.5\\textwidth]{Smart_Schematics}\n  \\caption{Schematics of the continuous \\textit{smart} dryer prototype with seven buckets which accommodates multiple drying  technologies to achieve better performance. In the example shown above, two DEP, two ultrasound, one IR, and two SJR modules are used in a specific order.}\n   \\label{fig:SmartD}\n\\end{figure}\n \nTo give an example, let us consider the \\textit{continuous} drying testbed shown in Fig.\\ref{fig:SmartD} which contains different drying modules such as ultrasound, DEP, SJR nozzle, and IR technologies. In this case, every drying module is controlled by a set of parameters influencing the amount of moisture removal. For example, ultrasound power and its duty cycle are the control parameters of the ultrasound technology, whereas electric field intensity is the control variable of the DEP module. Moreover, the control parameters of each dryer affect the amount of energy consumed during the process resulting in a tradeoff between energy consumption and moisture removal. Therefore, we can pose a combinatorial optimization problem to determine the optimal order by which the drying modules should be placed in the testbed and the optimal control parameters associated with them in order to minimize the total energy consumed by the testbed while reaching desired moisture removal by the end of the process. \n\nThis approach can also be extended to \\textit{batch-process} drying in a similar fashion, but with the slight difference that in this case, each technology can be used more than once. Fig. \\ref{fig:Batch_p} is an example of a testbed used for the batch drying process which consists of an ultrasonic module \\cite{US}, a drying chamber with a rectangular cross-section, a blower, and a heater. The food sample is located on a vibrating sheet attached to the ultrasonic transducer. In addition, it is exposed to the hot air coming from the heater which allows combined hot-air and ultrasound drying. In other words, the setup is designed in such a way that enables alternating between (a) pure hot-air (HA), and (b) combined hot-air and ultrasound (HA\/US) drying processes. In this case, the problem of interest is to reduce energy consumption (if possible) by dividing the process into consecutive pure HA and HA\/US sub-processes; where each of these sub-processes possibly have different operating conditions.\n\n\\captionsetup[figure]{font=small,labelfont=small}\n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=0.5\\textwidth]{Batch_p}\n  \\caption{Schematics of the convective\/ultrasound testbed for batch-process drying which can be used for both pure hot air (HA) and combined hot air and ultrasound (HA\/US) processes. The HA mechanism consists of a blower and a heater, whereas the US mechanism is a vibrating sheet attached to the ultrasound transducer. One can switch from HA\/US process to pure HA process by turning off the US transducer}\n   \\label{fig:Batch_p}\n\\end{figure}\n\nPrior works in drying literature majorly focus on either improving the efficiency of existing drying methods or developing new technologies \\cite{DEP},\\cite{US},\\cite{SJR}. The work done in \\cite{RSM3}, and \\cite{RSM2}, further utilizes optimization routines such as the response surface method (RSM) --- a statistical procedure --- to optimize the process control variables using experimental data \\cite{RSM}. To the best of our knowledge, there is scant literature that addresses optimization problems (sequencing and parameter values) related to unifying these different existing technologies. The main contribution of our work lies in the use of multiple existing technologies in a \\textit{modular} fashion towards achieving cost efficiency with desired performance levels.\nIn addition, this approach enables us to have optimal operating conditions that vary over time which can further improve the performance. Simulation results in section \\ref{sec:SR} show as large as 12\\% energy consumption reduction compared to the most efficient single-stage HA\/US process. The results also indicate an improvement of up to 63\\% in energy efficiency compared to the optimal commonly used hot air drying.\n\nSuch optimization problems appear in many industrial processes where a sequence of distinct devices with similar functions are used to form a unified process. Wood pulp industry with drying drums varying in radius and temperature, route optimization in multi-channel wireless networks with heterogeneous routers, and sensor network placement are examples of application areas where such optimization problems arise.\n \nThis paper presents a Maximum Entropy Principle (MEP)-based framework to model and subsequently optimize the different sub-processes in an industrial drying unit. Such optimization problems come with a lot of inherent complexities. For example, in the problem defined, the number of valid sequences of sub-processes is combinatorially large. Also, the discrete nature of these problems makes them difficult to deal with. In our proposed algorithm, this is addressed by assigning a probability distribution to the space of all possible configurations. In addition, the problem of determining the optimal operating conditions of sub-processes alone is broadly analogous to the resource allocation problem that is shown to be NP-hard \\cite{NP-hard} with a non-convex cost surface riddled with multiple poor local minima. Algorithms such as k-means, in this case, generally get trapped in the local minima and are very sensitive to initialization. A key concept of our algorithm is to develop a homotopy from an auxiliary function to the original non-convex cost function. This auxiliary function is a weighted linear combination of the original non-convex cost function and an appropriately chosen convex function. We choose this convex function to be the negative Shannon entropy of the probability distribution defined above. This concept is very similar to the Deterministic Annealing algorithm \\cite{rose1998deterministic} used for the clustering problem. At the beginning of the procedure, the weights are chosen such that the auxiliary objective function is dominated by the negative Shannon entropy term. Thus, the function is convex, and finding its global minimum is straightforward. After every iteration, the weight of the original non-convex cost gradually increases, and the obtained local minima are used to initialize the subsequent iteration. At the end of the procedure, the auxiliary function converges to the original non-convex cost function. This procedure is independent of the initialization and its main heuristic is that the global minimum is tracked by the algorithm as the convex cost function gradually changes to the desired non-convex cost function.\n\n\\begin{comment}\n \\section{Maximum Entropy Principle}\\label{sec:MEP}\n\nSince the framework we are presenting is built upon the Maximum Entropy Principle (MEP), we shall briefly review it. MEP states that given prior information about the random variable $X$, the most unbiased probability distribution for $X$ is the one maximizing its Shannon entropy. Had we chosen any other distribution, we would have imposed extra information on $X$. Let us assume the information we have about $X$ is in the form of constraints for expected values of the functions $f_k: X \\rightarrow \\mathbb{R}$. Then, according to MEP, the most unbiased probability distribution $p_X$ solves the optimization problem\n\n\\begin{equation} \\label{eq:MEP}\n\\begin{aligned}\n\\max_{p_X} \\hspace{0.25cm} &H(X) = -\\sum_{i=1}^n p_X(x_i) \\ln p_X(x_i) \\\\\n\\text{subject to}  \\hspace{0.25cm} &\\sum_{i=1}^n p_X(x_i)f_k(x_i) = F_k \\hspace{0.25cm} \\forall 1\\leq k \\leq m,\n\\end{aligned}\n\\end{equation}\nin which $F_k$, $1\\leq k \\leq m,$ are known expected values of the functions $f_k$. Solution of (\\ref{eq:MEP}) is given by  the following Gibbs' distribution\n\n\\begin{equation} \\label{eq:Gibbs}\n\\begin{aligned}\np_X(x_i) = \\frac{e^{-\\sum_k \\lambda_k f_k(x_i)}}{\\sum_{j=1}^n e^{-\\sum_k \\lambda_k f_k(x_j)}}\n\\end{aligned}\n\\end{equation}\nwhere $\\lambda_k$, $1 \\leq k \\leq m$, are the corresponding Lagrange multipliers of the equality constraints in (\\ref{eq:MEP}).\n\\end{comment}\n\n\n\\section{Problem Formulation}\\label{sec:PF}\n\nWe formulate the problem stated above as a \\textit{parameterized path-based optimization problem} \\cite{Kale}. Such problems are described by a tuple\n\\begin{align}\n\\mathcal{M}=\\langle M,\\gamma_1,\\hdots,\\gamma_M,\\eta_1,\\hdots,\\eta_M,D\\rangle,\n\\end{align}\nwhere $M$ is the number of stages allowed, and $\\gamma_k$ denotes the sub-process chosen to be used in the $k-$th stage. In particular,\n\\begin{align}\n\\gamma_k \\in \\Gamma_k := \\{f_{k1},\\hdots,f_{kL_k}\\} \\quad  \\forall \\quad 1 \\leq k \\leq M,\n\\end{align}\nwhere $\\Gamma_k$ is the set of all sub-processes permissible in the $k-$th stage. Moreover,\n\\begin{align}\n&\\eta_k \\in H(\\gamma_k) \\subseteq \\mathbb{R}^{d_{\\gamma_k}} \\quad  \\forall \\quad 1 \\leq k \\leq M,\n\\end{align}\nwhere $\\eta_k$ and $H(\\gamma_k)$ denote the control parameters associated with the $k-$th sub-process and its feasible set, respectively.\n$D(\\omega,\\eta_1,\\hdots,\\eta_M)$ denotes the cost incurred along a {\\em path} $\\omega$, where $\\omega\\in\\Omega := \\{(f_{1i_1},f_{2i_2},\\hdots,f_{Mi_M}):f_{ki_k}\\in\\Gamma_k\\}$ represents a sequence of sub-processes starting from the first stage to the terminal stage $M$. The objective of the underlying parameterized path-based optimization problem is to determine (a) the optimal path $\\omega^*\\in\\Omega$, and (b) the parameters $\\eta_k^*$ for all $1\\leq k \\leq M$ that solves the following optimization problem \n\\begin{equation}\n\\begin{aligned}\n\\min_{\\{\\eta_k\\},\\nu(\\omega)} \\label{eq: path-based}\n&\\sum_{\\omega\\in\\Omega}\\nu(\\omega)D(\\omega,\\eta_1,\\hdots,\\eta_M),\\\\\n\\text{subject to }& \\sum_{\\omega\\in\\Omega} \\nu(\\omega) = 1,~~ \\nu(\\omega)\\in\\{0,1\\}\\\\\n& \\eta_k \\in H(\\gamma_k) \\quad \\forall \\quad 1 \\leq k \\leq M\n\\end{aligned}\n\\end{equation}\nwhere $\\nu(\\omega)$ determines whether or not the path $\\omega$ has been taken. In other words,\n\\begin{align}\n\\begin{split} \\label{eq:nu}\n\\nu(\\omega) = \\begin{cases}\n1 \\quad \\text{if $\\omega$ is chosen}\\\\\n0 \\quad \\text{otherwise.}\n\\end{cases}\n\\centering\n\\end{split}\n\\end{align}\\\\\nFig. \\ref{fig:Seq_Dec} further illustrates all the notations defined, for the exemplary process shown in Fig. \\ref{fig:SmartD}.\n\nOne approach to address the optimization problem stated in (\\ref{eq: path-based}) is to solve each objective separately, namely, first identifying the optimal process configuration $\\omega^*$ and then trying to determine the optimal control variables $\\{\\eta^*_k\\}_{1 \\leq k \\leq M}$ for the selected configuration. However, in this approach, the coupledness of the two objectives is not taken into account which may result in a sub-optimal solution. On the other hand, our MEP-based approach aims for solving the two simultaneously.\n\nLet us reconsider the batch-process drying example described earlier, where the testbed allows up to $M$ different sub-processes, each could be either HA or HA\/US. To pose the problem of interest as a parameterized path-based optimization problem, we define\n\\begin{align}\n\\Gamma_k := \\{0,1\\} \\quad \\forall \\quad 1 \\leq k \\leq M,\n\\end{align}\nin which 0 and 1 indicate HA and HA\/US sub-processes, respectively. Thus, the process configuration $\\omega \\in \\Omega$ would become\n\\begin{equation} \\label{eq:gamma}\n\\begin{aligned}\n\\omega = (\\gamma_1, \\gamma_2, ... , \\gamma_M), \\quad \\gamma_k \\in \\{0,1\\} \\quad \\forall \\quad 1 \\leq k \\leq M,\n\\end{aligned}\n\\end{equation}\n\nThe control parameters of both sub-processes, in this case, are residence time $t$ and air temperature $T$. The heater of the setup in Fig. \\ref{fig:Batch_p} is designed to keep the air temperature between $30^{\\circ}C$ and $70^{\\circ}C$. Also, considering the settling time of the air temperature, it is required for all sub-processes to take at least $t_0 = 2$ minutes. Hence,\n\\begin{align}\n&\\eta_k = (t_k,T_k) \\quad &&\\forall \\quad 1 \\leq k \\leq M,\\\\\n& 2 \\leq t_k \\quad &&\\forall \\quad 1 \\leq k \\leq M,\\\\\n& 30^{\\circ}C \\leq T_k \\leq 70^{\\circ}C \\quad &&\\forall \\quad 1 \\leq k \\leq M,\n\\end{align}\nA key assumption here is that all the samples within a batch are similar in properties \n such as porosity and initial moisture content.\n\n\\captionsetup[figure]{font=small,labelfont=small}\n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=0.5\\textwidth]{Seq_Dec}\n  \\caption{Diagram corresponding to the process shown in Fig. \\ref{fig:SmartD} in which $\\Gamma = \\{\\text{DEP (1), SJR (2), US (3), IR (4)}\\}$. In the sequence shown by the arrows, $\\gamma_1 =1, \\gamma_2 = 3, ... , \\gamma_7 = 2$ which defines the process configuration $\\omega = (\\gamma_1,\\gamma_2,...,\\gamma_7)$. Moreover, $\\eta_i (1 \\leq i \\leq7)$ determines the control variables of the technology used in $i-$th stage.}\n   \\label{fig:Seq_Dec}\n\\end{figure}\n\nIn order to define the cost of a process, we need to first identify the desired properties of dried food products such as wet basis moisture content (ratio of the weight of water to the total weight of the material) and color.\nFor simplicity, we will consider the wet basis moisture content as the only property that needs to lie within a prespecified range ($\\leq x_d$) at the end of the process. Throughout this paper, we denote the wet basis moisture content of the food sample at the end of the $k-$th stage under process configuration $\\omega$ by $x^{(\\omega)}_k$. To include the cost associated with the final moisture content, it is required to model the corresponding dynamics as\n\\begin{equation} \\label{eq:Dynamics}\n\\begin{aligned}\nx_{k+1}^{(\\omega)} = f_{\\gamma_k}(x_k^{(\\omega)},\\eta_k) , \\quad \\forall \\quad \\quad 0 \\leq k \\leq M-1,\n\\end{aligned}\n\\end{equation}\nwhere $x_0^{(\\omega)}$ denotes the initial wet basis moisture content of the food sample. The semi-empirical drying curves (moisture content versus time) of distillers dried grains (DDG) were derived and evaluated in \\cite{US} for $T=25^{\\circ}C$, $T=50^{\\circ}C$, and $T=70^{\\circ}C$. Interpolation between drying curves obtained for various temperatures is commonly used in drying literature to predict the kinetics of drying. For the sake of our simulations, we use interpolation and rewrite the kinetics of moisture removal in the following way\n\\begin{equation*} \\label{eq:Stat_M}\n\\begin{aligned}\nx_{k+1}^{(\\omega)} = \\frac{e^{-K_{\\gamma_k}(T_k)t_k^*(T_k,x_k^{(\\omega)},t_k)} (3.16 - M_{\\gamma_k}(T_k)) + M_{\\gamma_k}(T_k)}{1 + e^{-K_{\\gamma_k}(T_k)t_k^*(T_k,x_k^{(\\omega)},t_k)} (3.16 - M_{\\gamma_k}(T_k)) + M_{\\gamma_k}(T_k)}\n\\end{aligned}\n\\end{equation*}\nin which\\\\\n\\begin{equation} \\label{eq:K}\n\\begin{aligned}\nK_0(T_k) = &(0.074493T_k^2 - 45.5058T_k + 6839.9)\/1000\\\\\nK_1(T_k) = &(-0.05811T_k^2 + 39.962T_k - 6680.1)\/1000\\\\\n&(T_k \\hspace{0.1cm} \\text{in Kelvin})\n\\end{aligned}\n\\end{equation}\ndenote the Lewis model constants \\cite{Lewis} for HA and HA\/US sub-processes, respectively. Also,\\\\\n\\begin{equation} \\label{eq:Meq}\n\\begin{aligned}\nM_0(T_k) = &(0.2479T_k^2 - 172.09T_k + 30133)\/10000\\\\\nM_1(T_k) = &(0.1468T_k^2 - 107.27T_k + 19720)\/10000\\\\\n&(T_k \\hspace{0.1cm} \\text{in Kelvin})\n\\end{aligned}\n\\end{equation}\nrepresent the equilibrium dry basis moisture content (ratio of the weight of water to the weight of the solid material) of the DDG products. Moreover, $t_k^*(T_k,x_k,t_k)$ is defined to be\n\\begin{equation} \\label{eq:Meq}\n\\begin{aligned}\nt_k^*(T_k,x_k^{(\\omega)},t_k) = \\frac{1}{K_{\\gamma_k}(T_k)}\\log(\\frac{3.16-M_{\\gamma_k}(T_k)}{\\frac{x_k^{(\\omega)}}{1 - x_k^{(\\omega)}} - M_{\\gamma_k}(T_k)}) + t_k\n\\end{aligned}\n\\end{equation}\nTherefore, we define the process cost $D(\\omega,\\eta_1,\\hdots,\\eta_M)$ as\n\\begin{equation} \\label{eq:Cost}\n\\begin{aligned}\nD = \\sum_{k=1}^M g_{\\gamma_k}(\\eta_k) + G(x_M^{(\\omega)},x_d) \\hspace{0.1cm},\n\\end{aligned}\n\\end{equation}\nin which $g_{\\gamma_k}:\\mathbb{R}^{d_{\\gamma_k}} \\rightarrow \\mathbb{R}$ is the cost (e.g. energy consumption) of the $k-$th sub-process, and $G: \\mathbb{R}^2 \\rightarrow \\mathbb{R}$ is a function penalizing the violation of the constraint. Fig. \\ref{fig:Penalty} shows an example of a penalty function used in our simulations. \nIn batch process drying example, $g_0(.)$ is the energy consumed for HA drying which can be approximated using the following:\n\\begin{equation} \\label{eq:Cost_HA}\n\\begin{aligned}\ng_0(\\eta_k) = g_0(t_k,T_k) \\propto & \\hspace{0.1cm} \\dot{m}_{air} c_p (T_k - T_0) t_k \\\\\n= & \\hspace{0.1cm} \\rho_{air} A V_{air} c_p (T_k - T_0) t_k \\hspace{0.1cm},\n\\end{aligned}\n\\end{equation}\nwhere $\\dot{m}_{air}$ is the mass flow rate of the inlet air, $T_0$ is ambient air temperature, $A$ is the cross-section area of the chamber, $V_{air}$ is air velocity, $c_p$ and $\\rho$ are  average specific heat capacity and density of air in the temperature operating range of the testbed. Therefore, we can use a weighting coefficient $\\alpha$ to adjust the cost of the HA sub-process.\n\\captionsetup[figure]{font=small,labelfont=small}\n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=0.35\\textwidth]{MC_penalty.png}\n  \\caption{The penalty function $G(x_M^{(\\omega)},x_d)$ used in our simulations in which $x_d = 0.075$. The function outputs large values for $x_M^{(\\omega)} > x_d$ and zero for $x_M^{(\\omega)} \\leq x_d$.}\n   \\label{fig:Penalty}\n\\end{figure}\n\\begin{equation} \\label{eq:Cost_HA}\n\\begin{aligned}\ng_0(t_k,T_k) = \\hspace{0.1cm} \\alpha \\rho_{air} A V_{air} c_p (T_k - T_0) t_k \\hspace{0.1cm},\n\\end{aligned}\n\\end{equation}\nIn the setup we used (shown in Fig. \\ref{fig:Batch_p}), choosing $\\alpha = 0.5$ matches well with the experimental data. On the other hand, $g_1(.)$ is the energy consumed for HA\/US sub-processes and can be computed using\n\\begin{equation} \\label{eq:Cost_HAUS}\n\\begin{aligned}\ng_1(\\eta_k) = g_1(t_k,T_k) = g_0(t_k,T_k) + P_{US}t_k \\hspace{0.1cm},\n\\end{aligned}\n\\end{equation}\nin which $P_{US}$ is the power consumption of the ultrasound transducer. Therefore, the total cost of the process can be written in the following way:\n\\begin{equation} \\label{eq:TotalCost}\n\\begin{aligned}\nD = \\sum_{k=1}^M (\\alpha \\rho_{air} A V_{air} c_p (T_k - T_0) + \\gamma_k P_{US})t_k + G(x_M^{(\\omega)},x_d)\n\\end{aligned}\n\\end{equation}\nAs a result, we write the corresponding combinatorial optimization problem below\n\\begin{align}\n\\begin{split} \\label{eq:constraints1}\n\\min_{\\{\\gamma_k\\},\\{t_k\\},\\{T_k\\}}&~ \\sum_{k=1}^M (\\alpha \\dot{m}_{air} c_p (T_k - T_0) + \\gamma_k P_{US})t_k + G(x_M^{(\\omega)},x_d)\\\\\n\\text{subject to }~ &2 \\leq t_k\\quad \\forall \\quad 1 \\leq k \\leq M \\\\\n&30^{\\circ}C \\leq T_k \\leq 70^{\\circ}C \\quad \\forall\\quad 1 \\leq k \\leq M\n\\end{split}\n\\end{align}\nTo adapt the above problem to the form of the parameterized path-based optimization problem described in (\\ref{eq: path-based}), we rewrite it as below\n\\begin{dmath}  \\label{eq:Optimization2}\n\\min_{\\{t_k\\},\\{T_k\\},v(\\omega)} \\sum_{\\omega \\in \\Omega} \\nu(\\omega) \\left( \\sum_{k=1}^M (\\alpha \\dot{m}_{air} c_p (T_k - T_0) + \\gamma_k P_{US})t_k + G(x_M^{(\\omega)},x_d)\\right)\n\\end{dmath}\n{subject to\n\\begin{align*}\n\\begin{split}\n~ &2 \\leq t_k\\quad \\forall \\quad 1 \\leq k \\leq M \\\\\n&30^{\\circ}C \\leq T_k \\leq 70^{\\circ}C \\quad \\forall\\quad 1 \\leq k \\leq M\\\\\n&\\nu(\\omega) \\in \\{0,1\\} \\quad \\forall\\quad \\omega \\in \\Omega\\\\\n&\\sum_{\\omega \\in \\Omega} \\nu(\\omega) = 1    \n\\end{split}\n\\end{align*}\n\n\n\\section{Problem Solution}\\label{sec:PS}\n\nCombinatorial optimization techniques can be used to solve the optimization problem stated in (\\ref{eq:Optimization2}). Intuitively, we can view it as a clustering problem in which the goal is to assign a particular sequence of sub-processes to every food sample with known initial moisture content. In this case, the location of the cluster centers can be thought of as the control parameters associated with that sequence. This work, similar to \\cite{Kale} and \\cite{9096570}, utilizes the idea of the Maximum Entropy Principle (MEP) \\cite{jaynes1957information}\\cite{jaynes2003probability}. To be able to invoke MEP, we relax the constraint $v(\\omega) \\in \\{0,1\\}$ in (\\ref{eq:Optimization2}) and let it take any value in $[0,1]$. We denote this new weighting parameter by\n\\begin{align}\np(\\omega) \\in [0,1] \\quad \\forall \\quad \\omega \\in \\Omega\n\\end{align}\nIn other words, we allow partial assignment of process configurations to the food sample. Note that this relaxation is only used in the intermediate stages of our proposed approach. The final solution still satisfies $p(\\omega) \\in \\{0,1\\}$. Without loss of generality, we assume that $\\sum_{\\omega \\in \\Omega}p(\\omega) = 1$. Hence, we can rewrite (\\ref{eq:Optimization2}) as:\n\\begin{dmath}  \\label{eq:Optimization_relaxed}\n\\min_{\\{t_k\\},\\{T_k\\},p(\\omega)} \\sum_{\\omega \\in \\Omega} p(\\omega) \\left( \\sum_{k=1}^M (\\alpha \\dot{m}_{air} c_p (T_k - T_0) + \\gamma_k P_{US})t_k + G(x_M^{(\\omega)},x_d)\\right)\n\\end{dmath}\n{subject to\n\\begin{align*}\n\\begin{split}\n~ &2 \\leq t_k\\quad \\forall \\quad 1 \\leq k \\leq M \\\\\n&30^{\\circ}C \\leq T_k \\leq 70^{\\circ}C \\quad \\forall\\quad 1 \\leq k \\leq M\\\\\n&p(\\omega) \\in [0,1] \\quad \\forall\\quad \\omega \\in \\Omega\\\\\n&\\sum_{\\omega \\in \\Omega} p(\\omega) = 1    \n\\end{split}\n\\end{align*}\n\nSince the framework we are presenting is built upon MEP, let us briefly review it in the context of this problem. MEP states that given prior information about the process, the most unbiased set of weights is the one that has the maximum Shannon entropy. Assume the information we have about the process is the expected value of the process cost ($\\mathbb{E}(D) = D_0$). Then, according to MEP, the most unbiased weighting parameters solve the optimization problem\\\\\n\\begin{align}  \\label{eq:MEP}\n\\begin{split}\n\\max_{p} &~ H(X) = -\\sum_{\\omega \\in \\Omega} p(\\omega) \\ln p(\\omega)  \\\\\n\\text{subject to} &~ \\Bar{D} = D_0\n\\end{split}\n\\end{align}\nwhere $\\Bar{D}$ is the expected value of the cost $D$, namely,\n\\begin{dmath*} \\label{Expected_D}\n \\Bar{D} = \\sum_{\\omega \\in \\Omega} p(\\omega) \\left( \\sum_{k=1}^M (\\alpha \\dot{m}_{air} c_p (T_k - T_0) + \\gamma_k P_{US})t_k + G(x_M^{(\\omega)},x_d) \\right)    \n\\end{dmath*} \\label{Expected_D}\nThe Lagrangian corresponding to (\\ref{eq:MEP}) is given by the maximization of $H - \\beta \\Bar{D}$, or equivalently, minimization of $F = \\Bar{D} - \\frac{1}{\\beta}H$, where $\\beta$ is the Lagrange multiplier. Therefore, the problem reduces to minimizing $F$ with respect to $p(\\omega)$, $t_k$s, and $T_k$s such that $\\sum_{\\omega \\in \\Omega} p(\\omega) = 1$. We add the last constraint with the corresponding Lagrange multiplier $\\mu$ to the objective function $F$ and rewrite the problem as below:\n\\begin{align}\n\\begin{split} \\label{eq:Unc_Lagrangian}\n\\min_{\\{t_k\\},\\{T_k\\},p(\\omega)} &~ \\Bar{D} - \\frac{1}{\\beta}H + \\mu (\\sum_{\\omega \\in \\Omega}p(\\omega) - 1)\\\\\n\\text{subject to }~ &t_0 \\leq t_k\\quad \\forall \\quad 1 \\leq k \\leq M \\\\\n&30^{\\circ}C \\leq T_k \\leq 70^{\\circ}C \\quad \\forall\\quad 1 \\leq k \\leq M\\\\\n\\end{split}\n\\end{align}\nWe denote the new objective function in (\\ref{eq:Unc_Lagrangian}) by $\\Bar{F}$. Note that $\\Bar{F}$ is convex in $p$, and the constraints in (\\ref{eq:Unc_Lagrangian}) do not depend on $p$. Hence, we determine the optimal weights by solving $\\frac{\\partial \\Bar{F}}{\\partial p}=0$ for fixed $t_k$s and $T_k$s. This gives the Gibbs distribution\n\\begin{align}\n\\begin{split} \\label{eq:p_opt}\np^*(\\omega) = \\frac{e^{-\\beta \\sum_{k=1}^M (\\alpha \\dot{m}_{air} c_p (T_k - T_0) + \\gamma_k P_{US})t_k + G(x_M^{(\\omega)},x_d) }}{\\sum_{\\omega^\\prime \\in \\Omega}e^{-\\beta \\sum_{k=1}^M (\\alpha \\dot{m}_{air} c_p (T_k - T_0) + \\gamma_k^\\prime P_{US})t_k + G(x_M^{(\\omega^\\prime)},x_d) }}\n\\end{split}\n\\end{align}\nTherefore, by  plugging (\\ref{eq:p_opt}) into $\\Bar{F}$, we obtain its corresponding minimum $\\Bar{F}^*$.\n\\begin{align}\n\\begin{split} \\label{eq:F_star}\n \\Bar{F}^* &~= \\min_{p(\\omega)} \\Bar{F}\\\\\n &~= -\\frac{1}{\\beta} \\log \\sum_{\\omega \\in \\Omega} e^{-\\beta \\alpha \\dot{m}_{air} c_p (T_k - T_0) + \\gamma_k P_{US})t_k + G(x_M^{(\\omega)},x_d) }\n\\end{split}\n\\end{align}\nSubsequently, to determine the optimal process parameters, we minimize $\\Bar{F}^*$ with respect to $\\eta_k$s. In other words, solving the constrained optimization problem\n\\begin{align}\n\\begin{split} \\label{eq:eta_opt}\n\\min_{\\{t_k\\},\\{T_k\\}} &~ \\Bar{F}^*\\\\\n\\text{subject to }~ &t_0 \\leq t_k\\quad \\forall \\quad 1 \\leq k \\leq M \\\\\n&30^{\\circ}C \\leq T_k \\leq 70^{\\circ}C \\quad \\forall\\quad 1 \\leq k \\leq M\\\\\n\\end{split}\n\\end{align}\nresults in finding the optimal residence times $t_k^*$ and Temperatures $T_k^*$ for all the sub-processes. Any constrained optimization algorithm can be used to solve (\\ref{eq:eta_opt}). As an example, we used the interior point algorithm in our simulations.\\\\\nThe proposed algorithm, thus, consists of iterations with the following two steps:\n\\begin{enumerate}\n    \\item Use parameters $t_k$ and $T_k$ ($\\forall 1\\leq k \\leq M$) obtained in the previous iteration to find the optimal weights according to (\\ref{eq:p_opt}).\n    \\item Numerically solve the constrained optimization in (\\ref{eq:eta_opt}) to find the optimal parameters $t_k^*$ and $T_k^*$ for all $1 \\leq k \\leq M$ using $t_k$s and $T_k$s as the initial guess for the algorithm.\n\\end{enumerate}\n\n\\captionsetup[figure]{font=small,labelfont=small}\n\\begin{figure*}[t]\n\\centering\n  \\includegraphics[width=1\\textwidth]{gamma0.5}\n  \\caption{Figures (a1), (b1), (c1), and (d1) represent the solution obtained using our proposed algorithm for $M=2$, $M=3$, $M=4$, and $M=5$ accordingly, where $M$ indicates the maximum number of stages allowed. In all simulations, $\\alpha$ is chosen to be 0.5. Figures (a2), (b2), (c2), and (d2) are the wet basis moisture content profiles corresponding to temperature profiles (a1), (b1), (c1), and (d1) respectively. Figure (e1) compares the temperature profile of the solution for $M=6$ and optimal single-stage HA and single-stage HA\/US processes. The results show a 12.09\\% energy consumption reduction compared to the single-stage HA\/US process and a 63.19\\% improvement compared to the single-stage pure HA process. Figure (e2) shows the wet basis moisture content profiles corresponding to the temperature profiles shown in figure (e1). }\n   \\label{fig:gamma0.5}\n\\end{figure*}\n\n\nIn both steps, the value of $\\Bar{F}$ is reduced. Therefore, the algorithm converges. Furthermore, we can adjust the relative weight of the entropy term $-H$ to the average cost $\\Bar{D}$ using the Lagrange multiplier $\\beta$. For $\\beta \\rightarrow 0$, maximizing the entropy term dominates minimizing the expected cost. In this case, the optimal weights derived in (\\ref{eq:p_opt}) would be equal for all the valid process configurations. On the other hand, when $\\beta \\rightarrow \\infty$, more importance is given to $\\Bar{D}$. In other words, for very large values of $\\beta$, we have:\n\\begin{align}\n\\begin{split} \\label{eq:p_lim}\n\\lim_{\\beta \\rightarrow \\infty}p(\\omega) = \\begin{cases}\n1 \\quad \\text{if} ~\\omega = \\underset{\\omega^\\prime \\in \\Omega}{\\mathrm{\\text{argmin}}}\n &\\sum_{k=1}^M \\{  (\\alpha \\dot{m}_{air} c_p (T_k - T_0) \\\\&+\\gamma_k^\\prime P_{US})t_k + G(x_M^{(\\omega^\\prime)},x_d) \\}\\\\\n 0 \\quad \\text{otherwise}\n\\end{cases}\n\\centering\n\\end{split}\n\\end{align}\nThe idea behind the algorithm is to start with $\\beta$ values close to zero, where the objective function $\\Bar{F}$ is convex and the global minimum can be found. Then, we keep track of this global minimum by gradually increasing $\\beta$ until $\\max_{\\omega \\in \\Omega} p(\\omega) \\rightarrow 1$. This procedure helps us avoid poor local minima. The proposed algorithm is as follows:\n\\begin{algorithm}[H]\n\\caption{Calculate optimal $p(\\omega)$, $t_k$, $T_k$ $(\\forall 1 \\leq k \\leq M)$}\n\\begin{algorithmic} \\label{alg:Main}\n\\STATE \\textbf{Initialize:} $T_k \\in [30,70]$,  $t_0 = 2 \\leq t_k$,  $\\beta = \\beta_{min}$,  $p_{max}$, $\\zeta \\geq 1$\n\\STATE Compute $p(\\omega)$ according to (\\ref{eq:p_opt}), $\\forall \\omega \\in \\Omega$.\n\\WHILE{$\\max_{\\omega}p(\\omega) \\leq p_{max}$}\n\\STATE Solve (\\ref{eq:eta_opt}) using any constrained optimization algorithm to find\n$(t_k^*,T_k^*)$ with $(t_k,T_k)$ as initial guess.\n\\STATE $(t_k,T_k) \\leftarrow (t_k^*,T_k^*) \\quad \\forall \\quad 1 \\leq k \\leq M$\n\\STATE Update $p(\\omega)$ according to (\\ref{eq:p_opt}), $\\forall \\omega \\in \\Omega$.\n\\STATE $\\beta \\leftarrow \\zeta \\beta$\n\\ENDWHILE\n\\end{algorithmic}\n\\end{algorithm}\nNote that it is better to initialize $(t_k, T_k)$ in such a way that the constraint $x_M^{(\\omega)} \\leq x_d$ is satisfied for all $\\omega \\in \\Omega$ since it results in faster convergence and fewer numbers of iterations.\n\n\\section{Simulations and Results}\\label{sec:SR}\n\n\n\n\nIn this section, we simulate our proposed algorithm for drying DDG products using multiple sub-processes and compare the results with the commonly-used single-stage drying process. We also change the number of sub-processes allowed ($M$), to see how additional sub-processes affect efficiency. Moreover, we can assign weights to the energy consumed by different sub-processes in order to include their additional costs (e.g. maintenance).  In our problem formulation, the coefficient $\\alpha$ defined in (\\ref{eq:Cost_HA}) can accommodate this relative weight.\n\n\\textbf {Effect of the permissible number of stages ($M$):} In simulations shown in Fig. \\ref{fig:gamma0.5}, we have considered drying fresh DDG products with roughly $75.5\\%$ initial wet basis moisture content to around $7.5\\%$ at the end of the process. We have used our proposed algorithm for $\\alpha = 0.5$ with different numbers of allowable sub-processes from two to six (Fig. \\ref{fig:Er_gamma0.5}) and compared the results with the optimal single-stage HA\/US and HA processes. The results show 11.96\\% ($M=2$), 12.07\\% ($M=3$), and 12.09\\% ( $M=4$, $M=5$, and $M=6$) improvement in energy consumption compared to the most efficient single-stage HA\/US drying process. In addition, it reduced the energy consumption by  63.13\\%  ($M=2$), 63.18\\% ($M=3$), and 63.19\\% ($M=4$, $M=5$, and $M=6$), in comparison with the optimal pure HA process.\n\n\\captionsetup[figure]{font=small,labelfont=small}\n\\begin{figure}[H]\n\\centering\n  \\includegraphics[width=0.27\\textwidth]{Er_gamma0.5}\n  \\caption{Energy consumption of the algorithm solution for different numbers of allowable stages.}\n   \\label{fig:Er_gamma0.5}\n\\end{figure}\n\n\\captionsetup[figure]{font=small,labelfont=small}\n\\begin{figure*}[h]\n\\centering\n  \\includegraphics[width=1\\textwidth]{gamma_vary}\n  \\caption{Figures (a1), (b1), (c1), (d1), and (e1) represent the solution of Algorithm \\ref{alg:Main} for $M=5$ and $\\alpha = 0.2$, $\\alpha = 0.1$, $\\alpha = \\frac{1}{15}$, $\\alpha = 0.05$ and $\\alpha = 0.04$, respectively. Here, $\\alpha$ denotes the relative cost weight of the pure HA process. Figures (a2), (b2), (c2), (d2), and (e2) are the wet basis moisture content profiles corresponding to temperature profiles (a1), (b1), (c1), (d1), and (e1) respectively.}\n   \\label{fig:gamma_vary}\n\\end{figure*}\n\n\nAs shown in Fig. \\ref{fig:Er_gamma0.5}, the cost of the solution given by Algorithm \\ref{alg:Main} decreases as the number of allowable sub-processes increases from two to four. However, for $M \\geq 4$, increasing $M$ does not impact energy consumption reduction. In general, by increasing $M$, the cost of the process either decreases or remains constant. The reason is that the space of all valid process configurations with $M$ stages is a subspace of process configurations with $N > M$ stages. Consequently, we can choose $M$ in such a way that further increasing it does not significantly affect the cost value. In this case, as an example, $M=4$ can be chosen as the optimal number of stages.\n\n\\textbf{Effect of the relative weight of HA process ($\\alpha$):} For $\\alpha = 0.5$, the HA\/US process is significantly more efficient than the pure HA process. Therefore, the pure HA process does not appear in the optimal solutions shown in Fig. \\ref{fig:gamma0.5}. In other words, in this case, the optimal process only consists of HA\/US sub-processes with different operating conditions. However, according to Fig.\\ref{fig:gamma_vary}, by reducing $\\alpha$, less weight is given to the cost of HA, and therefore, the pure HA process starts showing up in the optimal solution. Fig.\\ref{fig:gamma_vary} plots the solutions resulting from Algorithm \\ref{alg:Main} for $M=5$, and varying $\\alpha$ from 0.2 to 0.04. For $\\alpha = 0.2$ , $\\alpha = 0.1$ , and $\\alpha = \\frac{1}{15}$ , the optimal configuration is only comprised of HA\/US sub-processes. On the other hand, for $\\alpha = 0.05$, the pure HA sub-process is used in one stage of the optimal process, and finally, for $\\alpha = 0.04$, the optimal solution is entirely a pure HA process with a constant temperature profile. Processes (a), (b), (c), and (d) in Fig.\\ref{fig:gamma_vary} resulted in 9.95\\%, 5.75\\%, 3.62\\%, and 2.08\\% reduction in energy consumption compared to the most efficient single-stage processes.\n\n\n\n\n\n\\section{Conclusion and Future Work}\n\nIn this paper, we formulated a class of combinatorial optimization problems that arise in many industrial processes comprising sub-processes with the same end goals. We examined industrial drying in continuous and batch processes in more detail and applied our proposed algorithm to a batch process drying prototype which allows both HA and HA\/US drying. In addition, we discussed the advantage of solving for both the optimal process configuration and the optimal values of the control parameters simultaneously as compared to treating them as separate problems. \nThough in the example used to illustrate our proposed algorithm only two technologies were permitted to be used, the framework can be extended to $|\\Gamma_k| \\in \\mathbb{N}$ for all $1 \\leq k \\leq M$. Moreover, our algorithm is capable of accommodating more control parameters and quality constraints. In our future works, we aim for adding  air velocity, ultrasound power, and its duty cycle to the control variables and quantitative color to the constraints representing the desired features. \n\nIn the methodology presented, the space of all valid configurations of up to M sub-processes out of N available technologies is combinatorially large $O(\\sum_{k=1}^{M} {N \\choose k})$. In our future work, we are planning to use the Principle of Optimality to reduce the space of decision variables to polynomial size. According to the Principle of Optimality, along an optimal sequence of sub-processes, the next technology to be used and its operating conditions are decided only by the current state (e.g. moisture content), independent of the prior sub-processes. Therefore, successful utilization of this fact makes the proposed algorithm more scalable. Furthermore, the algorithm is flexible in adapting new constraints that are specific to the setup and technologies being used.\n\n\n\n\n\n\n\n\n\n\n\n \n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\nOver the last decades, diverse quantum computing components from application to physical device have been actively researched and developed.\nSome of them already have theoretically optimal performance~\\cite{Jones:2013ii,Muralidharan:2016ky,Ross:2016uz,Bombin:2015hi,Napp:2012vn,Bombin:2007ed}.\nGigantic IT corporations have announced that they succeeded in the development of dozens of qubit system~\\cite{IBM:Iidq8v80,Kelly:2018uc,Intel:1vYziePb}.\nEven they expect to see a thousand qubit system within ten years.\nBesides, several startups also started to develop a quantum computing hardware and software~\\cite{QuantumComputingReport:FR0ihxsu}.\nIt seems that the era of a quantum computing is gradually coming close to us.\n\n\nSo far, to see the feasibility of a practical large-scale quantum computing, some efforts have been dedicated to investigation into the quantity of the required quantum resource~\\cite{Suchara:2013ho,Smith:2014ux,Reiher:2017cv,Goto:2016jo,Suchara:2013tg,JavadiAbhari:2015jf,Steiger:2018to} and the expected performance~\\cite{Fowler:2012fi,Jones:2012kc,VanMeter:2009ut,Suchara:2013tg}.\nBased on such analysis, the security of conventional cryptography against a theoretically super-fast quantum computing also has been analyzed~\\cite{Grassl:2015wa}.\nHowever, we think that the analysis reported so far are not fully satisfactory because they usually have considered only a few components of a quantum computing and\/or focused on statistical examination. \nIn particular, the analysis results with the statistical examination are nothing more than simple calculation based on individual data of several quantum computing components~\\cite{Suchara:2013ho,Suchara:2013tg}.\n\n\nTo achieve more exact analysis, we believe the practical operation of a quantum computing needs to be considered.\nFor that, one first has to consider quantum computing components in terms of the practical operation.\nFor example, we have to prepare a quantum algorithm decomposed into individual gates instead of only the dominant part of a quantum algorithm.\nSecond, the above components have to be properly integrated with a target quantum computer system architecture.\nFor example, we have to find a path for qubit movements (or braiding) on the target system architecture without raising any conflicts with other paths.\nBy doing so, the quantity of qubit movements which cannot be considered in the statistical analysis can be exactly measured, and therefore we can say that the analysis results are realistic.\n\n\nIn this work, to perform the integrated analysis more efficiently, we propose and develop a quantum computing framework composed of three functional layers (compile, system and building block), where each layer has a well-defined role for a quantum computing as follows.\n\\begin{itemize}\n\\item Compile: Decomposition of a quantum algorithm into a sequence of quantum gates called a \\textit{quantum assembly code}\n\\item System: Integration of a quantum algorithm and building blocks under a quantum computer system architecture\n\\item Building Block: Implementation of logical qubits and gates according to a quantum error-correcting code and a fault-tolerant scheme\n\\end{itemize}\nWith the proposed framework, we can conduct the analysis in the most fine-grained manner.\nFor example, we even count a qubit movement one by one, and the quantity of physical qubits based on the provided quantum computer architecture and the ancilla qubits required for error correction and magic state.\nFurthermore, the framework examines the performance and the resource with the criterion, 100\\% fidelity quantum computing~\\cite{Whitney:2009wh}.\nBy doing so, we can provide full quantum resource and compare various quantum computing configurations fairly.\n\n\nThe objective of this work is to estimate the most accurate quantum computing performance and quantum resource by the help of the framework we developed.\nFor that, we first need to configure a quantum computing by choosing specific protocols and device properties. \nBy doing so, as mentioned above, we can see the quantum resource and the performance of a quantum computing based on most of quantum computing components from algorithm to hardware.\nBesides, it is also possible to see which component affects a quantum computing most seriously by applying realistic technology from the top layer to the bottom layer gradually.\nIn this regards, we can see that a theoretically optimal protocol really leads to optimal quantum computing or instead works suboptimally in composition with other components.\n\n\nBy exploiting the accurate analysis function, we can show numerical data for some theoretical conjectures in the fault-tolerant quantum computing.\nIn Steane code based quantum computing, increasing a concatenation level makes it possible to achieve more reliable quantum computing, but we guess that the more does not lead to the better all the time because a higher concatenation level requires more quantum resource and longer execution time.\nTherefore, it is reasonable to guess that there may exist a trade-off between the reliability and the performance (resource).\nIn this work, we show there exists such trade-off, and in surface code based quantum computing too.\n\n\nThe remainder of this paper is organized as follows.\nSection~\\ref{sec:related_works} reviews some related works.\nSection~\\ref{sec:configuration} overviews the proposed quantum computing framework and describes how to configure and analyze a quantum computing with the proposed framework.\nSection~\\ref{sec:proposed_framework} describes each layer of the proposed framework in detail.\nSection~\\ref{sec:performance_metric} describes the performance metric we evaluated in this work, and the analysis results will be shown in section~\\ref{sec:performance_evaluation}.\nSection~\\ref{sec:performance_improvement} discusses, by exploiting the proposed framework, what happens in a quantum computing if an individual component is improved.\nThis paper concludes with discussions in section~\\ref{sec:discussion}.\n\n\n\\section{Related Works}\\label{sec:related_works}\n\n\\begin{table*}[t]\n\\caption{\nSummary of the related works.\nNote that ``$\\bigtriangleup$\" indicates the corresponding component is partially applied.\nNamely, as mentioned before, Refs.~\\cite{Fowler:2012fi} and~\\cite{Jones:2012kc} perform the performance analysis based on only the dominant part of a quantum algorithm, not covering all quantum gates.\nBesides, surface code basically requires that physical qubits are arranged on the 2-dimensional nearest neighbor layout.\nTherefore, the related works applying surface code error correction implicitly consider a 2D layout of physical qubits in spite of explicitly no mentioning about the qubit layout.\n\\newline\n}\n\\small\n\\centering\n\\begin{tabular}{c|c|c|c|c|c} \\hline \n& \\multirow{2}{*}{Compile} & \\multirow{2}{*}{FTQC} & Micro-Architecture & System & Analysis \\\\ \n&  &  & (Qubit Layout) & Synthesis & Criterion\\\\ \\hline\nQuipper~\\cite{Smith:2014ux,Green:2013ha,Green:2013wb} & \\multirow{2}{*}{$\\bigcirc$} & \\multirow{2}{*}{X} & \\multirow{2}{*}{X} & \\multirow{2}{*}{X} & {One Time}\\\\ \nScaffCC~\\cite{JavadiAbhari:2015jf,JavadiAbhari:2014eu,ScaffCCScaffCC:vm} & & & & & Computing \\\\ \\hline\nFowler \\textit{et al.}~\\cite{Fowler:2012fi} & \\multirow{2}{*}{$\\bigtriangleup$} & \\multirow{2}{*}{Surface} & \\multirow{2}{*}{$\\bigtriangleup$} & \\multirow{2}{*}{X} & {One Time}\\\\ \nJones \\textit{et al.}~\\cite{Jones:2012kc} &  &  &  & & Computing\\\\ \\hline\n\\multirow{2}{*}{Reiher \\textit{et al.}~\\cite{Reiher:2017cv}} & \\multirow{2}{*}{$\\bigcirc$} & \\multirow{2}{*}{Surface} & \\multirow{2}{*}{$\\bigtriangleup$} & \\multirow{2}{*}{X} & {One Time}\\\\ \n &  &  &  & & Computing \\\\ \\hline\n\\multirow{3}{*}{QuRE~\\cite{Suchara:2013ho,Suchara:2013tg}} & \\multirow{3}{*}{$\\bigcirc$} & Steane, & \\multirow{3}{*}{Layout of Physical Qubits} &\\multirow{3}{*}{X} &\\multirow{3}{*}{\\minitab[c]{One Time\\\\Computing}}\\\\ \n &  & Bacon-Shor, &  & \\\\ \n &  & Surface &  & \\\\ \\hline \n\\multirow{2}{*}{Present Work} & \\multirow{2}{*}{$\\bigcirc$} & Steane, & Layouts of Physical and Logical Qubits, & \\multirow{2}{*}{$\\bigcirc$} & \\multirow{2}{*}{Fidelity 100\\%}\\\\ \n &  & Surface & Communication Bus, Computing Regions & \\\\ \\hline\n\\end{tabular}\n\\label{tab:summary_relatedworks}\n\\end{table*} \n\nWe review several quantum computing frameworks discussing the performance and the quantum resource of a quantum computing.\nTable~\\ref{tab:summary_relatedworks} briefly summarizes the related works.\n\n\nQuipper~\\cite{Smith:2014ux,Green:2013ha,Green:2013wb,Green:2013gi} and ScaffCC~\\cite{JavadiAbhari:2015jf,JavadiAbhari:2014eu,ScaffCCScaffCC:vm} are frameworks for quantum compile and resource estimation of a quantum algorithm.\nThey basically compile a programmed quantum algorithm into a sequence of quantum instructions of a quantum gate and target qubit(s).\nFrom the compile results, they statistically analyze the quantum resource such as the quantities of gates and qubits.\nAll the data come from a quantum algorithm only.\nWe believe, without considering physical implementation of the algorithm, the analysis results cannot be used for the reference to the feasibility of a quantum computing.\n\n\n\nFowler \\textit{et al.}~\\cite{Fowler:2012fi} approximated the quantum computer size and the execution time of Shor algorithm ($N=2000$) with a surface code quantum computing.\nHowever, their analysis is based on the dominant part of the algorithm such as the modular exponentiation circuit only.\nTherefore, the execution time completely depends on the quantity of \\textit{Toffoli} gates in the circuit, the decomposition of Toffoli gates and measurement gate execution time.\nThe size of a quantum computer was also calculated based on the quantity of magic states to implement logical \\textit{T} gates and the quantity of logical (algorithm) qubits.\nJones~\\textit{et al.}~\\cite{Jones:2012kc} also estimated the performance and resource of a quantum algorithm within their layered architecture for a quantum computing.\nBut, their analysis methods are almost the same with Fowler \\textit{et al.}~\\cite{Fowler:2012fi}.\n\n\nOn the other hand, Reiher \\textit{et al.}~\\cite{Reiher:2017cv} applied a quantum compile and surface code error correction to estimate the performance and the resource for quantum simulation of complex chemical system.\nSince the authors did not consider any quantum computer system architecture and the system integration on the architecture, their analysis corresponds to the statistical calculation based on the data from compiled algorithm and fault-tolerant scheme.\nNote that in the literature, the authors claimed that despite overhead of quantum error correction and compile, quantum computations can be performed within reasonable time.\n\n\nThe toolbox for estimating quantum resource \\textit{QuRE}~\\cite{Suchara:2013ho,Suchara:2013tg} considers most of quantum computing components such as quantum compile, quantum error correction, physical qubit layout and quantum computing device.\nBy taking quantum compile, they prepared all quantum gates of a quantum algorithm.\nFurthermore, they covered both of block-type quantum codes (Steane code and Bacon-Shor code) and surface code and employed 2-dimensional qubit layout for their logical qubits.\nIn this regards, this toolbox performs the analysis based on more quantum computing components than before.\nHowever, their analysis method is still statistical investigation.\nFor example, their estimation on the execution time of a quantum computing simply depends on the quantity of quantum gates and its running time, $\\sum 1\/g_{parallel} \\times \\#g \\times g_T$.\nNote that $g_{parallel}$ denotes the quantity of gates $g$ executed in maximally parallel and $g_T$ is the execution time of the gate.\nAs mentioned before, it is difficult to say their analysis results coincide with the practical situation.\n\n\n\n\\section{Configuration of Quantum Computing}\\label{sec:configuration}\nWe describe how to configure a quantum computing with the proposed quantum computing framework.\nWe first overview the structure and functionality of our quantum computing framework, and second describe how to configure a quantum computing.\nWe also describe which schemes and protocols are currently supported by the framework.\n\n\n\\subsection{Overview of Framework}\\label{subsec:overview_framework}\nWe describe the structure and functionality of the proposed quantum computing framework.\nIt deals with quantum computing programming, quantum compile, quantum computer architecture, fault-tolerant quantum computing scheme and quantum computing device.\nTo this end the proposed framework is composed of three functional layers: \\textit{compile}, \\textit{system} and \\textit{building block}.\nEach layer has a well-defined role and provides several options to conduct their functions.\nAll layers are closely related to each other.\nFIG.~\\ref{fig:data_flow_components} shows the data flow on the framework for performance analysis and quantum computing.\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=data_flow_performance.eps, scale=0.3}\n}\n\\subfigure[]{\n\t\\epsfig{file=data_flow_execution.eps, scale=0.3}\n}\n\\caption\n{\nData flow over the layers.\n(a) For the performance analysis, the compile layer and the building block layer provide a quantum assembly code and the performance of logical operations to the system layer.\nThe system layer then performs the performance analysis during the system mapping.\n(b) The platform can be utilized to perform a practical quantum computing. \nFor that, all data is flowing sequentially from the most top layer to the bottom layer.\nIn the bottom layer control signal for physical device has to be generated for the practical computing.\n}\n\\label{fig:data_flow_components}\n\\end{figure}\n\n\nThe compile layer covers programming and compiling a quantum algorithm. \nA quantum algorithm is written in a high-level programming language, and then compiled into a sequence of quantum gates, called a \\textit{quantum assembly code}.\nTo compile a programmed quantum algorithm, target quantum gates and compile algorithm have to be determined beforehand.\nFIG.~\\ref{fig:io_compile_layer} shows the input and output of the compile layer.\nIn the present work, we hire an open-source quantum computing compiler \\textit{ScaffCC}~\\cite{JavadiAbhari:2015jf,JavadiAbhari:2014eu,ScaffCCScaffCC:vm} that supports programming language \\textit{Scaffold}~\\cite{JavadiAbhari:2012wx}.\nThe details about quantum compile, quantum gates and a quantum assembly code are described in section~\\ref{subsec:compile}.\nWhy we exploit ScaffCC (Scaffold) in this work is also discussed there. \n\n\n\\begin{figure}[t]\n\\centering\n\\epsfig{file=compile_layer.eps, scale=0.3}\n\\caption{\nThe input and output in the compile layer.\nBy a compile, a programmed quantum algorithm is decomposed into a quantum assembly code.\nFor the compile, a compile algorithm and target gates have to be determined beforehand.\nTarget gates can be varied according to a quantum computing type such as a fault-tolerant quantum computing or a non-fault-tolerant (physical) computing.\nThe selection of target gates can also be influenced by a qubit technology.\n}\n\\label{fig:io_compile_layer}\n\\end{figure}\n\n\nThe system layer synthesizes a quantum computing by integrating quantum algorithm, quantum computer architecture and building block (qubit\/gate and quantum computing device).\nIt deals with everything required to run a quantum algorithm on a quantum computer system architecture.\nFor example, it recasts a quantum algorithm for a target quantum computer architecture via \\textit{system synthesis} (also called \\textit{system mapping}).\nFor the system synthesis, this layer first takes a quantum computing system architecture.\nA qubit connectivity must be definitely defined there, and a communication bus may be included for an efficient interaction over distant qubits.\nIssues related to a quantum computer architecture depend on a chosen quantum error-correcting code and a fault-tolerant quantum computing scheme.\nFIG.~\\ref{fig:io_system_layer} describes the input and output in the system layer.\nIn terms of the performance analysis, the output of the system layer is the performance analysis result, but to run a quantum algorithm this layer generates an architecture-specific description of a quantum algorithm as shown in FIG.~\\ref{fig:data_flow_components}.\n\n\n\n\\begin{figure}[t]\n\\centering\n\\epsfig{file=system_layer.eps, scale=0.3}\n\\caption{\nThe input and output in the system layer.\nQuantum assembly code and logical gates are passed from the compile layer and the building block layer respectively.\nQuantum computer architecture describes the layout of logical\/physical qubits and a communication bus over qubits.\nSystem mapping algorithm describes how to integrate a quantum algorithm (quantum assembly code) and a quantum computer architecture.\nIt includes the qubit placement, the gate scheduling and so on.\n}\n\\label{fig:io_system_layer}\n\\end{figure}\n\n\nThe building block layer is associated with qubits and gates of quantum algorithm.\nThis work basically assumes a fault-tolerant quantum computing, and therefore such qubits and gates are related to logical qubits and gates encoded in a quantum error-correcting code.\nIn this regards, the main functionality of this layer is to assemble physical qubits and gates to implement a logical qubit and gate.\nFor that, a quantum error-correcting code should be determined first, and then logical qubit and gate will be implemented according to a fault-tolerant quantum computing scheme of the chosen quantum code.\nDuring the implementation, it is able to generate the performance of logical qubit and gate based on the properties of physical operations, time and fidelity (or \\textit{error rate}).\nFIG.~\\ref{fig:io_building_block_layer} shows the input and output at the building block layer. \nIn terms of the performance analysis, the output of the building block layer is the performance of logical quantum operations, \\textit{time} and \\textit{fidelity}.\nNote that the proposed platform supports $[[7,1,3]]$ Steane code and a surface code. \n\n\n\\begin{figure}[t]\n\\centering\n\\epsfig{file=building_block_layer.eps, scale=0.3}\n\\caption{\nThe input and output in the building block layer.\nTo make logical qubits\/gates, a quantum error-correcting code should be determined first.\nBesides, to derive the performance of logical operations, the properties of physical device, time and fidelity, have to be considered.\nThen, based on the physical device property and the fault-tolerant quantum computing scheme, logical operations with a specific performance will be generated.\n}\n\\label{fig:io_building_block_layer}\n\\end{figure}\n\n\n\\subsection{Configuration of Quantum Computing}\\label{subsec:configuration}\n\nWe describe how to configure a quantum computing with the proposed framework.\nAs mentioned above, it is possible to configure a quantum computing by selectively choosing specific protocols or the properties of physical device. \nFor example, you can configure a Steane code based fault-tolerant quantum computing to run Shor algorithm.\nFor that, through compile you generate a quantum assembly code. \nLogical building blocks are built up based on the chosen quantum error-correcting code, quantum device and the size of the quantum algorithm.\nThe next task is to determine a quantum computer architecture of a certain physical and\/or logical qubit layout.\nThe architecture depends on the chosen fault-tolerant quantum computing scheme.\nNote that it is also possible to choose a physical non-fault-tolerant quantum computing by assuming high-fidelity quantum computing device (see Section~\\ref{sec:accurate_gates}).\n\n\n\\begin{table*}[t]\n\\caption{\nList of protocols and layouts currently supported by the framework.\nIn the compile layer, we choose a compile type and a target gate set.\nThe compile type is the format of a quantum assembly code.\nThe compile type is closely related to the mapping type and qubit layout.\nA specific qubit layout can be chosen within a selection.\nIn the building block layer, the scheme for a fault-tolerant quantum computing is determined.\nAccording to the scheme, the protocol of logical operations are fixed.\n\\newline\n}\n\\small\n\\centering\n\\begin{tabular}{c|c|c} \\hline \nLayer & Options & Values \\\\ \\hline\n\\multirow{3}{*}{Compile} & Compile Type & Structured code, Non-Structured code \\\\  \\cline{2-3}\n & \\multirow{2}{*}{Target Gate Set} & $\\{X, Z, H, S (S^{\\dag}), T (T^{\\dag}), R_Z(\\theta), CNOT\\}$, \\\\ \n & & $\\{X, Z, H, S (S^{\\dag}), T (T^{\\dag}),  CNOT\\}$ \\\\ \\hline\n\\multirow{3}{*}{System} & Mapping Type & Structured, Non-structured \\\\ \\cline{2-3}\n & \\multirow{2}{*}{Qubit Layout} & (Non-structured) All-to-All, 1D, 2D, Arbitrary \\\\ \n  &   & (Structured) All-to-All, (1D, 1D), (1D, 2D), (2D, 2D) \\\\ \\cline{2-3} \\hline\nBuilding & FTQC Scheme & Steane code, Surface code\\\\ \\cline{2-3}\nBlock  & Device & Time, Fidelity \\\\ \\hline\n\\end{tabular}\n\\label{tab:options}\n\\end{table*} \n\n\nIn Table~\\ref{tab:options}, we show some options our framework currently supports.\nIn the table, the compile type and the mapping type (with qubit layout) completely depend on the type of a quantum assembly code, a \\textit{structured} code and a \\textit{non-structured} code.\nA structured quantum assembly code is processed by a structured system mapping for a structured quantum computer architecture.\nSimilarly, a non-structured quantum assembly code is taken by a non-structured system mapping based on a simple qubit layout such as a regular 1- or 2-dimensional lattice.\nThe details of the structured and non-structured quantum assembly codes will be described in Section~\\ref{subsec:compile}.\nIn Sections~\\ref{sec:performance_evaluation} and~\\ref{sec:performance_improvement}, we configure quantum computings with such options and analyze the performance and the quantum resource.\n\n\nWith the proposed framework, a quantum computing can be configured as follows.\nThe first step is to determine the type of a quantum computing, a fault-tolerant quantum computing or a non-fault-tolerant physical quantum computing.\nIn general, such determination depends on the size of a quantum algorithm and the assumption on the reliability of physical device.\nThe larger quantum algorithm requires the more reliable qubits and gates.\nIf you decided a fault-tolerant quantum computing, you need to choose a quantum error-correcting code.\nOur platform supports $[[7,1,3]]$ Steane code and a surface code now.\nAccording to a quantum algorithm and the reliability (error rate) of physical device, the concatenation level for Steane code or the code distance for a surface code will be determined.\n\n\n\n\n\nThe choice of the quantum computing type affects a succeeding quantum compile step. \nExactly, target quantum gates for a compile completely depends on the chosen quantum computing type.\nFor example, $R_Z(\\theta)$ gate for a rotational angle $\\theta$ is very exploited in a quantum algorithm, but its logical version is not generally implemented in a fault-tolerant manner. \nTherefore, to realize a fault-tolerant quantum computing, you have to decompose such $R_Z(\\theta)$ gate into a sequence of $H$, $S$ and $T$ gates those are fault-tolerantly implementable.\nOn the other hand, since the physical $R_Z(\\theta)$ gate can be easily performed on physical qubits, there is no problem with quantum gates including the rotational gate for a non-fault-tolerant physical quantum computing.\n\n\nThe second step is to make a quantum computing program and compile it into a quantum assembly code.\nAs mentioned above, for the compile, you have to exploit target quantum gates determined in the previous step.\nThe proposed framework exploits open-source programming language \\textit{Scaffold} and compiler \\textit{ScaffCC}.\nYou can see how to make a Scaffold program in Ref.~\\cite{JavadiAbhari:2012wx} and how to use ScaffCC compiler in Ref.~\\cite{JavadiAbhari:2015jf,ScaffCCScaffCC:vm}.\nIn section~\\ref{subsec:compile}, we show a simple example of a Scaffold program and the associated quantum assembly code. \nNote that ScaffCC decomposes an arbitrary one-qubit gate into a sequence of $H$, $S$ and $T$ by exploiting \\textit{gridsynth}~\\cite{Ross:2016uz} or \\textit{sqct}~\\cite{Kliuchnikov:2013tr}.\n\n\nThe third step is to choose a quantum computing architecture, in particular a qubit array. \nThe proposed platform takes not only a simple regular 1- or 2-dimensional lattice, but also a hierarchically structured qubit array.\nA communication bus also should be considered to make an efficient interaction over distant qubits.\nFIG.~\\ref{fig:layout_example} shows an example of a hierarchically structured quantum computing architectures.\nIn case of Steane code quantum computing, the structure of a qubit array seriously affects a quantum computing due to the limited local qubit interaction.\nWe will show such limitation raises highly nontrivial temporal overhead in section~\\ref{sec:performance_local_qec}.\nOn the other hand, surface code quantum computing scheme is fundamentally established based on local qubit interaction on the 2-dimensional lattice.\nFIG.~\\ref{fig:surface_code_architecture} shows a quantum computing architecture for a surface code quantum computing.\n\n\nIf the configuration is completed, we can perform the system synthesis.\nDuring the system synthesis, the quantum algorithm is reformulated for the target quantum computer architecture. \nAs will be discussed later, from the system synthesis, the expected performance (circuit depth, execution time, fidelity, KQ, and so on) and the required quantum resource (qubits and gates) of a quantum computing are evaluated.\n\n\n\\subsection{Analysis of Quantum Computing}\\label{subsection:analysis}\n\nWe briefly mention what performance are evaluated by the framework, but the detailed analysis method will be described in section~\\ref{sec:performance_metric}. \nThe framework first inspects the quantities of qubits and gates.\nThose figures are usually analyzed by a quantum compile without considering a quantum computer architecture~\\cite{Smith:2014ux,JavadiAbhari:2015jf}.\nBut, our framework examines such quantities with the consideration for a quantum computer system architecture. \nIn the system synthesis, a fault-tolerant quantum computing scheme in particular a magic state factory is taken into consideration.\nTherefore, it is possible to estimate the temporal and spatial overhead caused by factors those are veiled in a quantum algorithm, and therefore we believe our estimation nearly coincides with the resource to perform a real quantum computing.\n\n\nThe framework examines the expected quantum computing execution time (circuit depth, fidelity, KQ and so on) of a quantum algorithm based on quantum computer architecture, fault-tolerant protocol and physical device.\nBy applying the properties of physical device and fault-tolerant protocol, we deduce the performance of logical gates and quantum error correction.\nBesides, by conducting a system integration, we are able to obtain the single-round execution time of a quantum algorithm by applying the performance of logical gates and error correction above.\nOur framework goes further for more detailed analysis.\nDuring the system integration, as mentioned above, the fidelity of a quantum computing can be calculated. \nBy reflecting the fidelity, it is possible to estimate the execution time for a quantum computing achieving a fidelity 100$\\%$.\nIn doing so, we fairly compare fault-tolerant quantum computing and non-fault-tolerant quantum computing.\nThe quantity of qubits is also based on this performance criterion.\n\n\nBesides the above-mentioned, the framework can generate various performance data.\nBased on the data, we can estimate the temporal and spatial overhead of a quantum computing.\nFor example, as mentioned above limited local interaction between nearest neighbor qubits sometimes requires additional qubit movements to perform 2-qubit CNOT gate.\nThe quantity of such qubit movements corresponds to the temporal overhead.\nThe platform evaluates such temporal overhead as a ration of SWAP gates to total quantum gates.\nAs will be shown in section~\\ref{sec:performance_local_qec}, a quantum computing requires highly nontrivial temporal overhead.\n\n\n\\section{Proposed Quantum Computing Framework}\\label{sec:proposed_framework}\n\\subsection{Compile Layer}\\label{subsec:compile}\n\nA quantum compile is a process that decomposes a quantum algorithm into a sequence of quantum gates.\nHere a quantum algorithm is in entirely programmed form by using a high-level abstract programming language.\nRecently, several research groups have developed programming environments for a quantum computing by modifying conventional classical programming languages such as python and C\/C++~\\cite{Steiger:2018to,JavadiAbhari:2012wx,Microsoft:wz,Ying:2017exa,Green:2013wb,Green:2013gi}.\n\n\nIt is well known that an arbitrary quantum algorithm can be decomposed into a combination of 1-qubit rotational gates and 2-qubit $CNOT$ gate~\\cite{Nielsen:2000ga}.\nThe target quantum gates for a compile can be varied according to situation.\nFor example, the set of $H$, $T$, and $CNOT$ is de facto standard for a universal fault-tolerant quantum computing.\nBut, to reduce the complexity in compile or to provide a flexibility to a programmer, one usually add more quantum gates to target gates.\nFurthermore, according to specific quantum mechanical system, physically implementable quantum gates slightly differ~\\cite{Nielsen:2000ga,Lin:2014il}. \nIn this work, we utilize two sets of quantum gates $\\{X, Z, H, S (S^{\\dag}), T (T^{\\dag}), R_Z(\\theta), CNOT\\}$ and $\\{X, Z, H, S (S^{\\dag}), T (T^{\\dag}), CNOT\\}$.\nThe difference between both is $R_Z(\\theta)$ gate.\nAs mentioned before, since the rotational gate for an arbitrary rotational angle $\\theta$ can not be implemented in the fault-tolerant manner, the first set is exploited for a physical quantum computing and the second is used for a fault-tolerant quantum computing.\n\n\nThe output of the quantum compile, the sequence of quantum gates, is called a \\textit{quantum assembly code}.\nA quantum assembly code is a list of quantum instructions that is a combination of a quantum gate and its target qubit(s).\nIt is an intermediate representation of a quantum algorithm between a mathematical description and a physical machine instruction description~\\cite{Cross:2017ud,JavadiAbhari:2015jf,Svore:2006iw}.\nThere is not any standard for a quantum assembly code, and therefore a specific representation and a structure of which slightly differ according to literature.\nFor example, a quantum instruction to apply a Hadamard gate to a qubit $q$ is represented as ``$hadamard$ $q$\" or ``$H$($q$)\".\nBesides, a certain quantum assembly code has its own special structure.\nFor example, Open QASM by IBM~\\cite{Cross:2017ud} provides a conditional statement ``\\textit{if... then... else}\" usually supported by conventional programming languages.\n\n\nIn the present work, we hire open-source quantum compiler \\textit{ScaffCC}~\\cite{JavadiAbhari:2015jf,ScaffCCScaffCC:vm}. \nIt compiles a quantum computing program written in quantum computing programing language \\textit{Scaffold}~\\cite{JavadiAbhari:2012wx}.\nA Scaffold program consists of one \\textit{main} module and multiple sub-modules (see FIG.~\\ref{fig:scaffold_code}).\nA module generally seems like a function or a procedure of conventional programming languages such as $C\/C++$, Python and so on.\nThe module is composed of instructions calling quantum gates and\/or other modules.\nThe execution of a Scaffold program begins with the first instruction of the main module, and terminates by conducting the last instruction of the same module.\nDuring the execution of the main module, other sub-modules may be called.\n\n\nPreviously, we mentioned that some quantum assembly codes have unique feature in their structure.\nSo does the output of ScaffCC.\nThe compiler generates a hierarchically structured quantum assembly code, in which a quantum algorithm is composed of multiple modules.\nA module consists of performing quantum gates and\/or other modules. \nIn the previous paragraph, we also mentioned that a Scaffold program consists of modules.\nTo avoid any ambiguity, we need to distinguish both.\nA module in a Scaffold program is completely defined and written by a programmer, and through a compile it is converted to a module in a quantum assembly code.\nTherefore, both are technically identical.\nFIG.~\\ref{fig:module_model} shows an example of a module in a quantum assembly code. \n\n\n\\begin{figure}[t]\n\\centering\n\\epsfig{file=module_model.eps, scale=0.45}\n\\caption{\nAn example of a module. \nParameter qubits passed from external modules are clearly specified at the beginning.\nA module is defined by the preparation of local qubits and classical bits, calling quantum gates and other sub-modules, and measurement of local qubits.\n}\n\\label{fig:module_model}\n\\end{figure}\n\n\n\n\nWe need to say why we exploit ScaffCC in this work.\nAs mentioned above, a quantum assembly code is a list of quantum instructions.\nAs the size of a quantum algorithm increases, the size of a quantum assembly code nontrivially increases.\nIt completely follows the complexity of a quantum algorithm.\nObviously, the size of meaningful quantum algorithm in reality is beyond the capability of conventional super-computing.\nIn other words, the size of a quantum assembly code for our interested algorithm will be very huge, and such scalability will cause a practical problem in classical control of a quantum computing.\nFor example, the size of a quantum assembly code of Shor algorithm to factorize 512 bit integer is around 39 TB (see FIG.~\\ref{fig:qasm_size_comparison}).\nTherefore, we have had trouble in generating and managing such a huge code with a classical computing system.\nWe could not even attempt to generate more larger-sized quantum algorithm than the algorithm above due to the lack of classical storage and memory.\n\n\nOn the other hand, it is believed that the hierarchy provided by ScaffCC can suppress the scalability.\nFor example, to perform a composite quantum operation of $N$ gates as much as $M$ times, a normal (non-structured) quantum assembly code requires $MN$ quantum instructions ($\\# gates \\times \\# iteration$).\nHowever, the hierarchical assembly code requires only $M+N$ instructions ($\\# gates + \\# iteration$) by defining the operation as a module.\nBy doing this, the hierarchical quantum assembly code is much smaller than the non-structured code as shown in FIG.~\\ref{fig:qasm_size_comparison}.\nTo the best of our knowledge, only ScaffCC supports such hierarchically structured quantum assembly code.\nThis is the main reason whey we exploit ScaffCC in the proposed platform.\n\n\n\n\n\n\\begin{figure}[t]\n\\centering\n\\epsfig{file=Shor_file_size_base.eps, scale=0.23}\n\\caption{\nThe size of the quantum assembly codes for Shor algorithm $n=128$, $256$, and $512$.\nBoth codes are generated by ScaffCC.\nAs the input increases, the size of quantum assembly code in the non-structured format increases over dozens TB.\n}\n\\label{fig:qasm_size_comparison}\n\\end{figure}\n\n\nTo conclude this section, we show a simple example of a Scaffold program to make a $5$-qubit CAT state $\\frac{1}{\\sqrt{2}}(|0\\rangle^{\\otimes 5} + |1\\rangle^{\\otimes 5})$ and a corresponding quantum assembly code.\nReaders can see how to make a Scaffold program in Ref.~\\cite{JavadiAbhari:2012wx} and how to use ScaffCC compiler in Ref.~\\cite{ScaffCCScaffCC:vm}.\nFIG.~\\ref{fig:scaffold_code} shows a quantum circuit to implement a $5$-qubit CAT state and a corresponding Scaffold program. \nA structured and a non-structured quantum assembly code are respectively shown in FIG.~\\ref{fig:cat_qasm}.\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=cat_state_circuit.eps, scale=0.8}\n}\n\\subfigure[]{\n\t\\epsfig{file=scaffold_code.eps, scale=0.4}\t\n}\n\\caption{\n\tAn example of a Scaffold program to implement a $5$-qubit CAT state.\n\t(a) A quantum circuit and (b) its Scaffold program.\n\tThe module \\textit{MakeCAT} makes the CAT state.\n}\n\\label{fig:scaffold_code}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=cat_state_qasmh.eps, scale=0.4}\n}\n\\subfigure[]{\n\t\\epsfig{file=cat_state_qasmf.eps, scale=0.45}\n}\n\\caption\n{\n\tQuantum assembly codes to generate a $5$-qubit CAT state.\n\t(a) A structured code and (b) a non-structured code.\n}\n\\label{fig:cat_qasm}\n\\end{figure}\n\n\n\\subsection{System Layer}\\label{subsec:system}\n\n\nA quantum algorithm (quantum assembly code) is a logic of how to solve a given problem.\nIt is based on ideal physical situation where noiseless physical gates and arbitrary long qubit interaction are allowed.\nIn other words, a quantum algorithm is developed without considering any physical implementation.\n\n\nOn the other hand, a quantum computer where a quantum algorithm is executed has a certain logical and physical architecture such as a qubit layout. \nTherefore, to run a quantum algorithm on such quantum computer, we need to reformulate the algorithm to be compatible with the given quantum computer architecture.\nFor example, real quantum computing devices in IBM Quantum Experience~\\cite{IBM:Iidq8v80} have very limited qubit layout and allow only one-directional CNOT.\nTherefore, the quantum assembly codes shown in FIG.~\\ref{fig:cat_qasm} can not be directly executed on the IBM QX4 device (see FIG.~\\ref{fig:mapping_simple_example} (a)) because the codes include unallowable CNOT gates. \nTherefore, for the execution, we have to recast a quantum assembly code for IBM QX4. \nIn FIG.~\\ref{fig:mapping_simple_example} (b), we show the recasted (non-structured) quantum assembly code for the device.\nThis is the motivation of a quantum computing system mapping.\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=ibmqx4_layout.eps, scale=0.4}\n}\n\\subfigure[]{\n\t\\epsfig{file=ibmqx4_mapping.eps, scale=0.35}\n}\n\\caption\n{\n(a) The qubit layout of IBM QX4 device~\\cite{IBM:Iidq8v80}. \nA node indicates a qubit, and an edge with a direction implies that the application of a controlled-CNOT gate is possible, where the control qubit and the target qubit are the root and end of the arrow. \nTherefore, as you see no bi-directional CNOT is allowed on the IBM QX4.\n(b) The recasted assembly code from FIG. 7 (b). \nSince the instruction ``\\textit{CNOT data0,data1}`` is not allowed directly on the IBM QX4, Hadamard gates, ``\\textit{H data1}\" and ``\\textit{H data2}\", have to add before and after of the instruction.\nNote that the node index $k$ indicates the qubit data$k$.\nWe have not cancel out repetitive Hadamard gates.\nBy cancelling out those gates, the circuit depth can be reduced from 12 to 9.\n}\n\\label{fig:mapping_simple_example}\n\\end{figure}\n\n\nThe principle of a quantum computing system mapping is very simple, 1) \\textit{set up} a quantum computer architecture and 2) \\textit{recast} a quantum assembly code for the architecture.\nIn what follows, we first describe a quantum computer architecture, and then show how to actualize a quantum algorithm on the target architecture.\n\n\n\\subsubsection{Quantum Computer Architecture}\nWe discuss a hierarchically structured quantum computer architecture for the proposed framework.\nIn general, there is no restriction to the architecture.\nIn other literature, a regular 1- or 2-dimensional lattice is usually exploited.\nBut, in this work we assume that it is hierarchically structured.\nA quantum computer is composed of several computing regions called a module and a communication bus connecting to the modules.\nBy assuming such structured architecture, the system mapping with a hierarchically structured quantum assembly code can be done efficiently.\n\n\n\nA computing region is completely associated with a module in a quantum assembly code.\nIt consists of multiple cells for logical (or physical) qubits described in the module. \nSome cells are allocated for parameter qubits passed from other modules, and others are allocated for local qubits which are temporarily used within a module.\nAdditional space may be sometimes required to form the rectangular shape of a module.\nFIG.~\\ref{fig:example_module_layout} shows an example of a module in a quantum assembly code and its associated computing region.\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=module_parsenodeeven.eps, scale=0.35}\n}\n\\subfigure[]{\n\t\\epsfig{file=module_parsenodeeven_qubit_layout.eps, scale=0.35}\n}\n\\caption\n{\n(a) An example of a module in a quantum assembly code and (b) the associated computing region on a quantum computer architecture.\nIn (a), the qubits \\textit{a} and \\textit{even} are parameter qubits passed from other modules, and the qubit \\textit{scratch} is a local qubit locally used within the module.\nIn (b), the dark grey cells are for the parameter qubits, the white cells are for the local qubits and the light grey cells are just empty space or null qubits not working anything.\nWhile the size of the parameter qubits \\textit{a} and \\textit{even} are not specified in the module definition, it can be determined by tracing all modules that calls the module.\n}\n\\label{fig:example_module_layout}\n\\end{figure}\n\n\nFIG.~\\ref{fig:layout_example} shows examples of the above-mentioned quantum computer architecture.\nThe box labelled $M_{i,j}$ ($M_i$) indicates a module (computing region).\nWe call the arrangement of modules a \\textit{global} layout and the arrangement of qubits within a module a \\textit{local} layout.\nAll modules communicate with each other via a communication bus.\nIn the figure, the bus is depicted as a white space outside of modules.\nWe will discuss the bandwidth of the communication bus in Section~\\ref{sec:performance_metric}.\n\n\nQubit that resides inside a module supports universal quantum operation.\nThe logical qubit is composed of data qubits for holding data and ancilla qubits for error correction and logical operations.\nOn the other hand, qubits for a communication bus only perform error correction and logical Clifford operations.\nTherefore, the composition of logical qubits for a communication bus can be differ from that for modules according to a quantum error-correcting code and a fault-tolerant quantum computing scheme.\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=2d_global_layout.eps, scale=0.25}\n}\n\\subfigure[]{\n\t\\epsfig{file=1d_global_layout.eps, scale=0.25}\n}\n\\caption{\nAn example of a proposed quantum computer architecture.\n(a) 2D global layout and 2D local layout.\n(b) 1D global layout and 2D local layout.\ntLogical qubits of different color play different role in a module; parameter qubits (dark grey), local qubits (white) and dummy qubits (light grey).\n}\n\\label{fig:layout_example}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\subsubsection{System Mapping}\nFIG.~\\ref{fig:data_flow_components} shows that all data are collected in the system layer and the performance of a quantum computing is evaluated there.\nIn this section, we will describe the system mapping in terms of the gate reformulation for the target quantum computer structure and the performance evaluation.\nA specific mapping process definitely depends on the type of quantum instructions.\nQuantum instructions in the hierarchically structured quantum assembly code are classified into three types: 1-, 2-qubit gate and module.\n\n\nThe set of 1-qubit gates includes $X$, $Z$, $H$, $S$ ($S^{\\dag}$), $T$ ($T^{\\dag}$), $R_Z(\\theta)$ and a preparation and a measurement in the Z basis.\nThe mapping process for such gates is straightforward and can be done independently. \nSuppose that a Hadamard gate is applied to a qubit $q$.\nIf the application of a certain quantum operation to the qubit was scheduled previously, the application of Hadamard gate will be performed after the previous operation.\nIf the previous operation was over at time $t(q)$, then Hadamard operation will start at $t(q)$ and finish at $t(q)+H_t$, where $H_t$ is the execution time of the gate.\nThis is everything for the mapping of an 1-qubit gate.\nNote that the execution time and fidelity of a quantum gate is provided from the building block layer. \n\n\nThe present work deals with a CNOT gate only for a 2-qubit gate.\nEven though there is a case when a SWAP gate is required, it is possible to implement a SWAP gate with three CNOT gates.\nWe deal with a CNOT gate as a local gate acting on two qubits located in nearest neighbor.\nSuppose that a CNOT gate is applied to qubits $q_a$ and $q_b$.\nThen, to execute the gate, both qubits have to be in temporary and spatially ready status.\nIf the are apart, we have to move both qubits to be in neighbor via SWAP operations.\nIf one qubit is being manipulated by other operation, we have to delay the CNOT operation until both qubits are in idle status.\nThen, the CNOT operation definitely begins at $\\max \\{t(q_a), t(q_b) \\}$, and finishes at time $\\max \\{t(q_a), t(q_b) \\} + CNOT_t$.\nNote that $\\max \\{t(q_a), t(q_b\\}$ is the time both qubits become idle status, and $CNOT_t$ is the execution time of a CNOT gate.\n\n\n\n\n\n\nThe third type quantum instruction, a module, seems like a multi-qubit composite quantum operation.\nTherefore, on the surface, it seems that the mapping of a module is very similar with the mapping of a 2-qubit CNOT gate.\nFor the mapping of a module, argument qubits for a module should be temporally and spatially ready.\nThe critical difference from the case of a CNOT gate is that a distinguished physical space\\footnote{The physical space for a module is the computing region we described before.} is allocated for a module.\nTherefore, to perform the mapping of a module, we have to consider qubit movements between a present module and a target module. \n\n\nSuppose that a module $A$ is being mapped now, and we have to treat a quantum instruction ``$M$ ($q_a$, $q_b$, $q_c$)\" that calls a module $M$ with argument qubits $q_a$, $q_b$ and $q_c$. \nThen, we have to move the argument qubits to the designated area of the module $M$. \nThe qubit movements are achieved by SWAP operations through a communication bus.\nWe call this movement a \\textit{forward} qubit passing.\nAfter the forward qubit passing, the qubits will be placed at the parameter qubit section of the module $M$.\nPlease see FIG.~\\ref{fig:example_module_layout} (b).\nQuantum instructions of the module $M$ will then be executed.\nIf it is faced with a quantum instruction calling other module, then some qubits in the module $M$ will be passed to the designated space of the newly called module and manipulated there by following the quantum instructions of the module.\nAfter executing all quantum instructions of the module $M$, the passed qubits have to be back to the original module $A$. \nWe call this returning movement a \\textit{backward} qubit passing.\nFIG.~\\ref{fig:parameter_passing} shows the module operation including the forward and backward qubit passings.\n\n\n\n\n\\begin{figure}[t]\n\\centering\n\\epsfig{file=parameter_passing.eps, scale=0.3}\n\\caption{\nAn example of the module operation that consists of seven steps: 1. (forward) move qubits to the bus, 2. (forward) move to the target module, 3. (forward) move to the parameter qubit cells (dark grey cells), 4. module operations, 5. (backward) move qubits to the bus, 6. (backward) move to the original module, and 7. (backward) move to the original qubit positions.\n}\n\\label{fig:parameter_passing}\n\\end{figure}\n\n\nWe perform the mapping for all modules sequentially as they appear in a quantum assembly code.\nFor that, we have to keep two lookup tables, a \\textit{global} lookup table and a \\textit{local} lookup table.\nFor each module, we first initialize a local lookup table for all qubits, and update the manipulation time of each qubit as we process each quantum instruction.\nAfter processing all instruction of a module, we determine the execution time of the module by picking up the maximum time among all qubits.\nThe performance of a module is recorded in the global lookup table.\nDuring the mapping of a module, if a module which was already mapped is called then we can refer the performance information of the called module from the global lookup table.\n\n\nAfter mapping all modules, we can determine the execution time of a quantum algorithm as the maximum time among the qubits in the \\textit{main} module. \nFor example, in FIG.~\\ref{fig:cat_qasm} (a), the execution time of the algorithm is $PrepZ_t$ + $FP_{main\\rightarrow MakeCAT}$ + $BP_{main\\leftarrow MakeCAT}$ + $MakeCAT_t$, where $MakeCAT_t=H_t + 4CNOT_t$.\nNote that $FP_{main\\rightarrow MakeCAT}$ ($BP_{main\\leftarrow MakeCAT}$) is the time for the forward (backward) qubit passing.\nThe execution time of the qubit passing depends on the distance between modules.\n\n\n\nSo far, we have described a system mapping algorithm for a hierarchically structured quantum assembly code.\nBy the way, the presented algorithm can be applied to a non-structured quantum assembly code.\nIn such code, there are two types of quantum instructions: 1- and 2-qubit gate.\nRegardless of the type of a quantum assembly code, as mentioned before the heart of the system mapping is 1) \\textit{set up} a quantum computer architecture and 2) \\textit{recast} quantum algorithm for the architecture.\nTo be compatible with the quantum assembly code, a simple qubit array such as a regular 2-dimensional lattice may be enough.\nThe proposed framework supports a system mapping on an arbitrary qubit layout as shown in FIG.~\\ref{fig:mapping_simple_example} (a).\n\n\n\n\n\n\\subsection{Building Block Layer}\\label{subsec:building_block}\n\n\nWe apply fault-tolerant quantum computing protocols based on $[[7,1,3]]$ Steane code~\\cite{Steane:2006wn} and surface code~\\cite{Fowler:2009ep,Fowler:2012fi}.\nBoth codes have well-studied logical gate protocols.\nThe concatenation level for Steane code and the code distance for a surface code are completely determined by a given quantum algorithm~\\cite{Jones:2012kc,Suchara:2013tg}.\nIn this work, we set both figures by using KQ formalism~\\cite{Steane:2003gp}.\n\n\n\\subsubsection{Steane code}\\label{sec:steane_code}\n\n\n$[[7, 1, 3]]$ Steane code encodes logical quantum information in a qubit into seven physical qubits, and protects it from an arbitrary 1-qubit quantum noise.\nSince the transversal implementations for a logical Hadamard and a logical CNOT gate are supported, many studies on the fault-tolerant quantum computing based on the Steane code have been conducted.\nIn Ref.~\\cite{Svore:2007tb}, an optimal design of a logical qubit for Steane code under the 2-dimensional nearest neighbor qubit interaction was proposed.\nThey achieved the threshold $O(10^{-5})$ with 48 physical qubits and modified quantum error correction.\n\n\nIn this work, we have redesigned a logical qubit with 30 physical qubits.\nSeven among them are used for holding data, and the others are temporarily used for logical operations and error correction.\nIn particular, we applied the Shor quantum error correction~\\cite{Shor:1996wi} that exploits $4$-qubit Shor state for the syndrome measurement.\nFor that, we prepare and verify the Shor state~\\cite{Weinstein:2012jr}. \nWe implemented the preparation of a logical state by following Ref.~\\cite{Goto:2016jo}.\nMost of logical gates are implemented as transversal gates, and the non-Clifford $T$ gate is implemented by exploiting a magic state.\nWe generate magic states by employing a $7$-qubit Shor state without magic state distillation~\\cite{Weinstein:2015ce}.\n\n\nAccuracy threshold theorem~\\cite{Knill:1996tm,Aharonov:2008jn} says that if we have a quantum device of physical error rate below a code threshold, it is possible to achieve an arbitrarily reliable quantum computing. \nBut, for a very large-sized quantum algorithm, encoding only once may not be enough.\nFortunately, by encoding a qubit recursively~\\cite{Knill:1996ty}, we can lower the effective error rate to the degree where a reliable quantum computing is possible.\n\n\nGiven a quantum algorithm, we can calculate $KQ$ and determine the maximum tolerable error rate $P_{max}$ as $1\/KQ$.\nWe then determine the concatenation level $l$ by the following inequalities satisfies\n\\begin{equation}\nP_{max} \\geq \\frac{{(c_{op}p^2)}^{2^{l}}}{c_{op}},\n\\end{equation}\nwhere $op$ is quantum error correction and logical operations, and $c_{op}$ is the constant factor of a specific logical operation $op$.\nWe obtained the constant values of each logical operation from $KQ$ of a quantum circuit for the operation.\nFor example, $c_{QEC}$ corresponds to $KQ$ of the QEC quantum circuit. \nIn this work, we have not optimized the arrange of qubits (see Table~\\ref{tab:logical_qubit_steane}), and therefore the quantum error correction and a logical operation work sub-optimally, and therefore the threshold is lower than the optimal value\\footnote{Please note that the objective of our work is not to increase a code threshold, but to configure a quantum computing and analyze its performance accurately.}.\n\n\nSuppose that a concatenation level for a quantum computing is determined as $l$.\nThe implementation of a logical $T$ gate in the level $l$ consists of only Clifford operations in a lower level $k<l$.\nThen, in the level $k$, the implementation of a logical $T$ gate is not necessary and therefore the qubits to implement a magic state, the 7-qubit Shor state, are not strongly required.\nTherefore only 23 qubits are required to implement a logical qubit in the lower level $k$.\nBut, to form a rectangular shape of a logical qubit, we require 25 qubits ($5\\times 5$ layout) for a lower level qubit in the level $k$.\nIn Table~\\ref{tab:logical_qubit_steane}, the qubits denoted by $7Sh [i]$ is not required in the lower level qubit $k$.\nOn the other hand, the qubits $V_{7Sh} [i]$, the main role of which is to verify the $7$-qubit Shor state, are used for the other purpose, logical measurement.\n\n\n\\begin{table}[t]\n\\caption{\nThe arrangement of qubits to implement a logical qubit in the concatenation level $l$.\nThe component qubits are in the concatenation level $l-1$.\nThe qubit denoted by $D[i]$ indicates a $i$-th data qubit.\nThe qubits $4Sh [i]$ and $7Sh [j]$ are for $4$- and $7$-qubit Shor states for syndrome measurement and a logical $T$ gate, and $V_{4Sh}$ and $V_{7Sh}$ are used to verify the $4$- and $7$-qubit Shor states respectively.\nThe qubit $M [i]$ is also used to implement a logical $T$ gate.\n\\newline\n}\n\\small\n\\centering\n\\begin{tabular}{|c|c|c|c|c|} \\hline\n$V_{4Sh} [1]$ & $V_{4Sh} [2]$ & $D [1]$ & $D [2]$ & $D [3]$ \\\\ \\hline\n$4Sh [1]$ & $4Sh [2]$ & $D [4]$ & $D [5]$ & $D [6]$ \\\\ \\hline\n$4Sh [4]$ & $4Sh [3]$ & $D [7]$ & $V_{7Sh} [1]$ & $V_{7Sh} [2]$ \\\\ \\hline\n$M [1]$ & $M [2]$ & $M [3]$ & $M [4]$ & $M [5]$ \\\\ \\hline\n$M [6]$ & $M [7]$ & $V_{7Sh} [3]$ & $7Sh [1]$ & $7Sh [2]$ \\\\ \\hline\n$7Sh [3]$ & $7Sh [4]$ & $7Sh [5]$ & $7Sh [6]$ & $7Sh [7]$ \\\\ \\hline\n\\end{tabular}\n\\label{tab:logical_qubit_steane}\n\\end{table} \n\n\n\\subsubsection{Surface code}\\label{sec:surface_code}\n\n2D surface code based fault-tolerant quantum computing is recognized as the most promising fault-tolerant quantum computing scheme due to physically less challenging requirements.\nThe code has a high threshold around $O(10^{-3})$~\\cite{Wang:2010ts,Raussendorf:2007dr,Fowler:2009ep}, and its structure is well suited to nearest neighbor interacted qubits arranged on the 2-dimensional lattice.\nIn this work, we have implemented double defect based logical qubits and logical gates described in Refs.~\\cite{Fowler:2009ep,Fowler:2012fi,Suchara:2013tg}.\nThe detailed protocols are beyond the scope of the present work, and we will describe performance parameters only. \n\n\nWe use the $KQ$ formalism to determine a code distance $d$~\\cite{Jones:2012kc,Suchara:2013tg}.\nThe objective error rate of a quantum computing is determined by $P_{fail} \\approx KQ\\epsilon_L$, where $\\epsilon_L$ is a logical error rate.\nThe code distance $d$ is determined as\n\\begin{equation}\\label{eq:surface_code_distance}\nd \\approx \\frac{2 \\bigl(\\log {\\epsilon_L} - \\log {C_1}\\bigr)}{\\log{C_{2}} + \\log{\\frac{\\epsilon_p}{\\epsilon_{th}}}} - 1,\n\\end{equation}\nwhere $\\epsilon_p$ and $\\epsilon_{th}$ are physical error rate and the threshold of the surface code respectively.\n$C_1$ and $C_2$ are code parameters, and we use the specific figures, $C_{1}\\approx 0.13$, $C_{2}\\approx 0.61$, from Ref.~\\cite{Jones:2012kc}.\nWe apply the code threshold $\\epsilon_{th}=0.009$. \n\n\nNow it is possible to determine the execution time of surface code logical gates.\nAbove all, we need to stress that the surface code error correction has to iterate $d$ rounds of a syndrome measurement.\nWe assume that logical Pauli operators are performed in classical control software by updating logical Pauli frame~\\cite{Fowler:2012fi}.\nA logical CNOT gate between the same type ($X$-cut or $Z$-cut) logical qubits consists of three CNOT gates between different type logical qubits.\nFor that, we have to prepare a pair of different type logical ancilla qubits~\\cite{Fowler:2012fi}.\nA logical Hadamard gate protocol consists of cutting and reconnecting a target logical qubit from\/to a whole qubit array and performing transversal physical Hadamard and SWAP gates~\\cite{Fowler:2012uz,Fowler:2012fi}.\nThe Hadamard gate makes the role of syndrome qubits interchanged, and the syndrome qubit reverts to the original position (role) via the SWAP operation.\n\n\nWe now turn the attention to the non-transversal gates $S$ and $T$.\nA logical $S$ gate is deterministically implemented by using a high fidelity magic state $|Y_L\\rangle = \\frac{1}{\\sqrt{2}} \\bigl(|0_L\\rangle + i |1_L\\rangle \\bigr)$~\\cite{Fowler:2012fi,Jones:2012kc}.\nSince the magic state is not destroyed during the gate protocol, it is possible to reuse it if plenty of high fidelity magic states are prepared at the beginning.\nTherefore before running a quantum algorithm, we prepare a number of $|Y_L\\rangle$ states.\nWe include the duration of preparing the states in the execution time of a quantum computing.\nHow many $|Y_L\\rangle$ states should be prepared will be discussed later.\n\n\nA logical $T$ gate is implemented by consuming a high fidelity magic state $|A_L\\rangle = \\frac{1}{\\sqrt{2}} \\bigl(|0_L\\rangle + \\exp^{i\\pi\/4} |1_L\\rangle \\bigr)$~\\cite{Fowler:2012fi}.\nA magic state has to be prepared for every $T$ gate.\nWe assumed that a magic state is prepared and supplied in offline. \nIn other words, the preparation of a high fidelity $|A_L\\rangle$ is not included in the quantum computing time.\nOn the other hand, the logical $T$ gate operation is probabilistically achieved up to the logical $S$ gate correction.\nTherefore to implement a logical $T$ gate, a $|Y_L\\rangle$ state is probabilistically required.\n\n\nWe now discuss the quantity of the required $|Y_L\\rangle$ states.\nIt depends on the quantity of the states maximally required at one time.\nSince a logical $T$ gate probably requires a $|Y_L\\rangle$, we have to prepare $|Y_L\\rangle$ as much as $\\max \\{ parallel T, parallel S\\}$ where $parallel T$ ($parallel S$) is the number of $T$ ($S$) gates executed in parallel.\nNote that the quantities of $parallel T$ and $parallel S$ can be found from the system mapping process.\n\n\nThe preparation of a high fidelity magic state takes two steps, \\textit{state injection} and \\textit{state distillation}~\\cite{Fowler:2012fi,Fowler:2013ht}.\nThe state injection in the surface code quantum computing injects an arbitrary logical state into the distance 1 logical qubit called a short qubit and makes the logical qubit larger~\\cite{Fowler:2012fi}.\nEnlarging a double defect logical qubit consists of multi-cell qubit movements and measurement on data qubits.\nThe state distillation protocol takes $m$ noisy states and generates $k$ less noisy states, where $m>k$.\nBy performing multiple rounds of the distillation, the magic spreaded over many states are concentrated on only a few states and therefore we can obtain high fidelity magic states.\nIn this work, we deal with the magic state distillation protocols described in Refs~\\cite{Fowler:2009ep,Fowler:2012fi}.\nThe required iteration of the protocol is completely determined by the objective fidelity and a physical error rate~\\cite{Suchara:2013tg}.\nWe set the objective error rate of the magic states as $10^{-12}$ to achieve high fidelity for the configured quantum computing\\footnote{$1\/ \\# \\textrm{T gates}$}, and empirically the 2-round distillation achieved the objective error rate in the physical error rate $10^{-3}\\sim 10^{-5}$.\n\n\nWe determine the capacity of a magic state factory that prepares and supplies $|A_L\\rangle$ states.\nThe capacity depends on a quantum algorithm and the durations of a state distillation and a logical $T$ gate.\nSuppose that a logical $T$ gate is applied consecutively to a qubit.\nThen, a magic state factory has to supply high fidelity magic states continuously.\nIf a magic state factory generates only one magic state at a time, there may happen a latency for the supply of the magic states if the magic state distillation takes more time than the duration of a logical $T$ gate.\nTherefore, the magic state factory has the capacity to prepare at least $\\max \\{parallel T\\} \\times time(MSD)\/ time(T)$ states at a time where $time(MSD)$ and $time(T)$ are the durations of the magic state distillation and the logical $T$ gate protocol.\nEmpirically, the $time(MSD)\/time(T)$ is approximately 20 in our estimation.\n\n\n \nTo conclude, the required physical qubits for $|A_L\\rangle$ and $|Y_L\\rangle$ are respectively\n\\begin{equation}\n\\max \\{parallel T\\} \\cdot \\frac{time(MSD)}{time(T)} \\cdot \\bigl(15 \\cdot Q_L)^{r-1} \\cdot (16\\cdot Q_L),\n\\end{equation}\nand \n\\begin{equation}\n\\max \\{parallel T, parallel S \\} \\cdot \\bigl(7\\cdot Q_L\\bigr)^{r-1}\\cdot (8\\cdot Q_L),\n\\end{equation}\nwhere $Q_L$ is the number of physical qubits to implement a logical qubit and $r$ is the required iterations.\nThe last distillation round requires one more logical qubit from the Bell state~\\cite{Fowler:2012fi}.\nAbove this, the ancilla qubits to perform CNOT gates during the distillation protocol also should be included.\n\n\n\n\\section{Performance Metric}\\label{sec:performance_metric}\nWe describe how to evaluate the quantum computing metrics, execution time, fidelity and the quantity of qubits.\n\n\\subsection{Execution Time}\\label{subsec:execution_time}\n\nWe examine the quantum computing time in two steps.\nIn the system mapping, we obtain the single round execution time $T_{one}$ of a quantum algorithm.\nAt the same time, the fidelity $F_{alg}$ of a quantum computing can be determined.\nNote that how to calculate the fidelity of a quantum computing is described in the following section. \nSince $T_{one}$ is the time for running a quantum algorithm once, and there is no guarantee about a reliable quantum computing.\nNoisy components may make a quantum computing broken.\nTo overcome the problem, we calculate the average execution time $T_{avg}$ by reflecting the number of the required iterations to achieve the fidelity 1 as\n\\begin{equation}\nT_{avg} = T_{one}\/F_{alg}.\n\\end{equation}\nWe believe this averaged time shows the time required for getting a reliable answer from a quantum computing\\footnote{This does not indicate that the output from a quantum computing is an exact solution. We do not consider the probabilistic nature of a quantum algorithm.}.\n\n\n\n\\subsection{Fidelity}\\label{subsec:fidelity}\nThe fidelity of a quantum computing can be calculated based on the fidelity of logical quantum gates as follows~\\cite{Suchara:2013tg},\n\\begin{equation}\nF_{alg} = \\prod_{g} {F_g}^{N_g},\n\\end{equation}\nwhere $g$ is a quantum gate utilized in the algorithm.\n$F_g$ is the fidelity of the gate $g$, and $N_g$ is the total count of the gate in the algorithm.\nThe value $N_g$ can be found from the system mapping and $F_g$ is determined in the building block layer. \nBy the way, this fidelity calculation is only applicable to Steane code based quantum computing.\nAs shown in Section~\\ref{sec:surface_code}, the final fidelity of a surface code based quantum computing is given by $F_{alg} = 1- KQ\\cdot \\epsilon_L$.\n\n\n\\subsection{The Number of Physical Qubit}\\label{subsec:qubits}\n\nWe examine the quantity of physical qubits required to run a quantum algorithm.\nSince the quantity of the required qubits differs according to a fault-tolerant quantum computing scheme, we first identify the common factor, the qubits in a quantum algorithm, and then go inside specific cases later.\n\nThe proposed hierarchical quantum computer structure consists of multiple modules and a communication bus connecting all modules.\nIn the quantum assembly code, we can find the quantity of logical (or physical) qubits for a module.\n\\begin{equation}\nQ_{comp} = \\sum_{M} \\bigl(Q^{M}_{local} + Q^{M}_{param} \\bigr), \n\\end{equation}\nwhere $Q^{M}_{local}$ ($Q^{M}_{param}$) is the number of local (parameter) qubits of a (computing) module $M$.\n\n\n\\subsubsection{Steane Code Quantum Computing}\nWe consider the Steane code quantum computing.\nThe structure of a communication bus depends on the chosen global layout over all modules.\nOn the 1D global layout, the number of the qubits can be simply calculated as $Q_{comm} = bandwidth \\times length$, where $length$ is obtained as \n\\begin{equation}\nlength = \\sum_{M} M_{width},\n\\end{equation}\nwhere $M_{width}$ is the width of a module, which is 1 for 1D local layout in common and $\\lfloor \\sqrt{Q^{M} } \\rfloor$ for 2D local layout.\nNote that $Q^{M} = Q^{M}_{local} + Q^{M}_{param}$.\n\n\nOn the other hand, on the 2D global layout, the number of qubits can be calculated as follows.\nLet us suppose that the number of modules is $|M|$.\nThen, $\\lfloor \\sqrt{|M|} \\rfloor \\times \\lfloor \\sqrt{|M|} \\rfloor$-sized 2D layout is necessary.\nTo keep the shape of a module on the 2D layout, all modules have the same size cells $n\\times n$, where $n = \\lfloor \\sqrt{\\max \\{Q^{M}}\\} \\rfloor$.\nThen, the required logical qubits for the communication bus is \n\\begin{equation}\nQ_{comm} = 2\\cdot bandwidth\\cdot n \\cdot A\\cdot B + \\bigl(n \\cdot A\\bigr)^2\n\\end{equation}\nwhere $A = \\lfloor \\sqrt{|M|} \\rfloor - 1$ and $B = \\lfloor \\sqrt{|M|} \\rfloor$.\nIn this work, we determine the bandwidth of a bus as the maximum number of parameter qubits, $bandwidth = \\max_{M} \\{Q^{M}_{param}\\}$.\n\n\nSo far, we have identified the number of logical qubits required for a quantum computing.\nAs we mentioned in Section~\\ref{sec:steane_code}, a logical qubit in the concatenation levels $k=1\\sim l-1$ is composed of 25 qubits and the qubit in the level $l$ consists of $30$ qubits.\nAccording to a physical error rate, we need to apply recursive encoding.\nTherefore, the number of total physical qubit in the Steane code quantum computing is \n\\begin{equation}\nQ_{steane} = 25^{r-1} \\cdot 30 \\cdot Q_{comp} + 25^r \\cdot Q_{comm} \n\\end{equation}\nwhere $r$ is the determined concatenation level.\n\n\n\\subsubsection{Surface Code Quantum Computing}\nWe implement double defect based logical qubits. \nFor a double defect logical qubit with code distance $d$, each defect has to be apart from a boundary as much as $d$ data qubits, and double defects also should be separated as much as $d$ data qubits.\nOn the other hand, to perform a braiding operation in a fault tolerant manner, the space between double defects has to be at least $2d+\\lfloor d\/4 \\rfloor$ rather than only $d$.\nTherefore, to implement a double defect logical qubit of code distance $d$, $(2A+1)(2B+1)$ physical qubits are required where $A=\\bigl(2d-2 + \\lfloor d\/4 \\rfloor \\bigr)$ and $B=\\bigl(4d-4 + 3\\lfloor d\/4 \\rfloor  \\bigr)$.\nFIG.~\\ref{fig:double_defect_distance_3} shows a double defect logical qubit of code distance 3.\nA total of 253 physical qubits, 126 data qubits and 125 syndrome qubits, are required.\n\n\nTwo neighboring logical qubits also have to be separated as much as $\\lfloor d\/4 \\rfloor$ data qubits to keep the code distance between both during the braid transformation.\nIn this regards, if $N$ logical qubits are arranged on the 2-dimensional layout of $n_h\\times n_w$, we need \n\\begin{equation}\n\\Bigl(2\\bigl(n_w A + (n_w-1)\\lfloor d\/4 \\rfloor\\Bigr)+1\\Bigr) \\Bigl(2\\bigl(n_h B + (n_h -1)\\lfloor d\/4\\rfloor \\bigr)+1\\Bigr)\n\\end{equation}\nqubits are necessary, where $A$ and $B$ are what we mentioned above.\n\n\n\n\nWe have to consider ancilla qubits required for a CNOT gate. \nAs mentioned before, the CNOT gate between the same type logical qubits consists of three CNOT gates between different logical qubits.\nFor that two ancilla qubits, $X$-cut qubit $|g_L\\rangle$ and $Z$-cut qubit $|+_L\\rangle$ are required.\nWe allocate a pair of both ancilla qubits to every module where a CNOT gate is performed.\nIn that case, the number of logical qubits for a module is the sum over parameter qubits, local qubits and two ancilla qubits.\n\n\\begin{figure}[t]\n\\centering\n\\epsfig{file=double_defect_distance_3.eps, scale=0.28, angle=90}\n\\caption{\n\tA double $Z$-cut qubit of a code distance 3.\n\tThe blue dots indicate data qubits.\n\tOne of the green chains indicates a logical $Z$ operator, and the red chain indicates a logical $X$ operator.\n\tThrough the yellow line, it is possible to perform a fault-tolerant braiding operation from other $X$-cut qubit to this $Z$-\tcut qubit. \n\tEach defect has to be away from boundary as much as 3 data qubits, and both defects have to be separated 6 data qubits.\n\t126 data qubits and 125 syndrome qubits are required to implement a distance-3 logical qubit.\n}\n\\label{fig:double_defect_distance_3}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\epsfig{file=surface_code_architecture.eps, scale=0.3}\n\\caption{\n\tA quantum computer structure based on the surface code quantum computing.\n\tDark green dots indicate defects and yellow cells are used as a path for the braiding transformations.\n\tBlue cells can be used for the forward\/backward qubit passings over distance modules.\n\tAn enclosed section by dotted red line is a computing region, a module.\n\tA defect has to be away from the boundary of a logical qubit or a braiding path as much as $d$ data qubits, and two logical qubits are mutually separated as much as $\\lfloor d\/4 \\rfloor$ data qubits.\n}\n\\label{fig:surface_code_architecture}\n\\end{figure}\n\nIn case of a surface code quantum computing, the communication bus may be also necessary to efficiently perform forward\/backward qubit passings\\footnote{It is possible to perform a fault-tolerant braiding between distant logical qubits in different modules. On considering that, the qubit passings may not be required. By the way, a braiding between distant logical qubits requires so many physical measurements. Which makes a quantum computing unreliable. In this regards, we believe that moving logical qubits to nearby location (a target module) and performing a braiding between close qubits is more reliable. In this regards, we also perform qubit passing in the surface code quantum computing.}.\nBy the way, unlike the Steane code quantum computing that performs a sequence of SWAP operations as much as the passing distance, the qubit movement in the surface code quantum computing is much efficient.\nIt can be achieved by only performing multi-cell qubit movements~\\cite{Suchara:2013tg,Fowler:2012fi}.\nTherefore, we assumed that the surface code quantum computing performs the qubit passing in sequentially on the bus of the narrow bandwidth.\nWe set the bandwidth of the bus as $\\lfloor d\/4 \\rfloor$, and additionally the movement path should be away from a boundary as much as $d$ data qubits.\nFIG.~\\ref{fig:surface_code_architecture} shows the quantum computer architecture based on a surface code and a structured quantum assembly code, where all the modules are arranged on the 1-dimensional layout by keeping the space as much as $\\lfloor d\/4 \\rfloor$ data qubits between both modules.\n\n\nWe conclude this section by repeating the physical qubits for the magic state factory. \nThe required physical qubits for $|Y_L\\rangle$ are $\\max \\{parallel T, parallel S \\}\\times (7Q_L)^{r-1}\\times (8Q_L)$ and for $|A_L\\rangle$ are $\\max \\{parallel T\\} \\times (15Q_L)^{r-1}\\times (16Q_L)\\times time(MSD)\/time(T)$.\nNote that $Q_L$ is the physical qubits to implement a logical double defect qubit, and $r$ is the iteration of the magic state distillation. \n\n\n\\section{Analysis of Performance and Resource}\\label{sec:performance_evaluation}\n\nWe show the performance analysis of quantum computings we configured.\nFor that, we set the objective fidelity of a quantum computing as 0.7.\nThat is, a single round quantum computing time $T_{one}$ is the time of a quantum computing that achieves a fidelity at least 0.7.\nWe set the error rate of physical operation for Steane code quantum computing as $10^{-9}$, but for surface code quantum computing we apply the physical error rate $10^{-3}$.\nWe also assume that the duration of a physical operation is 1 $\\mu s$ conservatively.\nThis assumption may be pessimistic than other literature assuming tens $\\sim$ hundreds nano seconds for a physical operation.\nThe Shor algorithm we test comes from the benchmark~\\cite{ScaffCCScaffCC:vm,Suchara:2013tg}.\n\n\n\n\n\n\\subsection{Case of applying Compile}\\label{sec:performance_compile}\n\nWe show the performance changes by applying a quantum compile, i.e., decomposition of a $R_Z(\\theta)$ gate for an arbitrary angle $\\theta$.\nEven though such decomposition is required to implement a fault-tolerant quantum computing, in this section we perform physical quantum computing without error correction to see the effect of the compile only.\nFor that, the components of the other layers are completely fixed.\n\n\nFor the decomposition, we set the precision of the decomposition as $10^{-2}$, which means that a decomposition of $R_Z(\\theta)$ gate achieves the $R_Z(\\theta)$ operation with an error probability $10^{-2}$.\nConsequentially, both $R_Z(\\theta_1)$ and $R_Z(\\theta_2)$ can be decomposed into the same sequence of $H$, $S$ and $T$ if $|\\theta_1-\\theta_2| \\leq 0.01$.\nUnder such precision, a $R_Z(\\theta)$ gate is usually decomposed into a sequence of $40\\sim 50$ $H$, $S$ and $T$ gates.\\footnote{If we set the precision degree with a smaller number, we will get a longer sequence of $H$, $S$ and $T$ gates for a $R_Z(\\theta)$ gate. On the one hand, such sequence can achieve the target $R_Z(\\theta)$ gate more exactly. On the other hand, the quantum computing time will be larger than the time shown in this work. Besides, practically the duration to conduct the performance analysis also increase nontrivially. In this regards, we have set $10^{-2}$ for the precision.}\nPlease note that the decomposition algorithm in the compile works probabilistically.\n\n\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=noiseless_nonlocal_gate_time.eps, scale=0.22}\n}\n\\subfigure[]{\n\t\\epsfig{file=noiseless_nonlocal_gate_qubits.eps, scale=0.22}\n}\n\\caption\n{\nWe show the quantum computing performance change by the compile effect. \n(a) Quantum computing time and (b) Qubits.\n}\n\\label{fig:noiseless_nonlocal_gate}\n\\end{figure*}\n\n\nFIG.~\\ref{fig:noiseless_nonlocal_gate} compares the performance. \nBy decomposing $R_Z(\\theta)$ gate, the quantum computing time increases as much as $4\\sim 5$ times, but the number of physical qubits stays equivalently.\nBy the way, in general $R_Z(\\theta)$ gate takes more than half of all quantum gates in Shor's factoring algorithm (see Table~\\ref{tab:RZ_ratio}).\nOn considering that $R_Z(\\theta)$ gate is decomposed into a sequence of dozens of $H$, $S$ and $T$ gates as we mentioned above, readers may guess that the performance difference between both cases should be more larger than the shown in the figure.\n\n\nAs mentioned above, we have set the precision of the decomposition as $10^{-2}$.\nMost $\\theta$ in the Shor algorithm are very small ($< 0.01$),\\footnote{$\\theta = \\pi\/2^{n-1}$ with $n=1\\sim N$ for $N$-bit integer factoring.} and therefore the decomposition of such rotation operation works as the identity operation. \nWe show the top dominant $\\theta$ used in Shor N=128 algorithm in Table~\\ref{tab:dominant_theta}.\nAll the angles are less than $0.01$.\nWhile we have not described all $\\theta$ in the algorithm in the table, empirically $75\\%$ of the angles applied in the algorithm are less than $0.01$.\nIn this regards, the performance degradation by decomposing $R_Z(\\theta)$ gates is not so remarkable regardless of the quantity of $R_Z(\\theta)$ gates in the algorithm.\n\n\n\\begin{table}[t]\n\\caption{\nThe proportion of $R_Z(\\theta)$ gate in Shor $N=128$.\n\\newline\n}\n\\small\n\\centering\n\\begin{tabular}{c|c|c|c} \\hline \nInput Size & $R_Z(\\theta)$ & Total Gates & Proportion\\\\ \\hline\n128 & $2.036 \\times 10^{9}$ & $3.399\\times 10^{9}$ & 59.90\\%\\\\ \\hline\n256 & $1.630\\times 10^{10}$ & $2.719\\times 10^{10}$ & 59.94\\%\\\\ \\hline\n512 & $1.304\\times 10^{11}$ & $2.175\\times 10^{11}$ & 59.95\\%\\\\ \\hline\n\\end{tabular}\n\\label{tab:RZ_ratio}\n\\end{table} \n\n\n\\begin{table}[t]\n\\caption{\nList of top 10 dominant angles in Shor's factoring algorithm, N=128.\nThe $\\theta$ listed in this table is less than $0.01$ and therefore $R_Z(\\theta)$ works as an identity operator.\nThe rotational angle $\\theta$ of the gate is from $\\pi\/2^{n-1}$ in Quantum Fourier Transform, and the exact representation of the angle is limited by a classical computer precision.\n\\newline\n}\n\\small\n\\centering\n\\begin{tabular}{c|c|c} \\hline \n$\\theta$ & Count & Proportion\\\\ \\hline\n$0.000000\\times 10^{0}$ & $6.88\\times 10^{8}$ & $0.3381$ \\\\ \\hline\n$-0.000000\\times 10^{0}$ & $3.44\\times 10^{8}$ & $0.1691$ \\\\ \\hline\n$-5.000000\\times 10^{-5}$ & $3.13\\times 10^{7}$ & $0.0154$ \\\\ \\hline\n$-1.000000\\times 10^{-4}$ & $3.11\\times 10^{7}$ & $0.0153$ \\\\ \\hline\n$5.000000\\times 10^{-5}$ & $3.10\\times 10^{7}$ & $0.0152$ \\\\ \\hline\n$-2.000000\\times 10^{-4}$ & $3.09\\times 10^{7}$ & $0.0152$ \\\\ \\hline\n$1.000000\\times 10^{-4}$ & $3.08\\times 10^{7}$ & $0.0151$ \\\\ \\hline\n$-4.000000\\times 10^{-4}$ & $3.08\\times 10^{7}$ & $0.0151$ \\\\ \\hline\n$2.000000\\times 10^{-4}$ & $3.07\\times 10^{7}$ & $0.0151$ \\\\ \\hline\n$-7.500000\\times 10^{-4}$ & $3.06\\times 10^{6}$ & $0.0150$ \\\\ \\hline\n\\end{tabular}\n\\label{tab:dominant_theta}\n\\end{table} \n\n\n\\subsection{Case of applying Compile and Error Correction}\\label{sec:performance_qec}\n\n\nWe show the performance of a fault-tolerant quantum computing but without considering local qubit interaction on a quantum computer architecture.\nWe assume that all qubits are directly interacted with an arbitrary qubit regardless of its position, and therefore a communication bus is not required in the architecture.\nThis setting is to see the effect of quantum error correction.\nFor that, we only configured Steane code based quantum computing because surface code computing inherently takes 2-dimensional nearest neighbored qubit layout into consideration.\nAs mentioned above, for the fault-tolerant quantum computing, we compile the quantum algorithm by decomposing $R_Z(\\theta)$ gate into $H$, $S$ and $T$ gates. \n\n\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=noisy_nonlocal_gate_time.eps, scale=0.22}\n}\n\\subfigure[]{\n\t\\epsfig{file=noisy_nonlocal_gate_qubits.eps, scale=0.22}\n}\n\\caption\n{\nQuantum computing performance based on quantum error correction.\nFor the fault-tolerant operation, we have compiled a quantum algorithm by decomposing $R_Z(\\theta)$ gates into a sequence of $H$, $S$ and $T$ gates.\nIn this evaluation, the quantum computer architecture and local qubit interaction are not completely considered.\n(a) Quantum computing time and (b) Qubits.\nThe concatenation level for the input size 128 is 1, and 2 for the other cases.\n}\n\\label{fig:noisy_nonlocal_gate}\n\\end{figure*}\n\n\nFIG.~\\ref{fig:noisy_nonlocal_gate} shows the quantum computing performance.\nThe increase of the execution time and the number of qubits is very remarkable when the input size increases from 128 to 256.\nThis is because the required concatenation level increases from 1 to 2 there to satisfy the objective fidelity 0.7.\nBut, as the concatenation level stays when the input increases from 256 to 512, the increases of a quantum computing time and the number of qubits are rather modest.\n\n\nSince in this section we assume a fault-tolerant quantum computing but with non-local qubit interaction, the performance change shown in the figure is only caused by the fault-tolerant quantum computing protocol.\nFor example, in FIG.~\\ref{fig:noisy_nonlocal_gate} (b), the numbers of qubits in the Steane code quantum computing are bigger than physical computing as much as respectively 30, 900 and 900 times.\nPlease recall that we have designed a logical qubit by assembling 30 physical qubits.\n\n\n\\subsection{Case of applying Compile, Error Correction and System Architecture}\\label{sec:performance_local_qec}\n\nIn this section, we show the quantum computing performance by considering all the realistic factors we have described previously.\nWe apply fault-tolerant quantum computings based on certain quantum computer architectures where only local qubit interaction is permitted.\n\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=noisy_local_gate_time.eps, scale=0.22}\n}\n\\subfigure[]{\n\t\\epsfig{file=noisy_local_gate_qubits.eps, scale=0.22}\n}\n\\caption\n{\nQuantum computing performance of the Steane code based local fault-tolerant quantum computing.\nToo see the influence by the architectural limitation, we also add the performance when the arbitrary long qubit interaction is allowed.\n(a) Quantum computing time and (b) Qubits.\nAll concatenation levels for the local qubit interaction cases (black, red and blue lines) are 3 in common. \nBut, in case of the non-local qubit interaction (green line), the concatenation level is 1 for the input size 128 and 2 for the others.\nPlease see FIG.~\\ref{fig:layout_example} about the quantum computer architectures.\n}\n\\label{fig:noisy_local_gate_steane}\n\\end{figure*}\n\nFIG.~\\ref{fig:noisy_local_gate_steane} shows the performance analysis of the Steane code quantum computing.\nWe have used the quantum computer architectures of the layouts; (1D global, 1D local), (1D global, 2D local) and (2D global, 2D local).\nPlease see FIG.~\\ref{fig:layout_example} for the quantum computer architectures.\nTo see the influence of local qubit interaction, we also compare the performance of the quantum computing based on non-local qubit interaction shown in the previous section.\n\n\nAs shown in the figure, the performance degradation by the local qubit interaction on a quantum computer architecture is highly nontrivial.\nThis is because many modules are spreaded over the quantum computer, and the communication (qubit passing) are performed frequently.\nTable~\\ref{tab:SWAP_ratio} shows the proportion of SWAP gates in the implementation of Shor algorithm.\nSurprisingly, on the proposed quantum computer architecture with a nearest neighbor qubit interaction, most of quantum operations in the Steane code quantum computing are qubit movements.\nWe think the quantity of the qubit movements is a temporal overhead to implement a quantum algorithm on a quantum computer.\nSuch large overhead caused by the qubit movements can be reduced by improving a quantum computer structure, a fault-tolerant quantum computing scheme or a system mapping algorithm.\n\n\n\\begin{table}[t]\n\\caption{\nThe proportion of \\textit{SWAP} gate in Shor's factoring algorithm.\nThe layout indicates a combination of Global Layout and Local Layout.\n\\newline\n}\n\\small\n\\centering\n\\begin{tabular}{c|c|c|c|c} \\hline \nInput Size & Layout & SWAP & Total Gates & Proportion\\\\ \\hline\n\\multirow{3}{*}{128} & (1D, 1D) & $ 7.371\\times 10^{11}$ & $ 7.405\\times 10^{11}$ & 99.54\\%\\\\ \\cline{2-5}\n & (1D, 2D) & $1.262\\times 10^{12}$ & $1.266\\times 10^{12}$ & 99.73\\%\\\\ \\cline{2-5}\n & (2D, 2D) & $1.527\\times 10^{13}$ & $1.527\\times 10^{13}$ & 99.97\\%\\\\ \\hline\n\\multirow{3}{*}{256} & (1D, 1D) & $1.116\\times 10^{13}$ & $1.118\\times 10^{13}$ & 99.76\\%\\\\ \\cline{2-5}\n & (1D, 2D) & $1.856\\times 10^{13}$ & $1.859\\times 10^{13}$ & 99.85\\%\\\\ \\cline{2-5}\n & (2D, 2D) & $2.719\\times 10^{14}$ & $2.720\\times 10^{14}$ & 99.99\\%\\\\ \\hline\n\\multirow{3}{*}{512} & (1D, 1D) & $1.068\\times 10^{14}$ & $1.070\\times 10^{14}$ & 99.80\\%\\\\ \\cline{2-5}\n & (1D, 2D) & $1.811\\times 10^{14}$ & $1.813\\times 10^{14}$ & 99.88\\%\\\\ \\cline{2-5}\n & (2D, 2D) & $5.789\\times 10^{15}$ & $5.789\\times 10^{15}$ & 99.99\\%\\\\ \\hline\n\\end{tabular}\n\\label{tab:SWAP_ratio}\n\\end{table} \n\nThe figure shows that a quantum computer architecture of the 1D global layout provides the better performance than a quantum computer of the 2D global layout.\nHowever, it may not always be the case.\nIt completely depends on the number of modules in a quantum computing program (see FIG.~\\ref{fig:scaffold_code}), and the arrangement of the modules on the architecture.\nIn general, the 2D global layout is a better architecture in terms of a qubit communication when the number of modules is very large.\nOn average, the arrangement of the modules on the 2D global layout can reduce the distance between modules than the 1D global layout.\nTherefore the communication cost of the qubit passing is less than the 1D global layout.\nAs an example, FIG.~\\ref{fig:noiseless_local_gate_gse} shows that a ground state estimation algorithm~\\cite{Suchara:2013tg,JavadiAbhari:2015jf,ScaffCCScaffCC:vm} works better on a quantum computer architecture with 2D global layout\\footnote{In the benchmark program, the algorithm is composed of at least tens of thousands modules, but the program of Shor algorithm consists of thousands modules.}.\n\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=noiseless_local_gate_gse_time.eps, scale=0.22}\n}\n\\subfigure[]{\n\t\\epsfig{file=noiseless_local_gate_gse_qubits.eps, scale=0.22}\n}\n\\caption\n{\nThe quantum computing performance of a ground state estimation algorithms over input size $M=20, 40, 60, 80$.\nQuantum gates are noiseless, and only nearest neighbor qubits are mutually interacted on the quantum computer architectures.\n(a) Quantum computing time and (b) Qubits.\n}\n\\label{fig:noiseless_local_gate_gse}\n\\end{figure*}\n\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=surface_code_time.eps, scale=0.22}\n}\n\\subfigure[]{\n\t\\epsfig{file=surface_code_qubits.eps, scale=0.22}\n}\n\\caption{\nThe quantum computing performance based on the surface code quantum computing.\n(a) Time and (b) Qubits.\nThe code distances are respectively 25, 27 and 30.\nIn (b), we also show the required physical qubits for a magic state factory.\n}\n\\label{fig:surface_code}\n\\end{figure*}\n\n\nFIG.~\\ref{fig:surface_code} shows the quantum computing performance of the surface code quantum computing.\nThe quantum computer architecture for a surface code quantum computing is shown in FIG.~\\ref{fig:surface_code_architecture}.\nIn the error rate $10^{-3}$, as the input size increases, the required code distance is raising 25, 27 and 30 to satisfy the objective fidelity.\nIn the figure, we also show the quantity of physical qubits to run a magic state factory that supplies $|A_L\\rangle$ states during the quantum computing.\nAs shown in the figure, in this work, the capacity of a magic state factory stays almost the same regardless of the input size of Shor algorithm.\nIt increases as much as the code distance.\n\n\nIn Ref.~\\cite{Fowler:2012fi}, the authors estimated the surface code quantum computing execution time of Shor algorithm for factoring a 2000-bit integer.\nThey did the estimation only by focusing on the quantity of the sequential Toffoli gates in the modular exponentiation circuit.\nBy following their estimation method, the quantum computing time to factorize a 512-bit integer will be only 0.45 hours ($40\\times 512^3 \\times 3\\times100~ns$) regardless of physical error rate.\nHowever, as shown in the figure, our estimation shows $8.78\\times 10^ 5$ hours are required for the same task when the physical error rate is $10^{-3}$.\nThe algorithm execution time, as shown in Section~\\ref{sec:accurate_gates}, is reduced as the physical error rate decreases.\n\n\nWe believe the reasons for such enormous gap between both estimations are as follows.\nFirst, our analysis is based on all quantum gates included in Shor algorithm but their analysis focuses on the Toffoli gates only.\nIn our estimation, the proportion of transversal gates takes about 60 \\% (\\# transversal gates\/\\# total gates), in other words, their estimation ignores the execution time by such huge transversal gates and the involved error correction.\nSecond, we have assumed that a physical gate operates in 1 $\\mu s$ conservatively while their estimation is based on 100 $ns$ measurement gates\\footnote{Their implementation of a Toffoli gate completes the operation within three measurement operations.}.\nThird, while they used an efficient decomposition of a Toffoli gate, we have used the de facto standard decomposition~\\cite{Nielsen:2000ga}.\nFourth, they might assume that braiding operations for parallel CNOT gates can be performed in parallel without any path conflicts, but we applied a braiding operation at a time to avoid conflicts between other braiding paths.\nPlease note that for Shor algorithm of $N=m$, $m$ CNOT gates can be performed in parallel in ideal case.\nFifth, we have considered architecture related issues such as qubit passing over distant modules, but they do not.\nLastly, we have applied $d$ round quantum error correction where the code distance $d$ is determined based on the physical error rate, but they did not.\n\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=steane_surface_10-9_time.eps, scale=0.22}\n}\n\\subfigure[]{\n\t\\epsfig{file=steane_surface_10-9_qubits.eps, scale=0.22}\n}\n\\caption\n{\nWe simply compare the quantum resource in the Steane code quantum computing and the surface code quantum computing.\nThe physical error rate is $10^{-9}$.\nThe quantum computer architecture for the Steane code quantum computing is the 1D global layout and the 2D local layout because as shown in FIG.~\\ref{fig:noisy_local_gate_steane} the architecture shows the best performance in this work.\nWe also compare the Steane code quantum computing with non-local qubit interaction because the performance of the Steane code quantum computing significantly depends on the quantum computer architecture.\n}\n\\label{fig:steane_surface_compare}\n\\end{figure*}\n\n\nOne of the reasons why a surface code has attracted much attention is it requires relatively less overhead.\nIn what follows, we simply compare the Steane code and surface code in light of quantum resource without considering their theoretical foundation.\nFor the fair comparison, we assumed the error rate of physical device is $10^{-9}$ for both cases. \nFIG.~\\ref{fig:steane_surface_compare} shows the quantum computing time and qubits to run Shor algorithm.\nAs we mentioned before, the performance of the Steane code quantum computing completely depends on a quantum computer architecture.\nTherefore to focus on the difference in quantum resource only by a fault-tolerant quantum computing, we also compare the situation where a non-local qubit interaction is allowed.\nAs shown in the figure, in the small input size, the Steane code quantum computing requires less time and qubits than the surface code quantum computing.\nThis is because of the non-locality of multi-qubit operation.\nBut, as the input size increases, the surface code quantum computing shows the better performance than the Steane code quantum computing even non-local qubit interaction is allowed.\n\n\n\n\n\n\\section{Usability of the proposed framework}\\label{sec:performance_improvement}\n\nThe objective of the proposed framework is to help to design and analyze a quantum computing.\nIn this regards, in this section, we show how to use it for analyzing previously proposed high performance quantum computing methods with realistic quantum computer. \nThe first is an efficient compile (Section~\\ref{sec:efficient_CRN}), and the second is an improved physical gate (Section~\\ref{sec:accurate_gates}) and the last is the strategy for the fault-tolerance (Section~\\ref{sec:degree_fault_tolerance}).\n\n\n\\subsection{Efficient Decomposition of Controlled-$R_n$}\\label{sec:efficient_CRN}\nAuthors proposed an efficient decomposition algorithm for a controlled-$R_n$ gate~\\cite{Kim:2018hs}.\nBy hiring an ancilla qubit, they achieved that the total number of quantum gates $\\{H, S, T\\}$ is reduced to from 35 (Ref.~\\cite{Kliuchnikov:2013tr}) to 21.\nWe show how the proposed decomposition affects the execution time of Shor algorithm. \nEven though the proposed compile algorithm itself requires more qubits, by reducing the algorithm execution time and increasing the fidelity of a quantum computing simultaneously, in total less qubits are required.\n\n\nFIG.~\\ref{fig:compile_improvement_steane} shows the performance improvement by the efficient compile in the Steane code quantum computing.\nAt the input size $N=128$, the improved decomposition lowers the quantum computing time as much as over 400-fold and the qubits as much as 30-fold.\nThe degree of the performance improvement depends on the input size.\nAs shown in the figure, at the input size where the required concatenation level lowers by applying the proposed decomposition, the performance improvement is remarkable.\n\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=base_option_time_10-9.eps, scale=0.22}\n}\n\\subfigure[]{\n\t\\epsfig{file=base_option_qubits_10-9.eps, scale=0.22}\n}\n\\caption\n{\nThe performance comparison between a naive compile and the proposed compile under Steane code based quantum computing.\n(a) The algorithm execution time. (b) The required qubits.\n}\n\\label{fig:compile_improvement_steane}\n\\end{figure*}\n\nFIG.~\\ref{fig:compile_improvement_surface} shows the performance improvement by the efficient compile in the surface code quantum computing.\nUnlike the Steane code quantum computing, the performance improvement in the quantum computing time increases gradually as the input size increases.\nThis is because there always exists the difference in the code distance.\nBy the improved decomposition, the required code distance lowers to 22, 24, 27 from 25, 27, 30 respectively.\n\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=base_option_surface_time.eps, scale=0.22}\n}\n\\subfigure[]{\n\t\\epsfig{file=base_option_surface_qubits.eps, scale=0.22}\n}\n\\caption\n{\nThe performance comparison between a naive compile and the proposed compile under surface code based quantum computing.\n(a) The algorithm execution time and (b) The required qubits.\n}\n\\label{fig:compile_improvement_surface}\n\\end{figure*}\n\n\n\\subsection{Accurate Quantum Gates}\\label{sec:accurate_gates}\nPreviously, we basically assumed that physical error rate is $10^{-9}$ for Steane code quantum computing and $10^{-3}$ for surface code quantum computing.\nIn this section, we show what happens in a quantum computing if we have more accurate quantum device.\nFor that we show the performance evaluations based on the physical error rates $10^{-9}\\sim 10^{-15}$ for Steane code quantum computing and based on the physical error rates over $10^{-3}\\sim 10^{-9}$.\n\n\nFIG.~\\ref{fig:robust_gate_steane} shows the Steane code quantum computing performance over physical error rates $10^{-9}\\sim 10^{-15}$.\nWe also compare a physical quantum computing to those fault-tolerant quantum computings at the physical error rate $10^{-15}$.\nThe performance improvement by lowering the error rate from $10^{-9}$ to $10^{-12}$ is highly nontrivial because the required concatenation level is reduced from 2 and 3 to 1 in both case respectively.\nBut, lowering the error rate more does not lead to the better fault-tolerant quantum computing performance.\nIn other words, the fault-tolerant quantum computing in the physical error rate $10^{-15}$ does not show any advantage against the quantum computing in the physical error rate $10^{-12}$.\nThis is because as the physical error rate lowers the fault-tolerant quantum computings with the same concatenation level achieves very high fidelity ($>0.9$).\nIf both quantum computings are performed with the same concatenation level, both have the same single round quantum computing time.\nIn that case, if there is no big difference between fidelities, the average quantum computing $T_{avg}$ will be very similar.\n\n\n\nIn the same reason, in the physical error rate $10^{-15}$, physical quantum computing shows the better performance than a fault-tolerant quantum computing because the physical quantum computing already achieves high fidelity.\nEmpirically, $T_{one}$ of the physical quantum computing in the error rate $10^{-15}$ is $6.89 \\times 10^{8}$  with the fidelity $0.6433$.\nOn the other hand, $T_{one}$ of the fault-tolerant quantum computing is $7.78 \\times 10^{10}$ with the fidelity $0.9999$.\nObviously, physical quantum computing shows the better performance in terms of $T_{avg}$, $\\frac{6.89\\times 10^{8}}{0.6433} > \\frac{7.78\\times 10^{10}}{0.9999}$.\n\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=robust_gates_steane_time.eps, scale=0.22}\n}\n\\subfigure[]{\n\t\\epsfig{file=robust_gates_steane_qubits.eps, scale=0.22}\n}\n\\caption\n{\nThe performance comparison over physical error rates $10^{-9}\\sim 10^{-15}$ under Steane code based quantum computing. \n(a) The algorithm execution time and (b) The required qubits.\nAt the error rate $10^{-15}$, as shown in this figure, a fault-tolerant quantum computing is not required.\n}\n\\label{fig:robust_gate_steane}\n\\end{figure*}\n\nFIG.~\\ref{fig:robust_gate_surface} shows the performance improvement in the surface code quantum computing over physical error rates $10^{-3}\\sim 10^{-9}$.\nIn the figure, we also compare the required code distance.\nAs shown in the figure, as the physical error rates lowers, the required code distance decreases and therefore the performance increases.\nBut, since the code distance is already too low, 4 or 5, there is not enough room for the performance improvement as the gate is improved more.\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\n\t\\epsfig{file=robust_gates_surface_distance.eps, scale=0.22}\n}\n\\subfigure[]{\n\t\\epsfig{file=robust_gates_surface_time.eps, scale=0.22}\n}\n\\subfigure[]{\n\t\\epsfig{file=robust_gates_surface_qubits.eps, scale=0.22}\n}\n\\caption\n{\nThe performance comparison over physical error rates $10^{-3}\\sim 10^{-9}$ under surface code based quantum computing.\n(a) The code distance, (b) The algorithm execution time and (c) The required qubits. \n}\n\\label{fig:robust_gate_surface}\n\\end{figure}\n\n\n\\subsection{Degree of Fault Tolerance}\\label{sec:degree_fault_tolerance}\nAccuracy threshold theorem~\\cite{Aharonov:2008jn,Knill:1996tm} says that if we have quantum device of physical error rates below a threshold, it is possible to achieve an arbitrary long quantum computation is possible.\nBy applying a recursive concatenated coding~\\cite{Knill:1996ty}, we can lower the effective error rate to where a reliable quantum computing is possible.\nAs we increase the concatenation level, the fidelity of a quantum computing also definitely improves.\nBut, the duration of a quantum computing is also increased by raising a concatenation level.\nTherefore, the higher concatenation level does not always make the better quantum computing possible.\nFIG.~\\ref{fig:degree_ft_steane_code} shows that there exists a recommendation for the concatenation level in the Steane code quantum computing, in particular for a quantum computing time.\nThe number of qubits unconditionally becomes larger as the concatenation level increases.\n\n\n\\begin{figure}[t]\n\\centering\n\\epsfig{file=degree_ft_steane_code.eps, scale=0.23}\n\\caption{\n\tThe quantum computing time of Shor algorithm with input size $N=512$.\n\tWe have evaluated the quantum computing performance $T_{one}$ and $T_{avg}$ according to the concatenation levels $1\\sim 5$.\n\tAfter the concatenation level 3, the fidelity of a quantum computing is almost 1 and therefore the average computing time closely approaches to the single round computing time.\n\tWhen, the concatenation level is 1, the fidelity of a quantum computing is almost vanishing and therefore the average computing goes to almost infinity.\n}\n\\label{fig:degree_ft_steane_code}\n\\end{figure}\n\n\nIn case of a surface code quantum computing, the performance completely depends on the code distance.\nThe code distance is determined to satisfy the objective fidelity of a quantum computing, but in most case the accuracy of the quantum computing by the chosen code distance exceeds than the target fidelity.\nIn this regards, on considering the averaged quantum computing time $T_{avg}$, the chosen code distance may not bring the best quantum computing performance as shown in the Steane code case.\n\n\nFIG.~\\ref{fig:degree_ft_surface_code} shows that the surface code quantum computing has the best performance with a code distance $31$, but the code distance determined by the equation is $30$.\nEven though the code distance determined from the target fidelity 0.7 is 30, the goal of a quantum computing is to find an exact answer, not a probable answer.\nBy applying the code distance 31, we can reduce the quantum computing time as much as 1400 days than the case of the code distance 30 at the cost of qubits.\n\n\n\\begin{figure}[t]\n\\centering\n\\epsfig{file=degree_ft_surface_code.eps, scale=0.23}\n\\caption{\n\tThe quantum computing time of Shor algorithm with input size $N=512$.\n\tWe have evaluated the quantum computing performance $T_{one}$ and $T_{avg}$ by varying the code distance from 29 to 35.\n\tThe calculated code distance for the objective fidelity is 30. \n\tAs shown in the figure, the code distance 31 introduces the best quantum computing performance.\n\tBy taking the code distance 31, we can reduce the quantum computing time as much as 1400 days than the case of the distance 30 at the cost of qubits.\n}\n\\label{fig:degree_ft_surface_code}\n\\end{figure}\n\n\n\n\\section{Discussion}\\label{sec:discussion}\n\nWe have proposed an integrated method for analyzing the performance and the resource of a quantum computing.\nIn particular, by considering practically running a quantum algorithm on a quantum computer hardware of a specific system architecture, we have obtained the most realistic performance and resource where the effects by all of fully decomposed algorithm, fault-tolerant scheme and system architecture are involved.\nFor that, we have proposed and developed a quantum computing framework composed of three functional layers where each layer plays a definite role of a quantum computing.\nBy exploiting the framework, we can configure a quantum computing model by selecting specific protocol and\/or properties.\nBy doing so, we can analyze not only the performance and resource of a quantum computing, but also the impact of specific components on the entire quantum computing.\nFor example, we have discussed optimal concatenation level and code distance of fault-tolerant quantum computing.\nWe believe such discussion was possible due to the proposed framework.\n\n\n\nThe analysis results completely depend on the protocols and properties of physical device we employed.\nAs shown in the figures, the quantity of the required qubits is too enormous and the execution time is too long.\nThe feasibility of a quantum computing seems too bad from our analysis results.\nHowever, we need to emphasize that the very those figures are not so important now.\nReaders need to see the tendency of the analysis results as the input size of a quantum algorithm increases.\nAs quantum computing components are being improved, the analysis results will be better than the shown in this work, but the tendency may be stayed.\n\n\nAs mentioned above, the objective of the present work is to provide the most realistic performance and resource of a quantum computing.\nOn the other hand, we believe the proposed software framework can play a significant role in practically running a quantum computing with a real quantum computing hardware later if some components are added (see FIG.~\\ref{fig:data_flow_components}).\nFor example, components to control a real quantum device have to be added in the building block layer.\nThe system layer also requires functions that execute a quantum algorithm and a quantum error correction efficiently.\nBesides, the existing components have to be improved as much as possible.\n\n\\begin{acknowledgements}\nThis work was supported by Electronics and Telecommunications Research Institute (ETRI) grant funded by the Korean government [18ZH1400, Research and Development of Quantum Computing Platform and its Cost-Effectiveness Improvement].\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n    Supermassive black holes (SMBHs) are thought to inhabit the center of every massive galaxy (e.g., \\citealp{1998AJ....115.2285M}). Throughout its lifetime, a SMBH can accrete significant amounts of matter, creating a very luminous and compact source of photons through the formation of an accretion disk \\citep{1973A&A....24..337S}. When a SMBH is undergoing this accreting phase, it is classified as an active galactic nucleus (AGN). AGN can then be subdivided based on their optical, radio, and X-ray properties. In order to explain the differences in the observational properties of accreting SMBHs, particularly in the optical and X-ray band, the unified model of AGN proposes a large scale dusty torus which sourrounds the broad line region and the accreting SMBH \\citep{1993ARA&A..31..473A,1995PASP..107..803U, 2015ARA&A..53..365N, 2017NatAs...1..679R}. The differences in AGN are then explained by the observers viewing angle with respect to the torus. When viewing an AGN through the torus, the central SMBH and accretion disk are obscured, and narrow emission lines are present. Observing an AGN perpendicular to the plane of the torus leads to an unobscured line of sight of the inner regions of the accreting system, which allows for the detection of both broad and narrow emission lines. \n\n    In AGN, X-rays are produced in a small region close to the SMBH (e.g., \\citealp{2009ApJ...693..174C, 2014A&ARv..22...72U}). The accretion disk cannot be responsible for the observed X-ray radiation because the temperature of the disk is not sufficient to create X-ray photons \\citep{1973A&A....24..337S}. Instead, it is now widely accepted that optical\/UV photons from the disk are up-scattered through the process of inverse Compton scattering when interacting with heated electrons in the X--ray corona \\citep{1991ApJ...380L..51H, 1993ApJ...413..507H}. The process in which these electrons become heated is still debated and could be caused by magnetic flares (e.g., \\citealp{1994ApJ...432L..95H,2000MNRAS.318L..15M}), clumpy disks (e.g., \\citealp{1988MNRAS.233..475G}), or aborted jets \\citep{1997A&A...326...87H}. The primary component of the X-ray continuum observed in AGN can be approximated by a power-law with an exponential cutoff at $\\sim 200$ keV (e.g., \\citealp{10.1093\/mnras\/sty1879}). The slope of this power-law is determined by the photon index $\\Gamma$, which typically has a value between 1.8 and 1.9 \\citep{Nandra7469,10.1111\/j.1365-2966.2011.18207.x, 2017ApJS..233...17R}. Absorption in the X-rays is driven by photoelectric absorption and Compton scattering, the latter becoming important at column densities of $~1\/\\sigma_{\\mathrm{T}}\\approx 10^{24}$ cm$^{-2}$, where $\\sigma_{\\text{T}}$ is the Thomson cross section. AGN that are observed through column densities higher than this threshold are dubbed Compton thick (CT; e.g., \\citealp{10.1046\/j.1365-8711.2000.03721.x,10.1111\/j.1365-2966.2010.17902.x,10.1111\/j.1365-2966.2012.20908.x,2015ApJ...815L..13R, 10.1093\/mnras\/stw1764, Hikitani_2018}). Reprocessed radiation is responsible for many characteristic features in the X-ray spectra of AGN, such as the reflection hump at $\\sim 30$\\,keV and the Iron K$\\alpha$ line at 6.4\\,keV. \n    \n    Until recently, most of the mid-infrared (MIR) emission associated to AGN has been thought to have its origin in the torus \\citep{2008ApJ...685..147N,2012MNRAS.420.2756S}. \\citet{Jaffe2004} reported MIR observations of NGC-1068 showing a warm dusty structure on scales of parsecs which surrounded a smaller hot structure. This was a significant confirmation of the presence of a torus-like structure in an AGN. Now, recent observations from the MID-infrared Interferometric instrument (MIDI) on the Very Large Telescope (VLT) show a large portion of the MIR spectrum of AGN stemming from the polar regions on scales of tens to hundreds of parsecs (e.g., \\citealp{H_nig_2012}) and not only from the torus as previously thought (\\citealp{10.1093\/mnras\/stx2227, 10.1093\/mnras\/stz220} and references therein). These observations have been made both by MIR interferometric (e.g., \\citealp{2012ApJ...755..149H, 2013A&A...558A.149B, refId03, Leftley_2019}) and single dish observations (e.g., \\citealp{Asmus_2016}). In addition to this, ALMA is also revealing polar outflows on the same spatial scales (e.g., \\citealp{2016ApJ...829L...7G}) which suggests the torus may be an obscuring outflow. This is in contrast with the traditional unified model, which does not predict this elongated polar MIR emission. This polar gas (or component) has a significant effect on the MIR SED, but it is likely optically thin, otherwise the X-ray emission would be obscured for type 1 AGN \\citep{Liu2019XraySO}. The origin of this polar component is still unknown. A plausible explanation is that this polar gas is the result of radiation pressure on dust grains from strong UV emission in the polar region from the accretion disk causing a dusty wind \\citep{Ricci2017, Leftley_2019, 2019ApJ...884..171H, 2020ApJ...900..174V}, a scenario also consistent with hydrodynamical simulations (e.g., \\citealp{2012ApJ...758...66W}).   \n\n    The two nearest sources in which this polar component has been confirmed by interferometric studies are NGC-1068 \\citep{refId01} and the Circinus galaxy \\citep{refId02}, although such elongated emission has also been observed in several other sources (e.g., \\citealp{refId03, 10.1117\/12.2231077}). There has been extensive literature studying the effect of this polar gas on the MIR spectrum, as well as the effect of different distributions of the polar gas itself (e.g., \\citealp{10.1093\/mnras\/stx2227, 10.1093\/mnras\/stz220, 10.1093\/mnras\/stz2289}). These studies concluded that the best model for the distribution of this polar gas to fit the MIR spectrum is a hollow cone or a hyperbolic cone on large scales, accompanied by a disk component on smaller scales. The hollow nature of these geometries is naturally consistent with the absence of obscuring material along the line of sight to Type 1 AGNs (e.g., \\citealp{Ricci2017, 2017ApJ...850...74K}). \\cite{Liu2019XraySO} studied the effect of polar gas on the X-ray spectrum of AGN by considering an X-ray source surrounded by an equatorial disk and a hollow cone perpendicular to the plane of the disk. \\cite{Liu2019XraySO} found that the polar gas has a significant effect on the X-ray spectrum of AGN, and argued that the scattered fluorescence line features can act as a potential probe for the kinematics of the polar gas.\n\n    So far, most studies of reprocessed X-ray radiation in AGN have been carried out assuming a torus \/ disk and no polar component. Thus, adding this polar component in the study of scattered X-ray emission can provide an important look into the kinematics of the polar gas and thus its origin. Since the polar gas is likely optically thin, it would only affect the soft (0.3--5\\,keV) part of the X-ray spectrum and not the hard portion ($>5$\\,keV). In this paper, we use the ray-tracing simulation software \\textsc{RefleX}\\footnote{\\url{https:\/\/www.astro.unige.ch\/reflex\/}} \\citep{RefleX} to simulate the spectra from an AGN with a torus accompanied by a large scale polar filled and hollow cone corresponding to the best fit model of the MIR data. We study the effect of the polar gas on the X-ray spectrum considering multiple inclination angles and column densities.\n    \n    \\begin{figure*}\n        \\centering\n       \n        \\includegraphics[scale = 0.14]{TorusD.png}\n        \\includegraphics[scale = 0.14]{ConeD.png}\n        \\includegraphics[scale = 0.14]{HConeD.png}\n        \\caption{Cross-sections for the three geometries considered. (Left panel) Geometry for torus with radius $r = 1.235$\\,pc a distance $R = 1.756$\\,pc from the X-ray source. (Middle panel) Geometry for the torus + filled cone. (Right panel) Geometry for the torus + hollow cone. The cones have an opening angle of 30$^{\\circ}$ with slant lengths of 40\\,pc. The hollow cone has a small angular width of $1^{\\circ}$ corresponding to the best fit MIR data for the galaxy Circinus.}\n        \\label{fig1}\n    \\end{figure*}\n\n    This paper is laid out as follows. In $\\S$2 we give an overview of the parameters and models of our simulations. Then we discuss the results of our simulations in $\\S3$. Next, in $\\S$4 we lay out the equivalent widths of the spectral lines to be compared to current and future observations from X-ray missions as well as consider observational differences in the resulting spectra of our simulations. We also add many components seen from the Narrow Line Region (NLR) such as a scattered power-law and many photoionised lines seen in obscured AGNs. $\\S5$ investigates the difference in the spectra when the opening angle and abundances of the hollow cone is changed. Finally, in $\\S$6 we summarize our findings and present our conclusions. \n\n\\section{Simulation Setup}\n\n    \\subsection{\\textsc{RefleX}: a ray-tracing simulation platform}\n    \n        In this paper we use the ray-tracing simulation platform \\textsc{RefleX} \\citep{RefleX} in order to study the emission from this new polar region. \\textsc{RefleX} is not the first  ray-tracing code to be used in studying reprocessed radiation from AGN. Indeed some of the first ray-tracing simulation platforms date to the early 1990s with Monte Carlo methods (e.g., \\citealp{1991MNRAS.249..352G}). Monte Carlo simulations became instrumental in simulating the X-ray spectra of heavily obscured AGN. They were used, for example, to calculate the equivalent width of the Iron K$\\alpha$ complex produced in neutral matter considering various column densities and Iron abundances \\citep{2002MNRAS.337..147M}. Newer models, such as \\textsc{MYTorus} \\citep{10.1111\/j.1365-2966.2009.15025.x}, consider a toroidal geometry with arbitrary column densities, while codes such as \\cite{2011MNRAS.413.1206B} and \\cite{2018ApJ...854...42B} also include varying covering factors. Clumpy torus models have also been developed recently (e.g., \\citealp{2014ApJ...787...52L, 2019A&A...629A..16B}). However, most of these models limit the users input in the allowed geometries. Alternatively, \\textsc{RefleX} allows the user to specify an X-ray source with different geometries and various spectral shapes in the 0.1\\,keV--1\\,MeV energy range. After the geometry and spectral shapes are determined, \\textsc{RefleX} allows the user to build quasi-arbitrary gas distributions around the X-ray source, with user specified densities. When the simulation is run, \\textsc{RefleX} tracks the life of every X-ray photon as it is propagated though the distribution of gas. The photons can undergo several physical processes when propagating through the gas, such as fluorescence, Compton scattering, and Rayleigh scattering. Once a photon escapes the gas and reaches the detector it is collected in energy bins in terms of flux (keV\\,cm$^{-2}$\\,s$^{-1}$\\,keV$^{-1}$), counts (integer number of photons), or photons (cm$^{-2}$\\,s$^{-1}$\\,keV$^{-1}$), depending on the specifications of the user.\n        \n        \\textsc{RefleX} also allows the user to specify the composition of the material surrounding the X-ray source. This includes setting the gas metallicity, the fraction of Hydrogen in molecular form (H$_{2}$ fraction), as well as the composition of the material which can be set, for example, to follow what was reported by \\cite{1989GeCoA..53..197A} or \\cite{2003ApJ...591.1220L}. All simulations in $\\S$3 are run using the gas composition from \\cite{1989GeCoA..53..197A}, solar metallicity, and a H$_{2}$ fraction of 1. In $\\S$5 we will consider the composition from \\cite{2003ApJ...591.1220L}, as well as an H$_{2}$ fraction of 0.2, showing that no significant difference is expected in the features arising from the polar gas.    \n    \n    \\subsection{Geometries}\n    \\subsubsection{Torus model}\n    \n        In order to study the effect polar gas has on the X-ray spectrum of AGN, we implement three simulation setups. The first is a torus with a Compton thick equatorial column density of $N_{\\rm H,\\, eq} =10^{24.5}$\\,cm$^{-2}$ surrounding an isotropic X-ray source emitting a power-law distribution of photons with an exponential cutoff at 200\\,keV and a photon index of $\\Gamma = 1.9$. This is equivalent to an AGN with no polar component and thus will provide a comparison to our later simulations with the polar component. The distance and size of the torus is characterized by two parameters in \\textsc{RefleX}: $R$ and $r$ as seen on the left panel of Figure\\,\\ref{fig1}. The values of $R$ and $r$ are kept consistent in all the simulations and have values of $R = 1.756$\\,pc and $r = 1.235$\\,pc, corresponding to the best fit model of the Circinus galaxy (Andoine et al. submitted), leading to a covering fraction of $CF = 0.7$. The gas in the torus and all other geometries is smooth. For the torus (as well as all other geometries), we selected the spectra from four inclination angle ranges: $0^{\\circ} \\leq i  \\leq 5^{\\circ}$ ($N_{\\text{H}} = 0$\\,cm$^{-2}$), $45.6^{\\circ} \\leq i \\leq 50.6^{\\circ}$ ($N_{\\text{H}} = 10^{22.49}$\\,cm$^{-2}$), and $85^{\\circ} \\leq i \\leq 90^{\\circ}$ ($N_{\\text{H}} \\approx 10^{24.50}$\\,cm$^{-2}$) where $i$ is the angle measured from the normal to the plane of the torus. The column densities are calculated from the center of the range. The range $45.6^{\\circ} \\leq i \\leq 50.6^{\\circ}$ will be referred to as the intermediate case and the ranges $0^{\\circ} \\leq i \\leq 5^{\\circ}$ and $85^{\\circ} \\leq i \\leq 90^{\\circ}$ will be referred to as the pole-on and edge-on cases respectively. We also ran simulations at an inclination range of $60^{\\circ} \\leq i \\leq 65^{\\circ}$, but with no significant change from the $45.6^{\\circ} \\leq i \\leq 50.6^{\\circ}$ case. We note in our simulations we use a linear resolution of 5\\,eV for $E \\leq 10$\\,keV and a logarithmic resolution of 0.001\\,eV for $E > 10$\\,keV   \n        \n        \\begin{figure} \n            \\centering\n            \\includegraphics[scale = 0.55]{Tor8560450ScattContBRUH.png}\n            \\caption{Simulated 0.3$-$200\\,keV spectrum obtained considering a torus with an equatorial column density $N_{\\text{H, eq}} = 10^{24.5}$\\,cm$^{-2}$ at different inclination angles. All the fluxes are normalized as discussed in section 3.1. The edge-on case (black line) has a high photoelectric cutoff due to the strong absorption from the torus. The intermediate case (blue line) displays many fluorescence lines in the soft portion of the spectrum due to the decreased column density. The pole-on inclination angle just shows strong continuum due to the complete lack of line of sight absorbing material along with the Fe K$\\alpha$ complex.}\n            \\label{fig2}\n        \\end{figure}\n        \n        \\begin{figure*}\n            \\ContinuedFloat*\n            \\includegraphics[scale = 0.384]{004.png}\n            \\includegraphics[scale = 0.384]{005.png}\n            \\includegraphics[scale = 0.384]{006.png}\n            \\caption{\\label{fig3a}Simulated spectra for a torus + hollow cone geometry (shown in blue) along with the torus only simulations (shown in black). Moving from the left panel to the right panel, inclination angle decreases as the slant column density of the hollow cone remains constant ($\\log{N_{\\text{H}}}\\big\/\\text{cm}^{-2} = 21$). The torus only simulations shown are linearly shifted to higher energies by 0.02\\,keV to better view the differences between spectral lines in the soft spectrum. The largest difference between the torus + hollow cone simulations and the previous torus simulations comes at the edge-on case where the previous torus simulations contain a photoelectric cutoff (left panel). The intermediate case (middle panel) shows little deviation from the torus only case. This is also the case for the pole-on case (right panel) due to the continuum domination.}\n        \\end{figure*}\n        \\begin{figure*}\n            \\ContinuedFloat\n            \\includegraphics[scale = 0.384]{003.png}\n            \\includegraphics[scale = 0.384]{002.png}\n            \\includegraphics[scale = 0.384]{001.png}\n            \\caption{\\label{fig3b}Same as top, however, moving from the left panel to the right panel represents simulations with constant inclination angle ($85^{\\circ} \\leq i \\leq 90^{\\circ}$) and increasing slant column densities for the polar component. The black line again represents the previous torus only simulation for the edge-on case. As the density of the polar component increases, the flux when compared with the torus only simulations increase as there is now more gas for photons to interact with.} \n        \\end{figure*}\n    \n    \\subsubsection{Torus + Hollow\/Filled Cone Model}\n    \n         The geometry for the torus along with the hollow cone can be seen on the right panel of Figure\\,\\ref{fig1}. The slant length of the cone was chosen to be 40\\,pc with an opening angle of $\\alpha = 30^{\\circ}$ and small angular width of 1$^{\\circ}$, consistent with the best MIR model for the Circinus galaxy \\citep{10.1093\/mnras\/stx2227, 10.1093\/mnras\/stz220}. The same three inclination angles are considered as in the torus-only model. However, to investigate the effect of the density of the polar gas on the X-ray spectrum, four slant column densities\\footnote{the column density as seen looking along the slant length of the cone} are considered for the hollow cone at each inclination angle. These are $\\log{(N_{\\rm H}\\big \/\\text{\\,cm}^{-2})} = 21, 21.5, 22,$ and $22.5$. The cone starts at a height of 0.1\\,pc above the X-ray source, corresponding to the sublimation radius of dust in AGN (e.g., \\citealp{2007A&A...476..713K}). The setup for the simulations including the torus along with the filled cone can be seen in the middle panel of Figure\\,\\ref{fig1}. The filled cone is equivalent to the hollow cone except for the fact that the outer wall of the hollow cone is removed and the inner portion of the cone is filled with gas. The same three inclination angles and column densities are considered for both the hollow and filled cone. It should be noted that it is unlikely that the polar gas has such a filled geometry in AGN. Otherwise, we would expect a large fraction of type 1 AGN spectra to show obscured X-ray spectral features, which is not the case \\citep{Ricci2017, 2017ApJ...850...74K}. The material in these cones is also thought to be dusty (i.e., \\citealp{10.1093\/mnras\/stx2227, 10.1093\/mnras\/stz220}) and thus must have a low level of ionisation. Therefore, we assume neutral material for the cones.  \n\n    \\begin{figure*}\n        \\centering\n        \\includegraphics[scale = 0.35]{images.png}\n        \\caption{Images of the torus + hollow cone simulations for the edge-on case at the highest polar gas column density ($85^{\\circ} \\leq i \\leq 90^{\\circ}$, $\\log{N_{\\text{H}}}\\big\/\\text{cm}^{-2} = 22.5$). The colour bars represent the number of total interactions (scattering + fluorescence) undergone in that particular pixel. (Left panel) Image showing the most interactions in the 0.3--4\\,keV range takes place is in the polar component. (Middle panel) 5--6\\,keV range which is mostly devoid of interactions due to the fact that this range is dominated by continuum. (Right panel) 6.3--6.5\\,keV range which is dominated by the torus due to the production of the Iron K$\\alpha$ line at $\\sim6.4$\\,keV.}\n        \\label{fig4}\n    \\end{figure*}\n    \n    \\begin{figure}\n        \\centering\n        \\includegraphics[scale = 0.55]{nophotons.png}\n        \\caption{Simulation showing the result of increasing the equatorial column density of the torus to $N_{\\text{H, eq}} = 10^{25}$\\,cm$^{-2}$ for the edge-on inclination range and the highest polar gas column density. The torus only component (black) shows very poor photon transmission due to the increased column density allowing most of the emission to stem from the polar component in the total spectrum (purple).}\n        \\label{fig4.5}\n    \\end{figure}\n    \n\\section{Results}\n\n    \\subsection{Torus}\n    \n        First we present the results for the torus only geometry as to provide a comparison for our later simulations with the polar component. The results for the torus simulations at each inclination angle are shown in Figure\\,\\ref{fig2}. The the edge-on case ($85^{\\circ} \\leq i \\leq 90^{\\circ}$) is characterized by a high photoelectric absorption due to the large column density in the torus and a more prominent Iron K$\\alpha$ line at $\\sim$6.4\\,keV which is also seen at all other inclination ranges. The Compton shoulder is also visible at $\\sim$ 6.2--6.4\\,keV. The simulation run at the intermediate inclination angle ($45.6^{\\circ} \\leq i \\leq 50.6^{\\circ}$) has a much lower column density ($N_{\\rm{H}} = 10^{22.49}$\\,cm$^{-2}$) which allows for the detection of more fluorescence lines in the soft portion of the spectrum, due to the decreased importance of photoelectric absorption. The fluorescence lines implemented in \\textsc{RefleX} can be found in Appendix A of \\cite{RefleX}. The hard portion of the spectrum in all simulations shows a reflection hump peaking $\\sim$ 20--30 keV. The pole-on case shows no fluorescence lines in the soft band as the line of sight is no longer obscured by the torus and the spectral lines are washed out by the dominating X-ray continuum.\n    \n    \\subsection{Torus + Hollow Cone}\n    \n        The results for all the torus + hollow cone simulations for the different inclination angles and polar gas column densities are shown in Figures\\,\\ref{fig3a} and \\ref{fig3b}. Moving from left to right, Figure\\,\\ref{fig3a} shows how the X-ray spectra are affected by the polar gas as different inclination angles are considered while keeping column density constant (in this case, the column density shown is $\\log{N_{\\mathrm{H}}}\\big\/\\text{cm}^{-2} = 21$). Figure\\,\\ref{fig3b} shows how the X-ray spectra are affected by the polar gas as different slant column densities are considered while keeping inclination angle constant. The inclination angle shown in Figure\\,\\ref{fig3b} is $85^{\\circ} \\leq i \\leq 90^{\\circ}$. The black spectra shown in each panel shows the torus only simulation at the same inclination angle. This gives the ability to clearly distinguish which parts of the X-ray spectrum are most affected by adding a polar component to the simulations. The simulations with a hollow cone clearly show the most deviations from the torus at the edge-on case in the 0.3--4\\,keV portion of the spectrum at the edge-on inclination range. The appearance of the polar gas allows many strong fluorescence line features to dominate the 0.3--4\\,keV range due to its optically thin nature. The strongest of these lines being O, Ne, Mg, and Si, with the O line dominating in equivalent width when the polar gas is at its highest column density and the largest inclination angle ($\\log{N_{\\mathrm{H}}}\\big\/\\text{cm}^{-2} = 22.5$, $85^{\\circ} \\leq i \\leq 90^{\\circ}$; see Table \\ref{table1}). The flux of these fluorescence lines increase with the density of the polar gas. However, for the intermediate and pole-on cases, little deviation from the torus model occurs. The deviations that do occur in the intermediate case only appears in the very soft (0.3--1.2\\,keV) portion of the spectrum.\n        \n        In addition to these simulations, we also considered how the new polar component would be effected if the torus considered had a much higher equatorial column density, such as $N_{\\text{H, eq}} = 10^{25}$\\,cm$^{-2}$. The results of this for the edge-on case can be seen in Figure\\,\\ref{fig4.5} with the torus component shown in black and the torus + hollow cone shown in purple. In this case the X-ray emission is much weaker, due to the higher column density. This allows us to attribute most of the scattered radiation below $\\sim$ 6\\,keV to the polar component.\n\n        \\subsubsection{Imaging the system}\n        \n            To better understand the morphology of the torus + hollow cone simulations, we took advantage of a feature of \\textsc{RefleX} which allows the user to create simulated images. The user inputs the location of a detector, its aperture size, field of view, number of desired pixels, and creates an image. The user can also select the energy range of the photons which are collected. Using this feature, we created three images of the simulations of a torus + hollow cone at the largest inclination angle and highest polar gas density [$85^{\\circ} \\leq i \\leq 90^{\\circ}$, $\\log{(N_{\\text{H}}\\big \/\\text{\\,cm$^{-2}$}}) = 22.5$]. We selected three energy ranges to create the images: 0.3--4\\,keV, 5--6\\,keV, and 6.3--6.5\\,keV. In addition to this, we isolated only photons which undergo at least one interaction (scattering or fluorescence) in order to remove the direct emission from the central source. These three images are shown in Figure\\,\\ref{fig4}. The left panel of Figure\\,\\ref{fig4} shows a large amount of interactions due to the fluorescent line features in the soft X-rays occurring in the polar region as expected due to the low density nature of the polar gas. The middle panel shows the range from 5--6\\,keV which contains less interactions than the other ranges due to the fact that this range is dominated by the scattered continuum and no strong fluorescent line is present. In this range there is emission from both regions, however, more interactions occur in the torus due to its higher density. Finally, the right panel shows the 6.3--6.5\\,keV range. This range contains fluorescent emission such as the Iron K$\\alpha$ line, and a dominating torus component due to the fact that the hollow cone is not dense enough to scatter those high energy photons. \n           \n            \n        \\begin{figure}\n            \\centering\n            \\includegraphics[scale = 0.55]{Cone.png}\n            \\caption{This figure shows the pole-on spectra for the torus + filled cone for the four slant column densities considered. All the column densities show absorption in the soft portion of the spectrum which shows that this geometry for the polar component is unlikely, as Type 1 AGN spectra would be obscured if this was the true geometry of the polar component.}\n            \\label{figCone}\n        \\end{figure}\n        \n    \\subsection{Torus + Filled Cone}\n    \n        When considering the geometry of a torus + filled cone, the spectra show many similarities to the simulations with the torus + hollow cone. These similarities are a large increase in flux and fluorescence line features in the soft portion of the spectrum while the hard portion of the spectrum remains the same as the simulations for a torus-only geometry, with a large reflection hump, Fe K$\\alpha$ line, and continuum domination. In addition to this, the intermediate case shows little deviation in flux from the torus-only case. However, the filled cone deviates from the hollow cone in a few important ways. First, the flux increase in the soft portion of the spectrum is not as large as in the hollow cone case. This is due to the fact that in this geometry it is more likely for the photons to be absorbed as there is more gas to interact with. Also, Figure\\,\\ref{figCone} shows at the pole-on case, the filled cone does not show only continuum, as the X-ray source is obscured by the polar gas, unlike the pole-on case for the hollow cone. This allows for some line features to appear in the soft spectrum for the pole-on case. It was noted earlier that this geometry is unlikely for the polar component as many Type 1 AGN spectra would be obscured. Simulations were also performed considering the same total number of atoms in the filled cone as in the hollow cone and showed that the spectra still showed a clear photoelectric cutoff at low energies, giving further evidence that this geometry is unlikely for the polar component.  \n        \n        \\begin{figure}\n            \\centering\n            \\includegraphics[scale = 0.55]{EWvNH85HIGH.png}\n            \\includegraphics[scale = 0.55]{EWvNH85LOW.png}\n            \\caption{Equivalent widths of spectral lines for a torus + hollow cone as a function of polar gas column density for the edge-on simulations ($85^{\\circ} \\leq i \\leq 90^{\\circ}$). (Top panel) EWs for the Si, S, Ar, and Ca lines as a function of polar gas column density. (Bottom panel) Same as top but for the O, F, Ne, and Mg lines. These plots show a clear energy dependence with the EWs of lines of lower energy (O, F, Ne, Mg, Si) increasing while lines with higher energy (S, Ar, Mg) decrease in EW due to the increasing continuum at higher energies. The equivalent widths for all simulations are reported in Tables \\ref{table1}-\\ref{table6} in Appendix A.}\n            \\label{fig6}\n        \\end{figure} \n        \n    \\section{Detecting Polar Gas in Obscured AGN}\n        \n        \\subsection{Equivalent Width of Spectral Lines}\n    \n            We discuss here the equivalent widths of the strongest spectral lines in the 0.3$-$4\\,keV range for the torus + hollow cone simulations (see Figures\\,\\ref{fig3a} and \\ref{fig3b}). The simulations for the pole-on case ($0^{\\circ} \\leq i \\leq 5^{\\circ}$) were ignored since the continuum is not suppressed and thus contrast between the lines and the continuum is not enough for detection. Therefore, these equivalent width measurements are achievable only in an absorbed AGN. The equivalent widths for these simulations are reported in Tables \\ref{table1}-\\ref{table6} in Appendix A. We also note that, when considering a galactic absorption of 10$^{21}$\\,cm$^{-2}$, no significant difference in the lines occurred. However, significant deviations in ISM abundances may cause challenges in attributing these EWs to any particular geometry.     \n        \n            A plot of the equivalent widths (EWs) as a function of polar gas density of the chosen spectral lines for a torus + hollow cone for the edge-on case is shown in Figure\\,\\ref{fig6}. The O, F, Ne, Mg, and Si lines all show an increase in equivalent widths as the column density of the polar gas increases, with the O line increasing by the largest amount ($\\Delta\\text{EW}=$ 437\\,eV) followed by the Ne ($\\Delta\\text{EW}=$ 158\\,eV) and Florine ($\\Delta\\text{EW}=$ 82\\,eV) lines. This increase is due to the fact that, as the density of the polar gas increases, there is more material for which to scatter the soft X-rays and create larger fluorescence lines. An interesting feature seen in this plot is that the equivalent widths of the S, Ar and Ca lines decrease with the polar gas column density. To investigate this further, we noted that the flux of the continuum around these lines increases at a larger rate than the line fluxes themselves as we increase the column density. This explains the behaviour of those two lines, since the equivalent width is proportional to the ratio between the flux of the line and that of the continuum.   \n\n        \\subsection{Observational Differences in Simulations}\n        \n            \\begin{figure}\n                \\centering\n                \\includegraphics[scale = 0.55]{Torus60HCone85.png}\n                \\caption{Flux in the soft X-ray band for the edge-on case of a torus + hollow cone (purple) with the torus-only (black) intermediate case ($45.6^{\\circ} \\leq i \\leq 50.6^{\\circ}$). This shows the comparison of the slopes of the power-laws in the soft X-ray bands for these two geometries and inclinations with the intermediate torus only case having a power-law slope of $\\Gamma = -1.63$ and the edge-on torus + hollow cone having a slope of $\\Gamma = -0.8$.}\n                \\label{fig8.5}\n            \\end{figure}\n        \n            \\begin{figure}\n                \\centering\n                \\includegraphics[scale = 0.365]{MyModel4.pdf}\n                \\caption{Plot showing the additional components which will most likely be observed stemming from the NLR added to the edge-on spectrum for a torus + hollow cone at the highest polar gas column density considered (original simulation shown as the dash-dotted black line). This includes a power-law model with 0.1\\% of the flux of the primary continuum (solid black line), as well as many photoionized lines seen in the NLR such as the ones from the soft X-ray spectrum of NGC 3393 (see Table 4 in \\citealp{2006A&A...448..499B}), as well as the Circinus galaxy (see Tables 1, A1 of \\citealp{2001ApJ...546L..13S, 2014ApJ...791...81A}, Andonie et al. submitted, respectively). These photoionized lines are plotted as dashed lines in the figure and are labeled with their transitions.}\n               \n                \\label{fig10.5}\n            \\end{figure}\n            \n            \\begin{figure}\n                \\centering\n                \\includegraphics[scale = 0.42]{res5.pdf}\n                \\caption{(Top panel) Spectrum with a torus + hollow cone for the edge-on case ($85^{\\circ} \\leq i \\leq 90^{\\circ}$) with the highest polar gas column density ($\\log{N_{\\mathrm{H}}}\\big\/\\text{cm}^{-2}$ = 22.5) with \\textit{XRISM} \\citep{xrismscienceteam2020science} response and background files. (Bottom panel) Same as top but with \\textit{Athena} \\citep{2016SPIE.9905E..2FB} response and background files (Barret et al. 2021 in prep.). For both simulations, we used an exposure time of 200\\,ks and considered a $2-10$\\,keV flux of $F_{2-10} = 12.7\\times10^{-12}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$, as in the Circinus galaxy.}\n                \\label{figres}\n            \\end{figure}\n            \n            \\begin{figure*}\n                \\centering\n                \\includegraphics[scale = 0.55]{10v30.png}\n                \\includegraphics[scale = 0.55]{LvAG85.png}\n                \\caption{(Left panel) Resulting spectra when considering an opening angle of $\\alpha = 10^{\\circ}$ for an edge-on torus + hollow cone at the highest polar gas column density (shown in black). The soft spectra (0.3$-$4\\,keV) for this new opening angle is best modeled with a power-law with a photon index of $\\Gamma = -0.71$ while the $\\alpha = 30^{\\circ}$ case (shown in red) is best modeled with a photon index of $\\Gamma = -0.80$. The EWs for the $\\alpha = 10^{\\circ}$ case are shown in Table \\ref{table7}. (Right panel) Comparison of simulations with a torus + hollow cone at the edge-on inclination with polar gas column density of $N_{\\text{H}} = 10^{22.5}$\\,cm$^{-2}$ using matter compositions from \\citeauthor{1989GeCoA..53..197A} (\\citeyear{1989GeCoA..53..197A}; red) and \\citeauthor{2003ApJ...591.1220L} (\\citeyear{2003ApJ...591.1220L}; black) with H$_{2}$ fractions of 1 and 0.2 respectively. The simulations obtained by considering an opening angle of $\\alpha = 30^{\\circ}$, as well as a matter composition from \\protect\\cite{2003ApJ...591.1220L} are shifted to the right by 4$\\%$ of the energy in order to clearly see the difference in the spectral lines. There is little spectral difference either in the lines or continuum between the two compositions.}\n                \\label{fig10}\n            \\end{figure*}\n        \n            Looking at the simulations in the soft X-ray band for the torus with a hollow cone gives rise to a question: would an observer be able to distinguish between the edge-on case with a torus + hollow cone at the highest polar gas column density ($N_{\\text{H}} = 10^{22.5}$\\,cm$^{-2}$) against the intermediate case ($45.6^{\\circ} \\leq i \\leq 50.6^{\\circ}$) with a torus alone (blue line in Figure\\,\\ref{fig2})? Both cases show strong fluorescence lines in the soft X-ray band along with very similar features in the hard X-ray band.\n            \n            In order to distinguish between the spectra obtained by these two geometries, we fit the slope of the continuum in the soft X-ray band (0.3$-$3\\,keV) with simple power-laws. When fitting the spectra, fluorescence lines were not masked when computing the slope, since most instruments cannot mask these lines. The torus alone was best fit with a photon index of $\\Gamma = -1.63$. However, the power-law  fit for the edge-on case with a torus + hollow cone gave a photon index of $\\Gamma = -0.80$. A plot of the soft spectrum for both simulations considered is shown in Figure\\,\\ref{fig8.5}. Another approach to distinguish between these two spectra is compare the difference in equivalent widths of the strongest spectral lines in the soft X-ray band for intermediate case with a torus alone and the edge-on case with a torus + hollow cone (the soft spectra for both these cases are shown in Figure\\,\\ref{fig8.5}). The equivalent widths of the O, Ne, and Si lines were compared for the two geometries. The equivalent widths of these lines are 569, 171, and 173\\,eV respectively for the torus alone and 944, 321, and 203\\,eV respectively for the torus + hollow cone. All these lines are stronger in the torus + hollow cone scenario with the O line showing the strongest difference, increasing by 375\\,eV, compared to the torus-only case.\n            \n            However, It should be noted that this X-ray emission from the polar gas might be combined with other features, such as those stemming from the NLR on scales of hundreds of parsecs (e,g., \\citealp{2006A&A...448..499B, 2007MNRAS.374.1290G}). These contributions include an extra scattered power-law component, as well as the addition of many narrow photoionized lines. The contribution from the NLR to the scattered flux is typically $\\sim$0.1--1\\% \\citep{Ueda_2007,2017ApJS..233...17R, 2021MNRAS.504..428G}, with the lowest values typically associated to the most obscured objects \\citep{2021MNRAS.504..428G}. In Figure\\,\\ref{fig10.5} we add, to the edge-on simulation with a torus + hollow cone at the highest polar gas column density, a power-law component to reproduce a 0.1\\% scattered fraction of the intrinsic flux of Circinus, as well as photoionized lines from the obscured AGN NGC 3393 \\citep{2006A&A...448..499B} and the Circinus galaxy (\\citealp{2001ApJ...546L..13S, 2014ApJ...791...81A}, Andonie et al. submitted), in order to verify how visible the lines in the soft X-ray band will be. The normalisation of these lines were set as to keep the ratio of the flux of line to the power-law the same as in their original source. The value of the extra scattered fraction was selected in order to be consistent with an AGN obscured by CT column densities \\citep{2021MNRAS.504..428G}. The resulting spectrum is shown in blue, with the original simulation in dash-dotted black, the scattered power-law as solid black, and the extra photoionized lines dashed and labeled with their transitions. These photoionized lines were chosen from Tables 1, 4, A1 of \\citealp{2001ApJ...546L..13S, 2014ApJ...791...81A}, Andonie et al. submitted, as well as Table 4 in \\cite{2006A&A...448..499B} based on their proximity to our simulated lines. These new photoionized lines can be distinguished from the O, Ne, and Mg lines. However, the Mg XII - Si II-VI, S II-X, and Ar II-XI transitions have the potential to interfere with the detection of the Si, S, and Ar lines from the polar component. But if the bulk motion of the polar component is blue-shifted enough, these lines could still be distinguished. With this in mind, we calculated the EWs of the O, Ne, Mg and Si (assuming no interference from the photoionized lines) lines which had values of 7\\,eV, 12\\,eV, 33\\,eV, and 87\\,eV respectively. \n            \n            Additionally, X-ray emission in obscured AGN below $\\sim$ 2\\,keV caused by populations of X-ray binaries and collisionally ionised plazma in star forming regions (e.g., \\citealp{2008MNRAS.386.1464R}) could also pose a challenge in detecting the true emission from the polar component. To consider this additional contribution, we added to the model shown in Figure\\,\\ref{fig10.5} a thermal plasma model with a temperature of 0.5\\,keV which corresponds to the median temperature in nearby obscured AGN \\citep{2017ApJS..233...17R}. We found the addition of this component should not add additional difficulty in detecting the spectral lines from the polar component in obscured AGN.\n            \n        \\subsection{Observations with \\textit{XRISM} and \\textit{Athena}}\n        \n            Here we consider how our previous simulations of the polar component would be observed with instruments on board \\textit{XRISM} (Resolve; \\citealp{xrismscienceteam2020science}) and \\textit{Athena} (X-IFU; \\citealp{2016SPIE.9905E..2FB}). \\textit{XRISM}\/Resolve boasts a $\\sim 5-7$\\,eV FWHM spectral resolution over the entire bandpass, while \\textit{Athena}\/X-IFU in expected to operate under a $\\sim 2.5$\\,eV spectral resolution up to 7\\,keV. Based on results of our spectral simulations, these instruments could be extremely well suited to identify the X-ray signature of the polar component. We use the X-ray spectral-fitting program software \\textsc{XSPEC} \\citep{1996ASPC..101...17A} along with the latest \\textit{XRISM}\/Resolve and \\textit{Athena}\/X-IFU (Barret et al. 2021 in prep.) response and background files to simulate the spectra that would be expected from these two instruments. The spectral simulations are illustrated in Figure\\,\\ref{figres}, where we consider the edge-on torus + hollow cone geometry, assuming the highest polar gas column density ($\\log{N_{\\mathrm{H}}\\big\/\\text{cm}^{-2}} = 22.5$; left panel of Figure\\,\\ref{fig3b}), along with the components discussed in section 4.2 (power-law with 0.1\\% of the scattered flux with various photoionized lines from the NLR). We assumed an exposure time of 200\\,ks with an observed flux in the $2-10$\\,keV range of $F_{2-10} = 12.7\\times10^{-12}$\\,erg s$^{-1}$ cm$^{-2}$, as expected for the Circinus galaxy \\citep{2017ApJS..233...17R}. The top panel displays the \\textit{XRISM}\/Resolve response while the bottom shows the \\textit{Athena}\/X-IFU response. Based on this figure, \\textit{Athena}\/X-IFU will be better suited to detect lines from the polar component, since \\textit{XRISM}\/Resolve is unable to distinguish between some of the lines from the polar component and and photoionized lines from the NLR. However, \\textit{XRISM}\/Resolve is still able to resolve the Ca line at $\\sim$ 3.69\\,keV. \\textit{Athena}\/X-IFU is able to clearly distinguish the O, Ne, Mg, and Ca lines from the NLR lines. It should be stressed that, if this polar component in caused by out-flowing gas \\citep{Ricci2017, Leftley_2019, 2019ApJ...884..171H, 2020ApJ...900..174V}, it is likely that these lines will be shifted to different energies. We expect, given the resolutions of \\textit{XRISM}\/Resolve and \\textit{Athena}\/X-IFU, to be able to recover velocity shifts of $\\sim$ 1000\\,km\/s and 375\\,km\/s for the two instruments respectively for the O, F, Ne, Mg, and Si lines.     \n            \n    \\section{Testing Different Cone Geometries and Abundances}\n    \n        \\subsection{Changing the Opening Angle}\n    \n            It is possible that the opening angle of the polar component could be controlled through collimation by the absorbing material surrounding the BLR and acccretion disk (i.e., the torus), as is the case for the NLR \\citep{2013ApJS..209....1F}. With this in mind, we considered a new opening angle of $\\alpha = 10^{\\circ}$ to simulate a nucleus with higher collimation of ionizing radiation which corresponds to a torus with a larger covering factor. The resulting spectra compared with the previous opening angle of $\\alpha = 30^{\\circ}$ for an edge-on torus + hollow cone at the highest column density can be seen on the left panel of in Figure\\,\\ref{fig10}. Fitting the soft spectra with a simple power-law for the $\\alpha = 10^{\\circ}$ case gives a photon index of $\\Gamma = -0.71$ compared to an index of $\\Gamma = -0.80$ for the $\\alpha = 30^{\\circ}$ case. Table \\ref{table7} shows all the EWs of the lines in the soft spectra for this new opening angle for different densities of the polar gas as well. We note that, while the O line has a higher EW for the  $\\alpha = 30^{\\circ}$ case (an average ratio of 1.29 when averaging with all four column densities), the Ne and Si lines tend to have larger EWs in the  $\\alpha = 10^{\\circ}$ case with average ratios of 0.79 and 0.99 respectively.\n            \n        \\subsection{Changing Abundances}\n        \n            The simulations carried out so far considered a matter composition from \\cite{1989GeCoA..53..197A} as well as a molecular hydrogen fraction of one. In this section, we consider the more recent gas composition from \\cite{2003ApJ...591.1220L} as well as a Hydrogen fraction of 0.2 (H$_{2} = 0.2$). We ran simulations using this new matter composition for the edge-on case at the highest polar gas column density in order to compare with the previous simulations. The results are seen on the right panel of Figure\\,\\ref{fig10} with the black plot representing the composition from \\cite{2003ApJ...591.1220L} (H$_{2}$ = 0.2) and the red plot being from the previous simulations using \\cite{1989GeCoA..53..197A} (H$_{2}$ = 1). As can be seen, there is no significant difference in spectral or continuum features. \n    \n    \\section{Summary and Conclusion}\n    \n        In this paper we have investigated the effect polar gas has on the X-ray spectra of accreting supermassive black holes. Three geometries (torus alone, torus + hollow cone, torus + filled cone), along with three inclination angles and four slant length column densities for the filled and hollow cones were considered. We summarize our main results below:\n        \n        \\begin{itemize}\n            \\item The polar gas was found to have a significant impact on the soft X-ray spectrum producing several fluorescence lines in the 0.3--5\\,keV band such as the O ($524.9$\\,eV), Ne ($848.6$\\,eV), Mg ($1253.6$\\,eV), and Si ($1739.38$\\,eV) lines. The most significant impact on the X-ray spectra from the polar gas occurs at the edge-on inclination angle where photons which would normally be absorbed in the torus are allowed to scatter from the less dense polar component (see Figures\\,\\ref{fig2}, \\ref{fig3a} and \\ref{fig3b}, as well as Figure\\,\\ref{fig6} and Tables \\ref{table1}-\\ref{table6} for the EWs of selected spectral lines). \n            \\item The most significant impact the polar component has is when it is in the form of a hollow cone, as the filled cone causes more self-absorption as there is more gas to interact with. Simulations were also run using the same number of atoms in the filled as in the hollow cone. These showed that the soft spectrum of the pole-on case was still obscured and, therefore, is likely not the geometry for the polar component (see Figure \\,\\ref{figCone}). \n            \\item When considering 0.1\\% of extra scattered continuum stemming from the NLR in addition to the new polar gas emission, as well as extra photoionized lines from NGC 3393 and the Circinus galaxy, we find that the spectral lines from the polar component would still be detectable, particularly by future X-ray calorimeters such as those onboard \\textit{XRISM} (Resolve; \\citealp{xrismscienceteam2020science}) and \\textit{Athena} (X-IFU; \\citealp{2016SPIE.9905E..2FB}) (see Figure\\,\\ref{fig10.5}). We also simulate spectra using \\textit{XRISM} and \\textit{Athena} (Barret et al. 2021 in prep.) to see how our simulated data would be observed (see Figure\\,\\ref{figres}). This figure shows that \\textit{Athena}\/X-IFU is better suited to observe more spectral features from the polar component than \\textit{XRISM}\/Resolve, given the extra features stemming from the NLR.  \n           \n           \n            \\item When considering a smaller opening angle for the polar component we find that, while the EWs of many spectral lines change, the general trends as seen in Figure\\,\\ref{fig6} remain the same with the EWs of lower energy lines (O, F, Ne, Mg, Si) increasing with increasing column density of the polar component and the EWs of the higher energy lines (Ar, Ca) decreasing with column density. This is with the exception of the S line which increases in EW, contrary to what is seen in Figure\\,\\ref{fig6} (see Table \\ref{table7}). In addition to this, when considering an abundance from \\cite{2003ApJ...591.1220L}, we find little difference in spectral or continuum features (see the right panel of Figure\\,\\ref{fig10}).  \n        \\end{itemize}\n        \n        These simulations show that low-energy fluorescent lines could be an important tracer of the polar gas in AGN. Observations of nearby AGN with future high-resolution X-ray instruments, such as those on-board {\\it XRISM} \\citep{xrismscienceteam2020science} and {\\it Athena} X-IFU \\citep{2016SPIE.9905E..2FB}, will allow to search for signatures of polar gas. Comparison with simulations, such as those reported in this paper, will allow further constrain the kinematics, geometry, and origins of the polar gas.  \n        \n\n\n\n\n\\newpage\n\n\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Limitations}\n\n\nSocial media does not necessarily represent offline events, \nand unlike the surveys we use for comparison, we cannot control for demographics.\nWe study texts written in English and platforms that attract a U.S. audience, as the authors are all most familiar with this setting.\nWe focus on one specific community out of many, and we do not expect that the findings in this paper will necessarily generalize outside of the U.S. or to other online spaces; \nindeed, one of our results indicates that different platforms display different patterns.\nFinally, averaging over posts allows us to track patterns and make comparisons but can also reduce nuance.\nOur work is best read in conjunction with ethnographic studies like \\citet{homewood2017turned} and \\citet{daley2014influences}, which highlight individual voices of birth control users.\n\\section{Introduction}\n\n\nBirth control users include nearly two thirds of women in the U.S. between the ages of 15-49---almost 50 million women \\citep{daniels2020current}.\nIn addition to providing a means of family planning, birth control methods can also be used to treat and manage many medical conditions, including acne, endometriosis, cancer risks, gender dysphoria, and irregular and painful menstruation \\citep{schindler2013non, nahata2017gender}.\nHowever, birth control methods are not one-size fits all, and the choices of whether to use birth control and which method to select are complicated by personal beliefs, cost and accessibility, and a wide array of side effects that are difficult to predict and identify \\citep{manzer2022did,yoost2014understanding,polis2016typical}.\n\n\nWhen navigating this decision, birth control users face a sensemaking challenge that involves dealing with social stigma, understanding contraceptive methods and potential side effects, and seeking normalcy after upsetting experiences seeking and using contraception.\nThis challenge leads many birth control users to the internet \\citep{yee2010role,russo2013women}, where they can engage in community sensemaking activities, e.g., seeking and sharing information, reviews, advice, hypotheses, and stories \\citep{carron2015help,yang2019seekers}.\n\n\nPrior work has shown that these various sensemaking activities can vary in frequency across different online platforms \\citep{choudhury2014seeking,rivas2020classification,zhang2021distress}.\nDifferent healthcare conditions \\citep{sannon2019really} and side effects \\citep{choudhury2014seeking} can result in different behavior,\nand recent work has demonstrated differences in the online sensemaking practices surrounding reproductive healthcare topics like pregnancy and vulvodynia \\citep{young2019girl,andalibi2021sensemaking,chopra2021living}.\nHowever, while birth control researchers have examined the frequencies of different method discussions over time \\citep{nobles2019repeal,merz2020population}, a comparison across platforms of the difficult decision-making of birth control users and their sensemaking practices has so far been unexplored. \n\n\nWe study birth control users' unique combination of sensemaking activities---deciding between methods, interpreting side effects, calculating risks, etc.---and we use computational text analysis methods to discover the intersections between these activities and online platforms.\nBecause different methods and side effects could lead to different activities, we first\nexplore \\textit{which birth control methods and side effects are more likely to be discussed on different online platforms} (\\textbf{RQ1}).\nUsing lexicons crafted for our online settings, we compare these discussion frequencies to the distribution of experienced side effects as reported via a nationally representative survey by \\citet{nelson2018women}, indicating increased or decreased interest in discussing these concerns online.\nNext, using topic modeling and hand-annotations, we discover and characterize \\textit{the kinds of sensemaking activities that birth control users engage in online and how these differ by platform} (\\textbf{RQ2}).\nWe contrast these activities with prior work on related online healthcare communities, and we highlight the particular strategies of birth control users.\n\n\n\\paragraph{Contributions.}\nWe identify and characterize how birth control users make sense of their experiences and options via three large English-language online platforms (Reddit, Twitter, and WebMD).\nTheir unique combination of sensemaking strategies included \\textit{storytelling}, \\textit{risk analysis}, \\textit{timing and calculations}, \\textit{causal reasoning}, \\textit{method and hormone comparison}, and \\textit{information and explanations}; in particular, \\textit{storytelling} is used to prepare for and overcome painful insertion experiences.\nWe examine these patterns across birth control methods, side effects, and platforms, unlike prior work, and we compare our observations of side effect discussions to survey data, finding large variations and highlighting the importance of integrating different methods.\nThroughout this study, we employ a mixture of lexicons (which we provide as a community resource\\footnote{\\url{https:\/\/github.com\/maria-antoniak\/birth-control-across-platforms}}), topic modeling, and hand annotation, demonstrating how a detailed and diverse analysis across online platforms can expand our understandings of social sensemaking, web and social media activity, and vital healthcare issues.\n\n\n\\section{Related Work}\n\\label{section-related-work}\n\n\n\n\n\n\\smallskip\n\\noindent\\textbf{Sensemaking in online communities.}\nAs explained by \\citet{weick2005organizing}, sensemaking necessarily involves communication and is an ``\\textit{activity that talks events and organizations into existence},'' in part through retrospective storytelling.\nIt is a process that \nrelies on collaborative problem solving \\citep{pirolli2005sensemaking}, and more recently, it has been defined in closely related work by \\citet{andalibi2021sensemaking} \nas ``\\textit{how individuals make sense of complex phenomena by constructing mental models that draw on new or existing experiences, information, emotions, ideas, and memories}.''\nIn online healthcare communities, the individual works to make sense of their healthcare experience, and the community collectively gathers information, compares stories, and makes sense of a shared experience \\citep{mamykina2015collective}. \nSharing experiences can help narrators make sense of their stories \\citep{tangherlini2000heroes} and can transfer important information to others without firsthand experience \\citep{Bietti2019StorytellingAA}, leading users through a transformative process via the gathering and organizing of information \\citep{genuis2017looking,patel2019feel}.\nThis process is contextual; each community can rely on different strategies \\citep{young2019girl}.\n\n\n\n\\smallskip\n\\noindent\\textbf{Healthcare activity across platforms.}\nSpecific platform affordances can facilitate different types of healthcare-related disclosure and interactions.\n\\citet{zhang2021distress} formulate a \\textit{social media disclosure ecology} in which platform affordances like anonymity, persistence, and visibility control can predict, e.g., pandemic-related disclosures.\nHealth information seeking behavior can differ across platforms, with search engines more frequently used for serious and stigmatized conditions and Twitter more frequently used for symptoms \\citep{choudhury2014seeking}, and community affordances such as threaded conversations and hashtags can influence participation decisions of those with invisible chronic health conditions \\citep{sannon2019really}.\nThe distribution of content type can also differ by platform; e.g., prior work found that \\textit{sharing experiences} is less frequent on social networks than forums, although these rates can depend on healthcare topic \\citep{rivas2020classification}.\n\n\n\n\n\\smallskip\n\\noindent\\textbf{Reproductive healthcare and online sensemaking.}\nStudies of reproductive healthcare communities have emphasized some unique sensemaking practices.\nFor example, in a study of sensemaking practices after pregnancy loss, \\citet{andalibi2021sensemaking} highlighted the importance of seeking emotional validation, rather than just information, when trying to re-discover normalcy.\n\\citet{young2019girl} similarly supported these dual information management and emotional needs in a study of a vulvodynia Facebook group,\nand \\citet{chopra2021living} discussed polycystic ovary syndrome and why personal tracking is an important communal activity that involves comparison to others' experiences.\nWhether these themes hold for online birth control communities is not currently known.\n\n\nTwo recent studies have focused specifically on online discussions of birth control.\nFirst, \\citet{nobles2019repeal} used search queries to track interest in different birth control methods, finding that U.S. political events correlate strongly with increased information seeking behavior online.\nSecond, \\citet{merz2020population} examined the prevalence and sentiment of tweets mentioning different birth control methods. \nThe authors found that long-acting methods like the IUD were mentioned more often on Twitter, and the proportion of these tweets increased over time.\nWhile most tweets expressing sentiment about contraception were negative, tweets about long-acting methods were more likely to express positive sentiment.\n\n\n\\section{Discussion}\n\n\n\n \\paragraph{Methods and side effects across platforms.}\nIn response to our first research question, we find that birth control discussions on Twitter, WebMD, and Reddit substantially differ in their distributions of methods and side effect mentions.\nWe cannot claim that using a method or experiencing a side effect \\textit{cause} people to choose a specific platform, but our observations add detail and sometimes contradict prior findings.\nFor example, in a study of general healthcare information seeking, \\citet{choudhury2014seeking} found that people more often use search engines for serious and stigmatized conditions and more often use Twitter to discuss symptoms.\nWe do not include search engines in our study, but in our comparison across platforms, we find that Twitter has a higher frequency of severe side effect discussions for birth control, while WebMD and Reddit have higher frequencies of general side effect discussions.\nOur focus on birth control might explain these variations, as many of the birth control side effects are themselves highly stigmatized (e.g., menstrual bleeding, vaginal discharge). \nThe discussion of severe side effects on Twitter is likely related to Twitter's tendency toward sensational content, and could also explain the relative frequency of IUD discussion on Twitter; the IUD has been associated with both severe side effects and potential legal bans\n\\citep{nobles2019repeal}.\nThese findings emphasize the importance of considering platform setting when studying online patterns related to specific health conditions, medications, and side effects.\n\n\n\n\\paragraph{Online sensemaking practices of birth control users.}\nIn response to our second research question, our identification and comparison of topics that align with sensemaking themes reveals important activities specific to birth control and to the different platforms and methods.\nPrior work has found that online communities focused on reproductive healthcare (e.g., pregnancy, vulvodynia) employ strategies related to validation \\citep{andalibi2021sensemaking}, information management \\citep{patel2019feel,young2019girl,andalibi2021sensemaking}, personal tracking \\citep{chopra2021living}, and identifying causation \\citep{patel2019feel}.\nWe found these themes again in our birth control communities, with variations; for example, personal tracking is also a prominent theme in birth control discussions, but it is focused on pill timings and calculations and on self-observations of side effects. \nIdentifying causation is a common concern for those experiencing infertility \\citep{patel2019feel}, and we find this theme again in our datasets but focused on side effects like weight gain.\nThis combination of themes reflects the unique problems facing birth control users, as they choose between ``least bad'' options, struggle to identify and treat side effects, determine risk of pregnancy given their circumstances, and avoid rare but alarming outcomes like stroke and heart attack.\n\n\n\n\n\\paragraph{Storytelling, pain, and the IUD.}\nWe would like to particularly highlight storytelling as an important sensemaking strategy for birth control users.\nStorytelling is known to help communities work through trauma \\citep{tangherlini2000heroes}, but birth control users employ storytelling specifically to address physical pain.\nPain is a major cross-cutting theme across the platforms; it is the most frequently and consistently discussed side effect, and it is most often mentioned alongside the IUD.\nWhile \\citet{barr2010managing} identifies pain as the second most common side effect (after bleeding changes) and \\citet{dickerson2013satisfaction} identifies pain as the most common side effect for the IUD and second most common side effect for the implant, we find a much larger gap between pain and the next most frequent side effects in our analysis.\nPain is mentioned frequently in posts about the IUD and in posts whose probable topics are about seeking and sharing IUD insertion stories.\nPain is inherently a subjective experience that cannot be precisely communicated \\citep{scarry1987body}, but sharing of personal experiences provides one way for a community to build a sense of what is \\textit{normal} or \\textit{to be expected}  \\citep{patel2019feel,andalibi2021sensemaking}. \nWe suggest that there is an urgent unmet need for (a) honest education and preparation before IUD insertions and (b) pain treatment options during this procedure.\n\n\n\n\n\\paragraph{Online discussions differ from survey reports.}\nIn comparison to the survey results in \\citet{nelson2018women}, we find not only large differences in the frequency at which different side effects are discussed but also differences across the platforms.\nIt could be that the differences from the survey are due to the demographic distribution opting into the survey versus those opting to post online; future work surveying the demographic distribution of these users would address this question, but our results take a first step at measuring differences between these settings. \nWe find, e,g., while mood changes are discussed at similar frequencies in comparison to the survey data across the platforms, other side effects like strokes, bloating, fatigue, bleeding, and dizziness might be discussed more, less, or at equal frequency ranks to the reports in \\citet{nelson2018women}, depending on the platform.\nBloating is less frequently mentioned on all the platforms and for all the methods in comparison to the survey data, perhaps indicating a general lack of concern about this side effect in contrast to its reported frequency.\nBut bloating is much less frequent on Twitter for the pill and implant, which could also indicate that this particular setting is less suited to discussing this side effect, perhaps because of the embarrassing or intimate nature of this side effect.\n\n\n\n\\paragraph{Stigma, privacy, and contextual disclosures.}\nBirth control can be a controversial, stigmatized, and intimate topic.\nThis can lead birth control users to seek out additional information privately.\nFor example, in a set of interviews of young Black and Hispanic women, \\citet{yee2010role} found that a greater number reported seeking decision-marking support on the internet, citing its privacy, in comparison to other sources of information (e.g., talking to physicians, reading provided information).\nInterpreting side effects, analyzing the risk of pregnancy, or normalizing a painful experience require disclosing personal details and stories.\nBirth control users might analyze the risk and benefit of making these disclosures in certain settings and to certain audiences.\nWhile social penetration theory \\citep{altman1973social} posits that more disclosures are possible as social bonds deepen, prior work has also found that intimate language can be frequent among both close connections and strangers (but not in between) \\citep{pei-jurgens-2020-quantifying}.\nThe variations we observe across methods, side effects, and sensemaking practices could be indicative of platform affordances for privacy, audience size, and anonymity, each of which can affect decisions to self-disclose.\nFor example, we found that side effect discussions are less frequent on Twitter, where users are facing a much larger and non-specialized audience, unlike the other platforms.\nGiving users more platform-specific tools to control their audience size and membership \\citep{,mondal2014understanding} could allow for more productive discussions of this sensitive topic.\n\n\n\n\n\\paragraph{Other factors influencing platform decisions.}\nIt is important to note that decisions to seek information online can correlate with demographic characteristics.\nFor example, in a survey of U.S. young adults, those with a sexual risk history (early sexual activity, involvement in an unintended pregnancy) less frequently reported using the internet as a source and more frequently reported seeking information from a doctor\/nurse, and men more frequently reported using the internet than women \\cite{khurana2015young}.\nIt is also possible that users follow a \\textit{birth control journey}, where different needs at different points in one's journey can lead one to different platforms, as has been reported for other healthcare topics \\citep{sannon2019really,andalibi2018announcing}.\nThese journeys can intersect with methods; for example, while the pill is a popular first method, many people report switching to the IUD as they gain more experience with birth control \\citep{nelson2018women}.\nThis would mirror the journeys of those with invisible chronic illnesses who move from one platform to another as their needs evolve and as they grow more comfortable with self-disclosure \\citep{sannon2019really}.\nThis is mirrored in intra-community research that models user trajectories in online cancer support groups, finding that users often transition from information-seeking to information-sharing roles over time \\citep{yang2019seekers}. \n\n\n\n\n\\paragraph{Recommendations.}\nOur results show that patterns found in one platform are not necessarily replicable on other online platforms, even when grounded in the same health topic.\nWe recommend that social media researchers compare results across multiple platforms to increase confidence in shared patterns.\nIf relying on a single platform, its choice should be guided by an understanding of which platforms might be more less or conducive to discussions of the desired methods, side effects, or sensemaking activities.\nOur work also highlights benefits of mixing computational tools like topic modeling with lexicon-based methods and hand-annotation, and how these methods can be used to characterize difficult-to-identify themes like sensemaking practices; we recommend taking this careful approach, putting unsupervised results in context with fine-grained measurements.\n\n\n\\section{Methods}\n\n\n\\subsection{Birth Control Method Lexicon}\n\\label{section:method-labeling}\n\n\nTo explore the prevalence across platforms of different birth control methods, we develop a lexicon to match each document to the primary method being discussed.\nDue to data scarcity and space limitations, we do not consider methods that are not available on all three platforms (e.g., WebMD does not include the male condom), and of the remaining methods, we limit our analysis to the three most commonly discussed reversible, non-emergency methods, mirroring prior work such as \\citet{merz2020population}.\\footnote{Using the lexicons described above for Reddit, we find that the implant (8,578 posts) is discussed more than twice as often as the shot (3,529 posts) or barrier methods like the male condom (3,029 posts). This contrasts with reported rates of contraception use in the U.S., where the pill is used by 14.0\\%, the IUD by 8.4\\%, the male condom by 8.4\\%, the implant by 2.0\\%, and the shot by 2.0\\% of women aged 15-49 \\citep{daniels2020current}.}\nOur goal is to measure \\textit{prevalence of discussions} rather than \\textit{prevalence of usage} because (1) we are interested in the level of interest and concern of birth control users regardless of actual usage and (2) other study designs (e.g., surveys) are better suited to studying usage rates.\n\n\n\nWe rely on sets of keywords to assign each text a primary birth control method.\nBecause of their different structures, each platform requires its own set of keywords and matching techniques.\nAll of these lexicons are made available for future research.\nTo estimate the lexicon performance, we checked 450 documents balanced across Reddit posts, Twitter posts, and Twitter replies.\\footnote{We omitted Reddit comments because of our method detection strategy, which defaulted to the method in the parent post if no method was found in the comment; and we omitted WebMD reviews because our method detection strategy already used hand-labeling of medication tags.}\nOur precision and recall scores were perfect (1.0).\n\n\n\n\n\\subsubsection{WebMD.}\nThe URLs for WebMD reviews include a \\emph{drugname} parameter.\nThis parameter includes the medication name followed by its class. \nWe were able to classify 99.2\\% of the WebMD reviews by mapping these names to their corresponding method, and the remaining reviews were for methods outside our target three. \n\n\n\\subsubsection{Reddit.}\n\\label{subsubsection:reddit-method-labeling}\nTo assign Reddit posts and comments to the birth control methods, we use a custom set of keywords.\nIn all cases, the text was assigned to the method that had the highest keyword count (of all the methods).\\footnote{We could instead have assigned each Reddit post and comment to as many methods as were mentioned in its text, rather than assigning each post and comment only to the method mentioned most frequently. \nWe chose the latter assignment because it better aligns with the WebMD assignments; WebMD reviews are explicitly assigned by the user to a single method (even if other methods are also mentioned).\nWe do not find that these different assignment techniques affect our distributional results.}\nIf the highest keyword count was not for one of our target three methods, or if multiple methods were mentioned with equal frequency, we discarded the text.\nWe follow a similar procedure for Reddit comments, but in cases where the comments do not contain a birth control keyword, we assign the comment to the birth control method of its parent post.\n\n\nWe developed the Reddit methods keyword set by (a) drawing terms from the WebMD keyword set, (b) drawing terms from the Twitter keyword set,\n(c) supplementing the WebMD and Twitter sets with general terms that are overloaded on Twitter but usable on Reddit (e.g., ``pill''), and (d) iteratively running the assignments and examining the unassigned posts and comments.\nWe were able to assign all but 4.6\\% of the Reddit posts and 32.6\\% of the Reddit comments to a birth control method.\nManual examination of the unassigned posts reveals discussions of pregnancy scares, access (e.g., online appointments and prescriptions), and treatment of side effects with non-contraceptive medications.\nThe remainder of these unassigned posts discussed topics that are adjacent to birth control, e.g., insurance, menstruation, but are not explicitly connected to a birth control method.\n\n\n\\subsubsection{Twitter.}\nWe use a more limited keyword set, derived from a similar survey of birth control tweets by \\citet{merz2020population}, to query for tweets about each of our target birth control methods.\nOur focus was on precision rather than recall, as the Twitter API requires a keyword match to retrieve tweets on a particular topic; if we included keywords like ``pill'', our results would contain many false positives, unlike Reddit and WebMD where the topic is already constrained to birth control.\nAfter gathering our initial Twitter dataset, we then apply the full Reddit keyword set, using the same methods described for Reddit above.\nAs with the Reddit data, we find that more texts can be assigned to only the pill, IUD, or implant than to a combination of methods; 12.7\\% of posts and 17.6\\% of the gathered tweets either mentioned multiple methods with equal frequency or most frequently mentioned a method not in our set of three target methods.\n\n\n\n\n\n\\subsection{Side Effects Lexicon}\n\\label{section:side-effects}\n\n\nTo measure the frequency of discussions about side effect birth control, we develop a lexicon of terms and patterns.\nBecause we do not use the side effects lexicon for data collection (unlike the methods lexicon), we can rely on one lexicon across our three datasets.\nAs with the methods lexicon, we focus specifically on measuring the \\textit{prevalence of discussions} rather than the \\textit{prevalence of experiences}, as these are better measured via surveys \\citep{nelson2018women}.\nThe frequency of discussions could be influenced by the prevalence of experiences and the level of \\textit{concern} birth control users on this platforms have about particular side effects, as well as platform setting and a user's individual goals.\nWe select patterns that match any discussion of the side effect, whether or not it is mentioned in the affirmative. \n\n\nWe grounded our development of the side effects lexicon in prior work.\nWe matched the side effect categories from \\citet{nelson2018women} as closely as possible; this study conducted a nationally representative survey of U.S. birth control experiences and reported prevalences of side effect experiences across different birth control methods.\nIn addition to these, we added lexicon categories for pain, skin changes, PMS, appetite changes, sexual partner feeling IUD strings, and heart attack.\nWe identified these additional categories and patterns from topic models trained on our datasets and from other work; for example, we also drew on work by \\citet{barr2010managing} that discusses a variety of side effects and their known frequencies and associations with different birth control methods.\nFor each side effect, we then iteratively queried and made updates to the lexicon when we encountered false positives or false negatives.\n\n\n\n\\input{tables\/lexicon_coverage}\n\n\n\\subsubsection{Lexicon evaluation.}\n\nTo evaluate our lexicon, for each side effect we randomly selected a set of matching and non-matching texts, balanced across platforms and match status, resulting in a set of 1,040 texts.\nWe then manually checked whether each text does or does not contain a discussion of the target side effect.\nAcross the side effects, we found precision of 0.98 and recall of 0.96.\nWe show the lexicon coverage in Table \\ref{table:lexicon-coverage}.\n\n\\subsubsection{Comparing observations with survey responses.}\n\nWe compare the distributions of side effect mentions to prior work surveying people in the U.S. about their experiences with birth control side effects.\nThis allows us to compare the frequency of side effect experiences with the frequency of online discussions.\nDifferences between these distributions can indicate healthcare needs that users are addressing via the internet.\nWe compare to surveys (rather than controlled studies) because surveys better approximate the self-reported anecdotes and personal experiences shared online.\nWe first converted each distribution to a ranking, where lower ranks represent greater percents of discussions $r_{platforms}$ or experiences $r_{survey}$.\nWe then find the difference between the ranks for each side effect, $d_{ranks}$.\n\\begin{equation}\n    d_{ranks} =  r_{survey} - r_{platforms}\n\\end{equation}\nThese ranks avoid issues in directly comparing percents, which might on average be higher or lower.\nWhen $d_{ranks}$ is positive, it indicates more online discussion than expected given the frequency of reported side effect experiences. \n\n\n\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{figures\/final\/barplot.side_effects_distributions.pdf}\n    \\includegraphics[width=0.49\\textwidth]{figures\/final\/barplot.bc_type_x_side_effect.pdf}\n    \\caption{Distribution of the side effect mentions across the platforms and methods. Bars represent the percents of documents for the specified platform or method mentioning \\textit{any} side effect that also mention the \\textit{specified} side effect. \n    See Table \\ref{table:lexicon-coverage} for the denominator sizes. \n    Error bars indicate the standard deviation over 20 bootstrapped samples of the datasets.\n    } \n    \\label{figure:barplot-side-effects-distribution}\n\\end{figure*}\n\n\n\n\n\\begin{figure}[t]\n    \\centering\n\n        \\centering\n        \\includegraphics[width=0.49\\linewidth]{figures\/final\/barplot.bc_type_dist.reddit_posts.normalized_by_year.pdf}\n        \\includegraphics[width=0.49\\linewidth]{figures\/final\/barplot.bc_type_dist.twitter_posts.normalized_by_year.pdf}\n        \\includegraphics[width=0.49\\linewidth]{figures\/final\/barplot.bc_type_dist.reddit_comments.normalized_by_year.pdf}\n        \\includegraphics[width=0.49\\linewidth]{figures\/final\/barplot.bc_type_dist.twitter_replies.normalized_by_year.pdf}\n        \\includegraphics[width=0.49\\linewidth]{figures\/final\/barplot.bc_type_dist.webmd_reviews.normalized_by_year.pdf}\n    \n    \\caption{Document distributions of methods over time.}\n    \n    \\label{figure:bc-type-distributions}\n\\end{figure}\n\n\n\n\\begin{figure}[!ht]\n\n    \\centering\n    \n    \\includegraphics[width=0.1805\\textwidth]{figures\/final\/barplot.side_effects_distributions.compare_to_nelson.ranks.pill.v.pdf}\n    \\includegraphics[width=0.135\\textwidth]{figures\/final\/barplot.side_effects_distributions.compare_to_nelson.ranks.iud.v.pdf}\n    \\includegraphics[width=0.135\\textwidth]{figures\/final\/barplot.side_effects_distributions.compare_to_nelson.ranks.implant.v.pdf}\n    \\\\\n    \\includegraphics[width=0.25\\textwidth]{figures\/final\/legend.png}\n    \n    \\caption{Distribution of the side effect mentions across the platforms. Bars represent the \\textbf{differences} between the rank reported in the survey results from \\citet{nelson2018women} and the rank on the specified platform. Platform ranks are determined by first finding the frequency of side effect mentions; these are the percent of documents mentioning \\textit{any} side effect that also mention the \\textit{specified} side effect. See Table \\ref{table:lexicon-coverage} for the denominator sizes. Bars to the \\textbf{right} of the x-axis represent side effects that are mentioned \\textbf{more frequently} on the platforms than are reported in the survey.}\n    \n    \\label{figure:barplot-side-effects-distribution-comparison}\n    \n\\end{figure}\n\n\n\n\\begin{figure*}[!ht]\n\n    \\centering\n    \\begin{subfigure}[b]{0.24\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/final\/heatmap.topic_x_methods.percents.storytelling.pdf}\n        \\caption{Storytelling}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.24\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/final\/heatmap.topic_x_methods.percents.risk_analysis.pdf}\n        \\caption{Risk Analysis}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.24\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/final\/heatmap.topic_x_methods.percents.timing_and_calculations.pdf}\n        \\caption{Timing \\& Calculations}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.24\\textwidth}\n        \\includegraphics[width=\\textwidth]{figures\/final\/heatmap.topic_x_methods.percents.method_and_hormone_comparison.pdf} \n        \\caption{Method \\& Dose Comparison}\n    \\end{subfigure}\n    \n\n    \\caption{Mean topic probabilities across methods for four of our sensemaking clusters. Rows are normalized to sum to one.}\n    \n    \\label{figure:heatmaps-topics-methods}\n    \n\\end{figure*}\n\n\n\n\n\n\n\\subsection{Sensemaking Topic Model}\n\n\n\nWe compare sensemaking activity distributions through an unsupervised, bottom-up analysis using topic modeling.\nTopic modeling is an automatic method to identify prominent themes and discourses in a dataset \\citep{blei2003latent} and is a popular unsupervised technique for the analysis of online health communities \\citep{nobles2018std,abebe2020quantifying}.\nWhile some sensemaking themes might be shared across settings, other themes might be more prevalent in certain platforms and side effects.\nThis process helps us to avoid biases in our data coding (which might overlook certain themes while over-representing others) by quickly revealing frequent themes in each dataset \\citep{nelson2020computational}.\n\n\n\n\\subsubsection{Model training.}\n\n\nWe trained a latent Dirichlet allocation (LDA) topic model \\citep{blei2003latent} on each of our datasets, by combining the Reddit posts and comments and the Twitter posts and replies for training, resulting in three models, using MALLET\\footnote{\\url{http:\/\/mallet.cs.umass.edu\/}} for training.\nLDA remains a highly consistent and reliable model \\citep{harrando-etal-2021-apples,hoyle2022AreNT}, especially when trained via Gibbs sampling for smaller datasets.\nWe balanced each training set, sampling 8,000 documents for each method for each of the Reddit and Twitter datasets and 800 documents for each method for the WebMD dataset (see Figure \\ref{figure:bc-type-distributions} for method distributions).\nBy balancing the training set, we avoid weighting the topics toward a certain method.\n\n\nWe removed a set of frequent stopwords,\nnumbers, punctuation, and duplicate documents from the training sets.\nRemoving stopwords and duplicate documents has been shown to improve the legibility of the final topics, whereas stemming can be harmful \\citep{schofield-mimno-2016-comparing,schofield-etal-2017-pulling,schofield-etal-2017-quantifying}.\nTo avoid capitalization discrepancies,\nwe lowercase all text.\nWe experimented with different numbers of topics and found 35 to be interpretable across the datasets; at this number of topics, we observed topics that were neither too fine-grained (e.g., splitting a single theme across multiple topics) nor too high level (e.g., combining themes that should be separated), and that produced reasonable evaluation scores (see below).\nHowever, we emphasize that there is no ``correct'' number of topics and that this method is used for exploration and interpretation. \n\n\n\n\n\n\\subsubsection{Model evaluation.}\n\\label{topic-model-evaluation}\n\n\nWhile we do not have space here to list the full sets of topics for each dataset, we provide the topics, most probable words, and example paraphrased documents in our code repository for manual examination.\nWe report human evaluation of our topics following the recommendations in \\citet{hoyle2021automated}.\nUsing the \\textit{word intrusion task} \\citep{chang2009reading}, we show two expert annotators (the first two authors) and one non-expert annotator a set of the four most probable words plus an ``intruder'' word that has low probability for the current topic but high probability for another topic.\nWe report the proportion of topics for which the annotators identified the intruder.\n\n\nWe found that our Reddit and WebMD topics have performance much higher than a random baseline of 0.2 (annotator accuracies for Reddit: 0.71, 0.74, 0.77, WebMD: 0.54, 0.66, 0.77) while the Twitter topics have lower performance but are still substantially above the random baseline (Twitter: 0.46, 0.46, 0.51).\nThe first (non-expert) annotator consistently had lower scores.\nThe lower performance on Twitter is expected, as text processing methods are notoriously challenged by the short text lengths and non-standard language \\citep{gimpel-etal-2011-part}, and the short tweets require contextual knowledge to interpret.\nThis vulnerability of the word intrusion task to esoteric topics is noted by \\citet{hoyle2021automated}.\n\n\nWe also calculate the ``UMass'' Coherence: the log probability that a document containing at least one instance of a higher-ranked word also contains at least one instance of a lower-ranked word \\citep{roder2015exploring}.\nWe find the highest mean coherence scores across topics for Reddit ($-416$), lower scores for WebMD ($-577$), and lowest scores for Twitter ($-712$).\nWe note the criticisms of this and other automatic metrics in \\citet{hoyle2021automated}; in comparison to human evaluation, automatic metrics can exaggerate differences between models.\n\n\n\n\\subsubsection{Identifying sensemaking themes.}\n\n\nAcross the datasets, we find that many of the topics discuss birth control methods (including one method in their 10 most probable words), side effects, pregnancy, and access (costs, appointments).\nWe then identify a series of cross-cutting sensemaking topics.\nThese topics include discussions of information seeking\/sharing, educational links and resources, how-to explanations, experience seeking\/sharing, emotional support, and other sensemaking-related discussions.\nTwo researchers independently coded the topics as more or less related to sensemaking.\nThe researchers then conferred and agreed upon a final set of topics and assigned them to thematic clusters.\nDuring this coding, we relied on the sensemaking definition from \\citet{andalibi2021sensemaking}: ``\\textit{sensemaking describes how individuals make sense of complex phenomena by constructing mental models that draw on new or existing experiences, information, emotions, ideas, and memories}.''\nWe also drew inspiration from prior work studying sensemaking in online healthcare communities, particularly those works also focused on reproductive healthcare (see \\S Related Work).\nWe further explore and validate these sensemaking topics by hand-labeling a small subsection of the data with social support goals \\citep{yang2019seekers}.\nAfter coding 150 documents for each dataset, we measured the agreement between the annotators using Krippendorff $\\alpha$, as each document could receive zero, one, or more labels.\nOur agreement was acceptable, with a score across the labels of 0.74.\\footnote{Lower scores are not surprising for subjective language labeling tasks \\citep{artstein2008inter,godwin2016collecting}, and our scores are substantially higher than the agreement scores for a very similar classification task in \\citet{rivas2020classification}.}\n\n\n\nTo compare the topics across the platforms, we aligned the topics from the different models using Jensen-Shannon divergence (JSD) for the word distributions associated with each pair of topics. \nJSD is frequently used in prior work to compare topic distributions \\citep{hall-etal-2008-studying,fang2012mining}.\nAfter manual examination of the ranked topic pairs, we categorized topics with JSD scores below 0.6 as aligned across the datasets, and those with scores above 0.8 we considered diverging.\n\n\n\\section{Data}\n\\label{section:data}\n\n\nWe collected data related to birth control from three prominent online platforms: posts and comments on \\textit{r\/BirthControl}, birth control reviews on \\textit{WebMD}, and \\textit{Twitter} posts and replies related to birth control.\nThese are all English-language subcommunities of larger websites, and while we have limited demographic information, we observe that the majority of location-specific posts (e.g., politics, insurance) are about the U.S.\nTable \\ref{table:data} summarizes these datasets.\n\n\n\n\\input{tables\/data}\n\n\n\n\\paragraph{Reddit.}\nr\/BirthControl\\footnote{\\url{https:\/\/www.reddit.com\/r\/birthcontrol\/}} is a user-created and user-moderated online forum dedicated to birth control.\nUsers are pseudonymous and range from one-time questioners to experienced question answerers. \nAs of November 20, 2022, r\/BirthControl had 107,537 members.\nAccording to a Pew Research Center survey of U.S. residents, more men (15\\%) than women (8\\%) use Reddit, and more White (12\\%) and Hispanic (14\\%) than Black (4\\%) survey respondents use Reddit \\citep{perrin2019share},\nbut these platform-wide distributions are unlikely to be representative of r\/BirthControl, for which more detailed demographics are unavailable.\nPrior work on a related subreddit found that 81\\% of the users identified themselves as white \\citep{nobles2020examining}.\n\n\nPosts tend to be longer than comments, there are more comments than posts in our dataset, and both posts and comments have increased in frequency over time.\nFor each document, we collected the title (for posts), text, date, and user-assigned tag (for posts).\nWe removed comments written by the parent post's author, and we also removed stickied comments (auto-generated or mod-written comments).\nWe did not include documents deleted by the time of collection.\n\n\n\n\n\\paragraph{WebMD.}\nWebMD\\footnote{\\url{https:\/\/www.webmd.com\/sex\/birth-control\/birth-control-pills}} is a healthcare website that has hosted news, information, prescription discounts, community forums, and user-written medication reviews.\nWe focused on the medication reviews as a contrast to the Reddit and Twitter datasets.\nThe majority of reviews are written by women (71\\%), and unlike Twitter and Reddit, the website has declined in visitors and reviewers since 2009 \\citep{yun2019decline}.\nUnlike Twitter and Reddit, WebMD reviews do not allow users to interact directly, though reviews sometimes mention other reviews or the general trends observed in other reviews.\n\n\nWe gathered reviews by querying \\emph{birth control} as a condition on the drugs and supplements portion of the site and selecting medications that had more than 5 reviews.\\footnote{At the time of data collection, this search term yielded both preventative and emergency contraception, but the terms must be queried separately as of January 2021.}\nFor each WebMD review, we collect the birth control name, the date of the review, and the review text. \n\n\n\n\\paragraph{Twitter.}\nTwitter is a large social network where discussions range widely from personal to global topics.\nCompared to the general public, Twitter users are more likely to be Democrats, and they also skew younger than users of YouTube, Facebook, or Instagram \\citep{perrin2019share}.\nThe gender and racial distribution on Twitter are close to uniform; out of a set of U.S. survey respondents, 24\\% of men and 21\\% women report using Twitter, while 24\\% of Black, 25\\% of Hispanic, and  21\\% of White respondents report using Twitter \\citep{perrin2019share}.\nOn Twitter, birth control discussions take place in the context of many other discussion topics, and while users can group their conversations using hashtags, there are not separated communities like subreddits.\nModeration is organized by Twitter, rather by users as on Reddit.\n\n\nUsing the Twitter Academic API (v2), we collected all the English Twitter posts containing a set of keywords corresponding to our three target birth control methods.\nSeparately, we collected all the Twitter replies containing the same set of keywords.\nRecent work has shown that this API returns reliable representations of the full tweet space \\citep{Pfeffer2022ThisSS}.\nSee \\S Methods below for the design of the Twitter-specific keywords.\nWe removed Twitter handles from tweet texts.\nThe high vocabulary size for Twitter posts (Table \\ref{table:data}) partly reflects many unique URLs shared in these documents, which can be indicative of information-providing behavior.\nThere are many possible design decisions when gathering Twitter data, and we chose not to collect the full conversations (replies and parent posts of the keyword-containing posts and replies) to (a) limit the size of our collection, (b) target texts explicitly discussing birth control, and (c) more closely replicate prior work on birth control tweets \\citep{merz2020population}.\n\n\n\\section{Broader Impacts, Ethics, and Limitations}\n\\label{section:ethics}\n\n\nOur study was considered exempt under our institution's IRB.\nHowever, while Reddit, Twitter, and WebMD posts and replies are ``public,'' they can contain highly personal information,\nrequiring a balance between potential harms and potential benefits to the community, as described in the guiding principles of the Belmont Report: \\textit{respect for persons}, \\textit{beneficence}, and \\textit{justice}.\\footnote{\\url{https:\/\/www.hhs.gov\/ohrp\/regulations-and-policy}}\nConsidering possible harms, e.g., re-identification of those using stigmatized medications, we do not collect any user-specific information, and we do not infer medical conditions for individual users; instead, we rely on patterns averaged across many users.\nWe also anonymize and paraphrase any direct quotations.\nTo protect users' agency to edit or delete their data at its original source, we release our data collection lexicons but not copies of the collected data.\nWe balance these concerns and protective actions against the benefits of this research; among other benefits, our work highlights the unmet pain treatment needs of a population known to be discriminated against by physicians \\citep{Samulowitz2018BraveMA} and examines the kinds of support needed by those facing difficult healthcare choices.\n\\section{Results}\n\n\n\n\n\\subsection{Results: Birth Control Methods Across Platforms}\n\n\nWe compare the rates of discussion of the different birth control methods across platforms and over time.\nWe find that the types of questions that users ask and the type of information they share differs based on birth control method.\nWe measure the rise and fall of discussions of different birth control methods over time, and we compare these distributions across platforms.\nImportantly, these patterns do not necessarily indicate real-world increases or decreases in use of different methods, but they do show the concerns and interests of users on these platforms. \n\n\n\n\nFigure \\ref{figure:bc-type-distributions} shows the frequency of documents for each medication method by platform over time. \nThe pill is the most popular method in both \\textit{Reddit Posts} and \\textit{WebMD Reviews}, while the pill and IUD are tied in \\textit{Reddit Comments}.\nDiscussions of the pill on WebMD are much more frequent that discussions of other birth control methods and do not show a rise in interest for the IUD, counter to both the other datasets and our original hypothesis.\nOn Twitter, the pill begins as the most popular but is replaced by the IUD in posts, while replies always center on the IUD.\nTwitter replies also increase sharply after 2016, differing from the Twitter posts (so is unlikely to be a symptom of our keywords).\nThis suggests that the IUD generates more discussion on Twitter, especially post-2016, compared to the other birth control methods.\nOn Reddit, the number of posts discussing the pill and IUD are similar, with slightly more posts discussing the pill, though this difference is erased in the comments.\nThe implant is always the least discussed of the three methods.\n\n\n\n\\subsection{Results: Side Effects Across Platforms}\n\n\nFigure \\ref{figure:bc-type-distributions} shows the distributions of the side effects across the different platforms, where frequency proportions are calculated by dividing the number of texts mentioning the specified side effect by the total number of texts mentioning any side effect (i.e., $p(s_i|a)$ where $s_i$ is the specific side effect and $a$ is any side effect).\nDiscussions of different methods have changed dramatically over time on Twitter, with discussions of the IUD rising and the pill falling.\nThe pill remains most most mentioned in Reddit posts.\n\n\nFigure \\ref{figure:barplot-side-effects-distribution} shows the distribution of side effects across platforms.\nWe find that pain, cramps, menstrual bleeding irregularities, mood changes, and skin conditions are the most commonly discussed side effects.\nIn particular, \\textit{pain is consistently and frequently discussed across the platforms} and is an outlier among the side effects.\nDiscussions of stroke are more frequent in Twitter posts than in the other datasets, but for all other side effects, Twitter has the lowest discussion frequencies (perhaps because of stigma around publicly sharing such information compared to the more frequently pseudonymous settings on Reddit and WebMD).\nExcept for pain, menstrual bleeding, and severe effects, WebMD has the highest frequency of side effects discussions compared to the other platforms.\nIn particular, WebMD has higher frequencies for discussions of mood changes, skin conditions, and headache discussions.\nReddit has has the highest frequency of discussions for menstrual bleeding irregularities.\n\n\n\n\n\nFigure \\ref{figure:barplot-side-effects-distribution} also compares the side effects by birth control method.\nAcross the birth control methods, we again find that pain is a frequently mentioned side effect, but it is most frequently mentioned in discussions that also mention the IUD rather than the pill or implant.\nThe implant is the only method shown to cause weight gain \\citep{barr2010managing}, so this association is expected---but the discussions of weight gain for the other methods are less expected and could indicate that this potential side effect is a concern across methods.\nMenstrual bleeding, mood changes, headache, and libido are least often discussed in with the IUD, as are general mentions of side effects,  while nausea, stroke, and skin conditions are most often discussed in discussions of the pill.\n\n\n\n\n\n\n\\subsection{Results: Comparison to Observed Distributions}\n\n\nFigure \\ref{figure:barplot-side-effects-distribution-comparison} shows the differences between our observed distributions of side effect mentions and the reported distributions of side effect experiences in \\citet{nelson2018women} (described above).\nWe find large differences between how frequently side effects are discussed online compared with how frequently they are reported in the survey from \\citet{nelson2018women}.\nFor example, strokes are rarely experienced according to \\citet{nelson2018women}, but when mentioned with the pill, they are discussed more frequently on Twitter in comparison to other side effects; this is unsurprising, since sensational topics are frequently discussed on Twitter.\nMood changes are more likely to be discussed across the platforms for the implant in comparison to the survey data.\nDizziness is more likely to be discussed for the IUD than expected, while weight gain is universally discussed less frequently than expected.\nThese patterns indicate cases in which users turn to the internet more or less frequently than expected based on their self-reported experiences in the survey.\n\n\n\n\n\\subsection{Results: Sensemaking Across Platforms}\n\n\nOur final set of sensemaking themes included: \\textbf{storytelling} (e.g., \\textit{implant insertion and removal} on Twitter), \\textbf{risk analysis} (e.g., \\textit{risks of serious side effects} on WebMD), \\textbf{timing and calculations} (e.g., \\textit{daily timing of pill} on WebMD), \\textbf{method and dose comparison} (e.g., \\textit{hormone dosages and comparison} on Reddit), \\textbf{causal reasoning} (e.g., \\textit{weight changes and causes} on Reddit), and \\textbf{information and explanations} (e.g., \\textit{research on the male pill} on Twitter).\nWe show four of these themes (others omitted for space) in Figure \\ref{figure:heatmaps-topics-methods}, with the topics from each platform included in that theme and their probabilities for each birth control method.\nWe observe a greater number of storytelling topics for the IUD and implant than for the pill, and examination of these topics shows that insertion and removal experiences fuel this pattern.\nEach method is associated with risks and is included in method comparisons, but the pill in particular is included in discussions of timing and calculations.\n\n\n\nUsing Jensen-Shannon divergence to compare topics across the platforms, we find that the most \\textbf{aligned} ($JSD < 0.6$) topic categories were: \\textit{weight changes}, \\textit{general side effects}, \\textit{IUD insertion experiences}, \\textit{implant insertion experiences}, \\textit{menstrual timing and cycles}, and \\textit{bleeding changes}.\nEach of our three datasets included at least one representative topic from these categories.\n\n\nThe most \\textbf{diverging} ($JSD > 0.8$) topics were: \\textit{causes and side effects of vaginal infections} and \\textit{explanations of how birth control works} on Reddit; \\textit{IUD jokes and random}, \\textit{viral folk stories}, and \\textit{implant news about unexpected experiences} on Twitter; and \\textit{treating or causing other medical issues} and \\textit{secondhand advice and experiences} on WebMD.\nThese topics were the most unique to their training dataset, without directly comparable topics in the other datasets.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTo control the occuracy of perturbative QCD predictions for physical\nobservables it becomes important to have a regular way of \nestimating corrections suppressed by powers of large momentum scale,\n$\\Lambda_p\/Q^p$, and if not calculate explicitly the nonperturbative \nscale $\\Lambda_p$ then find the leading exponent $p$ at least. \nAt present, the understanding of power corrections has a solid\ntheoretical status only for the special class of processes like\n$\\mbox{e}^+ \\mbox{e}^-\\to\\mbox{hadrons}$ and deeply inelastic\nscattering (DIS) for which the analysis based on the operator product\nexpansion (OPE) is applicable. The OPE fixes the structure of power\ncorrections (as $1\/Q^4$ for the total cross-section of $\\mbox{e}^+\\mbox{e}^-$ \nannihilation and $1\/Q^2$ for the structure functions of DIS) and \nallows to identify the corresponding scales $\\Lambda_p$ as matrix elements \nof higher twist composite operators in QCD. The understanding of power \ncorrections in the processes which do not admit the OPE is the subject \nof intensive discussions (see review~\\ci{B} and references therein). \n\nAn important example of the process, which does not admit the OPE is the \nDrell-Yan (DY) process $h_A+h_B\\to \\mu^+\\mu^- +X$. Here, the lepton pair\nwith invariant mass $Q^2$ is created in the partonic subprocess\n$q + \\bar q\\to\\mu^+\\mu^-(Q^2)+X$ of annihilation of two quarks with \ninvariant energy $\\hat s$ and we choose the kinematics such that \n$\\tau=Q^2\/\\hat s \\sim 1$. One of the reasons for this is that as $\\tau\\to\n1$, the final state in the partonic subprocess consists of the lepton\npair with the energy $Q$ accompanied by a soft gluon radiation with the \ntotal energy $(1-\\tau)Q\/2$. Due to the presence of two different scales\nthe partonic cross-section gets large perturbative (Sudakov logs in \n$\\mbox{ln}(1-\\tau)$) and nonperturbative (as inverse powers of $(1-\\tau)Q$) \ncorrections induced by soft gluons \\ci{CS,KS}. \nCalculating the moments $\\sigma_N=\\int_0^1 d\\tau \\tau^N d\\sigma_{{\\rm _{DY}}}\n\/d Q^2$ at large $N\\sim\\frac1{1-\\tau}$ and subtracting collinear \ndivergences in the DIS-scheme one can represent the leading term \nof the expansion of $\\sigma_N$ in powers of $1\/Q$ as \n\\begin{equation}\n\\mbox{ln}\\,\\sigma_N=\\sum_k \\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^k(Q\/N) C_k(\\mbox{ln} N) + \\lr{\\Lambda_{{\\rm _{DY}}} N\/Q}^p,\n\\lab{moment}\n\\end{equation}\nwhere the coupling constant is defined at the characteristic soft \ngluon scale, $C_k$ contains the powers of Sudakov logs plus finite, \n$\\mbox{ln}^0 N$, contributions and the second term represents the leading\npower correction. As $N$ increases, the energy of soft gluons in the\nfinal state decreases as $Q\/N$ toward the infrared region in which\nperturbative expressions should fail. Indeed, it is well-known\nthat the perturbative expansion in \\re{moment} is not well-defined due to\nfactorial growth of coefficients $C_k \\sim \\beta_0^k k! p^{-k}$\nat higher orders in $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi$. This makes the perturbative series non Borel \nsummable and induces an IR renormalon ambiguity into perturbative \ncontribution to \\re{moment} at the level of power corrections,\n$\\delta_{{\\rm _{IR}}} \\sigma_N\\sim \\exp(-\\frac{p}{\\beta_0\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi(Q\/N)})\n=(\\Lambda_{\\rm QCD} N\/Q)^p$. Then, IR renormalon ambiguity of \nperturbative expression is compensated in \\re{moment} by\nambiguity in the definition of the scale $\\Lambda_{{\\rm _{DY}}}$.\nThus, examing the large order behaviour of the perturbative series \none can determine the level $p$ at which the leading power correction\nappear in \\re{moment}. However, since it is impossible to calculate \nthe coefficients $C_k$ exactly to higher orders in $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi$ one has to \nfind a reasonable approximation to $C_k$ which would allow to identify \nthe $k!$ factorial growth of the coefficients. To this end different \nschemes were proposed~\\ci{BB,DMW,AZ}. Being applied to the Drell-Yan \ncross-section they predict that there are no $N\/Q$ corrections to \n\\re{moment} and the leading renormalon contribution has the following \nform\n\\begin{equation}\n\\delta_{{\\rm _{IR}}}\\sigma_N = 0\\cdot (N\/Q) + \\lr{A_{{\\rm _{DY}}} N\/Q}^2\\,.\n\\lab{zero}\n\\end{equation}\nWe would like to stress that these schemes are {\\it approximate\\\/}\nand although they were tested in the case of $\\mbox{e}^+\\mbox{e}^-$ annihilation \nand DIS, it is not clear yet whether they are applicable in the \nDrell-Yan process and whether they really predict the {\\it leading\\\/}\npower correction to the cross-section. \n\n\\section{Wilson line approach}\n\nA different approach to the analysis of power corrections has been proposed \nin Ref.~\\ci{KS}. It is based on the remarkable property of soft gluons that\ntheir contribution to the Drell-Yan cross-section can be factorized with\na power occuracy into universal factor having the form of a Wilson line.\nSimilar to the OPE approach, the power corrections can be identified\nwith the matrix elements of certain operators built from the Wilson line. \n\nThe Wilson line operator appears in the Drell-Yan partonic subprocess as an\neikonal phase of quarks interacting with soft gluon radiation. Combining the\neikonal phases of quark and antiquark together one gets the expression for\npartonic cross-section as\n$$ \n\\frac{d\\sigma_{{\\rm _{DY}}}}{d Q^2}\n=\\sum_n |\\langle n | T U_{{\\rm _{DY}}}(0) |0\\rangle|^2\n\\delta(E_n-(1-\\tau)Q\/2)\n$$\nwhere summation is performed over all possible final states consisting\nof an arbitrary number, $n$, of soft gluons with the total energy\n$E_n=(1-\\tau)Q\/2$ and $T$ stands for time-ordering of gauge fields. \nHere, $U_{{\\rm _{DY}}}(0)=P\\exp(i\\int_{C(0)} dx \\cdot A(x))$ is the eikonal \nphase of quark + antiquark and\n$A_\\mu(x)$ is the gauge field operator describing soft gluons.  \nThe contour $C(0)$ in Minkowski space-time corresponds to the classical\ntrajectory of quark and antiquark. It goes from $-\\infty$ to the\nannihilation point $0$ along the quark momentum and then returns \nto $-\\infty$ in the direction opposite to the antiquark momentum.\nPerforming summation over final states in the last relation one can \nobtain the representation for the Drell-Yan cross section for $\\tau\\to\n1$ as a Fourier transformed Wilson line expectation value~\\ci{KM}\n\\begin{eqnarray}\n\\frac{d\\sigma_{{\\rm _{DY}}}}{d Q^2}\n&=&\n\\frac{Q}2\\int_{-\\infty}^\\infty\\frac{dy_0}{2\\pi}\\mbox{e}^{-iy_0(1-\\tau)Q\/2}\n\\langle 0| W(y_0) |0 \\rangle\n\\nonumber\n\\\\\nW(y_0)&\\equiv&\n\\widebar T U^\\dagger_{{\\rm _{DY}}}(0)\\, T U_{{\\rm _{DY}}}(y_0)\\,,\n\\lab{W}\n\\end{eqnarray}\nwhere $U_{{\\rm _{DY}}}(y_0)$ is defined at the point $y=(y_0,\\vec 0)$. \nCalculating the moments of this expression one obtains the following \nrelation \\footnote{This identity is an analog of the relation between \nthe moments of the structure functions of DIS and matrix elements of \ncomposite operators in the OPE.}\n\\begin{equation}\n\\sigma_N= \n\\frac1{N_c} \\mbox{tr\\,}\n\\langle 0 | W(y_0) |0 \\rangle\\,,\\quad\n\\mbox{for $y_0=iN\/Q$}\\,,\n\\lab{rel}\n\\end{equation}\nwhich is valid up to corrections vanishing as $N\\to \\infty$ and which \ntakes into account all Sudakov logs and power corrections in $N\/Q$. \nThe scale $Q\/N$ enters into \\re{rel} just as a parameter of \nthe integration contour in $U_{{\\rm _{DY}}}(y_0)$. Eq.\\re{rel}\nstates that the asymptotic expansion of\nthe moments of the Drell-Yan cross section is related to the behaviour\nof the Wilson line expectation value at small $y_0=\\frac{iN}{Q}$. \n\nPerturbative expansion of \\re{rel} involves standard ``short-distance'' \nlogarithms $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^n\\mbox{ln}^m(|y_0|\\mu)$, which are transformed into Sudakov logs\nfor $y_0=iN\/Q$. The factorization scale $\\mu\\approx Q$ has a meaning of \nthe maximal energy of soft gluons contributing to the partonic\ncross-section and after factorization of soft gluon emission it\nappears in \\re{W} and \\re{rel} as a {\\it ultraviolet\\\/} renormalization \nscale of the Wilson line operators. As a result, all perturbative Sudakov\ncorrections to the Drell-Yan cross-section can be effectively resummed \nusing the remarkable renormalization properties of Wilson lines. They\nare summarized by the following RG equation\n\\begin{equation}\n\\frac{d\\mbox{ln} \\sigma_N}{d\\mbox{ln}\\mu^2}= 2 \\Gamma_{\\rm cusp}(\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi) \\mbox{ln}(N\\mu\/Q)\n+ \\Gamma_{{\\rm _{DY}}}(\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi)\\,, \n\\lab{RG}\n\\end{equation}\nwith $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi=\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi(\\mu)$ and $\\Gamma_{\\rm cusp\/DY}$ being some anomalous \ndimensions. The evolution of $\\sigma_N$ from $\\mu=Q\/N$ up to $\\mu=Q$ \ngenerates Sudakov logs while the initial condition for $\\sigma_N$\nat $\\mu=N\/Q$ takes into account nonperturbative power corrections in\n$N\/Q$. To understand their structure we perform the operator expansion \nof the Wilson line $W(y_0)$ in powers of $y_0$\n\\begin{equation}\n\\vev{W(y_0)}=\\vev{W(0)} + y_0 \\vev{\\partial W(0)} + ... \\ \\,.\n\\lab{ope}\n\\end{equation}\nSubstituting this expansion into \\re{rel} we find that the power \ncorrections to the cross-section are described by matrix elements \nof the operators defined as derivatives of the Wilson line~\\ci{KS}. \nAlthough these operators are nonlocal in QCD, one can represent them as \nlocal operators in an effective nonabelian Bloch-Nordsieck theory in \nwhich Wilson line appears as a result of integration over effective\nquark fields. Solving the RG \nequation \\re{RG} and using \\re{ope} as a boundary condition we obtain \nthat the leading power correction to the Drell-Yan cross-section appears\nas~\\ci{KS}\n\\begin{equation}\n\\sigma_N^{\\rm power} = \\vev{i\\partial W(0)}N\/Q + {\\cal O}(N^2\/Q^2)\\,,\n\\lab{fin}\n\\end{equation}\nprovided that there is no any hidden symmetry which would prohibit the \nappearence of the linear $\\sim y_0$ term in the expansion of the Wilson \nline \\re{ope} and would enforce the matrix element\n$\\vev{i\\partial W(0)}$ to vanish. \n\n\\section{IR renormalon analysis}\n\nOne possibility to test the presence of the linear term in \\re{ope} is to\napply the IR renormalon approach and identify the source of linear\nterms inside perturbative series for the Wilson line. At the leading\norder of renormalon calculus~\\ci{B}, one can perform the explicit one-loop \ncalculation of $\\vev{W(y_0)}$ with a gluon mass $\\lambda$ and identify \nthe power corrections as nonanalytical terms in the small $(y_0\\,\\lambda)$ \nexpansion. One finds~\\ci{BB} that $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi \\mbox{ln} (y_0\\lambda)$ term is absent in \nagreement with the Bloch-Nordsieck cancellation, the linear term \n$\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi (\\lambda y_0)$ also vanishes and the leading term is \n$\\sim\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi (\\lambda y_0)^2$. This means, that at the leading order the\ncontribution of IR renormalons has the form \\re{zero}.\nThe same result can be obtained in the \nlimit of large number of light quark flavours in which one performs 1-loop \ncalculation of $\\vev{W(y_0)}$ with the gluon propagator dressed by a \nchain of quark loops~\\ci{BB}. We notice that in both cases one calculates \nessentially abelian diagrams containing the color factors $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi C_F$ and \n$\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi C_F (\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi N_f)^{n}$, respectively. It is well-known that the \ncontribution of abelian diagrams to the Wilson line exponentiates to \nhigher orders in $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi$ and one can generalize the above statement \nabout the absence of linear $\\sim y_0$ term to a much larger class of \nabelian diagrams beyond the leading order of the renormalon \ncalculus. However, the natural question rises whether the abelian\ndiagrams provide a meaningful approximation to the exact expression \nfor the Wilson line or may be there is the leading contribution\nto $W(y_0)$ coming only from nonabelian diagrams. To give an example\nwhere the second possibility is realized we mention the breakdown of\nthe Bloch-Nordsieck cancellation of IR logs in the Drell-Yan\ncross-section at twist $1\/Q^4$. The IR divergent part of \nthe cross-section has the form~\\ci{IR} \n$\\sim\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^2 C_A C_F Q^{-4}\\mbox{ln} \\lambda^2$ with $C_A (C_F)$ \nbeing gluon (quark) Casimir operators. It does not appear at order \n$\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi$ and at order $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^2$ it comes only from diagrams with nonabelian \ncolor structure. One might expect that similar phenomenon may happen when \none considers linear in $y_0$ contributions to the Wilson line beyond \nthe leading order. \n\nThe straighforward way to check this possibility would be to perform \n2-loop calculation of the Wilson line with a power occuracy in\n$\\lambda$. However, even without going through this complicated\ncalculation one can employ the operator methods of the QCD asymptotic \ndynamics~\\ci{CCM} in order to analyse the properties of the matrix element\n\\begin{equation}\n\\vev{\\partial W(0)}=\\frac1{N_c}\\mbox{tr\\,}\n\\vev{\n\\widebar T U_{{\\rm _{DY}}}^\\dagger(0|A)\\,\\partial\\, T U_{{\\rm _{DY}}}(0|A)\n}\\,,\n\\lab{dW}\n\\end{equation}\nparameterizing the leading $N\/Q$ power correction to the Drell-Yan\ncross-section. The operator $U_{{\\rm _{DY}}}(y_0|A)$ describes the eikonal\nphase of fast quark and antiquark annihilating at the space-time \npoint $y_0$ and $T-$ordering of gauge field operators $A_\\mu^a(x)$\nis needed in \\re{W} and \\re{dW} to describe properly the possibility \nfor quark and antiquark to exchange by a virtual soft gluon in the\ninitial state. As a result, due to effects of the initial state \ninteraction~\\ci{CCM}\n$$\nTU_{{\\rm _{DY}}}(y_0|A)\\neq U_{{\\rm _{DY}}}(y_0|A)\\,.\n$$\nThis does not happen however for single-parton initiated process like \nDIS and it is this phenomenon that makes different the properties of \npower corrections in DY and DIS. Indeed, if we omit the $T-$ordering \nin \\re{dW} then it is easy to show by direct calculation that the derivative\n$U_{{\\rm _{DY}}}^\\dagger\\partial U_{{\\rm _{DY}}}$ is proportional to the \ngenerators of the $SU(N_c)$ group and it does not contribute to \n\\re{dW}. To work out the effects of quark-antiquark correlations in\n$\\vev{W(y_0)}$ one defines two gauge field operators\n$A_\\mu^\\pm(x)=\\frac12(A_\\mu(x)\\pm A_\\mu(-x))$ and notices~\\ci{CCM}\nthat the field $A_\\mu^+$ describes the Coulomb gluons while the field \n$A_\\mu^-$ is associated with radiative gluons forming quark and\nantiquark coherent states. Since $[A^{a,\\pm}_\\mu(x),A^{b,\\pm}_\\nu(y)]=0$ and\n$[A^{a,\\pm}_\\mu(x),A^{b,\\mp}_\\nu(y)]\\neq 0$, the expansion of the matrix\nelement $\\vev{n|TU_{{\\rm _{DY}}}(y_0|A^++A^-)|0}$ takes the following form~\\ci{CCM}\n\\begin{eqnarray}\n&&\\hspace*{-5mm}\n\\vev{n|TU_{{\\rm _{DY}}}^{ij}\n(y_0|A^++A^-)|0}\n=[\\mbox{e}^{i\\Phi_C}]^{ij}_{i'j'}\n\\lab{V}\n\\\\\n&&\\hspace*{5mm}\n\\times\n\\vev{n|\nU_{{\\rm _{DY}}}^{i''j''}(y_0|A^-)\nV_{i''j''}^{i'j'}(y_0|A^+,A^-)\n|0} \\,,\n\\nonumber\n\\end{eqnarray}\nwhere the flow of quark color is indicated explicitly.\nHere, in the r.h.s.\\ the first factor is the nonabelian unitary\nCoulomb phase matrix and the eikonal phase $U_{{\\rm _{DY}}}$ depends\nonly on commutative operators $A^-$. The residual\nfactor $V$ is unitary, $V^\\dagger V=\\hbox{{1}\\kern-.25em\\hbox{l}}$, and it\ndescribes the correlations between Coulomb and radiative\ngluons. The perturbative expansion of $V$ looks like~\\ci{CCM}\n$$\nV=\\hbox{{1}\\kern-.25em\\hbox{l}}+ig^3 f^{abc}t^a\\otimes t^b\\!\\int\\!\\! d^4x\\, \nG(x,y_0) A_\\mu^{c,+}(x)+{\\cal O}(g^4)\n$$\nwhere effective nonlocal coupling $G(x,y_0)$ resembles the \nFadin-Kuraev-Lipatov vertex. In abelian theory $V=\\hbox{{1}\\kern-.25em\\hbox{l}}$ to all orders, \nbut in QCD it gets {\\it nonabelian} corrections starting from order \n$g^3$ in the strong coupling constant. \n\nSubstituting \\re{V} into \\re{dW} one observes that \nnonabelian Coulomb phase is canceled. Moreover, if we neglect \n$g^3-$corrections to $V$, then due to unitarity and commutativity of \n$U_{{\\rm _{DY}}}(y_0|A^-)$, the operator $U_{{\\rm _{DY}}}^\\dagger\\partial U_{{\\rm _{DY}}}$\nis again proportional to the $SU(N_c)$ generators and its matrix element\nvanishes identically in Eq.\\re{dW}. Thus, a nonzero contribution to \n\\re{dW} comes only from higher order terms in $V$. It immediately \nfollows from the explicit form of $V$ that the corresponding Feynman \ndiagrams should have a nonabelian color structure and they arise \nstarting from $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^2$ order. The set of relevant 2-loop Feynman \ndiagrams was identified in Ref.~\\ci{CCM}. This also explains why \nlinear term did not appear in 1-loop calculation of the Wilson line \nperformed in Ref.~\\ci{BB}.\n\nThe fact that the contribution of the residue factor $V$ to the Wilson \nline $W(y_0)$ is nonzero was demonstrated by a direct calculation \nin Ref.~\\ci{CCM}, in which a small gluon mass $\\lambda$ was used as an \nIR cutoff and only logarithmic terms, $\\sim\\mbox{ln}^k\\lambda$, were kept. \nIt was shown that the leading term $W(0)$ of the expansion of the\nWilson line in Eq.\\re{ope} gets a higher twist correction \n$\\sim\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^2 C_FC_A Q^{-4}\\mbox{ln}\\lambda$, which leads to the breakdown of \nBloch-Nordsieck theorem for the Drell-Yan cross-section but is \nnevertheless consistent with the Kinoshita-Lee-Nauenberg (KLN) theorem. \nTo find the linear term in \\re{ope} one has to perform the\ncalculation of the same diagrams with a power \naccuracy in $\\lambda$, although interpretation of the linear term\n$\\sim\\lambda$ to order $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^2$ as a ``true'' power correction is not \nobvious. One may dress instead the gluon propagators by a single \nchain of quark loops, resum all 2-loop diagrams of order \n$\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^2 C_FC_A(\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi N_f)^k$ in the large $N_f$ limit and identify the \ncontribution of the leading IR renormalon.  \n\n\\section{Conclusions}\n\nWe conclude that the effects of the initial state interaction, that is\ncorrelations between Coulomb and radiative gluons, provide a new source \nof $N\/Q$ power corrections in the Drell-Yan process. In the Wilson\nline approach they are described by the matrix element of the  \nderivative of the Wilson line operator in Eq.\\re{fin}. In the IR\nrenormalon analysis these effects could manifest themselves only  \nin nonabelian Feynman diagrams at higher orders in $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi$.\nThe presence of $N\/Q$ corrections in the Drell-Yan cross-section is\nconsistent with the KLN theorem. Summation over degenerate initial \npartonic states in \\re{W} and \\re{rel} \nleads to the expression for the KLN cross-section, $\\sigma_N^{\\rm _{KLN}}=\n\\sum_k \\vev{k|W(iN\/Q)|k}$, where sum goes over an arbitrary number of\nincoming soft gluons. As was shown in Ref.~\\ci{CCM}, the initial \nstate interaction effects can be analysed in $\\sigma_N^{\\rm _{KLN}}$ on \nthe same footing as the final state effects thus resulting in the \ncancellation of $N\/Q$ corrections in $\\sigma_N^{\\rm _{KLN}}$.\n\n\\section*{Acknowledgements}\n\nThe author is most grateful to George Sterman for useful discussions.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{section.01.introduction}\n{\\it \nWe plan to start from the refined topological vertex and obtain Virasoro $A$-series minimal model \nconformal blocks times Heisenberg factors.\n}\n\n\\medskip\n\n\\subsection{Background and basic concepts}\n\n\\subsubsection{Topological strings} Perturbative string theory, seen from a world-sheet point of view, \nis a 2D conformal field theory, coupled to 2D gravity, on Riemann surfaces. If the 2D conformal field \ntheory is topological, in the sense that the correlation functions are independent of the metrics on \nthe Riemann surfaces, the resulting string theory is {\\it topological}. For an introduction, we refer \nto \\cite{vafa.neitzke}, and references therein.\nThere are two constructions of topological string theories, the A-model and the B-model. For the \npurposes of this note, it suffices to say that we work are in the context of the A-model, and that \nthe target space, that the strings propagate in, is $\\mathbb{R}^{\\, 1, 3} \\times \\mathcal{X}$, where $\\mathcal{X}$ \nis a toric Calabi-Yau complex 3-manifold. For suitable choices of $\\mathcal{X}$, topological strings are \nnon-trivial but tractable, and one can compute their partition functions. For further details, we \nrefer to the reviews \\cite{marino.2005, marino.book}, and references therein.\n\n\\subsubsection{From topological strings to 5D gauge theories} Topological string partition functions \nare interesting in themselves, and for yet another, an {\\it apriori} unexpected reason. Namely, for \nsuitable choices of the Calabi-Yau manifold $\\mathcal{X}$, one can compute the topological string partition \nfunction $\\mathcal{Z}^{\\, top}$, and moreover, identify the result with the instanton partition function \n$\\mathcal{Z}^{\\, 5D}_{instanton}$ of a corresponding 5D $\\mathcal{N} \\! = \\! 2$ supersymmetric quiver gauge theory,\nthereby {\\it \\lq geometrically-engineering\\rq} the latter theory \n\\cite{geometric.engineering.01, geometric.engineering.02}. \n\n\\subsubsection{Refined partition functions} A 5D instanton partition function depends, in general, \non two deformation parameters, $\\epsilon_1$ and $\\epsilon_2$. \nFor $\\epsilon_1 + \\epsilon_2 = 0$, the instanton partition function is unrefined. \nFor $\\epsilon_1 + \\epsilon_2 \\neq 0$, the instanton partition function is refined. \nGiven the identification of 5D \ninstanton partition function and topological strings, the latter also depend, in general on \nthe deformation parameters,  $\\epsilon_1$ and $\\epsilon_2$.\nFor $\\epsilon_1 + \\epsilon_2 = 0$, the topological string partition function is unrefined.\nFor $\\epsilon_1 + \\epsilon_2 \\neq 0$, the instanton partition function is refined.  \n\n\\subsubsection{The refined topological vertex} To compute topological string partition functions,\none splits the full partition function into basic building blocks, computes the contribution of each \nbuilding block, then combines the contributions to obtain the required result.\nIn the unrefined case, $\\epsilon_1 + \\epsilon_2 = 0$, this was achieved in \\cite{topological.vertex}. \nIn this case, the building blocks are copies of the original, unrefined {\\it \\lq topological vertex\\rq} \nintroduced in \\cite{topological.vertex}.\nIn the   refined case, $\\epsilon_1 + \\epsilon_2 \\neq 0$, this was achieved in \n\\cite{awata.kanno.01, awata.kanno.02, ikv}\nIn this case, the building blocks are copies of the {\\it \\lq refined topological vertex\\rq} \n\\cite{awata.kanno.01, awata.kanno.02, ikv}. \n\n\\subsubsection{From 5D to 4D gauge theories} \nA 5D quiver gauge theory instanton partition function, $\\mathcal{Z}^{\\, 5D}_{instanton}$, depends on the radius \n$R$ of a space-like circle. In the limit $R \\rightarrow 0$, $\\mathcal{Z}^{\\, 5D}_{instanton}$ reduces to \na corresponding 4D instanton partition function $\\mathcal{Z}^{\\, 4D}_{instanton}$ \n\\cite{nekrasov, hollowood.iqbal.vafa, ikv, eguchi.kanno, zhou, taki.01}.\nThe deformation parameters $\\epsilon_1$ and $\\epsilon_2$ of the 5D theory are inherited by the 4D theory.\n\n\\subsubsection{From 4D gauge theories to generic 2D conformal field theories} The 4D instanton partition \nfunctions $\\mathcal{Z}^{\\, 4D}_{instanton}$ are identified {\\it via} the AGT correspondence with 2D Virasoro \ngeneric conformal blocks times Heisenberg factors, $\\mathcal{B}^{\\, gen, \\, \\mathcal{H}}$ \\cite{agt}. In its original \nformulation, the AGT correspondence applies only to generic, that is non-minimal conformal field \ntheories. From 4D instanton partition functions with $\\epsilon_1 + \\epsilon_2 = 0$, one obtains\nconformal blocks of the conformal field theory of a Gaussian free field at the free fermion point.\nFrom 4D instanton partition functions with $\\epsilon_1 + \\epsilon_2 \\neq 0$, one obtains\nconformal blocks of non-Gaussian, but also non-minimal conformal field theories.\n\n\\subsubsection{From generic conformal field theories to minimal models} One can choose the parameters \nthat appear in $\\mathcal{Z}^{\\, 4D}_{instanton}$, and restrict the states that are allowed as intermediate states, \nin such a way that one obtains restricted 4D instanton partition functions $\\mathcal{Z}^{\\, 4D, \\, min}_{instanton}$ \nthat can be identified with conformal blocks in Virasoro $A$-series minimal models times Heisenberg factors \n\\cite{alkalaev.belavin, bershtein.foda}. In particular, as we will show in the sequel, choosing \n$\\epsilon_1 = - \\sqrt{   p \/ p^{\\prime}}$, and \n$\\epsilon_2 =   \\sqrt{ p^{\\prime} \/   p}$, \nwhere $p$ and $p^{\\prime}$ are two co-prime positive integers, $0 < p < p^{\\prime}$, one obtains conformal blocks in \na minimal model parameterised by $p$ and $p^{\\prime}$. \n\n\\subsection{Plan of this work}\nFrom the above chain of connections, it is expected that one can start from $\\mathcal{Z}^{\\, ref\\, top}$, \nchoose the parameters and restrict the intermediate states to obtain 5D instanton partition functions \n$\\mathcal{Z}^{\\, 5D, \\, min}_{instanton}$, reduce to the corresponding $\\mathcal{Z}^{\\, 4D, \\, min}_{instanton}$, and \ncompute minimal model conformal blocks from the latter. \nIn this note, we work out the above chain of connections, which amounts to extending the result \nof \\cite{alkalaev.belavin, bershtein.foda} by starting from topological strings and topological \nvertices rather than from 4D instanton partition functions. We glue four refined topological \nvertices to obtain the building block of the 5D instanton partition functions, and take the 4D \nlimit of the latter to obtain the building block of the 4D instanton partition functions\n$\\mathcal{Z}^{\\, 4D}_{building.block}$. From that point on, we use the results of \\cite{alkalaev.belavin, \nbershtein.foda} to obtain conformal blocks in minimal models.\n\nSince $\\mathcal{Z}^{\\, 4D}_{building.block}$, which can be regarded as the starting point of the results \nin \\cite{alkalaev.belavin, bershtein.foda}, is constructed here from refined topological vertices, \nthe construction in this note is, in this sense, more basic than that in \\cite{alkalaev.belavin, \nbershtein.foda}. \n\nThe Virasoro $A$-series minimal conformal blocks that we obtain can be computed using other methods, \nbut we view this note as an accessible introduction to one more approach to the minimal conformal \nblocks that, hopefully, can be extended to minimal blocks beyond what can currently be computed, \nincluding the $W_N$ minimal blocks that do not satisfy the conditions of \\cite{fateev.litvinov, wyllard}\n\\footnote{\\,\nExpressions for the 3-point functions of $W_N$ Toda theory were proposed, starting from topological \nstrings, in \\cite{mitev.01}, and checked in \\cite{mitev.02}. These results should be useful in \nsubsequent studies of $W_N$ minimal models.\n}. \n\n\\subsection{Outline of contents}\nTo simplify the presentation, we divide the content into short sections.\nIn {\\bf \\ref{section.02.refined.topological.vertex}}, we recall basic definitions related to Young \ndiagrams and Schur functions, followed by\nthe definition of the refined topological vertex $\\mathcal{C}_{\\, \\lambda \\, \\mu \\, \\nu}[q, t]$ of Iqbal, \nKozcaz and Vafa \\cite{ikv}, where $\\lambda$, $\\mu$, and $\\nu$ are Young diagrams, $q$ and $t$ are \nparameters.\nIn {\\bf \\ref{section.03.5d.web}}, we glue four copies of $\\mathcal{C}_{\\, \\lambda \\, \\mu \\, \\nu} [q, t]$ \nto build a $U(2)$ basic web partition function $\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta} [q, t, R]$, where \neach of ${\\bf V}$ and ${\\bf W}$ is a pair of Young diagrams, and \n${\\bf \\Delta} = \\{\\Delta_1, \\Delta_{ {    \\scriptstyle M} }, \\Delta_2 \\}$ are K{\\\"a}hler parameters.\nIn {\\bf \\ref{section.04.4d.web}}, we set $q= e^{\\, R \\epsilon_2}$, $t= e^{\\, - R \\epsilon_1}$, where $\\epsilon_1$ and \n$\\epsilon_2$ are Nekrasov's regularisation parameters, then take the limit $R \\rightarrow 0$, to obtain \n$\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta} [\\epsilon_1, \\epsilon_2, R \\! \\rightarrow \\! 0]$.\nIn {\\bf \\ref{section.05.nekrasov.partition.functions}}, we recall the definition of the normalised \ncontribution of the bifundamental hypermultiplet, \n$\\mathcal{Z}_{building.block}^{\\, 4D}$ to  $\\mathcal{Z}^{\\, 4D}_{instanton}$. \nIn {\\bf \\ref{section.06.identification.numerators}}, we compare the numerators of \n$\\mathcal{W}^{\\, norm}_{\\bf V, W, \\Delta}$ and $\\mathcal{Z}^{\\, 4D}_{building.block}$, and identify the K{\\\"a}hler \nparameters with gauge theory parameters.\nIn {\\bf \\ref{section.07.identification.denominators}}, we compare the denominators of \n$\\mathcal{W}^{\\, norm}_{\\bf V, W, \\Delta}$ and $\\mathcal{Z}^{\\, 4D}_{building.block}$, and show that there is \na normalisation such that the two denominators agree, without changing the results of the computations \nof the instanton partition functions.\nIn {\\bf \\ref{section.08.restricted.instanton.partition.functions}}, we recall the choice of parameters \nthat allows us to use $\\mathcal{Z}^{\\, 4D}_{building.block}$ to build 4D instanton partition functions that can \nbe identified with Virasoro $A$-series minimal model conformal blocks times Heisenberg factors.\nIn {\\bf \\ref{section.09.from.gauge.theory.to.minimal.model}}, we identify the parameters of \n$\\mathcal{W}^{\\, norm}_{\\bf V \\, W \\, \\Delta}$ and of $\\mathcal{Z}^{\\, 4D}_{building.block}$. \nIn {\\bf \\ref{section.10.burge.pairs}}, we outline a proof, following \\cite{bershtein.foda}, of \nthe statement that we need to impose Burge conditions on the partition pairs that appear in our \nconstructions, to make the topological string partition functions free of non-physical singularities, \nonce we choose the parameters to coincide with those of the minimal model.\nSection {\\bf \\ref{section.11.comments}} includes comments and remarks. \n\\subsection*{Abbreviations and notation} \nWe focus on $U(2)$ quiver gauge theories, and simply say \n{\\it \\lq the instanton partition functions\\rq\\,}.\nWe assume that every conformal block, whether Liouville or minimal, \nincludes a factor from a field theory \nof a free boson on a line, and omit {\\it \\lq times Heisenberg factors\\rq}. \nThe normalised contribution of the bifundamental hypermultiplet is understood as the \nbuilding block of the instanton partition function, as all other contributions can be \nobtained from it. We simply say {\\it \\lq the bifundamental partition function\\rq\\,}.\n\n\\subsubsection*{Topological string-related notation}\n$\\mathcal{C}_{\\,     \\lambda \\, \\mu \\, \\nu} [q, t]$ is the refined topological vertex.\n$\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta} [q, $ $t, R]$ is a basic web refined topological \nstring partition function.\n${\\bf V}$ and ${\\bf W}$ are two pairs of Young diagrams. \n${\\bf \\Delta}$ is a set of three K{\\\"a}hler parameters. \n$q$ and $t$ are deformation parameters. \n$R$ is the radius of a space-like circle\n\\footnote{\\,\nThe 5D $U(2)$ quiver theories in this note live on D5-branes in time-like $x^0$, and \nspace-like $x^1, x^2, x^3, x^5$ and $x^6$ dimensions. Following \\cite{bao.01}, we take \n$x^5$ to be a circle of radius $R$, $\\beta$ in \\cite{bao.01}, such that setting \n$R \\rightarrow 0$ is equivalent to taking the 4D limit. For a complete discussion, we \nrefer to \\cite{bao.01}.\n}. \n$\\mathcal{Z}^{\\, ref\\, top}$ is the topological string partition function.\n\n\\subsubsection*{Gauge theory-related notation}\n$\\mathcal{Z}^{\\, 5D}_{instanton}$ is the 5D $U(2)$ quiver gauge theory instanton partition functions.\n$\\mathcal{Z}^{\\, 4D}_{instanton}$ is the 4D $U(2)$ quiver gauge theory instanton partition functions.\n$\\mathcal{Z}_{building.block}^{\\, 4D}$ is the normalised contribution of the bifundamental hypermultiplet.\nThe parameters $\\epsilon_1$ and $\\epsilon_2$ are deformation parameters.\n\n\\subsubsection*{Conformal field theory-related notation}\n$\\mathcal{B}^{\\, gen, \\,  \\mathcal{H}}$ are 2D Virasoro generic conformal blocks times Heisenberg factors.\n$\\mathcal{B}^{\\, gen, \\,  min}$ are 2D Virasoro A-series minimal model conformal blocks times Heisenberg \nfactors.\nThe parameters $p$ and $p^{\\prime}$ are coprime positive integers that label a Virasoro minimal model. \nThe parameters $r$ and $s$ are integers that satisfy $0 < r < p$, $0 < s < p^{\\prime}$ and label Virasoro \nminimal model highest weight representations.\n\n\\section{The refined topological vertex}\n\\label{section.02.refined.topological.vertex}\n{\\it We recall basic definitions related to Young diagrams and Schur functions, followed by the \ndefinition of the refined topological vertex.}\n\n\\subsection{Partitions and Young diagrams}\n\n\\subsubsection{Partitions}\n\\label{partitions} A partition $\\pi$ of a non-negative integer $|\\pi|$\nis a set of non-negative integers $\\{\\pi_{1},\\pi_{2},$ $\\cdots,\\pi_{p}\\}$,\nwhere $p$ is the number of parts, $\\pi_{i}\\geqslant\\pi_{i+1}$, and $\\sum_{i=1}^{p} \\pi_{i}= | \\pi |$.\n\n\\subsubsection{Young diagrams}\n\\label{young.diagrams} A partition $\\pi$ is represented as a Young diagram $Y$, as in \nFigure {\\bf \\ref{A.Young.diagram}}, which is a set of $p$ rows $\\{y_1, y_2, \\cdots, y_p \\}$, such \nthat row-$i$ has $y_i = \\pi_{i}$ cells\n\\footnote{\\, \nWe use $y_i$ for $i$-th row as well as for the number of cells in that row.\n}, \n$y_i \\geqslant y_{i+1}$, and $\\sum_i y_i = |Y| = |\\pi|$. We use $Y^{\\intercal}$ for the transpose of $Y$.\n\n\\subsubsection{Cells}\n\nWe use $\\square$ for a cell, or a square in the south-east quadrant\nof the plane, and refer to the coordinates of $\\square$ as $\\{ {\\tt \\scriptstyle R} , {\\tt \\scriptstyle C} \\}$.\nIf $\\square$ is inside a Young diagram $Y$, then $ {\\tt \\scriptstyle R} $ is the\n$Y$-row-number, counted from top to bottom, and $ {\\tt \\scriptstyle C} $ is the\n$Y$-column-number, counted from left to right, that $\\square$ lies\nin. If $\\square$ is outside $Y$, we still regard $ {\\tt \\scriptstyle R} $ as a $Y$-row-number,\nalbeit the length of this row is zero, and we still regard $ {\\tt \\scriptstyle C} $\nas a $Y$-column-number, albeit the length of this column is zero.\n\n\\subsubsection{Arms and legs, half-extended and extended}\n\\label{arms.legs} Consider a cell $\\square$ that has coordinates $\\{  {\\tt \\scriptstyle R} ,  {\\tt \\scriptstyle C}  \\}$. \nWe define the lengths of \nthe           arm $A^{  }_{\\square, Y}$, \nhalf-extended arm $A^{+ }_{\\square, Y}$, \n     extended arm $A^{++}_{\\square, Y}$, \nthe           leg $L^{  }_{\\square, Y}$,\nhalf-extended leg $L^{+ }_{\\square, Y}$, \n     extended leg $L^{++}_{\\square, Y}$, of $\\square$  \nwith respect to the Young diagram $Y$, to be\n\n\\begin{eqnarray}\nA^{  }_{\\square, Y} & = & y_{ {\\tt \\scriptstyle R} }               - \\,  {\\tt \\scriptstyle C} , \\quad\nA^{+ }_{\\square, Y}   =   A^{  }_{\\square, Y} + \\frac12,            \\quad \nA^{++}_{\\square, Y}   =   A^{  }_{\\square, Y} +      1,  \n\\\\\nL^{  }_{\\square, Y} & = & y_{ {\\tt \\scriptstyle C} }^{\\intercal} - \\,  {\\tt \\scriptstyle R} ,   \\quad\nL^{+ }_{\\square, Y}   =   L^{  }_{\\square, Y} + \\frac12,            \\quad        \nL^{++}_{\\square, Y}   =   L^{  }_{\\square, Y} +      1,\n\\end{eqnarray}\n\n\\subsubsection{Remark} $A_{\\square,Y}$ and $L_{\\square,Y}$ can be negative when $\\square$ lies \noutside $Y$. \n\n\\begin{figure}\n\\begin{tikzpicture}[scale=.8]\n\\draw [thick] (0, 0) rectangle (1,1);\n\\draw [thick] (1, 0) rectangle (1,1);\n\\draw [thick] (2, 0) rectangle (1,1);\n\\draw [thick] (3, 0) rectangle (1,1);\n\\draw [thick] (4, 0) rectangle (1,1);\n\\draw [thick] (5, 0) rectangle (1,1);\n\\draw [thick] (0,-1) rectangle (1,1);\n\\draw [thick] (1,-1) rectangle (1,1);\n\\draw [thick] (2,-1) rectangle (1,1);\n\\draw [thick] (3,-1) rectangle (1,1);\n\\draw [thick] (4,-1) rectangle (1,1);\n\\draw [thick] (0,-2) rectangle (1,1);\n\\draw [thick] (1,-2) rectangle (1,1);\n\\draw [thick] (2,-2) rectangle (1,1);\n\\node at (1.5,-0.5) {$\\checkmark$};\n\\end{tikzpicture}\n\\caption{\n{\\it\nThe Young diagram $Y$ of $5 + 4 + 2$.\nThe rows are numbered from top to bottom. \nThe columns are numbered from left to right. \nThe checkmarked cell has \n$A^{  } = 2$, \n$A^{+ } = \\frac52$, \n$A^{++} = 3$, \n$L^{  } = 1$, {\\it etc.}\n}\n}\n\\label{A.Young.diagram}\n\\end{figure}\n\n\\subsubsection{Partition pairs}\nA partition pair ${\\bf Y}$ is a set of two Young diagrams, $\\{ Y^{\\, 1}, Y^{\\, 2} \\}$, as in \nFigure {\\bf \\ref{A.partition.pair}}, where $|\\bf Y| = |Y^{\\, 1}| + |Y^{\\, 2}|$ is the total \nnumber of cells in ${\\bf Y}$.\n\n\\begin{figure}\n\\begin{tikzpicture}[scale=.8] \n\\draw [thick] (0, 0) rectangle (1, 1); \n\\draw [thick] (1, 0) rectangle (2, 1);\n\\draw [thick] (2, 0) rectangle (3, 1);\n\\draw [thick] (3, 0) rectangle (4, 1);\n\\draw [thick] (4, 0) rectangle (5, 1);\n\\draw [thick] (0,-1) rectangle (1, 0);\n\\draw [thick] (1,-1) rectangle (2, 0);\n\\draw [thick] (2,-1) rectangle (3, 0);\n\\draw [thick] (3,-1) rectangle (4, 0);\n\\draw [thick] (0,-2) rectangle (1,-1);\n\\draw [thick] (1,-2) rectangle (2,-1);\n\\draw [thick] (7, 0) rectangle (8, 1); \n\\draw [thick] (8, 0) rectangle (9, 1); \n\\draw [thick] (9, 0) rectangle (10,1);   \n\\draw [thick] (10,0) rectangle (11,1);  \n\\draw [thick] (7,-1) rectangle (8, 0); \n\\draw [thick] (7,-2) rectangle (8,-1);       \n\\node at (3.5,-0.5) {$\\checkmark$};\n\\node at (10.5,-0.5) {$\\checkmark$};\n\\end{tikzpicture}\n\\caption{\n{\\it \nA partition pair $\\{\\mu, \\nu\\}$. $\\mu$ is on the left, $\\nu$ is on the right. \nThe checkmarked cell at coordinates $(2, 4)$, in the lower right quadrant of \nthe plane, is in $\\mu$, but not in $\\nu$, and has\n$A^{  }_{\\mu} = 0$,\n$A^{+ }_{\\mu} = \\frac12$,\n$A^{++}_{\\mu} = 1$,\n$L^{  }_{\\mu} = 0$,\n$L^{+ }_{\\mu} = \\frac12$,\n$L^{++}_{\\mu} = 1$,\nas well as\n$A^{  }_{\\nu} = -3$, {\\it etc.} and\n$L^{  }_{\\nu} = -1$, {\\it etc.} \n}\n}\n\\label{A.partition.pair}\n\\end{figure}\n\\bigskip\n\n\\subsubsection{Sum of row-lengths and squares of row-lengths} Given \na Young diagram $Y$, with rows of length $y_1 \\geqslant y_2 \\geqslant \\cdots$, \nwe define\n\n\\begin{equation}\n| Y |                 = \\sum_{i=1} y_i, \n\\quad\n\\parallel Y \\parallel = \\sum_{i=1} y_i^{\\, 2}\n\\end{equation}\n\n\\noindent where the sum is over all parts of $Y$\n\\footnote{\\\nIn \\cite{ikv}, Iqbal {\\it et al.} use \n$\\parallel Y \\parallel^{\\, 2} = \\sum_{i=1} y_i^{\\, 2}$. We define $\\parallel Y \\parallel = \\sum_{i=1} y_i^{\\, 2}$ \nto simplify the notation. This is consistent with other notation and should cause no confusion.\n}.\nFurther, to simplify the equations, we define\n\n\\begin{equation}\n| Y |^{\\prime}                 = \\frac12 \\sum_{i=1} y_i,\n\\quad\n\\parallel Y \\parallel^{\\prime} = \\frac12 \\sum_{i=1} y_i^{\\, 2}\n\\end{equation}\n\n\\subsubsection{Sequences} Given the sequence of row-lengths \n$  \\lambda   = \\{   \\lambda_1,    \\lambda_2, \\cdots \\}$, the sequence of half-integers\n$  \\rho      = \\{ - \\frac12,    - \\frac32,   \\cdots \\}$, and two variables $x$ and $y$, \nwe define the \\lq exponentiated\\rq\\, sequences \n\n\\begin{equation}\nx^{-\\lambda}= \\{ x^{-\\lambda_1}, x^{-\\lambda_2}, \\cdots \\}, \n\\quad\ny^{\\, -\\rho}= \\{ y^{\\, \\frac12}, y^{\\, \\frac32}, \\cdots \\},\n\\quad\n\\textit{and}\n\\quad\nx^{\\, -\\lambda} y^{\\, -\\rho} = \n\\{\nx^{\\, -\\lambda_1} \\, y^{\\, \\frac12}, \nx^{\\, -\\lambda_2} \\, y^{\\, \\frac32}, \n\\cdots \\}, \n\\end{equation}\n\n\\subsubsection{A function of the arm-lengths and the leg-lengths} Given a Young diagram \n$\\lambda$, we define the function\n\n\\begin{equation}\n\\label{z.factor}\nZ_{\\lambda} [q, t]\n= \\prod_{\\square \\in \\lambda} \n\\frac{1}{ \\left\\lgroup 1 -  \\left\\lgroup \\frac{q}{t} \\right\\rgroup^{\\, \\frac12} \\, \nq^{\\, A_{\\square, \\lambda}^{+ }} \\, \nt^{\\, L_{\\square, \\lambda}^{+ }} \n\\right\\rgroup}\n= \\prod_{\\square \\in \\lambda} \\frac{1}{ \\left\\lgroup 1- \\, q^{\\, A_{\\square, \\lambda}^{++}}\\, \n                                             t^{\\, L_{\\square, \\lambda}^{  }} \\right\\rgroup}\n\\end{equation}\n\n\\subsubsection{Remark} \nThe expression in the middle of Equation {\\bf \\ref{z.factor}} corresponds to \nsplitting the cell $\\square$ at the corner of a hook in the partition $\\lambda$ into two halves, then\nattaching one       half to the arm of that hook to form {\\it a half-extended arm} of length \n$A^{+ }_{\\square, \\lambda} = A^{  }_{\\square, \\lambda} + \\frac12$, and \nattaching the other half to the leg of that hook to form {\\it a half-extended leg} of length \n$L^{+ }_{\\square, \\lambda} = L^{  }_{\\square, \\lambda} + \\frac12$. \nThe expression on the right corresponds to \nattaching the cell $\\square$ at the corner of a hook to the arm of that hook to form \n{\\it an extended arm} of length \n$A^{++}_{\\square, \\lambda} = A^{  }_{\\square, \\lambda} + 1$. The length of the leg of that hook remains \n$L^{  }_{\\square, \\lambda}$. \n\n\\subsection{Schur and skew-Schur functions}\nGiven an $n$-row Young diagram $\\lambda$, with parts $\\lambda_1 \\geqslant \\lambda_2 \\geqslant \\cdots$, \nand a set of $n$ variables $\\{x_1, x_2, \\cdots, x_n \\}$, the Schur function $s_{\\lambda} [{\\bf x}]$\nis defined as\n\\begin{equation}\ns_{\\lambda}[{\\bf x}] =\n\\frac{\ndet  \\left\\lgroup x_i^{\\, \\lambda_j+n-j} \\right\\rgroup_{1 \\leqslant i, j \\leqslant n}\n}\n{\n\\prod_{1 \\leqslant i < j \\leqslant n}  \\left\\lgroup x_i - x_j \\right\\rgroup\n}\n\\end{equation}\nThe skew-Schur function $s_{\\lambda\/\\mu}[{\\bf x}]$ is defined as\n\\begin{equation}\ns_{\\lambda\/\\mu}[{\\bf x}]=\\sum_{\\nu}c_{\\mu \\nu}^{\\, \\lambda}\\ \ns_{\\lambda}[{\\bf x}]\n\\end{equation}\n\\noindent where $c_{\\mu \\nu}^{\\, \\lambda}$ are Littlewood-Richardson coefficients defined by\n\\begin{equation}\ns_{\\mu}\\ \ns_{\\nu}=\\sum_{\\lambda}\\ c_{\\mu \\nu}^{\\, \\lambda}\\ \ns_{\\lambda}\n\\end{equation}\n\n\\subsection{Topological vertices}\nOur story starts from A-model closed topological string theory on non-compact Calabi-Yau \nthreefolds.\nWe cannot afford to review this vast subject and refer the reader to excellent available \nintroduction, including \\cite{marino.2005, marino.book}.\n\n\\subsubsection{The topological vertex of Aganagic {\\it et al}} \nIn \\cite{topological.vertex}, Aganagic, Klemm, Marino and Vafa introduced a systematic procedure \nto calculate A-model topological string partition functions on resolved conifolds. The main \ningredient of this procedure is the topological vertex, which has a combinatorial representation \nin terms of plane partitions with three boundaries specified by three Young diagrams, and can be \nschematically represented as a trivalent vertex, with bonds labelled by Young diagrams, as in \nFigure {\\bf \\ref{A.topological.vertex}}. For details, we refer to \\cite{topological.vertex}.\n\n\\subsubsection{The refined topological vertex}\nIn \\cite{ikv}, Iqbal, Kozcaz and Vafa defined the refined topological vertex, up to simple \nre-arrangements\n\\footnote{\\,\nWe use \n$\\mathcal{C}_{\\,     \\lambda \\, \\mu \\, \\nu} [q, t]$, and\n$Z_Y [q, t]$, while Iqbal {\\it et al.} use\n$\\mathcal{C}_{\\, \\lambda \\, \\mu \\, \\nu} [t, q]$, and\n$\\tilde{Z}_{\\mu} [t, q]$.\n}, as \n\n\\begin{multline}\n\\label{refined.topological.vertex}\nC_{\\lambda \\, \\mu \\, \\nu} [q, t] =\n\\\\\nq^{\\,  \\left\\lgroup \n                | \\lambda     |^{\\prime} \n      -         | \\mu         |^{\\prime} \n      + \\parallel \\mu \\parallel^{\\prime} \n      + \\parallel \\nu \\parallel^{\\prime} \n    \\right\\rgroup} \\, \nt^{\\,  \\left\\lgroup -     | {\\lambda}^{         }         |^{\\prime} \n      +         | {\\mu    }^{         }         |^{\\prime} \n      - \\parallel {\\mu    }^{\\intercal} \\parallel^{\\prime}\n\\right\\rgroup} \\, \nZ_{\\nu}[q, t]\n \\left\\lgroup\n\\sum_{\\eta} \n \\left\\lgroup \n\\frac{q}{t} \n\\right\\rgroup^{\\,| \\eta |^{\\prime}}\n\\ s_{ {\\lambda}^{\\intercal} \/ \\eta}[ q^{\\,- \\nu } \\ t^{\\,- {\\rho}^{         }}]\n\\ s_{ {\\mu    }^{         } \/ \\eta}[ q^{\\,- \\rho} \\ t^{\\,- {\\nu }^{\\intercal}}]\n\\right\\rgroup\n\\end{multline}\n\n\\noindent Note that the refined vertex is not manifestly cyclically-symmetric, in the sense\nthat the partitions that label the external legs do not appear on equal footing. In particular, \nthe partition $\\nu$ is distinguished from the other two. The external leg labelled by $\\nu$ is\nreferred to as {\\it \\lq preferred\\rq\\,}. The original, unrefined topological vertex of Aganagic \n{\\it et al.} is recovered by setting $q=t$.\n\n\\begin{figure}\n\\begin{tikzpicture}\n\\draw [ultra thick]  (0.0,2.0)--(2.0,2.0);\n\\draw [ultra thick]  (2.0,2.0)--(2.0,4.0);\n\\draw [ultra thick]  (2.0,2.0)--(3.5,0.5);\n\\node [left]  at (2.0,3.0) {$\\lambda$};\n\\node [left]  at (2.8,1.0) {$\\mu$};\n\\node [above] at (1.0,2.0) {$\\nu$};\n\\end{tikzpicture}\n\\caption{\n{\\it \nThe topological vertex $C_{\\lambda \\, \\mu \\, \\nu} [q, t]$ is trivalent and depends \non two parameters $q$ and $t$. \nThe segments are labelled by three partitions $\\lambda, \\, \\mu$ and $\\nu$, such that \n$\\lambda$ is assigned to the vertical segment, $\\mu$ is assigned to the segment that \nfollows in a clockwise direction, and $\\nu$ to the segment that follows. \nThe \\lq preferred leg\\rq\\, is labelled by $\\nu$. \n} \n}\n\\label{A.topological.vertex}\n\\end{figure} \n\n\\subsubsection{On framing vectors and framing factors} \nIn defining the refined topological vertex, one labels each of the three boundaries of a vertex \nby {\\it a framing vector} that indicates a possible twisting of the boundary. On gluing two \nvertices along a common boundary, there is in general {\\it a framing factor} that accounts for \na possible mismatch in the orientations of the relevant framing vectors. In this note, we glue \nvertices such that we do not require framing factors.  \n\n\\section{A 5D $U(2)$ basic web partition function}\n\\label{section.03.5d.web}\n{\\it We glue four copies of the refined topological vertex to obtain a 5D basic web that can be \nused as a building block of $U(2)$ topological string partition functions.}\n\n\\subsection{Geometric engineering} Following \\cite{geometric.engineering.01, geometric.engineering.02}, \nwe write the normalised $U(2)$ basic web partition function as\n\n\\begin{equation}\n\\label{W.norm}\n\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}^{\\,  norm} [q, t, R] = \n\\frac{\n\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}^{        } [q, t, R]\n}\n{\n\\mathcal{W}_{\\, \\bf \\emptyset\\, \\emptyset\\, \\Delta}[q, t, R]\n}\n\\end{equation}\n\n\\noindent where $\\emptyset$ is the trivial, or empty partition with no cells. The numerator is\n\n\\begin{multline}\n\\label{the.numerator}\n\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}[q, t, R]\n=\n\\sum_{\\xi_1, \\, \\xi_{ {    \\scriptstyle M} }, \\, \\xi_2}\n \\left\\lgroup -Q_1     \\right\\rgroup^{\\, |\\xi_1     |} \\,\n \\left\\lgroup -Q_{ {    \\scriptstyle M} } \\right\\rgroup^{\\, |\\xi_{ {    \\scriptstyle M} } |} \\,\n \\left\\lgroup -Q_2     \\right\\rgroup^{\\, |\\xi_2  |} \n\\\\\n\\times\nC_{\\emptyset                  \\, \\xi_1             \\, V^{\\, 1}             } [q, t] \\ \nC_{\\xi_{ {    \\scriptstyle M} }               \\, \\xi_1^{\\intercal} \\, W^{\\, 1 \\, \\intercal}} [t, q] \\ \nC_{\\xi_{ {    \\scriptstyle M} }^{\\intercal}   \\, \\xi_2             \\, V^{\\, 2}             } [q, t] \\ \nC_{\\emptyset                  \\, \\xi_2^{\\intercal} \\, W^{\\, 2 \\, \\intercal}} [t, q]\n\\end{multline}\n\n\\noindent where we use\n\n\\begin{equation}\n\\boxed{\nQ_i = e^{\\, - R \\Delta_i}, \\quad i = 1, { {    \\scriptstyle M} }, 2\n}\n\\end{equation}\n\n\\noindent and the denominator \n$\\mathcal{W}_{\\, \\bf \\emptyset \\, \\emptyset \\, \\Delta}[q, t, R]$ \nis identical to the numerator \n$\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}[q, t, R]$\nbut with all external partition pairs empty.\n\n\\begin{figure}\n\\begin{tikzpicture}[scale=.7]\n\\draw [ultra thick] (0,4)--(2,4);\n\\draw [ultra thick] (2,6)--(2,4);\n\\draw [ultra thick] (2,4)--(4,2);\n\\node [left, red] at (0,4) {$V^{\\, 1}$};\n\\node [right] at (3,3) {$\\xi_1, \\, \\Delta_1$};\n\\node [above] at (2,6) {$\\emptyset$};\n\\draw [ultra thick] (4,2)--(8,2);\n\\draw [ultra thick] (4,0)--(4,2);\n\\node [right, blue] at (8,2) {$W^{\\, 1}$};\n\\node [right] at (4,1) {$\\xi_{ {    \\scriptstyle M} }, \\, \\Delta_{ {    \\scriptstyle M} }$};\n\\draw [ultra thick] (0,0)--(4,0);\n\\node [left, red] at (0,0) {$V^{\\, 2}$};\n\\draw [ultra thick] (4,0)--(6,-2);\n\\node [right] at (5,-1) {$\\xi_2, \\, \\Delta_2$};\n\\draw [ultra thick] (6,-2)--(6,-4);\n\\draw [ultra thick] (6,-2)--(8,-2);\n\\node [right,blue] at (8,-2) {$W^{\\, 2}$};\n\\node [below] at (6,-4) {$\\emptyset$};\n\\end{tikzpicture}\n\\caption{\n{\\it \nThe $U(2)$ basic web diagram. Each external line is labeled by a partition. \nEach internal line is labeled by a partition and a K{\\\"a}hler parameter. This basic \nweb can be glued to form topological partition functions. The preferred legs are all\nexternal, and labelled by the partitions $V^1, V^2, W^1$ and $W^2$.\n}\n}\n\\label{basic.web}\n\\end{figure}\n\nAs shown in Figure {\\bf \\ref{basic.web}}, the basic web has two external horizontal legs \ncoming in from the left, two external horizontal legs going out to the right, and a pair \nof vertical legs, one going up and one down.\nThe horizontal external legs on the left  are assigned partitions $\\{V^{\\, 1}, V^{\\, 2}\\}$, \nthe horizontal external legs on the right are assigned partitions $\\{W^{\\, 1}, W^{\\, 2}\\}$. \nThe internal lines are assigned parameters $  Q_1$, $  Q_{ {    \\scriptstyle M} }$ and $  Q_2$, and partitions \n                                           $\\xi_1$, $\\xi_{ {    \\scriptstyle M} }$ and $\\xi_2$,\nfrom top to bottom. \nThe vertical external legs are assigned empty partitions. \n\nEach trivalent vertex corresponds to a refined topological vertex $\\mathcal{C}_{\\lambda \\, \\mu \\, \\nu}$. \nOur convention is such that each vertex has one vertical leg, we associate $\\lambda$ to that \nvertical leg, regardless of whether it is internal or external, pointing upwards or downwards, \nthen $\\mu$ and $\\nu$ to the remaining two legs, encountered sequentially as we start from the \nvertical leg and move around the vertex clockwise. \nUsing Equation {\\bf \\ref{refined.topological.vertex}}, we re-write the numerator in Equation \n{\\bf \\ref{the.numerator}} as\n\n\\begin{multline}\n\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}^{        } [q, t, R]\n\\\\\nq^{\\,  \\left\\lgroup \\parallel    V^{\\, 1}              \\parallel^{\\prime} + \\parallel   V^{\\, 2}              \\parallel^{\\prime} \\right\\rgroup}\nt^{\\,  \\left\\lgroup \\parallel    W^{\\, 1 \\, \\intercal} \\parallel^{\\prime} + \\parallel   W^{\\, 2 \\, \\intercal} \\parallel^{\\prime} \\right\\rgroup}\nZ_{  V^{\\, 1}             } [q, t]\\\nZ_{  W^{\\, 1 \\, \\intercal}} [t, q]\\\nZ_{  V^{\\, 2}             } [q, t]\\\nZ_{  W^{\\, 2 \\, \\intercal}} [t, q]\n\\\\\n\\times \\,\n\\sum_{\\, \\xi_1, \\, \\xi_{ {    \\scriptstyle M} }, \\, \\xi_2, \\, \\eta_1, \\, \\eta_2}\n \\left\\lgroup - Q_1     \\right\\rgroup^{\\, |\\xi_1     |} \n \\left\\lgroup - Q_{ {    \\scriptstyle M} } \\right\\rgroup^{\\, |\\xi_{ {    \\scriptstyle M} } |}\n \\left\\lgroup - Q_2     \\right\\rgroup^{\\, |\\xi_2     |}\n \\left\\lgroup \\frac{q}{t} \\right\\rgroup^{\\,  \\left\\lgroup | \\eta_2 |^{\\prime} - |\\eta_1 |^{\\prime} \\right\\rgroup}\n\\\\\n\\times\ns_{\\xi_1                         } \\, [q^{\\, -\\rho} \\, t^{\\, -V^{\\, 1 \\, \\intercal}}]     \\ \ns_{\\xi_1^{\\intercal}     \/ \\eta_1} \\, [q^{\\, - W^{\\, 1}} \\, t^{\\, -\\rho                 }]\\\ns_{\\xi_{ {    \\scriptstyle M} }^{\\intercal} \/ \\eta_1} \\, [q^{\\, -\\rho     } \\, t^{\\, -W^{\\, 1 \\, \\intercal}}]\\ \n\\\\\n\\times\ns_{\\xi_{ {    \\scriptstyle M} }             \/ \\eta_2} \\, [q^{\\, - V^{\\, 2}} \\, t^{\\, -\\rho           }]      \\\ns_{\\xi_2                 \/ \\eta_2} \\, [q^{\\, -\\rho}      \\, t^{\\, -V^{\\, 2 \\, \\intercal}}]\\ \ns_{\\xi_2^{\\intercal}             } \\, [q^{\\, - W^{\\, 2}} \\, t^{\\, -\\rho           }]      \\ \n\\label{web.02}\n\\end{multline}\n\n\\noindent where we used the fact that for an empty partition $\\emptyset$, the skew \npartition $\\emptyset \/ \\eta$ exists only for $\\eta = \\emptyset$, the sum over $\\eta$ \ntrivialises, and the skew Schur function $s_{\\emptyset \/ \\eta} = s_{\\emptyset} = 1$.\n\n\\subsubsection{Two skew Schur function identities}\nTo evaluate the sums in Equation {\\bf \\ref{web.02}} for $w^{\\, num}$, we need the two identities\n\\footnote{\\,\nExercise {\\bf 26}, page 93 of \\cite{macdonald.book}.}\n\n\\begin{eqnarray}\n\\sum_{\\lambda} \\, s_{\\lambda \/ \\eta_1}[x] \\, s_{\\lambda \/ \\eta_2}[y] \n& = & \n\\prod_{i, j}  \\left\\lgroup 1-x_i y_j \\right\\rgroup^{-1} \n\\sum_{\\tau}  \\, s_{\\eta_1 \/ \\tau}[y] \\, s_{\\eta_2 \/ \\tau} [x]\n\\label{schur.01}\n\\\\\n\\sum_{\\lambda} \\, s_{\\lambda \/ \\eta_1}[x] \\, s_{\\lambda^{\\intercal} \/ \\eta_2}[y] & = & \n\\prod_{i, j}  \\left\\lgroup 1+x_i y_j \\right\\rgroup      \n\\sum_{\\tau}  \\, s_{\\eta_1^{\\intercal} \/ \\tau^{\\intercal}}[y] \\, s_{\\eta_2^{\\intercal}\/ \\tau} [x]\n\\label{schur.02}\n\\end{eqnarray}\n\n\\noindent as well as the property that\n\n\\begin{equation}\nQ^{\\, |\\lambda| - |\\eta| } \\, s_{\\lambda\/\\eta}[x]= s_{\\lambda \/ \\eta} \\, [Q x]\n\\end{equation}\n\n\\noindent which follows from the definition of the skew Schur function. \n\n\\subsubsection{The basic web in product form}\nWe evaluate the sums over the right hand side of Equation {\\bf \\ref{web.02}}, using \n$\\rho_i = - i + \\frac12$.\n\n\\subsubsection{The sums over $\\xi_1$ and $\\xi_2$}\n\n\\begin{multline}\n\\sum_{\\xi_1}  \\left\\lgroup - Q_1 \\right\\rgroup^{\\,|\\xi_1|}  \\, \ns_{\\xi_1                     } [q^{\\, -\\rho} \\, t^{\\, -V^{\\, 1 \\, \\intercal}}]\\, \ns_{\\xi_1^{\\intercal} \/ \\eta_1} [q^{\\, - W^{\\, 1}} \\, t^{\\, -\\rho           }]\n\\\\\n= \\prod_{i, j = 1}^{\\infty}\n \\left\\lgroup\n                      1 - Q_1 \\, q^{\\, -W_i^{\\, 1} + j - \\frac12} \\, t^{\\, -V_j^{\\, 1 \\, \\intercal} + i - \\frac12}\n\\right\\rgroup \\, \ns_{\\eta_1^{\\intercal}} [- Q_1 \\, q^{\\, -\\rho}                \\, t^{\\, -V^{\\, 1 \\, \\intercal}}],\n\\label{4.3}\n\\end{multline}\n\n\\begin{multline}\n\\sum_{\\xi_2}  \\,\n \\left\\lgroup -Q_2 \\right\\rgroup^{\\, |\\xi_2|} \\, \ns_{\\xi_2 \/ \\eta_{2}} [ q^{\\, -\\rho} \\, t^{\\, - V^{\\, 2 \\, \\intercal}}] \\, \ns_{\\xi_2^{\\intercal}}[ q^{\\, - W^{\\, 2}} \\, t^{\\, - \\rho           }]\n\\\\\n=\\prod_{i,j=1}^{\\infty} \\,\n \\left\\lgroup\n1-Q_2 \\, q^{\\, -W_i^{\\, 2             }+j-\\frac12} \n      \\, t^{\\, -V_j^{\\, 2 \\, \\intercal}+i-\\frac12}\n\\right\\rgroup \\ \ns_{\\eta_2^{\\intercal} }[-Q_2 q^{\\, -W^{\\, 2}} \\, t^{\\, -\\rho} ]\n\\label{4.4}\n\\end{multline}\n\n\\subsubsection{The sum over $\\xi_{ {    \\scriptstyle M} }$} We re-write this in terms of a sum over \na new set of partition $\\tau$,\n\n\\begin{multline}\n\\sum_{\\xi_{ {    \\scriptstyle M} } } \\, \n \\left\\lgroup -Q_{ {    \\scriptstyle M} } \\right\\rgroup^{\\, | \\xi_{ {    \\scriptstyle M} }  | - | \\eta_1 | } \\,\ns_{\\xi_{ {    \\scriptstyle M} }^{\\intercal} \/ \\eta_1} [ \\, q^{\\, -\\rho} \\, t^{\\, -W^{\\, 1, \\, \\intercal}} ] \\, \ns_{\\xi_{ {    \\scriptstyle M} }             \/ \\eta_2} [ \\, q^{\\, -V^{\\, 2}}  \\, t^{\\, -\\rho}            ]\n\\\\\n=\\prod_{i, j}\n \\left\\lgroup\n1 - Q_{ {    \\scriptstyle M} } \\, \nq^{\\, -V_i^{\\, 2             } + j - \\frac12} \\,\nt^{\\, -W_j^{\\, 1 \\, \\intercal} + i - \\frac12}\n\\right\\rgroup\n\\sum_{\\tau}\n\\ \ns_{\\eta_1^{\\intercal}\/ \\tau^{\\intercal}}[             q^{\\, -V^{\\, 2} } \\, t^{\\, -\\rho}            ] \\, \ns_{\\eta_2^{\\intercal}\/ \\tau            }[-Q_{ {    \\scriptstyle M} }  \\, q^{\\, -\\rho} \\, t^{\\, -W^{\\, 1 \\, \\intercal}} ]\n\\label{4.5}\n\\end{multline}\n\n\\subsubsection{The sums over $\\eta_1$ and $\\eta_2$} \n\n\\begin{multline}\n\\sum_{\\eta_1}\n \\left\\lgroup \\frac{q}{t} \\right\\rgroup^{\\, - |\\eta_1|^{\\prime}} \\, \n \\left\\lgroup - Q_{ {    \\scriptstyle M} }\\right\\rgroup^{  |\\eta_1|         } \\,\ns_{\\eta_1^{\\intercal}}                     [      -Q_1 \\, q^{\\, -\\rho} \\, t^{\\, -V^{\\, 1 \\, \\intercal}}]\\ \ns_{\\eta_1^{\\intercal} \/ \\tau^{\\intercal}} \\, [            q^{\\, - V^{\\, 2}} \\, t^{\\, -\\rho           }]\n\\\\\n=\\prod_{i,j}\n \\left\\lgroup\n1-Q_1 \\, Q_{ {    \\scriptstyle M} }\\, q^{\\, -V_i^{\\, 2} + j - 1} \\, t^{\\, -V_j^{\\, 1 \\, \\intercal}+i} \n\\right\\rgroup^{-1}\n\\ \ns_{\\tau^\\intercal}[Q_1 \\, Q_{ {    \\scriptstyle M} } \\, q^{\\, j - \\frac12} \\, t^{\\, -V_j^{\\, 1 \\, \\intercal}}]\n\\label{4.6}\n\\end{multline}\n\n\\begin{multline}\n\\sum_{\\eta_2} \n \\left\\lgroup \\frac{q}{t} \\right\\rgroup^{\\,  |\\eta_2|^{\\prime}} \\,\ns_{\\eta_2^\\intercal       }[-Q_2     \\, q^{\\, - W^{\\, 2}} \\, t^{\\, -\\rho                 } ] \\, \ns_{\\eta_2^\\intercal \/ \\tau}[-Q_{ {    \\scriptstyle M} } \\, q^{\\, -\\rho}      \\, t^{\\, -W^{\\, 1 \\, \\intercal}} ]\n\\\\\n=\\prod_{i,j}\n \\left\\lgroup      1 - Q_{ {    \\scriptstyle M} } \\, Q_2 \\,  q^{\\, -W_i^{\\, 2} + j} \\, t^{\\, -W_j^{\\, 1 \\, \\intercal} + i - 1} \\right\\rgroup^{-1} \\, \ns_{\\tau}[  - Q_2     \\,         q^{\\, -W_i^{\\, 2}    } \\, t^{\\, i - \\frac12} ]\n\\label{4.7}\n\\end{multline}\n\n\\subsubsection{The sum over $\\tau$} We finally evaluate the sum over the partitions \nthat were introduced in an intermediate step above, \n\n\\begin{equation}\n\\sum_{\\tau}\\ \ns_{\\tau}[-Q_2 \\,                   q^{\\, -W_i^{\\, 2}} \\, t^{\\, i - \\frac12         }] \\, \ns_{\\tau^{\\intercal}}[Q_1 \\, Q_{ {    \\scriptstyle M} } \\, q^{\\, j - \\frac12} \\, t^{\\, - V_j^{\\, 1 \\, \\intercal} }]\n=\\prod_{i, j}\n \\left\\lgroup 1 - Q_1 \\, Q_{ {    \\scriptstyle M} } \\, Q_2 \\, q^{\\, - W_i^{\\, 2} + j - \\frac12} t^{\\, -V_j^{\\, 1 \\, \\intercal} + i - \\frac12} \\right\\rgroup\n\\label{4.8}\n\\end{equation}\n\n\\noindent to obtain \n\n\\begin{multline}\n\\label{w.num}\n\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}^{        } [q, t, R]\n= \n\\\\\nq^{\\,  \\left\\lgroup \\parallel    V^{\\, 1             } \\parallel^{\\prime} + \\parallel   V^{\\, 2             } \\parallel^{\\prime} \\right\\rgroup}\nt^{\\,  \\left\\lgroup \\parallel    W^{\\, 1 \\, \\intercal} \\parallel^{\\prime} + \\parallel   W^{\\, 2 \\, \\intercal} \\parallel^{\\prime} \\right\\rgroup} \\, \nZ_{  V^{\\, 1}             }[q, t]\\\nZ_{  W^{\\, 1 \\, \\intercal}}[t, q]\\\nZ_{  V^{\\, 2}             }[q, t]\\\nZ_{  W^{\\, 2 \\, \\intercal}}[t, q]\n\\\\\n\\times \\,\n\\prod_{i, j = 1}^{\\infty} \\,  \\left\\lgroup 1  - Q_1         \\, \nq^{\\, -W_i^{\\, 1} + j - \\frac12}                  \\, \nt^{\\, -V_j^{\\, 1 \\, \\intercal} + i - \\frac12} \\right\\rgroup \\, \n\\prod_{i, j = 1}^{\\infty} \\,  \\left\\lgroup 1 - Q_2          \\, \nq^{\\, -W_i^{\\, 2}             + j - \\frac12}      \\, \nt^{\\, -V_j^{\\, 2 \\, \\intercal} + i - \\frac12} \\right\\rgroup \\,\n\\\\\n\\times \\,\n\\prod_{i, j = 1}^{\\infty} \\,  \\left\\lgroup 1 -        Q_{ {    \\scriptstyle M} }        \\,               \nq^{\\, -V_i^{\\, 2} + j - \\frac12} \\, t^{\\, - W_j^{\\, 1 \\, \\intercal} + i - \\frac12} \\right\\rgroup \\,\n\\\n\\prod_{i, j = 1}^{\\infty} \\,  \\left\\lgroup 1 - Q_1 \\, Q_{ {    \\scriptstyle M} } \\, Q_2 \\, \nq^{\\, -W_i^{\\, 2} + j - \\frac12} \\, t^{\\, - V_j^{\\, 1 \\, \\intercal} + i - \\frac12} \\right\\rgroup\n\\\\\n\\times \\,\n\\prod_{i, j = 1}^{\\infty} \\,  \\left\\lgroup 1 - Q_1 \\, Q_{ {    \\scriptstyle M} } \\,        \nq^{\\, - V_i^{\\, 2}             + j - 1} \\, \nt^{\\, - V_j^{\\, 1 \\, \\intercal} + i    } \\right\\rgroup^{-1} \n\\\n\\prod_{i, j = 1}^{\\infty} \\,  \\left\\lgroup 1 -        Q_{ {    \\scriptstyle M} } \\, Q_2 \\,        \nq^{\\, - W_i^{\\, 2}             + j    } \\, \nt^{\\, - W_j^{\\, 1 \\, \\intercal} + i - 1} \\right\\rgroup^{-1} \n\\end{multline}\n\n\\noindent Note that regarding the right hand side of Equation {\\bf \\ref{w.num}} as a rational function, \nthe initial four products are in the numerator, while the latter two are in the denominator. \n\n\\subsubsection{Normalised products}\nGiven two partitions, $V$ with parts $v_i$, $i = 1, 2, \\cdots$, and \n                      $W$ with parts $w_i$, $i = 1, 2, \\cdots$, \nand two sequences of integers $\\alpha_k$ and $\\beta_k$, $k = 1, 2, \\cdots$, we define \n\n\\begin{equation}\n\\label{normalised.product}\n{\\prod_{i, j=1}^{\\infty}}^{^\\prime} \n \\left\\lgroup 1-Q \\, q^{\\, - v_i + \\alpha_j} \\, t^{\\, - w_j + \\beta_i} \\right\\rgroup\n=\n\\prod_{i, j=1}^{\\infty}\n \\left\\lgroup\n\\frac{\n1-Q \\, q^{\\, - v_i + \\alpha_j} \\, t^{\\, - w_j + \\beta_i}\n}{\n1-Q \\, q^{\\,         \\alpha_i} \\, t^{\\,         \\beta_i}\n}\n\\right\\rgroup\n\\end{equation}\n\n\\noindent In this notation, the expression for \n$\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}^{\\, norm} [q, t, R]$ \nin Equation {\\bf \\ref{W.norm}}, is identical to that for \n$\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}[q, t, R]$, \nin Equation {\\bf \\ref{web.02}}, up to replacing each product $\\prod_{i, j}$ \nby a normalised product $\\prod_{i, j}^{^\\prime}$.\n\n\\subsubsection{From infinite to finite products}\n\n\\noindent Using Equation {\\bf \\ref{normalised.product}}, we have the following identities \n\\cite{nakajima.yoshioka, awata.kanno.02}\n\\footnote{\\\nFor an excellent reference and compendium of relevant combinatorial identities, including \nproofs, see \\cite{awata.kanno.02}. Equation {\\bf \\ref{nakajima.identity}} in this note \nfollows from Equations {\\bf 2.8--2.11} in \\cite{awata.kanno.02}.\n}. Firstly,\n\n\\begin{eqnarray}\n\\label{nakajima.identity}\n{\\prod_{i, j=1}^{\\infty}}^{^\\prime} \n \\left\\lgroup 1-Q \\, q^{\\, -w_i + j - 1} \\, t^{\\, -v_j^{\\intercal} + i} \\right\\rgroup\n& = & \n\\prod_{\\square \\in V}\n \\left\\lgroup\n1-Q\\, q^{\\, -A_{\\square, V}^{++}} \\, t^{\\, -L_{\\square, W}^{  }}\n\\right\\rgroup\n\\prod_{\\blacksquare \\in W}\n \\left\\lgroup\n1-Q\\, q^{\\,  A_{\\blacksquare, W}^{  }} \\, t^{\\,  L_{\\blacksquare, V}^{++}}\n\\right\\rgroup\n\\nonumber\n\\\\\n& = & \n\\prod_{\\blacksquare \\in W}\n \\left\\lgroup\n1-Q\\, q^{\\, -A_{\\blacksquare, V}^{++}} \\, t^{\\, -L_{\\blacksquare, W}^{  }}\n\\right\\rgroup\n\\prod_{\\square \\in V}\n \\left\\lgroup\n1-Q\\, q^{\\,  A_{\\square, W}^{  }} \\, t^{\\,  L_{\\square, V}^{++}}\n\\right\\rgroup\n\\end{eqnarray}\n\n\\noindent Note that while the product on the left of Equation {\\bf \\ref{nakajima.identity}} is \nnormalised in the sense of Equation {\\bf \\ref{normalised.product}}, the remaining products are \nnot.  \n\n\\subsubsection{The 5D basic web in product form}\n\nUsing the identities {\\bf \\ref{nakajima.identity}} in Equation {\\bf \\ref{web.02}} for the numerator, \nwe write the normalised $U(2)$ basic web partition function in Equation {\\bf \\ref{W.norm}} as\n\n\\begin{equation}\n\\label{W.5D.norm}\n\\mathcal{W}^{\\, norm}_{\\, \\bf V \\, W \\, \\Delta}[q, t, R]\n=\nq^{\\,  \\left\\lgroup \\parallel    V^{\\, 1}              \\parallel^{\\prime} + \\parallel   V^{\\, 2}              \\parallel^{\\prime} \\right\\rgroup}\nt^{\\,  \\left\\lgroup \\parallel    W^{\\, 1 \\, \\intercal} \\parallel^{\\prime} + \\parallel   W^{\\, 2 \\, \\intercal} \\parallel^{\\prime} \\right\\rgroup}\n\\\n\\frac{\nw^{\\, num} [{\\bf V, \\, W, \\, \\Delta}, q, t, R]\n}{\nw_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, q, t, R]\n}\n\\end{equation}\n\n\\noindent where, using $Q_3 = Q_1 Q_{ {    \\scriptstyle M} } Q_2$, $w^{num}$ is \n\n\\begin{eqnarray}\n\\label{w.numerator.5D}\nw^{\\, num} [{\\bf V, \\, W, \\, \\Delta}, q, t, R] \n& = & \n\\prod_{\\square \\in V^{\\, 1}}  \\left\\lgroup 1-Q_1    \\, q^{\\, -A_{\\square, W^{\\, 1}}^+} \\, t^{\\, -L_{\\square, V^{\\, 1}}^+} \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 1}}  \\left\\lgroup 1-Q_1    \\, q^{\\,  A_{\\blacksquare, V^{\\, 1}}^+} \\, t^{\\,  L_{\\blacksquare, W^{\\, 1}}^+} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 2}}  \\left\\lgroup 1-Q_2    \\, q^{\\, -A_{\\square, W^{\\, 2}}^+} \\, t^{\\, -L_{\\square, V^{\\, 2}}^+} \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 2}}  \\left\\lgroup 1-Q_2    \\, q^{\\,  A_{\\blacksquare, V^{\\, 2}}^+} \\, t^{\\,  L_{\\blacksquare, W^{\\, 2}}^+} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\blacksquare \\in W^{\\, 1}}  \\left\\lgroup 1-Q_{ {    \\scriptstyle M} } \\, q^{\\, -A_{\\blacksquare, V^{\\, 2}}^+} \\, t^{\\, -L_{\\blacksquare, W^{\\, 1}}^+} \\right\\rgroup\n\\prod_{\\square \\in V^{\\, 2}}  \\left\\lgroup 1-Q_{ {    \\scriptstyle M} } \\, q^{\\,  A_{\\square, W^{\\, 1}}^+} \\, t^{\\,  L_{\\square, V^{\\, 2}}^+} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 1}}  \\left\\lgroup 1- Q_3    \\, q^{\\, -A_{\\square, W^{\\, 2}}^+} \\, t^{\\, -L_{\\square, V^{\\, 1}}^+} \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 2}}  \\left\\lgroup 1- Q_3    \\, q^{\\,  A_{\\blacksquare, V^{\\, 1}}^+} \\, t^{\\,  L_{\\blacksquare, W^{\\, 2}}^+} \\right\\rgroup, \n\\end{eqnarray}\n\n\\noindent where we have used the second equality in Equation {\\bf \\ref{nakajima.identity}} \nto put the products in the above uniform form. The denominator $w_{den}$ is \n\n\\begin{eqnarray}\n\\label{w.denominator.5D}\nw_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, q, t, R] \n& = &\n\\prod_{\\square \\in V^{\\, 1}}  \\left\\lgroup 1-                 q^{\\, A_{\\square,   V^{\\, 1}}^{++}} \\, t^{\\, L_{\\square, V^{\\, 1}}^{  }} \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 1}}  \\left\\lgroup 1-                 q^{\\, A_{\\blacksquare,   W^{\\, 1}}     } \\, t^{\\, L_{\\blacksquare, W^{\\, 1}}^{++}} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 2}}  \\left\\lgroup 1-                 q^{\\, A_{\\square,   V^{\\, 2}}^{++}} \\, t^{\\, L_{\\square, V^{\\, 2}}^{  }} \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 2}}  \\left\\lgroup 1-                 q^{\\, A_{\\blacksquare,   W^{\\, 2}}     } \\, t^{\\, L_{\\blacksquare, W^{\\, 2}}^{++}} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 1}}  \\left\\lgroup 1-Q_1 \\, Q_{ {    \\scriptstyle M} } \\, q^{\\, -A_{\\square, V^{\\, 2}}^{++}} \\, t^{\\, -L_{\\square, V^{\\, 1}}     } \\right\\rgroup\n\\prod_{\\square \\in V^{\\, 2}}  \\left\\lgroup 1-Q_1 \\, Q_{ {    \\scriptstyle M} } \\, q^{\\,  A_{\\square, V^{\\, 1}}     } \\, t^{\\,  L_{\\square, V^{\\, 2}}^{++}} \\right\\rgroup \n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\blacksquare \\in W^{\\, 1}}  \\left\\lgroup 1-Q_{ {    \\scriptstyle M} } \\, Q_2 \\, q^{\\, -A_{\\blacksquare, W^{\\, 2}}     } \\, t^{\\, -L_{\\blacksquare, W^{\\, 1}}^{++}} \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 2}}  \\left\\lgroup 1-Q_{ {    \\scriptstyle M} } \\, Q_2 \\, q^{\\,  A_{\\blacksquare, W^{\\, 1}}^{++}} \\, t^{\\,  L_{\\blacksquare, W^{\\, 2}}     } \\right\\rgroup\n\\end{eqnarray}\n\n\\noindent where the first four products on the right hand side of Equation {\\bf \\ref{w.denominator.5D}} \nare due to the product \n$ Z_{  V^{\\, 1}                  }[t, q]\\,\n  Z_{  W^{\\, 1 \\, \\intercal}     }[q, t]\\, \n  Z_{  V^{\\, 2}                  }[t, q]\\, \n  Z_{  W^{\\, 2 \\, \\intercal}     }[q, t]$ in Equation {\\bf \\ref{web.02}}.\n$w_{den}$ is equal to the denominator $\\mathcal{W}_{\\, \\bf \\emptyset \\, \\emptyset \\, \\Delta} [q, t, R]$ on the right hand \nside of Equation {\\bf \\ref{W.norm}}.\n\n\\subsubsection{Remark} One can glue copies of the basic web partition function \n$\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}[q, t, R]$ in several ways. In this note, we \nrestrict our attention to gluing linearly or cyclically, to form linear or \ncyclic $U(2)$ quiver gauge theories, as described in paragraphs \n{\\bf \\ref{linear.conformal.blocks}} and {\\bf \\ref{cyclic.conformal.blocks}}.\nWe do not, for example, glue basic webs to form a Hirzbruch surface.\n\n\\section{A 4D $U(2)$ basic web partition function}\n\\label{section.04.4d.web}\n{\\it \nWe take the $R \\! \\rightarrow \\! 0$ limit of the 5D basic web partition function to obtain \nits 4D analogue. \n}\n\n\\subsection{Two parameters} We take the relationship between the parameters $q$ and $t$ of \nthe refined vertex and the parameters $\\epsilon_1$ and $\\epsilon_2$ of the instanton \npartition function to be,\n\\begin{equation}\n\\boxed{\nq=e^{\\,   R \\epsilon_2},\n\\quad\nt=e^{\\, - R \\epsilon_1}\n}\n\\end{equation} \n\n\\noindent where $R$, the radius of a space-like circle, plays the role of a deformation parameter. \nWe write $\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}^{\\, norm} [\\epsilon_1, \\epsilon_2, R]$, then take the limit \n$R \\! \\rightarrow \\! 0$. The prefactor on the left hand side of Equation {\\bf \\ref{W.5D.norm}} tends \nto 1 in the limit $R \\rightarrow 0$, and we obtain\n\n\\begin{equation}\n\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}^{\\, norm} [\\epsilon_1, \\epsilon_2, R \\rightarrow 0] \n=\n\\frac{\nw^{\\, num} [{\\bf V, \\, W, \\, \\Delta}, \\epsilon_1, \\epsilon_2, R \\rightarrow 0] \n}{\nw_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\epsilon_1, \\epsilon_2, R \\rightarrow 0] \n}\n\\label{4.15}\n\\end{equation}\n\n\\noindent where, using $\\Delta_3 = \\Delta_1 + \\Delta_{ {    \\scriptstyle M} } + \\Delta_2$, we have\n\n\\begin{multline}\n\\label{w.numerator.4D}\nw^{\\, num} [{\\bf V}, {\\bf W}, {\\bf \\Delta}, \\epsilon_1, \\epsilon_2, R \\rightarrow 0] \n= \n\\\\\n\\prod_{\\square \\in V^{\\, 1}}  \\left\\lgroup \\Delta_1     + A^+_{\\square, W^{\\, 1}} \\epsilon_2 - L^+_{\\square, V^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 1}}  \\left\\lgroup \\Delta_1     - A^+_{\\blacksquare, V^{\\, 1}} \\epsilon_2 + L^+_{\\blacksquare, W^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\\\\n\\times   \n\\prod_{\\square \\in V^{\\, 2}}  \\left\\lgroup \\Delta_2     + A^+_{\\square, W^{\\, 2}} \\epsilon_2 - L^+_{\\square, V^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 2}}  \\left\\lgroup \\Delta_2     - A^+_{\\blacksquare, V^{\\, 2}} \\epsilon_2 + L^+_{\\blacksquare, W^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\\\\n\\times   \n\\prod_{\\square \\in V^{\\, 2}}  \\left\\lgroup \\Delta_{ {    \\scriptstyle M} } - A^+_{\\square, W^{\\, 1}} \\epsilon_2 + L^+_{\\square, V^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 1}}  \\left\\lgroup \\Delta_{ {    \\scriptstyle M} } + A^+_{\\blacksquare, V^{\\, 2}} \\epsilon_2 - L^+_{\\blacksquare, W^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\\\\n\\times \n\\prod_{\\square \\in V^{\\, 1}}  \\left\\lgroup \\Delta_3     + A^+_{\\square, W^{\\, 2}} \\epsilon_2 - L^+_{\\square, V^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 2}}  \\left\\lgroup \\Delta_3     - A^+_{\\blacksquare, V^{\\, 1}} \\epsilon_2 + L^+_{\\blacksquare, W^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\end{multline}\n\n\\noindent and\n\n\\begin{multline}\n\\label{w.denominator.4D}\nw_{\\, den} [{\\bf V}, {\\bf W}, {\\bf \\Delta}, \\epsilon_1, \\epsilon_2, R \\rightarrow 0] \n=   \n\\\\\n\\prod_{\\square \\in V^{\\, 1}}  \\left\\lgroup                        - A^{++}_{\\square, V^{\\, 1}} \\epsilon_2 + L^{  }_{\\square, V^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\square \\in V^{\\, 2}}  \\left\\lgroup                        - A^{++}_{\\square, V^{\\, 2}} \\epsilon_2 + L^{  }_{\\square, V^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\\\\n\\times   \n\\prod_{\\blacksquare \\in W^{\\, 1}}  \\left\\lgroup                        - A^{  }_{\\blacksquare, W^{\\, 1}} \\epsilon_2 + L^{++}_{\\blacksquare, W^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 2}}  \\left\\lgroup                        - A^{  }_{\\blacksquare, W^{\\, 2}} \\epsilon_2 + L^{++}_{\\blacksquare, W^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\\\\n\\times   \n\\prod_{\\square \\in V^{\\, 1}}  \\left\\lgroup \\Delta_1 + \\Delta_{ {    \\scriptstyle M} } + A^{++}_{\\square, V^{\\, 2}} \\epsilon_2 - L^{  }_{\\square, V^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\square \\in V^{\\, 2}}  \\left\\lgroup \\Delta_1 + \\Delta_{ {    \\scriptstyle M} } - A^{  }_{\\square, V^{\\, 1}} \\epsilon_2 + L^{++}_{\\square, V^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\\\\n\\times   \n\\prod_{\\blacksquare \\in W^{\\, 1}}  \\left\\lgroup \\Delta_2 + \\Delta_{ {    \\scriptstyle M} } + A^{  }_{\\blacksquare, W^{\\, 2}} \\epsilon_2 - L^{++}_{\\blacksquare, W^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 2}}  \\left\\lgroup \\Delta_2 + \\Delta_{ {    \\scriptstyle M} } - A^{++}_{\\blacksquare, W^{\\, 1}} \\epsilon_2 + L^{  }_{\\blacksquare, W^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\end{multline}\n\n\\section{The building block of the 4D $U(2)$ quiver instanton partition function}\n\\label{section.05.nekrasov.partition.functions}\n{\\it \nWe recall the normalised contribution of the bifundamental hypermultiplet which acts as a building block \nof the instanton partition function.\n}\n\nIn the notation of Equation {\\bf 6} and Section {\\bf 2} of \\cite{bershtein.foda}, the normalised bifundamental \npartition function \n$\\mathcal{Z}^{\\, 4D}_{building.block}$ is \n\\begin{equation}\n\\label{z.bb}\n\\mathcal{Z}^{\\, 4D}_{building.block}   \\left\\lgroup {\\bf a}, {\\bf V^{\\prime}} \\ | \\ \\mu \\ | \\ {\\bf b}, {\\bf W^{\\prime}} \\right\\rgroup = \n\\frac{\nz^{\\, num}  \\left\\lgroup {\\bf a}, {\\bf V^{\\prime}} \\ | \\ \\mu \\ | \\ {\\bf b}, {\\bf W^{\\prime}} \\right\\rgroup\n}\n{\nz_{\\, den}  \\left\\lgroup {\\bf a}, {\\bf V^{\\prime}} \\ |           \\ {\\bf b}, {\\bf W^{\\prime}} \\right\\rgroup\n} \n\\end{equation}\n\n\\noindent where ${\\bf a} = \\{a, -a\\}$, ${\\bf b} = \\{b, -b\\}$, $a, b$ and $\\mu$ are linear combinations \nof $\\epsilon_1$ and $\\epsilon_2$, as will be explained in \nSection {\\bf \\ref{section.08.restricted.instanton.partition.functions}} below, and we use \n${\\bf V^{\\prime}}$ and \n${\\bf W^{\\prime}}$ for partition pairs that we will relate in \nSection {\\bf \\ref{section.07.identification.denominators}} to the pairs that appear in the 4D basic web. \nWe refer to \\cite{bershtein.foda} for brief explanations of the parameters that appear in \n$\\mathcal{Z}^{\\, 4D}_{building.block}$. Defining\n\n\\begin{equation}\n\\alpha_0 = \\frac12  \\left\\lgroup \\epsilon_1 + \\epsilon_2 \\right\\rgroup\n\\end{equation}\n\n\\noindent the numerator $z^{\\, num}$, as given in Equation {\\bf 9} of \\cite{bershtein.foda}, is\n\n\\begin{multline}\n\\label{z.numerator}\nz^{\\, num}  \\left\\lgroup {\\bf a}, {\\bf V^{\\prime}}\\  | \\ \\mu \\ | \\ {\\bf b}, {\\bf W^{\\prime}} \\right\\rgroup  = \n\\\\\n\\prod_{\\square \\in V^{\\, 1 \\, \\prime}}  \\left\\lgroup [  a - b - \\mu + \\alpha_0] + A^+_{\\square, V^{\\, 1 \\, \\prime}} \\epsilon_1 \n                                                                  - L^+_{\\square, W^{\\, 1 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 1 \\, \\prime}}  \\left\\lgroup [  a - b - \\mu + \\alpha_0] - A^+_{\\blacksquare, W^{\\, 1 \\, \\prime}} \\epsilon_1 \n                                                                  + L^+_{\\blacksquare, V^{\\, 1 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\\\\n\\times \n\\prod_{\\square \\in V^{\\, 2 \\, \\prime}}  \\left\\lgroup [- a + b - \\mu + \\alpha_0] + A^+_{\\square, V^{\\, 2 \\, \\prime}} \\epsilon_1 \n                                                                  - L^+_{\\square, W^{\\, 2 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 2 \\, \\prime}}  \\left\\lgroup [- a + b - \\mu + \\alpha_0] - A^+_{\\blacksquare, W^{\\, 2 \\, \\prime}} \\epsilon_1 \n                                                                  + L^+_{\\blacksquare, V^{\\, 2 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\\\\n\\times \n\\prod_{\\square \\in V^{\\, 2 \\, \\prime}}  \\left\\lgroup [- a - b - \\mu + \\alpha_0] + A^+_{\\square, V^{\\, 2 \\, \\prime}} \\epsilon_1 \n                                                                  - L^+_{\\square, W^{\\, 1 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 1 \\, \\prime}}  \\left\\lgroup [- a - b - \\mu + \\alpha_0] - A^+_{\\blacksquare, W^{\\, 1 \\, \\prime}} \\epsilon_1 \n                                                                  + L^+_{\\blacksquare, V^{\\, 2 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\\\\n\\times\n\\prod_{\\square \\in V^{\\, 1 \\, \\prime}}  \\left\\lgroup [  a + b - \\mu + \\alpha_0] + A^+_{\\square, V^{\\, 1 \\, \\prime}} \\epsilon_1 \n                                                                  - L^+_{\\square, W^{\\, 2 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 2 \\, \\prime}}  \\left\\lgroup [  a + b - \\mu + \\alpha_0] - A^+_{\\blacksquare, W^{\\, 2 \\, \\prime}} \\epsilon_1 \n                                                                  + L^+_{\\blacksquare, V^{\\, 1 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\end{multline}\n\nThe denominator $z_{\\, den}$, as given in Equation {\\bf 7} of \\cite{bershtein.foda}, is\n\n\\begin{equation}\n\\label{z.denominator}\nz_{\\, den}  \\left\\lgroup {\\bf a}, {\\bf V^{\\prime}} \\ | \\ {\\bf b}, {\\bf W^{\\prime}} \\right\\rgroup\n= \n \\left\\lgroup \nz^{\\, num}  \\left\\lgroup {\\bf a}, {\\bf V^{\\prime}} \\ | \\  0 \\ | \\ {\\bf a}, {\\bf V^{\\prime}} \\right\\rgroup \\ \nz^{\\, num}  \\left\\lgroup {\\bf b}, {\\bf W^{\\prime}} \\ | \\  0 \\ | \\ {\\bf b}, {\\bf W^{\\prime}} \\right\\rgroup \\ \n\\right\\rgroup^{\\frac{1}{2}}\n\\end{equation}\n\n\\section{Identification of $\\mathcal{W}_{\\bf \\, V \\, W \\, \\Delta}^{\\, norm}$ and $\\mathcal{Z}^{\\, 4D}_{building.block}$. The numerators}\n\\label{section.06.identification.numerators}\n\nComparing Equation {\\bf \\ref{w.numerator.4D}} and Equation {\\bf \\ref{z.numerator}}, we find that if \nwe set  \n$V^{\\, i \\, \\prime} = V^{\\, i \\, \\intercal}$, \n$W^{\\, i \\, \\prime} = W^{\\, i \\, \\intercal}$, and multiply each factor by $-1$, which is possible \nsince the number of factors is even by construction, we obtain\n\n\\begin{eqnarray}\n\\label{numerator.nekrasov.02}\n&        & \n\\\\\n&        & z^{\\, num}  \\left\\lgroup {\\bf a}, {\\bf V}\\,  | \\ \\mu \\ | \\, {\\bf b}, {\\bf W} \\right\\rgroup =\n\\nonumber\n\\\\\n&        &\n\\prod_{\\square \\in V^{\\, 1}}  \\left\\lgroup [- a + b + \\mu - \\alpha_0] + A^+_{\\square, W^{\\, 1}} \\epsilon_2 - L^+_{\\square, V^{\\, 1}} \\epsilon_1 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 1}}  \\left\\lgroup [- a + b + \\mu - \\alpha_0] - A^+_{\\blacksquare, V^{\\, 1}} \\epsilon_2 + L^+_{\\blacksquare, W^{\\, 1}} \\epsilon_1 \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 2}}  \\left\\lgroup [  a - b + \\mu - \\alpha_0] + A^+_{\\square, W^{\\, 2}} \\epsilon_2 - L^+_{\\square, V^{\\, 2}} \\epsilon_1 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 2}}  \\left\\lgroup [  a - b + \\mu - \\alpha_0] - A^+_{\\blacksquare, V^{\\, 2}} \\epsilon_2 + L^+_{\\blacksquare, W^{\\, 2}} \\epsilon_1 \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 2}}  \\left\\lgroup [  a + b + \\mu - \\alpha_0] + A^+_{\\square, W^{\\, 1}} \\epsilon_2 - L^+_{\\square, V^{\\, 2}} \\epsilon_1 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 1}}  \\left\\lgroup [  a + b + \\mu - \\alpha_0] - A^+_{\\blacksquare, V^{\\, 2}} \\epsilon_2 + L^+_{\\blacksquare, W^{\\, 1}} \\epsilon_1 \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 1}}  \\left\\lgroup [- a - b + \\mu - \\alpha_0] + A^+_{\\square, W^{\\, 2}} \\epsilon_2 - L^+_{\\square, V^{\\, 1}} \\epsilon_1 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 2}}  \\left\\lgroup [- a - b + \\mu - \\alpha_0] - A^+_{\\blacksquare, V^{\\, 1}} \\epsilon_2 + L^+_{\\blacksquare, W^{\\, 2}} \\epsilon_1 \\right\\rgroup\n\\nonumber\n\\end{eqnarray}\n\n\\noindent which leads to the identification\n\n\\begin{equation}\n\\label{identification.12}\n\\boxed{\n\\Delta_1     =          -   a + b + \\mu - \\alpha_0, \\quad\n\\Delta_2     = \\phantom{-}  a - b + \\mu - \\alpha_0, \\quad\n\\Delta_{ {    \\scriptstyle M} } =          -   a - b - \\mu + \\alpha_0\n}\n\\end{equation}\n\n\\section{Identification of $\\mathcal{W}_{\\bf \\, V \\, W \\, \\Delta}^{\\, norm}$ and $\\mathcal{Z}^{\\, 4D}_{building.block}$. The denominators}\n\\label{section.07.identification.denominators}\n\nUsing the identification of parameters obtained in Equation {\\bf \\ref{identification.12}} in\n$w_{\\, den} [{\\bf V, \\, W, \\, \\Delta}]$ \nand $z_{\\, den}$ as given in Equations {\\bf \\ref{w.denominator.4D}} and {\\bf \\ref{z.denominator}}, \nit is clear that these two functions are not the same. However, what matters is not the denominator \nof s single factor, but the product of all denominators, as we explain below.\n\nThe denominator $w_{\\, den} [{\\bf V, \\, W, \\, \\Delta}]$ is a natural object, as we can see in the \nderivation in Section {\\bf \\ref{section.03.5d.web}}. \nOn the other hand, the denominator $z_{\\, den}$ was obtained in \\cite{bershtein.foda} by taking the \nfull denominator that appears in expressions for the 4D $U(1)$ linear and cyclic quiver instanton \npartition functions and factoring that into denominators for the contributions of the bifundamental \nhypermultiplets.  Such a factorisation is not unique and any factorisation is allowed for as long as \nthe product of all factors is equal to the full denominator of the original expression. \n\nIn this work, to identify $\\mathcal{W}^{\\, 4D}$ and $\\mathcal{Z}^{\\, 4D}_{building.block}$, we need \na factor $\\mathcal{F}$, such that the product of all normalisation factors that appear in a conformal block \nis equal to 1. Consider the abbreviations\n\\begin{eqnarray}\n\\mathcal{A}_{V^{ij}} [x] = \\mathcal{A}_{V^i, V^j} [x] & = & \\prod_{\\square \\in V^i}  \\left\\lgroup x + A^{++}_{\\square, V^i} \\epsilon_2\n- L^{  }_{\\square, V^j} \\epsilon_1 \\right\\rgroup,\n\\quad\n\\mathcal{A}_{\\emptyset, \\emptyset} [x] = 1,\n\\\\\n\\mathcal{L}_{W^{ij}} [x] = \\mathcal{L}_{W^i, W^j} [x] & = & \\prod_{\\blacksquare \\in W^j}  \\left\\lgroup x - A^{  }_{\\blacksquare, W^j} \\epsilon_2\n+ L^{++}_{\\blacksquare, W^i} \\epsilon_1 \\right\\rgroup,\n\\quad\n\\mathcal{L}_{\\emptyset, \\emptyset} [x] = 1,\n\\nonumber\n\\\\\n\\mathcal{H}_{Y^{ij}} [x] & = & \\mathcal{A}_{Y^{ij}} [x] \\, \\mathcal{L}_{Y^{ij}} [x]\n\\nonumber\n\\end{eqnarray}\nIn this notation, $z_{\\, den}$ and $w_{\\, den}$ are \n\\begin{multline}\nz_{\\, den}  \\left\\lgroup {\\bf a}, {\\bf V} \\ | \\ 0 \\ | \\ {\\bf b}, {\\bf W} \\right\\rgroup =\n\\\\\n \\left\\lgroup\n\\mathcal{H}_{V^{11}} [0] \\, \\mathcal{H}_{V^{12}} [2a] \\, \\mathcal{H}_{V^{12}} [-2a] \\, \\mathcal{H}_{V^{22}} [0]\n\\mathcal{H}_{W^{11}} [0] \\, \\mathcal{H}_{W^{12}} [2b] \\, \\mathcal{H}_{W^{12}} [-2b] \\, \\mathcal{H}_{W^{22}} [0]\n\\right\\rgroup^{\\frac12}\n\\end{multline}\n\n\\noindent and \n\\begin{multline}\nw_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\epsilon_1, \\epsilon_2, R \\rightarrow 0] = \nw_{\\, den}  \\left\\lgroup {\\bf a}, {\\bf V} \\, | \\, {\\bf b}, {\\bf W} \\right\\rgroup =\n\\\\\n \\left\\lgroup - \\right\\rgroup^{\\, | V^{\\, 1} | + | V^{\\, 2} | + | W^{\\, 1} | + | W^{\\, 2} |}\n\\mathcal{L}_{V^{11}} [  0] \\,\n\\mathcal{L}_{V^{22}} [  0] \\,\n\\mathcal{H}_{V^{12}} [-2a] \\,\n\\mathcal{H}_{W^{12}} [ 2b] \\,\n\\mathcal{A}_{W^{11}} [  0] \\,\n\\mathcal{A}_{W^{22}} [  0]\n\\end{multline}\n\nNow consider the factor\n\n\\begin{equation}\n\\mathcal{F}  \\left\\lgroup {\\bf a}, {\\bf V} \\, | \\, {\\bf b}, {\\bf W} \\right\\rgroup\n=\n\\frac{\nz_{\\, den} \\left[ {\\bf a}, {\\bf V} \\, | \\, {\\bf b}, {\\bf W} \\right]\n}{\nw_{\\, den} \\left[ {\\bf a}, {\\bf V} \\, | \\, {\\bf b}, {\\bf W} \\right]\n}\n\\end{equation}\n\n\\noindent and define \n\n\\begin{equation}\nz_{\\, den}^{\\prime} = {\\mathcal{F}}^{-1} \\, z_{\\, den} = w_{\\, den}, \\quad\n\\mathcal{Z}_{building.block }^{\\prime} =  \\mathcal{F}       \\, \\mathcal{Z}_{building.block}\n\\end{equation}\n\n\\noindent $\\mathcal{Z}^{\\, \\prime}_{building.block}$ is constructed such that \n{\\bf 1.} It has the same numerator as $\\mathcal{Z}^{\\, 4D}_{building.block}$, which \nis the same as that of $\\mathcal{W}^{\\, norm}_{\\bf V W \\Delta}$, \nwhen we choose the parameters as in Equation {\\bf \\ref{identification.12}} and\n{\\bf 2.} It has the same denominator as $\\mathcal{W}^{\\, norm}_{\\bf V W \\Delta}$, also \nwhen we choose the parameters as in Equation {\\bf \\ref{identification.12}}.\n\nSince the denominator of $\\mathcal{Z}^{\\, \\prime}_{building.block}$ is not manifestly \nthe same as that of $\\mathcal{Z}^{\\,     4D}_{building.block}$, we need to show that \ngluing copies    of $\\mathcal{Z}^{\\, \\prime}_{building.block}$ to build a topological \npartition function, leads to the same result obtained by gluing copies of the \noriginal            $\\mathcal{Z}^{         }_{building.block}$. $\\mathcal{F}$ can be written \nin a simpler form as follows,\n\n\\begin{equation}\n \\left\\lgroup \\mathcal{F}  \\left\\lgroup {\\bf a}, {\\bf V} \\, | \\, {\\bf b}, {\\bf W} \\right\\rgroup \\right\\rgroup^{\\, 2} \n\\\\\n=\nF^{\\,  left}_{\\bf V} [a] \\,\nF^{\\, right}_{\\bf W} [b]\n\\end{equation}\n\n\\noindent where \n\n\\begin{eqnarray}\nF^{\\, left}_{\\bf V} [a] & = &\n \\left\\lgroup - \\right\\rgroup^{\\, | V^{\\, 1} | + | V^{\\, 2} | }\n \\left\\lgroup\n\\frac{\n\\mathcal{A}_{V^{11}} [0] \\,\n\\mathcal{A}_{V^{22}} [0]\n}{\n\\mathcal{L}_{V^{11}} [0] \\,\n\\mathcal{L}_{V^{22}} [0]\n}\n\\right\\rgroup \\, \n \\left\\lgroup\n\\frac{\n\\mathcal{H}_{V^{12}} [ 2a]\n}{\n\\mathcal{H}_{V^{12}} [-2a]\n}\n\\right\\rgroup \n\\\\\nF^{\\, right}_{\\bf W} [b] & = &\n \\left\\lgroup - \\right\\rgroup^{\\, | W^{\\, 1} | + | W^{\\, 2} |}\n \\left\\lgroup\n\\frac{\n\\mathcal{H}_{W^{12}} [-2b]\n}{\n\\mathcal{H}_{W^{12}} [ 2b]\n}\n\\right\\rgroup \\, \n \\left\\lgroup\n\\frac{\n\\mathcal{L}_{W^{11}} [0] \\,\n\\mathcal{L}_{W^{22}} [0]\n}{\n\\mathcal{A}_{W^{11}} [0] \\,\n\\mathcal{A}_{W^{22}} [0]\n}\n\\right\\rgroup\n\\end{eqnarray}\n\n\\noindent $F^{\\, right}$ and $F^{\\, left }$ satisfy the obvious properties \n\n\\begin{equation}\n\\label{property.00}\nF^{\\, right}_{\\bf \\emptyset} [x] =\nF^{\\, left }_{\\bf \\emptyset} [x] = 1\n\\end{equation}\n\n\\noindent and\n\n\\begin{equation}\n\\label{property.01}\nF^{\\, right}_{\\bf Y} [x] \\, F^{\\, left }_{\\bf Y} [x] = 1\n\\end{equation}\n\nThe physical objects that we are interested in are the conformal blocks which are constructed \nby gluing copies of $\\mathcal{Z}_{building.block}$ \\cite{bershtein.foda}. We need to show that gluing \ncopies of $\\mathcal{Z}_{building.block}^{\\prime}$ leads to the same result, which will be the case if \nproducts of the normalisation factors trivialise. This will follow directly from Equations \n{\\bf \\ref{property.00}} and {\\bf \\ref{property.01}}. There are two cases to consider, the \nlinear conformal block case and the cyclic conformal block case.\n\n\\subsubsection{Linear conformal blocks}\n\\label{linear.conformal.blocks}\nConsider the linear conformal block obtained by gluing $n$ copies of $\\mathcal{Z}_{building.block}$, \nthat is $\\mathcal{Z}_{building.block.1}$, $\\mathcal{Z}_{building.block.2}$, $\\cdots$, $\\mathcal{Z}_{building.block.n}$, \nsequentially. Using copies of $\\mathcal{Z}_{building.block}^{\\prime}$, we obtain the same result as \nusing copies of $\\mathcal{Z}_{building.block}$ up to a factor \n\n\\begin{equation}\nF^{\\, left}_{\\bf  \\emptyset} [x_0    ] \\, F^{\\, right}_{\\bf    Y^{\\, 1}} [x_1] \nF^{\\, left}_{\\bf     Y^{\\, 1}} [x_1    ] \\, F^{\\, right}_{\\bf    Y^{\\, 2}} [x_2] \n\\cdots\nF^{\\, left}_{\\bf Y^{n-1}} [x_{n-1}] \\, F^{\\, right}_{\\bf \\emptyset} [x_n] \n= 1\n\\end{equation}\n\n\\subsubsection{Cyclic conformal blocks}\n\\label{cyclic.conformal.blocks}\n\n\\begin{equation}\nF^{\\, left}_{\\bf  Y^0} [x_0] \\, F^{\\, right}_{\\bf    Y^{\\, 1}} [x_1]\nF^{\\, left}_{\\bf  Y^{\\, 1}} [x_1] \\, F^{\\, right}_{\\bf    Y^{\\, 2}} [x_2]\n\\cdots\nF^{\\, left}_{\\bf Y^{n-1}} [x_{n-1}] \\, F^{\\, right}_{\\bf Y^0} [x_0]\n= 1\n\\end{equation}\n\nWe conclude that $\\mathcal{Z}_{building.block}^{\\prime}$ leads to the same conformal blocks as $\\mathcal{Z}_{building.block}$.  \n\n\\subsubsection{The denominator $z_{\\, den}^{\\prime}$ and the Burge conditions}\nIn \\cite{alkalaev.belavin, bershtein.foda}, it was shown that for the choice of parameters that \nleads to Virasoro $A$-series minimal conformal blocks, the denominator $z_{\\, den}$ will contain non-physical \nzeros, unless we restrict the partition pairs, that $\\mathcal{Z}^{\\, 4D, \\, min}_{building.block}$ depends on, to obey \nBurge conditions. These conditions were derived in \\cite{bershtein.foda} using $z_{\\, den}$ rather than \n$z_{\\, den}^{\\prime}$. Using $z_{\\, den}^{\\prime}$ leads to the same conditions, since the product of all \n$z_{\\, den}^{\\prime}$ is the same as the product of all $z_{\\, den}$ that show up in the conformal block\n\\footnote{\\,\nIn Section {\\bf \\ref{section.10.burge.pairs}}, we outline the derivation of the Burge conditions from \n$z_{\\, den}^{\\prime}$.}.\n\n\\section{Restricted instanton partition functions for $\\mathcal{M}^{\\, p, \\, p^{\\prime}, \\, \\mathcal{H}}$. The parameters}\n\\label{section.08.restricted.instanton.partition.functions}\n\n{\\it \nWe recall the choice of parameters such that $\\mathcal{Z}^{\\, 4D}_{building.block}$ reduces to \n$\\mathcal{Z}^{\\, 4D, \\, min}_{building.block}$ which is the building block of Virasoro $A$-series minimal \nmodel conformal blocks times Heisenberg factors.\n}\n\n\\subsection{AGT parameterisation of minimal models}\n\\label{parameterisation.minimal.models}\n\nA minimal model $\\mathcal{M}^{\\, p, \\, p^{\\prime}}$, based on a Virasoro algebra $\\mathcal{V}^{\\, p, \\, p^{\\prime}}$, characterised \nby a central charge $c_{p, \\, p^{\\prime}} < 1$, that we parameterise as\n\n\\begin{equation}\n\\label{central.charge.minimal}\nc_{p, \\, p^{\\prime}} = 1 - 6  \\left\\lgroup a_{p, \\, p^{\\prime}} - \\frac{1}{a_{p, \\, p^{\\prime}}} \\right\\rgroup^{\\, 2},\n\\quad\na_{p, \\, p^{\\prime}} =  \\left\\lgroup \\frac{p^{\\prime}}{p} \\right\\rgroup^{\\frac12},\n\\end{equation}\n\n\\noindent where $\\{p, \\, p^{\\prime}\\}$ are co-prime integers that satisfy $0 < p < p^{\\prime}$. In the Coulomb \ngas approach to computing conformal blocks in minimal models with $c_{p, \\, p^{\\prime}} < 1$ \n\\cite{nienhuis, dotsenko.fateev}, the screening charges $\\{\\alpha_{+}, \\alpha_{-}\\}$, and the background \ncharge $\\alpha_{back.ground}$, satisfy \n\n\\begin{equation}\n\\label{minimal.model.charges}\n\\boxed{\n\\alpha_{+} =             a_{p, \\, p^{\\prime}}, \n\\quad\n\\alpha_{-} = -  \\frac{1}{a_{p, \\, p^{\\prime}}},\n\\quad\n\\alpha_{back.ground} = - 2 \\alpha_0, \n\\quad \n\\alpha_0  = \\frac12  \\left\\lgroup \\alpha_{+} + \\alpha_{-} \\right\\rgroup \n}\n\\end{equation}\n\n\\noindent The AGT parameterisation of $\\mathcal{M}^{\\, p, \\, p^{\\prime}, \\, \\mathcal{H}}$ is obtained by choosing \n\n\\begin{equation}\n\\label{neg.0.pos}\n\\boxed{\n\\epsilon_1 < 0 < \\epsilon_2, \n\\quad\n\\epsilon_1 = \\alpha_{-}, \n\\quad\n\\epsilon_2 = \\alpha_{+} \n}\n\\end{equation}\n\n\\noindent so that $\\alpha_{-} < 0 < \\alpha_{+}$. Since we focus on $\\mathcal{M}^{\\, p, \\, p^{\\prime}, \\, \\mathcal{H}}$, \nwe work in terms of $\\{\\alpha_{-}, \\alpha_{+}\\}$ instead of $\\{ \\epsilon_1, \\epsilon_2\\}$.\n\n\\subsection{Two sets of charges in minimal models}\n\\label{charge.content}\nWe consider two types of charges that, in Coulomb gas terms, are expressed in terms of \nthe screening charges $\\{\\alpha_{+}, \\alpha_{-}\\}$. \n{\\bf 1.} The charge $a_{r, s}$ of the highest weight $|a_{r, s} \\rangle$ of the irreducible\nhighest weight representation $\\mathcal{H}^{\\, p, \\, p^{\\prime}}_{r, s}$, and\n{\\bf 2.} The charge $\\mu_{r, s}$ of the vertex operator ${\\mathcal{O}}_{\\mu}$ that intertwines \ntwo highest weight ireducible representations \n$\\mathcal{H}^{\\, p, \\, p^{\\prime}}_{r_1, s_1}$  and \n$\\mathcal{H}^{\\, p, \\, p^{\\prime}}_{r_2, s_2}$.\nThese charges are parameterised in terms of $\\alpha_{+}$ and $\\alpha_{-}$ as follows \n\n\\begin{equation}\n\\label{parameters.03}\n\\boxed{\n  a_{r, s} = - \\frac{r}{2} \\, \\alpha_{+} - \\frac{s}{2} \\, \\alpha_{-},            \\quad \n\\mu_{r, s} = - \\frac{r}{2} \\, \\alpha_{+} - \\frac{s}{2} \\, \\alpha_{-} + \\alpha_0, \\quad \n1 \\leqslant r \\leqslant   p - 1,\n1 \\leqslant s \\leqslant p^{\\prime} - 1\n}\n\\end{equation}\n\n\\section{From gauge theory parameters to minimal model parameters}\n\\label{section.09.from.gauge.theory.to.minimal.model}\n{\\it We compare the parameters of $\\mathcal{W}^{\\, 4D}$ and the parameters of $\\mathcal{Z}^{\\, 4D}_{building.block}$.}\n\nWe set\n\n\\begin{equation}\n\\label{notation.02.01}\na = -  \\left\\lgroup \\frac{r_a}{2}        \\right\\rgroup \\, \\alpha_{+} -  \\left\\lgroup \\frac{s_a}{2}     \\right\\rgroup \\, \\alpha_{-}, \n\\quad\nr_a \\in \\{1, 2, \\cdots,    p - 1\\}, \n\\quad\ns_a \\in \\{1, 2, \\cdots,  p^{\\prime} - 1\\}, \n\\end{equation}\n\n\\begin{equation}\n\\label{notation.02.02}\nb    = -  \\left\\lgroup \\frac{r_b}{2}     \\right\\rgroup \\, \\alpha_{+} -  \\left\\lgroup \\frac{s_b}{2}     \\right\\rgroup \\, \\alpha_{-}, \n\\quad\nr_b \\in \\{1, 2, \\cdots,   p - 1\\},\n\\quad\ns_b \\in \\{1, 2, \\cdots, p^{\\prime} - 1\\},\n\\end{equation}\n\n\\begin{equation}\n\\label{notation.02.03}\n\\mu  = -  \\left\\lgroup \\frac{r_{\\mu}}{2} \\right\\rgroup \\, \\alpha_{+} -  \\left\\lgroup \\frac{s_{\\mu}}{2} \\right\\rgroup \\, \\alpha_{-} + \\alpha_0,\n\\quad\nr_{\\mu} \\in \\{1, 2, \\cdots,   p - 1\\}, \n\\quad\ns_{\\mu} \\in \\{1, 2, \\cdots, p^{\\prime} - 1\\}\n\\end{equation}\n\n\\subsection{The fusion rules}\nFor completeness, let us mention the fusion rules. In the notation\n\n\\begin{equation}\nm_i = r_i - 1, \n\\quad \nn_i = s_i - 1\n\\end{equation}\n\n\\noindent the fusion rules take the simple form\n\n\\begin{equation}\n\\label{fusion.rules}\nm_a + m_b + m_{\\mu} = 0  \\ \\textit{mod}  \\ 2, \\quad\nn_a + n_b + n_{\\mu} = 0  \\ \\textit{mod}  \\ 2,\n\\end{equation}\n\n\\noindent where the triple $\\{m_a, m_b, m_{\\mu}\\}$ satisfies the triangular conditions\n\n\\begin{equation}\n\\label{triangular.conditions}\nm_a + m_b     \\geqslant m_{\\mu}, \\quad \nm_b + m_{\\mu} \\geqslant m_a,     \\quad \nm_{\\mu} + m_a \\geqslant m_b\n\\end{equation}\n\n\\noindent with analogous conditions for the triple $\\{n_a, n_b, n_{\\mu}\\}$. \n\n\\section{Restricted instanton partition functions for $\\mathcal{M}^{\\, p, \\, p^{\\prime}, \\, \\mathcal{H}}$. \nThe partition pairs}\n\\label{section.10.burge.pairs}\n{\\it We write the denominator \n$w_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\epsilon_1,   \\epsilon_2,   g \\rightarrow 0]$ as\n$w_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\alpha_{+}, \\alpha_{-}                 ]$, and, following \n\\cite{bershtein.foda}, we check the conditions required so that that it has no zeros.}\n\nUsing $\\alpha_{+}$ and $\\alpha_{-}$, let us write \n$w_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\epsilon_1,   \\epsilon_2, R \\rightarrow 0]$ as\n$w_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\alpha_{+}, \\alpha_{-}]$, that is \n\n\\begin{eqnarray}\n\\label{denominator.4D.02}\nw_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\alpha_{+}, \\alpha_{-}]\n& = &\n\\prod_{\\square \\in V^{\\, 1}}  \\left\\lgroup                                 A^{++}_{\\square, V^{\\, 1}} \\, \\alpha_{+} - L^{  }_{\\square, V^{\\, 1}} \\, \\alpha_{-} \\right\\rgroup\n\\prod_{\\square \\in V^{\\, 2}}  \\left\\lgroup                                 A^{++}_{\\square, V^{\\, 2}} \\, \\alpha_{+} - L^{  }_{\\square, V^{\\, 2}} \\, \\alpha_{-} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\blacksquare \\in W^{\\, 1}}  \\left\\lgroup                                 A^{  }_{\\blacksquare, W^{\\, 1}} \\, \\alpha_{+} - L^{++}_{\\blacksquare, W^{\\, 1}} \\, \\alpha_{-} \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 2}}  \\left\\lgroup                                 A^{  }_{\\blacksquare, W^{\\, 2}} \\, \\alpha_{+} - L^{++}_{\\blacksquare, W^{\\, 2}} \\, \\alpha_{-} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times & \n\\prod_{\\square \\in V^{\\, 1}}  \\left\\lgroup r_a \\, \\alpha_{+} + s_a \\, \\alpha_{-}  +  A^{++}_{\\square, V^{\\, 2}} \\, \\alpha_{+} - L^{  }_{\\square, V^{\\, 1}} \\, \\alpha_{-} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 2}}  \\left\\lgroup r_a \\, \\alpha_{+} + s_a \\, \\alpha_{-}  -  A^{  }_{\\square, V^{\\, 1}} \\, \\alpha_{+} + L^{++}_{\\square, V^{\\, 2}} \\, \\alpha_{-} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times & \n\\prod_{\\blacksquare \\in W^{\\, 1}}  \\left\\lgroup r_b \\, \\alpha_{+} + s_b \\, \\alpha_{-}  +  A^{  }_{\\blacksquare, W^{\\, 2}} \\, \\alpha_{+} - L^{++}_{\\blacksquare, W^{\\, 1}} \\, \\alpha_{-} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\blacksquare \\in W^{\\, 2}}  \\left\\lgroup r_b \\, \\alpha_{+} + s_b \\, \\alpha_{-}  -  A^{++}_{\\blacksquare, W^{\\, 1}} \\, \\alpha_{+} + L^{  }_{\\blacksquare, W^{\\, 2}} \\, \\alpha_{-} \\right\\rgroup\n\\end{eqnarray}\n\n\\noindent where we have used \n\n\\begin{eqnarray}\n-2a & = & r_a \\, \\alpha_{+} + s_a \\, \\alpha_{-}, \\quad r = 1, 2, \\cdots, \n\\\\\n-2b & = & r_b \\, \\alpha_{+} + s_b \\, \\alpha_{-}, \\quad s = 1, 2, \\cdots\n\\end{eqnarray}\n\nConsider the denominator $w_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\alpha_{+}, \\alpha_{-}]$ in \nEquation {\\bf \\ref{denominator.4D.02}}, \non a product by product basis. We need to check the conditions under which any of these products has a zero, then\nfind the restriction that are necessary and sufficient to remove these zeros. The reasoning that we use to obtain\nthese conditions is the same as that in \\cite{bershtein.foda}. There are eight products to consider.\n\n\\subsection{The initial four products}  In each of the initial four products, the product is over the cells inside \na single diagram, thus the arm length $A$ and the leg length $L$ in each of these factors is non-negative. Since \n$\\alpha_{-} < 0 < \\alpha_{+}$, and there is a term $\\alpha_0 > 0$ in each factor, the minimal value of each of these factors \nis greater than zero. Thus there can be no zeros from these factors. \nTo consider the remaining four factors, we require some preparation.\n\n\\subsection{Two zero-conditions}\n\\label{to.vanish}\nFollowing \\cite{bershtein.foda}, we note that, since $\\alpha_{-} < 0 < \\alpha_{+}$, any factor of the type that \nappears in Equation {\\bf \\ref{denominator.4D.02}} has a zero when an equation of type\n\n\\begin{equation}\n\\label{zeros.example.01}\nC_{+} \\, \\alpha_{+} + C_{-} \\, \\alpha_{-} = 0,\n\\end{equation}\n\n\\noindent where $C_+$, $C_-$ $\\in \\mathbbm{Z}$, is satisfied. Since $p$ and $p^{\\prime}$ are coprime, $\\alpha_{-}$ \nand $\\alpha_{+}$ are $\\not \\in \\mathbbm{Q}$, the condition in Equation {\\bf \\ref{zeros.example.01}} is \nequivalent to the two conditions\n\n\\begin{equation}\n\\label{zeros.example.02}\nC_+ =  c   \\ p, \n\\quad\nC_- =  c \\ p^{\\prime}\n\\end{equation}\n\n\\noindent are satisfied, where $c$ is a proportionality constant that needs to be determined. \n\n\\subsection{From two zero-conditions to one zero-condition} \n\\label{from.2.to.1}\n\nConsider the two conditions\n\n\\begin{equation}\n\\label{conditions.example.01}\n\\boxed{\n- A^{ }_{\\square, i}   =  A^{\\prime} \\geqslant 0,\n\\quad \n  L^{ }_{\\square, j}   =  L^{\\prime} \\geqslant 0\n}\n\\end{equation}\n\n\\noindent which are satisfied if $i \\neq j$, $\\square \\not \\in Y^i$, and $\\square \\in Y^j$.\nIf $\\square$ is in row-$ {\\tt \\scriptstyle R} $ and column-$ {\\tt \\scriptstyle C} $ in $Y^j$, then the second condition in \n(\\ref{conditions.example.01}) implies that there is a cell $\\boxplus \\in Y^{\\, 1}$, strictly \nbelow $\\square$, with coordinates $\\{ {\\tt \\scriptstyle R}  + L^{\\prime},  {\\tt \\scriptstyle C} \\}$, such that there are \nno cells strictly below $\\boxplus$. Since there may, or may not, be cells to the right of \n$\\boxplus$, row-$( {\\tt \\scriptstyle R}  + L^{\\prime})$ in $Y^j$ has length {\\it at least} $ {\\tt \\scriptstyle C} $,\n\n\\begin{equation}\n\\label{conditions.example.02}\ny^{\\, j}_{\\,  {\\tt \\scriptstyle R}  + L^{\\prime}} \\geqslant  {\\tt \\scriptstyle C} \n\\end{equation}\n\n\\noindent From the definition of $A_{\\square, \\, i}$, we write the first condition in \n(\\ref{conditions.example.01}) as \n$- A_{\\square, \\, i}   = A^{\\prime} =  {\\tt \\scriptstyle C}  - y^{\\, i}_{\\,  {\\tt \\scriptstyle R} }$,\nthat is, \n$ {\\tt \\scriptstyle C}  = A^{\\prime} + y^{\\, i}_{\\,  {\\tt \\scriptstyle R} }$, \nand using (\\ref{conditions.example.02}), we obtain \n$y^{\\, j}_{\\,  {\\tt \\scriptstyle R}  + L^{\\prime}} \\geqslant A^{\\prime} + y^{\\, i}_{\\,  {\\tt \\scriptstyle R} }$,\nwhich we choose to write as\n\n\\begin{equation}\n\\label{conditions.example.04}\ny^{\\, j}_{\\,  {\\tt \\scriptstyle R}  + L^{\\prime}} - y^{\\, i}_{\\,  {\\tt \\scriptstyle R} } \\geqslant A^{\\prime}\n\\end{equation}\n\n\\noindent The condition in Equation {\\bf \\ref{conditions.example.04}} is equivalent to \nthe two conditions in Equation {\\bf \\ref{conditions.example.01}}. \n\n\\subsection{One non-zero condition}\n\\label{1.not.to.vanish}\nConsider a function $f_{Y^{\\, i}, Y^{\\, j}}$, of a pair of Young diagrams $Y^{\\, i}$ and $Y^{\\, j}$, \n$i \\neq j$, such that \n$f_{Y^{\\, i}, Y^{\\, j}} = 0$, {\\it if and only if} (\\ref{conditions.example.04}) is satisfied. This \nimplies that $f_{Y^{\\, i}, Y^{\\, j}} \\neq 0$, {\\it if and only if} $Y^{\\, i}$ and $Y^{\\, j}$ satisfies \nthe complementary condition\n\n\\begin{equation}\n\\label{conditions.example.05}\ny^{\\, j}_{\\,  {\\tt \\scriptstyle R}  + L^{\\prime}} - y^{\\, i}_{\\,  {\\tt \\scriptstyle R} } < A^{\\prime}\n\\end{equation}\n\n\\noindent which we choose to write as \n\n\\begin{equation}\n\\label{conditions.example.06}\n\\boxed{\ny^{\\, i}_{\\,  {\\tt \\scriptstyle R} } - y^{\\, j}_{\\,  {\\tt \\scriptstyle R}  + L^{\\prime}} \\geqslant 1 - A^{\\prime}\n}\n\\end{equation}\n\n\\subsubsection{Remark} \nIn the sequel, we refer \nto Equation {\\bf \\ref{conditions.example.01}} as {\\it \\lq       zero-conditions\\rq\\,}, and \nto Equation {\\bf \\ref{conditions.example.06}} as {\\it \\lq a non-zero-condition\\rq\\,}. \n\nNext we consider the latter four products on the right hand side of Equation \n{\\bf \\ref{denominator.4D.02}}.\n\n\\subsection{The first product} \n\\label{den.12}\n\n\\begin{equation}\n\\prod_{\\square \\in V^{\\, 1}}  \\left\\lgroup  r_a \\, \\alpha_{+} + \n                         s_a \\, \\alpha_{-} + A^{++}_{\\square, \\, V^{\\, 2}} \\, \\alpha_{+} \n                                      - L^{  }_{\\square, \\, V^{\\, 1}} \\, \\alpha_{-} \\right\\rgroup\n\\end{equation}\n\n\\noindent vanishes if any factor satisfies \n\n\\begin{equation}\n\\label{zeros.12}\n \\left\\lgroup  r_a + A^{++}_{\\square, \\, V^{\\, 2}} \\right\\rgroup \\, \\alpha_{+} +\n \\left\\lgroup  s_a - L^{  }_{\\square, \\, V^{\\, 1}} \\right\\rgroup \\, \\alpha_{-} = 0\n\\end{equation}\n\\noindent which leads to the conditions\n\\begin{equation}\n\\label{two.zero.conditions.12}\n- A^{ }_{\\square, \\, V^{\\, 2}} = r_a + 1 + c \\,    p, \\quad \n  L^{ }_{\\square, \\, V^{\\, 1}} = s_a     + c \\,  p^{\\prime} \n\\end{equation}\n\n\\noindent Since $\\square \\in V^{\\, 1}$, $L_{\\square, V^{\\, 1}} \\geqslant 0$. Given that $s$ and $p^{\\prime}$ are \nnon-zero positive integers, \nthe second equation in Equation {\\bf \\ref{two.zero.conditions.12}} admits a solution \nonly if $c = 0, 1, \\cdots$ \nThe first  equation in Equation {\\bf \\ref{two.zero.conditions.12}} admits a solution \nif $\\square \\not \\in V^{\\, 2}$.\n\n\\subsubsection{From two zero-conditions to one non-zero-condition} \n\\label{translating.01}\nFollowing paragraphs {\\bf \\ref{from.2.to.1}} and {\\bf \\ref{1.not.to.vanish}}, the two \nzero-conditions in (\\ref{two.zero.conditions.12}) are equivalent to one non-zero-condition, \n\n\\begin{equation}\n\\label{one.non.zero.condition.12}\nV^{\\, 2}_{ {\\tt \\scriptstyle R} } - V^{\\, 1}_{ {\\tt \\scriptstyle R}  + s + c \\, p^{\\prime}} \\geqslant - r - c \\, p \n\\end{equation}\n\n\\subsubsection{The stronger condition} \n\\label{the.stronger.condition.01}\nEquation {\\bf \\ref{one.non.zero.condition.12}} is the statement that to eliminate the zeros, \nwe want \n$V^{\\, 2}_{ {\\tt \\scriptstyle R} } - V^{\\, 1}_{ {\\tt \\scriptstyle R}  + s + c \\, p^{\\prime}} \\geqslant - r - c \\, p$, \nwhere $c = \\{0, 1, \\cdots\\}$\nSince the row-lengths of a partition are by definition weakly decreasing, and \n$c = \\{0, 1, \\cdots\\}$, this is the case if \n$V^{\\, 2}_{ {\\tt \\scriptstyle R} } - V^{\\, 1}_{ {\\tt \\scriptstyle R}  + s} \\geqslant - r - c \\, p$, \nwhich is the case if \n$V^{\\, 2}_{ {\\tt \\scriptstyle R} } - V^{\\, 1}_{ {\\tt \\scriptstyle R}  + s} \\geqslant - r$. \nThus, we should set $c=0$, and obtain \n\n\\begin{equation}\n\\label{weak.Burge.condition.02}\nV^{\\, 2}_{ {\\tt \\scriptstyle R} } - V^{\\, 1}_{ {\\tt \\scriptstyle R}  + s} \\geqslant - r \n\\end{equation}\n\n\\subsection{The second product} \n\\label{den.21}\n\n\\begin{equation}\n\\prod_{\\square \\in V^{\\, 2}}  \\left\\lgroup  r_a \\, \\alpha_{+} + s_a \\, \\alpha_{-}  -  A^{  }_{\\square, V^{\\, 1}} \\, \\alpha_{+} \n                                                           +  L^{++}_{\\square, V^{\\, 2}} \\, \\alpha_{-} \\right\\rgroup\n\\end{equation}\n\n\\noindent vanishes if any factor satisfies \n\\begin{equation}\n\\label{zeros.21}\n \\left\\lgroup r_a - A^{  }_{\\square, V^{\\, 1}} \\right\\rgroup \\, \\alpha_{+} \n+ \n \\left\\lgroup s_a + L^{++}_{\\square, V^{\\, 2}} \\right\\rgroup \\, \\alpha_{-} = 0\n\\end{equation}\n\\noindent which leads to the conditions\n\\begin{equation}\n\\label{two.zero.conditions.21}\n- A^{ }_{\\square, V^{\\, 1}} = - r_a     + c \\,   p, \n\\quad\n  L^{ }_{\\square, V^{\\, 2}} = - s_a - 1 + c \\, p^{\\prime},\n\\end{equation}\n\n\\noindent Since $\\square \\in V^{\\, 2}$, $L_{\\square, V^{\\, 2}} \\geqslant 0$. Given that $s_a$ and $p^{\\prime}$ are \nnon-zero positive integers,\nthe second equation in Equation {\\bf \\ref{two.zero.conditions.21}} admits a solution \nonly if $c = 1, \\cdots$ \nThe first  equation in Equation {\\bf \\ref{two.zero.conditions.21}} admits a solution  \nif $\\square \\not \\in V^{\\, 1}$.\n\n\n\\subsubsection{From two zero-conditions to one non-zero-condition} \n\\label{translating.02}\nFollowing paragraphs {\\bf \\ref{from.2.to.1}} and {\\bf \\ref{1.not.to.vanish}}, the two \nzero-conditions in Equation {\\bf \\ref{two.zero.conditions.21}} are equivalent to one non-zero-condition, \n\n\\begin{equation}\n\\label{condition.21.04}\n V^{\\, 1}_{ {\\tt \\scriptstyle R} } - V^{\\, 2}_{ {\\tt \\scriptstyle R}  - 1 - s_a + c \\, p^{\\prime}} \\geqslant 1 + r_a - c \\, p \n\\end{equation}\n\n\\subsubsection{The stronger condition} \n\\label{the.stronger.condition.02}\nEquation ({\\bf \\ref{condition.21.04}}) is the statement that to eliminate the zeros, \nwe want \n$V^{\\, 1}_{ {\\tt \\scriptstyle R} } - V^{\\, 2}_{ {\\tt \\scriptstyle R}  - 1 - s_a + c \\, p^{\\prime}} \\geqslant 1 + r_a - c \\, p$, \nwhere $c = \\{1, 2, \\cdots\\}$.\nSince the row-lengths of a partition are by definition weakly decreasing, and \n$c = \\{1, 2, \\cdots\\}$, this is the case if \n$V^{\\, 1}_{ {\\tt \\scriptstyle R} } - V^{2,  {\\tt \\scriptstyle R}  + p^{\\prime} - s_a - 1}      \\geqslant 1 + r_a - c \\, p$.\nIn turn, is the case if \n$V^{\\, 1}_{ {\\tt \\scriptstyle R} } - V^{\\, 2}_{ {\\tt \\scriptstyle R}  + p^{\\prime} - s_a - 1}      \\geqslant 1 + r_a -      p$. \nThus, we should set $c=1$, to obtain \n\\begin{equation}\n\\label{Burge.condition.01}\nV^{\\, 1}_ { {\\tt \\scriptstyle R} } - V^{\\, 2}_{ {\\tt \\scriptstyle R}  + [p^{\\prime} - s_a] - 1} \\geqslant 1 -  \\left\\lgroup p - r_a \\right\\rgroup\n\\end{equation}\n\n\\subsection{The third product} \n\\label{den.conjugate.12}\n\n\\begin{equation}\n\\prod_{\\blacksquare \\in W^{\\, 1}}  \\left\\lgroup  r_b \\, \\alpha_{+} + s_b \\, \\alpha_{-}  + A^{  }_{\\blacksquare, W^{\\, 2}} \\, \\alpha_{+} \n                                                           - L^{++}_{\\blacksquare, W^{\\, 1}} \\, \\alpha_{-} \\right\\rgroup\n\\end{equation}\n\n\\noindent vanishes if any factor satisfies\n\n\\begin{equation}\n\\label{zeros.conjugate.12}\n \\left\\lgroup r_b + A^{  }_{\\blacksquare, W^{\\, 2}} \\right\\rgroup \\, \\alpha_{+} \n+\n \\left\\lgroup s_b - L^{++}_{\\blacksquare, W^{\\, 1}} \\right\\rgroup \\, \\alpha_{-} = 0\n\\end{equation}\n\n\\noindent which leads to the conditions\n\n\\begin{equation}\n\\label{conditions.conjugate.12.01}\n-  A_{\\blacksquare, W^{\\, 2}}   =       r_b + c \\,   p,  \n\\quad\n   L_{\\blacksquare, W^{\\, 1}}   = - 1 + s_b + c \\, p^{\\prime},\n\\end{equation}\n\n\\subsubsection{The stronger condition} Using the same arguments as in subsections {\\bf \\ref{den.12}} \nand {\\bf \\ref{den.21}}, are possible for $c = 0, 1, \\dots$, $\\blacksquare \\in W^{\\, 1}$, \n$\\blacksquare \\not \\in W^{\\, 2}$, and we should choose $c = 0$ to obtain  \n\n\\begin{equation}\n\\label{Burge.condition.02}\nW^{\\, 2}_{ {\\tt \\scriptstyle R} } - W^{\\, 1}_{ {\\tt \\scriptstyle R}  + s - 1} \\geqslant 1 - r \n\\end{equation}\n\n\\subsection{The fourth product} \n\\label{den.conjugate.21}\n\n\\begin{equation}\n\\prod_{\\blacksquare \\in W^{\\, 2}}  \\left\\lgroup r_b \\, \\alpha_{+} + s_b \\, \\alpha_{-} - A^{++}_{\\blacksquare, W^{\\, 1}} \\, \\alpha_{+} \n                                                         + L^{  }_{\\blacksquare, W^{\\, 2}} \\, \\alpha_{-} \\right\\rgroup\n\\end{equation}\n\n\\noindent vanishes if any factor satisfies  \n\n\\begin{equation}\n\\label{zeros.conjugate.21}\n   \\left\\lgroup r_b - A^{++}_{\\blacksquare, W^{\\, 1}} \\right\\rgroup \\, \\alpha_{+} \n+  \\left\\lgroup s_b + L^{  }_{\\blacksquare, W^{\\, 2}} \\right\\rgroup \\, \\alpha_{-}  = 0,\n\\end{equation}\n\n\\noindent which leads to the conditions\n\n\\begin{equation}\n\\label{conditions.conjugate.21.01}\n- A^{ }_{\\blacksquare, W^{\\, 1}} = 1 - r_b + c\\,   p, \n\\quad\n  L^{ }_{\\blacksquare, W^{\\, 2}} =   - s_b + c\\, p^{\\prime} \n\\end{equation}\n\n\\subsubsection{The stronger condition} Using the same arguments as in subsections {\\bf \\ref{den.12}} \nand {\\bf \\ref{den.21}}, are possible for $c = 1, 2, \\cdots$, $\\square \\in W^{\\, 2}$ and $\\square \\not \\in W^{\\, 1}$, \nand we should choose $c=0$ to obtain\n\n\\begin{equation}\n\\label{weak.Burge.condition.01}\nW^{\\, 1}_{ {\\tt \\scriptstyle R} } - W^{\\, 2}_{ {\\tt \\scriptstyle R}  + [p^{\\prime} - s_b]} \\geqslant -  \\left\\lgroup p - r_b \\right\\rgroup\n\\end{equation}\n\n\\subsection{The Burge conditions}\n\nEquations {\\bf \\ref{weak.Burge.condition.02}} and {\\bf \\ref{Burge.condition.01}} are \nconditions on the partition pair {\\bf V} on one side of $\\mathcal{Z}_{building.block}$, while \nEquations {\\bf \\ref{Burge.condition.02}} and {\\bf \\ref{weak.Burge.condition.01}} are \nconditions on the partition pair {\\bf W} on the other side of $\\mathcal{Z}_{building.block}$.\nWhen copies of $\\mathcal{Z}_{building.block}$ are glued to form conformal blocks, partition pairs \non one side are identified with partition pairs on the other side. Thus each partition \npair must satisfy all conditions. However, these conditions are not independent as two \nof them are satisfied when the other two are satisfied. More specifically, following \nparagraphs {\\bf \\ref{from.2.to.1}} and {\\bf \\ref{1.not.to.vanish}}, we can see that it \nis sufficient to enforce the two conditions in Equations {\\bf \\ref{Burge.condition.01}} \nand {\\bf \\ref{Burge.condition.02}}, \n\n\\begin{equation}\n\\label{Burge.conditions}\n\\boxed{\n\\mu_{ {\\tt \\scriptstyle R} } - \\mu_{ {\\tt \\scriptstyle R}  + [p^{\\prime} - s] - 1} \\geqslant 1 -  \\left\\lgroup p - r \\right\\rgroup, \\quad\n\\nu_{ {\\tt \\scriptstyle R} } - \\nu_{ {\\tt \\scriptstyle R}  +        s  - 1} \\geqslant 1         - r\n}\n\\end{equation}\n\n\\noindent which we write in terms of a partition pair $\\{\\mu, \\nu\\}$ that could be \non either side of $\\mathcal{Z}_{building.block}$. Partition pairs that satisfy the conditions \nin Equation {\\bf \\ref{Burge.conditions}} first appeared in the work of W H Burge on \nRogers-Ramanujan-type identities \\cite{burge}. The appeared in earlier studies of Virasoro \ncharacters in \\cite{foda.lee.welsh, foda.welsh, welsh} and more recently in the context \nof the AGT correspondence in \\cite{alkalaev.belavin, bershtein.foda}.\n\n\\section{Comments and open questions}\n\\label{section.11.comments}\n\n\\subsubsection*{Outline of result} We can generate conformal blocks of Virasoro $A$-series \nminimal models, labelled by the co-primes $p$ and $p^{\\prime}$, times a Heisenberg factor, as follows. \n{\\bf 1.} Start from the refined topological vertex of \\cite{ikv} defined in Equation \n{\\bf \\ref{refined.topological.vertex}}, \n{\\bf 2.} Glue four copies of the refined topological vertex to produce a 5D $U(2)$ basic web \npartition function, then take the $R \\rightarrow 0$ limit, to obtain its 4D counterpart \n$\\mathcal{W}_{\\bf V W \\Delta}^{\\, 4D}$ as in Equations {\\bf \\ref{W.5D.norm}, \\ref{w.numerator.5D}, \\ref{w.denominator.5D}}. \n{\\bf 3.} Set the K{\\\"a}hler parameters ${\\bf \\Delta}$, and the deformation parameters $q$ and $t$ in \n$\\mathcal{W}_{\\bf V W \\Delta}^{\\, 4D}$ as in Equations {\\bf \\ref{identification.12}}, \n                                               {\\bf \\ref{minimal.model.charges}},\n\t\t\t\t\t       {\\bf \\ref{neg.0.pos}}, and\n\t\t\t\t\t       {\\bf \\ref{parameters.03}},\nand\n{\\bf 4.} require each of the partition pairs {\\bf V} and {\\bf W} to satisfy the Burge conditions\nin Equation {\\bf \\ref{Burge.conditions}}.\n\n\\subsubsection*{The refined topological vertex of Awata and Kanno} We have used the refined topological \nvertex of Iqbal, Kozcaz and Vafa \\cite{ikv}, but could have equally well used that of Awata and Kanno \n\\cite{awata.kanno.01, awata.kanno.02}. The two vertices are equivalent as explained in \n\\cite{awata.feigin.shiraishi}. \n\n\\subsubsection*{Layers} The topic discussed in this note is vast and consists of many layers. \nWe could have started our discussion from M-theory and used the language of M5 branes, but we \ndecided to stay away from this, in this short note. \nInstead, we started from A-model topological strings, which live in a corner of the M-theory. \nFrom the refined topological vertex and topological strings, we obtained the building block of \nthe instanton partition function of a 5D quiver gauge theory. We could have used the K-theoretic \nversion of the AGT correspondence to obtain the minimal model analogues of the $q$-deformed \nLiouville conformal blocks discussed in \\cite{awata.yamada.01, awata.yamada.02, aganagic.01, \naganagic.02, nieri.01, nieri.02}. \nInstead, we skipped the $q$-deformed blocks, took the 4D limit, and used the 4D version of AGT \nto obtain minimal model conformal blocks. The M-theoretic origins of the minimal conformal blocks \nand their $q$-deformations should be topics of separate studies.\n\n\\subsubsection*{Interpretation} Missing from this note is an interpretation of the minimal model \nparameters in topological string or gauge theory terms. A complete interpretation will require \nworking in M-theory terms which lies outside the scope of this work. In particular, missing is\nan interpretation of the Burge conditions in topological string or gauge theory terms. \nWe conjecture that such interpretations require re-derivations of existing results in ways that \nallow {\\it ab initio} for minimal model parameters. Current derivations do not do that, and for\nthat reason, one obtains results that are not well-defined upon substitution of minimal-model \nparameters and that require {\\it aposteriori} restrictions as in \n\\cite{alkalaev.belavin, bershtein.foda} \n\\footnote{\\,\nOF wishes to thank J-E Bourgine for discussions on this point.\n}. We plan to address this topic in future work. \n\n\n\\subsubsection*{Postscript} Following the completion of work on this note, we became aware of \nthe fact that $U(N)$ versions of Equations {\\bf 2.14} and {\\bf 2.17} in this note, were obtained \nin Equation {\\bf 4.67} in \\cite{bao.01}, and in Equation {\\bf 5.1} and subsequent equations in \n\\cite{mitev.03}\n\\footnote{\\, \nThe {\\it basic web} in this note is related, by {\\it \\lq a flop\\rq\\,} to the {\\it a strip} in \n\\cite{bao.01}. This term was introduced in \\cite{iqbal.kashani.poor.01}, where strip partition \nfunctions were studied in the context of the original, unrefined topological vertex.\n}. \n\n\\section*{Acknowledgements}\nOF would like to thank M Bershtein for collaboration on \\cite{bershtein.foda}, results of which \nwere used in this note, and H Awata, J-E Bourgine, D Krefl, V Mitev, E Pomoni, A Tanzini and Y \nZenkevich for discussions and useful remarks at various stages of work on this note. We thank \nthe anonymous referee for many useful comments that helped us improve the presentationh. OF is \nsupported by the Australian Research Council [ARC].\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nPeriodic oscillations abound in the atmosphere of sunspots as revealed by the temporal variations in, say, Doppler velocities and intensities. These oscillations are usually considered as slow magnetoacoustic waves. The magnetic field orientation is different in the umbra from that in the penumbra; however, the propagation of slow magnetoacoustic waves is anisotropic to the orientation of magnetic fields. So, different types of magnetohydrodynamic waves are displayed inside and above sunspots at different solar atmospheric layers, such as oscillations above light bridges \\citep{2013ApJ...767..169L,2013A&A...560A..84S,2014ApJ...792...41Y, 2015MNRAS.452L..16B, 2016ApJ...816...30S,  2016A&A...594A.101Y}, umbral flashes \\citep{2014RAA....14.1001F}, penumbral running waves \\citep{2000A&A...354..305C,2013ApJ...779..168J,2015A&A...580A..53L} and coronal fan structures \\citep{2011A&A...533A.116Y,2012ApJ...757..160J,2016ApJ...823L..16T}. Slow magnetoacoustic waves are assumed to result from the interaction between the photospheric $p$-modes and magnetic fields, and can propagate upward along magnetic field lines, and finally reach coronal heights \\citep{2012ApJ...757..160J,2016NatPh..12..179J, 2015ApJ...812L..15K, 2017ApJ...847....5K,2008ApJ...682L..65Y}. Studies on the multiple oscillations above the sunspot atmosphere prove useful for understanding the physical conditions of oscillations and waves, and their formation and propagation mechanisms \\citep{2015LRSP...12....6K,2016ApJ...830L..17Z,2016NatPh..12..179J}. \n\nThe frequency distributions of sunspot oscillations usually tend to be concentrated in the bandwidth of 3--6 mHz (i.e., 3--5 minutes). It is believed that three-minute umbral oscillations exist in the chromosphere, and five-minute oscillations exist in the photosphere due to a cutoff frequency. Recently, thanks to the high spatial and temporal observations acquired with ground- and space-based instruments, it has been found that the oscillations below the typical acoustic cutoff frequency can also reach chromospheric heights, and even propagate into the corona along the magnetic field lines. For example, \\citet{2014ApJ...792...41Y} detected five-minute oscillations at a light bridge observed at chromospheric heights. Employing a multi-wavelength approach, \\citet{2012ApJ...757..160J} investigated three-minute magnetoacoustic waves, and revealed the propagation from the photosphere, through the chromosphere, and into the corona. \\citet{2012ApJ...746..119R} studied the perturbations in the 5--9 mHz frequency range in different umbra atmospheres, and examined its propagation along inclined magnetic field lines. \\citet{2013A&A...554A.146K} found a high-frequency mode with 8 mHz (2 minutes) located at a peculiar location inside an umbra, and concluded that the frequency probably relates to the oscillation of umbral dots in the photosphere. \\citet{2014A&A...569A..72S} further revealed that the spectra of umbral oscillations contain distinct peaks at 1.9, 2.3, and 2.8~minutes, and the oscillations in the higher atmospheric layers occur later than in lower ones, and claimed that they are upward propagating waves. \n\nSo far, the high-frequency perturbation modes around one minute have been reported at a particular atmosphere height \\citep{2016A&A...594A.101Y,2017ApJ...847....5K}. Utilizing the multiple optical and UV wavelength observations obtained by the Dunn Solar Telescope (DST) and the \\textit{Interface Region Imaging Spectrograph (IRIS)}, \\citet{2016A&A...594A.101Y} studied the oscillations in the emission intensity of the light bridge plasmas at different temperatures, and revealed that the light bridge exhibits a perturbation with some period shorter than one minute in a particular channel at the solar atmosphere. \n\\citet{2017ApJ...847....5K} confirmed that a high-frequency oscillation mode with 13.1~mHz is located near an umbral center in the DST Ca \\textsc{ii} K and \\textit{IRIS} 2796 \\AA\\ channels, and does not seem to be related to the magnetic field inclination angle effects. A similar period is also found in a flare loop observed by the \\textit{Geostationary Operational Environmental Satellites} \\citep{Ning:2017eh}. However, a periodicity around one minute has not been found in imaging observations at other umbral atmospheric layers. \n\nIn this letter, we use ground- and space-based imaging instruments, with high spatial and temporal resolution, to investigate the high-frequency mode with a period around one minute at different heights above a sunspot umbra. For representing, decomposing, and reconstructing high-frequency oscillation modes, we employ a novel time--frequency analysis technique named synchrosqueezing transform \\citep[SST;][]{Daubechies:2011bg,Thakur:2013kc}. This technique has been used \\citep{2013ApJ...771...33S,2017ApJ...845...11F} to represent intrinsic modes of solar cycles. Furthermore, \\citet{2017ApJ...845...11F} demonstrated that it can decompose and reconstruct intrinsic modes with a high spectral resolution without mode mixing. In Section~\\ref{sec_valid}, we first conduct a simulation experiment to evaluate the performance of SST. We then describe, in Section~\\ref{sec_data}, our observations and data processing procedures as well as an error analysis. Finally, Section~\\ref{sec_disc} concludes this study.  \n\n\\begin{figure*}\n\t\\centering\n        \\includegraphics[width=18cm]{fig1.eps}\n\t\\caption{Simulation experiment for evaluating the performance of SST. (a) The synthetic signal without noise (black curve) and with noise (blue curve). The noise follows a Poisson distribution. Its standard error is 6.3, and the S\/N ratio is about 3. The synthetic signal consists of three frequency components within 3--4, 6--7, and 9--14 mHz, and their amplitudes are approximately 40, 10, and 4. These are generated by a random process. (b) Power spectrum of Fourier transform. Gray regions denote the frequency ranges of the three components. (c) Power spectrum of the SST. (d) Original components (black curves) and the ones reconstructed from the noisy signal (blue curves).}\n\t\t\\label{fig1}\n\\end{figure*}\n\n\\section{Performance evaluation of the synchrosqueezing transform}\n\\label{sec_valid}\n\nSST is a time--frequency analysis technique based on the continuous wavelet (CWT) and the spectral reassignment method. The reassignment method compensates for the spreading effects inherent to CWT. SST concentrates the frequency content only along the frequency direction, and preserves the time resolution of a quasi-periodic signal. Moreover, the inverse synchrosqueezing algorithm can reconstruct instantaneous frequencies of the signal. More detailed descriptions on SST can be found in \\citet{Daubechies:2011bg} and \\citet{Thakur:2013kc}. \n\nFor evaluating the performance of SST, we synthesize a signal $s(t)$ with three frequency components, and each component varies around the central values of 4, 6, and 12 mHz along the time direction, respectively. Moreover, their amplitudes also randomly fluctuate around specific values. The amplitudes of the 4, 6, and 12 mHz components are approximately 40, 10, and 4, respectively. It should be noted that all the frequency and amplitude fluctuations are generated by a random process around the given value for a near ``real'' signal. \n\nFor an observational image, its counting noise is mainly generated by Poisson processes. So, a Poisson noise with a signal-to-noise ratio (S\/N) being 3 and standard error being 6.3 is considered. Figure 1a shows the signal we actually analyze (blue) and its noise-free counterpart (black). In the noise-free signal, the power is located in the ranges with 3--4 mHz, 6--7 mHz, and 9--14 mHz, respectively. The sampling interval and the total number of data points are 12 s and 300, respectively. The amplitude of the 12 mHz component is only 10 \\% of the 4 mHz component. In absolute values, this amplitude ($\\sim 4$) is less than the standard error ($\\sim 6.3$) of the Poisson noise. \n\n\\begin{figure*}\n\t\\centering\n        \\includegraphics[width=8cm]{fig2a.eps}\n        \\includegraphics[width=8cm]{fig2b.eps}\n\t\\caption{(a) An AIA 171 \\AA\\ image. (b) The co-temporal and co-spatial NVST TiO image, which is also superimposed on the 171\\AA\\ image. The pixel size of TiO images is 0\\arcsec.041. The umbral region is outlined by a red curve. A region marked ``B\" is used to compare the spectra to the umbral region in Section 4. (c) A co-spatial stacked image. From bottom to top, the images represent observations from a variety of ground- and space-based instruments: NVST TiO, AIA 1700 \\AA, AIA 1600 \\AA, NVST H$\\alpha$, AIA 304 \\AA, and AIA 171 \\AA. The white curves mark the umbral region that is the region of interest in this work.}\n\t\\label{fig2}\n\\end{figure*}\n\nTo evaluate the spectrum representation performance based on SST, the power spectra of the noisy signal (the blue curve in Figure 1a) based on SST and the Fourier transform are shown in Figures 1c and 1b, respectively. The gray regions in Figure 1b denote the ranges of the three frequency components. One can see that the noise floor appears at about 8 mHz, and the component between 9 and 14 mHz is heavily contaminated by the high-frequency noise. In Figure 1c, there are some spectral contents, although weak, in the range 9--15 mHz, demonstrating that the SST has better ability to suppress random noise.  Figure 1d shows the original frequency components (black) and the corresponding reconstructed modes (blue). Compared with the original components, the amplitudes of the reconstructed modes are slightly different. However, both the frequency and the phase are almost the same as in the original ones, which is particularly true for the weakest component (12 mHz). To sum up at this point, SST performs well for resolving the physical signals that are heavily contaminated by a Poison noise. \n\n\n\\begin{figure*}\n\t\\centering\n        \\includegraphics[width=18cm]{fig3.eps}\n\\caption{Left column: intensity curves averaged over the entire umbra region in different channels. Right column: the corresponding Fourier spectra of the de-trended curves. Red dashed curves denote the 95 \\% confidence levels. The shaded regions denotes the frequency range of interest in this work.}\n\t\t\\label{fig3}\n\\end{figure*}\n\n\\section{Observations and Data Reduction}\n\\label{sec_data}\n\n\\begin{figure*}\n\t\\centering\n        \\includegraphics[width=18cm]{fig4.eps}\n\\caption{Left column: SST power spectra in each channel. Black dashed curves denote the influence of cone, and the contours present 95 \\% confidence levels in the 12 mHz components. Right column: the corresponding reconstructed components. }\n\t\t\\label{fig4}\n\\end{figure*}\n\nThis study focuses on active region NOAA 11081, located in the northern hemisphere on 2013 August 1. The New Vacuum Solar Telescope \\citep[NVST;][]{2014RAA....14..705L,2014IAUS..300..117X}, located at the Fuxian Solar Observatory of the Yunnan Observatories, China, was employed to obtain the high-resolution ground-based imaging observations. The two optical channels, centered at the H$\\alpha$ line core (6562.8$\\pm$0.25 \\AA) and TiO (7058$\\pm$10~\\AA), were used to capture the chromospheric and photospheric images between 03:38 and 04:35 UT. The pixel scale is 0\\arcsec.162 for the H$\\alpha$ images, and 0\\arcsec.041 for the TiO ones. The sampling interval is 12 s for the H$\\alpha$ sequences, and 20 s for the TiO ones. \n\n\\begin{figure}\n\t\\centering\n        \\includegraphics[width=8cm]{fig5.eps}\n\t\\caption{(a) A reconstructed running difference based on every pixel in the AIA 171 \\AA\\, whose spectra only includes the frequency contents from 10 to 14 mHz. Its original image is observed on 2013 August 1 at 04:04:59 UT. The red and yellow curves outline the umbra and penumbra edge, and the dashed blue lines highlight expanding annulus. A cut taken to make the time--distance diagram is indicated with a green line. (b) The time--distance diagram started at 04:00:35 UT and finished at 04:08:59 UT covers about six cycles of the propagating features. Its propagating velocity is about 49 $\\pm$ 4 km s$^{-1}$. (c) The power spectrum of region ``B\" marked in Figures 2a and b. Two horizontal dashed lines indicate the range between 10 and 14 mHz. }\n\t\t\\label{fig5}\n\\end{figure}\n\nThe co-temporal and co-spatial observations were also obtained from the Atmospheric Imaging Assembly \\citep[AIA;][]{2012SoPh..275...17L} on board the \\textit{Solar Dynamics Observatory} \\citep[SDO;][]{2012SoPh..275....3P}. Our analysis concentrates on the 171 \\AA, 304 \\AA, 1600 \\AA\\, and 1700 \\AA\\ channels. The formation height of the 1700 \\AA\\ channel is located at the temperature minimum, 1600 \\AA\\ at the lower chromosphere, 304 \\AA\\ at the transition region, and 171 \\AA\\ at the corona. To get the level 1.5 data, the level 1.0 data were processed using the routine $aia\\_prep.pro$. The pre-process brings all the data to a common center and plate scale with $\\approx$ 0\\arcsec.6, and the cadence is 12 s for the AIA 171\\AA\\ and 304 \\AA\\ channels, and 24 s for the 1600 \\AA\\ and 1700 \\AA. Furthermore, the images were co-aligned by the sub-pixel registration algorithm \\citep{Feng:2012hk}. Similarly, all the NVST images were aligned to the AIA data with a scale reduction and a rotation transform. Finally, we truncated the size of the data sets into a set of 54\\arcsec\\ $\\times$ 54\\arcsec, which contains the entire sunspot. \n\nFigure 2a shows an AIA 171 \\AA\\ image and its co-temporal and co-spatial TiO thumbnail. The original TiO image is shown in Figure 2b. The umbral edge is marked with a red curve. The entire umbra is employed to analyze high-frequency oscillation modes. The images stacked in Figure 2c represent the observations at different formation heights. \n\n\nFigure~3 presents the average intensity variations of the entire umbra in every channel (left column).  We removed long-term trend greater than 2.5 minutes in all data with frequency filtering for concentrating on high-frequency component analysis,  and the power spectra of the Fourier transform of the de-trended curves are presented in the right column. Red dashed lines indicate the 95\\% confidence levels, and the shaded regions denote a frequency range between 10 and 14 mHz. We can find that the spectral contents between 10 and 14 mHz exist in each  channel, and are higher than the 95 \\% confidence level. There are significant peaks within the shaded regions.  In what follows, we use ``the 12~mHz component'' to label the oscillations in this frequency range for brevity. Subsequently, the 12~mHz components are represented and reconstructed by the SST, respectively.  The SST spectra are represented in the left column of Figure 4, in which the black dashed curves denote the cone of influence and those contours present 95 \\% of the confidence level. The shaded regions are decomposed and reconstructed by the SST and are shown in the right column of Figure 4.\n\n\nFor estimating the amplitude error of the reconstructed components, we analyze their noise level. The error estimation proposed by \\citet[see Section 3 by][]{2012A&A...543A...9Y} is used to calculate the image noise in each AIA channel. For a pixel with an intensity value $F$ (in units of DN), the standard error of the noise can be approximated by $\\sqrt{2.3+0.06F}$. Because the light curve in the AIA 171 \\AA\\ channel incorporates the entire umbra region that amounts to 190 pixels in total, the estimated data noise should be multiplied by a factor of 1\/$\\sqrt{190}$. Thus, the error of the AIA 171 \\AA\\ channel is about 0.6~DN for the selected data. The error can be estimated similarly for the rest of the EUV\/UV channels. \n\nFor the NVST data, we assume that the noise is primarily due to photon counting, which follows a Poisson distribution. Therefore, its amplitude is the square root of the signal intensity. Because the pixel resolution of the original image of the TiO data is reduced from 0\\arcsec .041 to 0\\arcsec .6, and a 190-pixel region is selected to generate the light curve, the noise is about 1\/202 (1\/50 for the H$\\alpha$ channel) of the square root of the original intensity. The end result is that the amplitude error of the TiO channel is about 0.6, and that of the H$\\alpha$ is about 1.5. So, for AIA 1700, 1600, and 304 \\AA, the amplitudes of the reconstructed mode are two times larger than the estimated noise. Likewise, for AIA 171 \\AA, the NVST TiO, and H$\\alpha$, the amplitudes are at least three times larger. From this error analysis, together with the confidence levels in the Fourier and SST spectra, we conclude that the reconstructed components of 12 mHz are physical rather than high-frequency noise. \n\n\\begin{figure*}\n\t\\centering\n        \\includegraphics[width=0.9\\textwidth]{fig6anim.eps}\n\t\\caption{Online animation of the the AIA 171 \\AA\\ images (left) and the 12~mHz modes reconstructed by every pixel in the AIA 171 \\AA\\ channel (right).}\n\t\t\\label{fig6anim}\n\\end{figure*}\n\nWe reconstructed the 12~mHz modes for every pixel in the AIA 171 \\AA\\ channel, and then generated a running-difference animation (Figure 6). This animation is restricted to the time interval between 03:59:59 and 04:09:47 UT, during which the 12~mHz signal is the most evident (see the uppermost panel in Figure~4). In the animation, the footpoint of the coronal fan structures is marked with a white cross. The umbral and penumbral borders as determined with the NVST TiO images are marked with the red and yellow curves, respectively. The dashed blue lines highlight a series of expanding annuli. The movie exhibits an intermittent and outward propagating wave near the footpoint of the coronal fan structures. It emerges first near the footpoint and then propagates outward along the fan structures before disappearing. A running-difference image, extracted from the animation, is shown in Figure 5a, where we also display an artificial slit extending from the footpoint and following the coronal fan structure (the thick green line). A time--distance plot along this slit, 24 pixels long and 3 pixels wide, is shown in Figure 5b. The propagating perturbations are featured by the repeating diagonal ridges, which we outline with a series of red dashed lines to help guide the eye. Standard practice yields that the disturbances propagate at a speed of 49 $\\pm$ 4 km s$^{-1}$. \n\n\nTo further understand where the 12~mHz signal in the TiO image sequences comes from, we also examined the power spectrum in the TiO observations for the region in the quiet photosphere that surrounds the sunspot. Shown in Figure 5c is a spectrum at an arbitrarily chosen location (marked ``B'' in the left column of Figure~2). Evidently, this spectrum does not exhibit any power in the range of 10--14 mHz, delineated by the horizontal dashed lines. This means that the 12~mHz signal originates from within the umbra rather than coming from the surrounding quiet photosphere. \n\n\\section{discussion and conclusion}\n\\label{sec_disc}\n\nA high-frequency oscillation with a frequency range 10--14 mHz (about one minute) above a sunspot umbra atmosphere was found using simultaneous observations captured by the NVST and SDO\/AIA instruments. A novel time--frequency analysis method, called SST, was used to decompose and reconstruct the high-frequency components. To ensure that the reconstructed components are physical rather than high-frequency noise, a synthetic signal (the blue curve in Figure 1a), the noise floor appears at even lower frequencies (the spectral break now occurs at 8 mHz) was analyzed. The results decomposed and reconstructed demonstrate that the SST has no problem in resolving the physical signal even if the contamination from high-frequency noise is even stronger in the synthetic time series. Analyzing Fourier and SST spectra of the intensity curves at six channels: the AIA 171, 304, 1600, 1700 \\AA\\, the NVST H$\\alpha$, and TiO, the spectral contents between 10 and 14 mHz not only have significant peaks, but also higher than 95 \\% confidence levels.  Furthermore, error analysis demonstrates that all the reconstructed amplitudes of the 12 mHz signals are at least twice larger than the corresponding error. The time--distance diagram, coupled with a subsonic speed ($\\sim$ 49 km s$^{-1}$), highlights the fact that these coronal perturbations are best described as upwardly propagating magnetoacoustic slow-mode waves.\n\nCompared with the spectra from the region surrounding the sunspot, the 12~mHz periodicity exists only inside the umbra, meaning that the modes at the coronal height are probably related to the perturbations from the photospheric umbra. We note that \\citet{2017ApJ...847....5K} have already found a similar periodicity at chromospheric heights above an umbra. However, to our knowledge, the present study is the first to reveal the existence of such a periodicity at different heights above an umbra in imaging observations.  \n\nBefore closing, we would like to point out that our findings benefit from the capability of SST for suppressing spectral smearing. As such, SST is likely to find some wider use for representing and decomposing intrinsic modes in non-stationary signals.\n\n\n\\acknowledgments\nThe authors thank the referee for useful comments. We thank the NVST team for their high-resolution observations and the use of \\textit{SDO\/AIA} image obtained courtesy of NASA\/SDO and the AIA science teams. This work is supported by the National Natural Science Foundation of China (Numbers: 11463003, 11503080, 11761141002, 41474149, 41674172) and the Joint Research Fund in Astronomy (U1531132, U1531140, U1631129) under coperative agreement between the NSFC, the National Key Research and Development Program of China (Number: 2016YFE0100300), and the CAS Key Laboratory of Solar Activity of National Astronomical Observatories (KLSA201715).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nFinding numerical approximations of fractional integrals or fractional derivatives\nof a given function is one of the most important problems in theory of numerical\nfractional calculus. The operators of fractional integration and fractional\ndifferentiation are more complicated than the classical ones, so their evaluation \nis also more difficult than the integer order case. Li et al. use spectral \napproximations for computing the fractional integral and the Liouville--Caputo \nderivative \\cite{sal2}. They also developed\nnumerical algorithms to compute fractional integrals and Liouville--Caputo derivatives\nand for solving fractional differential equations based on piecewise polynomial\ninterpolation \\cite{sal3}. In \\cite{sal4}, Pooseh et al. presented two approximations\nderived from continuous expansions of Riemann--Liouville fractional derivatives\ninto series involving integer order derivatives and they present application\nof such approximations to fractional differential equations and fractional problems\nof the calculus of variations. Some other computational algorithms\nare also introduced in \\cite{sal7,sal5,sal6}.\nFor increasing the accuracy of the calculation, using the Gauss--Jacobi quadrature rule\nis appropriate for removing the singularity of the integrand. So considering\nthe nodes and weights of the quadrature rule is an important problem.\nThere are many good papers in the literature addressing the question of how to\nfind the nodes and weights of the Gauss quadrature rule---see \\cite{sal10,sal9,sal8}\nand references therein. The more applicable and developed method\nis the Golub--Welsch (GW) algorithm \\cite{sal12,sal11}, that is used by many\nof the mathematicians who work in numerical analysis. This method takes $O(n^2)$\noperations to solve the problem of finding the nodes and weights.\nHere we use a new method introduced by Hale and  Townsend \\cite{sal13},\nwhich is based on the Glasier--Liu--Rokhlin (GLR) algorithm \\cite{sal14}.\nIt computes all the nodes and weights of the $n$-point quadrature rule\nin a total of $O(n)$ operations.\n\nThe structure of the paper is as follows.\nIn Section~\\ref{sec2}, we introduce the definitions of fractional operators\nand some relations between them. Section~\\ref{sec3} discusses the Gauss--Jacobi\nquadrature rule of fractional order and its application to approximate\nthe fractional operators. In Section~\\ref{sec4} we present two methods\nfor finding the nodes and weights of Gauss--Jacobi and discuss their advantages\nand disadvantages. Two illustrative examples are solved. In Section~\\ref{sec5}\napplications to ordinary fractional differential equations are presented.\nIn Section~\\ref{sec6} we investigate problems of the calculus of variations\nof fractional order and present a new algorithm for solving boundary\nvalue problems of fractional order. We end with Section~\\ref{sec7} \nof conclusions and possible directions of future work.\n\n\n\\section{Preliminaries and notations about fractional calculus}\n\\label{sec2}\n\nIn this section we give some necessary preliminaries of the\nfractional calculus theory \\cite{MR2218073,sal15},\nwhich will be used throughout the paper.\n\n\\begin{definition}\nThe left and right Riemann--Liouville fractional integrals\nof order $\\alpha$ of a given function $f$ are defined by\n\\begin{equation*}\n_aI_x^\\alpha f(x)=\\frac{1}{\\Gamma(\\alpha)}\n\\int_a^x(x-t)^{\\alpha-1}f(t)dt\n\\end{equation*}\nand\n\\begin{equation*}\n_xI_b^{\\alpha}f(x)\n=\\frac{1}{\\Gamma(\\alpha)}\\int_x^b(t-x)^{\\alpha-1}f(t)dt,\n\\end{equation*}\nrespectively, where $\\Gamma$ is Euler's gamma function, that is,\n\\begin{equation*}\n\\Gamma(x)=\\int_0^\\infty t^{x-1}e^{-t}dt,\n\\end{equation*}\n$\\alpha>0$ with $n-1<\\alpha\\leq n$, $n\\in\\mathbb{N}$, and $a<x<b$.\nThe left Riemann--Liouville fractional operator has the following properties:\n\\begin{equation*}\n{_aI_x^\\alpha} {_aI_x^\\beta}\n= {_aI_x}^{\\alpha+\\beta},\n\\quad {_aI_x^\\alpha} {_aI_x^\\beta}\n= {_aI_x^\\beta} {_aI_x^\\alpha},\n\\end{equation*}\n\\begin{equation*}\n_aI_x^\\alpha x^\\mu=\\frac{\\Gamma(\\mu+1)}{\\Gamma(\\alpha+\\mu+1)}x^{\\alpha+\\mu},\n\\end{equation*}\nwhere $\\alpha,\\beta\\geq 0$ and $\\mu>-1$. Similar relations hold for the right\nRiemann--Liouville fractional operator. On the other hand, we have the left\nand right Riemann--Liouville fractional derivatives of order $\\alpha>0$ \nthat are defined by\n\\begin{equation*}\n_aD_x^{\\alpha}f(x)=\\frac{1}{\\Gamma(n-\\alpha)}\\frac{d^n}{dx^n}\\int_a^x(x-t)^{n-\\alpha-1}f(t)dt\n\\end{equation*}\nand\n\\begin{equation*}\n_xD_b^\\alpha f(x)=\\frac{(-1)^n}{\\Gamma(n-\\alpha)}\\frac{d^n}{dx^n}\\int_x^b(t-x)^{n-\\alpha-1}f(t)dt,\n\\end{equation*}\nrespectively.\n\\end{definition}\n\nThere are some disadvantages when trying to model real world phenomena with fractional\ndifferential equations, when fractional derivatives are taken in Riemann--Liouville sense.\nOne of them is that the Riemann--Liouville derivative of the constant function \nis not zero. Therefore, a modified definition of the fractional differential operator,\nwhich was first considered by Liouville and many decades later\nproposed by Caputo \\cite{sal16}, is considered.\n\n\\begin{definition}\nThe left and right fractional differential operators in Liouville--Caputo \nsense are given by\n\\begin{equation*}\n_a^CD_x^\\alpha f(x)\n=\\frac{1}{\\Gamma(n-\\alpha)}\\int_a^x(x-t)^{n-\\alpha-1}f^{(n)}(t)dt\n\\end{equation*}\nand\n\\begin{equation*}\n_x^CD_b^\\alpha f(x)\n=\\frac{(-1)^n}{\\Gamma(n-\\alpha)}\\int_x^b(t-x)^{n-\\alpha-1}f^{(n)}(t)dt,\n\\end{equation*}\nrespectively.\n\\end{definition}\nThe Liouville--Caputo derivative has the following two properties\nfor $n-1<\\alpha\\leq n$ and $f\\in L_1[a,b]$:\n\\begin{equation*}\n({_a^CD_x^\\alpha} {_aI_x^\\alpha} f)(x)=f(x)\n\\end{equation*}\nand\n\\begin{equation*}\n({_aI_x^\\alpha} {_a^CD_x^\\alpha} f)(x)\n=f(x)-\\sum_{k=0}^{n-1}f^{(k)}(0^{+})\\frac{(x-a)^k}{k!},\\quad t>0.\n\\end{equation*}\n\n\\begin{remark}\nUsing the linearity property of the ordinary integral operator, one deduces\nthat left and right Riemann--Liouville integrals, left and right\nRiemann--Liouville derivatives and left and right\nLiouville--Caputo derivatives are linear operators.\n\\end{remark}\n\nAnother definition of a fractional differential operator, that is useful for numerical\napproximations, is the Gr\\\"{u}nwald--Letnikov derivative, which is a generalization \nof the ordinary derivative. It is defined as follows:\n\\begin{equation*}\nD_{GL}^{\\alpha}=\\lim_{n\\to\\infty}\\frac{(\\frac{t}{N})^{-\\alpha}}{\\Gamma(-\\alpha)}\n\\sum_{j=0}^{n-1}\\frac{\\Gamma(j-\\alpha)}{\\Gamma(j+1)}f\\left(t-\\frac{tj}{n}\\right).\n\\end{equation*}\n\n\n\\section{Fractional Gauss--Jacobi quadrature rule}\n\\label{sec3}\n\nIt is well known that the Jacobi polynomials\n$\\{P_n^{(\\lambda,\\nu)}(x)\\}_{n=0}^{\\infty}$, $\\lambda, \\nu >-1$, $x\\in [-1,1]$,\nare the orthogonal system of polynomials\nwith respect to the weight function\n$$\n(1-x)^\\lambda (1+x)^\\nu,\n\\quad \\lambda, \\nu >-1,\n$$\non the segment $[-1,1]$:\n\\begin{equation}\n\\label{eq3.1}\n\\int_{-1}^{1}(1-x)^\\lambda (1+x)^\\nu P_n^{(\\lambda,\\nu)}(x)\nP_m^{(\\lambda,\\nu)}(x) dx=\\partial_n^{\\lambda,\\nu}\\delta_{mn},\n\\end{equation}\nwhere\n\\begin{equation*}\n\\partial_n^{\\lambda,\\nu}=\\Vert P_n^{(\\lambda,\\nu)}\\Vert_{w^{\\lambda,\\nu}}^2,\n\\quad w^{\\lambda,\\nu}(x)=(1-x)^\\lambda (1+x)^\\nu,\n\\end{equation*}\n$$\n\\delta_{mn}=\\left\\{\n\\begin{array}{ll}\n1, & m=n,\\\\\n0, & m\\neq n\n\\end{array} \\right.\n$$\n(see, e.g., \\cite{sal17}). These polynomials satisfy\nthe three-term recurrence relation\n\\begin{equation}\n\\label{eq3.4}\n\\begin{gathered}\nP_0^{(\\lambda,\\nu)}(x)=1,\n\\quad P_1^{(\\lambda,\\nu)}(x)=\\frac{1}{2} (\\lambda+\\nu+2) x +\\frac{1}{2} (\\lambda-\\nu),\\\\\nP_{n+1}^{(\\lambda,\\nu)}(x)=\\left( a_n^{\\lambda,\\nu} x- b_n^{\\lambda,\\nu}\\right)\nP_n^{(\\lambda,\\nu)}(x) - c_n^{\\lambda,\\nu} P_{n-1}^{(\\lambda,\\nu)}(x), \n\\quad n\\geq 2,\n\\end{gathered}\n\\end{equation}\nwhere\n\\begin{align*}\na_n^{\\lambda,\\nu}\n&=\\dfrac{(2n+\\lambda+\\nu+1) (2n+\\lambda+\\nu+2)}{2(n+1) (n+\\lambda+\\nu+1)},\\\\\n&\\quad\\\\\nb_n^{\\lambda,\\nu}\n&=\\dfrac{(\\nu^2-\\lambda^2) (2n+\\lambda+\\nu+1)}{2(n+1) \n(n+\\lambda+\\nu+1) (2n+\\lambda+\\nu)},\\\\\n&\\quad\\\\\nc_n^{\\lambda,\\nu}\n&=\\dfrac{(n+\\lambda) (n+\\nu) (2n+\\lambda+\\nu+2)}{(n+1) \n(n+\\lambda +\\nu +1) (2n+\\lambda +\\nu)}.\n\\end{align*}\nThe explicit form of the Jacobi polynomials is\n\\begin{equation}\n\\label{eq3.5}\nP_n^{(\\lambda,\\nu)}(x)=\\sum_{k=0}^{n}\n\\frac{2^{n-k}n!(k+\\nu+1)_{n-k}}{(n-k)!(n+k+\\lambda+\\nu+1)_{n-k}k!}(t-1)^k,\n\\end{equation}\nwhere we use Pchhammer's notation:\n$$\n(a)_l=a(a+1)(a+2)\\cdots (a+l-1)\n$$\n(see \\cite{sal25}). Furthermore, the Jacobi polynomials\nsatisfy in the following relations:\n\\begin{equation*}\nP_n^{(\\lambda,\\nu)}(-x)=(-1)^n  P_n^{(\\lambda,\\nu)}(x),\n\\end{equation*}\n\\begin{equation*}\n\\frac{d}{dx}P_n^{(\\lambda,\\nu)} (-x)=(-1)^n \\frac{d}{dx}P_n^{(\\lambda,\\nu)}(x),\n\\end{equation*}\n\\begin{multline}\n\\label{equ3.4}\n(2n+\\lambda+\\nu) (1-x^2) \\frac{d}{dx} P_n^{(\\lambda,\\nu)}(x)\\\\\n=n\\left( \\lambda-\\nu-(2n+\\lambda+\\nu)x\\right) P_n^{(\\lambda,\\nu)}(x)\n+2(n+\\lambda) (n+\\nu)P_{n-1}^{(\\lambda,\\nu)}(x),\n\\end{multline}\n\\begin{equation*}\nP_n^{(\\lambda,\\nu)}(1)=\\binom{n+\\lambda}{n},\n\\quad P_n^{(\\lambda,\\nu)}(-1)\n=(-1)^j \\,\\dfrac{\\Gamma (n+\\nu+1)}{n!\\Gamma(\\nu+1)}.\n\\end{equation*}\nOn the other hand, the Gauss--Jacobi quadrature\nformula with remainder term is given by\n\\begin{equation}\n\\label{eq:GJqf}\n\\int_{-1}^1 (1-x)^\\lambda (1+x)^\\nu f(x) dx\n= \\sum_{k=1}^n A_k f(x_k)+R_n(f),\n\\end{equation}\nwhere $x_k$, $k=1,\\ldots, n$, are the zeros of\n$P_n^{(\\lambda,\\nu)}$, the weights $A_k$ are given by\n\\begin{equation*}\nA_k = \\dfrac{2^{\\lambda+\\nu+1} \\Gamma(\\lambda+n+1)\n\\Gamma(\\nu+n+1)}{n! \\Gamma (\\lambda+\\nu+n+1)\n\\left(\\frac{d}{dx} P_n^{(\\lambda,\\nu)}(x_k)\\right)^2 (1-x_k^2)}\n\\end{equation*}\nand the error is given by\n\\begin{equation*}\nR_n(f) = \\dfrac{f^{(2n)}(\\eta)}{(2n)!}\n\\,\\dfrac{2^{\\lambda+\\nu+2n+1} n! \\Gamma(\\lambda+\\eta+1)\n\\Gamma(\\nu +\\eta +1)\\Gamma(\\nu+\\lambda+\\eta\n+1)}{(\\lambda+\\nu+2n+1)\\left(\\Gamma(\\lambda+\\nu+2n+1)\\right)^2}\n\\end{equation*}\n(see \\cite{sal17}). We know that formula \\eqref{eq:GJqf}\nis exact for all polynomials of degree up to $2n-1$\nand is valid if $f(x)$ possesses no singularity in $[-1,1]$\nexcept at points $\\pm 1$.\n\nNow, as a special case, we introduce the fractional Jacobi polynomials\n$f_n^{\\alpha-1}(x)$, that is, $\\lambda=\\alpha-1$ and $\\nu=0$ in \\eqref{eq3.1}.\nThe set of fractional Jacobi polynomials\n$\\{f_n^{\\alpha-1}\\}_{n=0}^{\\infty}$\nis an orthogonal system of polynomials that are orthogonal\nwith respect to the weight function $(1-x)^{\\alpha-1}$,\n$\\alpha>0$, on the segment $[-1,1]$. It means that\n\\begin{equation*}\n\\int_{-1}^1w^{\\alpha-1}(x) f_n^{\\alpha-1}(x) f_m^{\\alpha-1}(x) dx\n= \\zeta_n^{\\alpha-1} \\delta_{mn},\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\zeta_n^{\\alpha-1}\n=\\Vert f_n^{\\alpha-1}\\Vert_{w^{\\alpha-1}}^2,\n\\quad w^{\\alpha-1}(x)= (1-x)^{\\alpha-1}.\n\\end{equation*}\nThe three-term recurrence relation for fractional Jacobi polynomial is given by\n\\begin{gather*}\nf_0^{\\alpha-1}(x)=1,\n\\quad f_1^{\\alpha-1}(x)=\\frac{\\alpha-1}{2} (x+1),\\\\\nf_{n+1}^{\\alpha-1} (x)= \\left(A_n^\\alpha x+ B_n^\\alpha \\right)\nf_n^{\\alpha-1}-C_n^\\alpha f_{n-1}^{\\alpha-1}(x),\\quad n\\geq 2,\n\\end{gather*}\nwhere\n\\begin{align*}\nA_n^\\alpha\n&= \\dfrac{(2n+\\alpha) (2n+\\alpha+1)}{2(n+1) (n+\\alpha)},\\\\\n&\\quad \\\\\nB_n^\\alpha\n&= \\dfrac{(\\alpha-1)^2 (2n+\\alpha)}{2(n+1)(2n+\\alpha-1) (n+\\alpha)},\\\\\n&\\quad \\\\\nC_n^\\alpha\n&=\\dfrac{n(n+\\alpha-1) (2n +\\alpha +1)}{(n+1) (n+\\alpha) (2n+\\alpha-1)}.\n\\end{align*}\nUsing \\eqref{equ3.4}, we have that\n\\begin{multline}\n\\label{equ3.5}\n(2n+1-\\alpha) (1-x^2) \\frac{d}{dx} f_n^{\\alpha -1} (x)\\\\\n=n\\left(1-\\alpha-(2n+1-\\alpha)x\\right) f_n^{\\alpha -1}(x)\n+2n(n+1-\\alpha) f_{n-1}^{\\alpha -1}(x).\n\\end{multline}\nFurthermore, the fractional Gauss--Jacobi quadrature rule is\n\\begin{equation}\n\\label{eq3.19}\n\\int_{-1}^1 (1-x)^{\\alpha-1} f(x) dx = \\sum_{k=1}^n l_k f(x_k) + E_n^\\alpha (f),\n\\end{equation}\nwhere $x_k$, $k=1,2,\\ldots, n$, are the zeros\nof $f_n^{\\alpha-1}$, the weights $l_k$ are given by\n\\begin{equation*}\nl_k=\\dfrac{2^\\alpha}{(1-x_k^2)[\\frac{d}{dx}f_n^{\\alpha-1} (x_k)]^2}\n\\end{equation*}\nand\n\\begin{equation*}\nE_n^\\alpha (f)=\\dfrac{2^n f^{(2n)}(\\eta)}{(2n)! (2n+\\alpha)}\n\\, \\left(\\dfrac{2^n n! \\Gamma (n+\\alpha)}{\\Gamma (2n+\\alpha)}\\right)^2.\n\\end{equation*}\nConsider the left Riemann--Liouville fractional integral\nof order $\\alpha>0$ for $x>0$:\n\\begin{equation}\n\\label{eq3.22}\n_0I_x^\\alpha f(x)=\\dfrac{1}{\\Gamma (\\alpha)}\n\\int_0^x (x-t)^{\\alpha-1} f(t) dt.\n\\end{equation}\nFor calculating the above integral at the collocation nodes,\nfirst we transform the interval $[0,x]$ into the segment\n$[-1,1]$ using the changing of variable\n$$\ns=2\\left(\\frac{t}{x}\\right)-1,\n\\quad ds=\\frac{2}{x} dt.\n$$\nTherefore, we can rewrite \\eqref{eq3.22} as\n\\begin{equation*}\n\\dfrac{1}{\\Gamma (\\alpha)} \\int_0^x (x-t)^{\\alpha-1} f(t) dt\n=\\left(\\frac{x}{2}\\right)^\\alpha \\dfrac{1}{\\Gamma(\\alpha)}\n\\int_{-1}^1 \\dfrac{\\rho (s) ds }{(1-s)^{1-\\alpha}},\n\\end{equation*}\nwhere\n$$\n\\rho(s)=f\\left(\\frac{x}{2} (s+1)\\right).\n$$\nSo, we have\n\\begin{equation}\n\\label{eq3.24}\n_0I_x^\\alpha f(x)=\\dfrac{1}{\\Gamma (\\alpha)}\\left(\\frac{x}{2}\\right)^\\alpha\n\\int_{-1}^1 (1-s)^{\\alpha-1} \\rho(s) ds,\n\\end{equation}\nand using the $n$ point fractional Gauss--Jacobi quadrature rule\n\\eqref{eq3.19} for \\eqref{eq3.24} we get\n\\begin{equation}\n\\label{eq3.25}\n_0I_{s_i}^\\alpha f (x) = \\dfrac{(\\frac{s_i}{2})^\\alpha}{\\Gamma (\\alpha)}\n\\sum_{k=1}^n l_k f \\left( \\frac{s_i}{2} (x_k+1)\\right)+ E_n^\\alpha (f),\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq3.26}\nl_k=\\dfrac{2^\\alpha}{(1-x_k)^2 \\left(\\frac{d}{dx} f_n^{\\alpha-1} (x_k)^2\\right)},\n\\end{equation}\n\\begin{equation*}\nE_n^\\alpha(f) = \\left(\\frac{s_i}{2}\\right)^\\alpha \\frac{1}{\\Gamma (\\alpha)}\n\\, \\dfrac{f^{(2n)} \\left(\\frac{s_i}{2}(\\eta+1)\\right) }{(2n)!}\n\\, \\dfrac{2^{2n} (n!)^2}{(2n+\\alpha)} \\, \\left(\n\\dfrac{\\Gamma (n+\\alpha)}{\\Gamma (2n+\\alpha)}\\right)^2\n\\end{equation*}\nand $x_k$, $k=1, \\ldots, n$, are quadrature nodes. In this way\nwe just need to compute the nodes and weights of the\nfractional Gauss--Jacobi quadrature rule using a fast and accurate algorithm.\nSimilarly, an approximation formula for computing the left Liouville--Caputo \nfractional derivative\nof a smooth function $f$, using the suggested method, is given by\n\\begin{equation}\n\\label{eq3.28}\n{_0^CD_{s_i}^\\alpha} f(x)=\\dfrac{(\\frac{s_i}{2})^{1-\\alpha}}{\\Gamma(1-\\alpha)}\n\\sum_{k=1}^n l_k f' \\left( \\frac{s_i}{2} (x_k+1)\\right)+ E_n^\\alpha (f),\n\\quad \\alpha \\in (0,1),\n\\end{equation}\nwhere $x_k$, $k=1,2,\\ldots,n$, are the zeros of $f_n^{-\\alpha}$,\n$$\nl_k=\\dfrac{2^{1-\\alpha}}{(1-x_k)^2 \\left(\\frac{d}{dx} f_n^{-\\alpha} (x_k)^2\\right)}\n$$\nand\n$$\nE_n^\\alpha (f) = \\dfrac{(\\frac{s_i}{2})^{1-\\alpha}}{\\Gamma(1-\\alpha)}\n\\dfrac{f^{(2n+1)}\\left(\\frac{s_i}{2} (\\eta +1)\\right)}{(2n+1)!}\n\\, \\dfrac{2^{2n} (n!)^2}{(2n+1-\\alpha)}\n\\, \\left(\\dfrac{\\Gamma (n+1-\\alpha)}{\\Gamma (2n+1-\\alpha)}\\right)^2.\n$$\nAnalogous formulas hold for the right Liouville--Caputo fractional derivative.\nNext we present two methods for computing the nodes and weights \nof the fractional Gauss--Jacobi quadrature rule \\eqref{eq3.19}.\nThe first one is based on Golub--Welsch algorithm, while the second\nis a recent method introduced by Hale and Townsend \\cite{sal13},\nwhich is based on Newton's iteration for finding roots\nand a good asymptotic formula for the weights.\n\n\n\\section{Two methods for calculating nodes and weights}\n\\label{sec4}\n\nPang et al. \\cite{sal1} use Gauss--Jacobi and Gauss--Jacobi--Lobatto quadrature\nrules for computing fractional directional integrals, together with \nthe Golub--Welsch (GW) algorithm for computing nodes and weights \nof quadrature rules. The GW algorithm exploits the three-term recurrence \nrelations \\eqref{eq3.4} satisfied by all real orthogonal polynomials.\nThis relation gives rise to a symmetric tridiagonal matrix,\n\\begin{equation*}\nx\\left[\n\\begin{array}{c}\n{J}_0^{\\lambda,\\nu} (x) \\\\\n{J}_1^{\\lambda,\\nu} (x) \\\\\n\\vdots \\\\\n{J}_{N-2}^{\\lambda,\\nu} (x)\\\\\n{J}_{N-1}^{\\lambda,\\nu} (x)\n\\end{array}\\right]\n=\\left[ \\begin{array}{ccccc}\nA_1 & B_1 & 0 & \\cdots & 0\\\\\nB_1 & A_2 & B_2 & \\cdots & 0 \\\\\n\\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\\n0 & \\cdots & \\ddots & A_{N-1} & B_{N-1}\\\\\n0 & \\cdots & 0  & B_{N-1} & A_N\\\\\n\\end{array}\n\\right] \\,\n\\left[ \\begin{array}{c}\n{J}_0^{\\lambda,\\nu} (x)\\\\\n{J}_1^{\\lambda,\\nu} (x)\\\\\n\\vdots \\\\\n{J}_{N-2}^{\\lambda,\\nu} (x)\\\\\n{J}_{N-1}^{\\lambda,\\nu} (x)\n\\end{array}\\right]+\n\\left[  \\begin{array}{c}\n0\\\\\n0\\\\\n\\vdots\\\\\n0\\\\\nB_N{J}_{N}^{\\lambda,\\nu} (x)\n\\end{array}\\right] ,\n\\end{equation*}\nwhere\n\\begin{equation*}\nA_n=\\frac{-b_n}{a_n},\\quad B_n=\\sqrt{\\frac{c_{n+1}}{a_n a_{n+1}}},\n\\quad {J}_{n}^{\\lambda,\\nu}(x)=\\dfrac{{J}_{n}^{\\lambda,\\nu}}{\\sqrt{c_{n,\\lambda,\\nu}}},\n\\end{equation*}\nin which\n\\begin{equation*}\na_n = \\left\\{\n\\begin{array}{ll}\n\\dfrac{(2n+\\lambda+\\nu-1) (2n+\\lambda+\\nu)}{2n (n+\\lambda+\\nu)},\n& \\lambda+\\nu \\neq -1,\\\\\n\\dfrac{2n-1}{n}, &  n\\geq 2, \\lambda+\\nu =-1,\\\\\n\\dfrac{1}{2}, & n=1, \\lambda+\\nu=-1,\n\\end{array} \\right.\n\\end{equation*}\n\\begin{equation*}\nb_n = \\left\\{\n\\begin{array}{ll}\n-\\dfrac{(2n+\\lambda+\\nu+1) (\\lambda^2-\\nu^2)}{2n (n+\\lambda+\\nu) (2n+\\lambda+\\nu -2)},\n& \\lambda+\\nu \\neq -1,\\\\\n\\,&\\,\\\\\n\\dfrac{\\nu^2-\\lambda^2}{n(2n-3)}, & n\\geq 2, \\lambda+\\nu=-1,\\\\\n\\,&\\,\\\\\n\\dfrac{\\nu^2-\\lambda^2}{2}, & n=1, \\lambda+\\nu=-1,\n\\end{array} \\right.\n\\end{equation*}\n\\begin{equation*}\nc_n=\\dfrac{(n+\\lambda-1) (n+\\nu-1) (2n+\\lambda+\\nu)}{n(n+\\lambda+\\nu) \n(2n+\\lambda+\\nu-2)}, \\, n\\geq2,\n\\end{equation*}\n$$\nc_{n,\\lambda,\\nu}=\\partial^{\\lambda,\\nu}.\n$$\nIt is easy to prove that $x_i$ is the zero of\n$P_n^{(\\lambda,\\nu)}$ if and only if $x_i$\nis the eigenvalue of the tridiagonal matrix \\cite{sal1}.\nMoreover, Golub and Welcsh proved that the weights of quadrature\nare the first component of corresponding eigenvectors \\cite{sal11}.\nThis algorithm takes $O(n^2)$ operations to solve the eigenvalue problem\nby taking advantage of the structure of the matrix and noting that only\nthe first component of the normalized eigenvector need to be computed.\nAn alternative approach for computing the nodes and weights of the Gauss--Jacobi rule\nis to use the same three-term recurrence relation in order to compute Newton's iterates,\nwhich converge to the zeros of the orthogonal polynomial \\cite{sal18}.\nSince the recurrence requires $O(n)$ operations for each evaluation of the polynomial\nand its derivative, we have the total complexity of order $O(n^2)$ \nfor all nodes and weights.\nFurthermore, the relative maximum error in the weights is of order $O(n)$\nand for the nodes is independent of $n$. Here we develop the new technique\nintroduced by Hale and Townsend, which utilizes asymptotic formulas\nfor both accurate initial guesses of the roots and efficient evaluation\nof the fractional Jacobi polynomial $f_n^{\\alpha-1}$ inside Newton's method \\cite{sal13}.\nWith this method it is possible to compute the nodes and weights\nof the fractional Gauss--Jacobi rule in just $O(n)$ operations\nand almost full double precision of order $O(1)$. For performance of the method,\nfirst put $\\theta_k=\\cos^{-1}x_k$ to avoid the existing clustering in computing\n$w_k$, because of presence of the term $(1-x_k^2)$ in the denominator of\n\\eqref{eq3.26} that lead to cancellation error for $x_k = \\pm 1$. \nThen, using a simple method, like the bisection one, \nfor finding $\\theta_k^{[0]}$ as the initial guess of the $k$th root,\nwe construct the Newton iterates for finding the nodes as follows:\n\\begin{equation*}\n\\theta_k^{[j+1]}=\\theta_k^{[j]}-\\frac{f_n^{\\alpha-1}\\left(\n\\cos\\theta_k^{[j]}\\right)}{\\left[-\\sin \\theta_k^{[j]}\n\\frac{d}{d\\theta}f_n^{\\alpha-1} (\\cos \\theta_k^{[j]}) \\right]},\n\\quad j=0,1,2,\\ldots\n\\end{equation*}\nOnce the iterates have converged, the nodes are given by\n$x_k=\\cos\\theta_k$ and using \\eqref{eq3.26} the weights are given by\n\\begin{equation*}\nl_k=\\left( \\frac{d}{d\\theta} f_n (\\cos \\theta_k)\\right)^{-2}.\n\\end{equation*}\nSince the zeros of the orthogonal fractional Jacobi polynomial are simple \\cite{sal17},\nwe conclude that the Newton iterates converge quadratically \\cite{sal22}. Furthermore,\nsince all fractional Jacobi polynomials satisfy the relation\n\\eqref{eq3.5}, we just need to consider all calculations for\n$x\\in [0,1]$, i.e., $\\theta \\in [0,\\frac{\\pi}{2}]$. There are three asymptotic\nformulas for finding the nodes.\n\\begin{enumerate}\n\\item[1)] One is given by Gatteschi and Pittaluga \\cite{sal19}\nfor Jacobi polynomials and here to the case of fractional Jacobi polynomials:\n\\begin{equation*}\n\\tilde{x}_k=\\cos\\left[f_k+\\frac{1}{4\\rho^2} \\left( \\frac{1}{4}\n- (\\alpha-1)^2\\right) \\cot\\left(\\frac{f_k}{2}\\right)\n-\\frac{1}{4}\\tan\\frac{f_k}{2}\\right]+O(n^{-4}),\n\\end{equation*}\n$|\\alpha|\\leq \\frac{1}{2}$, where $\\rho=n+\\frac{\\alpha}{2}$ and\n$f_k=(k+\\frac{\\alpha}{2}-\\frac{3}{4})\\frac{\\pi}{\\rho}$.\n\n\\item[2)] Let $j_{\\alpha-1,k}$ denote the $k$th root of Bessel's function\n$J_{\\alpha-1}(z)$. Then the approximation given by Gatteschi and Pittaluga \\cite{sal19}\nfor the nodes near $x=1$ becomes\n\\begin{gather*}\n\\tilde{x_k}=\\cos \\left[\\dfrac{j_{\\alpha-1,k}}{\\gamma}\\left(1\n-\\dfrac{4-(\\alpha-1)^2}{720 \\gamma^4}\\right)\\left(\\frac{ j_{\\alpha-1, k}^2}{2}\n+\\alpha^2 -2\\alpha\\right)\\right]\\\\\n+j_{\\alpha-1,k}^5 O(n^{-7}),\n\\end{gather*}\n$\\alpha \\in [-\\frac{1}{2}, \\frac{1}{2}]$, where\n$\\gamma=\\sqrt{\\rho^2+(2-\\alpha^2-2\\alpha)\/12}$.\n\n\\item[3)] Other asymptotic formulas for the nodes near\n$x=1$ are given by Olver et al. \\cite{sal20}:\n\\begin{gather*}\n\\tilde{x}_k=\\cos \\left[\\psi_k+ (\\alpha^2-2\\alpha+3\/4)\\,\n\\dfrac{\\psi_k \\cot (\\psi_k)-1}{2\\rho^2 \\psi_k}\n- (\\alpha-1)^2 \\dfrac{\\tan (\\frac{\\psi_k}{2})}{4\\rho^2}\\right]\\\\\n+ j_{\\alpha-1 ,k}^2 O (n^5),\n\\end{gather*}\n$\\alpha>-\\frac{1}{2}$, where $\\psi_k=\\dfrac{j_{\\alpha-1 ,k}}{\\rho}$.\n\\end{enumerate}\nFor computing the weights $l_k$ we just need to evaluate\n$\\dfrac{d}{d\\theta} f_n^{\\alpha-1} (\\cos \\theta)$\nin the $\\theta_k$. To this end, we use relation \\eqref{equ3.5}\nbetween $f_n^{\\alpha-1}$ and $(f_n^{\\alpha-1})'$.\nSo we just need to compute the value of $ f_n^{\\alpha-1}$ at\n$x = \\cos\\theta$. This work is done by using an asymptotic formula introduced in\n\\cite{sal13}, which takes the following form for fractional Jacobi polynomials:\n\\begin{multline*}\n\\sin^{\\alpha-1\/2}(\\theta\/2)\\cos^{1\/2}(\\theta\/2)f_n^{\\alpha-1} (\\cos \\theta)\\\\\n=\\dfrac{2^{2\\rho} B (n+\\alpha,n+1)}{\\pi}\\sum_{m=0}^{M-1}\n\\dfrac{f_m(\\theta)}{2^m (2\\rho+1)_m}+V_{M,n}^{\\alpha-1}(\\theta),\n\\end{multline*}\nwhere $\\rho=n+\\alpha\/2$, $B(\\alpha,\\beta)$ is the Beta function,\n\\begin{equation*}\nf_m(\\theta)=\\sum_{l=0}^m \\dfrac{c_{m,l}^{\\alpha-1}}{l! (m-l)!}\n\\,\\dfrac{\\cos (\\theta_{n,m,l})}{\\sin^l (\\theta\/2) \\cos ^{m-l} (\\theta\/2)},\n\\end{equation*}\n\\begin{equation*}\n\\theta_{n,m,l}=\\frac{1}{2} (2\\rho+m)\\theta -\\frac{1}{2}(\\alpha+l-a\/2)\\pi\n\\end{equation*}\nand\n\\begin{equation*}\nC_{m,l}^\\alpha= \\left(\\alpha-\\frac{1}{2}\\right)_l \\left(\\frac{3}{2}\n-\\alpha\\right)_l \\left(\\frac{1}{4}\\right)_{m-l},\n\\end{equation*}\nwhere for $\\alpha\\in(-1\/2,1\/2)$ the error term $V_{M,n}$\nis less than twice the magnitude of the first neglected term.\nWe give two examples illustrating the usefulness of the method.\n\n\\begin{example}\n\\label{exa4.1}\nThe fractional integral of the function $f(t)=\\sin(t)$ is defined as\n\\begin{equation*}\n{_0I_t}^\\alpha \\sin(t)=\\frac{1}{\\Gamma(\\alpha)}\\int_{0}^{t}(t-x)^{\\alpha-1} \\sin(x)dx,\n\\quad t\\in[0,2\\pi],\\quad \\alpha\\in(0,1).\n\\end{equation*}\nThe explicit expression of the integral, presented in Table 9.1 of \\cite{sal21}, is\n\\begin{equation*}\n{_0I_t}^\\alpha \\sin(t)=\\frac{t^\\alpha}{2i\\Gamma(\\alpha+1)}[_1F_1(1;\\alpha+1;it)\n- {_1F_1}(1;\\alpha+1;-it)],\n\\end{equation*}\nwhere $i=\\sqrt{-1}$ and $_1F_1$ is the generalized hyper-geometric function defined by\n\\begin{equation*}\n_1F_1(p;q;z)=\\sum_{k=0}^{\\infty}\\frac{(p)_k z}{(q)_k}.\n\\end{equation*}\nChoosing the test points in the set $\\{s_k=\\frac{k\\pi}{8}, k=0,1,\\ldots,16\\}$,\nthe error is given by\n\\begin{equation}\n\\label{eq5.4}\n\\sqrt{\\frac{\\sum_{k=0}^{16} (E_k-A_k)^2}{\\sum_{k=0}^{16} E_k}},\n\\end{equation}\nwhere $E_k$ is the exact result at $s_k$ and $A_k$ is the approximated result by our method.\nThe values of \\eqref{eq5.4} for $\\alpha=0.25$, $0.5$ and $0.75$ are listed in the Table~\\ref{Table 1}.\nWe used the command \\texttt{jacpts} of the \\texttt{chebfun} software \\cite{sal13}\nfor finding the nodes and weights of the fractional Gauss--Jacobi quadrature (\\texttt{fgjq}) rule.\nThe errors show that the method is more accurate than the method introduced in \\cite{sal1}.\n\\begin{table}[ht]\n\\caption{The errors obtained for Example \\ref{exa4.1} from \\eqref{eq5.4},\nwith $\\alpha= 0.25, 0.5, 0.75$ and  $n=5,6,7,8,16$ in \\eqref{eq3.25}.}\n\\label{Table 1}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n$\\alpha$ & $n=5$&$n=6$&$n=7$&$n=8$&$n=16$\\\\\\hline\n$0.25$ & $3.22\\times10^{-6}$&$5.14\\times10^{-8}$\n&$6.1\\times10^{-10}$& $5.58\\times10^{-12}$&$2.81\\times10^{-15}$\\\\\n$0.5$ & $4.85\\times10^{-6}$ & $7.75\\times10^{-8}$ & $9.18\\times10^{-10}$\n& $8.37\\times10^{-12}$ & $7.12\\times10^{-16}$ \\\\\n$0.75$ & $5.35\\times10^{-6}$ & $8.35\\times10^{-8}$\n& $9.65\\times10^{-10}$ & $8.6\\times10^{-12}$ & $1.39\\times10^{-15}$ \\\\\\hline\n\\end{tabular}\n\\end{table}\n\\end{example}\n\n\\begin{example}\n\\label{exam:5.2.}\nIn this example we consider the function $f(t)=t^4$. The Liouville--Caputo\nfractional derivative of $f$ of order $0<\\alpha<1$ is defined as\n\\begin{equation*}\n_0^CD_t ^{\\alpha}t^4=\\frac{1}{\\Gamma(1-\\alpha)}\n\\int_{0}^{t}4(t-x)^{1-\\alpha}t^3dx,\\quad \\alpha\\in(0,1).\n\\end{equation*}\nThe explicit form of the result is\n\\begin{equation*}\n_0^CD_t ^{\\alpha} t^4=\\frac{\\Gamma(5)}{\\Gamma(4.5)}t^{3.5}\n\\end{equation*}\n(see \\cite{sal4}). Using \\eqref{eq3.28}, we can write\n\\begin{equation*}\n_0^CD_{s_i} ^{\\alpha}t^4\\simeq \\dfrac{(\\frac{s_i}{2})^{1-\\alpha}}{\\Gamma (1-\\alpha)}\n\\sum_{k=1}^n 4 l_k \\left(\\frac{s_i}{2} (x_k+1)\\right)^3.\n\\end{equation*}\nHere the nodes and weights coincide with the nodes and weights of Example~\\ref{exa4.1}\nfor $\\alpha=0.5$. With the test point set $\\{s_k=\\frac{k}{10}, k=0,1,2,\\ldots,10\\}$\nthe error defined by \\eqref{eq5.4} is listed in Table~\\ref{Table 2}. Looking to Table~\\ref{Table 2},\none can see that the best approximation is obtained for $n=2$\nand that by increasing $n$ we do not get better results.\n\\begin{table}[ht]\n\\caption{The errors obtained for Example~\\ref{exam:5.2.} from \\eqref{eq5.4}\nwith $n=2,3,\\ldots,7$ in \\eqref{eq3.28} for approximating the Liouville--Caputo \nfractional derivative of order $\\alpha= 0.5$ of function $f(t)=t^4$.}\n\\label{Table 2}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \\hline\n$n=2$&$n=3$&$n=4$&$n=5$&$n=6$&$n=7$\\\\\\hline\n$4.0\\times10^{-17}$ & $8.8\\times10^{-16}$ & $1.9\\times10^{-16}$\n& $6.4\\times10^{-16}$ & $3.8\\times10^{-16}$ & $3.8\\times10^{-16}$\\\\\\hline\n\\end{tabular}\n\\end{table}\n\\end{example}\n\n\n\\section{Application to fractional ordinary differential equations}\n\\label{sec5}\n\nConsider the fractional initial value problem\n\\begin{equation}\n\\label{eq6.1}\n\\left\\{\n\\begin{array}{ll}\n_0^CD_t^\\alpha y(t)=g(t,y(t)),\n& n-1<\\alpha<n,\\quad n\\in\\mathbb{N},\\quad 0<t<b,\\\\\ny^{(k)}(0)=b_k, & k=0,1,\\ldots,n-1,\n\\end{array} \\right.\n\\end{equation}\nwhere $y(t)$ is the unknown function, $g:\\mathbb{R}^2\\to\\mathbb{R}$\nis given and $b_k$, $k=0,1,\\ldots,n-1$, are real constants.\nThe following result establishes existence and uniqueness\nof solution for \\eqref{eq6.1}.\n\n\\begin{theorem}[See \\cite{MR2218073,sal15}]\n\\label{thm:6.1}\nConsider the fractional differential equation \\eqref{eq6.1}.\nLet $K>0$, $h^*>0$, $b_0,b_1,\\ldots,b_{n-1}\\in\\mathbb{R}$,\n$G:=[0,h^*]\\times[b_0-K,b_0+K]$,\nand function $g:G\\to\\mathbb{R}$ be continuous.\nThen, there exists $h > 0$ and a function\n$y\\in C[0, h]$ solving the Liouville--Caputo fractional initial\nvalue problem \\eqref{eq6.1}. If $0<\\alpha<1$,\nthen the parameter $h$ is given by\n$$\nh:=\\min\\left\\{h^*,\\left(\\frac{K\\Gamma(\\alpha+1)}{M}\\right)^{1\/\\alpha}\\right\\},\n\\quad M:=\\sup_{(t,z)\\in G}|g(t,z)|.\n$$\nFurthermore, if $g$ fulfils a Lipschitz condition with respect to the second variable,\nthat is,\n$$\n|g(t,y_1)-g(t,y_2)|\\leq L(y_1-y_2)\n$$\nfor some constant $L > 0$ independent of $t, y_1$ and $y_2$,\nthen the function $y\\in C[0, h]$ is unique.\n\\end{theorem}\n\nIn order to solve problem \\eqref{eq6.1} using our method, we need the following theorem.\n\n\\begin{theorem}\n\\label{thm:6.2}\nUnder the assumptions of Theorem~\\ref{thm:6.1},\nfunction $y\\in C[0, h]$ is a solution to the Liouville--Caputo\nfractional differential equation \\eqref{eq6.1}\nif and only if it is a solution to the Volterra\nintegral equation of second kind\n\\begin{equation*}\ny(t)=\\sum_{k=0}^{n-1}\\frac{t^k}{k!}b_k\n+\\frac{1}{\\Gamma(\\alpha)}\\int_0^t(t-x)^{\\alpha-1}g(x,y(x))dx.\n\\end{equation*}\n\\end{theorem}\n\n\\begin{proof}\nUsing the Laplace transform formula\nfor the Liouville--Caputo fractional derivative,\n\\begin{equation*}\n\\mathcal{L}\\{{_0^CD_t^\\alpha} y(t)\\}\n= s^\\alpha Y(s)-\\sum_{k=0}^{n-1} s^{\\alpha-k-1}y^{(k)}(0)\n\\end{equation*}\n(see \\cite{MR2218073,sal15}). Thus, using \\eqref{eq6.1}, we can write that\n\\begin{equation*}\ns^\\alpha Y(s)-\\sum_{k=0}^{n-1}s^{\\alpha-k-1}y^{(k)}(0)=G(s,y(s))\n\\end{equation*}\nor\n\\begin{equation*}\nY(s)=s^{-\\alpha}G(s,y(s))+\\sum_{k=0}^{n-1}s^{-k-1}b_k,\n\\end{equation*}\nwhere $G(s,y(s))= \\mathcal{L}\\{g(t,y(t))\\}$. \nApplying the inverse Laplace transform gives\n$$\ny(t)=\\sum_{k=0}^{n-1}\\frac{t^k}{k!}b_k\n+\\frac{1}{\\Gamma(\\alpha)}\\int_0^t(t-x)^{\\alpha-1}g(x,y(x))dx,\n$$\nwhere we used the relations\n\\begin{equation*}\n\\mathcal{L}\\{_0I_t^\\alpha y(t)\\} = \\mathcal{L}\\left\\{\\frac{1}{\\Gamma(\\alpha)}\n\\int_0^t(t-x)^{\\alpha-1}y(x)dx\\right\\}\n=\\mathcal{L}\\left\\{\\frac{t^{\\alpha-1}}{\\Gamma(\\alpha)}\n*y(t)\\right\\}=\\frac{Y(s)}{s^\\alpha}\n\\end{equation*}\nand\n$$\n\\mathcal{L}\\{t^{\\alpha-1}\\}=\\frac{\\Gamma(\\alpha)}{s^\\alpha}.\n$$\nThe proof is complete.\n\\end{proof}\n\nNow, using \\eqref{eq3.24}, \\eqref{eq3.25} and Theorem~\\ref{thm:6.2}, we have\n\\begin{equation*}\n\\begin{split}\ny(t)&=\\sum_{k=0}^{n-1}\\frac{t^k}{k!}b_k\n+\\frac{1}{\\Gamma(\\alpha)}\\int_0^t(t-x)^{\\alpha-1}g(x,y(x))dx\\\\\n&=\\sum_{k=0}^{n-1}\\frac{t^k}{k!}b_k+\\frac{1}{\\Gamma(\\alpha)}\\left(\n\\frac{t}{2}\\right)^\\alpha\\int_{-1}^1(1-s)^{\\alpha-1}\\tilde{g} (s,\\tilde{y}(s))ds,\n\\end{split}\n\\end{equation*}\nwhere\n$$\n\\tilde{g}(s,\\tilde{y}(s))=g\\left(\\frac{t}{2}(1+s),\ny\\left(\\frac{t}{2}(1+s)\\right)\\right),\n\\quad -1\\leq s\\leq 1;\n$$\n$$\n\\tilde{y}(s)=y\\left(\\frac{t}{2}(1+s)\\right),\\quad -1\\leq s\\leq 1.\n$$\nSo, using the fractional Gauss--Jacobi quadrature rule introduced in Section~\\ref{sec2},\nwe can approximate $y(t)$ by the following formula:\n\\begin{equation}\n\\label{eq6.7}\ny(t)\\simeq\\sum_{k=0}^{n-1}\\frac{t^k}{k!}b_k\n+\\frac{1}{\\Gamma(\\alpha)}\\left(\\frac{t}{2}\\right)^\\alpha\n\\sum_{k=1}^N l_k \\tilde{g}(x_k,\\tilde{y}(x_k)),\n\\end{equation}\nwhere $l_k$ and $x_k$ are the weights and nodes of our quadrature rule, respectively.\nAlso, let $t\\in[0,1]$ and $t_j=\\frac{j}{n}$ for $j=0,1,\\ldots,n$. Furthermore,\nlet the numerical values of $y(t)$ at $t_0,t_1,\\ldots,t_n$ have been obtained,\nand let these values be $y_0,y_1,\\ldots,y_n$, respectively, with $y_0=b_0$.\nNow we are going to compute the value of $y(t)$ at $t_{n+1}$, i.e., $y_{n+1}$.\nTo this end we use the predictor-corrector method introduced\nby Diethelm et al. in \\cite{sal26}, but where for the prediction part we use\ninterpolation at equispaced nodes instead of piecewise linear interpolation.\nUsing first \\eqref{eq6.7}, we know that\n\\begin{equation}\n\\label{eq10}\ny(t_{n+1})\\simeq\\sum_{k=0}^{n-1}\\frac{t_{n+1}^k}{k!}b_k\n+\\frac{1}{\\Gamma(\\alpha)}\\left(\\frac{t_{n+1}}{2}\\right)^\\alpha\\sum_{k=1}^N\nl_k g\\left(\\frac{t_{n+1}}{2}(1+x_k),y\\left(\\frac{t_{n+1}}{2}(1+x_k)\\right)\\right),\n\\end{equation}\nwhere $l_k$ and $x_k$, $k=1,2,\\ldots,N$, are the weights and nodes of\nthe fractional Gauss--Jacobi quadrature rule introduced in Section~\\ref{sec3}.\nLooking to \\eqref{eq10} we see that for calculating the value of $y(t_{n+1})$\nwe need to know the value of $g$ at point\n$\\left(\\frac{t_{n+1}}{2}(1+x_k),y\\left(\\frac{t_{n+1}}{2}(1+x_k)\\right)\\right)$.\nFor obtaining these values we act as follows.\nFirst, using the predictor-corrector method introduced in \\cite{sal28},\nwe calculate the values of $y_1(t_i)$, $i=0,1,\\ldots,n+1$,\nusing the following formulas:\n\\begin{equation*}\ny_1(t_{n+1})= \\left\\{\n\\begin{array}{ll}\ny_0+\\frac{k^\\alpha}{\\Gamma(\\alpha+2)}\\left(g(t_1,y^{pr}(t_1))\n+\\alpha g(t_0,y_0)\\right), & n=0,\\\\\ny_0+\\frac{k^\\alpha}{\\Gamma(\\alpha+2)}\\left(g(t_{n+1},\ny^{pr}(t_{n+1}))+(2^{\\alpha+1}-2) g(t_n,y(t_n))\\right)\\\\\n+\\frac{k^\\alpha}{\\Gamma(\\alpha+2)}\n\\sum_{j=0}^{n-1} a_j g(t_j,y^{pr}(t_j)), &  n\\geq 1,\n\\end{array} \\right.\n\\end{equation*}\nwhere\n\\begin{equation*}\na_j= \\left\\{\n\\begin{array}{ll}\nn^{\\alpha+1}-(n-\\alpha)(n+1)^\\alpha, & j=0,\\\\\n(n-j+2)^{\\alpha+1}+(n-j)^{\\alpha+1}-2(n-j+1)^{\\alpha+1},\n&  1\\leq j \\leq n,\\\\\n1, & j=n+1\n\\end{array} \\right.\n\\end{equation*}\nand\n\\begin{equation*}\ny^{pr}(t_{n+1})= \\left\\{\n\\begin{array}{ll}\ny_0+\\frac{k^\\alpha}{\\Gamma(\\alpha+1)}g(t_0,y_0), & n=0,\\\\\ny_0+\\frac{k^\\alpha}{\\Gamma(\\alpha+2)} \\cdot (2^{\\alpha h+1}-1) \\cdot g(t_n,y(t_n))\\\\\n+\\frac{k^\\alpha}{\\Gamma(\\alpha+2)}\\sum_{j=0}^{n-1}a_j g(t_j,y(t_j)), &  n\\geq1\n\\end{array} \\right.\n\\end{equation*}\nand $k$ is the step length, i.e., $k=t_{j+1}-t_j$ (for more details, see \\cite{sal28}).\nFinally, if $P_n$ is the interpolating polynomial passing through $\\{(t_i,y1(t_i)\\}_{i=0}^{n+1}$,\nthen we have the following formula for calculating $y(t_{n+1})$:\n\\begin{equation}\n\\label{eq6.12}\ny(t_{n+1})\\simeq\\sum_{k=0}^{n-1}\\frac{t_{n+1}^k}{k!}b_k\n+\\frac{1}{\\Gamma(\\alpha)}\\left(\\frac{t_{n+1}}{2}\\right)^\\alpha\n\\sum_{k=1}^N l_k g\\left(\\frac{t_{n+1}}{2}(1+x_k),\nP_n\\left(\\frac{t_{n+1}}{2}(1+x_k)\\right)\\right).\n\\end{equation}\n\nWe solve two examples illustrating the applicability of our method.\n\n\\begin{example}\n\\label{exam5.3}\nConsider the nonlinear ordinary differential equation\n\\begin{equation*}\n{_0^CD_t ^{\\alpha}}y(t)+y^2(t)=f(t)\n\\end{equation*}\nsubject to $y(0)=0$ and $y'(0)=0$, where\n$$\nf(t)=\\frac{120t^{5-\\alpha}}{\\Gamma(6-\\alpha)}-\\frac{72t^{4-\\alpha}}{\\Gamma(5-\\alpha)}\n+\\frac{12t^{3-\\alpha}}{\\Gamma(4-\\alpha)}+k(t^5-3t^4+2t^3)^2.\n$$\nFollowing \\cite{sal29}, we take $k=1$\nand $\\alpha=\\frac{3}{2}$. The exact solution is then given by\n$$\ny(t)=(t^5-3t^4+2t^3)^2.\n$$\nFigure~\\ref{Fig:6.1} plots the results obtained for $N=16$ in \\eqref{eq6.12}.\n\\begin{figure}\n\\includegraphics[scale=.75]{Figure1.eps}\n\\caption{The exact solution (the solid line) versus the approximated solution \n(the dashed line) of the fractional initial value problem of Example~\\ref{exam5.3}.}\n\\label{Fig:6.1}\n\\end{figure}\n\\end{example}\n\n\\begin{example}\n\\label{exam5.4}\nConsider the fractional oscillation equation\n\\begin{equation*}\n{_0^CD_t ^{\\alpha}}y(t)+y(t)=t e ^{-t}\n\\end{equation*}\nsubject to initial conditions  $y(0)=0$ and $y'(0)=0$. The exact solution is\n$$\ny=\\int_0^t G(t-x)xe^{-x}dx,\n\\quad G(t)=t^{\\alpha-1} E_{\\alpha,\\alpha}(t),\n$$\nwhere $E_{\\alpha,\\beta}(t)$ is the generalized Mittag--Lefller \nfunction \\cite{sal29}. Results are shown in Figure~\\ref{Fig:6.2} \nfor $N=16$ in \\eqref{eq6.12} and $\\alpha=\\frac{3}{2}$.\n\\begin{figure}\n\\includegraphics[scale=.75]{Figure2.eps}\n\\caption{The exact solution (the solid line) versus the approximated solution \n(the dashed line) of the fractional initial value problem of Example~\\ref{exam5.4}.}\n\\label{Fig:6.2}\n\\end{figure}\n\\end{example}\n\n\n\\section{Application to fractional variational problems}\n\\label{sec6}\n\nIn this section we apply our method to solve, numerically, \nsome problems of the calculus of variations of fractional order.\nThe calculus of variations is a rich branch of classical mathematics \ndealing with the optimization of physical quantities \n(such as time, area, or distance). It has applications in many\ndiverse fields, including aeronautics (maximizing the lift of an aircraft wing), \nsporting equipment design (minimizing air resistance on a bicycle helmet \nor optimizing the shape of a ski), mechanical engineering \n(maximizing the strength of a column, a dam, or an arch), \nboat design (optimizing the shape of a boat hull) and physics (calculating trajectories\nand geodesics, in both classical mechanics and general relativity) \\cite{sal32}.\nThe problem of the calculus of variations of fractional order is more recent\n(see \\cite{sal34,sal35,sal33} and references therein) and consists, typically,\nto find a function $y$ that is a minimizer of a functional\n\\begin{equation}\n\\label{eq7.1}\nJ[y]=\\int_{a}^{b}L\\left(t,y(t),(_0^CD_t ^{\\alpha})y(t)\\right)dt,\n\\quad \\alpha\\in(n-1,n),\n\\quad n\\in \\mathbb{N},\n\\end{equation}\nsubject to boundary conditions\n$$\ny(a)=u_a,\\quad y(b)=u_b.\n$$\nOne can deduce fractional necessary optimality equations to problem \\eqref{eq7.1}\nof Euler--Lagrange type: if function $y$ is a solution to problem \\eqref{eq7.1},\nthen it satisfies the fractional Euler--Lagrange differential equation\n\\begin{equation}\n\\label{Euler}\n\\frac{\\partial L}{\\partial y} + {_t D_b^\\alpha}\n\\frac{\\partial L}{\\partial (_0^CD_t ^{\\alpha}) }=0.\n\\end{equation}\nIt is often hard to find the analytic solution to problems of the calculus of variations\nof fractional order by solving \\eqref{Euler}. Therefore, \nit is natural to use some numerical method to find approximations \nof the solution of \\eqref{Euler}. Here we are going to apply our method\nfor solving some examples of such fractional variational problems. To this end,\nat first we find a class of fractional Lagrangians\n\\begin{equation*}\nL\\left(t,y(t),(_0^CD_t ^{\\alpha})y(t)\\right),\n\\quad 1<\\alpha <2,\n\\end{equation*}\nsuch that the corresponding Euler--Lagrange equation \\eqref{Euler} takes the form\n\\begin{equation}\n\\label{eq7.2}\n\\frac{\\partial L}{\\partial y}+ {_t D_b^\\alpha} \\frac{\\partial L}{\\partial (_0^CD_t ^{\\alpha})}\n={_t D_b^\\alpha}\\left((_0^CD_t ^{\\alpha})y(t)\\right)+g(t,y(t)) = 0.\n\\end{equation}\nWe assume a solution to this problem as follows:\n\\begin{equation}\n\\label{eq7.3}\nL\\left(t,y(t),(_0^CD_t ^{\\alpha})y(t)\\right)\n=\\frac{1}{2}\\left((_0^CD_t ^{\\alpha})y(t)\\right)^2+f(t,y(t)).\n\\end{equation}\nThen, we evaluate the left-hand side of \\eqref{eq7.2} for \\eqref{eq7.3} and we obtain\n\\begin{equation*}\n{_t D_b^\\alpha}\\left((_0^CD_t ^{\\alpha})y(t)\\right)\n+\\frac{\\partial f(t,y(t))}{\\partial y} = 0.\n\\end{equation*}\nSo, if the Lagrangian of the variational problem \\eqref{eq7.1} is of form \\eqref{eq7.3},\nthen we just need to solve the following boundary value problem of fractional order:\n\\begin{equation}\n\\label{eq7.4}\n{_t D_b^\\alpha}\\left((_a^CD_t ^{\\alpha})y(t)\\right)+g(t,y(t))=0\n\\end{equation}\nsubject to the boundary conditions $y(a)=u_a$ and $ y(b)=u_b$, where\n\\begin{equation}\n\\label{eq:m:g}\ng(t,y(t))=\\frac{\\partial f(t,y(t))}{\\partial y}.\n\\end{equation}\nNow, we apply the operator ${_tI_b^\\alpha}$ to both sides of  equation \\eqref{eq7.4}.\nWe get the following boundary value problem of fractional order:\n\\begin{equation}\n\\label{eq7.5}\n\\begin{cases}\n(_a^CD_t^\\alpha) y(t)=-{_tI_b^\\alpha}g(t,y(t)), & 1<\\alpha<2, \\quad a<t<b,\\\\\ny(a)=u_a, \\quad y(b)=u_b.\n\\end{cases}\n\\end{equation}\n\n\\begin{theorem}\n\\label{theorem6.1}\nConsider the Lagrangian \\eqref{eq7.3}\nbe such that $g(t,y(t))$ given by \\eqref{eq:m:g}\nis continuous and let  $-{_tI_b^\\alpha}g(t,y(t))=h(t,y(t))$.\nThen the function $h: [a,b]\\times \\mathbb{R}\\to \\mathbb{R}$\nis continuous and the problem \\eqref{eq7.5}\nis equivalent to the integral equation\n\\begin{multline}\n\\label{eq7.6}\ny(t)=u_a+(u_b-u_a)t+\\frac{1}{\\Gamma(\\alpha)}\n\\int_0^t (t-s)^{\\alpha-1} h(s,y(s))ds\\\\\n-\\frac{t}{\\Gamma(\\alpha)}\\int_0^1 (1-s)^{\\alpha-1}h(s,y(s))ds.\n\\end{multline}\n\\end{theorem}\n\n\\begin{proof}\nThe continuity of $h$ is one of the properties of the Riemann--Liuoville\nfractional integral operator ${_tI_b^\\alpha}$ (see, for example, \n\\cite{MR2218073,sal15}). The second part of the theorem \nis the main result of \\cite{sal30}.\n\\end{proof}\n\nNow, let $t_j=jk$, $j=0,1,\\ldots,n$, and $k=\\frac{b-a}{n}$.\nFurthermore, let the numerical values of $y(t)$ at $t_0,t_1,\\ldots,t_n$\nhave been obtained and let these values be $y_0,y_1,\\ldots,y_n$, respectively,\nand $y_0=u_a$. Now we are going to compute the value of $y(t)$ at $t_{n+1}$,\ni.e., $y_{n+1}$. To this end, first we use the same idea of \\cite{sal31}\nto predict the values of $y_0,y_1,\\ldots,y_n$. \nThen, we correct them using our method. Let\n\\begin{equation*}\na_j= \\left\\{\n\\begin{array}{ll}\n\\frac{k^\\alpha}{\\alpha(\\alpha+1)}(n^{\\alpha+1}-(n-\\alpha)(n+1)^\\alpha),\n& j=0,\\\\\n\\frac{k^\\alpha}{\\alpha(\\alpha+1)}((n-j+2)^{\\alpha+1}\n+(n-j)^{\\alpha+1}-2(n-j+1)^{\\alpha+1}),\n&  1\\leq j \\leq n,\\\\\n\\frac{k^\\alpha}{\\alpha(\\alpha+1)},\n& j=n+1\n\\end{array} \\right.\n\\end{equation*}\nand\n\\begin{equation*}\nb_j=\\frac{k^\\alpha}{\\alpha}((n+1-j)^\\alpha-(n-j)^\\alpha.\n\\end{equation*}\nThen, using \\eqref{eq7.6}, we have\n\\begin{multline*}\ny(t_{n+1})=u_a+(u_b-u_a)t_{n+1}\n+\\frac{1}{\\Gamma(\\alpha)}\\int_0^{t_{n+1}}\n(t_{n+1}-s)^{\\alpha-1} h(s,y(s))ds\\\\\n-\\frac{t_{n+1}}{\\Gamma(\\alpha)}\n\\int_0^1 (1-s)^{\\alpha-1}h(s,y(s))ds.\n\\end{multline*}\nTherefore,\n\\begin{equation*}\ny_{n+1}=u_a+(u_b-u_a)t_{n+1}+\\frac{1}{\\Gamma(\\alpha)}\n\\sum_{i=0}^n a_i h(t_i,y_i)-\\frac{t_{n+1}}{\\Gamma(\\alpha)}\n\\sum_{i=0}^n b_i h(t_i,y_i).\n\\end{equation*}\nFinally, if $P_n$ is the interpolation polynomial of the nodes\n$\\{(t_i,y_i)\\}_{i=0}^{n+1}$, then the following correction formula\nis an approximation of the solution to problem \\eqref{eq7.5} at the point $t_{n+1}$:\n\\begin{multline*}\ny(t_{n+1})\\simeq u_a+(u_b-u_a)t_{n+1}\\\\\n+\\frac{1}{\\Gamma(\\alpha)}\\left(\\frac{t_{n+1}}{2}\\right)^\\alpha\n\\sum_{k=1}^N l_k g\\left(\\frac{t_{n+1}}{2}(1+x_k),P_n(\\frac{t_{n+1}}{2}(1+x_k)\\right)\n-\\frac{t_{n+1}}{\\Gamma(\\alpha)}\\sum_{i=0}^{n+1} b_i h(t_i,y_i),\n\\end{multline*}\nwhere $x_k$ and $l_k$, $k=1,2,\\ldots,N$, are the nodes and the weights\nof the fractional Gauss--Jacobi quadrature rule. Now we are going to apply\nthis method to solve a concrete fractional problem of the calculus of the variations.\n\n\\begin{example}\n\\label{example6.2}\nConsider the following Fractional Variational Problem (FVP):\nto find the minimizer of the functional\n\\begin{equation}\n\\label{eq:fvp:a}\nJ[y]=\\int_0^1 \\left((_0^CD_t ^{\\alpha}y(t))^2+y^2(t)\\right)dt,\n\\quad 1 < \\alpha <2,\n\\end{equation}\nsubject to\n\\begin{equation}\n\\label{eq:fvp:bc}\ny(0)=1,\\quad y(1)=\\frac{2 e}{1 + e^2}.\n\\end{equation}\nWhen $\\alpha \\rightarrow 1$, this FVP tends to the classical problem\nof the calculus of variations\n\\begin{equation}\n\\label{eq:p:a1}\nJ[y]=\\int_0^1 \\left(y'(t)^2+y^2(t)\\right)dt \\longrightarrow \\min,\n\\quad y(0)=1, \\quad y(1)=\\frac{2 e}{1 + e^2},\n\\end{equation}\nwhich has the solution\n\\begin{equation}\n\\label{eq:s:a1}\ny(t)=\\frac{e^t + e^{2-t}}{1+e^2}.\n\\end{equation}\nOn the other hand, the fractional Euler--Lagrange equation\nassociated with \\eqref{eq:fvp:a} is\n\\begin{equation}\n\\label{eq7.8}\n2 \\, {_t D_1^\\alpha}\\left(_0^CD_t ^{\\alpha}y(t)\\right)+2y(t)=0.\n\\end{equation}\nLooking to Figure~\\ref{Fig:3} and Figure~\\ref{Fig:5},\nwe can see that the numerical solutions to the fractional Euler--Lagrange\nequation \\eqref{eq7.8} subject to boundary conditions \\eqref{eq:fvp:bc}\nare approximating the solution \\eqref{eq:s:a1} to the classical variational\nproblem \\eqref{eq:p:a1}. Note that the exact solution to the FVP\n\\eqref{eq:fvp:a}--\\eqref{eq:fvp:bc} is unknown.\n\\begin{figure}\n\\includegraphics[scale=.75]{Figure3.eps}\n\\caption{The exact solution to \\eqref{eq:p:a1} (solid line)\nand the approximated solution to the FVP \\eqref{eq:fvp:a}--\\eqref{eq:fvp:bc}\nwith $\\alpha=1.5$ (dashed line).}\n\\label{Fig:3}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[scale=.75]{Figure4.eps}\n\\caption{The exact solution to \\eqref{eq:p:a1} (solid line)\nand the approximated solution to the FVP \\eqref{eq:fvp:a}--\\eqref{eq:fvp:bc}\nwith $\\alpha=1.1$ (dashed line).}\n\\label{Fig:5}\n\\end{figure}\n\\end{example}\n\n\n\\section{Conclusions}\n\\label{sec7}\n\nWe presented a new numerical method for approximating Riemann--Liouvile\nfractional integrals and Liouville--Caputo fractional derivatives. \nWe applied the method for solving fractional linear and nonlinear \ninitial value problems. Furthermore, we introduced a new method \nfor solving fractional boundary value problems and applied this new method\nfor solving fractional Euler--Lagrange equations. Finally, by considering \na concrete problem of the calculus of variations, we showed that the method \nis useful to solve fractional variational problems.\n\nThe results of the paper can be extended in several directions.\nAs an example of further possible developments in the field, \nwe can mention the new fractional derivative with non-local \nand non-singular kernel proposed by Atangana and Baleanu \\cite{Atangana:Bal}.\nSuch derivative seems to answer some outstanding questions that were posed \nwithin the field of fractional calculus, giving origin \nto chaotic behaviors \\cite{MR3507052}. The importance to develop\nnew numerical methods to deal with such fractional calculus is well\npresented in \\cite{Alkahtani}. We claim that our results can be generalized\nto cover the Atangana--Baleanu calculus. Similarly can be said\nabout numerical simulations with the Caputo--Fabrizio fractional \norder derivative \\cite{MR3427809,Cap:Fab}.\n\n\n\\section*{Acknowledgements}\n\nThis work is part of first author's PhD project.\nPartially supported by Islamic Azad University\n(Science and Research branch of Tehran), Iran;\nand CIDMA--FCT, Portugal, within project UID\/MAT\/04106\/2013.\nJahanshahi was supported by a scholarship from the Ministry of Science,\nResearch and Technology of the Islamic Republic of Iran,\nto visit the University of Aveiro, Portugal.\nThe hospitality and the excellent working conditions\nat the University of Aveiro are here gratefully acknowledged.\nThe authors are grateful to two anonymous referees, for several comments \nand suggestions, which helped them to improve the manuscript. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nDespite many efforts, solving an interacting gauge theory in 4D still remains an unsolved problem. At the same time there is a number of exact non-perturbative results available in the maximally supersymmetric gauge theory in 4D.\nEven though this theory has a large number of symmetries, it is still a highly nontrivial interacting\ntheory, which generates physically significant observables interpolating   between a free 4D QFT at $\\lambda=0$ and the dynamics of a classically integrable 2D string worldsheet at strong coupling. In fact, integrability was also noticed on the QFT side first in two different regimes, \\cite{Lipatov:1993yb,Faddeev:1994zg} and \\cite{Minahan:2002ve}, which are now understood to be related by a unified non-perturbative quantum integrable structure known as Quantum Spectral Curve (QSC) \\cite{Gromov:2013pga,Gromov:2014caa}. \nIt is believed that integrability  governs all observables at least in the planar limit, and could be used to solve the full theory. \n\nAt the moment, the status of this program is the following. The QSC unlocks the  non-perturbative spectrum of anomalous dimensions of all single trace operators. In order to solve the planar theory, one would also need to understand how to compute all correlation functions at finite $\\lambda$. At weak coupling, the Hexagon formalism~\\cite{Basso:2015zoa,Fleury:2016ykk,Eden:2016xvg,Bargheer:2017nne} allows to effectively reformulate the computation of Feynman diagrams into a simpler diagrammatic description involving the scattering of ``magnons''. For generic correlators, the complexity of this method still grows exponentially with the order in perturbation theory. Nevertheless, in some regimes of infinitely heavy operators there was some progress in resumming the series recently, e.g.~\\cite{Coronado:2018cxj,Kostov:2019stn,Bargheer:2019kxb,Bargheer:2019exp}.  \nThere are two other approaches trying to tackle the finite coupling region for general operators -- one based on the Separation of Variables method~(see \\cite{Jiang:2015lda,Gromov:2016itr,Cavaglia:2018lxi,Giombi:2018hsx,Cavaglia:2019pow,Gromov:2019wmz,Cavaglia:2021mft,Gromov:2022waj} for recent progress), and  another based on a combination of  Integrability with the Numerical Conformal Bootstrap (NCB)~\\cite{Rattazzi:2008pe,El-Showk:2012cjh}, started in our previous paper \\cite{Cavaglia:2021bnz}\\footnote{In different contexts unrelated to gauge theories, the integration of NCB with exact spectral data coming from integrability was also used, see e.g. \\cite{Picco:2016ilr,He:2020rfk} and similar ideas in \\cite{Nakayama:2016cim,Gliozzi:2015qsa,Gliozzi:2016cmg}. \n}.\nIn this work we continue developing this approach -- which we call ``Bootstrability''. It incorporates  both new exact analytical results and new numerical techniques. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.82\\linewidth]{finalC1V2.png}\n    \\caption{\n    The square OPE coefficient $C_1^2$ of two protected line deformation operators $\\Phi_{\\perp}$ into the non-protected one $\\Phi_{||}$, as a function of the coupling $g$. The\nthickness of the solid lines gives the allowed regions for $C_1^2$ for different methods. The yellow dashed lines give the weak coupling \\eqref{eq:C1finalresult} and strong coupling \\cite{Ferrero:2021bsb} analytic expansion. The four tiles are magnification of the previous one of a factor of $25$. The blue domain is the previous result of~\\cite{Cavaglia:2021bnz} where only two states were included. The green domain includes $10$ states and one integrated correlator while the red one includes both the constraints.}\n    \\label{fig:zoom}\n\\end{figure}\n\n\nIn this paper we study the same observables as in our previous paper \\cite{Cavaglia:2021bnz} -- namely operator insertions on an infinite, straight $\\frac{1}{2}$-BPS supersymmetric Wilson line  in $\\mathcal{N}$=4 SYM. They define a set of  correlators respecting the properties of a 1D CFT~\\cite{Drukker:2006xg}. These physical observables have been investigated intensively in the past few years with a wide variety of methods, from string and gauge theory computations to localisation, integrability  and the conformal bootstrap~\\cite{Drukker:2006xg,Correa:2012hh,Drukker:2012de,Giombi:2017cqn,Kim:2017sju,Cooke:2017qgm,Liendo:2018ukf,Giombi:2018qox,Giombi:2018hsx,Giombi:2020amn,Grabner:2020nis,Giombi:2021zfb,Barrat:2021yvp,Ferrero:2021bsb,Barrat:2021tpn,Giombi:2022anm} (for less supersymmetric setups see e.g. \\cite{Polchinski:2011im,Beccaria:2017rbe,Gimenez-Grau:2019hez}). Beside being interesting for the connection to  the study of general Wilson loops, this setting is also a  convenient laboratory to develop exact methods. For example, operators inserted on Wilson lines are simpler to study in the SoV approach~\\cite{Cavaglia:2018lxi}, since they naturally have twisted boundary conditions. Moreover, the 1D defect CFT is a very nice setup to start experimenting with Bootstrability, since we can just restrict to the 't Hooft large N limit and it is still described by a consistent 1D CFT. \n\nA recent important advance was the adaptation of the QSC to describe the spectrum of the  defect CFT in the planar limit~\\cite{Grabner:2020nis,Julius:2021uka,spec1DCFT}. \nThe approach we started in \\cite{Cavaglia:2021bnz} is to exploit\nthe knowledge of the exact spectrum to extract maximal information from the NCB. \nIn particular, we showed how  the knowledge of only two non-protected states in the spectrum allows to deduce (with rigorous NCB methods) a very narrow numerical estimate for an OPE coefficient involving the simplest non-protected operator.\\footnote{One could argue that we use some extra input, beyond the spectrum coming from integrability. In fact, we also use the knowledge of an OPE coefficient involving a supersymmetric operator (see section \\ref{sec:bootstrapsetup}), which was first obtained using information from localisation in~\\cite{Liendo:2018ukf}. However, we show in this paper that it is also possible to obtain the same result for $C_{BPS}$ using purely a comparison with integrability data, see appendix \\ref{app:CBPS}.}\n\nThe main new ingredient of this paper are two new integrated correlator  constraints~\\cite{upcomingAJMNderivation}, which greatly increase the precision of our estimates.\nThis new insight comes from the knowledge about the spectrum of the deformed observable with a defect -- such as cusp or a parameter change.\nConceptually, this is similar to what was observed for the bulk $\\mathcal{N}$=4 SYM theory, where integrated correlator relations were deduced from localisation~\\cite{Binder:2019jwn} in a deformed model, and proved to be very constraining in the bootstrap~\\cite{Chester:2021aun}. \n\nWe reserve the derivation of these integral relations to~\\cite{upcomingAJMNderivation}, where we consider the deformation obtained by forming a \\emph{cusp}  (in this case a discontinuity) in R-space on the  line.  This is associated to the cusp anomalous dimension~\\cite{Polyakov:1980ca,Korchemsky:1987wg}, which can be studied with integrability in this context~\\cite{Correa:2012hh,Drukker:2012de,Gromov:2015dfa}. It was showed in~\\cite{Cooke:2017qgm} that deformations of the contour of a Wilson loop -- in physical or $R$-space -- are equivalent to summing over integrated correlators on the undeformed loop (see for instance \\cite{Polyakov:2000ti,Semenoff:2004qr,Zarembo:2016bbk,Cooke:2017qgm}). In a certain limit, we obtain a nontrivial identity relating an integrated 4-point function in the 1D CFT to quantities in the cusp setup known from integrability.  \nConsidering similar arguments but a different deformation, we obtained a second independent relation, giving a total of two new constraints.\n\nWe  show that including the new constraints in the Bootstrability approach leads to a huge gain in precision, illustrated in Figure \\ref{fig:zoom}. We also introduce new tricks on the numerical side of the analysis, allowing  us to incorporate knowledge from more states of the spectrum and to deduce new results for the  OPE coefficients involving excited states. \n\n\n\nFurthermore, we also develop an analytic bootstrap approach at weak coupling (inspired by a similar strategy at strong coupling~\\cite{Ferrero:2021bsb}, but now including input from integrability as well as the new integrated correlator constraints). \nWith this method, we obtain a 4-loop prediction for the leading OPE coefficient, fusing two protected line deformation operators $\\Phi_{\\perp}^i$ into one non-protected operator $\\Phi_{||}$, \n\\begin{equation}\\begin{split}\\nonumber\nC_1^2(g) = &2 g^2 - \\left(24-\\frac{4 \\pi ^2}{3}\\right)g^4+\n\\left(320-16 \\pi\n   ^2+48 \\zeta_3-\\frac{76 \\pi ^4}{45}\\right)g^6\\\\\n   &-\\left(4480-\\frac{832 \\pi ^2}{3}+256 \\zeta_3-\\frac{224 \\pi ^4}{15}+880 \\zeta_5-\\frac{64 \\pi ^6}{45}\\right)g^8+\nO(g^{10})\\;.\n\\end{split}\\end{equation}\nWe also give the analytic expression at 2 loops for a 4-point correlation function~\\eqref{eqn:Gweakg4}.\n\nThe rest of this paper is organised as follows. In section \\ref{sec:setup}, we describe the main setup, and the new integrated correlator constraints are presented in section \\ref{sec:integrated}. Section \\ref{sec:numerical} contains a detailed discussion of numerical Bootstrability illustrating our main results bounding three OPE coefficients, while section \\ref{sec:analytical} develops the analytical approach at weak coupling. Finally, in section \\ref{sec:discussion} we summarise the results and discuss   future directions. The Appendices contain technical details as well as data for the spectrum of the first 10 states and numerical bounds for three structure constants. Appendix \\ref{app:CBPS} contains a new non-perturbative derivation of the form of an  OPE coefficient involving a supersymmetric operator, using a comparison with integrability data. \n\n\\section{Setup}\\label{sec:setup}\nIn this section we describe in detail the setup. First, we introduce the supersymmetric Wilson line, and its important deformation obtained by forming a cusp, defining two crucial quantities -- the Bremsstrahlung and Curvature functions. \nWe then introduce the 1D defect CFT, \n and present the conformal data for the spectrum from \\cite{Cavaglia:2021bnz} to be used in the following. Finally, we describe the conformal bootstrap problem considered in the rest of the paper. \n \n\\subsection{The line CFT and the cusp as its deformation}\\label{sec:linecusp}\n\\paragraph{The line defect.}\nThe defect CFT we consider lives on an infinite supersymmetric Wilson line in four dimensional ${\\cal N} = 4$ SYM. \n This is the so-called  Maldacena--Wilson line (MWL), defined as\n\\begin{align}\n\\label{eqn:MWLdefine}\n    {\\cal W} = \\operatorname{Tr}W_{-\\infty}^{+\\infty}, \\qquad \\text{ with }  W_{-\\infty}^{+\\infty}\\equiv \\operatorname{P}\\exp\\int_{-\\infty}^{+\\infty}dt\\,(i\\, A_{\\mu}\\dot{x}^\\mu + \\Phi_{||}|\\dot{x}|)\\;,\n\\end{align}\nwhere $\\operatorname{P}$ is the path-ordering and $x^\\mu(t)$ is the parametrisation of a straight line.\nHere $\\Phi_{||}$ denotes one scalar field out of the six real scalars of the theory. The five remaining scalars, that do not couple to the line, are denoted as $\\Phi_{\\perp}^{i}$ with $i = 1,\\cdots, 5$.\nThe MWL preserves the following symmetries of the full theory:\n\\begin{itemize}\n    \\item \\texttt{Spacetime symmetries.} The 4D conformal symmetry is broken in presence of the MWL. However, there is an ${\\rm SO}(3)$ symmetry of physical rotations about the line. Additionally, the 1D conformal group ${\\rm SO}(1,2)$ is preserved along the line. \n    \\item \\texttt{$R$-symmetry.} Since the MWL couples to only one of the six scalars, the ${\\rm SO}(6)$ $R$-symmetry of the parent theory is broken to ${\\rm SO}(5)$ in the defect theory. This symmetry represents the flavour symmetry which allows one to rotate the five scalars $\\Phi_{\\perp}^{i}$ orthogonal to the line. \n    \\item \\texttt{Supersymmetry.} The MWL is a $1\/2$-BPS observable since preserves half of the supersymmetries. This implies that its expectation value is trivial and it is given by $\\langle \\mathcal{W} \\rangle = 1$~\\cite{Drukker:1999zq,Erickson:2000af,Zarembo:2002an}. \n\\end{itemize}\nAltogether, the $1\/2$-BPS MWL preserves a ${\\rm OSp}(2,2|4)$ subgroup of the superconformal symmetry ${\\rm PSU}(2,2|4)$ of the full theory. \n\n\n\\paragraph{The cusp.}\nAn important integrable deformation of the $\\frac{1}{2}$-BPS MWL can be constructed by introducing a cusp, see Fig. \\ref{fig:cusp}, where two semi-infinite lines connect. In general this is defined as\n\\begin{align}\n    {\\cal W}_< \\equiv \\operatorname{Tr}\\left[ W_{-\\infty}^0(0,0)\n    W_{0}^{+\\infty}(\\phi,\\theta)\\right]\n    \\;,\n\\end{align}\nwhere the second infinite segment is rotated both in space-time with angle $\\phi$, as well as in the space of scalar couplings with an internal angle $\\theta$. Without loss of generality, choosing planes for these rotations we can write\n\\begin{align}\\label{eqn:thetaphinotation}\n    {W}_{t_1}^{t_2}(\\phi,\\theta) &= \\operatorname{P}\\exp\\int_{t_1}^{t_2}dt\\bigg[i\\, A_\\mu \\dot{x}^{\\mu}(t) + (\\Phi_{||}\\cos\\theta + \\Phi^1_\\perp\\sin\\theta)\\,|\\dot{x}(t)|\\bigg]\\;,\\\\\n    x(t) &= \\big(t\\cos\\phi,t\\sin\\phi,0,0\\big)\\; .\n\\end{align}\n\n\nThe MWL with a cusp is no longer finite and scales with both UV and IR regulator. In other words it is associated to an anomalous dimension (the \\emph{cusp anomalous dimension}) defined through its divergence,  \n\n\\begin{align}\n   \\langle\\,{\\cal W}_<\\,\\rangle\\sim \\left(\\frac{\\Lambda_{\\rm IR}}{\\Lambda_{\\rm UV}}\\right)^{- \\Gamma(g,\\phi,\\theta)}\\;,\n\\end{align}\nwhere  $\\Lambda_{\\text{UV}}$, $\\Lambda_{\\text{IR}}$ are UV and IR cutoffs, respectively, screening the points $0$ and $\\infty$.  $\\Gamma(g,\\phi,\\theta)$ is called the \\textit{generalised cusp anomalous dimension}. It was introduced and studied  at weak and strong coupling in~\\cite{Drukker:2011za}. A set of TBA equations for it was introduced  in~\\cite{Correa:2012hh,Drukker:2012de} and reformulated using the QSC in~\\cite{Gromov:2015dfa}, which allowed for its non-perturbative analysis.\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale = 1]{cusp2.pdf}\n    \\caption{Two semi-infinite Wilson rays meet at the origin at a cusp. This is a cusp both in physical as well as $R$-space. The physical cusp is parametrised by the angle $\\phi$ subtended by the second ray on the first. The ``internal'' angle $\\theta$ represents a change in the scalar polarisation.}\n    \\label{fig:cusp}\n\\end{figure}\nIn the near-BPS limit $\\phi\\to\\pm\\theta$, the generalised cusp anomalous dimension at order $(\\phi-\\theta)^2$ is given by \n\\begin{equation}\n\\label{dexpphi}\n\t\\Gamma(g,\\phi,\\theta)=\\frac{\\cos\\phi-\\cos\\theta}{\\sin\\phi}\\Gamma^{(1)}(g,\\phi)\n\t+\\left(\\frac{\\cos\\phi-\\cos\\theta}{\\sin\\phi}\\right)^2\\Gamma^{(2)}(g,\\phi)+{O}((\\cos\\phi-\\cos\\theta)^3)\\;.\n\\end{equation}\nThe first coefficient corresponds to the \\textit{generalised Bremsstrahlung function} originally computed at any coupling in \\cite{Correa:2012at,Fiol:2012sg} by relating this observable to expectation values of $\\frac{1}{4}$-BPS Wilson loops  accessible with  localisation techniques~\\cite{Erickson:2000af,Drukker:2000rr,Drukker:2006ga,Pestun:2009nn}.  The result was later reproduced and extended to an infinite family of observables from integrability in  \\cite{Gromov:2012eu,Gromov:2013qga} and  then checked at strong and weak coupling in \\cite{Sizov:2013joa,Bonini:2015fng}.\nIt reads\n\\begin{equation}\\label{eq:Gamma1}\n\t\\Gamma^{(1)}(g,\\phi)\\equiv \\frac{2\\phi}{1-\\frac{\\phi^2}{\\pi^2}} \\mathbb{B}_\\phi(g)\\ =\n\t\\frac{2\\phi g}{\\sqrt{\\pi^2-\\phi^2}}\n\t\\frac{I_2\\(4 g\\sqrt{\\pi^2-\\phi^2}\\)}{I_1\\(4 g\\sqrt{\\pi^2-{\\phi^2}}\\)} \\;,\n\\end{equation}\nwhere $I_n(x)$ is the modified Bessel function of the first kind. \n\nThe second coefficient in \\eqref{dexpphi} was computed analytically in \\cite{Gromov:2015dfa} using the QSC formalism. It can be written in terms of the following double contour integral \n\\begin{equation}\\begin{split}\n\\label{Gamma2res}\n\t\\Gamma^{(2)}(g,\\phi)&=\\!-\\frac{1}{2}\\!\\oint\\!\\frac{du_x}{2\\pi i}\\!\\oint\\!\\frac{du_y}{2\\pi i}\n\tu_x u_y\n{\\Big [ }D_+(x,y)K_{+\\phi}(u_x-u_y)\n\\!+\\!D_0(x,y)K_{0}(u_x-u_y)\n\\\\\n&\\qquad \\qquad\\qquad \\qquad\\qquad \\qquad \\qquad  \\qquad \\qquad \\qquad \\!+\\!D_-(x,y) K_{-\\phi}(u_x-u_y){\\Big ] }\\\\\n\\equiv &\\phi^2 \\mathbb{C}_\\phi(g) \\;,\n\\end{split}\\end{equation}\nwhere both integrals run clockwise around the cut $[-2g,2g]$, and we are using the Zhukovsky parametrisation $u_x = g ( x + 1\/x)$, and similarly for $u_y$ and $y$. The kernels $D$ and $K$ are given in Appendix A. In the following we will refer to this observable as \\textit{generalised Curvature function} $\\mathbb{C}_\\phi(g)$.\n\nIn this paper we will focus on the case in which the euclidean angle $\\phi$ is set to zero and $\\theta$ is small. This corresponds to an expansion close to 1\/2 BPS Wilson line. Considering that the coefficients appearing in \\eqref{dexpphi} scale at small $\\phi$ as follows,\n\\begin{equation}\n    \\Gamma^{(1)}\\sim 2\\phi\\mathbb{B}(g)+O(\\phi^3)\\quad\\text{and}\\quad\n    \\Gamma^{(2)}\\sim \\phi^2\\mathbb{C}(g)+O(\\phi^4) \\; ,\n\\end{equation}\nwe have\n\\begin{equation}\n    \\Gamma(g,\\phi=0,\\theta\\rightarrow 0)=\n     \\mathbb{B}(g)\\,\\sin^2\\theta\n    +\\frac{1}{4}\\left(\\mathbb{B}(g)+\\mathbb{C}(g)\\right)\\,\\sin^4\\theta +O(\\sin^6\\theta)\\;,\n\\end{equation}\nwhere $\\mathbb{B}$, simply called the Bremsstrahlung function, is given by\n\\begin{equation}\\label{eq:B0}\n\\mathbb{B}(g) = \\frac{g}{\\pi}\\frac{I_2(4 \\pi g)}{I_1(4 \\pi g)}\\;,\n\\end{equation}\nand $\\mathbb{C}$ is the Curvature function\\footnote{It is denoted by $f_2(g)$ in \\cite{Gromov:2015dfa}.}~\\cite{Gromov:2015dfa} \n\\begin{equation}\\label{curvaturedef}\n\\mathbb{C}(g) = -4\\,\\mathbb{B}^2(g)\n\t-\\frac{1}{2}\\oint\\frac{du_x}{2\\pi i}\\oint\\frac{d\n u_y}{2\\pi i}\n    K_0(u_x-u_y)F[x, y]\\;.\n\\end{equation}\nThis result is obtained by  carefully taking the $\\phi\\rightarrow 0$ limit of the generalised expression  \\eqref{Gamma2res}, with the integrands  defined in Appendix \\ref{app:curvature}. \nSolving the integral \\eqref{curvaturedef}, one can compute the perturbative expansion of the curvature function at weak coupling, \n\\begin{equation}\\begin{split}\\label{eq:weakC0}\n\\mathbb{C}&=\n4 g^4-\\left(24 \\zeta_3+\\frac{16 \\pi ^2}{3}\\right)g^6+ \\left(\\frac{64 \\pi ^2 \\zeta_3}{3}+360 \\zeta_5+\\frac{64 \\pi ^4}{9}\\right)g^8\\\\\n&- \\left(\\frac{112 \\pi ^4 \\zeta_3}{5}+272 \\pi ^2 \\zeta_5+4816 \\zeta_7+\\frac{416 \\pi ^6}{45}\\right)g^{10}\\\\\n&+\\left(\\frac{3488 \\pi ^6 \\zeta_3}{135}+\\frac{2192 \\pi ^4 \\zeta_5}{9}+\\frac{9184 \\pi ^2 \\zeta_7}{3}+63504 \\zeta_9+\\frac{176 \\pi\n  ^8}{15}\\right)g^{12} +O\\left(g^{14}\\right)\n\\end{split}\\; , \\end{equation}\nand at strong coupling\n\\begin{equation}\\begin{split}\n\\mathbb{C}&=\n\\frac{\\left(2 \\pi ^2-3\\right) g}{6 \\pi ^3}+\\frac{-24 \\zeta_3+5-4 \\pi\n   ^2}{32 \\pi ^4}+\\frac{11+2 \\pi ^2}{256 \\pi ^5 g}+\\frac{96 \\zeta_3+75+8 \\pi ^2}{4096 \\pi ^6 g^2}\\\\\n   &+\\frac{3 \\left(408 \\zeta_3-240\n   \\zeta_5+213+14 \\pi ^2\\right)}{65536 \\pi ^7 g^3}+\\frac{3 \\left(315\n   \\zeta_3-240 \\zeta_5+149+6 \\pi ^2\\right)}{65536 \\pi ^8\n   g^4}+O\\left(\\frac{1}{g^5}\\right)\\label{eq:strongC}\n\\end{split}\\;,\\end{equation}\nmatching the first two perturbative orders obtained by direct field theory and string theory computations in \\cite{Drukker:2011za}.\nThe expression at strong coupling is a new result of this paper. In order to compute it, we solved the integral \\eqref{curvaturedef} numerically with very high precision and then we converted the result in a linear combination of Riemann Zetas.\n\n\n\\subsection{1D CFT on the line}\n\nThe focus of our study is the 1D CFT that lives on the $1\/2$-BPS MWL~\\cite{Drukker:2006xg}.  Its correlation functions are defined by the expectation values of \\emph{operator insertions on the line}\n\\begin{equation}\n    \\left\\langle\\left\\langle O_{1}\\left(t_{1}\\right) O_{2}\\left(t_{2}\\right) \\cdots O_{n}\\left(t_{n}\\right)\\right\\rangle\\right\\rangle\n    \\equiv \\langle\n    \\operatorname{Tr}\n   \\operatorname{P}{ O}_1(t_1)\n \\,{ O}_{2}(t_{2})\n \\,   \\ldots \n\\, O_n(t_n)\\, W_{-\\infty}^{+\\infty}\n    \\rangle\/\\langle  W_{-\\infty}^{+\\infty} \\rangle\n    \\; ,\n\\end{equation}\nwhere $O_i$ are composite fields transforming in the adjoint representation of the gauge group. \n It descends from the embedding in $\\mathcal{N}$=4 SYM that such correlators satisfy the properties of $n$-point functions of a 1D conformal field  theory~\\cite{Drukker:2006xg}.  \n \n  The states of the  CFT live in unitary representations of the ${\\rm OSp}(2,2|4)$ symmetry left unbroken by the defect. Thus, they can be organised in superconformal multiplets labelled by four quantum numbers --- the scaling dimension $\\Delta$ associated with the 1D conformal group, the ${\\rm sp}(4)\\cong {\\rm so}(5)$ Dynkin lables $\\[a,b\\]$ associated with the $R$-symmetry, and the spin $s$ associated with ${\\rm su}(2) \\cong {\\rm so}(3)$ symmetry of rotations about the line. These representations were classified in \\cite{Gunaydin:1990ag,Liendo:2016ymz}.\n\n\nIn particular, the CFT admits $\\frac{1}{2}$-BPS multiplets denoted as ${\\cal B}_{k}$, whose superconformal primaries have the quantum numbers\n\\begin{align}\n    \\{\\Delta, \\[a,b\\], s\\} = \\{k,\\[0,k\\],0\\}\\;, \\quad k\\in \\mathbb{Z}\\; .\n\\end{align}\nThe conformal dimension of these operators is protected by supersymmetry. Two of such multiplets play an important role in our\nsetup, which is the same considered in \\cite{Giombi:2017cqn,Liendo:2016ymz,Liendo:2018ukf,Ferrero:2021bsb}. \n\nFirst is the multiplet $\\mathcal{B}_1$. This contains the simplest superconformal primary operators of the theory, corresponding to the orthogonal scalars $\\Phi^i_\\perp$, $i = 1,\\dots, 5$, and is also known as the displacement multiplet. In fact, it contains the components of the field-strength corresponding to the displacement operator associated to translations perpendicular to the Wilson line. Furthermore, the primary operators $\\Phi_{\\perp}^i$ can be seen as displacement operators corresponding to broken symmetries in R-space.\n\nThe other important multiplet for our analysis is $\\mathcal{B}_2$, which contains the superconformal primary operators $\\Phi_{\\perp}^{\\left\\{i  \\right.} \\Phi_{\\perp}^{\\left.\\;j\\right\\}} - \\frac{1}{5}\\delta^{ij} (\\Phi_{\\perp}\\cdot  \\Phi_{\\perp} )$ of protected dimension $2$.\n\n\nIn addition there are the long multiplets which in principle preserve no supercharges. They are denoted by ${\\cal L}^{\\Delta}_{s,\\[a,b\\]}$ with the indices corresponding to the quantum numbers of the primary. For long operators, $\\Delta$ is a non-trivial function of the coupling $g$. \n\nIn \\cite{Liendo:2018ukf}, several selection rules for the OPE in the CFT were deduced. In particular, the following will be relevant for us: at finite coupling,\n\\begin{equation}\\label{eq:OPEfusion}\n\\underbrace{\\mathcal{B}_1 \\times \\mathcal{B}_1 }_{\\text{OPE}} = \\mathcal{I} + \\mathcal{B}_2 + \\sum_{\\Delta>1} \\mathcal{L}_{[0,0]}^{\\Delta} \\;,\n\\end{equation}\nwhere $\\mathcal{I}$ denotes the identity multiplet, and ${\\cal L}_{0,\\[0,0\\]}^\\Delta$ are the non-protected multiplets transforming as singlets under the global ${\\rm SO}(5)\\times{\\rm SO}(3)$ symmetry. There are infinitely many multiplets with such quantum numbers, all with unprotected scaling dimensions. As shown in (\\ref{eq:OPEfusion}), the whole infinite set of such multiplets can appear in the fusion of two operators in the displacement multiplet, and this is the basis of the bootstrap problem discussed in section \\ref{sec:bootstrapsetup}. \n\n\n\\subsubsection{Spectrum}\n\n\n\nThe QSC method was shown to be applicable to compute the spectrum of such neutral operators in~\\cite{Grabner:2020nis}, where the non-perturbative scaling dimension of the lightest state was obtained. \nThe QSC equations relevant to this case descend from the ones written down for the cusped Wilson line in~\\cite{Gromov:2015dfa}. In fact, as first noticed in a special limit in \\cite{Cavaglia:2018lxi}, the cusp QSC equations admit infinitely many solutions corresponding to operator insertions at the cusp with the same quantum numbers as the vacuum. Then, in~\\cite{Grabner:2020nis}  it was found how to take the non-trivial straight-line limit to describe states of the defect CFT. \n\nIt is currently understood that solutions of the QSC equations in \\cite{Grabner:2020nis} are in one-to-one correspondence with the multiplets $\\mathcal{L}_{0,[0,0]}^{\\Delta}$. \nA systematic way to find solutions with $\\Delta > 1$, by first solving the equations at weak coupling, was developed in~\\cite{Julius:2021uka}, giving access to an infinite family of states, and a generalisation of these techniques to the entire singlet sector of the defect CFT was developed shortly after~\\cite{spec1DCFT,Cavaglia:2021bnz}. A plot of 35 states in the spectrum in the singlet sector was presented in~\\cite{Cavaglia:2021bnz}. Details of  the setup and of the nontrivial techniques to find excited state solutions of the QSC will be provided elsewhere, along with a generalisation to long operators which carry a non-zero $R$-charge~\\cite{spec1DCFT}.\n\nIn the analysis of this paper, we will need the values of the lowest lying 10 states in the singlet sector~{\\it cf.}~figure~\\ref{fig:spec}. Perturbative data for these states are given in appendix~\\ref{apd:anylSpec}. Their numerical values with at least 12 digits precision are listed for several values of the coupling constant  in appendix~\\ref{apd:numSpec}. \nBoth the perturbative and numerical data are also shared in a~\\texttt{Mathematica} notebook attached to this paper.\nInclusion of the other states does not seem to lead to a significant improvement of the bounds obtained in this paper.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=1.5]{spec.png}\n    \\caption{The first 10 lowest-lying states of the spectrum computed with the QSC. These levels will be the input in the bootstrap algorithms of this paper. For a plot including 35 states, see \\cite{Cavaglia:2021bnz}. In this paper, we label these states, for any given value of the coupling, as $\\left\\{\\Delta_n\\right\\}$,  ordered as $\\Delta_{n}<\\Delta_{n+1}$. }\n    \\label{fig:spec}\n\\end{figure}\n\n\n\\subsubsection{Conformal bootstrap setup}\\label{sec:bootstrapsetup}\nFollowing \\cite{Giombi:2017cqn,Liendo:2018ukf}, we  study the 4-point function of four identical scalars polarised in the same direction, which we take for definiteness to be $\\Phi_{\\perp}^1$. Due to conformal symmetry, their correlator can be written in terms of a function of a single variable\\footnote{See appendix \\ref{app:covariant} for a covariant expression in the R-symmetry indices.}\n\\begin{equation}\n\\langle \\langle \\Phi_{\\perp}^1(x_1) \\Phi_{\\perp}^1(x_2) \\Phi_{\\perp}^1(x_3) \\Phi_{\\perp}^1(x_4)\\rangle\\rangle = {G}(x)\\; \\left( \\langle \\langle \\Phi_{\\perp}^1(x_1) \\Phi_{\\perp}^1(x_2) \\rangle \\rangle\\, \\langle \\langle \\Phi_{\\perp}^1(x_3) \\Phi_{\\perp}^1(x_4) \\rangle \\rangle \\right)\\;\\;, \n\\end{equation}\nwhere we normalised by the 2-point functions $\\langle \\langle \\Phi_{\\perp}^i(x_i) \\Phi_{\\perp}^j(x_j) \\rangle \\rangle \\propto x_{ij}^{-2}\\,\\delta_{ij}$,  and introduced the cross ratio:\n\\begin{equation}\n x\\equiv \\frac{x_{12} x_{34}}{x_{13} x_{24}}, \\;\\;\\;\\;x_{ij}\\equiv x_i - x_j .\n\\end{equation}\nThe invariance under the cyclic relabeling $(1234)\\rightarrow (2341)$ gives the crossing equation:\n\\begin{equation}\nx^2 {G}(1 - x) - (1-x)^2 {G}(x) = 0.\n\\end{equation}\nTo setup a bootstrap problem we decompose the correlator using the OPE. To take into account supersymmetry, it is best to use superconformal blocks. We follow the results of \\cite{Liendo:2018ukf}. \nTo write down the superconformal OPE, it is convenient to parametrise the 4-point function as\n\\begin{equation}\\la{pt4}\n{G}(x) = \\mathbb{F} \\;x^2 +  (2 x^{-1} - 1)f(x) -\\left(x^2 - x +1\\right)f'(x)\\; ,\n\\end{equation}\nwhere the function $f(x)$ satisfies crossing in the form\n\\begin{equation}\\label{eq:fcrossing}\nx^2 f(1 - x) + (1-x)^2 f(x) = 0\\; .\n\\end{equation}\nIn the following it is convenient to notice that the crossing-invariant combination $G(x)\/x^2$ is a total derivative:\n\\begin{equation}\\label{eq:totalder}\n\\frac{G(x)}{x^2} = \\partial_{x}\\left( \\mathbb{F} x - \\( 1 - \\frac{1}{x} +\\frac{1}{x^2} \\) f(x) \\right) \\;.\n\\end{equation}\nIn this paper, we use the following notation for the spectrum of non-protected operators $\\left\\{\\Delta_n \\right\\}_{n=1}^{\\infty}$ -- each level denoting the lowest scaling dimension in a multiplet $\\mathcal{L}_{0,[0,0]}^{\\Delta_n}$. We will order the states according to $\\Delta_n < \\Delta_{n+1}$.\\footnote{Notice that there are some level crossings in the spectrum, see Figure \\ref{fig:spec}. We keep our naming convention  separately for each value of $g$.} We denote the corresponding OPE coefficients as \\begin{equation}\nC_n \\equiv C_{\\Phi_{\\perp}^1 ,\\; \\Phi_{\\perp}^1 ,\\;\\mathcal{L}_{0,[0,0]}^{\\Delta_n}}.\\end{equation}\nThe OPE decomposition is:\n\\begin{equation}\\label{eq:OPEf}\nf(x) = F_{\\mathcal{I}}(x) + { {C^2_{\\rm BPS} \\,  {F}_{\\mathcal{B}_2}(x)}}  + \\sum_{n } { {C^2_{n} \\,  {F}_{{\\Delta_n}}(x)}} \\; ,\n\\end{equation}\nwhere the superconformal blocks are \n\\beqa\\la{superblocks}\nF_{\\mathcal{I}}(x) &=& x\\;,\\\\\nF_{\\mathcal{B}_2}(x) &=& x - x\\, _2F_1(1,2,4;x ) \\;,\\\\\nF_{{\\Delta}}(x) &=& \\frac{x^{\\Delta+1}}{1-\\Delta}\\, _2F_1(\\Delta+1,\\Delta+2,2 \\Delta+4;x )\\; ,\n\\eeqa\nand $C_n$ are OPE coefficients for the non-protected states, which are nontrivial functions of the coupling and the main objective of our work. Finally, the constant $\\mathbb{F}$  and the OPE coefficients corresponding to the $\\mathcal{B}_2$ block are related as $\\mathbb{F}(g) = 1 + C^2_{BPS}(g)$, and this OPE coefficient was fixed by comparison with a topological observable computable with localisation~\\cite{Giombi:2017cqn,Liendo:2018ukf}. The same relation can also be derived using integrability as shown in Appendix \\ref{app:CBPS}. The result reads\n\\begin{equation}\n\\mathbb{F}(g) = 1 + C^2_{BPS}(g) = \\frac{3 I_1(4 g \\pi ) \\left(\\left(2 \\pi ^2 g^2+1\\right) I_1(4 g \\pi )-2 g \\pi  I_0(4 g \\pi )\\right)}{2 g^2 \\pi ^2 I_2(4 g \\pi ){}^2} \\;\\;, \\end{equation}\nwhich can also be recast in terms of the  Bremsstrahlung function:\n\\begin{equation}\\label{eq:F0}\n\\mathbb{F}(g) =  \\frac{3 (g^2 -\\mathbb{B}(g))}{\\pi^2 (\\mathbb{B}(g))^2}\\;\\;.\n\\end{equation}\nThe conformal bootstrap constraint, arising from the compatibility of the OPE decomposition with crossing, takes the form:\n\\begin{framed}\n\\begin{equation}\\label{eq:bootstrapeq}\n\\sum_{n} C^2_n \\;\\mathcal{G}_{\\Delta_n}(x) + \\mathcal{G}_{\\text{simple}}(g, x) = 0\\;,\n\\end{equation}\n\\end{framed}\n\\noindent\nwhere we introduced the \\emph{crossed superconformal blocks}\n\\begin{equation}\\label{eq:defGD}\n\\mathcal{G}_{\\bullet}(x) \\equiv (1 - x)^2 F_{\\bullet}(x) +  x^2 F_{\\bullet}(1 - x) , \\qquad \\bullet \\in \\left\\{ \\mathcal{I},\\mathcal{B}_2, \\Delta\\right\\}\\;\\;,\n\\end{equation}\nand $\\mathcal{G}_{\\text{simple}}(g, x)$ is an explicitly known function:\n\\begin{equation}\\label{eq:defGsimple}\n\\mathcal{G}_{\\text{simple}}(g, x) \\equiv \\mathcal{G}_{\\mathcal{I}}(x) + C^2_{BPS}(g)\\; \\mathcal{G}_{\\mathcal{B}_2}(x)\\;.\n\\end{equation}\nThe bootstrap constraint (\\ref{eq:bootstrapeq}) contains two sets of nontrivial quantities: the scaling dimensions and OPE coefficients $\\left\\{C_n\\right\\}$ for the nontrivial operators. In our approach, we take advantage of the fact that integrability effectively solves the problem of computing the spectrum, and focus on determining the OPE coefficients. In the following section we present two new exact relations involving this correlation function. \n\n\\section{Integrated\ncorrelators}\\label{sec:integrated}\nThe main new ingredient of this paper is the inclusion of the constraints on the 4-point function $G(x)$ arising from  integrable deformations of the straight line (in addition to the spectral data coming from the QSC which were already used  in~\\cite{Cavaglia:2021bnz}). \nIn this section we describe in detail these constraints, which take the form of integrals over the cross ratio for the amplitude $G(x)$ or, equivalently, $f(x)$. \n\\subsection{New integral constraints}\nWe claim that the 4-point correlator introduced in section \\ref{sec:bootstrapsetup} satisfies two integral identities involving the Bremsstrahlung and Curvature functions~\\cite{upcomingAJMNderivation}:\n\\begin{framed}\n\\begin{align}\n  &\\text{ Constraint 1:  }&&  \\int_0^1\n    \\delta G(x)\\frac{1+\\log x}{x^2}dx=\\frac{3\\mathbb{C}-\\mathbb{B}}{8 \\;\\mathbb{B}^2}\\; , \\la{eq:constr1}\\\\\n   & \\text{ Constraint 2:  }&& \\int_{0}^1 dx \\frac{\\delta f(x)}{x} = \n        \\frac{\\mathbb{C}}{4\\;\\mathbb{B}^2} + \\mathbb{F}-3 \\; ,\\label{eq:constr2}\n\\end{align}\n\\end{framed}\n\\noindent\nwhere $\n\\delta G(x) \\equiv G(x) - G_{\\text{weak}}^{(0)}(x)$, $ \\delta f(x) \\equiv f(x) - f_{\\text{weak}}^{(0)}(x)$, and $G_{\\text{weak}}^{(0)}$, $f_{\\text{weak}}^{(0)}$ are the zero-coupling values:\n\\begin{equation}\\label{fandGtree}\nf_{\\text{weak}}^{(0)}(x)=2 x+\\frac{x}{x-1}\\;, \\;\\;\\; G_{\\text{weak}}^{(0)}(x) = \\frac{2 (x-1) x+1}{(x-1)^2}\n\\; ,\n\\end{equation}\nwhich can be easily deduced from free field theory, i.e., \n\\begin{equation}\n\\frac{G_{\\text{weak}}^{(0)}(x)}{x_{12}^2 x_{34}^2 } = \\frac{1}{x_{12}^2 x_{34}^2 } + \\frac{1}{x_{14}^2 x_{23}^2 }\\;.  \n\\end{equation}\nWe notice that both integrals are convergent at finite $g$, as \\beqa\\la{limits}\n\\delta f(x)\\simeq \\frac{(3-\\mathbb{F} )}{2} \\, x^2 \n\\;\\;&,&\\;\\;\n\\delta G(x)\\simeq \\frac{(4 \\,\\mathbb{F}-1 )}{5} \\, x^2   \n\\;\\;,\\;\\;x\\to 0\\; \n,\\\\\n\\delta f(x)\\simeq \\frac{(\\mathbb{F}-3)}{2} \n\\;\\;&,&\\;\\;\n\\delta G(x)\\simeq \\frac{3(\\mathbb{F}-1 )}{2}\n\\;\\;,\\;\\;x\\to 1 \\; ,\\label{eq:limits2}\n\\eeqa\nwhere corrections are $O(x^{\\Delta_1+1})$ for $x\\sim0$, $O((1-x)^{\\Delta_1+1})$ for $x\\sim 1$, scaling with the power $\\Delta_1+1>2$, as follows from the OPE \\eq{eq:OPEf} and crossing symmetry.\n Using (\\ref{eq:totalder}) and integrating by parts,  (\\ref{eq:constr1}) can also be rewritten in terms of $f(x)$ as \n\\beqa\\la{eq:constr1b}\n    \\int_{\\delta_x}^1\n     \\delta f(x)\\( \n    \\frac{1}{x}+\\frac{1}{x^3}\n    \\)dx\n    -\\frac{1}{2} (\\mathbb{F}-3) \\log {\\delta_x}-\\mathbb{F}+3\n    &=&\\frac{3\\mathbb{C}-\\mathbb{B}}{8 \\;\\mathbb{B}^2}\\;,\n\\eeqa\nin the limit of $\\delta_x\\rightarrow 0^+$. \nWe see that the divergence in \\eq{eq:constr1b} cancels as a consequence of \\eq{limits}.\n\n\n\n\\paragraph{Heuristic explanation of the integral relations.} \nWhile a detailed proof will be presented in \\cite{upcomingAJMNderivation}, let us discuss here the main steps. The existence of the constraints is based on a general principle~\\cite{Cooke:2017qgm}:\n\n \\emph{Every deformation of a Wilson line can be parametrised in terms of integrated correlators on the original line.}\n \nThis follows from the fact that operator insertions represent infinitesimal, localised deformations. Any general  deformation (both in space-time as in R-space) can be approached through a series of integrated correlators (see \\cite{Cooke:2017qgm} for a systematic treatment of a spacetime deformation in perturbation theory). An example of this principle in action and an anticipation of the idea outlined below is presented in Appendix \\ref{app:CBPS}. \n\nTo deduce the constraints (\\ref{eq:constr1}) and (\\ref{eq:constr2}), \nwe analyse two special types of deformations in R-space, which, at leading order, are related to  integrated values of $\\langle\\langle \\Phi_{\\perp}^i \\Phi_{\\perp}^i \\rangle\\rangle$. This was shown in \\cite{Correa:2012at}, leading to the first determination of the Bremsstrahlung function $\\mathbb{B}$. The constraints (\\ref{eq:constr1}) and (\\ref{eq:constr2}) arise from the extension of this analysis to the next order in the deformation parameters, as we describe in more detail below. \n\nFirst, we consider the deformation obtained by forming a cusp in R-space: i.e., we switch on the internal angle $\\theta$ on half of the line, as discussed in section \\ref{sec:linecusp}, while keeping $\\phi = 0$. As shown in \\cite{Correa:2012at}, at order ${\\cal O}(\\theta^2)$ this connects the integrated values  of $\\langle\\langle \\Phi_{\\perp}^i \\Phi_{\\perp}^i \\rangle\\rangle$ to $\\mathbb{B}$ arising from the expansion of the cusp -- fixing in this way the normalisation the 2-point function. \n We extended the analysis to the next order ${\\cal O}(\\theta^4)$, finding a relation between the Curvature function \\eq{curvaturedef} and integrated 4-point functions of $\\Phi_{\\perp}^i$. \n It will be shown in  \\cite{upcomingAJMNderivation}  how this leads to a linear combination of the two constraints  (\\ref{eq:constr1}) and (\\ref{eq:constr2}). \n\nA second important deformation of the line is the  $\\frac{1}{4}$-BPS deformation  defined in \\cite{Drukker:2006ga}. This  is defined by an explicit parameter $\\vartheta$ in the gauge connection. On a circular contour, the expectation value is known explicitly for any value of $g$ and of the deformation parameter~\\cite{Drukker:2006ga,Pestun:2009nn}: it is equivalent to the vev on a $\\frac{1}{2}$-BPS MWL, with a redefinition of the coupling $g\\rightarrow g' = g \\cos\\vartheta$. The perturbative expansion in $O(\\vartheta)$ generates a  series of identities for  correlators of $\\Phi_{\\perp}^i$, now integrated on the circular $\\frac{1}{2}$-BPS line. \nAt order $O(\\vartheta^2)$, one again finds integrated 2-point functions, and this observation allowed the authors of  \\cite{Correa:2012at} to compute $ \\mathbb{B}$.  Extending the analysis to  the next order in $\\vartheta$ leads to a second independent linear combination of the constraints (\\ref{eq:constr1}), (\\ref{eq:constr2}). \n\nWhile the above discussion sketches the main physical intuition, the full derivation of the constraints at finite coupling requires careful treatment. In fact, integrated correlators generated in these expansions produce UV divergences, which should be removed through a consistent regularisation scheme while preserving the symmetries of the setup. The derivation at finite coupling requires the use of next-to-leading order conformal perturbation theory, and will be presented in \\cite{upcomingAJMNderivation}. \n\n\n\n\\subsection{Tests of the relations at strong coupling}\nStrong coupling results for the 4-point function were recently obtained using the functional analytic bootstrap in \\cite{Ferrero:2021bsb}, where results for $f(x)$ and $G(x)$ to the first five orders at large $g$, where these functions expand as \n\\begin{equation}\\label{fstrongexp}\nG(x) = \\sum_{\\ell=0}^{\\infty}\\frac{G_{\\text{strong}}^{(\\ell)}(x)}{(4\\pi g)^\\ell}\n\\qquad\n\\text{and}\n\\qquad\nf(x) = \\sum_{\\ell=0}^{\\infty}\\frac{f_{\\text{strong}}^{(\\ell)}(x)}{(4\\pi g)^\\ell}\n\\qquad g\\rightarrow\\infty\\;.\n\\end{equation}\nThe first terms for the four-point function $G$ are\n\\begin{equation}\\begin{split}\nG_{\\text{strong}}^{(0)}(x)&=\\frac{((x-1) x+1)^2}{(x-1)^2}\\;,    \\\\\nG_{\\text{strong}}^{(1)}(x)&=\\frac{x^2 ((x-1) x (x (2 x-5)+4)+2) \\log (x)-2 (x-1) ((x-1) x+1)^2}{(x-1)^3} \\\\\n&\\quad+\\frac{\\left(-2 x^4+x^3+x-2\\right) \\log (1-x)}{x}\\;,\n\\end{split}\\end{equation}\nand for the reduced correlator $f$ are\n\\beqa\\begin{split}\\label{fstron01}\nf_{\\text{strong}}^{(0)}(x)&\\equiv \\frac{x ^2}{x - 1 } + x \\;,\\\\ f_{\\text{strong}}^{(1)}(x) &\\equiv \\frac{2 (x +1) \\log (1-x ) (x -1)^3-2x  (2 x -1) (x -1)-2 (x -2) x ^3 \\log (x )}{ (x -1)^2} \\;.\n\\end{split}\\eeqa\nThe remaining orders up to $\\ell=4$ can be found in the supplementary material of \\cite{Ferrero:2021bsb}. Plugging these results into the l.h.s. of the constraints and performing the integrations, one finds, from  (\\ref{eq:constr1}),  \n\\begin{equation}\n-\\frac{3}{16 \\pi  g}-\\frac{3 (24 \\zeta_3+7)}{256 \\pi ^2 g^2}-\\frac{3 (144 \\zeta_3+7)}{2048 \\pi ^3 g^3}-\\frac{9 (352 \\zeta_3-49)}{32768 \\pi ^4 g^4}+\\dots\n= \\frac{3\\mathbb{C}-\\mathbb{B}}{8 \\;\\mathbb{B}^2} \\;,\n\\end{equation}\nwhile (\\ref{eq:constr2}) yields\n\\begin{equation}\n\\frac{2 \\pi ^2-21}{24 \\pi g}+\\frac{4 \\pi ^2-7-24 \\zeta_3}{128 \\pi ^2 g^2}+\\frac{10 \\pi ^2+83-144 \\zeta_3}{1024 \\pi ^3 g^3}+\\frac{40 \\pi ^2+867-1056 \\zeta_3}{16384 \\pi ^4 g^4}+\\dots=  \\frac{\\mathbb{C}}{4\\;\\mathbb{B}^2} + \\mathbb{F}-3 \\;.\n\\end{equation}\nThese relations can be easily verified using the strong coupling expansions of  (\\ref{eq:B0}),(\\ref{eq:F0}) and (\\ref{eq:strongC}). \n\n\n\\subsection{Tests at weak coupling}\\label{sec:weakcouplingintegrals}\nThe functions $G(x)$ and $f(x)$\n also have a regular expansion at weak coupling,\n\\begin{equation}\\label{fandG}\nG(x) = \\sum_{\\ell=0}^\\infty G^{(\\ell)}_{\\text{weak}}(x)\\,g^{2\\ell}\\qquad\\text{and}\\qquad\nf(x) = \\sum_{\\ell=0}^\\infty\nf^{(\\ell)}_{\\text{weak}}(x)\\,g^{2\\ell},\n\\qquad g\\rightarrow 0 \\;, \n\\end{equation}\nwhich furthermore should have a finite radius of convergence. Indeed, the general expectation for observables in planar $\\mathcal{N}$=4 SYM is that the radius of convergence is $|g|<\\frac{1}{4}$, see \\cite{Marboe:2014gma,Volin:2008kd,ShotaTalk}. The leading order of this expansion is given in (\\ref{fandGtree}), while the next-to-leading order is~\\cite{Kiryu:2018phb}\n\\beqa\\label{f1weak0}\nf_{\\text{weak}}^{(1)}(x) &=&\n\\frac{2 x}{3 (1-x)}  \\left(6 \\text{Li}_2(x ) +3 \\log (1-x ) \\log (x\n   )-\\pi ^2 x\\right)\\;,\\\\\n  G_{\\text{weak}}^{(1)}(x) &=& \\left(-2 x-\\frac{2}{x-1}\\right) \\log (1-x)+\\left(\\frac{2 x ((x-1) x+1)}{(x-1)^2}+\\frac{(2-4 x) \\log (1-x)}{(x-1)^2}\\right) \\log (x)\\nonumber\\\\\n  &&+\\frac{2 \\left(\\pi ^2 x^2+(6-12 x) \\text{Li}_2(x)\\right)}{3 (x-1)^2} \\;. \\label{eq:G1weak}\n\\eeqa\nPlugging (\\ref{fandGtree}), (\\ref{f1weak0}) into the second integral relation  (\\ref{eq:constr2}) and computing the integrals, we get a perfect match with the expansion of the r.h.s., \n\\begin{equation}\n 0 \\times g^0+ \\left(\\frac{2 \\pi ^2}{3}-6 \\zeta_3\\right)g^2+ O(g^4)\\simeq\\frac{\\mathbb{C}}{4\\;\\mathbb{B}^2} + \\mathbb{F}-3 \\;, \\qquad \\quad g\\rightarrow 0.\n\\end{equation}\nVerifying the first integral relation (\\ref{eq:constr1}) in this regime, however, is more subtle. \n In fact, plugging the weak coupling expansion of $G(x)$ into the l.h.s. of (\\ref{eq:constr1})  leads, order by order for $\\ell>0$,  to integrals of the form\n \\begin{equation}\\label{eq:pertintegr}\n \\int_0^1  G_{\\text{weak}}^{(\\ell)}(x) \\frac{1+\\log x}{x^2} dx \\;,\n \\end{equation}\n which are log-divergent. \n The reason is that at weak coupling $\\Delta_1\\to 1$, which creates additional divergences at each given order in $g\\to 0$ due to the correction terms in (\\ref{limits}).\n On the other hand, the integral $\\int_0^1 \\delta G(x) \\frac{1+\\log x}{x^2} dx$ is perfectly convergent at finite coupling, and -- \\emph{after computing the integral at the non-perturbative level} -- the result can be expanded giving rise to a well-behaved weak coupling expansion (which, however, contains also one negative power of the coupling). We will explain this point in detail in section~\\ref{sec:numerology}, where we will also use the integral relations for the analytic evaluation of the structure constants. \n \n \n \n \n \n \n \n \n \n \n\n\n\t\n\\section{Numerical Bootstrability}\\label{sec:numerical}\n\nThe bootstrap constraint  (\\ref{eq:bootstrapeq}) presents us with a functional equation linear in the OPE coefficients, depending parametrically on the spectrum. \n\nThe equation is linear in the OPE coefficients -- which are our only unknowns since we know the spectrum from the QSC. Nevertheless, we are still faced with two challenges.  First, while integrability allows us \\emph{in principle} to compute the scaling dimension of any state, in practice we can only focus on a finite number of them.\\footnote{It is an interesting challenge to understand how to use the QSC formalism to deduce global properties of a spectrum (e.g., the asymptotic density of states or expansions around large quantum numbers).} This means we need a controlled way to truncate the bootstrap constraint to a finite number of levels.\nSecondly, we need to deal with the functional nature of the equation. The Numerical Conformal Bootstrap (NCB) approach allows us to solve these challenges and obtain rigorous bounds. Here we describe a number of possible NCB algorithms which incorporate knowledge of the spectrum, in increasing degree of complexity. We will then describe how to include the new integral relations in this setup, which will lead us to the main numerical results of the paper. \n\\subsection{Basics}\\label{sec:algorithms0}\nLet us start by reviewing well known aspects of the modern  NCB. Excellent reviews are  availalble~\\cite{Rattazzi:2008pe,Chester:2019wfx,Poland:2018epd}, so we keep the discussion short.\n\\paragraph{The linear functionals approach. }\nThe 1D CFT we are dealing with is unitary (which descends from the ambient theory $\\mathcal{N}$=4 SYM), which implies that $C_n^2\\geq 0$. In the NCB approach for unitary theories, one exploits this fact to turn the bootstrap equation  into a set of inequalities, which bound the conformal data. \n\nA convenient way to deduce such constraints is to act on the bootstrap equation (\\ref{eq:bootstrapeq}) with a \\emph{linear functional}, which transforms  functions of the cross ratio $x$ into numbers. One typically considers functionals obtained as linear  combinations of derivatives acting on a specific point, \nfor example\\footnote{We consider only even derivatives, since we will act with these functionals on crossed conformal blocks $\\mathcal{G}_{\\Delta}$ at $x=\\frac{1}{2}$. For an odd number of derivatives the action is trivial: $\\partial_x^{2 n+1} \\left.\\mathcal{G}_{\\Delta}(x) \\right|_{x=\\frac{1}{2}}$ = 0. }\n\\begin{equation}\\label{eq:derspace}\n\\alpha \\left[ F(x) \\right]\\equiv \\sum_{n = 0}^{N_{\\text{der}\/2} } {A}_n \\;\\partial_x^{2 n} \\left. F(x) \\right|_{x=\\frac{1}{2}}\\;,\n\\end{equation}\nfor some coefficients $\\vec{A}$, where the point $x=\\frac{1}{2}$ is chosen because it guarantees maximal convergence of the OPE. The value of $N_{\\text{der}}\\in 2 \\mathbb{N}$ gives a truncation on the space of functionals and will be a parameter in the approach. \n\nActing with $\\alpha$ on the bootstrap constraint (\\ref{eq:bootstrapeq}), we get a linear equation for the OPE coefficients \n\\begin{equation}\\label{eq:bootstraplinear}\n\\sum_{n} C^2_n \\; \\alpha\\left[ \\mathcal{G}_{\\Delta_n}\\right] +\\alpha\\left[ \\mathcal{G}_{\\text{simple}} \\right]= 0 \\; ,\n\\end{equation}\nwhere the cross-ratio dependence is removed,\nand $\\mathcal{G}_{\\Delta}$, $\\mathcal{G}_{\\text{simple}}$ are defined in (\\ref{eq:defGD}),(\\ref{eq:defGsimple}). Given a functional of the form (\\ref{eq:derspace}), this concretely is rewritten as\n\\begin{equation}\\label{eq:vectorform}\n\\sum_{n} C^2_n \\; \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_n}\\right) + \\left(\\vec{A}\\cdot \\vec{V}_{\\text{simple}}\\right)= 0 \\;,\n\\end{equation}\nwhich is true for arbitrary coefficients $\\vec{A} = (A_0,A_1,\\dots A_{N_{\\text{der}}\/2})$ , and where\n\\beqa\n\\vec{V}_{\\Delta}&\\equiv& \\left. \\left(  \\mathcal{G}_{\\Delta}(x)\\,,\\, \\partial_x^2\\mathcal{G}_{\\Delta}(x), \\dots, \\partial_x^{2 n}\\mathcal{G}_{\\Delta}(x)\\,,\\, \\dots\\,,\\, \\partial_x^{N_{\\text{der}}}\\mathcal{G}_{\\Delta}(x) \\right)\\right|_{x = \\frac{1}{2}}\\;,\\label{eq:VDelta}\\\\\n\\vec{V}_{\\text{simple}} &\\equiv& \\left. \\left(  \\mathcal{G}_{\\text{simple}}(x)\\,,\\, \\partial_x^2\\mathcal{G}_{\\text{simple}}(x)\\,,\\, \\dots\\,, \\, \\partial_x^{2 n}\\mathcal{G}_{\\text{simple}}(x)\\,, \\,\\dots\\,,\\, \\partial_x^{N_{\\text{der}}}\\mathcal{G}_{\\text{simple}}(x) \\right)\\right|_{x = \\frac{1}{2}} \\;.\\label{eq:Vsimple}\n\\eeqa\nStarting from section \\ref{sec:incorporating}, we will get other systems of equations of the form (\\ref{eq:vectorform}), but for more more general definitions of $\\vec{V}_{\\Delta}$, $\\vec{V}_{\\text{simple}}$. \n\nThe main principle of the NCB is finding appropriate functionals  allowing to extract maximal information from the equation. In particular, we will look for $\\alpha$ which is positive semi-definite above a threshold $\\Delta_{\\ast}$:\n\\begin{equation}\\label{eq:posi}\n\\texttt{Positivity condition: } \\;\\;\\; \\vec{A}\\cdot \\vec{V}_{\\Delta}  \\geq 0 , \\;\\;\\;\\; \\forall \\Delta \\geq \\Delta_{\\ast} \\;, \n\\end{equation}\nwhich allows to deduce an inequality:\n\\begin{equation}\\label{eq:inequa}\n\\sum_{\\left\\{\\Delta_n\\right\\},| \\Delta_n < \\Delta_{\\ast} } \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_n}\\right) \\; C^2_n + \\left(\\vec{A}\\cdot \\vec{V}_{\\text{simple}}\\right) \\leq 0 \\;.\n\\end{equation}\nThis relation involves the sum over a finite number of states in the spectrum and gives a rigorous truncation of the bootstrap.  \n\n\\paragraph{OPE bounds algorithm. } \nBy further specifying the properties of the functional, one can deduce constraints on the OPE coefficients. Here we start by revising the algorithm used in our previous paper  \\cite{Cavaglia:2021bnz}. It is an adaptation of a well-known NCB algorithm (see e.g.\\cite{ Chester:2019wfx}) to our situation, where we know the spectrum exactly.\n\\begin{framed}\n\\texttt{Algorithm 1 (Upper Bound). }\nWe start from the bootstrap constraint written in the form (\\ref{eq:vectorform}). \nWe will scan among the functionals satisfying the \\emph{positivity condition} (\\ref{eq:posi}), with the threshold coinciding with the second state in the spectrum,  $\\Delta_{\\ast} \\equiv \\Delta_2$. Among such functionals, we search for $\\vec{A}^{\\text{up}}$ such that:\n\\begin{enumerate}[1)]\n\\item $(\\vec{A}^{\\text{up}}\\cdot \\vec{V}_{\\Delta_1}) = 1 $,\n\\item $(\\vec{A}^{\\text{up}}\\cdot \\vec{V}_{\\text{simple}}) $ is maximal.\n\\end{enumerate}\nThen, (\\ref{eq:inequa}) gives the optimal bound\n\\begin{equation}\nC_1^2 \\leq - (\\vec{A}^{\\text{up}}\\cdot \\vec{V}_{\\text{simple}})\\;.\n\\end{equation}\n\\end{framed}\n\\noindent\nA small tweaking of the algorithm gives us a lower bound. \n\\begin{framed}\n\\texttt{Algorithm 1 (Lower Bound). }\nAgain, choose the positivity threshold as the second state in the spectrum,  $\\Delta_{\\ast} \\equiv \\Delta_2$. Among the functionals satisfying the \\emph{Positivity Condition}  (\\ref{eq:posi}), search for $\\vec{A}^{\\text{low}} $ such that:\n\\begin{enumerate}[1)]\n\\item $ (\\vec{A}^{\\text{low}}  \\cdot \\vec{V}_{\\Delta_1} ) = -1 $,\n\\item $(\\vec{A}^{\\text{low}}  \\cdot \\vec{V}_{\\text{simple}} )$ is maximal.\n\\end{enumerate}\nThen, (\\ref{eq:inequa}) gives the optimal bound\n\\begin{equation}\nC_1^2 \\geq  (\\vec{A}^{\\text{low}}  \\cdot \\vec{V}_{\\text{simple}} ) \\;.\n\\end{equation}\n\\end{framed}\n\\noindent\nIn short, we can obtain bounds on the OPE coefficient of the lowest lying state, \n\\begin{equation}\n (\\vec{A}^{\\text{low}}  \\cdot \\vec{V}_{\\text{simple}} )\\leq C_1^2\\leq -   (\\vec{A}^{\\text{up}}  \\cdot \\vec{V}_{\\text{simple}} )\\;,\n\\end{equation}\nusing as input only the values of the scaling dimensions of the first two states, $\\Delta_1$ and $\\Delta_2$. \n\nThis method was employed in \\cite{Cavaglia:2021bnz} to obtain  a very narrow allowed region for $C_1$ as a function of the coupling. In the following sections we will see how this result can be dramatically improved by including the new integral relations, and how bounds for excited states can be obtained. Before moving to these new results, we discuss how the mathematical optimisation problem defined above can be solved efficiently using \\texttt{SDPB}~\\cite{Simmons-Duffin:2015qma}. \n\n\n\\paragraph{Implementation using SDPB. }\nThe space of  functionals satisfying positivity conditions such as (\\ref{eq:posi}) can be navigated efficiently using semi-definite programming algorithms. These methods were introduced in the NCB context in \\cite{Simmons-Duffin:2015qma} and implemented numerically in the software package \\texttt{SDPB}~\\cite{Simmons-Duffin:2015qma,Landry:2019qug}, which we used in our work. A full description of the internal algorithms can be found in the above references. Here, we briefly describe the main approximations involved and how they impact the results. \n\nFirst, in order to apply linear programming algorithms to impose the positivity conditions, we need to employ an approximation for the conformal blocks in terms of polynomials in $\\Delta$. Similar to the cases  in higher dimensions~\\cite{Chester:2019wfx}, this is  obtained using a truncated expansion of the form:\n \\begin{equation}\\label{eq:expandblocks}\nf_{\\Delta}(x) \\sim \\sum_{k = 0}^{\\text{N}_{\\text{blocks}}} a_k(\\Delta) \\; \\left( r(x)\\right)^k , \\;\\;\\;\\; r(x)\\equiv \\frac{x}{\\left(\\sqrt{1-x}+1\\right)^2} \\; .\n \\end{equation}\nUnder this approximation, the action of the elementary functionals on the blocks takes the form:\n\\begin{equation}\\label{eq:polyapprox}\n\\left. \\frac{\\partial^{2 n} }{\\partial x^{2 n} } \\mathcal{G}_{\\Delta}(x)\\right|_{x=\\frac{1}{2}} \\equiv \\texttt{Pos}(\\Delta) \\times \\texttt{Poly}_n(\\Delta) \\;,\n\\end{equation}\nwhere $\\texttt{Pos}(\\Delta)$ is a strictly positive function\\footnote{We  take $\\texttt{Pos}(\\Delta) = \\left(4 r(\\frac{1}{2}) \\right)^{\\Delta} \\;\\left((\\Delta^2 - 1)\\Delta (\\Delta - 1) \\prod_{n=1}^{\\frac{N_{\\text{blocks}}}{2} - 1}(\\Delta + \\frac{(3 + 2 n)}{2} )\\right)^{-1}$.  The same positive prefactor allows to write a polynomial approximation both for the action of derivatives, and for the action of integral operators introduced in section \\ref{sec:incorporating}. Notice that this is different from the case of the integrated correlator constraints in  \\cite{Chester:2021aun}, which required to use different methods (linear programming rather than semidefinite programming). } of $\\Delta$ and $\\texttt{Poly}_n(\\Delta)$ are polynomial in $\\Delta$. This  polynomial approximation is used to  impose positivity conditions such as (\\ref{eq:posi}) by treating $\\Delta$ as a continuous parameter. \n\nIn our work, we typically take $N_{\\text{blocks}}\\sim  30$. While this approximation introduces an error, in practice its impact is invisible on the scale of our bounds. We have verified  on selected points that $N_{\\text{blocks}} = 50$ and  $N_{\\text{blocks}} = 100$  give the same  bounds to the relevant digits. \n\nThe most important parameter of the method is the integer  $N_{\\text{der}}$ which characterises the dimension of the vector $\\vec{A}$ -- namely, it restricts the space of functionals we explore. The choice of $N_{\\text{der}}$ does affect significantly the bounds. However, importantly the bounds associated to the Numerical Conformal Bootstrap are rigorous, in the sense that they are true for any value of $N_{\\text{der}}$. Moreover, they can only get sharper and sharper by taking this value larger and larger. \n\nWhile in \\cite{Cavaglia:2021bnz} we presented an extrapolation to $N_{\\text{der}}\\rightarrow \\infty$, here we simply take this number to be $N_{\\text{der}} =  140$ for our main results (see section \\ref{sec:results}). Even at this fixed value, thanks to the new integral relations, we will improve  significantly our previous results. \n\n\\paragraph{Precision considerations related to the spectrum. } There is still one potential source of errors we have not discussed, namely the finite precision on the spectral data coming from the numerical solution of the QSC, which we are using as input. We checked that all methods discussed in this paper are quite stable with respect to this error:  an error on the spectrum propagates into a shift roughly of the same magnitude for the best estimate for the OPE coefficients. \nThis means that, in order not to contaminate the bounds for $C_i^2$, it is sufficient to use spectral data with an error a couple of orders smaller than the width of the bounds.\n\nWe also noticed that the numerical bootstrap algorithms are quite sensitive to the injection of wrong spectral data, and often they cease to converge  if one inputs a spectrum which deviates too much from the QSC answer. For instance, at $g=3$, using the method discussed in section \\ref{sec:incorporating} with spectral input from 10 states, it is enough to introduce an error on the spectrum on a scale of $5\\times10^{-7}$ and the method to bound $C_1^2$ would no longer converge. \n\nTo generate the results published in this paper, we used spectral data with at least 12 digits precision (we expect that the precision is actually  higher for most points, and exceeds 20 digits at weak coupling).\nWe estimate that, with such precision, errors on the  spectrum do not have a significant effect on the bounds for OPE coefficients, on the full range of the coupling we consider.\n\n\\subsection{Including more information on the spectrum}\\label{sec:algorithms1}\nThe method of \\texttt{Algorithm 1} uses as input only the scaling dimensions for the first two low-lying states. A natural way to improve the bounds is to include more information on the exact spectrum. \n\nFor instance, if we know the values of the next few states $\\Delta_3,\\Delta_4,\\dots, \\Delta_{N}$, we can impose a relaxed positivity condition, where the functional should satisfy\n\\begin{equation}\\label{eq:posigap}\n(\\vec{A}\\cdot \\vec{V}_{\\Delta})\\geq 0 \\,, \\;\\;\\;\\forall \\Delta \\geq \\Delta_{N}\\; ,\n\\end{equation}\ntogether with a discrete set of additional constraints\n\\begin{equation}\\label{eq:discreteposi}\n(\\vec{A}\\cdot \\vec{V}_{\\Delta_n})\\geq 0 , \\;\\;\\; n = 2,3,\\dots, N  -1 \\;.\n\\end{equation}\nSuch generalised conditions can be easily implemented in \\texttt{SDPB}\\footnote{We are grateful to Petr Kravchuk for discussing this point.}. \nThe functionals satisfying these relations now span a larger space, since they are allowed to assume negative values when $\\Delta$ lies in the gaps between the first $N$ states. This is illustrated in figures \\ref{fig:figfunc1} and \\ref{fig:figfunc2}. If we now run again the previous algorithms, the bounds will become sharper. This is simply because the bounds come from maximising some quantities -- and we are now looking in a larger space for the functional that maximises it. \n Moreover, this extension of the method also allows us to easily generate bounds for OPE coefficients for states other than the ground state (in fact, for any of the first $N-1$ states). We summarise concisely how this works below.\n\\begin{framed}\n\\texttt{Algorithm 2 -- Bounds including more states. } \nWe now use the knowledge of the first $N$ states. We describe how to obtain bounds for $C_m^2$, with $m \\leq N - 1$. \nWe restrict to functionals satisfying the \\emph{positivity conditions with gaps} \n\\beqa\n&&(\\vec{A}\\cdot \\vec{V}_{\\Delta} )\\geq 0,\\;\\;\\;\\forall \\Delta \\geq \\Delta_{N},\\\\  \n&& (\\vec{A}\\cdot \\vec{V}_{\\Delta_n} ) \\geq 0,\\;\\;\\; n \\in \\left\\{1,2,\\dots,N-1\\right\\} , \\; n\\neq m \\; .\n\\eeqa\nUnder these conditions, we search for  $\\vec{A}^{\\text{up},m}$ and $\\vec{A}^{\\text{low},m}$  such that:\n\\begin{enumerate}[1)]\n\\item $(\\vec{A}^{\\text{up},m}\\cdot \\vec{V}_{\\Delta_m} ) = 1 $,\n\\item $(\\vec{A}^{\\text{up},m}\\cdot \\vec{V}_{\\text{simple}} ) $ is maximal,\n\\item$(\\vec{A}^{\\text{low},m}\\cdot \\vec{V}_{\\Delta_m} )= -1 $,\n\\item $(\\vec{A}^{\\text{low},m}\\cdot \\vec{V}_{\\text{simple}} )  $ is maximal.\n\\end{enumerate}\nThen, (\\ref{eq:inequa}) give the  bounds\n\\begin{equation}\n(\\vec{A}^{\\text{low},m}\\cdot \\vec{V}_{\\text{simple}} )\\leq C_m^2 \\leq - (\\vec{A}^{\\text{up},m}\\cdot \\vec{V}_{\\text{simple}} )\\; .\n\\end{equation}\n\\end{framed}\n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=0.85\\linewidth]{Functionalplot3.png}\n  \\captionof{figure}{A depiction of  $\\vec{A}^{\\text{up}} \\cdot \\vec{V}_{\\Delta}$ as a function of $\\Delta$, where $\\vec{A}^{\\text{up}}$ is calculated using \\texttt{Algorithm 1} with $N_{\\text{der}} = 60$, for $g = \\frac{1}{4}$. The red points mark the position of the first 10 spectral levels (the levels $\\Delta_7$ and $\\Delta_8$ are very close and not distinguishable by eye). The value of the functional is positive for all levels and takes the value $\\vec{A}^{\\text{up}} \\cdot \\vec{V}_{\\Delta_1} = 1$ on the ground state (outside the scale of the figure). }\n  \\label{fig:figfunc1}\n\\centering\n  \\includegraphics[width=0.85\\linewidth]{Functionalplot4.png}\n  \\captionof{figure}{Here, we depict again  $\\vec{A}^{\\text{up}} \\cdot \\vec{V}_{\\Delta}$ as a function of $\\Delta$ but obtained with the more general \\texttt{Algorithm 2} which allows for it to take  negative values  between the exact values of the first 10 levels. As the optimal functional is drawn from a larger set, this leads to a stronger bound. As in Fig. \\ref{fig:figfunc1}, here $N_{\\text{der}} = 60$ and $g = \\frac{1}{4}$.  }\n  \\label{fig:figfunc2}\n\\end{figure}\nFor illustration, let us compare the bounds for $C_1$ obtained with \\texttt{Algorithm 1} (i.e., including only $\\Delta_1$ and $\\Delta_2$ as input)  or with \\texttt{Algorithm 2} with the input of the first $N=10$ states. The functional corresponding to the upper bound is shown in Figs. \\ref{fig:figfunc1} and \\ref{fig:figfunc2} for a specific choice of parameters. In the latter case, the functional is allowed to become negative in between the states. Notice that it assumes negative values between $\\Delta_3$ and $\\Delta_4$, while being essentially zero at these two points. In general, in the case illustrated in Fig. \\ref{fig:figfunc2} the functional takes \\emph{smaller} values for all levels $\\Delta_n$, $2 \\leq n \\leq 10$. The consequence is that the truncation of the bootstrap equation is more efficient and the resulting upper bound becomes stronger. In the case illustrated, we used $N_{\\text{der}} = 60$ and $g = \\frac{1}{4}$. While the lower and upper bounds obtained with the simpler algorithm are $C_1^2 \\in [0.0908377,0.0967516]$, with \\texttt{Algorithm 2} the bounds are $C_1^2 \\in [0.0908945,0.0949486 ]$, and the width of the bound reduces by 31 percent. Over the full range of values for the coupling, the gain in precision is at least $\\sim$ 16 percent up to $g \\sim 0.8$, and becomes significant at strong  coupling, e.g. for $g>3$ the error decreases by more than 75 percent of its original value. A comparison of the error with different methods will be presented in Figures \\ref{fig:ErrorLogScale}, \\ref{fig:RelativeErrorLogScale} at the end of the section. \n\n\nBounds for the OPE coefficients of the three lowest states obtained with \\texttt{Algorithm 2} are shown in Fig. \\ref{fig:bigplot1}, with the inclusion of 10 states as input and $N_{\\text{der}} = 60$.  One immediately sees that the bounds for excited states are much less precise than the one for the ground state. At low values of the coupling, in particular, the algorithm produces negative values for the lower bound for $C_2^2$ and $C_3^2$, which are worse than the obvious estimate $C_i^2>0$, so they are not shown. In section \\ref{sec:incorporating}, we will obtain significant improvements of these bounds by including the new integral relations. \n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=0.8\\linewidth]{image1.png}\n  \\captionof{figure}{Bounds for the first three OPE coefficients squared, \n  obtained with \\texttt{Algorithm 2} with input of the first 10 states, and with $N_{\\text{der}} = 60$. The allowed region is very narrow for the ground state (lower and upper bounds form a thin region which almost looks like a line), but the precision for excited states is much less with this method. The gain in precision achieved by including the new integral relations can be seen in Figures \\ref{fig:plotintcompare}, \\ref{fig:plotintboth2}.}\n  \\label{fig:bigplot1}\n\\end{figure}\n\n\\paragraph{Phase transitions in the optimal functionals. }\nAn interesting feature of Fig. \\ref{fig:bigplot1} is the corner in the upper bound for $C_2^2$. \nSuch point of non-analyticity is associated to a ``phase transition'' in the shape of the optimal functional corresponding to the bound. We plot the functional for $g = 4\/5$ and $g = 17\/20$  in Figs. \\ref{fig:plotphase1} and \\ref{fig:plotphase2}, respectively. These two points are very close to each other,  on different sides of the phase transition. Correspondingly, the shape of the functional changes sharply. In particular, in the case of Figure \\ref{fig:plotphase2} the functional starts exploiting the gap between $\\Delta_3$ and $\\Delta_4$, and this leads to an abrupt improvement of the bound. \n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=0.9\\linewidth]{Functionalplot7.png}\n  \\captionof{figure}{Values of $(\\vec{A}^{\\text{up}} \\cdot \\vec{V}_{\\Delta} )$, as a function of $\\Delta$, for the optimal functional giving the upper bound for $C_2^2$, using \\texttt{Algorithm 2} with $N_{\\text{der}} = 60$ and $N_{\\text{states}} = 10$. Here, $g = 4\/5$. Notice the functional is positive between $\\Delta_3$ and $\\Delta_4$. }\n  \\label{fig:plotphase1}\n\\centering\n  \\includegraphics[width=0.9\\linewidth]{Functionalplot8.png}\n  \\captionof{figure}{The optimal functional giving the upper bound for $C_2^2$, for the same parameters as in \\ref{fig:plotphase1}, but with $g = 17\/20$. Notice the abrupt change in the shape of the functional, with the opening up of the gap between $\\Delta_3$ and $\\Delta_4$, associated to the phase transition. }\n  \\label{fig:plotphase2}\n\\end{figure}\n\nWe have observed that such type of phase transitions occur not only across different values of $g$, but also as $N_{\\text{der}}$ is increased. Recall that this parameter controls the dimensionality of the linear space to which the functionals belong. As this  value is increased, the linear functionals acquire more degrees of freedom allowing them to penetrate the gaps between the states in the spectrum. We observed that the opening of a new gap is associated to a jump in the error.  \n\nOne may wonder if the error would shrink to zero provided we used some kind of limiting procedure, by incorporating more and more levels, and consistently using more and more derivatives. We suspect this is however not the case, and that a single correlator does not contain all information on the OPE coefficients, even provided we were able to input all the spectrum. The shrinking of the bounds to zero very likely requires the use of multiple correlators.\n\nIn any case, since some pairs of levels in the spectrum are very close to each other, it is in practice not possible to exploit all the gaps within computationally reasonable values for $N_{\\text{der}}$. The presence of phase transitions as $N_{\\text{der}}$ is increased also  means that the dependence of the bounds on this cutoff cannot be considered smooth. This prevents us from performing an extrapolation to $N_{\\text{der}}\\rightarrow \\infty$, as we did in the analysis of \\cite{Cavaglia:2021bnz}.\\footnote{ In that case, we believe the procedure to be justified, since we were using \\texttt{Algorithm 1}. We were not allowing for the functional to be negative between excited states, and therefore we expect no phase transitions. } Here, therefore, we will content ourselves with working at fixed large values for $N_{\\text{der}}$.  \n\nIn the rest of this section we discuss  some further generalisations of the algorithms. These, however, were not used to obtain our main results, and at this point the reader can safely choose to jump to section \\ref{sec:incorporating}, where we discuss the inclusion of the integrated correlator constraints.  \n\n\\paragraph{Incorporating existing bounds in the algorithm.} \nSuppose we know some bounds for the first OPE coefficient, $C^2_1\\in\\left[ C_{1,-}^2 , C_{1,+}^2  \\right]$ (for instance obtained by running one of the previous algorithms), and we know the first $N$ states in the spectrum. Then, we can split $C_1^2 = C_{1,-}^2 + \\delta_+ C^2_1$, where  $\\delta_+ C^2_1$ should be a positive quantity, and rewrite the  bootstrap constraint (\\ref{eq:vectorform}) as:\n\\beqa\n0&=&\\sum_{n\\geq 1} C^2_n \\; \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_n}\\right) + \\left(\\vec{A}\\cdot \\vec{{V}}_{\\text{simple}}\\right) \\\\\n&=&\\delta_+ C^2_1 \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_1}\\right) + \\sum_{n\\geq 2} C^2_n \\; \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_n}\\right) + \\left(\\vec{A}\\cdot \\vec{\\widetilde{V}}_{\\text{simple}}\\right) \\;,\n\\eeqa\nwhere we redefined\n\\begin{equation} \\label{eq:simpletilde}\n\\vec{\\widetilde{V}}_{\\text{simple}} \\equiv \\vec{V}_{\\text{simple}} + C_{1,-}^2\\;\\vec{V}_{\\Delta_1} \\;.\n\\end{equation}\nNow, when computing and upper or lower bound for $C_m^2$ with $1<m < N$, we use a simple update of \\texttt{Algorithm 2}: we search for a functional $\\vec{A}$ which satisfies the conditions (\\ref{eq:posigap}), (\\ref{eq:discreteposi}) of the previous algorithm, but which maximises its value on the vector $\\vec{\\widetilde{V}}_{\\text{simple}}$ defined in (\\ref{eq:simpletilde}), \n rather than $\\vec{{V}}_{\\text{simple}}$ if (\\ref{eq:Vsimple}).  We can also clearly repeat the argument but using the knowledge of the upper bound. In this case, we would have\n \\begin{equation}\\label{eq:upperreformulation}\n\\delta_- C^2_1 \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_1}\\right) + \\sum_{n\\geq 2} C^2_n \\; \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_n}\\right) + \\left(\\vec{A}\\cdot \\vec{\\widetilde{V}}_{\\text{simple}}\\right) = 0 \\;,\n\\end{equation}\nwhere \n\\begin{equation} \\label{eq:upperreformulation2}\n\\vec{\\widetilde{V}}_{\\text{simple}} \\equiv \\vec{V}_{\\text{simple}} + C_{1,+}^2\\;\\vec{V}_{\\Delta_1} \\; , \n\\end{equation}\nand where $\\delta_- C^2_1 \\equiv C^2_1 - C_{1,+}^2$ is now a negative quantity. In this case, to find bounds for $C_m^2$, $1<m<N$, one has to scan the space of functionals such that $\\vec{A}\\cdot \\vec{V}_{\\Delta_1} < 0$. \n\nWe summarise  these considerations in the following iteration of the algorithm:\n\\begin{framed}\n\\texttt{Algorithm 3 -- Bounds including several states and previous bounds. }\nWe use the input of the first $N$ states,  and we assume knowledge of some bounds for the OPE coefficients, which we denote as:\n\\begin{equation}\nC_k^2 \\in \\left[ C_{k, -}^2 \\,,\\; C_{k, +}^2  \\right] , \\;\\;\\; k = 1,\\dots, N-1\\;.\n\\end{equation}\nThese could come from previous iterations of the algorithms, or from other considerations. \n\n\\noindent We now focus on obtaining a new bound for the $m$-th state, $1\\leq m \\leq N-1$. \n\nTo do this, we choose a combination of \\emph{signs}:\n\\begin{equation}\n\\left\\{ \\sigma_k \\right\\} , \\;\\;\\; k \\in \\left\\{ 1,\\dots, N-1\\right\\}, k\\neq m \\;,\n\\end{equation}\nwhere $\\sigma_i \\in \\left\\{-1,1\\right\\}$. Then the bootstrap equation rewrites as\n\\begin{equation}\\label{eq:bootstraprewrite2}\n C^2_m \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_m}\\right) + \\sum_{n\\neq m, \\; n\\leq N-1} \\delta_{\\sigma_n} C^2_n \\; \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_n}\\right) + \\left(\\vec{A}\\cdot \\vec{\\widetilde{V}}_{\\text{simple}}\\right) + \\sum_{n\\geq N} C^2_n \\; \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_n}\\right) = 0 \\; ,\n\\end{equation}\nwhere we define $\\delta_{+1} C_n^2 \\equiv C_n^2 - C_{k,-}^2 > 0$, and $\\delta_{-1} C_n^2 \\equiv C_n^2 - C_{k,+}^2  < 0$, and\n\\begin{equation}\n\\vec{\\widetilde{V}}_{\\text{simple}} \\equiv \\vec{V}_{\\text{simple}} + \\sum_{n\\neq m, \\; n\\leq N}  C_{n, \\; - \\sigma_n}^2 \\;\\vec{V}_{\\Delta_n} \\; .\n\\end{equation}\n\\noindent We work in the space of functionals satisfying the conditions:\n\\beqa\n&&(\\vec{A}\\cdot \\vec{V}_{\\Delta} )\\geq 0,\\;\\;\\;\\forall \\Delta \\geq \\Delta_{N},\\\\  \n&& \\sigma_n\\; (\\vec{A}\\cdot \\vec{V}_{\\Delta_n} ) \\geq 0,\\;\\;\\; n \\in \\left\\{1,2,\\dots,N-1\\right\\} , \\; n\\neq m\\; ,\n\\eeqa\nso that (\\ref{eq:bootstraprewrite2}) becomes\n\\begin{equation}\\label{eq:bootstraprewrite3}\n C^2_m \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_m}\\right) + \\left(\\vec{A}\\cdot \\vec{\\widetilde{V}}_{\\text{simple}}\\right) \\leq 0 \\; .\n\\end{equation}\nProceeding as in the previous cases, we now search for  $\\vec{A}^{\\text{up},m}$ and $\\vec{A}^{\\text{low},m}$  such that:\n\\begin{enumerate}[1)]\n\\item $(\\vec{A}^{\\text{up},m}\\cdot \\vec{V}_{\\Delta_m} ) = 1 $,\n\\item $(\\vec{A}^{\\text{up},m}\\cdot \\vec{\\widetilde{V}}_{\\text{simple}}) $ is maximal,\n\\item$(\\vec{A}^{\\text{low},m}\\cdot \\vec{V}_{\\Delta_m} )= -1 $,\n\\item $(\\vec{A}^{\\text{low},m}\\cdot \\vec{\\widetilde{V}}_{\\text{simple}})  $ is maximal.\n\\end{enumerate}\nThen, (\\ref{eq:bootstraprewrite2}) gives the  bounds\n\\begin{equation}\n(\\vec{A}^{\\text{low},m}\\cdot \\vec{\\widetilde{V}}_{\\text{simple}} )\\leq C_m^2 \\leq - (\\vec{A}^{\\text{up},m}\\cdot \\vec{\\widetilde{V}}_{\\text{simple}} )\\;.\n\\end{equation}\n\\end{framed}\nWe found that in some cases these tricks are effective. In particular, one can sometimes improve significantly the bounds for the excited states, provided a rather precise bound for $C_1^2$ is exploited as input. To make an example, for $g = 1\/4$, using \\texttt{Algorithm 2} with  $N_{\\text{states}}= 10$, $N_{\\text{der}} = 60$, one finds  $C_3^2\\in[0,0.23779]$. \n Incorporating the bounds for $C_1^2 \\in [0.09313678, 0.09313996]$ as in \\texttt{Algorithm 3}, one can recalculate the allowed region  for $C_3^2$, finding now $C_3^2 \\in [0.0214, 0.1845]$, i.e. the allowed interval shrinks by 31 percent. The magnitude of this improvement is however very sensitive to the precision of the injected bounds for $C_1^2$. In this example in particular we used as input a bound for $C_1^2$  which is more precise than what could be achieved by the algorithms presented so far, and was obtained by including the integral relations into the game. \n\nWhile \\texttt{Algorithm 3} could be useful in some contexts, we found that, once we include  uniformly the new integral constraints as explained in the next section, the precision is not significantly affected by using this  upgrade of the method. Our main results in section \\ref{sec:results} were obtained simply with  \\texttt{Algorithm 2}. \n\n\\paragraph{Other approaches. }\nAlthough we do not explore them in this paper, it is worth mentioning that there are other techniques to study OPE coefficients within the Numerical Bootstrap (for some modern developments see e.g. \\cite{Chester:2019ifh}).  \n\nOne possible approach, for which preliminary results were presented in \\cite{Cavaglia:2021bnz}, is to treat some of the OPE coefficients as variational parameters in order to find their allowed region. For example, in the case presented in \\cite{Cavaglia:2021bnz}, we rewrote the bootstrap equation as\n\\begin{equation}\n C^2_1 \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_1}\\right) + \\sum_{n\\geq 4} C^2_n \\; \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_n}\\right) + \\left(\\vec{A}\\cdot \\vec{\\widetilde{\\widetilde{V}}}_{\\text{simple}}\\right) = 0 \\;,\n\\end{equation}\nwith \n$\\vec{\\widetilde{\\widetilde{V}}}_{\\text{simple}} \\equiv \\vec{V}_{\\text{simple}} + C^2_2 \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_2}\\right) + C^2_3 \\left(\\vec{A}\\cdot \\vec{V}_{\\Delta_3}\\right)$. One can then treat $C_2^2$ and $C_3^2$ as parameters, and use one of the algorithms presented above to obtain bounds on $C_1^2$, $C_1^2 \\in \\left[ C_{1,-}^2(C_2^2, C_3^2) , \\; C_{1,+}^2(C_2^2, C_3^2)\\right] $, which depend parametrically on $C_2^2$ and $C_3^2 $. Such bounds indirectly define an allowed region for $C_2^2$ and $C_3^2$, which is given by the condition that  $C_{1,-}^2(C_2^2, C_3^2) < C_{1,+}^2(C_2^2, C_3^2)$. \nOne can also use other types of bootstrap algorithm, where one does not impose optimisation conditions, but rather looks for a functional  with positivity conditions that exclude a certain set of conformal data.\n\nIt is definitely worth investigating if these techniques can lead to an improvement of our results. Exploring the higher dimensional parametric space of OPE coefficients, while computationally expensive, might reveal a finer structure than treating them individually as we do in this paper (for instance, in the case of \\cite{Cavaglia:2021bnz} we observed that there is a linear combination of $C_2$ and $C_3$ for which the bound is much narrower than for each of them individually). \n\n\n\\subsection{Incorporating the integral relations}\\label{sec:incorporating}\nHere we explain how the new integral relations can be embedded in the NCB  framework, similarly to what was done in  \\cite{Chester:2021aun} in 4D.\n\nFirst of all, we rewrite the constraints using the OPE decomposition of the 4-point function, as new linear relations for the OPE coefficients. As shown in appendix  \\ref{app:rewriteconstr}, using crossing symmetry and the OPE the two relations (\\ref{eq:constr1}), (\\ref{eq:constr2}) can be rewritten as\n\\beqa\n\\text{Constraint 1:  }&&\\;\\; \\sum_{\\Delta_n} C^2_n \\; \\texttt{Int}_1\\left[\\, f_{\\Delta_n} \\,\\right] + \\texttt{RHS}_1 = 0 \\;, \\label{eq:newconstr1}\\\\\n\\text{Constraint 2:  }&&\\;\\; \\sum_{\\Delta_n} C^2_n \\; \\texttt{Int}_2\\left[\\, f_{\\Delta_n} \\,\\right] + \\texttt{RHS}_2 = 0 \\;,\\label{eq:newconstr2}\n\\eeqa\nwhere we introduced the integral operators\n\\beqa\\la{ints}\n&&\\texttt{Int}_1\\left[ F(x) \\right] \\equiv - \\int_0^{\\frac{1}{2}} (x-1-x^2)\\frac{F(x)}{x^2} \\partial_x\\log\\left( x (1-x)\\right)\\, dx \\;,\\\\\n&&\\texttt{Int}_2\\left[ F(x)\\right]\\equiv  \\int_{0}^{\\frac{1}{2}} dx \\frac{ F(x) \\; ( 2 x - 1) }{x^2} \\;,\n\\eeqa\nand the explicit functions of the coupling constant:\n\\beqa\n\\texttt{RHS}_1&=&\\frac{\\mathbb{B}-3\n   \\mathbb{C}}{8\\mathbb{B}^2}+\\left(7 \\log(2) -\\frac{41}{8}\\right) (\\mathbb{F}-1)+  \\log\n   (2)\\; , \\\\\n   \\texttt{RHS}_2 &=& \\frac{1-\\mathbb{F}}{6}+(2-\\mathbb{F}\n   ) \\log (2)+1 -\\frac{\\mathbb{C}}{4\\;\\mathbb{B}^2}\\; .\\label{eq:RHS2}\n\\eeqa\nIn order to obtain \\eq{ints}, we used the crossing equation \\eq{eq:fcrossing} to rewrite the integral over the half range $x\\in [0,1\/2]$. Within this interval the OPE expansion \\eq{eq:OPEf}\nconverges rapidly, which also implies that the action of the integral operators on the blocks $f_{\\Delta}$ are rapidly decreasing with $\\Delta$. Further details on the derivation of (\\ref{ints})-(\\ref{eq:RHS2}) are contained in Appendix \\ref{app:rewriteconstr}. \n\n\n\n To incorporate these equations in the bootstrap setup, we consider a generic linear combination of  derivatives acting on the bootstrap equation with the new constraints (\\ref{eq:newconstr1}),(\\ref{eq:newconstr2}):\n \\beqa\\label{eq:newbootstrap}\n&& \\sum_{n} C^2_n \\; \\left[ \\sum_{k = 0}^{N_{\\text{der}}\/2} b_k \\left. \\partial_x^{2 k} \\mathcal{G}_{\\Delta_n}  \\right|_{x = \\frac{1}{2}} + b_{-1} \\texttt{Int}_1[f_{\\Delta_n}] + b_{-2} \\texttt{Int}_2[f_{\\Delta_n}]\\right]\\\\ &&+\\sum_{k = 0}^{N_{\\text{der}}\/2} b_k \\left. \\partial_x^{2 k} \\mathcal{G}_{\\text{simple}}  \\right|_{x = \\frac{1}{2}}+ b_{-1}  \\texttt{RHS}_1 + b_{-2}  \\texttt{RHS}_2= 0 \\;.\n\\eeqa\nThis equation is true for any choice of the coefficients  $\\left\\{b_{-1},b_{-2}, b_0, \\dots, b_{N_{\\text{der}}\/2}\\right\\}$.  It is now apparent that this more general equation takes the same form as (\\ref{eq:vectorform}), which was the starting points of our bootstrap algorithms, where we redefine $\\vec{A}$, $\\vec{V}_{\\text{simple}}$, $\\vec{V}_{\\Delta}$ as\n\\beqa\n\\vec{A} &\\equiv& ( b_{-1},b_{-2})\\oplus\\left( b_0,b_1, \\dots, b_{N_{\\text{der}}\/2}\\right), \\label{eq:speacialvectors1}\\\\\n\\vec{V}_{\\Delta} &\\equiv& (\\texttt{Int}_1[f_{\\Delta}], \\texttt{Int}_2[ f_{\\Delta}] )\\oplus \\left. \\left(  \\mathcal{G}_{\\Delta}(x),\\, \\partial_x^2\\mathcal{G}_{\\Delta}(x),  \\dots, \\partial_x^{N_{\\text{der}}}\\mathcal{G}_{\\Delta}(x) \\right)\\right|_{x = \\frac{1}{2}} \\; ,\\\\\n\\vec{V}_{\\text{simple}} &\\equiv & (\\texttt{RHS}_1, \\texttt{RHS}_2 )\\oplus \\left. \\left(  \\mathcal{G}_{\\text{simple}}(x),\\, \\partial_x^2\\mathcal{G}_{\\text{simple}}(x),  \\dots, \\partial_x^{N_{\\text{der}}}\\mathcal{G}_{\\text{simple}}(x) \\right)\\right|_{x = \\frac{1}{2}} \\; . \\label{eq:speacialvectors3}\n\\eeqa\nAfter these redefinitions, we can run the same algorithms described sections \\ref{sec:algorithms0} and \\ref{sec:algorithms1}.\n\nAt the level of implementation, using the expansion (\\ref{eq:expandblocks}) we can approximate our integral operators as\n\\begin{equation}\n\\texttt{Int}_1[ f_{\\Delta}] = \\texttt{Pos}(\\Delta) \\times \\texttt{Poly}_{(1)}(\\Delta) \\;, \\;\\;\\;\\; \\texttt{Int}_2[ f_{\\Delta}] =\\texttt{Pos}(\\Delta) \\times \\texttt{Poly}_{(2)}(\\Delta) \\; ,\n\\end{equation}\nwith polynomials $\\texttt{Poly}_{(i)}(\\Delta)$ and, crucially, the same positive prefactor as in (\\ref{eq:polyapprox}). This is a difference to what was observed in \\cite{Chester:2021aun}, and allows us to  include the integral constraints into the  \\texttt{SDPB} setup. \n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=0.8\\linewidth]{Detail_1vs2.png}\n  \\captionof{figure}{Bounds for the first three OPE coefficients squared. We use the same  parameters as in Fig.  \\ref{fig:bigplot1}, but including either the first or the second  integral relation (solid vs dashed lines). \n  The use of the constraints lead to a visible  improvement. \n  The two relations lead to comparable results, with the first integral relation being slightly more effective.}\n  \\label{fig:plotintcompare}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=0.8\\linewidth]{BothPlotsSuperimposed.png}\n  \\captionof{figure}{Bounds for the first three OPE coefficients obtained using two integral relations simultaneously, with $N_{\\text{der}} = 60$ and $N_{\\text{der}}=140$ (darker). This is our main result. The data for the bounds are reported in Appendix \\ref{app:boundsC}. Shown in the figures are also weak and strong coupling predictions (dotted lines), obtained in the next section \\ref{sec:analytical}. }\n  \\label{fig:plotintboth2}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=0.78\\linewidth]{ErrorComparePlot0New.png}\n  \\captionof{figure}{The value of the error for the first OPE coefficient in logarithmic scale, $\\log_{10}\\left( \\frac{1}{2} (C_{1,+}^2 - C_{1,-}^2 ) \\right)$, as function of the coupling, resulting from various methods. We show two results from \\cite{Cavaglia:2021bnz} (obtained with input from two states): the bounds at $N_{\\text{der}}= 60$  (\\fcolorbox{black}{color1}{\\rule{0pt}{3pt}\\rule{3pt}{0pt}}),  and the best result of \\cite{Cavaglia:2021bnz}, obtained with the extrapolation $N_{\\text{der}} \\rightarrow \\infty$ (\\fcolorbox{black}{color2}{\\rule{0pt}{3pt}\\rule{3pt}{0pt}}). \n  These are compared to the new results, at $N_{\\text{der}}= 60$, \n  obtained without using integral relations (\\fcolorbox{black}{color3}{\\rule{0pt}{3pt}\\rule{3pt}{0pt}}), using either the second (\\fcolorbox{black}{color4}{\\rule{0pt}{3pt}\\rule{3pt}{0pt}}) or the first integral relation (\\fcolorbox{black}{color5}{\\rule{0pt}{3pt}\\rule{3pt}{0pt}}), or using both of them (\\fcolorbox{black}{color6}{\\rule{0pt}{3pt}\\rule{3pt}{0pt}}). Our best result is obtained using both integral relations and $N_{\\text{der}} = 140$ (\\fcolorbox{black}{color7}{\\rule{0pt}{3pt}\\rule{3pt}{0pt}}). In all these new results we use \\texttt{Algorithm 2 } with input from $N = 10$ states.  }\n  \\label{fig:ErrorLogScale}\n\\centering\n  \\includegraphics[width=0.78\\linewidth]{ErrorComparePlot1New.png}\n  \\captionof{figure}{With the same colour scheme as in Figure \\ref{fig:ErrorLogScale}, we compare various methods for the value of the relative error on a logarithmic scale,  plotting $\\log_{10}\\left( \\frac{ C_{1,+}^2 - C_{1,-}^2 }{ C_{1,+}^2 + C_{1,-}^2 } \\right)$. }\n  \\label{fig:RelativeErrorLogScale}\n\\end{figure}\n\n\n\\subsection{Results}\\label{sec:results}\nIncluding the integral relations leads to a dramatic improvement of the bounds. \nTo quantify this effect, let us first describe some experiments where we add one relation at a time. This can be easily done by just dropping one component from (\\ref{eq:speacialvectors1})-(\\ref{eq:speacialvectors3}). \n For instance, with the same parameters as in Figure \\ref{fig:bigplot1}, but now including the second integral constraint (\\ref{eq:constr2}), we find the width of the bound for $C_1^2$ decreasing by at least a factor of $10$ over all range in the coupling (for $g>1.5$, the gain is a factor of 50). Adding the first integral constraint  (\\ref{eq:constr1}) on its own has an even stronger effect, with the bound decreasing by at least factor $30$, which becomes a factor $80$ for $g>1.9$. The improvement is also marked at weak coupling. A comparison of the error with various methods can be found in Figures \\ref{fig:ErrorLogScale} and \\ref{fig:RelativeErrorLogScale}. \nThe gain in precision for the excited states is clearly visible in Figure \\ref{fig:plotintcompare}. With either integral relation the bounds shrink approximately by a factor $2$ starting from $g \\sim 0.3$, and by a factor $\\sim 9$ at strong coupling. \nAs can be seen in the figure, the two integral relations, separately, lead to very similar new bounds for the excited states coefficients, with the first relation (\\ref{eq:constr1}) being slightly more constraining. One might even be suspicious that the two relations are not independent,  but one easily sees that this is not the case, as combining them reduces the error much further. \n\n Our best results, obtained using both integral relations together in the algorithm, are shown in Figure \\ref{fig:plotintboth2}. One can immediately see a significant improvement for the excited states, with the upper and lower bounds indistinguishable by eye  for a wide range of values of the coupling. Keeping fixed the value of $N_{\\text{der}} = 60$, remarkably for $C_1^2$ the bound shrinks by at least a factor $10^3$ for all values of the coupling. For $C_2^2$ and $C_3^2$, the bound reduces monotonically with the coupling -- for $g = 0.3$  by at least  factor  $10^1$, which becomes $10^2$ for $g\\sim 1.5$. At strong coupling, the gain is almost a factor of 200.  \nWe run the algorithm with $N_{\\text{der}} = 140$ to obtain our best results, which are reported in Appendix \\ref{app:boundsC}. \n\nThe bounds produced by the same algorithm for $C_i^2$, with $i>3$, are less precise. In particular, for these OPE coefficients the current setup only produces a nontrivial upper bound. This is compatible in magnitude with the strong coupling results of \\cite{Ferrero:2021bsb}. In particular, for the four states with $\\Delta_{\\text{strong}}^{(0)} = 6$, they found the strong coupling limit $\\langle a_{\\text{strong}}^{(0)} \\rangle_3  = C_4^2 + C_5^2 + C_6^2 + C_7^2 = 10\/429\\simeq 0.023$, and we found the upper bounds e.g. $C_4^2 < 0.0079$ and $C_5^2 < 0.0123$ at $g = 4$. We reserve further study of these excited states for the future.  We expect that the precision can be improved by including more states in the algorithm and especially by considering more general bootstrap setups as we discuss in section \\ref{sec:discussion}.\n\n\n\n\n\\section{Analytic Bootstrability}\\label{sec:analytical}\nIn this section we develop a functional analytic bootstrap approach at weak coupling, using input from the QSC solution of the spectrum at weak coupling (collected in Appendix \\ref{app:spectrum}). Additionally, we use the two new integral constraints, one of which -- relation (\\ref{eq:constr1}) -- is particularly powerful to extract weak coupling data for the first OPE coefficient. \n\nWe start by discussing the subtleties in the weak coupling treatment of (\\ref{eq:constr1}), and then  proceed to discuss the weak coupling Bootstrability method. Our main results are summarised at the end of this section. \n\n\\subsection{Weak coupling expansion of the first integral relation }\n We now explain how to interpret the constraint (\\ref{eq:constr1}) at weak coupling. \n In particular, as anticipated in section \\ref{sec:integrated}, we need to understand how to \n regularise correctly the integrals \n $\\int_0^1 dx G^{(\\ell)}(x) \\frac{\\log{x} +1}{x^2}$, $\\ell>1$, arising from the perturbative expansion of the integrand. \nThese integral has $\\log$-divergences, so we define an extraction scheme where the integral is defined by its finite part after dropping $\\log^n\\epsilon$ terms.\n \n We will now show that\n \\begin{equation}\\begin{split}\\label{int1weak}\n \\left.\\int_0^1 \\!\\delta G(x)\\, \\frac{1 \\!+\\! \\log(x)}{x^2}\\;dx   \\right|_{\\text{small }g } \\!\\sim& 1 \\!+\\!  \\underbrace{\\int_{\\epsilon}^{\\frac{1}{2}}dx \\left(\\sum_{\\ell=1}^{M} g^{2\\ell} G_{\\text{weak}}^{(\\ell)}(x)\\right)\\frac{\\log(x (1-x))}{x^2} }_{\\texttt{regularised, }\\log(\\epsilon)\\rightarrow 0}\n \\\\\n&- \\underbrace{\\left[\\frac{C_1^2}{(\\Delta _1-1)^2}\\right] }_{\\texttt{``anomaly''}} + O(g^{2 M + 2})\\;,\n\\end{split}\\end{equation}\nfor any order $M=1,2,\\dots$.\nThe meaning of this equation is the following. To reproduce the weak coupling expansion of the integral on the l.h.s., we should: i) expand the integrand up to the desired order and integrate term by term; ii) since this produces log-divergences, we introduce a cutoff $\\epsilon$ in the integration range and  regularise the result by the prescription $\\log^n\\epsilon\\rightarrow 0$; iii) finally, we should add the  ``anomalous term'' in the second line of  (\\ref{int1weak}), which contains the OPE coefficient and scaling dimension of the ground state, expanded to the relevant order. \n\nIn the following subsection we make a digression to prove \\ref{int1weak}. Next, we will show that (\\ref{eq:constr1}) can be used to deduce several orders of the weak coupling expansion of $C_1^2$ analytically.\n\n\\subsubsection{The weak coupling ``anomaly'' }\nTo understand the regularisation (\\ref{int1weak}), it is convenient to break the l.h.s. in the two objects $\\int_0^1\\frac{\\delta G(x)}{x^2}dx + \\int_0^1\\frac{\\delta G(x)}{x^2}\\,\\log(x)\\,dx $. \n\n\\paragraph{The first piece. }\nWe start by analysing the first of these terms,  which can in fact be evaluated exactly for any $g>0$:\n\\begin{framed}\n\\begin{equation}\\label{eq:specialide}\n\\int_0^1\\frac{\\delta G(x)}{x^2}dx = 1 .\n\\end{equation}\n\\end{framed}\n\\noindent\nTo prove this identity, notice that the integrand is in fact a total derivative (see  (\\ref{eq:totalder})), \n\\begin{equation}\n\\frac{\\delta G(x)}{x^2}=\\partial_x\\[\\left(-\\frac{1}{x^2}+\\frac{1}{x}-1\\right) \\delta f(x)\\]+1+C^2_{\\rm BPS}(g) -2 .\n\\end{equation}\nThis combination is regular both at $x=0$ and $x=1$, so we can apply the integration formula\n\\begin{equation}\n\\int_0^1\\frac{\\delta G(x)}{x^2}dx=\nC_{BPS}^2(g)-1-\n2\\[\\left(-\\frac{1}{x^2}+\\frac{1}{x}-1\\right) \\delta f(x)\\]_{x\\to 0} \\label{eq:firstli}\\;\\; ,\n\\end{equation}\nwhere we used the fact that, due to crossing (\\ref{eq:fcrossing}),\n\\begin{equation}\n\\[\\left(-\\frac{1}{x^2}+\\frac{1}{x}-1\\right) \\delta f(x)\\]_{x\\to 1} = -\\[\\left(-\\frac{1}{x^2}+\\frac{1}{x}-1\\right)\\delta f(x)\\]_{x\\to 0} .\n\\end{equation}\n Using the OPE decomposition (\\ref{eq:OPEf}), which converges around $x\\sim 0$, one can easily evaluate the r.h.s. of  (\\ref{eq:firstli}) obtaining $C^2_{BPS}-1 - (C^2_{BPS}-2) = 1$, which proves (\\ref{eq:specialide}). \n \nNote, however, that at $g=0$ the integrand of the l.h.s. of (\\ref{eq:specialide}) is strictly zero, as we defined $\\delta G(x)$ as a difference of $G(x)$ with its tree level value. The discrepancy with the value $1$ taken by the integral is a clear manifestation of the weak coupling anomaly. We will now explain how to  cure this mismatch. \n\n\n\n\nTo understand the origin of the anomaly, let us look again at original integral at finite coupling,  which can be rewritten, using crossing, as $\\int_0^{1}\n    \\frac{\\delta G(x)}{x^2}dx =2\\int_0^{\\frac{1}{2}}\n    \\frac{\\delta G(x)}{x^2}dx $. This allows us to use the OPE decomposition, which is fastly convergent on $[0,\\frac{1}{2}]$. The contribution of a single conformal block $\\mathcal{L}_{0,[0,0]}^{\\Delta_n}$ to the integrand is \n \\begin{equation}\n \\frac{ G(x) }{x^2} = \n ... - C_n^2(g) \\; \\partial_x \\left[ \\left(1-\\frac{1}{x} + \\frac{1}{x^2} \\right)f_{\\Delta_n}(x) \\right] + ... \\,, \n \\end{equation}\n which behaves, close to the limit of integration $x \\sim 0$, as\n \\begin{equation}\\label{eq:relevantsingular}\n \\frac{ G(x) }{x^2} = ... + C_n^2(g) x^{\\Delta_n(g)-2} \\left( 1 + O(x) \\right) + ...\\;.\n \\end{equation}\n Under the weak coupling expansion  $\\Delta_n = \\Delta_{n}^{(0)} + \\gamma_n(g)$, with $\\gamma_n(g) = O(g^2)$, we find\n\\begin{equation}\\label{eq:smallx}\n \\left.  \\frac{ G(x) }{x^2} \\right|_{\\text{small $g$, $x$}}\\!= ...+ C_n^2(g) \\! \\left[ x^{\\Delta_n^{(0)} -2 } ( 1 + g^2 \\gamma_n^{(0)} \\log(x) + O(g^4) ) \\!+\\! O(x^{\\Delta_n^{(0)}-1} ) \\right] \\!+...\\;.\n \\end{equation}\nAs long as $\\Delta_n^{(0)} > 1 $, the weak coupling expansion thus produces terms that can be safely integrated. The only exception is the block corresponding to the ground state $\\Delta_1$: in this case, since $\\Delta_1^{(0)} = 1$, we encounter a log-divergence at $x \\sim 0$. We can therefore zoom on this term to understand the correct perturbative regularisation of the integral. \n\nFocusing on the relevant singular behaviour (\\ref{eq:relevantsingular}) for $n=1$, we have the integral\n\\begin{equation}\nI=2 \\int_\\epsilon^{1\/2} x^{\\Delta_1-2}dx ,\n\\end{equation}\nwhere $\\Delta_1\\to 1$ at $g\\to 0$. Let us compare its non-perturbative treatment to a na\\\"ive finite-part regularisation of the weak coupling expansion. At finite coupling, $\\Delta_1>1$ and the integral is convergent  giving\n\\begin{equation}\\label{eq:Ifinite}\nI_{\\rm finite}=\\frac{2^{\\Delta_1 }}{\\Delta_1 -1}\\;.\n\\end{equation}\nAt weak coupling, $\\Delta_1 =1 + \\gamma_1$, we expand first at small $\\gamma_1$  and then integrate. Choosing the  prescription $\\log^n\\epsilon\\to 0$, and then resumming the result order by order in $\\gamma_1$  yields \n\\begin{equation}\nI_{\\rm weak}=\\frac{2^{\\Delta_1 }-2}{\\Delta_1 -1} .\n\\end{equation}\nThe discrepancy is\n\\begin{equation}\n\\delta I = \nI_{\\rm finite}-I_{\\rm weak} = \\frac{2}{\\Delta_1-1}\\;.\n\\end{equation}\nTaking into account that the integral (\\ref{eq:Ifinite}) comes from the OPE expansion and is multiplied by $C_1^2$, we have established  the weak-coupling identity\n\\begin{equation}\\label{fint1}\n \\left. \\int_0^1 \\!\\frac{\\delta G(x) }{x^2}\\;dx   \\right|_{ \\text{small }g } \\!\\sim    \\underbrace{2 \\int_{\\epsilon}^{\\frac{1}{2}}dx \\frac{ \\sum_{\\ell=1}^{M} g^{2 \\ell} G_{\\text{weak}}^{(\\ell)}(x)}{x^2} }_{\\texttt{regularised, }\\log(\\epsilon)\\rightarrow 0} + \\underbrace{\\left[\\frac{2 C_1^2}{\\Delta _1-1}\\right] }_{\\texttt{``anomaly''}} + O(g^{2 M + 2}) ,\n\\end{equation}\nwhich is to be interpreted as explained above.\nLet us check if this identity is now compatible with the exact result (\\ref{eq:specialide}). At leading order $O(1)$, the integral term drops out. This means that the anomaly term, on its own, should match the value $1$ of the integral on the l.h.s. We know that at leading order the anomalous dimension is $\\Delta_1 -1 = 4 g^2 + O(g^4)$, which implies that we must have\n\\begin{equation}\\label{eq:firstprediction}\nC_1^2(g) \\simeq 2 g^2 + O(g^4),\n\\end{equation}\nwhich is indeed confirmed by the numerical bootstrap data~\\cite{Cavaglia:2021bnz} and the analytic bootstrap computation of section \\ref{sec:numerology}. At the next-to-leading order, we can plug in (\\ref{eq:G1weak}) and fix  one more term in the OPE coefficient. Before doing this, however, we conclude the proof of (\\ref{int1weak}), which will allow us to use the constraint (\\ref{eq:constr1}) and will prove more powerful. \n\n\\paragraph{The second piece and its anomaly. }\nThe other integral term on the l.h.s. of  (\\ref{eq:constr1}) can be analyzed by repeating the argument above. Now, using crossing, we rewrite $\\int_0^1 \\frac{\\delta G(x)}{x^2} \\log(x)\\;dx = \\int_0^{\\frac{1}{2}} \\frac{\\delta G(x)}{x^2} \\log\\left(x (1-x)\\right)\\;dx .$ The troublesome term, again coming from the singular contribution of the block $f_{\\Delta_1}$, is the integral\n\\begin{equation}\nI'=\\int_\\epsilon^{1\/2} x^{\\Delta_1-2} \\log\\left(x (1-x)\\right)\\, dx ,\n\\end{equation}\nwhich evaluates at finite coupling to \n\\begin{equation}\nI_{\\rm finite}' = -\\frac{2^{1-\\Delta_1 } \\left(\\Delta_1 +(\\Delta_1 -1) (\\Delta_1  \\log (4)-\\, _2F_1(1,1;\\Delta_1 +1;-1))\\right)}{(\\Delta_1 -1)^2 \\Delta_1 }.\n\\end{equation}\nComparing this with the weak-coupling regularisation,\n\\begin{equation}\nI_{\\rm weak}' = \\sum_{n=0}^{\\infty} \\underbrace{ \\int_{\\epsilon}^{\\frac{1}{2} } \\frac{ (\\log x )^{n} (\\Delta_1-1)^n}{(n!)\\, x} \\log\\left(x (1-x)\\right) dx }_{\\texttt{regularised},\\;\\log\\epsilon\\rightarrow 0} ,\n\\end{equation}\nwe find, order by order,\n\\begin{equation}\nI_{\\rm finite}' -I_{\\rm weak}' = -\\frac{1}{(\\Delta_1 - 1)^2} ,\n\\end{equation}\nwhich proves the relation (\\ref{int1weak}).\n\n\n\n\n\\subsubsection{Analytic results for $C_1^2$}\\label{sec:predictC1}\nWe are now ready to study the first integral constraint (\\ref{eq:constr1}) at weak coupling. \nUsing (\\ref{int1weak}), it becomes\n\\begin{framed}\n\\beqa\\label{int1weak2}\n \\left. \\frac{3\\mathbb{C}-\\mathbb{B}}{8 \\;\\mathbb{B}^2} \\right|_{\\text{small }g}  &=& 1 \\!+\\!  \\underbrace{\\int_{\\epsilon}^{\\frac{1}{2}}dx \\left(\\sum_{\\ell=1}^{M} g^{2\\ell} G_{\\text{weak}}^{(\\ell)}(x)\\right)\\frac{\\log(x (1-x))}{x^2} }_{\\texttt{regularised, }\\log(\\epsilon)\\rightarrow 0} \n- \\left. \\left[\\frac{C_1^2}{(\\Delta _1-1)^2}\\right] \\right|_{\\text{small } g}\\nonumber \\\\ &&+ O(g^{2 M + 2})\\;.\n\\eeqa\n\\end{framed}\nDue to the presence of the anomalous term in this equation,  the knowledge of $G(x)$ at a fixed order at weak coupling allows to produce  nontrivial  predictions for the  leading OPE coefficient.\nUsing (\\ref{eq:B0}), (\\ref{eq:weakC0}), the term on the l.h.s. of (\\ref{int1weak2}) has the expansion\n\\begin{equation}\\begin{split}\\label{eq:lhsconstr1}\n\\frac{3\\mathbb{C}-\\mathbb{B}}{8 \\;\\mathbb{B}^2} \n=\n&-\\frac{1}{8 g^2}+\\left(\\frac{3}{2}-\\frac{\\pi ^2}{12}\\right)+\\left(\\frac{\\pi ^4}{36}-9 \\zeta_3\\right) g^2+\\left(-\\frac{2 \\pi ^6}{135}-4 \\pi ^2 \\zeta_3+135 \\zeta_5\\right) g^4\\\\\n&+\\left(\\frac{7 \\pi ^8}{810}+\\frac{34 \\pi ^4 \\zeta_3}{15}+78 \\pi ^2 \\zeta_5-1806 \\zeta_7\\right) g^6+O\\left(g^8\\right).\n\\end{split}\\end{equation}\nOn the other hand, for general OPE coefficients and scaling dimensions, we expect the weak coupling expansion\n\\begin{equation}\nC_i^2(g) = \\sum_{\\ell=0}^{\\infty}  a_i^{(\\ell)}  \\, g^{2 \\ell} ,\\qquad\n\\Delta_i(g) =  \\sum_{\\ell=0}^{\\infty} \\Delta_i^{(\\ell)} \\, g^{2 \\ell} ,\n\\end{equation}\n which implies that  ``anomaly term'' on the r.h.s. of  (\\ref{int1weak2})  starts as $-a_1^{(0)}\/(g^2\\Delta_1^{(1)})^2$. Since this is not matched on the l.h.s., we must have $a_1^{(0)} = 0$, and the anomaly term expands as\n\\begin{equation}\\label{eq:anomalyweak}\n-  \\left[\\frac{C_1^2}{(\\Delta _1-1)^2}\\right] = -\\frac{a_1^{(1)}}{ (\\Delta_1^{(1)})^2g^2} + \\left(\\frac{2 a_1^{(1)} \\Delta_1^{(2)} }{(\\Delta_1^{(1)})^3} -\\frac{a_1^{(2)} }{(\\Delta_1^{(1)})^2} \\right) + \\dots\n\\end{equation}\nUp to order $O(1)$, the integral on the r.h.s. drops out from (\\ref{int1weak2}), and we just need to match (\\ref{eq:lhsconstr1}) and (\\ref{eq:anomalyweak}). From the QSC, we know (see section  \\ref{apd:anylSpec}) $\\Delta_1^{(1)} = 4$, $\\Delta_1^{(2)} = -16$. The term $O(g^{-2})$ then fixes $a_1^{(1)} = 2$, consistent with  (\\ref{eq:firstprediction}),  while at order $O(1)$ we fix\n\\begin{equation}\\label{C1w2}\na_1^{(2)} = \\frac{4 \\pi ^2}{3}-24\\,.\n\\end{equation}\nWe can also study the next order $O(g^2)$, which involves the integral over $G_{\\text{weak}}^{(1)}$ given in (\\ref{eq:G1weak}). Computing the integral and using the next terms in the expansion of (\\ref{eq:anomalyweak}), we can now extract \n\\begin{equation}\\label{C1w3}\na_1^{(3)} = 48 \\zeta_3+320-16 \\pi\n   ^2-\\frac{76 \\pi ^4}{45}\\,.\n\\end{equation}\nIn the next section, we will see how, using a functional analytic bootstrap approach, one can push this analysis to one more loop, all in all determining a 4-loop prediction for the OPE coefficient. Our full result is reported below in equation (\\ref{eq:C1finalresult}). \n\n\\subsection{Functional bootstrap at weak coupling}\\label{sec:numerology}\n\nIn this section we use an analytic functional bootstrap  approach, combined with the integrability data, to obtain structure constants in perturbation theory at weak  coupling. \n\n\\subsubsection{General strategy}\n\\paragraph{Warm up at the first two orders. }\nThe first two orders of the weak coupling expansion of the reduced correlator $f(x)$ defined in \\eqref{fandG} are known. The tree level \\eqref{fandGtree} is obtained by free field theory, while the one loop \\eqref{f1weak0} was computed in \\cite{Kiryu:2018phb}. \n\nAn alternative way to present the perturbative expansion of the reduced correlator is in terms of Harmonic Polylogarithms~\\cite{Remiddi:1999ew} (HPL), which are natural functions appearing in the evaluation of Feynman integrals and are implemented in the Mathematica package \\texttt{HPL} \\cite{Maitre:2005uu,Maitre:2007kp}. \nTo do this, it is useful to introduce the new object\n\\begin{equation}\\label{defh}\nh(x)=\\frac{1 - x}{x} f(x) ,\n\\end{equation}\nfor which we have the following HPL representation for the first two orders at weak coupling\n\\beqa\nh_{\\text{weak}}^{(0)}(x)&=&1-2x , \\label{eq:h0}\\\\\nh_{\\text{weak}}^{(1)}(x)&=&-2 {H}_{1,0}+2 {H}_2-\\frac{2 \\pi ^2}{3}x\\label{h1},\\label{eq:h1}\n\\eeqa\nwith\n\\begin{equation}\n{H}_{1,0}=-\\text{Li}_2(x)-\\log(1-x)\\log(x)\\,,\\qquad\n{H}_2={H}_{0,1}=\\text{Li}_2(x) ,\n\\end{equation}\nwhere we used the HPL property\n\\begin{equation}\n{H}_{n_1,...,\\underbrace{0,...,0}_{k \\text{-times}},n_i,...}=\n{H}_{n_1,...,n_i+k,...}\\;.\n\\end{equation}\nBefore discussing how to infer an ansatz for the general perturbative order, let us discuss the implications of these results for the conformal data. \n\nFirst, one can compare the leading order $h_{\\text{weak}}^{(0)}(x)$ with the OPE expansion \\eqref{eq:OPEf}, using the leading order for the scaling dimensions, which become degenerate at tree level to the values $\\Delta^{(0)} \\equiv J = 1,2,3,\\dots.$ Comparing order by order the small-$x$ expansion of \\eqref{eq:OPEf} and  $h_{\\text{weak}}^{(0)}(x) $, one can fix the following constraints on the structure constants\n\\begin{equation}\\label{CLOavg}\n\\langle\\; a^{(0)}\\;\\rangle_J=\\frac{4^{-J-1} \\sqrt{\\pi } (J-1) \\Gamma (J+3)}{\\Gamma \\left(J+\\frac{3}{2}\\right)} ,\n\\end{equation}\nwhere $\\langle...\\rangle_J$ represents the sum over the state multiplicity at weak coupling for a given $\\Delta^{(0)} = J$. \nSuch averages are very common in analytic bootstrap approaches, which typically expand around points of degeneracy of the spectrum. For us, however, a big advantage will be the knowledge of the spectrum, which will give us more conditions and allow us to resolve some of these averages much more easily. In particular, at this stage we know the degeneracies, so that for instance $J=1$ counts only one state, and then (\\ref{CLOavg}) means $\\langle\\; a^{(0)}\\;\\rangle_1=a_1^{(0)}=0$, which is the same result we found in the previous section. \nAt the next level $J=2$ we have two states implying $\\langle\\; a^{(0)}\\;\\rangle_2=a_2^{(0)}+a_3^{(0)}=1\/5$, and so on. \n\nThe case of the ground state $\\Delta_1$, with $\\Delta_1^{(0)} = 1$, is special. In fact, due to the factor $1\/(\\Delta - 1)$ in the definition of the superconformal blocks (\\ref{superblocks}), the scaling dimension and OPE coefficient  \\emph{at one loop}  enter the OPE expansion of the correlator at tree level. In particular, the leading behaviour at small $x$ is determined by the block $C_1^2 f_{\\Delta_1}(x) \\sim x^{\\Delta_1 + 1}\/(\\Delta_1 - 1)\\sim  a_1^{(1)} x^{2} \/\\Delta_1^{(1)} + O(g^2)$. Matching this with the small-$x$ behaviour of $h_{\\text{weak}}^{(0)}(x)$, and using  $\\Delta_1^{(1)} = 4$ coming from the QSC, we  can read off $a^{(1)}_1=2$, in agreement with the independent derivation in \\eqref{eq:firstprediction}. This obviously extends to higher loops: the small-$x$ behaviour of $h_{\\text{weak}}^{(\\ell)}(x)$ is determined by conformal data of the ground state up to $\\ell + 1$ loops. \n\nRepeating the same procedure for $h_{\\text{weak}}^{(1)}(x)$, and including the results of the previous order, we can read $a_1^{(2)} = 4 \\pi ^2\/3-24$, matching  the result obtained by the integral relation in \\eqref{C1w2}. Expanding the conformal blocks at higher order in $x$, it is possible to disentangle the average appearing in \\eqref{CLOavg} obtaining \n\\begin{equation}\\begin{split}\\label{cleading}\na_2^{(0)}&=a_3^{(0)}=\\frac{1}{10},\\qquad\na_4^{(0)}=a_5^{(0)}=a_7^{(0)}=a_9^{(0)}=0,\\\\\na_6^{(0)}&=\\frac{1}{14}+\\frac{2}{7\\sqrt{37}},\\qquad\\;\\;\na_8^{(0)}=\\frac{1}{14}-\\frac{2}{7\\sqrt{37}} ,\n\\end{split}\\end{equation}\ntogether with the following constraints for the sub-leading orders\n\\begin{equation}\\label{csubleading}\n\\langle\\,a^{(1)}\\,\\rangle_2=\\frac{2\\pi^2}{15}-1,\\qquad\n\\langle\\,a^{(1)}\\,\\rangle_3=\\frac{2\\pi^2}{21}-\\frac{1159}{882},\\qquad\n\\langle\\,a^{(1)}\\,\\rangle_4=-\\frac{29}{36}+\\frac{\\pi ^2}{21}.\n\\end{equation}\n\\paragraph{General ansatz and strategy. }\nThe reduced correlator at order $g^4$ is unknown. To proceed, we will formulate an ansatz for the generic term based on the  form of the first two orders (and also inspired by the functional forms observed at strong coupling~\\cite{Ferrero:2021bsb}). Consistent with (\\ref{eq:h0}),(\\ref{eq:h1}), we assume that, at $\\ell$ loops, $h$ is given in terms of a complete basis of HPL's with transcendentality\\footnote{For an HPL function $H_{n_1,n_2,\\dots, n_{\\tau}}(x)$, with $n_i\\in\\left\\{0,1\\right\\}$, the transcendentality is the number of indices.} up to $\\tau = 2 \\ell$. Besides,  we assume the same transcendentality for all terms, i.e.,  \n\\begin{equation}\\label{basis}\nh_{\\text{weak}}^{(\\ell)}(x)=\\beta_0^{(2\\ell)}+\\beta_1^{(2\\ell)} x+\\sum_{\\tau=1}^{2\\ell}\\beta_{n_1,...,n_\\tau}^{(2\\ell-\\tau)}\\,H_{n_1,...,n_\\tau},\n\\end{equation}\nwhere the coefficients $\\beta^{(m)}$ are transcendental numbers of weight $m$ and the indices $n_i=0,1$. For example, the basis used for $\\ell=1$ according to this ansatz contains eight terms:\n\\begin{equation}\\label{eq:hansatz}\nh_{\\text{weak}}^{(1)}(x)=\\beta_0^{(2)}+\\beta_1^{(2)} x+\\beta_{0}^{(1)}H_{0}+\\beta_{1}^{(1)}H_{1}\n+\\beta_{0,0}^{(0)}H_{0,0}\n+\\beta_{1,0}^{(0)}H_{1,0}\n+\\beta_{0,1}^{(0)}H_{0,1}\n+\\beta_{1,1}^{(0)}H_{1,1} .\n\\end{equation}\nComparing it with \\eqref{f1weak0}, one can conclude that the only non-vanishing coefficients are the following\n\\begin{equation}\\label{eq:1loopfixing}\n\\beta_1^{(2)}=-\\frac{2}{3}\\pi^2\\,\\qquad\n\\beta_{1,0}^{(0)}=-2\\,\\qquad\n\\beta_{0,1}^{(0)}=2\\,,\n\\end{equation}\nleading to \\eqref{h1}. \n\nIn order to fix the coefficients of the ansatz \\eqref{basis} for $\\ell>1$, we apply the following strategy\n\\begin{itemize}\n    \\item \\texttt{Crossing equation.} In terms of $h(x)$, crossing symmetry translates to\n\\begin{equation}\\label{hcrossing}\nh(1-x)+h(x)=0 .\n\\end{equation}    \nTo impose this equation, in practice is enough to study some terms of its expansion around $x \\sim 0$. Once these terms are set to zero by fixing some of the $\\beta$ coefficients, the equation is satisfied for any $x$. \n    \\item \\texttt{Cancelling logarithms.} In order to come from an OPE expansion (\\ref{eq:OPEf}) at weak coupling, the function $h(x)$ has to satisfy certain constraints on its $x\\sim 0$ behaviour. At $\\ell$ loops, $h^{\\ell}(x)$ can only contain terms $\\log^{m}(x)\\, x^n$, with $n\\in \\mathbb{N}$ and $m = 1,\\dots, \\ell$. We compare this with the expansion of the ansatz, and  impose the cancellation of the logarithms with higher powers. \n    \\item \\texttt{Conformal data matching.} The OPE expansion gives  infinitely many relations between terms in a small $x$ expansion of $h^{\\ell}(x)$ and the conformal data. Equating these predictions with the expansion of the ansatz (\\ref{eq:hansatz}), we find relations between the $\\beta^{(\\ell)}$ coefficients and the conformal data $a_i^{(m)}$, $\\Delta_i^{(m)}$, with $m\\leq \\ell$ for $i>1$ and $m\\leq \\ell+1$ for $i = 1$. Using the \\emph{knowledge of the spectrum} from integrability, we can use these relations to fix some $\\beta$'s as well as some OPE coefficients. \n \\item \\texttt{Integral constraints. }\n We impose the two integral relations described in section \\ref{sec:integrated}.  At a given order, the constraint (\\ref{eq:constr2}) fixes further information on the coefficients of the ansatz. The constraint (\\ref{eq:constr1}), due to the ``weak coupling anomaly'' effect described in section \\ref{sec:weakcouplingintegrals}, can be used to extract the structure constant $a_1^{(\\ell+2)}$ in terms of the coefficients $\\beta^{(\\ell)}$ of the ansatz.  \n \\item \\texttt{Transcendentality. } Our assumption is that the coefficients $\\beta^{(2 \\ell - \\tau)}_{n_1,\\dots, n_{\\tau}}$ are combinations with rational coefficients\\footnote{For many of them it turns out the coefficients are integer.}\n  of numbers of uniform transcendentality   $2 \\ell -\\tau$. \n  In particular, up to the perturbative order we considered, we did not encounter Multiple Zeta numbers, but only numbers which are products of elements of the basis\n  \\begin{equation}\\label{eq:basis}\n  \\left\\{\n\\pi^2 , \\zeta_3, \\zeta_5, \\zeta_7, \\dots, \\zeta_{2n+1} , \\dots  \n  \\right\\}.\n \\end{equation}\n The element of this basis have transcendental weight $2$,$3$,$5$, $7,\\dots,2n+1,\\dots$, and their products have weight equal to the sum of the weights of the factors. \n There are no linear relations with rational coefficients between the numbers in the basis (\\ref{eq:basis}). \n  This implies that some linear relations generated by the other ``axioms'' listed above split into more constraints, as terms of different transcendentality should vanish individually.  \n\\end{itemize}\n\\subsubsection{Higher loops}\n\\paragraph{Fixing the correlator at 2 loops. }\nFor $\\ell=2$, the basis \\eqref{basis} with maximal transcendentality 4 counts $32$ elements. \nImposing the crossing equation \\eqref{hcrossing}\none can fix $16$ of the coefficients $\\beta^{(2)}$. Furthermore, requiring that in the small $x$ expansion there are no $\\log^3x$ and $\\log^4x$ terms, we obtain $3$ coefficients more.\nThe remaining constraints can be found by injecting in the analytical bootstrap method the integrability data, namely the spectrum obtained with the QSC, the structure constants fixed at previous orders using $h_{\\text{weak}}^{(0)}$ and $h_{\\text{weak}}^{(1)}$, as well as using the integrated correlator \\eqref{eq:constr1} at weak coupling,~{\\it cf.}~\\eqref{int1weak}. Let us describe these steps in more detail.\n\nFirst, we expand at small $x$ and compare the terms $x^n\\log^m x$ for $n,m=0,1,2$ with the same expansion of \\eqref{eq:OPEf}. At this order, only the first $3$ lowest non-trivial states (i.e., the ground state with $J=1$\nand the two states with $J=2$) contribute to these coefficients. Then, using the spectral data\\footnote{Because of the pole in the conformal block at $\\Delta=1$, we actually need to use all terms up to $O(g^6)$ for $\\Delta_1$.} for these states reported in \\eqref{D1w}-\\eqref{D3w},  \ntogether with their structure constants at leading~\\eqref{cleading} and subleading order~\\eqref{csubleading}, we fix $9$ more parameters.\\footnote{Of these, 3  parameters are reduced by using the assumption of fixed transcendentality of the $\\beta$ coefficients.}  At this stage, we are left with 4 unfixed coefficients $\\beta^{(2)}$.  \n\nAs a by-product of the procedure explained above, and also including data for states with $J>2$, we generate new constraints for the structure constants. One of these constraints is particularly useful, since it expresses the $O(g^6)$ term in the expansion of $C_1^2$ in terms of the 4 remaining parameters $\\beta$ as follows\n\\begin{equation}\\begin{split}\\label{a13coeff}\na_{1}^{(3)}&=-\\frac{28 \\pi ^4}{15}-16 \\pi ^2+320+64 \\zeta_3+ \\left(\\frac{2 \\pi ^2}{3}+8 \\zeta_3-\\frac{7 \\pi ^4}{45}\\right)\\beta^{(0)}_{0,1,0,1}\\\\\n&-\\!\\left(\\!4\\!+\\!4 \\zeta_3-\\frac{4 \\pi ^4}{45}\\right) \\!\\beta_{0,0,0,1}^{(0)}\n- \\left(\\frac{2 \\pi ^2}{3}+4 \\zeta_3\\!-\\!\\frac{\\pi ^4}{9}\\right)\\!\\beta_{0,1,1,0}^{(0)}\\!-\\!\\left(\\!4\\!+\\!\\frac{2 \\pi ^2}{3}-8 \\zeta_3\\right)\\!\\beta_{0,0,1}^{(1)} . \n\\end{split}\\end{equation}\nAs shown in section \\ref{sec:predictC1}, using the integral relation \\eqref{eq:constr1} in its weak coupling fashion one can generate  terms for $C_1^2$ at higher orders exploiting the anomaly. We have already used this method to give a prediction for $a_1^{(3)}$ from the knowledge of the correlator at 1 loop. Comparing this result~\\eqref{C1w3} with \\eqref{a13coeff}, and using that the $\\beta$ coefficients are rational, we conclude that \n\\begin{equation}\n\\beta_{0,0,1}^{(1)}=\\beta_{0,0,0,1}^{(0)}=0\\,,\\qquad\\text{and}\\qquad\n\\beta_{0,1,1,0}^{(0)}=\\beta_{0,1,0,1}^{(0)}=-4 ,\n\\end{equation}\nleading to the following expression for $h(x)$ at order $g^4$:\n\\begin{equation}\\begin{split}\\label{h2}\nh^{(2)}_{\\text{weak}}(x)&=4 H_{1,3}-4 H_{2,2}+8 H_{3,0}+8 H_{3,1}-8 H_{1,1,2}+4 H_{1,2,0}+8 H_{1,2,1}-8 H_{2,0,0}\\\\\n&-4 H_{2,1,0}-8 H_{1,1,0,0}+\\frac{4}{3} \\pi ^2 H_2-\\frac{4}{3} \\pi ^2 H_{1,0}+\\frac{4}{3} \\pi ^2 H_{1,1}-12 \\zeta_3 H_{1}+\\frac{8 \\pi ^4}{15}x .\n\\end{split}\\end{equation}\nAs a cross check, this formula can be tested using the integral relation \\eqref{eq:constr2}. Indeed, plugging \\eqref{h2} into \\eqref{defh} and then performing the integral, we obtain\n\\begin{equation}\n g^4\\int_{0}^1 dx \\frac{h_{\\text{weak}}^{(2)}(x)}{1-x} = \n g^4\\left(-\\frac{8 \\pi ^4}{15}-\\frac{8 \\pi ^2 \\zeta_3}{3}+90 \\zeta_5\\right)\n =        \\left.\\frac{\\mathbb{C}}{4\\;\\mathbb{B}^2} + \\mathbb{F}-3\\right|_{\\text{order }g^4} ,\n\\end{equation}\nin agreement with the expansion of the r.h.s. of \\eqref{eq:constr2}. \n\n\\paragraph{Structure constants from $h_{\\text{weak}}^{(2)}$. }\n\nHaving fixed the reduced correlator at 2 loops, we can mine new data for the structure constants. Indeed, comparing this answer to the OPE, and using the knowledge for the spectrum at one loop, one can disentagle the first relation of \\eqref{csubleading}, obtaining the two separate subleading terms as follows\n\\begin{equation}\na_{2}^{(1)}=\\frac{1}{150} \\left(10 \\pi ^2-75-9 \\sqrt{5}\\right)\\,,\\qquad\na_{3}^{(1)}=\\frac{1}{150} \\left(10 \\pi ^2-75+9 \\sqrt{5}\\right).\n\\end{equation}\nSimilarly, the 3-point functions associated to the leading twist states\\footnote{Leading twist refers to states with oscillator content $[2,2|3,3,2,2|2,2]$, in the classification introduced in \\cite{Cavaglia:2021bnz}. Notice that in this case the indices $11$, $12$ and $13$ do not represent the order in which they appear at weak coupling, and they differ from the notation used in \\cite{Cavaglia:2021bnz}. The scaling dimensions of those states are computed with the QSC and read\n\\begin{equation}\n\\Delta_{11}=4+w_1 g^2+z_3 g^4+O(g^6),\\qquad\n\\Delta_{12}=4+w_2 g^2+z_2 g^4+O(g^6),\\qquad\n\\Delta_{13}=4+w_3 g^2+z_1 g^4+O(g^6),\\nonumber\n\\end{equation}\nwith $w_{1,2,3}$ and $z_{1,2,3}$ the solutions of the following equations\n\\beqa\n&& 3 w_i^3-73 w_i^2+553 w_i-1274=0\\qquad\\text{with}\\qquad w_1<w_2<w_3,\\nonumber\\\\\n&& 32399568 z_j^3+3007792328 z_j^2+66692590965 z_j+57114350496=0\n\\qquad\\text{with}\\qquad z_1<z_2<z_3.\\nonumber\n\\eeqa} at $J=4$ reads\n\\begin{equation}\na_{11}^{(0)}=s_1,\\qquad\na_{12}^{(0)}=s_2,\\qquad\na_{13}^{(0)}=s_3,\n\\end{equation}\nwhere $s_{1,2,3}$ are the three solutions of the following polynomial equation\n\\begin{equation}\\label{eq:defpi}\n794584 \\,s_i^3 - 56756 \\,s_i^2 + 1066 \\,s_i - 3=0\\qquad \\text{with}\\qquad p_1<p_2<p_3 ,\n\\end{equation}\nwhich are numbers deriving from the form of the one loop anomalous dimensions. \nNotice that $a_{11}^{(0)}+a_{12}^{(0)}+a_{13}^{(0)}=1\/14$ as expected by \\eqref{CLOavg}, which implies that all the other states at $J=4$ have vanishing structure constants at leading order. \nTogether with the previous results, we obtain also new constraints on the averages for $J=2,3,4$, given by\n\\begin{equation}\\begin{split}\n\\langle \\,a^{(2)}\\,\\rangle_2&=9-\\frac{4 \\pi ^2}{3}+6 \\zeta_3-\\frac{8 \\pi ^4}{75},\\\\\n\\langle \\,a^{(2)}\\,\\rangle_3&=\\frac{166907}{9261}-\\frac{2041 \\pi ^2}{1323}+\\frac{38 \\zeta_3}{7}-\\frac{8 \\pi ^4}{105},\\\\\n\\langle \\,a^{(2)}\\,\\rangle_4&=\\frac{4919}{432}-\\frac{28 \\pi ^2}{27}+3 \\zeta_3-\\frac{4 \\pi ^4}{105}.\n\\end{split}\\end{equation}\n\nFinally, following the same logic of section \\ref{sec:predictC1}, we can use the integral relation \\eqref{eq:constr1} to generate more data for $C_1^2$. First, from our result \\eqref{h2} and \\eqref{defh}, \\eqref{pt4} \nwe assemble the 4-point function at order $g^4$ obtaining \n\\beqa\\label{eqn:Gweakg4}\n&&G_{\\text{weak}}^{(2)}(x)\\!=\\!\\frac{4(1\\!-\\!2x)}{(1\\!-\\!x)^2}\\biggl[\n \\frac{\\pi^2}{3}(  H_{2}\\!-\\! H_{1, 0} \\!+\\! H_{1, 1})\\! -\\! \n    3 \\zeta_3 H_{1}\\!+\\! 2 (H_{1, 2, 1}\\! - \\!\n  H_{1, 1, 2}\\!-\\! \n H_{2, 0, 0}\\!-\\! \n  H_{1, 1, 0, 0})\\nonumber\\\\\n &&\n + \\!\\frac {(x \\!-\\! 1)x \\!+\\! 1} {2 x \\!-\\! 1} \\!\\left(\\!2H_{2, 1}\\!-\\! 2H_{1, 0, 0}\\!+\\! (x \\!-\\! 1)H_{1, 1, 0}\\!-\\! (x\\! -\\! 2)H_{2, 0}\\!+\\!  xH_{3}\\!+\\!\\frac{\\pi^2}{3}(H_{1}\\!-\\!    xH_{0})\\!\\right)\\label{eq:G2loop}\\\\\n && +\\!\\frac{ (x^3 \\!+\\! 1)}{1\\! -\\! 2 x} H_{1, 2}  \\!+\\! \n  H_{1, 3}\\! -\\! \n H_{2, 2} \\!+\\! 2 H_{3, 0} \\!+\\! \n 2 H_{3, 1}\\! +\\! H_{1, 2, 0}   \\!-\\!  H_{2, 1, 0}  \\!+\\! x\\frac{\n  2 \\pi^4 x\\! -\\! 45 ((x \\!-\\! 1) x\\! +\\! 1) \\zeta_3}{15(2 x \\!- \\!1)}\\biggr].\\nonumber\n\\eeqa\nThen, plugging $G_{\\text{weak}}^{(2)}$ into the integral relation \\eqref{int1weak2}, performing the integrals and solving for $a_1^{(4)}$ we get the 4 loop prediction\n\\begin{equation}\\label{a14}\na_1^{(4)}=-4480+\\frac{832 \\pi ^2}{3}-256 \\zeta_3+\\frac{224 \\pi ^4}{15}-880 \\zeta_5+\\frac{64 \\pi ^6}{45} .\n\\end{equation}\n\n\\paragraph{The correlator at 3 loops. }\nFixing the correlator at 3 loops is way more involved than the previous case, and it is the first order at which the algorithm described above fails to fix the answer completely. \nFor $\\ell=3$, the basis \\eqref{basis} with maximal transcendentality 6 counts 128 elements. \nWe will assume, as suggested by the previous orders, that the  $\\beta^{(3)}$ constants are linear combinations of numbers built as products of the basis elements (\\ref{eq:basis}), with rational coefficients. This assumption reduces the ansatz to 100  unknown rational parameters. \n\nRequiring that the small $x$ expansion of $h_{\\text{weak}}^{(3)}$ does not contains terms proportional to $\\log^nx$ with $n>3$, we get 5 additional $\\beta$'s. Then, using the crossing equation \\eqref{hcrossing}, we further constrain the system obtaining 50 additional coefficients.\n\nThe next step is to include integrability data in the derivation.\nExpanding $h_{\\text{weak}}^{(3)}$ at small $x$, we compare terms proportional to $x^n\\log^m x$ with the same terms appearing in the expansion of \\eqref{eq:OPEf} where we have injected spectral data from the QSC (see appendix \\ref{apd:anylSpec}) and structure constants from the previous orders. Inspecting the contribution of the $J=1$ state fixes 15 coefficients, while the contributions of higher states are less constraining since they are more and more degenerate. Indeed, $J=2$ states gives 3 constraints and $J=3$ and $J=4$ only 1 each. As before, this procedure generates several constraints on structure constants functions and  in particular we obtain $a_1^{(4)}$ in terms of $\\beta^{(3)}$ coefficients. Since this quantity was previously computed in \\eqref{a14} exploiting the integral relation \\eqref{eq:constr1}, one can use it to fix 8 additional coefficients. Finally, plugging  $h_{\\text{weak}}^{(3)}$ in the second integral relation \\eqref{eq:constr2}, performing the integrals and comparing with the $O(g^6)$ term of the r.h.s., we get 3 more constraints\\footnote{A partial result for the $4$-point function at $3$ loops is available upon request.}.\n\nThe remaining 14 coefficients are unconstrained. \nIt is possible that more sophisticated analytical bootstrap techniques, such as the inversion formula developed in 1D in \\cite{Mazac:2018qmi}, might be helpful in this context. In any case, our finding  seems to suggest that using analytical bootstrap equations for a single correlators may not be enough to pinpoint the solution completely, even with a full knowledge of the spectrum. One expects that studying multiple correlators would have the biggest impact on fixing the solution at higher orders, as also observed at strong coupling~\\cite{Ferrero:2021bsb}. \n\n\n\n\n\n\n\n\\subsection{Results} To summarise, we collect here the analytic results for structure constants obtained with the above approach. For the ground state:\n\\begin{equation}\\begin{split}\\label{eq:C1finalresult}\nC_1^2(g) = &2 g^2 - \\left(24-\\frac{4 \\pi ^2}{3}\\right)g^4+\n\\left(320-16 \\pi\n   ^2+48 \\zeta_3-\\frac{76 \\pi ^4}{45}\\right)g^6\\\\\n   &-\\left(4480-\\frac{832 \\pi ^2}{3}+256 \\zeta_3-\\frac{224 \\pi ^4}{15}+880 \\zeta_5-\\frac{64 \\pi ^6}{45}\\right)g^8+\nO(g^{10}),\n\\end{split}\\end{equation}\nand for excited states at $J=2$\n\\beqa\nC_2^2&=&\\frac{1}{10}+\\frac{1}{150} \\left(10 \\pi ^2-75-9 \\sqrt{5}\\right)g^2+\nO(g^{4}),\\\\\nC_3^2&=&\\frac{1}{10}+\\frac{1}{150} \\left(10 \\pi ^2-75+9 \\sqrt{5}\\right)g^2+\nO(g^{4}),\n\\eeqa\nand the non-vanishing excited states at leading order at $J=3,4$\n\\beqa\nC_6^2=\\frac{1}{14}+\\frac{2}{7\\sqrt{37}}+\nO(g^{2})\\qquad\nC_8^2=\\frac{1}{14}-\\frac{2}{7\\sqrt{37}}+\nO(g^{2}),\n\\eeqa\n\\beqa\nC_{11}^2=s_1+\nO(g^{2})\\qquad\nC_{12}^2=s_2+\nO(g^{2})\\qquad\nC_{13}^2=s_3+\nO(g^{2}),\n\\eeqa\nwhere the constants $s_i$ are the roots of (\\ref{eq:defpi}). Our 2-loop result for the 4-point function is given in (\\ref{eq:G2loop}).  Weak coupling data for scaling dimensions, coming from the QSC, are collected  in Appendix \\ref{app:spectrum}.\n\n\n\n\n\\section{Discussion}\\label{sec:discussion}\nIn this paper we have continued experimenting with a combination of integrability and conformal bootstrap methods to study observables in $\\mathcal{N}$=4 SYM. This has led to the most accurate results to date for a non-supersymmetric  OPE coefficient of short operators at finite coupling. For instance,  as highlighted in fig. \\ref{fig:zoom}, with the methods of this paper we determine one such structure constant with error $10^{-8}$ for 't Hooft coupling $\\lambda \\sim 24 \\pi^2 $. Presently, this would  not be  achievable either with integrability or conformal bootstrap methods on their own. \n\n In this work, we have introduced new constraints on integrated correlators in the 1D defect CFT,  connecting them to another quantity available from integrability, the cusp anomalous dimension. \nThe addition of these constraints was shown to greatly enhance the precision of the numerical bootstrap algorithm, allowing us to reach at least 7 digits of precision for $C_1^2$ over a wide range of the coupling for $g>1$ (becoming 9 digits for $g\\gtrsim 3$), and at least 2 digits precision for the next two OPE coefficients for $g>1$ (see fig.~\\ref{fig:plotintboth2}). \n\nThese new constraints were also very powerful in an analytic functional bootstrap approach, which allowed us to fix the form of the 4-point function at 2 loops, fix 4 loops for $C_1^2$, and 2 loops for the next two excited states OPE coefficients.\n\nThis development resonates nicely with the recent discovery of integrated correlators constraints for the bulk 4D theory, in that case arising from localisation~\\cite{Binder:2019jwn} (see also \\cite{Dorigoni:2021bvj,Dorigoni:2021guq}), which were also shown to have a great impact on the bootstrap~\\cite{Chester:2021aun}. \n\nIt would be interesting to see if more general deformations of the MWL can lead to even more constraints of the type studied here. At the very least,  \n there should be generalisations of the present identities involving integrated $n$-point functions with $n>4$. Those would be related to higher-orders in the  near-BPS expansion of  the cusp anomalous dimension, which are also in principle accessible with integrability. Such generalised constraints might be useful in the bootstrap. \n\nAn important question is whether using Bootstrability it is  possible to compute the OPE coefficients with (ideally) arbitrary precision, or if there is a fundamental limit. One way to improve the precision is certainly to include input from more states in the spectrum in the Numerical Boostrap algorithms (here we used only 10 states). Another possibility is to use analytical bootstrap techniques such as the ones developed for 1D CFTs in \\cite{Mazac:2016qev,Mazac:2018mdx,Mazac:2018qmi,Mazac:2018ycv,Paulos:2019fkw,Ferrero:2019luz,Bianchi:2021piu}, which might  prove to be advantageous. \n\nHowever, we believe it is very unlikely that the study of a single correlator will be enough to fix all OPE coefficients, even from a completely known spectrum. This is in fact what we observe analytically at weak coupling, where the functional bootstrap approach did not fix completely the 4-point function at 3 loops. One can argue that this is in fact to be expected, as the CFT is defined not by one, but by all its correlation functions. Also in the Numerical Conformal Bootstrap, it is only the study of multiple correlators that allows to find small islands for allowed conformal data~\\cite{Kos:2016ysd}. Thus, we believe that multi-correlator Bootstrability is a very promising direction for the future. \n\nOur present setup is the 1D defect CFT, which presents some simplifications. In particular, it is a consistent CFT at the planar level. \nThis is not the case for the bulk 4D CFT, where there is intermingling of single and double traces contributing to planar 4-point functions. This presents a challenge, since the QSC does not know about the anomalous dimensions of double traces. However, we expect that analytic conformal bootstrap techniques such as the ones in \\cite{Caron-Huot:2017vep,Alday:2017vkk,Caron-Huot:2020adz,Bissi:2022mrs,Bissi:2021spj} might help to resolve this problem. In particular, it is inspiring that the double discontinuity of a 4-point function, which parametrises  the full correlator thanks to the Lorentzian inversion formula~\\cite{Caron-Huot:2017vep}, is determined only by single trace operators in a large $N$ theory~\\cite{Alday:2017vkk}. \nFor instance, at strong coupling $\\lambda \\sim \\infty$, the correlator was reconstructed, at several orders in $1\/N$, starting from information on single-trace protected operators, which are the only ones contributing in this regime~\\cite{Aharony:2016dwx,Alday:2017xua}. \nOur hope is that Bootstrability could be the way to extend these beautiful results to the full finite $\\lambda$ region. \n\nIt is also an interesting direction for the future to extend these techniques to other integrable gauge theories. A natural setup would be the one of the Wilson line defect CFTs living in ABJM theory defined in \\cite{Bianchi:2017ozk,Bianchi:2018scb}, recently studied from the Bootstrap approach in \\cite{Bianchi:2020hsz}. For this setup, the cusp is also intensively studied in \\cite{Griguolo:2012iq,Bonini:2016fnc} and the Bremsstrahlung function is known exactly \\cite{Bianchi:2014laa,Correa:2014aga,Bianchi:2017svd,Bianchi:2018scb} (see also \\cite{Drukker:2019bev}), although an integrability formulation is still lacking. The QSC for the spectrum of local operators was found for this theory in \\cite{Cavaglia:2014exa,Bombardelli:2017vhk} (for its numerical solution see \\cite{Bombardelli:2018bqz}). Finding its deformation capturing the  defect CFT would be very interesting and would open the way to applying the methods presented here. \nThe AdS$_3$\/CFT$_2$ duality for which a QSC was recently proposed in \\cite{Cavaglia:2021eqr,Ekhammar:2021pys} is also a fascinating laboratory to develop Bootstrability in the context of 2D CFTs. \n\n Finally, the fishnet limit of $\\mathcal{N}$=4 SYM ~\\cite{Gurdogan:2015csr} and ABJM \\cite{Caetano:2016ydc} theories are also very interesting playgrounds for combining integrability and bootstrap techniques, which present additional challenges due to the non-unitarity. We should mention that the Fishnet theories are a very  promising setting to understand correlation functions analytically as shown in many recent works \n \\cite{Grabner:2017pgm,Basso:2017jwq,Gromov:2018hut,Kazakov:2018gcy,Pittelli:2019ceq,Derkachov:2019tzo,Derkachov:2020zvv,Shahpo:2021xax}. The QSC is also under control in this limit  \\cite{Gromov:2017cja,Gromov:2019jfh,Cavaglia:2020hdb,Levkovich-Maslyuk:2020rlp} (for the open spin chain case, relevant for the fishnet limit of the present setup, see \\cite{Gromov:2021ahm}), and for their  simplicity the Fishnet theories appear to be the  \n perfect laboratory to understand the expected connection between QSC and  correlators. \n There has been progress in this direction~\\cite{Cavaglia:2018lxi,Cavaglia:2021mft} (for other limits see e.g. \\cite{Jiang:2015lda,Giombi:2018qox,Giombi:2022anm}) but a full solution is still missing. Having  non-perturbative data obtained with the help of bootstrap methods could be important to inform these efforts. Moreover, perhaps the study of these limits will reveal new ways in which integrability and bootstrap should be fused at a more fundamental level to study AdS\/CFT-related theories. \n\n\\acknowledgments\nWe thank Simon Caron-Huot, Nadav Drukker, Pietro Ferrero, Alessandro Georgoudis, Gregory Korchemsky, Petr Kravchuk, Andrea Manenti, Carlo Meneghelli, Amit Sever, Evgeny Sobko, Nika Sokolova, Andreas Stergiou, Roberto Tateo, Emilio Trevisani and Pedro Vieira for inspiring discussions. \nThe work of AC, NG and MP is\nsupported by European Research Council (ERC) under\nthe European Union's Horizon 2020 research and innovation programme (grant agreement No. 865075) EXACTC. NG is also partially supported by the STFC grant (ST\/P000258\/1).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Experimental results}\nEuFe$_2$(As$_{1-x}$P$_{x}$)$_2$ single crystals used in this\nstudy were synthesized by a Bridgman method and characterized as\npreviously described.\\cite{Jeevan11} Optical investigations on crystals with $x = 0$ and $x = 0.18$ have been already reported.\\cite{Wu09,Wu11} Here we describe the magnetic behavior of EuFe$_2$As$_2$ and\nEuFe$_2$\\-(As$_{0.88}$P$_{0.12}$)$_2$, measured with a\nQuantum Design MPMS-XL superconducting quantum interference device\n(SQUID). We provide a complete characterization as a function of temperature ($T$) and magnetic field ($H$) for the main crystallographic directions. EuFe$_2$As$_2$ has orthorhombic\nsymmetry with $a$ and $b$ axes virtually identical. Even though twinning of the crystals did not allow us a characterization in the $ab$-plane, neutron scattering data indicate that the Eu$^{2+}$ moments align along the $a$-direction. \n\\begin{figure}\n \\centering\n  \\includegraphics[width=0.6\\columnwidth]{Fig1.eps}\n  \\caption{Zero-field cooled (ZFC) and field-cooled (FC) magnetization curves (solid red circles and open black squares, respectively) for EuFe$_2$(As$_{0.88}$P$_{0.12}$)$_2$, measured in $H=20$~Oe parallel (a) and perpendicular (b) to the $ab$-plane.}\n  \\label{G_T}\n\\end{figure}\nIn Fig.~\\ref{G_T} the temperature-dependent magnetization of\nEuFe$_2$(As$_{0.88}$P$_{0.12}$)$_2$ is plotted for $H=20$~Oe, measured parallel and perpendicular to the $ab$-plane. While at elevated temperatures ($T<T_{\\rm SDW}$) the\nparamagnetic regime can be nicely described by the Curie-Weiss\nlaw, similar to the parent compound,\\cite{Wu09} the cusp at\n$T_N=16$~K evidences AFM order of the  Eu$^{2+}$ moments. At low\ntemperatures we find a distinct difference between the\nfield-cooled (FC) and zero-field cooled (ZFC) behavior (especially for\n$H\\parallel c$) that seems to be more pronounced as P substitution\nincreases.\\cite{Jeevan11}\n\n\\begin{figure}\n \\centering\n  \\includegraphics[width=0.6\\columnwidth]{Fig2.eps}\n  \\caption{Isothermal magnetization {\\em versus} external magnetic field applied parallel (black dashed lines) and perpendicular (red solid line) to the $ab$-plane for $T=2$~K.\n(a) While for EuFe$_2$As$_2$ no hysteresis  can be observed, (b)\nin the case of EuFe$_2$(As$_{0.88}$P$_{0.12}$)$_2$ the hysteresis\nis obvious for $H\\parallel c$ but can also be identified for\n$H\\parallel ab$.}\n  \\label{G_Hyst}\n\\end{figure}\nTo clarify this point, we measured the field dependence of\nthe magnetization parallel and perpendicular to the $ab$-plane. The complete magnetization {\\em versus} field curve has already been discussed\\cite{Jiang09, Nowik11} and here we concentrate on the low-field features.\nFig.~\\ref{G_Hyst} displays the results obtained at $T=2$~K for\nEuFe$_2$As$_2$ and EuFe$_2$(As$_{0.88}$P$_{0.12}$)$_2$. While we\nobserve no hysteresis for the parent compound,\nEuFe$_2$(As$_{0.88}$P$_{0.12}$)$_2$ exhibits clear evidence of FM\nbehavior when the magnetic field is applied parallel to the\n$c$-axis. For $H\\parallel ab$ only a much smaller hysteresis is observed. \nThe hysteresis becomes narrower with increasing\ntemperature and completely  vanishes at 16~K\n(Fig.~\\ref{G_Hyst_T}), which is the magnetic transition\ntemperature common to both crystal directions. To follow the\nopening of the hysteresis, we plot in Fig.~\\ref{G_Hyst_T} the difference $\\Delta M$ between the magnetization curves acquired by sweeping the field down and up ($i.e.$ the hysteresis height). The curves show a decrease of the hysteresis height on raising the temperature and it is instructive to compare this trend with the difference between the ZFC and FC curves at the same field. As shown in Fig.~\\ref{G_Hyst_T}(b) the two trends coincide, demonstrating the development of magnetic ordering.\n\\begin{figure}\n    \\centering\n        \\includegraphics[width=0.9\\columnwidth]{Fig3.eps}\n    \\caption{(a) Height of the hysteresis for EuFe$_2$(As$_{0.88}$P$_{0.12}$)$_2$, measured with $H\\parallel c$ for different fields and temperatures. (b) Temperature dependence of the difference between the ZFC and FC curves shown in Fig \\ref{G_T}(b). This difference can be compared to the hysteresis height at $H=$ 20~Oe shown in (a). The broken lines are a guide to the eye to stress the correlation between the two trends.}\n    \\label{G_Hyst_T}\n\\end{figure}\n\n\n\\section{Model}\nBased on these magnetization measurements and in agreement with other\nstudies, we propose that the magnetic\nmoments of the Eu$^{2+}$ ions change their alignment under\npressure as displayed in Fig.~\\ref{PhasenDiagrammNeu}. In\nthe parent compound, the spins orient along the $a$-direction, but basically reverse the direction in adjacent\n$ab$-planes.\\cite{Jiang09} More precisely, one observes canted\nAFM causing a small FM component of the moments in $c$-direction that\nis usually neglected in the discussion. M\\\"ossbauer\nstudies reveal a steep increase of this canting angle with\nincreasing P substitution\nuntil the Eu spins are aligned  almost perpendicular to the\n$ab$-plane.\\cite{Nowik11} We want to emphasize that this\nhas a huge impact on magnetic susceptibility and has to\nbe considered in the interpretation.\n\nDue to the non-negligible tilt angle out of the $ab$-plane, as\ndisplayed in Fig.~\\ref{PhasenDiagrammNeu}, one always observes a FM\nsignal along the $c$-axis,\nincluding a hysteretic behavior. This is exactly what we measured\nin our EuFe$_2$(As$_{0.88}$P$_{0.12}$)$_2$ single crystal. The small hysteresis visible for $H\\parallel\nab$ is likely caused by our limited precision in crystal alignment and we will not discuss it further. The\nfact that in the parent compound the hysteresis cannot be resolved\nmight be due to the extremely small spin canting. Since\nchemical and physical pressures have a similar effect\non the 122 compounds, we propose to examine  EuFe$_2$As$_2$ single\ncrystals by a high-pressure SQUID susceptometer to\nconfirm and  complete the phase diagram. These measurements will\nreveal  whether at higher pressure the AFM interlayer coupling finally turns into a FM one, as claimed for different Eu 122\ncompounds.\\cite{Ren09,Jeevan11,Nowik11}  In Fig.~\\ref{PhasenDiagrammNeu} we also sketch the arrangement of the Eu$^{2+}$ moments in the high-pressure limit ($x = 1$).\n\\begin{figure}[t]\n \\centering\n  \\includegraphics[width=0.85\\columnwidth]{Fig4.eps}\n   \\caption{Schematic phase diagram of\n   EuFe$_2$(As$_{1-x}$P$_x$)$_2$  indicating the SDW\n   and SC phase. In the preceding panels, we\n    distinguish between the Eu net magnetization along the $a$-axis\n    ($M\\parallel a$) and along the $c$-axis ($M\\parallel c$) resulting\n    from the present data and from Ref.~\\onlinecite{Jeevan11,Nowik11}.\n    The scheme of the Eu$^{2+}$ spin alignment is also shown.\n    In the parent compound, A-type AFM is found with the spins being slightly canted towards the $c$-axis with an\n    angle\\cite{Nowik11} of 13$^\\circ$ to the $ab$-plane. Upon the application of\n    pressure the canting angle increases until it saturates\\cite{Nowik11} at 68$^\\circ$ with respect to the $a$-axis ($x = 0.2$). At even higher pressure, the AFM interlayer coupling turns probably into a FM one.}\n    \\label{PhasenDiagrammNeu}\n\\end{figure}\n\n\\section{Discussion}\nThe hysteresis loop observed in our\nEuFe$_2$\\-(As$_{0.88}$P$_{0.12}$)$_2$ single crystal measurements\nconstitute the key to understand the various phase diagrams of\nEu 122 pnictides proposed in literature. Since contributions from\nthe $c$-direction are unavoidable in polycrystalline samples, a\nhysteresis is always seen as soon as the canting of the spins is\npronounced enough. In the following we compare the phase\ndiagram of Jeevan {\\it et al.} based on single crystal\nmeasurements\\cite{Jeevan11} with that of Nowik {\\it et al.} who\nperformed measurements on polycrystalline samples.\\cite{Nowik11}\nThe former observe AFM Eu$^{2+}$ ordering in the superconducting\nphase, whereas the latter group claims the coexistence of FM and\nsuperconductivity. The statements do not contradict each other\nsince Jeevan {\\it et al.} refer to the ordering of the Eu$^{2+}$\nmoments in the $ab$-plane. If they measure along the $c$-axis,\nthey will also yield a phase diagram with the Eu ordering along\nthe $c$-axis; for this direction superconductivity coexists with\nFM. We sketch such a phase diagram in Fig.~\\ref{PhasenDiagrammNeu},\nwhere we distinguish between the Eu net magnetization along the\n$a$- and $c$-axis. This causes implications that are in particular\ninteresting for theoretical considerations about the interplay of\nmagnetism and superconductivity, as superconductivity in pnictides\nis often considered as quasi-two-dimensional.\n\nOn the basis of in-plane magnetization measurements of\nEuFe$_2$\\-(As$_{1-x}$P$_x$)$_2$ single crystals, Jeevan {\\it et\nal.}\\cite{Jeevan11} proposed a phase diagram that includes FM\nEu$^{2+}$ ordering for high P content. Assuming that the\nsample was perfectly oriented with $H\\parallel ab$, this suggests the\nspin alignment displayed in Fig.~\\ref{PhasenDiagrammNeu} for $x = 1$. This is supported by the fact that the hysteresis continues to open  with increasing P\nsubstitution,\\cite{Jeevan11} while the canting angle\nalready saturates at $x = 0.2$.\\cite{Nowik11}\n\nThere is still some discussion about the actual coupling mechanism\nbetween the Eu layers. Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction might play an important\nrole (although it is not clear what happens in the superconducting\nstate), but other mediation via the Fe layers is also possible,\nsince direct exchange and dipolar interaction can be ruled out.\nWhen going from EuFe$_2$As$_2$ to EuAs$_2$P$_2$ the $a$ and $c$\naxes decrease by 2.3\\% and 7.3\\%, respectively.\\cite{Feng10} The reduction of\nthe unit-cell  volume and the change of the Fermi surface with\npressure\\cite{Wu11,Thirupathaiah11} could turn an interlayer RKKY coupling  from AFM to FM.\\cite{Ren09}\n\nField-dependent magnetization measurements ranging up to several\nTesla can be also explained by the Eu$^{2+}$ spin alignment\ndisplayed in Fig.~\\ref{PhasenDiagrammNeu} and indicate FM interlayer coupling at higher pressure. In EuFe$_2$As$_2$ the\nmagnetic behaviors for $H\\parallel ab$ and  $H\\parallel c$\ndiffer significantly.\\cite{Jiang09} While in both directions\nsaturation is achieved around 1~T, a step-like magnetization curve\nappears only in the case of $H\\parallel ab$. At low field\nstrength, the Eu$^{2+}$ spins start to realign, but keep their AFM\norientation in adjacent planes; above a kind of spin-flip field\nthe orientation of the spins parallel to the external field wins\nover the AFM interlayer coupling. In the case of $H\\parallel c$,\nhowever, we propose that such magnetization behavior does not\nappear since there is no AFM ordering in the $c$-direction that\nhas to be broken (Fig.~\\ref{PhasenDiagrammNeu}).\nAccording to our model, the step-like behavior of the\nmagnetization with $H\\parallel ab$ should become weaker as the canting angle of the\nEu$^{2+}$ spins out of the $ab$-plane grows with pressure. For\nlarger canting angles, it gets more and more energetically\nfavorable to orient the spins directly parallel to the external\nfield instead of keeping the AFM ordering. In the case of the FM\nordered phase (Fig.~\\ref{PhasenDiagrammNeu}, $x = 1$), the\nstep-like behavior vanishes. Actually, this\nmagnetization dependence was observed for\nEuFe$_{2-x}$Ni$_x$As$_2$, with a weak step in the\ncase of $x = 0.03$, which disappears for higher\ndopings.\\cite{Ren09}\n\nPressure-dependent susceptibility measurements would help to identify the correlations between the Eu$^{2+}$ ordering and other phase transitions.\nIt is quite surprising that the tilting angle of the Eu$^{2+}$ moments stays the same, when the SDW is completely suppressed (Fig.~\\ref{PhasenDiagrammNeu}).\\cite{Nowik11} This infers that the itinerant AFM phase of the FeAs layers, with the magnetic moments being aligned within the $ab$-plane,\\cite{Johnston10} does influence the Eu$^{2+}$ ordering. It is unlikely that the minor structural phase transition between the orthorhombic and tetragonal phase, taking place at approximately the same P content, has some influence on the canting angle.\n\nUnderstanding the alignment of the Eu$^{2+}$ magnetic moments will also\nreveal more information on the interplay between magnetism\nand superconductivity. One interesting feature appearing in the Eu\n122 pnictides is a resistivity re-entrance in the superconducting\nphase near the Eu$^{2+}$ magnetic ordering temperature.\\cite{Jiang09b,Ren09b,Jeevan11}\nIn Eu(Fe$_{0.89}$Co$_{0.11}$)$_2$As$_2$ the Eu$^{2+}$ are ordered AFM in the $a$-direction. The superconducting\nre-entrance can be suppressed with an\nexternal magnetic field in the $ab$-plane, while a magnetic field\n$H\\parallel c$ has no influence on the re-entrance.\\cite{Jiang09b}\nWe do not understand yet, why the AFM Eu$^{2+}$ ordering\ndestroys the superconductivity, while it can coexist with the field-induced FM.\nNevertheless our model allows us to explain why\na magnetic field applied along the $c$-direction has no appreciable influence: it\nonly weakens the spin components in the $ab$-plane without destroying\nthe AFM ordering itself. It should also be noted that for several\nEu 122 compounds, susceptibility measurements show additional\nfeatures below the pronounced AFM or FM Eu ordering, which could\nhave the same origin as the resistivity re-entrance.\\cite{Jiang09b, Nowik11, Ren09} Here we want\nto point out that Ahmed {\\it et al.}\\cite{Ahmed10} observe a significant\nenhancement of the spin moment on the Fe $3d$ electrons in\nEuFe$_2$(As$_{0.73}$P$_{0.27}$)$_2$ at $T=18$~K. Together with\nthe ordered Eu$^{2+}$ moments, the internal magnetic field can then exceed the superconducting upper critical field leading to the re-entrance of resistivity.\n\n\\section{Concluding Remarks}\nAnalyzing measurements of the magnetic properties of various Eu\n122 compounds we propose that the alignment of the localized\nEu$^{2+}$  magnetic moments in EuFe$_2$\\-(As$_{1-x}$P$_x$)$_2$\nchanges with pressure. In the parent compound\nEuFe$_2$As$_2$, the spins are not simply antiferromagnetically\naligned in adjacent $ab$-planes but canted by a few degrees,\ncausing a small ferromagnetic component in $c$-direction. When the\ninterlayer coupling changes with pressure, the canting increases until the SDW in the FeAs\nsubsystem is suppressed and superconductivity sets in. There is a\ncoexistence of superconductivity with antiferromagnetism along the\n$a$-direction and ferromagnetism along the $c$-axis. At even\nhigher pressure, the interlayer coupling switches to a ferromagnetic\none.\n\\\\\n\\begin{acknowledgments}\nWe thank: N. Bari\\v{s}i\\'{c} and R. Beyer  for helpful discussions; the\nAlexander von Humboldt Foundation, DFG (SPP1458) and the\nLan\\-des\\-gra\\-du\\-ier\\-ten\\-f\\\"or\\-derungs\\-ge\\-setz\nBaden-W\\\"urttemberg for financial support. \n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\subsection{Background}\nConsider the task of image classification over a set of classes $[K] := \\{1,2,\\ldots,K\\}$, we have tuples of images and labels: $({\\bm{x}}, {\\bm{y}}) \\in {\\mathcal{X}}\\times{\\mathcal{Y}}$, with ${\\bm{y}}\\in\\{0,1\\}^K$ to denote one-hot encoded label, and $t \\in [K]$ to denote the ground-truth class. The goal is to learn a parametric mapping function $f({\\bm{x}}; \\theta) : {\\mathcal{X}} \\mapsto {\\mathcal{Y}}$ where $\\theta \\in \\Theta$  is characterized by a neural network. We learn the parameters $\\theta$ via Empirical Risk Minimization of the surrogate loss function, typically optimized using some variants of Stochastic Gradient Descent:\n\\begin{equation}\n    \\label{eq:erm}\n    \\theta^* = \\argmin_{\\theta \\in \\Theta} {\\mathcal{L}}({\\bm{y}}, f({\\bm{x}}; \\theta)),\n\\end{equation}\nwhere ${\\mathcal{L}}$ is the cross-entropy loss ${\\mathcal{H}}({\\bm{y}}, {\\bm{q}})=\\sum_{i=1}^{K} -{y}_i \\log {q}_i$ between one-hot encoded label ${\\bm{y}}\\in{\\mathcal{Y}}$, and network output distribution ${\\bm{q}}=f({\\bm{x}}; \\theta)$ computed by applying $\\mathrm{softmax}$ function over the logits output ${\\bm{z}}$:\n\\begin{equation}\\label{eq:softmax}\n    {q}_i = \\mathrm{softmax}({z}_i) = \\frac{\\exp({z}_i)}{\\sum_{j=1}^K \\exp({z}_j)}.\n\\end{equation}\nWe could also apply temperature $T$ on the $\\mathrm{softmax}$ to get a more smooth output distribution $\\tilde{{q}}_i = \\mathrm{softmax}({z}_i\/T)$. \n\nGradient of a single-sample \\emph{w.r.t.} logit ${z}_i$ is given by:\n\\begin{equation}\n    \\label{eq:grad}\n    \\frac{\\partial{\\mathcal{L}}}{\\partial {z}_i} = {q}_i - {y}_i ~\\big(=\\partial_i\\big).\n\\end{equation}\n\\subsection{Benefit from Label Smoothing}\n\nLabel Smoothing~\\citep{szegedy2016rethinking} is a technique to soften one-hot encoded label ${\\bm{y}}$ by a factor of $\\epsilon$, such that the modified label becomes:  $\\tilde{{y}}_i^{\\mathrm{LS}}=(1-\\epsilon){y}_i + \\epsilon\/K$. Label smoothing mitigates the over-confidence issue of neural networks, and improves model calibration~\\citep{muller2019does}.\nKnowledge Distillation~($\\mathrm{KD}$) on the other hand, uses an additional teacher model's predictions ${\\bm{p}}$:\n\\begin{equation}\\label{eq:dist}\n\\begin{aligned}\n    \\theta^* &=\\argmin_{\\theta \\in \\Theta} {\\mathcal{L}}^{\\mathrm{KD}}({\\bm{y}}, {\\bm{p}}, f({\\bm{x}}; \\theta), \\lambda, T)\\\\\n    &=(1-\\lambda){\\mathcal{H}}({\\bm{y}}, {\\bm{q}}) + \\lambda {\\mathcal{H}}(\\tilde{{\\bm{p}}}, \\tilde{{\\bm{q}}}),\n\\end{aligned}\n\\end{equation}\nwhere $\\lambda\\in[0,1]$ is a hyper-parameter; $\\tilde{{\\bm{q}}}$ and $\\tilde{{\\bm{p}}}$ are softened student and teacher's predictions by applying temperature $T$ in~\\Eqref{eq:softmax}. Logits gradient for $\\mathrm{KD}$ is given by:\n\\begin{equation}\\label{eq:dist_grad}\n    \\frac{\\partial{\\mathcal{L}}^{\\mathrm{KD}}}{\\partial z_i} = (1-\\lambda)(q_i - y_i) + \\frac{\\lambda}{T}(\\tilde{{q}}_i - \\tilde{{p}}_i) ~\\big(=\\partial^{\\mathrm{KD}}_i\\big).\n\\end{equation}\n\\citet{yuan2019revisit} established the connection between $\\mathrm{KD}$ and label smoothing --\nIn terms of gradient propagation, Knowledge Distillation is equivalent to Label Smoothing, when temperature $T=1$, and teacher's probability distribution ${\\bm{p}}$ follows a uniform distribution.\nIn other words, we can view $\\mathrm{KD}$ as an adaptive version of label smoothing, suggesting it should inherit most of the benefits from label smoothing, such as model regularization and better calibration, not being over-confident~\\citep{muller2019does}. %\n\nIn the next two subsections, we analyze the unique characteristics of real teacher's output distribution %\n${\\bm{p}}$ over the uniform distribution, and demonstrate how they could potentially facilitate student model's training with distillation.\n\n\n\\subsection{Benefit from Teacher's Prediction Confidence}\\label{sec:analysis_re-weight}\nOne important characteristic of the teacher distribution ${\\bm{p}}$ is that the prediction (confidence) ${p}_t$ on the ground-truth class $t$ is not a constant and varies across examples. Compared to vanilla loss function in~\\Eqref{eq:erm}, we find that $\\mathrm{KD}$ performs gradient rescaling in the logits space. Following is the ratio of gradients (~\\cref{eq:grad,eq:dist_grad} ) from the two losses:\n\\begin{equation}\\label{eq:grad_scale}\n\\begin{aligned}\n\\omega_i = \\partial^{\\mathrm{KD}}_i \\big\/ \\partial_i = (1-\\lambda) + \\frac{\\lambda}{T}\\left( \\frac{\\tilde{{q}}_i - \\tilde{{p}}_i}{{q}_i - {y}_i}\\right).\n\\end{aligned}\n\\end{equation}\n\nNext, we show that $\\mathrm{KD}$ performs example re-weighting based on teacher model's prediction confidence ${p}_t$ on the ground-truth class. The gradient rescaling factor $\\omega_i$ is larger on average, when teacher is more confident on making the right prediction. More specifically, we state the following:\n\n\n\n\\begin{prop}[\\textbf{Example Re-weighting}]\\label{prop:conf}\nGiven any example $({\\bm{x}}, {\\bm{y}}) \\in {\\mathcal{X}} \\times {\\mathcal{Y}}$, let $\\tilde{{p}}_t = \\tilde{{q}}_t + \\tilde{{c}}_t + \\eta$, where $\\tilde{{c}}_t > 0$ is teacher's relative prediction confidence on the ground-truth $t\\in [K]$ and $\\eta$ is a zero-mean random noise. Then the gradient rescaling factor for all classes by applying Knowledge Distillation is given by:\n$$\\mathbb{E}_\\eta\\left[\\sum_{i\\in[K]}|\\partial^{KD}_i|\\Big\/\\sum_{i\\in[K]}|\\partial_i|\\right] = (1 - \\lambda) + \\frac{\\lambda}{T}\\left(\\frac{\\tilde{{c}}_t}{1 - {q}_t}\\right).$$\n\\end{prop}\n\n\\begin{proof}\nWe first consider the ground-truth class $t\\in[K]$. Using $\\tilde{{p}}_t = \\tilde{{q}}_t + \\tilde{{c}}_t + \\eta$ and $\\mathbb{E}[\\eta] = 0$ in \\eqref{eq:grad_scale}, we get:\n\\begin{align*}\n\\mathbb{E}_{\\eta}[|\\omega_t|] &= (1-\\lambda) + \\frac{\\lambda}{T}\\left(\\frac{\\tilde{{c}}_t}{1 - {q}_t}\\right)\n\\end{align*}\n\nNow, sum of the incorrect class gradients is given by\\footnote{Under the assumption of having a better quality teacher model, we assume $p_t > q_t$, and correspondingly $q_i \\ge p_i,~\\forall i \\in [K]\\backslash t$.}:\n\n\n\\begin{align*}\n\\sum_{i \\neq t} |\\partial^{\\mathrm{KD}}_i| &= \\sum_{i\\neq t}\\big[ (1-\\lambda){{q}}_i + \\frac{\\lambda}{T}(\\tilde{{q}}_i - \\tilde{{p}}_i)\\big]\\\\\n& = (1-\\lambda)(1-{{q}}_t) + \\frac{\\lambda}{T}(\\tilde{{p}}_t - \\tilde{{q}}_t) = |\\partial^{\\mathrm{KD}}_t|\n\\end{align*}\n\nPenultimate equality follows from ${\\bm{q}},~{\\bm{p}}$ and $\\tilde{{\\bm{q}}}$ being probability masses. Similarly applies for $\\partial_i$, and hence the proof.\n\n\n\n\n\n\n\n\\end{proof}\nAt a given snapshot during training, we could assume ${\\tilde{{c}}_t}$ to be a constant for all examples. Then for any pairs of examples $({\\bm{x}}, {\\bm{y}})$ and $({\\bm{x}}', {\\bm{y}}') \\in {\\mathcal{X}} \\times {\\mathcal{Y}}$, if the teacher is more confident on one of them, i.e., ${p} > {p}'$, then the average $\\omega$ for all classes will be greater than $\\omega'$ according to Proposition~\\ref{prop:conf}.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figures\/p_t-w_t.png}\n    \\caption{When applying $\\mathrm{KD}$ for ResNet-20 student model with ResNet-56 as the teacher on CIFAR-100, we plot $\\tilde{{p}}_t$ vs. $\\omega_t$ (in log scale) with 10K samples at the end of training. Clearly, we can see that the two are positively correlated.}\n    \\label{fig:p_t-w_t}\n\n\\end{figure}\n\n\n\nIn Figure~\\ref{fig:p_t-w_t}, we plot the relationship between $\\omega_t$ and ${p}_t$ at the end of training ResNet-20~\\citep{he2016deep} student model with ResNet-56 as the teacher on CIFAR-100~\\citep{krizhevsky2009learning} dataset (more details of this dataset can be found in Suppl. Section~\\hyperref[sec:si-detail]{A}). The plot shows a clear positive correlation between the two. Also, we found the correlation to be stronger at the beginning of training.\n\n\nIn~\\citep{furlanello2018born}, the authors conjecture that example weight is associated with the largest value in ${\\bm{p}}$. However, in the above proof, we show that once the teacher makes a wrong prediction, using the largest value gives a contradictory result. It is also trivial to prove that, when only having two classes, that is $\\omega_{i \\neq t} = \\omega_t$, the only effect of using $\\mathrm{KD}$ is example re-weighting. So we can regard the use of $\\mathrm{KD}$ on binary classification~\\citep{Anil2018Large} as taking the binary log-loss, and multiply with the weight $\\omega_t$.\n\nIn summary, we think incorporating teacher's supervision in knowledge distillation has an effect of example re-weighting, and the weight is associated with teacher's prediction confidence ${p}_t$ on the ground-truth. Weight will be higher when ${p}_t$ is larger. Alternatively, this suggests that $\\mathrm{KD}$ would assign larger weights to training examples that are considered easier from teacher's perspective, and vice versa; which has a similar flavor to Curriculum Learning. \\citet{bengio2009curriculum} suggests this may not only speedup training convergence, but also helps to reach a better local minima. Also, according to~\\cite{roux2016tighter}, re-weighting examples during training with model prediction confidence results in a tighter bound on the classification error, leading to better generalization.\n\n\\begin{figure*}[t]\n    \\centering\n    \\begin{subfigure}[b]{0.22\\textwidth}\n            \\includegraphics[width=\\textwidth]{figures\/heatmap-k=100_t=3.png}\n            \\subcaption{}\n            \\label{fig:heatmap-k=100_t=3}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.22\\textwidth}\n            \\includegraphics[width=\\textwidth]{figures\/heatmap-k=100_t=10.png}\n            \\subcaption{}\n            \\label{fig:heatmap-k=100_t=10}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.22\\textwidth}\n            \\includegraphics[width=\\textwidth]{figures\/heatmap-k=10-t=10.png}\n            \\subcaption{}\n            \\label{fig:heatmap-k=10-t=10}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.26\\textwidth}\n            \\includegraphics[width=\\textwidth]{figures\/heatmap-cosine-sims.png}\n            \\subcaption{}\n            \\label{fig:heatmap-cosine-sims}\n    \\end{subfigure}\n    \\caption{Using 10K samples from CIFAR-100 for ResNet-56, we plot Pearson correlations of output probability ${\\bm{p}}$ with varying $\\mathrm{softmax}$ temperature (a) $T=3$, (b) $T=10$, and (c) $T=10$ where only top-10 largest values in ${\\bm{p}}$ are preserved. In (d), we show cosine similarities computed from the weights of the final logits layer. Since every 5 classes are within the same super-class, class correlations can be interpreted as the color `squares' on the block-diagonal.}\n\\end{figure*}\n\n\\subsection{Prior of Optimal Geometry in Logit Layer}\\label{sec:analysis_class_rel}\nFor binary classification, we showed in Section~\\ref{sec:analysis_re-weight} that $\\mathrm{KD}$ is essentially performing example re-weighting. However, for multi-class classification, we argue that $\\mathrm{KD}$ also leverages relationship between classes as captured by teacher's probability mass distribution ${\\bm{p}}$ over the incorrect classes. As argued by~\\citet{hinton2015distilling}, on MNIST dataset, model assigns relatively high probability for class `7', when the ground-truth class is `2'. In this section, we first confirm their hypothesis using empirical studies. Then, we provide new insights to interpret the class relationship as a prior on optimal geometry in student's last logit layer.\n\nTo illustrate that the teacher's distribution ${\\bm{p}}$ captures class relationships, we train ResNet-56 on CIFAR-100 dataset. CIFAR-100 contains 100 classes over 20 super-classes, with each super-class containing 5 sub-classes. Figures~\\ref{fig:heatmap-k=100_t=3} and~\\ref{fig:heatmap-k=100_t=10} show the heatmap for Pearson correlation coefficient on teacher's distribution ${\\bm{p}}$ for different temperatures. We sort the class indexes to ensure that the 5 classes from the same super-class appear next to each other. We observe in Figure~\\ref{fig:heatmap-k=100_t=3} that with lower temperature, there's no pattern on the heatmap showing class relationships. But as we increase the temperature in Figure~\\ref{fig:heatmap-k=100_t=10}, classes within the same super-class clearly have high correlations to each other, as seen in the block diagonal structure. This observation verifies that teacher's distribution ${\\bm{p}}$ indeed reveals class relationships, with proper tuning on the softmax temperature. %\n\nBefore diving into the details of geometric interpretation, we recall the case of label smoothing~\\citep{szegedy2016rethinking}.\n\\begin{itemize}\n    \\item From an optimization point of view,~\\citet{he2019bag} showed that there is an optimal constant margin $\\log(K(1-\\epsilon)\/\\epsilon + 1)$, between the logit of the ground-truth ${z}_t$, and all other logits ${z}_{-t}$, using a label smoothing factor of $\\epsilon$. For fixed number of classes $K$, the margin is a monotonically decreasing function of $\\epsilon$.\n    \\item From geometry perspective, \\citet{muller2019does} showed the logit ${z}_k = {\\bm{h}}^\\top {\\bm{w}}_k$ for any class $k$ is a measure of squared Euclidean distance $\\|{\\bm{h}} - {\\bm{w}}_k\\|^2$ between the activations of the penultimate layer ${\\bm{h}}$\\footnote{Here ${\\bm{h}}$ can be concatenated with a ``1'' to account for the bias.}, and weights ${\\bm{w}}_k$ for class $k$ in the last logit layer. \n\\end{itemize}\nThe above results suggests that label smoothing encourages $\\|{\\bm{h}} - {\\bm{w}}_t\\|^2 \\ge \\|{\\bm{h}} - {\\bm{w}}_{-t}\\|^2$, and pushes all the other incorrect classes equally apart.\nFollowing a similar proof technique, we extend the above results to knowledge distillation:\n\\begin{prop}\\label{prop:distance}\nGiven $({\\bm{x}}, {\\bm{y}}) \\in {\\mathcal{X}} \\times {\\mathcal{Y}}$, at the optimal solution of the student for the final layer logits ${\\bm{w}}^*_k,~\\forall k\\in[K]$ at $T=1$, Knowledge Distillation enforces different inter-class distances based on teacher's probability distribution ${\\bm{p}}$: \n$$\\|{\\bm{h}} - {\\bm{w}}^*_i\\|^2 < \\|{\\bm{h}} - {\\bm{w}}^*_j\\|^2~~~\\text{iff}~~~{p}_i > {p}_j,~\\forall i,j \\in [K]\\backslash t$$\n\\end{prop}\n\n\\begin{proof}\nAt the optimal solution of the student, equating gradient in~\\Eqref{eq:dist_grad} to $0$, for $T=1$, we get:\n\\begin{equation}\n    {q}^*_k=\n    \\begin{cases}\n      \\lambda{p}_k + (1-\\lambda) & \\text{if}\\ k=t, \\\\\n      \\lambda{p}_k & \\text{otherwise}.\n    \\end{cases}\n\\label{eq:optimal_stud}\n\\end{equation}\n\nUsing a similar proof technique as~\\citet{muller2019does}, $\\|{\\bm{h}} - {\\bm{w}}^*_k\\|^2 = \\|{\\bm{h}}\\|^2 + \\|{\\bm{w}}^*_k\\|^2 - 2 {\\bm{h}}^\\top{\\bm{w}}^*_k$, where ${\\bm{h}}$ is the penultimate layer activations, and ${\\bm{w}}^*_k$ are the weights of the last logits layer for class $k \\in [K]$. Note that $\\|{\\bm{h}}\\|^2$  is factored out when computing the $\\mathrm{softmax}$, and $\\|{\\bm{w}}^*_k\\|^2$ is usually a (regularized) constant across all classes.\n\nEquating ${z}^*_k={\\bm{h}}^\\top{\\bm{w}}^*_k$, and using the property $\\mathrm{softmax}({\\bm{z}}) = \\mathrm{softmax}({\\bm{z}} + c),~\\forall c\\in{\\mathbb{R}}$, we get:\n\\begin{align*}\n{q}^*_k &= \\mathrm{softmax}({z}^*_k) = \\mathrm{softmax}({\\bm{h}}^\\top{\\bm{w}}^*_k)\\\\ &=\\mathrm{softmax}\\Big(-\\frac{1}{2}\\|{\\bm{h}} - {\\bm{w}}^*_k\\|^2\\Big)\n= \\lambda{p}_k,~\\forall k \\in [K]\\backslash t\n\\end{align*}\nThe last equality follows from \\eqref{eq:optimal_stud}, and thus proves the claim on class partition prior geometry at optimality.\n\\end{proof}\nFrom Figure~\\ref{fig:heatmap-k=100_t=10}, we know that the teacher assigns higher probability to the classes within the same super-class, and hence $\\mathrm{KD}$ encourages hierarchical clustering of the logits layer weights based on the class relationships. \n\n\n\\subsection{Summarizing Effects of Knowledge Distillation}\\label{sec:summary_effects}\nPrimarily, KD brings a regularization\/calibration effect by introducing smoothed teacher distribution, although this effect is not considered as knowledge.\nBesides, we believe there are two types of knowledge teacher model will distill to its student -- real teacher's probability distribution ${\\bm{p}}$ not only benefits the student via \\emph{confidence on ground-truth class} to re-weight examples, but also from its probability mass on the \\emph{incorrect classes}.\nIntuitively, these values reflect class relationships, therefore provide the student with more guidance.\nWe further interpret these values as applying different label smoothing factor $\\epsilon$ for the incorrect classes.\nThe difference in $\\epsilon$ encourages student's optimal inter-class distance to be different for different classes.\nIn other words, the distillation loss will get lower if more desired output logit layer's geometric inequalities hold.\nAs a result, two sources of knowledge complementary to each other, which could potentially facilitate the student model's training process and further improve model generalization.\n\\subsection*{A. Experimental details}\\label{sec:si-detail}\n\n\\begin{figure*}[t]\n    \\centering\n    \\begin{subfigure}[b]{0.3\\textwidth}\n            \\includegraphics[width=\\textwidth]{figures\/synthetic_vis-sim=00-m=0.png}\n            \\subcaption{}\n            \\label{fig:synthetic-sim=0.0-m=0}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.3\\textwidth}\n            \\includegraphics[width=\\textwidth]{figures\/synthetic_vis-sim=09-m=0.png}\n            \\subcaption{}\n            \\label{fig:synthetic-sim=0.9-m=0}\n    \\end{subfigure}\n    ~\n    \\begin{subfigure}[b]{0.3\\textwidth}\n            \\includegraphics[width=\\textwidth]{figures\/synthetic_vis-sim=09-m=2.png}\n            \\subcaption{}\n            \\label{fig:synthetic-sim=0.9-m=2}\n    \\end{subfigure}\n    \n    \\caption{Visualization of 5K synthetic data points (with input dimensionality $d=2$) on 2-D plane. We use $K=4$, $C=2$, means there are two super-classes, one associate with label \\{0,1\\} and the other one associate with label \\{2,3\\}. We vary $\\tau$ and $M$ and produce 3 plots: (a) $\\tau=0.0$, no sine function is used; (b) $\\tau=0.9$, no sine function is used and (c) $\\tau=0.9$, $M=2$.}\n    \\label{fig:synthetic-sim-m}\n\\end{figure*}\n\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[scale=0.515]{figures\/histogram-pt.pdf}\n\\caption{ On CIFAR-100, we plot the histograms of teacher's confidence on ground-truth ${p}_t$, here teacher model is ResNet-56 with different levels $\\epsilon$ of label smoothing. From left to right, we use: (a) $\\epsilon=0.0$ and $T=1$; (b) $\\epsilon=0.0$ and $T=5$; (c) $\\epsilon=0.1$ and $T=1$ and (d) $\\epsilon=0.1$ and $T=3$. The distribution of ${p}_t$ becomes skewed after enabling label smoothing.}\n\\label{fig:histogram-pt}\n\\end{figure*}\n\n\\begin{figure*}[t]\n    \\centering\n    \\begin{subfigure}[b]{0.3\\textwidth}\n            \\includegraphics[width=\\textwidth]{figures\/heatmap-k=100-t=10-LS=true.png}\n            \\subcaption{}\n    \\end{subfigure}\n    ~~~~~~~~~~~~~~~~\n    \\begin{subfigure}[b]{0.355\\textwidth}\n            \\includegraphics[width=\\textwidth]{figures\/heatmap-cosine-sims-LS=true.png}\n            \\subcaption{}\n    \\end{subfigure}\n    \n    \\caption{Using 10K samples from CIFAR-100 for ResNet-56 trained with smoothed labels ($\\epsilon=0.1$), we plot (a) Pearson correlations of ${\\bm{p}}$ when $T=10$ and (b) Cosine similarities computed from the weights of the final logits layer. Since every 5 classes are within the same super-class, class correlations can be interpreted as the color `squares' on the diagonal.}\n    \\label{fig:heatmap-label-smooth-failure}\n\\end{figure*}\n\n\\paragraph{Implementation of $\\mathrm{KD}$.} \nIn practice, the gradients from the RHS of~\\Eqref{eq:dist_grad} are much smaller compare to the gradients from LHS when temperature $T$ is large. Thus, it makes tuning the balancing hyper-parameter $\\lambda$ become non-trivial. To mitigate this and make the gradients from two parts in the similar scale, we multiply $T^2$ to the RHS of~\\Eqref{eq:dist_grad}, as suggested in~\\citep{hinton2015distilling}.\n\n\\paragraph{Implementation of $\\mathrm{KD}\\text{-sim}$.}\nWhen synthesizing the teacher distribution for $\\mathrm{KD}\\text{-sim}$, we use ${\\bm{\\rho}}^{\\text{sim}} = \\mathrm{softmax}(\\hat{{\\bm{w}}}_t \\Hat{{\\bm{W}}}^\\top)$, where $\\Hat{{\\bm{W}}}$ is the $l_2$-normalized logit layer weights and $\\hat{{\\bm{w}}}_t$ is the $t$-th row of $\\Hat{{\\bm{W}}}$. However, the cosine similarities computed for $\\mathrm{softmax}$ are limited in the range of $[0,1]$. Therefore the resulting distribution is highly likely to be uniform. To mitigate this and bring more resolution to be cosine similarities, we use the following:\n$${\\bm{\\rho}}^{\\text{sim}} = \\mathrm{softmax}((\\hat{{\\bm{w}}}_t \\Hat{{\\bm{W}}}^\\top)^{\\alpha}\/\\beta).$$\nHere $\\alpha<1$ is a hyper-parameter to amplify the resolution of cosine similarities, $\\beta$ is another hyper-parameter indicating the temperature for $\\mathrm{softmax}$.\n\n\\begin{table}[t!]\n    \\begin{subtable}[h]{0.45\\textwidth}\n        \\centering\n        \\begin{tabular}{l | l}\n        {Method} & {Hyper-parameter setting}\\\\\n        \\hline\n        LS & $\\epsilon=0.3$ for any $\\tau$.\\\\\n        \\hline\n        \\multirow{6}{*}{KD} & $\\lambda=0.7, T=3$ when $\\tau=0.0$\\\\\n        & $\\lambda=0.7, T=5$ when $\\tau=0.1$\\\\\n        & $\\lambda=0.7, T=2$ when $\\tau=0.2$\\\\\n        & $\\lambda=0.7, T=3$ when $\\tau=0.3$\\\\\n        & $\\lambda=0.7, T=10$ when $\\tau=0.4$\\\\\n        & $\\lambda=0.7, T=5$ when $\\tau=0.5$;\\\\\n        \\hline\n        \\multirow{6}{*}{KD-pt} & $\\lambda=0.7, T=3$ when $\\tau=0.0$\\\\\n        & $\\lambda=0.7, T=5$ when $\\tau=0.1$\\\\\n        & $\\lambda=0.7, T=2$ when $\\tau=0.2$\\\\\n        & $\\lambda=0.7, T=3$ when $\\tau=0.3$\\\\\n        & $\\lambda=0.7, T=10$ when $\\tau=0.4$\\\\\n        & $\\lambda=0.7, T=5$ when $\\tau=0.5$\\\\\n        \\hline\n        KD-sim & $\\lambda=0.7,\\alpha=0.5,\\beta=0.5$ for any $\\tau$\\\\\n\n       \\end{tabular}\n       \\caption{Synthetic}\n       \\label{tb:hparam_synthetic}\n    \\end{subtable}\n    \\hfill\n    \\begin{subtable}[h]{0.45\\textwidth}\n        \\centering\n        \\begin{tabular}{l | l}\n\n        {Method} & {Hyper-parameter setting}\\\\\n        \\hline\n        LS & $\\epsilon=0.1$\\\\\n        KD & $\\lambda=0.3,T=5$\\\\\n        KD-pt & $\\lambda=0.3,T=5$\\\\\n        KD-sim & $\\lambda=0.3,\\alpha=0.3,\\beta=0.3$\\\\\n        KD-topk & $k=25,\\lambda=0.5,T=5$\\\\\n\n       \\end{tabular}\n       \\caption{CIFAR-100}\n       \\label{tb:hparam_cifar}\n    \\end{subtable}\n    \\hfill\n    \\begin{subtable}[h]{0.45\\textwidth}\n        \\centering\n        \\begin{tabular}{l | l}\n\n        {Method} & {Hyper-parameter setting}\\\\\n        \\hline\n        LS & $\\epsilon=0.1$\\\\\n        KD & $\\lambda=0.7,T=20$\\\\\n        KD-pt & $\\lambda=0.2,T=25$\\\\\n        KD-sim & $\\lambda=0.3,\\alpha=0.5,\\beta=0.3$\\\\\n        KD-topk & $k=500,\\lambda=0.5,T=3$\\\\\n\n       \\end{tabular}\n       \\caption{ImageNet}\n       \\label{tb:hparam_imagenet}\n    \\end{subtable}\n    \\hfill\n    \\begin{subtable}[h]{0.45\\textwidth}\n        \\centering\n        \\begin{tabular}{l | l}\n\n        {Method} & {Hyper-parameter setting}\\\\\n        \\hline\n        KD & $\\lambda=0.1,T=50$\\\\\n        KD-topk & $k=100,\\lambda=0.1,T=50$\\\\\n\n       \\end{tabular}\n       \\caption{Penn Tree Bank (PTB)}\n       \\label{tb:hparam_ptb}\n    \\end{subtable}\n     \\caption{Hyper-parameter settings for different methods on different datasets.}\n     \\label{tab:temps}\n\\end{table}\n\n\\paragraph{Synthetic dataset.} \nFirst, we follow the data generation procedure described in Section~\\ref{sec:exp_synthetic}, we get a toy synthetic dataset where we only have input dimensionality $d=2$ with $K=4$ classes and $C=2$ super-classes. Figure~\\ref{fig:synthetic-sim-m} shows a series of scatter plots with different settings of class similarity $\\tau$ and task difficulty $M$. This visualization gives a better understanding of the synthetic dataset and helps us imagine what it will look like in high-dimensional setting that used in our experiments. For the model used in our experiments, besides they are 2-layer network activated by $\\tanh$, we use residual connection~\\citep{he2016deep} and and batch normalization~\\citep{ioffe2015batch} for each layer. Following~\\citep{ranjan2017l2, zhang2018heated}, we found using $l_2$-normalized logits layer weight $\\Hat{{\\bm{W}}}$ and penultimate layer $\\hat{{\\bm{h}}}$ provides more stable results. The model is trained by ADAM optimizer~\\citep{kingma2014adam} for a total of 3 million steps without weight decay and we report the best accuracy.\nPlease refer to Table~\\ref{tb:hparam_synthetic} for the best setting of hyper-parameters.\n\n\\paragraph{CIFAR-100 dataset.} \nCIFAR-100 is a relatively small dataset with low-resolution ($32\\times32$) images, containing $50k$ training images and $10k$ validation images, covering $100$ classes and $20$ super-classes. It is a perfectly balanced dataset -- we have the same number of images per class (\\emph{i.e.,} each class contains $500$ training set images) and $5$ classes per super-class.\nTo process the CIFAR-100 dataset, we use the official split from Tensorflow Dataset\\footnote{\\url{https:\/\/www.tensorflow.org\/datasets\/catalog\/cifar100}}. Both data augmentation \\footnote{\\url{https:\/\/github.com\/tensorflow\/models\/blob\/master\/research\/resnet\/cifar_input.py}}\\footnote{We turn on the random brightness\/saturation\/constrast for better model performance.} for CIFAR-100 and the ResNet model\\footnote{\\url{https:\/\/github.com\/tensorflow\/models\/blob\/master\/research\/resnet\/resnet_model.py}} are based on Tensorflow official implementations. Also, following the conventions, we train all models from scrach using Stochastic Gradient Descent (SGD) with a weight decay of 1e-3 and a Nesterov momentum of 0.9 for a total of 10K steps. The initial learning rate is 1e-1, it will become 1e-2 after 40K steps and become 1e-3 after 60K steps. We report the best accuracy for each model.\nPlease refer to Table~\\ref{tb:hparam_cifar} for the best setting of hyper-parameters.\n\n\\paragraph{ImageNet dataset.} \nImageNet contains about $1.3$M training images and $50k$ test images, all of which are high-resolution ($224\\times224$), covering $1000$ classes. The distribution over the classes is approximately uniform in the training set, and strictly uniform in the test set.\nOur data preprocessing and model on ImageNet dataset are follow Tensorflow TPU official implementations\\footnote{\\url{https:\/\/github.com\/tensorflow\/tpu\/tree\/master\/models\/official\/resnet}}. The Stochastic Gradient Descent (SGD) with a weight decay of 1e-4 and a Nesterov momentumof 0.9 is used. We train each model for 120 epochs, the mini-batch size is fixed to be 1024 and low precision (FP16) of model parameters is adopted. We didn't change the learning rate schedule scheme from the original implementation.\nPlease refer to Table~\\ref{tb:hparam_imagenet} for the best setting of hyper-parameters.\n\n\\paragraph{Penn Tree Bank dataset.}\nWe use Penn Tree Bank (PTB) dataset for word-level language modeling task using the standard train\/validation\/test split by~\\citep{mikolov2010recurrent}. The vocabulary is\ncapped at 10K unique words. We consider the state-of-the-art LSTM model called AWD-LSTM proposed by~\\citet{merity2017regularizing}. The model used several regularization tricks on top of a 3-layer LSTM, including DropConnect, embedding dropout, tied weight, etc. We use different capacity (indicated by hidden size and embedding size) as our Teacher and Student. To be specific, Teacher has a hidden size of 1150 and an embedding size of 400, while Student has a smaller hidden size of 600 and a smaller embedding size of 300. We follow the official implementation\\footnote{\\url{https:\/\/github.com\/salesforce\/awd-lstm-lm}} with simple changes for $\\mathrm{KD}\\text{-topk}$. Besides capacity, we keep the default hyper-parameter as in the official implementation to train our Teacher. However, when training smaller Student model, we follow another implementation\\footnote{\\url{https:\/\/github.com\/zihangdai\/mos}} to: (1) lower the learning rate to 0.2, (2) increase training epochs to 1000, (3) use 0.4 for embedding dropout rate and (4) use 0.225 for RNN layer dropout rate.\n\n\\subsection*{B. Additional Experiments}\\label{sec:si-exp}\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{l|ccc}\n\\toprule\n\\textbf{Method} & \\textbf{\\% top-1 accuracy}\\\\\n\\hline\nStudent & 72.51\\\\\nKD & 75.94\\\\\n\\midrule\nKD-rel & 74.14\\\\\nKD-sim & 74.30\\\\\n\\midrule\nKD-pt+rel & 75.07\\\\\nKD-pt+sim & 75.24\\\\\n\\bottomrule\n\n\\end{tabular}\n\\caption{Performance of $\\mathrm{KD}\\text{-rel}$ on CIFAR-100. We report the mean result for 4 individual runs with different initializations. We use $\\beta_1=0.6,\\beta_2=\\frac{0.1}{4},\\beta_3=\\frac{0.3}{95}$.}\n\\label{tb:exp_kdrel}\n\\end{table}\n\\paragraph{Examine optimal geometry prior effect with class hierarchy.}\nIn section~\\ref{sec:pseudo_kd}, we mentioned the optimal geometry prior effects of $\\mathrm{KD}$ can also be examined using existing class hierarchy. \nSuppose our data has a pre-defined class hierarchy (e.g., on CIFAR-100), we can also use it to examine the optimal geometry prior effects of $\\mathrm{KD}$. To be specific, let ${\\mathbb{S}}_t \\subset [K]\\backslash t$ denote the other classes that share same parent of $t$.\nWe simply assign different probability masses to different types of classes:\n\\begin{equation}\\label{eq:kdrel}\n\\rho_i^{\\textrm{rel}} =  \\begin{cases}\n    \\beta_1 & \\text{if } i = t,\\\\\n    \\beta_2 & \\text{if } i \\in {\\mathbb{S}}_t,\\\\\n    \\beta_3 & \\text{otherwise},\n        \\end{cases}\n\\end{equation}\nwhere $\\beta_1 > \\beta_2 > \\beta_3$ are a hyper-parameters we could search and optimize, and we name this method as $\\mathrm{KD}\\text{-rel}$. \nAs shown in Table~\\ref{tb:exp_kdrel}, we found $\\mathrm{KD}\\text{-rel}$ performs slightly worse  than $\\mathrm{KD}\\text{-sim}$ on CIFAR-100. The trend is still hold when we compound each effect with $\\mathrm{KD}\\text{-pt}$.\n\n\\subsection{Empirical Improvements for Distillation Quality}\\label{sec:improv_effective}\nThough $\\mathrm{KD}\\text{-sim}$ is able to capture class relationships using different probability masses over the incorrect classes, there's a major drawback -- all the examples from the same class will have the same relationships to the other classes. However, this is not a realistic assumption for most real-world datasets. For instance, on MNIST dataset, only some versions of `2' looks similar to `7'. To overcome this drawback, we propose $\\mathrm{KD}\\text{-topk}$, which simply borrows the top-$k$ largest values of teacher's probability ${\\bm{p}}$, and uniformly distributes the rest of the probability mass to the other classes.\nFigure~\\ref{fig:heatmap-k=10-t=10} shows that only preserving top-$10$ largest values could closely approximate the class correlations as in the full teacher's distribution ${\\bm{p}}$, and is also less noisy.\nThe above finding shows that only a few incorrect classes that are strongly correlated with the ground-truth class are useful for $\\mathrm{KD}$, and the probability mass on other classes are random noise (which is not negligible under high temperature $T$), and only has the effect of label smoothing in expectation.\n\nUsing the above intuition, we test $\\mathrm{KD}\\text{-topk}$ for image classification task on CIFAR-100 and ImageNet. Beyond computer vision, we also test $\\mathrm{KD}\\text{-topk}$ for language modeling task on Penn Tree Bank (PTB) dataset. We apply state-of-the-art LSTM model~\\citep{merity2017regularizing} with different capacities for the teacher and student. Details of PTB dataset and model specifications are in Section~\\hyperref[sec:si-detail]{A} of Suppl. For image classification, the performance of $\\mathrm{KD}\\text{-topk}$ is shown in the last row of Table~\\ref{tb:exp_real_data}. We see that $\\mathrm{KD}\\text{-topk}$ outperforms $\\mathrm{KD}$ in both the datasets. For language modeling, the results are shown in Table~\\ref{tb:exp_real_data_lm}, which shows a similar trend for $\\mathrm{KD}\\text{-topk}$.\nWe plot the performance uplift of $\\mathrm{KD}\\text{-topk}$ along with $k$ in Figure~\\ref{fig:exp_topk}, which suggests that the best performance is achieved with proper tuning of $k < K$, which captures class relationships and also reduces noise. Note that the improvements of $\\mathrm{KD}\\text{-topk}$ over $\\mathrm{KD}$ is simple and free with easy modifications. This is also the reason we omit a more sophisticated comparison with other advanced distillation techniques, such as \\citep{romero2014fitnets, yim2017gift}.\n\n\n\\subsection{Potential Improvements for Training Efficiency}\nVanilla $\\mathrm{KD}$ method requires loading a pre-trained teacher model in-memory, and computing forward-pass to get the full output distribution. While all of our proposed partial-$\\mathrm{KD}$ methods and $\\mathrm{KD}\\text{-topk}$ can be achieved with better computational efficiency when training the student model. In terms of computation cost, we can pre-compute $K\\times K$ class similarity matrix before student training, and directly use it for $\\mathrm{KD}\\text{-sim}$. For $\\mathrm{KD}\\text{-topk}$, we only need the top-k predictions from the teacher, which could be efficiently (approximately) computed using SVD-softmax~\\citep{svdsoftmax}, when the output space is large.\nAlternatively for vanilla $\\mathrm{KD}$, one could also save computation by storing teacher's predictions on-disk, which could again be optimized using our proposed methods. We only need to store a single value, i.e., teacher's confidence on ground-truth class for $\\mathrm{KD}\\text{-pt}$; and top-$k$ predictions for $\\mathrm{KD}\\text{-topk}$. As shown in our experiments in Figure~\\ref{fig:exp_topk}, using $k=10\\%K$, we can achieve comparable performance with vanilla $\\mathrm{KD}$. %\n\n\\subsection{Diagnosis of Failure Cases}\nA good understanding of $\\mathrm{KD}$ enables us to diagnose failure cases.\n\\citet{muller2019does} observed that although label smoothing improves teacher model's quality, it results in a worse student model when applying $\\mathrm{KD}$.\nVerified in CIFAR-100 (see Table~\\ref{tb:label-smooth-failure}), we found the unfavorable distillation performance could be attributed to two factors -- Firstly, as argued by the~\\citet{muller2019does} and illustrated in Figure~\\ref{fig:heatmap-label-smooth-failure} in Suppl., $\\mathrm{LS}$ destroys class relationships. Secondly, we found that the skewed teacher's prediction distribution on the ground-truth (see Figure~\\ref{fig:histogram-pt} in Suppl.) also hinders the effectiveness of $\\mathrm{KD}$, because the example re-weighting effect will be less useful. Results of $\\mathrm{KD}\\text{-sim}$ and $\\mathrm{KD}\\text{-pt}$ from last two columns of Table~\\ref{tb:label-smooth-failure} verify these findings.\nFor another failure case, \\citet{mirzadeh2019improved} showed that the `distilled' student model's quality gets worse as we continue to increase teacher model's capacity. The larger capacity teacher might overfit, and predict high (uniform) confidence on the ground truth on all the examples; thereby hindering the effectiveness of example re-weighting in knowledge distillation. Another explanation could be that there exists an optimal model capacity gap between the teacher and student, which could otherwise result in inconsistency between teacher's confidence on the ground-truth, and the desired example difficulty for the student.\nPerhaps, an `easy' example for larger capacity teacher is overly difficult for the student, due to the large difference in model expressiveness.\n\\section{Introduction} \\input{introduction}\n\n\\section{Related work} \\input{related_work}\n\n\\section{Analysis on mechanism of Knowledge Distillation} \\label{sec:analysis} \\input{analysis}\n\n\\section{Pseudo Knowledge Distillation} \\label{sec:pseudo_kd} \\input{pseudo_kd}\n\n\\section{Experiments} \\label{sec:experiments}  \\input{experiments}\n\n\n\\section{Applications} \\label{sec:applications} \\input{applications}\n\n\\section{Conclusion}  \\label{sec:conclusions} \\input{conclusions}\n\n\n\n\\section{Introduction} \\input{introduction}\n\n\\section{Related Work} \\input{related_work}\n\n\\section{Analyzing Mechanisms of Knowledge Distillation} \\label{sec:analysis} \\input{analysis}\n\n\\section{Isolating Effects by Partial Knowledge Distillation Methods} \\label{sec:pseudo_kd} \\input{pseudo_kd}\n\n\\section{Improving and Diagnosing Knowledge Distillation} \\label{sec:application}  \\input{experiments}\n\n\n\\section{Conclusion}  \\label{sec:conclusions} \\input{conclusions}\n\\newpage\n\n\n\\subsection{Proposed Partial KD Methods}\n\\textbf{Examine Example Re-weighting Effect by KD-pt.}\nLabel smoothing does not have either re-weighting or the class relationships effect, due to its uniform teacher distribution. However, if we borrow ${p}_t$ (prediction on ground truth class $t\\in[K]$) from the real teacher's probability distribution ${\\bm{p}}$, we can synthesize partial-teacher distribution that is able to incorporate example re-weighting effect. More specifically, we craft teacher probability distribution as follows:\n\\begin{equation}\\label{eq:kdpt}\n\\rho_i^{\\textrm{pt}} =  \\begin{cases}\n    {p}_t & \\text{if } i = t,\\\\\n    (1 - {p}_t)\/(K-1) & \\text{otherwise}.\n        \\end{cases}\n\\end{equation}\nFrom Proposition~\\ref{prop:conf}, it is trivial to see that $\\mathrm{KD}\\text{-pt}$ is capable of differentiating weights for different examples. However, it does not capture class relationships, since every class that is not the ground truth has the same probability mass. %\n\n\n\n\\textbf{Examine Optimal Prior Geometry Effect by KD-sim.}\nFollowing the same methodology, we synthesize a teacher distribution that only capture class relationships, and assign the same weight to each example.\nTo achieve this, we use the weights of the last logit layer ${\\bm{W}} \\in {\\mathbb{R}}^{K \\times d}$ from the teacher model to obtain class relationships. We believe the teacher, due to its larger capacity is able to encode class semantics in the weights of the last logit layer.\nThus, we create a distribution ${\\bm{\\rho}}^{\\text{sim}}$ as the $\\mathrm{softmax}$ over cosine similarity\\footnote{In practice, \nbesides tuning the temperature of the $\\mathrm{softmax}$, one could also raise the similarities to a power $<1$ to amplify the resolution of cosine similarities. Please refer to Section~\\hyperref[sec:si-detail]{A} in Suppl. for more details of our implementation.} of the weights: ${\\bm{\\rho}}^{\\text{sim}} = \\mathrm{softmax}(\\hat{{\\bm{w}}}_t \\Hat{{\\bm{W}}}^\\top)$, where $\\Hat{{\\bm{W}}}$ is the $l_2$-normalized logit layer weights, and $\\hat{{\\bm{w}}}_t$ is the $t$-th row of $\\Hat{{\\bm{W}}}$ corresponding to the ground truth class $t\\in[K]$. \nThough other distance metrics could be also used as a measure of class similarity, we leave the discussion of analysing the different choices as future work.\nTo verify our assumption, we check the heatmap of cosine similarities in Figure~\\ref{fig:heatmap-cosine-sims}, which clearly shows a similar pattern as the Pearson correlation of the teacher distribution ${\\bm{p}}$ in Figure~\\ref{fig:heatmap-k=100_t=10}.\nWe call this method $\\mathrm{KD}\\text{-sim}$. \nFrom Propositions~\\ref{prop:conf} and \\ref{prop:distance}, our proposed method, though simple and straightforward, can achieve our purpose of factoring out the class relationships effect.\n\nNote that $\\mathrm{KD}\\text{-sim}$ doesn't require the knowledge of class hierarchy. However, if the hierarchy is available (as in CIFAR-100), we could also synthesize a teacher distribution apriori.\nIn Suppl. Section~\\hyperref[sec:si-exp]{B}, we synthesize ${\\bm{\\rho}}$ by setting different values for (1) ground-truth class $t$, (2) classes within the same super-class of $t$, and (3) other incorrect classes. The quality of resulting method is slightly worse than $\\mathrm{KD}\\text{-sim}$ but can still improve student model's quality.\n\n\\textbf{Compounded Effects.}\nTo enjoy the benefits from multiple effects and approximate the functionality of $\\mathrm{KD}$, we could combine the two partial-KD methods introduced above. We explore simple linear combination of synthetic teacher's distribution -- $(1-\\alpha) {\\bm{\\rho}}^{\\text{pt}} + \\alpha {\\bm{\\rho}}^{\\text{sim}}$ and name the method $\\mathrm{KD}\\text{-pt+sim}$. %\nIt is easy to verify that this compounded method performs example re-weighting and injects optimal prior geometry through class relationships.\n\nIn the next section, we evaluate our proposed partial-distillation methods on synthetic and real-world datasets to better understand how much each of these effects could benefit the student. Based on our understanding, we propose a novel distillation method that only adopts top-$k$ largest values from the teacher distribution ${\\bm{p}}$. In Section~\\ref{sec:improv_effective}, we illustrate how this method could reduce noise in ${\\bm{p}}$ (see Figure~\\ref{fig:heatmap-k=10-t=10}), and also result in a better quality student model.\n\n\\subsection{Empirical Studies}\n\n\n\\subsubsection{Synthetic Dataset}\\label{sec:exp_synthetic}\nPerformance of $\\mathrm{KD}$ is dependent on the dataset properties. To this end, a natural question is -- \\emph{Does $\\mathrm{KD}$ perform only example re-weighting when all the classes are uncorrelated to each other?} We proved this to be true for binary classification (Section~\\ref{sec:analysis_re-weight}). To answer the same for multi-class classification task, we generate synthetic dataset, where we can control the class similarities within the same super-class.\n\n\\textbf{Setup.}\nInspired by~\\citep{ma2018modeling}, we synthesize a classification dataset with $K$ classes, and $C$ super-classes. Each super-class will have equal number of $K\/C$ classes, and each class will be assigned with a basis vector. These basis vectors are carefully generated, so that we could control the class correlations within the same super-class. More specifically, we generate a single data-point as follows:\n\\begin{enumerate}\n    \\item Randomly sample $C$ orthonormal basis vectors, denoted by ${\\bm{u}}_i \\in{\\mathbb{R}}^d~\\forall i\\in[C]$.\n    \\item For each orthonormal basis ${\\bm{u}}_i$, we sample $(K\/C-1)$ unit vectors ${\\bm{u}}_j\\in{\\mathbb{R}}^d$ that are $\\tau$ cosine similar to ${\\bm{u}}_i$.\n    \\item Randomly sample an input data point in $d$-dimensional feature space ${\\bm{x}}\\sim\\mathcal{N}_d(\\mathbf{0}, \\mathbf{I})$.\n    \\item Generate one-hot encoded label ${\\bm{y}}\\in{\\mathcal{Y}}$ with target: $t=\\argmax_{k\\in[K]} \\big({\\bm{u}}^\\top_k \\hat{{\\bm{x}}} + \\sum_{m=1}^{M} \\sin({a}_m {\\bm{u}}^T_k \\hat{{\\bm{x}}} + {b}_m) \\big)$, where $\\hat{{\\bm{x}}}$ is the $l_2$-normalized ${\\bm{x}}$;  ${\\bm{a}},{\\bm{b}}\\in{\\mathbb{R}}^M$ are arbitrary constants; and we refer to the controlled $\\sin$ complexity term $M \\in {\\mathbb{Z}}^+$ as \\emph{task difficulty}.\n\\end{enumerate}\n\nAfter producing basis vectors with procedure (1) and (2), we run procedure (3) and (4) for $|\\mathcal{D}|$ times with fixed basis vectors to generate a synthetic dataset $\\mathcal{D}=\\{({\\bm{x}},{\\bm{y}})\\}$. By tuning the cosine similarity parameter $\\tau$, we can control the classes correlations within the same super-class. Setting task-difficulty $M=0$ generates a linearly separable dataset; and for $M>0$, more non-linearities will be introduced by the $\\sin$ function (see Figure~\\ref{fig:synthetic-sim-m} in Suppl. for visualization on a toy example).\nIn the following experiments, we set input dimension $d=500$ with $K=50$ classes, and $C=10$ super-classes. We use $|\\mathcal{D}|=500k$ data-points for training, and $|\\mathcal{D_{\\mathrm{valid}}}|=50k$ for validation. We use a simple 2-layer fully-connected neural network with $\\tanh$ activation, and hidden layer dimensions $64$ for the student, and $128$ for the teacher. Finally, we set $M=10$, hoping that this is the right task difficulty trade-off (i.e., not too easy, but hard enough to have a large margin between the two models for $\\mathrm{KD}$).\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.7\\columnwidth]{figures\/synthetic_results_new.png}\n    \\caption{Best performance over 4 runs on a synthetic dataset with different levels of controlled class similarities.}\n    \\label{fig:exp_synthetic_data}\n\\end{figure}\n\n\n\\textbf{Results and Analysis.}\nFigure~\\ref{fig:exp_synthetic_data} shows the classification accuracy on the validation set of all the methods, when varying $\\tau$ from $0.0 \\rightarrow 0.4$.\nWe notice a large margin between the teacher and student.\nLabel Smoothing ($\\mathrm{LS}$) marginally helps the student, while Knowledge Distillation ($\\mathrm{KD}$) benefits significantly.\nInterestingly, regardless of the class similarity $\\tau$; $\\mathrm{KD}\\text{-pt}$ has comparable performance with $\\mathrm{KD}$, suggesting that example re-weighting effect plays a major role in distillation. Thus, even when the classes are uncorrelated, $\\mathrm{KD}$ could still benefit the student through example re-weighting.\nNote that for this task, the data-points which are close to the decision boundary are harder to classify, and can be regarded as difficult examples.\nFurthermore, when increasing $\\tau$, we see a significant improvement in performance of $\\mathrm{KD}\\text{-sim}$, suggesting that the injected prior knowledge of class relationships boosts student model's quality. %\n\n\\subsubsection{Real-world Datasets}\nWe use two popular image classification datasets -- CIFAR-100~\\citep{krizhevsky2009learning} and ImageNet~\\citep{russakovsky2015imagenet} to analyze the quality of our proposed partial-distillation methods, and also to verify if we could approximate the performance of $\\mathrm{KD}$ by compounding effects.\n\n\\textbf{Setup.}\nCIFAR-100 is a relatively small dataset with 100 classes. We use ResNet-20 as the student, and ResNet-56 as the teacher.\nImageNet on the other hand is a large-scale dataset covering 1000 classes. We use ResNet-50 as the student, and ResNet-152 as the teacher.\nFor more details, please refer to Section~\\hyperref[sec:si-detail]{A} in Suppl.\nNote that instead of using different model families as in~\\citep{furlanello2018born,yuan2019revisit}, we use same model architecture (i.e., ResNet) with different depths for the student and teacher to avoid unknown effects introduced by model family discrepancy.\n\n\\begin{table}[t!]\n\\centering\n\\begin{tabular}{l|ccc}\n\\toprule\n\\textbf{Method} & \\textbf{CIFAR-100} & \\textbf{ImageNet}\\\\\n\\hline\n\\hline\nTeacher & 75.68 & 77.98 \\\\\nStudent & 72.51 & 76.32\\\\\n\\midrule\nLS & 73.87& 76.83\\\\\nKD & 75.94& 77.49\\\\\n\\midrule\nKD-pt & 75.08 & 77.00\\\\\nKD-sim & 74.30& 76.95\\\\\nKD-pt+sim & 75.24& 77.17\\\\\n\\midrule\nKD-topk & \\textbf{76.17} & \\textbf{77.75}\\\\\n\\bottomrule\n\n\\end{tabular}\n\\caption{We report the mean Top-1 accuracy (\\%) over 4 individual runs with different initializations. Best $k$ for $\\mathrm{KD}\\text{-topk}$ is 25 and 500 for CIFAR-100 and ImageNet, resp.}\n\\label{tb:exp_real_data}\n\\end{table}\n\n\\begin{table}[t!]\n\\centering\n\\begin{tabular}{l|ccc}\n\\toprule\n \\textbf{Method} & \\textbf{\\#Params} & \\textbf{Validation} & \\textbf{Test}\\\\\n\\hline\n\\hline\nTeacher & 24.2M & 60.90 & 58.58\\\\\nStudent & 9.1M & 64.17 & 61.55\\\\\n\\midrule\nKD & 9.1M & 64.04 & 61.33\\\\\nKD-topk & 9.1M & \\textbf{63.59} & \\textbf{60.85}\\\\\n\\bottomrule\n\n\\end{tabular}\n\\caption{Validation and test Perplexity (lower is better) of compared methods on PTB language modeling. We report the best result over 4 individual runs with different initializations. Best $k$ value for $\\mathrm{KD}\\text{-topk}$ is 100.}\n\\label{tb:exp_real_data_lm}\n\\end{table}\n\n\\textbf{Results and Analysis.}\nTable~\\ref{tb:exp_real_data} shows the overall performance when using the best hyper-parameters for each of the methods. On both the datasets, teacher model is much better than the student, and $\\mathrm{LS}$ improves student model's generalization. $\\mathrm{KD}$ can further boost student model's quality by a large margin, especially on CIFAR-100, where $\\mathrm{KD}$ even outperforms the teacher. We try to uncover the different benefits from distillation using partial-$\\mathrm{KD}$ methods. Both $\\mathrm{KD}\\text{-pt}$, and $\\mathrm{KD}\\text{-sim}$ outperforms $\\mathrm{LS}$; especially $\\mathrm{KD}\\text{-pt}$ on CIFAR-100. This suggests that the different effects from distillation benefits the student in different aspects. \nFurthermore, by combining the two effects together in $\\mathrm{KD}\\text{-pt+sim}$ (using $\\alpha=0.5$), we see a further improvement in quality. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently a new class of cosmological models based on the string field\ntheory (SFT)~\\cite{review-sft} and the $p$-adic string theory emerges\nand attracts a lot of attention \\cite{IA1}--\\cite{GK}. It is known that\nthe SFT and the $p$-adic string theory are UV-complete ones. Thus, one\ncan expect that resulting (effective) models should be free of\npathologies. These models exhibit one general non-standard property,\nnamely, their actions have terms with infinitely many derivatives, i.e.\nnonlocal terms. The higher derivative terms usually produce phantom\nfields \\cite{Ostrogradski:1850,PaisU} (see also~\\cite{AV-NEC}).  Models\nthat includes phantoms violate the null energy condition (NEC), and,\ntherefore, are unstable. Models with higher derivative terms produce\nalso well-known problems with quantum instability~\\cite{AV-NEC}.\n\nTo obtain a stable model with the NEC violation (the state parameter\n$w_{\\mathrm{DE}}<-1$) one should construct this model as an effective\nmodel, connected with the fundamental theory, which is stable and\nadmits quantization. With the lack of quantum gravity, we can just\ntrust string theory or deal with an effective theory admitting the UV\ncompletion.\n\nThe purpose of this paper is to study $f(R)$ gravity models with a\nnonlocal scalar field. We consider a general form of nonlocal action\nfor the scalar field with a quadratic potential, keeping the main\ningredient, the analytic function $\\mathcal{F}(\\Box_g)$, which in fact produces\nthe nonlocality, almost unrestricted.\n\n\n\n\n\\section{Nonlocal gravitation models}\n\nThe SFT inspired nonlocal gravitation models~\\cite{IA1} are introduced\nas a sum of the SFT action of the tachyon field $\\phi$ plus the gravity\npart of the action. One cannot deduce this form of the action from the\nSFT. In this paper we study the $f(R)$ gravity, which is a\nstraightforward modification of the general relativity. We consider the\nfollowing action:\n\\begin{equation}\nS_f=\\int d^4x \\sqrt{-g}\\left(\\frac{f(L^2R)}{16\\pi\nG_NL^2}+\\frac{1}{\\alpha^{\\prime}g_o^2}\\left(\\frac{1}{2}\\phi\\,\\mathcal{F}\\left(\\alpha^{\\prime}\\Box_g\\right)\\phi\n-V(\\phi) \\right)-\\Lambda\\right),\n\\end{equation}\nwhere $f(L^2R)$ is an arbitrary differentiable function. We use the\nsignature $(-,+,+,+)$, $g_{\\mu\\nu}$ is the metric tensor, $G_N$ is the\nNewtonian constant. The potential $V(\\phi)$ is a quadratic polynomial\n$V(\\phi)=C_2\\phi^2+C_1\\phi+C_0$, where $C_2$, $C_1$, and $C_0$ are\narbitrary real constants.\n\nThe function $\\mathcal{F}$ is assumed to be analytic at all finite points of\nthe complex plane, in other words, to be an entire function. The\nfunction $\\mathcal{F}$ can be represented by the convergent series expansion:\n$\\mathcal{F}(\\Box_g)=\\sum\\limits_{n=0}^{\\infty}f_n\\Box_g^{\\;n}$. The\nWeierstrass factorization theorem asserts that the function $\\mathcal{F}$ can\nbe represented by a product involving its zeroes $J_k$:\n\\begin{equation}\n\\mathcal{F}(J)=J^me^{Y(J)}\\prod_{k=1}^\\infty\\left(1-\\frac{J}{J_k}\\right)e^{\\frac{J}{J_k}+\\frac{J^2}{2J_k^2}\n+\\dots+\\frac{1}{p_k}\\left(\\frac{J}{J_k}\\right)^{p_k}},\n\\end{equation}\nwhere $m$ is an order of the root $J=0$ ($m$ can be equal to zero),\n$Y(J)$ is an entire function, natural numbers $p_n$ are chosen such\nthat the series\n$\\sum\\limits_{n=1}^\\infty\\left(\\frac{J}{J_n}\\right)^{p_n+1}$\n is an absolutely and uniformly convergent one.\n\nScalar fields $\\phi$ (associated with the open string tachyon) is\ndimensionless, while $[\\alpha^{\\prime}]=\\mbox{length}^2$, $[L]=\\mbox{length}$ and\n$[g_o]=\\mbox{length}$. Let us introduce dimensionless coordinates\n$\\bar{x}_\\mu=x_\\mu\/\\sqrt{\\alpha'}$, the dimensionless Newtonian\nconstant $\\bar{G}_N=G_N\/\\alpha'$, the dimensionless parameter $\\bar\nL=L\/\\sqrt{\\alpha'}$, and the dimensionless open string coupling\nconstant $\\bar g_o=g_o\/\\sqrt{\\alpha^{\\prime}}$. The dimensionless cosmological\nconstant $\\bar\\Lambda=\\Lambda{\\alpha^{\\prime}}^2$, $\\bar{R}$ is the curvature\nscalar in the coordinates $\\bar{x}_\\mu$:\n\\begin{equation}\nS_f=\\int d^4 \\bar{x} \\sqrt{-g}\\left(\\frac{f(\\bar{L}^2 \\bar{R})}{16\\pi\n\\bar{G}_N\\bar{L}^2}+\\frac{1}{\\bar{g}_o^2}\\left(\\frac{1}{2}\\phi\\,\\mathcal{F}\\left(\\bar{\\Box}_g\\right)\\phi\n-V(\\phi) \\right)-\\bar{\\Lambda}\\right). \\label{action_model2}\n\\end{equation}\nIn the following formulae we omit bars, but use only dimensionless\ncoordinates and parameters.\n\nIt is well-known~\\cite{Mukhanov1} that at $f'(R)>0$ any $f(R)$ gravity\nmodels in the metric variational approach are equivalent to the\nEinstein gravity with a scalar field\\footnote{There are two types of\n$f(R)$ gravity: the metric variational approach and the Palatini\nformalism. In the first case the equations of motion are obtained by\nvariation with respect to metric. Connections are the function of\nmetric in this formalism. In the Palatini formalism one should vary the\naction independently with respect to metric and the connections.}. In\nthe metric variational approach the equations of gravity are as\nfollows:\n\\begin{equation}\n\\label{fr_equ} G_{\\mu\\nu}\\equiv f'(R)R_{\\mu\\nu}-\n\\frac{f(R)}{2}g_{\\mu\\nu}-D_\\mu\n\\partial_\\nu f'(R)+g_{\\mu\\nu}\\Box_g f'(R)=8\\pi\nG_N T_{\\mu\\nu}, \\quad\n  \\mathcal{F}(\\Box_g)\\phi=\\frac{dV}{d\\phi},\n\\end{equation}\nwhere the energy--momentum (stress) tensor $T_{\\mu\\nu}$ is:\n\\begin{equation}\n  \\label{TEV}\n  T_{\\mu\\nu}\\equiv{}-\\frac{2}{\\sqrt{-g}}\\frac{\\delta{S}}{\\delta\n    g^{\\mu\\nu}}\n  =\\frac{1}{g_o^2}\\Bigl(E_{\\mu\\nu}+E_{\\nu\\mu}-g_{\\mu\\nu}\\left(g^{\\rho\\sigma}\n    E_{\\rho\\sigma}+W\\right)\\Bigr),\n\\end{equation}\n\\begin{equation}\n  E_{\\mu\\nu}\\equiv\\frac{1}{2}\\sum_{n=1}^\\infty\n  f_n\\sum_{l=0}^{n-1}\\partial_\\mu\\Box_g^l\\phi\\partial_\\nu\\Box_g^{n-1-l}\\phi,\\quad\n  W\\equiv\\frac{1}{2}\\sum_{n=2}^\\infty\n  f_n\\sum_{l=1}^{n-1}\\Box_g^l\\phi\\Box_g^{n-l}\\phi-\\frac{f_0}{2}\\phi^2+C_1\\phi.\n\\end{equation}\n\n\\section{Localization of nonlocal gravitational actions}\n\nThe Ostrogradski representation has been proposed for polynomial\n$\\mathcal{F}(\\Box)$ in the Minkowski space-time~\\cite{Ostrogradski:1850,PaisU}.\nOur goal is to generalize this result on gravitational models with an\narbitrary analytic function $\\mathcal{F}(\\Box)$ with simple and double roots.\nWe also generalize the Ostrogradski representation on the models with a\nlinear potential. The nonlocal cosmological models with quadratic\npotentials have been studied\nin~\\cite{Koshelev07,AJV0701,AJV0711,MN,KV,Vernov2010,VernovSQS}.\n\n\nLet us start with the case $C_1=0$. We consider a function $\\mathcal{F}(J)$,\nwhich has simple roots $J_i$ and double roots $\\tilde{J}_k$, and the\nfunction\n\\begin{equation}\n  \\label{phi0}\n  \\phi_0=\\sum\\limits_{i=1}^{N_1}\\phi_i+\\sum\\limits_{k=1}^{N_2}\\tilde\\phi_k,\n\\end{equation}\nwhere\n\\begin{equation}\n  (\\Box_g-J_i)\\phi_i=0 \\quad\\mbox{and}\\quad (\\Box_g-\\tilde{J}_k)^2\\tilde\\phi_k=0\\quad\\Leftrightarrow\\quad\n(\\Box_g-\\tilde{J_k})\\tilde\\phi_k=\\varphi_k,\\quad\n  (\\Box_g-\\tilde{J_k})\\varphi_k=0.\n  \\label{equphi}\n\\end{equation}\nWithout loss of generality we assume that for any $i_1$ and $i_2\\neq\ni_1$ conditions $J_{i_1}\\neq J_{i_2}$ and\n${\\tilde{J}}_{i_1}\\neq{\\tilde{J}}_{i_2}$ are satisfied.\n\n\nThe energy--momentum tensor, which corresponds to $\\phi_0$, has the\nfollowing form:\n\\begin{equation}\n  T_{\\mu\\nu}\\left(\\phi_0\\right)=\n  T_{\\mu\\nu}\\left(\\sum\\limits_{i=1}^{N_1}\\phi_i+\\sum\\limits_{k=1}^{N_2}\\tilde\\phi_k\\right)=\n  \\sum\\limits_{i=1}^{N_1}T_{\\mu\\nu}(\\phi_i)+\\sum\\limits_{k=1}^{N_2}T_{\\mu\\nu}(\\tilde\\phi_k),\n  \\label{Tmunugen}\n\\end{equation}\nwhere all $T_{\\mu\\nu}$ are given by (\\ref{TEV}) and\n\\begin{equation}\n  E_{\\mu\\nu}(\\phi_i)=\\frac{{\n      \\mathcal{F}'(J_i)}}{2}\\partial_{\\mu}\\phi_i\\partial_{\\nu}\\phi_i,\\quad\n  E_{\\mu\\nu}(\\tilde\\phi_k)= \\frac{{\n      \\mathcal{F}''(\\tilde{J}_k)}}{4}\\left(\\partial_\\mu\\tilde\\phi_k\\partial_\\nu\\varphi_k\n      +\\partial_\\nu\\tilde\\phi_k\\partial_\\mu\\varphi_k\\right)+\n  \\frac{\\mathcal{F}'''(\\tilde{J}_k)}{12}\\partial_\\mu\\varphi_k\\partial_\\nu\\varphi_k,\n\\end{equation}\n\\begin{equation}\n  \\label{Vdr} W(\\phi_i)=\\frac{J_i \\mathcal{F}'(J_i)}{2}\\phi_i^2,\\quad W(\\tilde{\\phi}_k)=\\frac{\\tilde{J}_k\n    \\mathcal{F}''(\\tilde{J}_k)}{2}\\tilde\\phi_k\\varphi_k+ \\left(\\frac{{\\tilde{J}_k\n        \\mathcal{F}'''(\\tilde{J}_k)}}{12}+\\frac{{\n        \\mathcal{F}''(\\tilde{J}_k)}}{4}\\right)\\varphi_k^2,\n\\end{equation}\nwhere a prime denotes a derivative with respect to $J$: $\\mathcal{F}'\\equiv\n\\frac{d\\mathcal{F}}{dJ}$, \\ $\\mathcal{F}''\\equiv \\frac{d^2\\mathcal{F}}{dJ^2}$ and $\\mathcal{F}'''\\equiv\n\\frac{d^3 \\mathcal{F}}{dJ^3}$.\n\nConsidering the following local action\n\\begin{equation}\n  S_{loc}=\\int d^4x\\sqrt{-g}\\left(\\frac{f(R)}{16\\pi\n      G_N}-\\Lambda\\right)+\\sum_{i=1}^{N_1}S_i+\\sum_{k=1}^{N_2}\\tilde{S}_k,\n  \\label{Sloc}\n\\end{equation}\nwhere\n\\begin{equation}\n  S_i=\\!{}-\\frac{1}{g_o^2}\\int d^4x\\sqrt{-g}\n  \\frac{\\mathcal{F}'(J_i)}{2}\\left(g^{\\mu\\nu}\\partial_\\mu\\phi_i\\partial_\\nu\\phi_i\n    +J_i\\phi_i^2\\right),\n\\end{equation}\n\\begin{equation}\n  \\begin{array}{l}\n    \\!\\displaystyle\\tilde{S}_k=\\!\\displaystyle\\! {}-\\frac{1}{g_o^2}\\int\n    d^4x\\sqrt{-g}\\left(g^{\\mu\\nu}\\left(\\frac{{\n            \\mathcal{F}''(\\tilde{J}_k)}}{4}\\left(\\partial_\\mu\n          \\tilde{\\phi}_k\\partial_\\nu\\varphi_k+\\partial_\\nu\n          \\tilde{\\phi}_k\\partial_\\mu\\varphi_k\\right)+{}\\right.\\right.\\\\[2.7mm]\n    \\displaystyle + \\left.\\frac{\n        \\mathcal{F}'''(\\tilde{J}_k)}{12}\\partial_\\mu\\varphi_k\\partial_\\nu\\varphi_k\\right)+\n    \\left. \\frac{\\tilde{J}_k \\mathcal{F}''(\\tilde{J}_k)}{2}\\tilde\\phi_k\\varphi_k\n      +\\left(\\frac{{\\tilde{J}_k \\mathcal{F}'''(\\tilde{J}_k)}}{12}+\\frac{{\n            \\mathcal{F}''(\\tilde{J}_k)}}{4}\\right)\\varphi_k^2\\right), \\label{Slocdr}\n  \\end{array}\n\\end{equation}\nwe can see that solutions of the Einstein equations and equations in\n$\\phi_k$, $\\tilde{\\phi}_k$ and $\\varphi_k$, obtained from this action,\nsolve the initial  nonlocal equations (\\ref{fr_equ}). Thus, we\nobtain that special solutions to nonlocal equations can be found as\nsolutions to system of local (differential) equations. If $ \\mathcal{F}(J)$ has\nan infinity number of roots then one nonlocal model corresponds to\ninfinity number of different local models and the initial nonlocal\naction (\\ref{action_model2}) generates infinity number of local actions\n(\\ref{Sloc}).\n\nWe should prove that the way of localization is self-consistent.  To\nconstruct local action (\\ref{Sloc}) we assume that equations\n(\\ref{equphi}) are satisfied. Therefore, the method of localization is\ncorrect only if these equations can be obtained from the local action\n$S_{loc}$. The straightforward calculations show that the way of\nlocalization is self-consistent because:\n\\begin{equation}\n  \\frac{\\delta{S_{loc}}}{\\delta \\phi_i}=0 \\, \\Leftrightarrow \\,\n  \\Box_g\\phi_i=J_i\\phi_i; \\, \\frac{\\delta{S_{loc}}}{\\delta\n    \\tilde{\\phi}_k}=0 \\, \\Leftrightarrow \\,\n  \\Box_g\\varphi_k=\\tilde{J}_k\\varphi_k; \\,\n  \\frac{\\delta{S_{loc}}}{\\delta \\varphi_k}=0 \\, \\Leftrightarrow \\,\n  \\Box_g\\tilde{\\phi}_k=\\tilde{J}_k\\tilde{\\phi}_k+\\varphi_k.\n\\end{equation}\n\n\n\n\nIn spite of the above-mention equations we obtain from $S_{loc}$ the\nequations:\n\\begin{equation}\n  G_{\\mu\\nu}=8\\pi G_N\\left(T_{\\mu\\nu}(\\phi_0)-\\Lambda g_{\\mu\\nu}\\right),\n\\end{equation}\nwhere $\\phi_0$ is given by (\\ref{phi0}) and $T_{\\mu\\nu}(\\phi_0)$ can be\ncalculated by (\\ref{Tmunugen}). So, we get such systems of\ndifferential equations that any solutions of these systems are\nparticular solutions of the initial nonlocal equations (\\ref{fr_equ}).\n\nLet us consider functions $\\mathcal{F}(J)$ with two and only two simple roots.\nIf $\\mathcal{F}(J)$ has two real simple roots, then $\\mathcal{F}'(J)>0$ at one root and\n$\\mathcal{F}'(J)<0$ at another root, so we get a quintom\nmodel~\\cite{Quinmodrev1}, in other words, local model with one standard\nscalar field and one phantom scalar field. In the case of two complex\nconjugated simple roots $J_j$ and $J_j^*$ one gets the following\naction:\n\\begin{equation}\nS_c=\\!\\int\\!\\! d^4x\\frac{\\sqrt{-g}}{2g_o^2}\\left(\n  \\mathcal{F}'(J_j)\\left(g^{\\mu\\nu}\\partial_\\mu\\phi_j\\partial_\\nu\\phi_j\n    +J_j\\phi_j^2\\right)+{\\mathcal{F}'}^*(J_j)\\left(g^{\\mu\\nu}\\partial_\\mu\\phi^*_j\\partial_\\nu\\phi^*_j\n    +J^*_j{\\phi_i^*}^2\\right)\\right).\n\\end{equation}\nWe introduce real fields $\\xi$ and $\\eta$ such that $\\phi_j=\\xi+i\\eta$,\n\\ $\\phi_j^*=\\xi-i\\eta$, denote $d_r\\equiv\\Re e(\\mathcal{F}'(J))$, \\\n$d_i\\equiv\\Im m(\\mathcal{F}'(J))$, and obtain:\n\\begin{equation}\nS_c=\\int d^4x\\frac{\\sqrt{-g}}{2g_o^2}\\Bigl(d_r\ng^{\\mu\\nu}\\left(\\partial_\\mu\\xi\\partial_\\nu\\xi-\n\\partial_\\mu\\eta\\partial_\\nu\\eta\\right)+\nd_ig^{\\mu\\nu}(\\partial_\\mu\\xi\\partial_\\nu\\eta-\\partial_\\mu\\eta\\partial_\\nu\\xi)+V_1\\Bigr),\n\\end{equation}\nwhere $V_1$ is a potential term. In the case $d_i=0$ we get a quintom\nmodel, in opposite case the kinetic term in $S_c$ has a nondiagonal\nform. To diagonalize the kinetic term we make the transformation:\n$\\chi=\\upsilon+\\tilde{C}\\sigma$, $\\eta={}-\\tilde{C}\\upsilon+\\sigma$,\nwhere $\\tilde{C}\\equiv\\left(d_r+\\sqrt{d_r^2+d_i^2}\\right)\/d_i$, and get\na quintom model:\n\\begin{equation}\nS_c=\\int\nd^4x\\frac{\\sqrt{-g}}{2g_o^2}\\left(\\frac{2\\left(d_r^2+d_i^2\\right)}{d_i^2}\\left(d_r+\\sqrt{d_r^2+d_i^2}\\right)\n\\left(\\partial_\\mu\\upsilon\\partial_\\nu\\upsilon\n-\\partial_\\mu\\sigma\\partial_\\nu\\sigma\\right)+V_1\\right).\n\\end{equation}\n\nIn the case of a real double root $\\tilde{J}_k$ we express\n$\\tilde{\\phi}_k$ and $\\varphi_k$ in terms of new fields $\\xi_k$ and\n$\\chi_k$:\n\\begin{eqnarray}\n    \\tilde{\\phi}_k&=&\\frac{1}{2\\mathcal{F}''(\\tilde{J}_k)}\\left(\\left(\\mathcal{F}''(\\tilde{J}_k)-\\frac{2}{3}\\mathcal{F}'''(\\tilde{J}_k)\\right)\n    \\xi_k-\\left(\\mathcal{F}''(\\tilde{J}_k)+\\frac{2}{3}\\mathcal{F}'''(\\tilde{J}_k)\\right)\\chi_k\\right),\n\\quad     \\varphi_k=\\xi_k+\\chi_k,\\nonumber\n\\end{eqnarray}\nwe obtain the corresponding $\\tilde{S}_k$ in the following form:\n\\begin{eqnarray}\n\\tilde{S}_k&=&\\frac{{}-1}{2g_o^2}\\!\\int\\!\nd^4\\!x\\sqrt{-g}\\left(g^{\\mu\\nu}\\frac{\\mathcal{F}''(\\tilde{J}_k)}{4}(\\partial_\\mu\n\\xi_k\\partial_\\nu\\xi_k-\\partial_\\nu\n\\chi_k\\partial_\\mu\\chi_k)+\\left[\\frac{{\\tilde{J}_k\\mathcal{F}'''(\\tilde{J}_k)}}{12}+\\frac{{\\mathcal{F}''(\\tilde{J}_k)}}{4}\\right]\n(\\xi_k+\\chi_k)^2+\\right.\\nonumber\\\\\n &+&\\left.\\frac{\\tilde{J}_k}{4}\\left[(\\mathcal{F}''(\\tilde{J}_k)-\\frac{2}{3}\\mathcal{F}'''(\\tilde{J}_k))\n    \\xi_k-(\\mathcal{F}''(\\tilde{J}_k)+\\frac{2}{3}\\mathcal{F}'''(\\tilde{J}_k))\\chi_k\\right](\\xi_k+\\chi_k)\\right).\\nonumber\n\\end{eqnarray}\nIt is easy to see that each $\\tilde{S}_k$ includes one phantom scalar\nfield and one standard scalar field. So, in the case of one double root\nwe obtain a quintom model. In the Minkowski space appearance of phantom\nfields in models, when $\\mathcal{F}(J)$ has a double root, has been obtained\nin~\\cite{PaisU}. So, we come to conclusion that both two simple roots\nand one double root of $\\mathcal{F}(J)$ generate quintom models.\n\n\nThe model with action (\\ref{action_model2}) in the case $C_1\\neq 0$ has\nbeen considered in detail in~\\cite{VernovSQS}. Here we present only the\nobtained algorithm of localization for an arbitrary quadratic potential\n$V(\\phi)=C_2\\phi^2+C_1\\phi+C_0$:\n\n\\begin{itemize}\n\\item Change values of $f_0$ and $\\Lambda$ such that the potential\n  takes the form $V(\\phi)=C_1\\phi$.\n\\item Find roots of the function $ \\mathcal{F}(J)$ and calculate orders of\n them. Select an finite number of simple and double roots.\n\\item Construct the corresponding local action. In the case $C_1=0$\n  one should use formula (\\ref{Sloc}). In the case $C_1\\neq 0$ and\n  $f_0\\neq 0$ one should use (\\ref{Sloc}) with the replacement of the\n  scalar field $\\phi$ by $\\chi$ and the corresponding modification of\n  the cosmological constant. In the case $C_1\\neq 0$ and $f_0=0$ the\n  local action is the sum of (\\ref{Sloc}) and either\n  \\begin{equation}\n    S_{\\psi}={}-\\frac{1}{2g_o^2}\\int\\! d^4x\\sqrt{-g}\\left(\n      f_1g^{\\mu\\nu}\\partial_\\mu\\psi\\partial_\\nu\\psi+2C_1\\psi+\\frac{f_2C_1^2}{f_1^2}\\right),\n  \\end{equation}\n  in the case of simple root $J=0$, or\n \\begin{eqnarray}\n   S_{\\tilde{\\psi}}&=&{}-\\!\\int\\! d^4x\\frac{\\sqrt{-g}}{2g_o^2}\\left[\n      g^{\\mu\\nu}\\left(f_2(\\partial_\\mu\\tilde{\\psi}\\partial_\\nu\\tau\n        +\\partial_\\nu\\tilde{\\psi}\\partial_\\mu\\tau)+f_3\\partial_\\mu\\tau\\partial_\\nu\\tau\\right)\n      +f_2\\tau^2+2C_1\\tilde{\\psi}+\\frac{f_3C_1}{2f_2}\\tau\\right]\\nonumber\n  \\end{eqnarray}\n  in the case of double root $J=0$. Note that in the case $C_1\\neq 0$\n  and $f_0=0$ the local action (\\ref{Sloc}) has no term, which\n  corresponds to the root $J=0$.\n\\item Vary the obtained local action and get a system of the Einstein\n  equations and equations of motion. The obtained system is a finite\n  order system of differential equations, \\textit{i.e.} we get a local\n  system. Seek solutions of the obtained local system.\n\n\\end{itemize}\n\n\n\n\\section{Conclusion}\n\nThe main result of this paper is the generalization of the algorithm of\nlocalization on  the $f(R)$ gravity models with a nonlocal scalar\nfield. The algorithm of localization is proposed for an arbitrary\nanalytic function $\\mathcal{F}(\\Box_g)$, which has both simple and double\nroots. We have proved that the same functions solve the initial\nnonlocal Einstein equations and the obtained local Einstein equations.\nWe have found the corresponding local actions and proved the\nself-consistence of our approach. In the case of two simple roots as\nwell as in the case of one double root we get a quintom\nmodel~\\cite{Quinmodrev1}. The algorithm of localization does not depend\non metric, so it can be used to find solutions for any metric.\n\nThe author wishes to express his thanks to I.~Ya.~Aref'eva for useful\nand stimulating discussions. The research\n  has been supported in part by RFBR grant 08-01-00798, grant of Russian\n  Ministry of Education and Science NSh-4142.2010.2 and by Federal\n  Agency for Science and Innovation under state contract\n  02.740.11.0244.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThere is a body of literature on an ideal Bose gas in a uniform gravitational \nfield~\\cite{gersch57,widom68,baranov01,cavalcanti02,uncu13}.\n The gas may be contained in an external potential, or a large box, and \nsubjected to a uniform gravitational potential. Using the semiclassical \napproximation, its thermal properties have been calculated analytically \nin the grand canonical formalism~\\cite{liu09a,du12}. Our motivation for studying this \nsimple system is to check how closely does the semiclassical approximation follow \nthe results of the quantum calculation.  The authors of Ref.~\\cite{cavalcanti02} claim that, contrary to previous wisdom, in three dimensions an ideal Bose gas in a uniform gravitational field does not undergo BEC at a finite temperature.    They attribute that to the replacement of the discrete quantum energy spectrum with a smooth density of states.\nIn the semiclassical approximation, one replaces the discrete density of states by a smooth one, while treating the ground state exactly. In the quantum calculation, on the other hand, the exact discrete energy levels of the system are calculated to compute the grand canonical ensemble (GCE) \nand the resulting thermal properties. In realistic statistical mechanics problems, one generally follows  the semiclassical route. For the system at hand, the quantum \ncalculation is done for a finite number of particles. We find that as the number of particles is increased to larger and larger values, the quantum and semiclassical results  become very close, even across the BEC critical temperature.  \n\nIn this paper, we pay special \nattention to the calculation of the isothermal compressibility of the Bose gas.   \nRecent experimental work on the isothermal compressibility across the Bose-Einstein condensation has been reported in Ref.~\\cite{poveda-cuevas15} for a harmonically trapped gas.  The authors suggest that the isothermal compressibility around the critical pressure reveals a second-order nature of the phase transition.  On the other hand predictions based on a number of different mean-field approximations~\\cite{olivares-quiroz10} do not lead to second-order phase transitions, and the isothermal compressibility does not diverge at criticality.  In contrast to these authors, we discuss noninteracting systems only.\n It is well documented that in GCE, the isothermal compressibility diverges at the critical temperature $T_c$ in the absence of interparticle interactions~\\cite{\nlandau58}. \nIt is also known  that even a weak \ninterparticle interaction  removes this divergence~\\cite{bhaduri02}. In the present \nproblem, however, gravitation is  introduced as a one-body ramp potential, and it \nis not clear at the outset how it will affect the compressibility.  \n\nWe find that the semiclassical calculation in three dimensions of the ideal Bose gas with uniform gravity is equivalent to the analysis of a five-dimensional ideal Bose gas without gravity. We use this novel approach to obtain results for the specific heat and isothermal compressibility.   \n The resulting compressibility is divergence-free and continuous across $T_c$.  In the case of heat capacity, \n in the absence of the gravitational field, there is a discontinuity in its slope at \n$T_c$. Introducing gravitation, or, alternately five spatial dimensions, this  \ndiscontinuity is in the heat capacity itself. \n\nThe calculations were performed by \ntaking a cylindrical container, as shown in Fig.~\\ref{fig:01}. \nIn the $zx$ plane, we take a \ncircular disc, which is the bottom of the cylinder at $y=0$. The atoms in the Bose \ngas are not allowed to take negative values of $y$. The gravitational field is along \nthe $y$ direction, and the potential is a ramp along the positive $y$ axis.  \n\nThe plan of the paper is as follows. In Sec.~\\ref{sec:2}, the semiclassical calculation is \ndone using the phase space approach. It is established that one can describe the \nsystem under consideration in five spatial dimensions, but without the gravitation.\n The grand potential is calculated and the critical temperature $T_c$ is obtained. \nIn Sec.~\\ref{sec:3}, we give the results for isothermal compressibility and the \nheat capacity. In Sec.~\\ref{sec:4} a quantum calculation is done to show that BEC takes \nplace and the results agree with the semiclassical calculation.  \n\n\\section{Three-dimensional  gas in a uniform gravitational potential}         \n\\label{sec:2}\nIn this section, we show that an ideal Bose gas in three spatial dimensions,  \nsubjected to a uniform gravitational potential, may be looked upon as an \nideal five dimensional gravity-free gas.  We then use the semiclassical method to \ncalculate the critical temperature of BEC. \n\\begin{figure}[!h]\n\\begin{center}\n\\begin{tikzpicture}[scale = 0.5,line width = 1.2pt]\n\\draw[->](0,0)-- (5,0) node [right]{$x$};\n\\draw[->](0,0)--(0,5) node [above]{$y$};\n\\draw[->](0,0)--(-3,-3) node [below,left]{$z$}; \n\\draw(0,0) node [left]{$O$};\n\\draw (0,0) ellipse (3cm and 1cm);\n\\draw(0,3) ellipse(3cm and 1cm);\n\\draw(3,0)--(3,3);\n\\draw(-3,0)--(-3,3);\n\\draw(3,1.5) node [right]{$L$};\n\\draw[->](2,4.5)--(1,3) ;\n\\draw (2,4.5) node [above,right]{Area $=A=\\pi a^2$}; \n\\end{tikzpicture}\n\\caption{The Bose gas is confined to a cylindrical box with the downward gravitational force parallel to the $y$ axis.}\n\\label{fig:01}\n\\end{center}\n\\end{figure}\n\nUsing the geometry of Fig.~\\ref{fig:01}, the single particle energy is given by\n\\begin{equation}\n\\epsilon (p,y)=\\frac {p^2}{2m}+mgy ,\n\\end{equation}\nwhere  $m$ is the mass of each boson, and $g$ is the gravitational acceleration \non the earth's surface, and $p^2=(p_x^2+p_y^2+p_z^2)$. \n The grand potential is given by ($k_B=1$) \n\\begin{equation}\n\\Omega_b=T\\sum_n \\ln \\left (1-z\\exp(-\\epsilon_n \\beta)\\right)\n=-T\\sum_{l=1}^{\\infty} \\frac{(z)^l}{l} Z_1(l\\beta)~,\n\\label{omega}\n\\end{equation}\nwhere $\\beta=1\/T$, the fugacity $z=\\exp(\\beta\\mu)$, and  \n$Z_1(l\\beta)$ is the one-body partition function in the variable $l\\beta$.\nIn the semiclassical approximation, $Z_1(\\beta)$ in the variable $\\beta$, is \ngiven by \n\\begin{equation}\nZ_1(\\beta)=\\frac{1} {h^3} \\int d^3p~ e^{\\textstyle  -\\beta p^2\/2m} \\int d^2 r\n\\int_0^L dy ~e^{\\textstyle   -\\beta mgy}.\n\\end{equation}\nThe two-dimensional spatial integral gives the area $A$ of the disc, yielding  \n\\begin{equation}\nZ_1(\\beta)=\\frac{A}{\\lambda_T^3}\\frac{(1-\\exp{(-\\beta mgL)})} {\\beta mg}~.\n\\label{dime1}\n\\end{equation}\nNote that as $g\\rightarrow 0$, we recover the correct \n$Z_1(\\beta)=\\dfrac{V}{\\lambda_T^3}$, \nwhere $V=A L$ is the three-dimensional spatial volume.  \nThe thermal wavelength $\\lambda_T$ (obtained from the $p$ integration) is given by \n\\begin{equation}\n\\lambda_T=\\sqrt{\\frac{2\\pi\\hbar^2}{m T}}~.\n\\end{equation}\nFor our present problem with \\textit{nonzero} $g$ and low temperatures, we impose the condition that $k_B T \\ll mgL$, i.e. $\\beta mgL \\gg 1$.  Under this condition, Eq.~(\\ref{dime1}) reduces to  \n\\begin{equation}\\label{eq:6}\nZ_1(\\beta)=\\frac{A}{\\lambda_T^3}\\frac{1}{\\beta m g}~.\n\\end{equation} \nEquation~(\\ref{eq:6}) could be rewritten as an ideal five-dimensional partition function (without gravity)\n\\begin{equation}\n\\tilde{Z}_1(\\beta)=\\frac{V_5}{\\lambda_T^5}\n\\label{new}\n\\end{equation}\nwhere \n\\begin{equation}\nV_5=\\frac{2\\pi \\hbar^2 A}{m^2g}\n\\label{dime2}\n\\end{equation}\nhas the dimension of (length)$^5$.   We write $V_5=(A\\cdot V_3)$, where $V_3$ is a \nhypothetical 3-volume. Taking $m$ to be that of a $Rb^{87}$ atom, we find \n$V_3$ to be exceedingly small, of the order of $10^{-18}$ cubic meter. \nThis $V_3$ is not to be confused with the large three-dimensional volume \n$V=AL$ in which the atoms are confined. \nIn the following, we shall calculate the thermal properties of this noninteracting gas \nof bosons in 5-spatial dimensions . \n  \nSubstituting for $\\tilde{Z}_1(\\beta)$ from Eq.~(\\ref{new}) in Eq.~(\\ref{omega}), \nwe see that the grand potential may be written as \n\\begin{equation}\n\\Omega_b=-T \\frac{V_5}{\\lambda_T^5}\\sum_{l=1}^{\\infty}\\tilde{ b}_l z^l~\n=-T \\frac{V_5}{\\lambda_T^5}~ g_{7\/2}(z)~.\n\\label{cluster}\n\\end{equation}\nwhere $\\tilde{b}_l=1\/l^{7\/2}$ are the statistical ``cluster integrals''.  \nIn standard notation, $\\displaystyle \\sum_{l=1}^{\\infty} z^l\/l^{7\/2}=g_{7\/2}(z) $~.\n\nIn the gas phase, \n\\begin{equation}\n\\bar{n}_5=\\frac{\\bar{N}}{V_5}=-\\frac{\\partial\\Omega_b}{\\partial\\mu}\n=\\frac{1}{\\lambda_T^5\n}\\sum_{l=1}\\tilde{b}_l z^l~=\\frac{1}{\\lambda_T^5} g_{5\/2}(z)~.\n\\label{deriv}\n\\end{equation}\nOne puts in the constraint that $\\bar{N}=N$ , and this makes $z$ a function \nof $T$. The sum on the RHS converges at $z=1$, so the above relation is valid \nonly for $T\\geq T_c$. For lower temperatures, the ground state starts having \nmacroscopic occupancies. The critical temperature is given by \n\\begin{equation}\n(\\bar{n}_5\\lambda_{T_c}^5)=\\zeta(5\/2)~,\n\\label{kya}\n\\end{equation}\nwhere $\\lambda_T$ is at $T=T_c$, and  $\\zeta(5\/2)$ is the Riemann zeta function.  \nIt is straight forward to deduce from Eq.~(\\ref{kya}) that the critical temperature \nis given by  \n\\begin{equation}\nT_c=\\left(\\frac{\\bar{N}\\hbar^3g(2\\pi)^{3\/2}}{\\zeta(5\/2)A\\sqrt{m}}\\right)^{2\/5}~.\n\\label{cond}\n\\end{equation} \nThis agrees with the expression for $T_c$  as given by Du \\textit{et al.}~\\cite{du12}, that was \nobtained by the standard procedure in three spatial dimensions in the \npresence of the uniform gravitational field. Furthermore it follows  that \n\\begin{equation}\\label{eq:13}\n\\dfrac{N_{\\epsilon=0}}{N} = 1 - \\left(\\dfrac{T}{T_c}\\right)^{5\/2} \\ \\ \\ \\mathrm{when} \\ \\ \\ T < T_c,\n\\end{equation}\nwhere $N_{\\epsilon=0}$ refers to the number of particles in the ground state.\nIn Sec.~\\ref{sec:4}, a fully \nquantum mechanical calculation is performed to demonstrate that BEC \ndoes take place at a finite temperature that is consistent with the \nsemiclassical result as given by Eq. (\\ref{cond}).  \n \n\n\\section{Semiclassical isothermal compressibility and heat capacity}\n\\label{sec:3}\n\\subsection{Isothermal compressibility }\n\nQuite generally, the isothermal compressibility is defined, in any \ndimension, by \n\\begin{equation}\n\\kappa_T=-\\frac{1}{V}\\left(\\frac{\\partial V}{\\partial P}\\right)_T ~,\n\\end{equation}\nand is directly related to number fluctuation in GCE.\nNote that $\\kappa_T$ has different dimensionality in three and five \nspace dimensions. For this reason, we denote the compressibility in five dimensions \nby $\\tilde{\\kappa}_T$.\n\nOnce the grand potential $\\Omega_b$ has been calculated (see Eq.~(\\ref{cluster})),\nthe average particle number $\\bar{n}_5$ is obtained from \n$-\\dfrac{\\partial \\Omega_b}{\\partial\\mu}$  (see Eq.~(\\ref{deriv})), \nand its second derivative with respect to $\\mu$ is related to $\\kappa_T$ :\n\\begin{equation}\n-\\frac{\\partial^2 \\Omega_b}{\\partial\\mu^2}=\\frac{\\partial \\bar{N}}{\\partial\\mu}\n=V_5 \\bar{n}_5^2\\tilde{\\kappa}_T\n\\end{equation}\nNote that we fix $\\bar{N}=N$ using Eq.~(\\ref{deriv}), so that the number density of \nbosons remains a constant. \nA little algebra gives, for our five-dimensional ideal gas,\n\\begin{equation}\n\\tilde{\\kappa}_T=\\frac{1}{\\bar{n}_5^2}\\frac{\\beta}{\\lambda_T^5}{ \n\\sum_{l=1}^{\\infty}l^2\\tilde{b}_lz^l} = \\dfrac{1}{\\bar{n}_5k_BT}\\dfrac{g_{3\/2}(z)}{g_{5\/2}(z)},\n\\end{equation} \nwhere $\\tilde{b}_l=1\/l^{7\/2}$. Note that $\\tilde{\\kappa}_T$ is finite and continuous across \nthe critical temperature (with $z=1$).   At $T_c$ the compressibility is finite, i.e., $\\bar{n}_5k_BT_c\\tilde{\\kappa}_T=1.9474$. \nThis is in contrast to the ideal gas in three spatial dimensions discussed in Ref.~\\cite{\npathria11}.   In that case, the \ncompressibility is given as\n\\begin{equation}\\label{eq:16}\n\\kappa_T = \\dfrac{1}{\\bar{n}_3k_BT}\\dfrac{g_{1\/2}(z)}{g_{3\/2}(z)},\n\\end{equation}\nwhere $\\bar{n}_3=\\dfrac{N}{V} $ is the number density in three dimensions. \nThe compressibility  diverges at $T_c$. In the presence of uniform gravitation, \nwe find that the above expression is modified to \n\\begin{equation}\\label{eq:16}\n\\kappa_T = \\dfrac{1}{\\bar{n}_3k_BT}\\dfrac{g_{3\/2}(z)}{g_{5\/2}(z)},\n\\end{equation}\nwhich is not divergent at $T_c$.   Note that this expression is valid when $T\\geq T_c$.  \n\nIn order to calculate the compressibility below as well as above $T_c$ we use a modification of Eq.~(\\ref{deriv}) that includes the effect of the ground state in order to obtain $z$ as a function of $T$, i.e.,\n\\begin{equation}\\label{eq:n3}\n\\bar{n}_5 = \\dfrac{1}{\\lambda_T^5}g_{5\/2}(z) + \\dfrac{1}{V_5}\\dfrac{z}{1-z}.\n\\end{equation} \nThe isothermal compressibility is then\n\\begin{equation}\n\\label{eq:n20}\n\\tilde{\\kappa}_T = \\dfrac{1}{\\bar{n}_5k_BT}\\dfrac{V_5g_{3\/2}(z)+ \\lambda_T^5\\dfrac{z}{(1-z)^2}}{V_5g_{5\/2}(z) + {\\lambda_T^5}\\dfrac{z}{(1-z)}},\n\\end{equation}\nwhich diverges at $T=0$.\nWe use the corresponding equation in three dimensions, with $\\kappa_T$ replacing $\\tilde{\\kappa}_T$, \n$\\bar{n}_3$ replacing $\\bar{n}_5$, and $V$ replacing $V_5$ in Eq.~(\\ref{eq:n20}), to compare with the quantum results in\nthree dimensions in the presence of uniform gravity.\n As we shall see, this procedure gives good agreement with the quantum calculations for both the heat capacity and the compressibility as shown in Fig.~\\ref{fig:06} in  Sec.~\\ref{sec:4} of this paper.\n\n\\subsection{Heat capacity}\nTo calculate the heat capacity, we need to calculate the average energy $\\bar{E}$, \nwhich is $-\\dfrac{5}{2}\\Omega_b$, where $\\Omega_b$ is defined in Eq.~(\\ref{cluster}), \nand the average number of bosons $\\bar{N}$, defined in \nEq.~(\\ref{deriv}).\nThe energy is differentiated with respect to $T$, with the constraint that \n$\\bar{N}=N$, where $N$ is a constant. This gives the condition that \n$\\dfrac{d\\bar{N}}{dT}=0$, implying that the fugacity $z$ is  dependent on $T$.  \nThe derivation will not be given here since our result coincides with Du \\textit{et al.}~\\cite{du12}. We \ngive the final result below, so that this semiclassical result may be plotted \nnumerically, and compared with the quantum calculation in the next section. \n\\begin{equation}\\label{heat}\n\\def2.{2.}\n\\dfrac{C_V}{N} = \\left\\{ \\begin{array}{ll}\n\\dfrac{35}{4}\\dfrac{g_{7\/2}(z)}{g_{5\/2}(z)}-\n                \\dfrac{25}{4}\\dfrac{g_{5\/2}(z)}{g_{3\/2}(z)},~~ &T>T_c \\\\\n                       \\dfrac{35}{4}\\dfrac{\\zeta(7\/2)}{\\zeta(5\/2)}\\left(\\dfrac{T}{T_c}\\right)^{5\/2},  &T<T_c\n                       \\end{array} \\right.\n\\end{equation}                       \nNoting that for $T\\rightarrow \\infty$, $z\\rightarrow 0$, and in that limit \n$g_{l}(z)\\rightarrow z$, we deduce from Eq.~(\\ref{heat}) that in this limit \n$\\dfrac{C_V}{N}=2.5 k_B$, where we have inserted the Boltzmann constant $k_B$ \nthat was hitherto suppressed.  This is in accord with the classical equipartition \ntheorem that per degree of freedom the asymptotic heat capacity per particle is \n$k_B\/2$.  Therefore the system has five independent dimensions.  Note also the discontinuity at $T_c$, i.e., $[C_V(T_c^{(+)}) - C_V(T_c^{(-)})]\/Nk_B = -3.209$. \n\n\\section{Quantum calculation for a system of a finite number of particles}\n\\label{sec:4}\nQuantum calculations for systems with a finite number of particles have been done~\\cite{grossmann95b,ketterle96}, which indicate that for trapped particles such systems experience phase transitions even for one or two dimensions.  Such systems have properties that begin to approach even for a small number of particles those in the thermodynamic limit.   Cavalcanti \\textit{et al.}~\\cite{cavalcanti02} raise some doubts about the continuous state approximation in the BEC results of the present problem when compared with the quantum calculations.   For that reason we consider  the exact quantum calculation, but with a finite, but increasing, number of particles.\n\nWe consider a quantum calculation in three spatial dimensions with the bosons subject to uniform gravity, and compare the quantum results to the semiclassical results in five spatial dimensions.\nThe energies spectrum is obtained for a particle in a cylindrical space shown in Fig.~\\ref{fig:01} with $L$ very large.  In that case, setting $\\hbar^2\/2m=1$, the energies are\n\\begin{equation}\\label{eq:39}\n\\epsilon_i = E_{n\\nu s} = k^2_{\\nu s} +\\varepsilon_{n},\n\\end{equation}\nwhere $\\nu$ is the angular quantum number taking values $\\nu = 0,\\pm 1,\\pm 2,\\dots$.   The $k_{\\nu s}a$ are the zeros  of the Bessel function of the first kind $J_\\nu(ka)$,  $\\varepsilon_n=-\\alpha^{2\/3}\\eta_n$ where $\\eta_n$ are the zeros of the Airy function $\\mathrm{Ai}(\\eta)$ and $\\alpha^2= mg$.  The subscript $i$ refers to all three quantum numbers collectively.  \n\nFrom the perspective of the bosonic  grand canonical ensemble, the expression for the number of bosons in a system with energy $\\epsilon_i$ is\n\\begin{equation}\\label{eq:01}\nN_i = \\dfrac{g_i}{e^{\\textstyle \\beta(\\epsilon_i-\\mu)}-1} = \\dfrac{zg_i}{e^{\\textstyle \\beta\\epsilon_i}-z},\n\\end{equation}\nwhere $g_i$ is the degeneracy of the state with energy $\\epsilon_i$; $g_i=1$ when $\\nu=0$ and $g_i=2$ when $\\nu\\neq 0$. We use an energy scale such that $\\epsilon_0=0$. \nThe total number of bosons in the system is\n\\begin{equation}\\label{eq:01a}\n\\bar{N} \n= N_0 +  \\sum_{i=1}^\\infty \\dfrac{g_i}{e^{\\textstyle  \\beta(\\epsilon_i - \\mu)} - 1}.\n\\end{equation}\n     To ensure positive $N_i$'s we require that the chemical potential $\\mu\\leq 0$.  By setting $N_0=0$ and $\\mu=0$, the critical temperature $T_c$ can be defined ~\\cite{ketterle96}, and we solve\n\\begin{equation}\\label{eq:13a}\n\\bar{N} = \\sum_{i=1}^\\infty \\dfrac{g_i}{e^{\\textstyle  \\beta_c\\epsilon_i } - 1}\n\\end{equation}\nfor $\\beta_c$ to obtain $T_c$.\n\nThe chemical potential (or the fugacity) is determined by the constraint that the total number of particles is fixed as $N = \\bar{N}$.  We can obtain $z$ as a function of $T$ or $T\/T_c$ by solving \n\\begin{equation}\\label{eq:02}\nN = \\sum_{i=0}^\\infty \\dfrac{zg_i}{e^{\\textstyle \\beta\\epsilon_i}-z}\n\\end{equation}\nfor $z$ when $T$ is given.  The energy as a function of temperature is\n\\begin{equation}\\label{eq:03}\nE = \\sum_{i=1}^\\infty \\dfrac{zg_i\\epsilon_i e^{\\textstyle  -\\beta\\epsilon_i}}{1-ze^{\\textstyle  -\\beta\\epsilon_i }}.\n\\end{equation}\nThe specific heat is\n\\begin{equation}\\label{eq:35}\n\\begin{split}\n\\dfrac{C_V}{Nk_B} & = \\dfrac{1}{Nk_B}\\left(\\dfrac{\\partial E}\n{\\partial T}\\right)_{N,V} \\\\\n& = z\\left\\{\\dfrac{\\left[ \\displaystyle\\sum_{i=0}^\\infty\n \\dfrac{g_i\\epsilon_ie^{\\textstyle \\beta\\epsilon_i}}{(e^{\\textstyle  \\beta\\epsilon_i}-\n z)^2}\\right]^2}{\\displaystyle\\sum_{i=0}^\\infty \\dfrac{g_ie^{\\textstyle \n  \\beta\\epsilon_i}}{(e^{\\textstyle \\beta\\epsilon_i}-z)^2}} - \\displaystyle\\sum_{i=0}^\\infty \\dfrac{g_i\\epsilon_i^2e^{\\textstyle \\beta\\epsilon_i}}{(e^{\\textstyle \\beta\\epsilon_i}-z)^2}   \\right\\},\n\\end{split}\n\\end{equation}\nwhere in the derivation we have used the constancy of the particle number, $\\dfrac{\\textstyle  dN}{\\textstyle  dT}=0$, in order to obtain an expression for $\\dfrac{\\textstyle \\partial z}{\\textstyle \\partial T}$.  Finally we obtain the isothermal compressibility \n\\begin{equation}\\label{eq:38}\n\\kappa_T = \\dfrac{V}{N^2}\\left(\\dfrac{\\partial N}{\\partial\\mu}\\right)_{V,T} = \\dfrac{V}{N^2} \\beta z \\sum_{i=0}^\\infty \\dfrac{g_ie^{\\textstyle \\beta\\epsilon_i}}{(e^{\\textstyle \\beta\\epsilon_i} -z)^2}.\n\\end{equation} \n\n\\begin{figure}[!h]\n\\centering                                                                                      \n$\\begin{array}{c}\n\\subfigure[~Fugacity as a function of temperature.]{\n\\resizebox{3.2in}{!}{\\includegraphics[angle=-90]\n{fig_2a.eps}}\n\\label{fig:05a}}  \\\\\n\\subfigure[~Energy-state occupation as a function of temperature.]{\n\\resizebox{3.2in}{!}{\\includegraphics[angle=-90]\n{fig_2b.eps}}\n\\label{fig:05b}}\n\\end{array}$\n\\caption{Fugacity and state occupation of system including the ramp potential as a function of temperature.  The results follow from an exact quantum calculation with $a=5$, $\\alpha =1$, and $\\varepsilon_0=0$.  The curves converge as $N$ increases.}\n\\label{fig:05}\n\\end{figure}\nIn Figs.~\\ref{fig:05a} and \\ref{fig:05b} we show the fugacity and the number of particles distribution as a function of temperature.  In Figs.~\\ref{fig:06a} and \\ref{fig:06b} we show the specific heat and the isothermal compressibility as functions of the temperature.  In the ramp calculations care must be taken that a sufficient number of energy eigenstate states are included since the energy spacing decreases as the energies increase.  Thus we use 10,000  energy states from the zeros of the Airy functions and we let the maximum values of $\\nu$ and $s$ be 200.  A good check to see whether a sufficient number of energy states have been included is the high temperature limit of the specific heat which is 5\/2.   The parameters of the calculation are $a=5$ and $\\alpha=1$. \n\\begin{figure}[!t]\n\\centering                                                                                      \n$\\begin{array}{c} \n\\subfigure[~Specific heat as a function of temperature.  The curves converge to the semiclassical one (in thermodynamic limit) as $N$ increases.]{\n\\resizebox{3.2in}{!}{\\includegraphics[angle=-90]\n{fig_3a.eps}}\n\\label{fig:06a}}  \\\\\n\\subfigure[~Isothermal compressibility as a function of temperature. Note that the vertical axis of the inset is a logarithmic scale.]{\n\\resizebox{3.2in}{!}{\\includegraphics[angle=-90]\n{fig_3b.eps}}\n\\label{fig:06b}}\n\\end{array}$\n\\caption{Comparison of quantum and semiclassical calculations of the specific heat and isothermal compressibility.  For the quantum calculation $N= 500, 5,000$ and $10,000$.   Parameters are $a = 5$, $\\alpha = 1$, and $\\varepsilon_0 =0$.}\n\\label{fig:06}\n\\end{figure}\n\nFigures~\\ref{fig:05} and \\ref{fig:06} clearly show the onset of the BEC; the trend as the particle number increases is already nearly saturated for $N=500$.  The quantum calculations for finite numbers of particles shown in Fig.~\\ref{fig:06} approach the semiclassical results which are in thermodynamic limit. The isothermal compressibility has an interesting behaviour  in that the particle number sensitivity seems most pronounced around the critical temperature.  Furthermore the compressibility is finite at the critical temperature and increases rapidly with decreasing temperature.  This corresponds to what is observed in the semi-classical calculation for free bosons in five dimensions.  This is in contrast to the free ideal boson case in three dimensions for which the compressibility approaches infinity when $T$ approaches $T_c$ from above. \nThe semiclassical and quantum calculations for $N=500$ are very close.  We used the volume as a parameter in the semiclassical calculation, and it was chosen to be 350 in order that $z$ remain less than or equal to unity.  In all cases we chose $\\bar{n}_3=1$.  For consistency with the $N=1,000$ and 10,000 cases we should have chosen $V$ to be 1,000 and 10,000, respectively.  If we do the semiclassical calculations with these values of $V$, the compressibility graphs correspond to the quantum graphs with $N=1,000$ and 10,000, respectively.  In the last two cases the semiclassical values for $z$ slightly exceed unity (by say 0.5\\%).\n \n\n\n\\section{Summary}\nIn summary, we note that a five-dimensional  gravity-free ideal Bose gas appears to have \nthe same thermal properties as that of three-dimensional bosons in a uniform \ngravitational field. The semiclassical approximation using continuous density of \nstates is seen to agree with quantum calculations with a finite number of particles, especially as the number of bosons is increased. The divergence in the isothermal compressibility of an ideal Bose gas at $T_c$ is shown to be shifted to zero temperature in  uniform gravity . \n\n\n\\input{rajat_PLA.bbl}\n\\end{document} \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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\\normalsize\n}\n\n\\providecommand{\\highlightTerm}{}\n\\renewcommand{\\highlightTerm}[1]{\\emph{#1}}\n\\providecommand{\\highlightForReview}{}\n\\renewcommand{\\highlightForReview}[1]{%\n    \\bgroup\\relax%\n    \\color{blue}\\relax%\n    #1\\relax%\n    \\egroup\\relax%\n}\n\n\n\n\\def\\@adminfootnotes{%\n    \\let\\@makefnmark\\relax\n    \\let\\@thefnmark\\relax\n    \\ifx\\@empty\\@date\\else%\n        \\@footnotetext{\\@setdate}%\n    \\fi%\n    \\ifx\\@empty\\@subjclass\\else%\n        \\@footnotetext{\\@setsubjclass}%\n    \\fi\n    \\ifx\\@empty\\@keywords\\else%\n        \\@footnotetext{\\@setkeywords}%\n    \\fi\n    \\ifx\\@empty\\thankses\\else%\n        \\@footnotetext{\\def\\par{\\let\\par\\@par}\\@setthanks}%\n    \\fi\n}\n\n\\def\\@settitle{\\Large\\bfseries\\scshape\\@title}\n\n\\def\\@maketitle{%\n  \\normalfont\\normalsize\n  \\@adminfootnotes\n  \\@mkboth{\\@nx\\shortauthors}{\\@nx\\shorttitle}%\n  \\global\\topskip42\\p@\\relax\n  {\\centering\\@settitle}\n  \\ifx\\@empty\\authors\\else{\\centering\\small\\@setauthors}\\fi\n  \\ifx\\@empty\\@date\\else{\\vtop{\\centering\\small\\@date\\@@par}}\\fi\n  \\ifx\\@empty\\@dedicatory%\n  \\else%\n    \\baselineskip\\p@\n    \\vtop{\\centering{\\footnotesize\\itshape\\@dedicatory\\@@par}%\n    \\global\\dimen@i\\prevdepth}\\prevdepth\\dimen@i%\n  \\fi\n  \\@setabstract\n  \\normalsize\n  \\if@titlepage\n    \\newpage\n  \\else\n    \\dimen@34\\p@\\advance\\dimen@-\\baselineskip\n  \\fi\n}\n\n\\def\\addresseshere{%\n  \\bgroup\n  \\setlength{\\parindent}{0pt}\n  \\enddoc@text\n  \\egroup\n  \\let\\enddoc@text\\relax\n}\n\n\\makeatother\n\n\n\n\\begin{document}\n\\startdocumentlayoutoptions\n\n\\thispagestyle{plain}\n\n\n\n\\defAbstract{Abstract}\n\\begin{abstract}\n    We consider characterisations of unitary dilations and approximations\n    of irreversible classical dynamical systems on a Hilbert space.\n   \n    In the commutative case,\n    building on the work in \\cite{Chung1962exp}, one can express well known approximants\n    (\\exempli Hille- and Yosida-approximants)\n    via expectations over certain stochastic processes.\n    Using this, our first result characterises\n    the simultaneous regular unitary dilatability of commuting families of $\\Cnought$-semigroups\n    via the dilatability of such approximants as well as via regular polynomial bounds.\n    This extends the results in \\cite{Dahya2023dilation} to the unbounded setting.\n   \n    We secondly consider characterisations of unitary and regular unitary dilations\n    via continuous and discrete functional calculi.\n    Applying these tools to a large class of classical dynamical systems,\n    these two notions of dilation exactly characterise\n    when a system admits unitary approximations\n    under certain distinct notions of weak convergence.\n    This establishes a sharp topological distinction between the two notions of unitary dilations.\n    Our results are applicable to commutative systems as well as\n    non-commutative systems satisfying the \\emph{canonical commutation relations} (CCR) in the Weyl form.\n\\end{abstract}\n\n\n\n\\subjclass[2020]{47A20, 47D06, 60G55}\n\\keywords{Semigroups of operators; dilations; approximations; point processes; functional calculus; group $C^{\\ast}$-algebras.}\n\\title[Characterisations of dilations via approximants, expectations, and functional calculi]{Characterisations of dilations via approximants, expectations, and functional calculi}\n\\author{Raj Dahya}\n\\email{raj.dahya@web.de}\n\\address{Fakult\\\"at f\\\"ur Mathematik und Informatik\\newline\nUniversit\\\"at Leipzig, Augustusplatz 10, D-04109 Leipzig, Germany}\n\n\\maketitle\n\n\n\n\\setcounternach{section}{1}\n\n\n\n\\@startsection{section}{1}{\\z@}{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}{\\formatsection@text}[Introduction]{Introduction}\n\\label{sec:introduction:sig:article-stochastic-raj-dahya}\n\n\n\\noindent\nClassical dynamical systems on Hilbert or Banach spaces,\nwhich are in general irreversible,\ncan be studied in at least two natural ways\nin terms of more ideal systems:\n    via approximations\n    and\n    via embeddings into (or: \\usesinglequotes{dilations} to) larger reversible systems.\nTowards the former, see \\exempli \\cite{Hillephillips1957faAndSg,Butzer1967semiGrApproximationsBook,Chung1962exp,Krol2009}.\nThe study of the latter was in part inspired by a result from Halmos \\cite{Halmos1950dilation},\nand properly initiated by Sz.-Nagy and Foias in \\cite{Nagy1953,Nagy1970}\nwith their work on unitary (power) dilations of contractions\nand of $1$-parameter contractive $\\Cnought$-semigroups\nover Hilbert spaces.\nRemaining in the commutative setting,\nvarious results have been achieved for systems consisting of multiple operators\nas well as multi-parameter $\\Cnought$-semigroups\n(see \\exempli\n    \\cite{Ando1963pairContractions,Slocinski1974,Slocinski1982,Ptak1985,LeMerdy1996DilMultiParam,Shamovich2017dilationsMultiParam}).\nFor a good overview, see \\exempli \\cite{Averson2010DilationOverview,Shalit2021DilationBook}.\n\nIn the \\textbf{first part} of this paper\n(\\S{}\\ref{sec:stochastic:sig:article-stochastic-raj-dahya}--\\ref{sec:first-results:sig:article-stochastic-raj-dahya}),\nwe shall first consider commutative systems of $\\Cnought$-semigroups.\nNote that a commuting family, $\\{U_{i}\\}_{i=1}^{d}$,\nof unitary $\\Cnought$-semigroups on a Hilbert space $\\HilbertRaum$\ncan be uniquely extended to an \\topSOT-continuous unitary representation,\n$U$, of $(\\reals^{d},+,\\zerovector)$ on $\\HilbertRaum$\ndefined via\n    $\n        U(\\mathbf{t})\n        \\colonequals\n        (\n            \\prod_{\\mathclap{i\\in\\mathop{\\textup{supp}}(\\mathbf{t}^{-})}}\n            U_{i}(t_{i}^{-})\n        )^{\\ast}\n        (\n            \\prod_{\\mathclap{i\\in\\mathop{\\textup{supp}}(\\mathbf{t}^{+})}}\n            U_{i}(t_{i}^{+})\n        )\n        $\nfor all $\\mathbf{t} = (t_{i})_{i=1}^{d} \\in \\reals^{d}$,\nwhere\n    $t^{-}=\\max\\{-t,\\,0\\}$\n    and\n    $t^{+}=\\max\\{t,\\,0\\}$\nfor $t\\in\\reals$\n(\\cf Stone's theorem \\cite[Theorem~I.4.7]{Goldstein1985semigroups}).\nBearing this in mind, a commuting family,\n    $\\{T_{i}\\}_{i=1}^{d}$,\nof $\\Cnought$-semigroups on $\\HilbertRaum$\nis said to have a\n    \\highlightTerm{simultaneous regular unitary dilation}\nif\n\n    \\vspace{-3\\parskip}\n    $$\n        \\Big(\n            \\prod_{i=1}^{d}T(t_{i}^{-})\n        \\Big)^{\\ast}\n        \\Big(\n            \\prod_{i=1}^{d}T(t_{i}^{+})\n        \\Big)\n        =\n        r^{\\ast}\\,U(\\mathbf{t})\\,r\n    $$\n\n\\noindent\nholds for all $\\mathbf{t} = (t_{i})_{i=1}^{d} \\in \\reals^{d}$,\nfor\n    some $\\topSOT$-continuous unitary representation $U$ of $(\\reals^{d},+,0)$ on a Hilbert space $\\HilbertRaum_{1}$\n    and\n    some isometry ${r\\in\\BoundedOps{\\HilbertRaum}{\\HilbertRaum_{1}}}$\n(and in this case we shall refer to the data\n    $(\\HilbertRaum_{1},U,r)$\nas the simultaneous regular unitary dilation).\nA \\highlightTerm{simultaneous unitary dilation}\nis defined by the above condition restricted to $\\mathbf{t} \\in \\realsNonNeg^{d}$.\n\nFor the cases $d\\in\\{1,2\\}$ it was proved in\n    \\cite[Theorem~I.8.1]{Nagy1970},\n    \\cite{Slocinski1974},\n    \\cite[Theorem~2]{Slocinski1982},\n    and\n    \\cite[Theorem~2.3]{Ptak1985},\nthat all contractive commuting families $\\{T_{i}\\}_{i=1}^{d}$\nhave a simultaneous unitary dilation.\nIn the case of $d=1$, these are in fact regular unitary dilations.\nIn the general case of $d \\geq 1$,\nthe existence of simultaneous regular unitary dilations was fully characterised\nin \\cite[Theorem~3.2]{Ptak1985}\nvia a general condition which we may refer to as \\emph{Brehmer positivity}.\nLe~Merdy fully classified the existence of simultaneous unitary dilations\nas well as dilations \\emph{up to similarity}\nvia the complete boundedness of a certain functional calculus map\n(%\n    see \\cite[Theorems~2.2~and~3.1]{LeMerdy1996DilMultiParam},\n    which builds on \\cite[Corollaries~4.9~and~4.13]{Pisier2001bookCBmaps}%\n),\nand successfully applied the latter to commuting families of bounded analytic $\\Cnought$-semigroups.\nMoreover, Shamovich and Vinnivok provided\nsufficient embedding conditions on the generators\nfor the existence of simultaneous unitary dilations\n(see \\cite{Shamovich2017dilationsMultiParam}).\nRecent results contribute to this history by providing\ntwo further complete characterisations of the existence of simultaneous regular unitary dilations\nfor commuting families of $\\Cnought$-semigroups\nunder the assumption of bounded generators\n(see \\cite[Theorems~1.1~and~1.4]{Dahya2023dilation}).\nThe first characterisation was achieved\nvia the notion of \\emph{complete dissipativity}\n(see \\cite[Definition~2.8]{Dahya2023dilation}),\nwhich is a spectral-theoretic notion of a simple combinatorial expression involving the generators.\nThe second characterisation builds on the first\nand characterises the existence of regular unitary dilations\nvia \\emph{regular polynomial bounds}\n(see \\cite[Definition 6.2]{Dahya2023dilation}).\nFurthermore, analogue to \\cite{LeMerdy1996DilMultiParam},\nit was shown that all commuting families of $\\Cnought$-semigroups\nwith bounded generators have regular unitary dilations\nup to certain natural modification\n(see \\cite[Corollary~1.2]{Dahya2023dilation}).\n\nThe latter reference left open the question,\nwhether the characterisation via regular polynomial bounds\ncould be extended to the unbounded setting\n(see \\cite[Remark~6.6]{Dahya2023dilation}).\nMoreover, the characterisation via complete dissipativity involves a characterisation via approximants\nwhich raises the question, whether for certain \\emph{natural choices of approximants},\na commuting family of $\\Cnought$-semigroups has a simultaneous regular unitary dilation\nif and only if families of their approximants do.\nThe current paper shall answer both these questions positively\nin the general setting without the boundedness assumption\n(see \\S{}\\ref{sec:first-results:sig:article-stochastic-raj-dahya}).\n\nIn the \\textbf{second part} of this paper\n(\\S{}\\ref{sec:functional-calculus:sig:article-stochastic-raj-dahya}--\\ref{sec:second-results:sig:article-stochastic-raj-dahya}),\nwe consider classical dynamical systems more broadly described by\n\\emph{homomorphisms} defined over topological monoids.\nTo motivate this, observe that there is a natural correspondence between\ncommuting families, $\\{T_{i}\\}_{i=1}^{d}$, of\n(bounded\/contractive\/unitary) $\\Cnought$-semigroups\nand\n(bounded\/contractive\/unitary) \\topSOT-continuous homomorphisms,\n$T$, between $(\\realsNonNeg^{d},+,\\zerovector)$ and spaces of operators\n(\\cf \\cite[\\S{}1]{Dahya2023dilation})\nvia\n    $T(\\mathbf{t}) = \\prod_{i=1}^{d}T_{i}(t_{i})$\n    and\n    $T_{i}(t) = T_{i}(t) = T(0,0,\\ldots,\\underset{i}{t},\\ldots,0)$\nfor $\\mathbf{t}\\in\\realsNonNeg^{d}$, $t\\in\\realsNonNeg$,\n$i\\in\\{1,2,\\ldots,d\\}$.\nIt is thus natural to consider homomorphisms defined over topological monoids.\nIf $G$ is a topological group and $M \\subseteq G$ is a submonoid\nand ${T:M\\to\\BoundedOps{\\HilbertRaum}}$ is an \\topSOT-continuous homomorphism from $M$\nto bounded operators on a Hilbert space $\\HilbertRaum$,\nwe define a \\highlightTerm{unitary dilation} of $T$\nto be a tuple $(\\HilbertRaum_{1},U,r)$,\nwhere\n    $\\HilbertRaum_{1}$ is a Hilbert space,\n    $U$ is an $\\topSOT$-continuous unitary representation $G$ on $\\HilbertRaum_{1}$,\n    and\n    ${r\\in\\BoundedOps{\\HilbertRaum}{\\HilbertRaum_{1}}}$ is an isometry,\nsuch that\n\n    \\vspace{-3\\parskip}\n    $$\n        T(x) = r^{\\ast}\\,U(x)\\,r\n    $$\n\n\\noindent\nholds for all $x \\in M$.\nWe shall also consider monoids for which one can define\na \\emph{positivity structure},\nwhich consists of continuous maps ${\\cdot^{+},\\cdot^{-} \\colon G \\to M}$\n(see \\Cref{defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya}).\nUsing this concept, we say that $(\\HilbertRaum_{1},U,r)$\nis a \\highlightTerm{regular unitary dilation} of $T$,\nif\n\n    \\vspace{-3\\parskip}\n    $$\n        (T(x^{-}))^{\\ast}T(x^{+})\n        =\n        r^{\\ast}\\,U(x)\\,r\n    $$\n\n\\noindent\nholds for all $x \\in G$.\nConsidering $(G,M) = (\\reals^{d},\\realsNonNeg^{d})$, $d\\in\\naturals$,\nby the above mentioned correspondence one can see that these concepts\nagree with the definitions of\nsimultaneous (regular) unitary dilations of commuting families.\n\nWe present in \\S{}\\ref{sec:functional-calculus:sig:article-stochastic-raj-dahya}\ncharacterisations\nof unitary and regular unitary dilations\nvia the means of \\emph{functional calculi},\ninspired by Sz.-Nagy and le~Merdy.\nThese tools enable us to characterise the existence of unitary approximations\nfor a broad class of classical systems\n(see \\S{}\\ref{sec:second-results:sig:article-stochastic-raj-dahya}).\n\nBefore proceeding,\nwe recall the afore mentioned results from the bounded setting,\ndefine our terminology\nand state the main results of this paper.\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Characterisation via complete dissipativity]{Characterisation via complete dissipativity}\n\\label{sec:introduction:dissipativity:sig:article-stochastic-raj-dahya}\n\n\\noindent\nFor a $\\Cnought$-semigroup, $T$, on a Hilbert space $\\HilbertRaum$ with generator $A$,\nif $T$ is contractive then it has a regular unitary dilation\n(\\cf \\cite[Theorem~I.8.1]{Nagy1970}).\nAnd clearly, the latter necessarily requires $T$ to be contractive.\nOn the other hand, by the Lumer-Phillips form of the Hille-Yosida theorem,\n$T$ is contractive if and only if $A$ is dissipative\n(see \\cite[Theorem~I.3.3]{Goldstein1985semigroups}).\nThus the dissipativity of the generator characterises\nthe regular unitary dilatability of a $\\Cnought$-semigroup.\n\nIn the setting of commuting families of semigroups,\ndissipativity can be generalised as follows:\nFor each $k\\in\\naturalsZero$,\nthe \\highlightTerm{$k$\\textsuperscript{th}-order dissipation operators}\nare defined by\n\n    \\vspace{-3\\parskip}\n    $$\n        (-\\tfrac{1}{2})^{\\card{K}}\n        \\sum_{\\isPartition{(C_{1},C_{2})}{K}}\n            \\Big(\n                \\prod_{i\\in C_{1}}\n                    A_{i}\n            \\Big)^{\\ast}\n            \\prod_{j\\in C_{2}}\n                A_{j}\n    $$\n\n\\noindent\nfor $K \\subseteq \\{1,2,\\ldots,d\\}$ with $\\card{K} = k$,\nwhere $\\isPartition{(C_{1},C_{2})}{K}$ denotes\nthat $C_{1},C_{2} \\subseteq K$ form a partition of $K$.\nWe say that the generators $\\{A_{i}\\}_{i=1}^{d}$ are \\highlightTerm{completely dissipative},\nif the dissipation operators of all finite orders are positive\n(\\cf \\cite[Definition~2.8]{Dahya2023dilation}).\nThis notion leads to a characterisation result obtained in\n    \\cite{Dahya2023dilation},\nwhich for the purposes of this paper we restate as follows:\n\n\\begin{thm}[Characterisation via complete dissipativity]\n\\makelabel{thm:classification:dissipativity:sig:article-stochastic-raj-dahya}\n    Let\n        $d\\in\\naturals$,\n        $\\HilbertRaum$ be a Hilbert space,\n        and\n        $\\{T_{i}\\}_{i=1}^{d}$\n        be a commuting family of $\\Cnought$-semigroups on $\\HilbertRaum$,\n        with generators $\\{A_{i}\\}_{i=1}^{d}$.\n    If the semigroups have bounded generators,\n    then the following are equivalent:\n\n    \\begin{kompaktenum}{\\bfseries (a)}\n        \\item\\punktlabel{1}\n            The family $\\{T_{i}\\}_{i=1}^{d}$ has a simultaneous regular unitary dilation.\n        \\item\\punktlabel{2}\n            The generators $\\{A_{i}\\}_{i=1}^{d}$ are completely dissipative.\n        \\item\\punktlabel{3}\n            There exists a net $(\\{T^{(\\alpha)}_{i}\\}_{i=1}^{d})_{\\alpha \\in \\Lambda}$\n            consisting of\n            a commuting families of $\\Cnought$-semigroups on $\\HilbertRaum$,\n            which each have simultaneous regular unitary dilations,\n            such that\n                $$\n                    \\sup_{\\mathbf{t} \\in L}\n                        \\normLong{\n                            \\Big(\n                                \\prod_{i=1}^{d}T^{(\\alpha)}_{i}(t_{i})\n                                -\n                                \\prod_{i=1}^{d}T_{i}(t_{i})\n                            \\Big)\n                            \\xi\n                        }\n                    \\underset{\\alpha}{\\longrightarrow} 0\n                $$\n            for all\n                $\\xi \\in \\HilbertRaum$\n                and\n                compact $L \\subseteq \\realsNonNeg^{d}$.\n    \\end{kompaktenum}\n\n    \\noindent\n    Furthermore, the notion of convergence in \\punktcref{3}\n    can be replaced by pointwise \\topSOT-convergence.\n    The implication\n        \\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1}\n    also holds without the boundedness assumption.\n\\end{thm}\n\nSee \\cite[Theorem~1.1 and Remark~4.4]{Dahya2023dilation} for a proof.\nNote that in \\punktcref{3} the choice of nets of approximants\nis artificially constructed in \\cite{Dahya2023dilation}.\nThus it is not immediate that\n    \\punktcref{1}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}\nholds for any given choice (even in the bounded setting).\nThis leads to a natural question (in the general setting),\nwhich motivated the research in the present paper:\n\n\\begin{qstn}\n\\makelabel{qstn:classical-approximants:dilation:sig:article-stochastic-raj-dahya}\n    Let $\\{T_{i}\\}_{i=1}^{d}$ be a commuting family of\n    $\\Cnought$-semigroups on a Hilbert space $\\HilbertRaum$.\n    For which choices of approximants,\n        $(\\{T^{(\\alpha)}_{i}\\}_{i=1}^{d})_{\\alpha\\in\\Lambda}$,\n    does it hold that\n    the simultaneous regular unitary dilatability of $\\{T_{i}\\}_{i=1}^{d}$\n    implies that of each family\n        $\\{T^{(\\alpha)}_{i}\\}_{i=1}^{d}$?\n\\end{qstn}\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Characterisation via polynomial bounds]{Characterisation via polynomial bounds}\n\\label{sec:introduction:polynomial:sig:article-stochastic-raj-dahya}\n\n\\noindent\nFor commuting operators\n    $\\{S_{i}\\}_{i=1}^{d} \\subseteq \\BoundedOps{\\HilbertRaum}$,\n\\highlightTerm{regular polynomial evaluation}\nis defined as the unique linear map\n\n    \\vspace{-3\\parskip}\n    $$\n        \\complex[X_{1},X_{1}^{-1},X_{2},X_{2}^{-1},\\ldots,X_{d},X_{d}^{-1}] \\ni p\n        \\mapsto p(S_{1},S_{2},\\ldots,S_{d}) \\in \\BoundedOps{\\HilbertRaum}\n    $$\n\n\\noindent\nsatisfying\n\n    \\vspace{-3\\parskip}\n    $$\n        p(S_{1},S_{2},\\ldots,S_{d})\n        = \\Big(\\prod_{\\mathclap{i\\in\\mathop{\\textup{supp}}(\\mathbf{n}^{-})}}S_{i}^{-n_{i}}\\Big)^{\\ast}\n        \\Big(\\prod_{\\mathclap{i\\in\\mathop{\\textup{supp}}(\\mathbf{n}^{+})}}S_{i}^{n_{i}}\\Big)\n    $$\n\n\\noindent\nfor all monomials of the form\n    $p = \\prod_{i=1}^{d}X_{i}^{n_{i}}$\nwith $\\mathbf{n} \\in \\integers^{d}$\n(\\cf \\cite[Definition~6.1]{Dahya2023dilation}).\nWe say that $\\{T_{i}\\}_{i=1}^{d}$ satisfies\n\\highlightTerm{regular polynomial bounds}\nif\n\n    \\vspace{-3\\parskip}\n    $$\n        \\norm{\n            p(T_{1}(t_{1}),T_{2}(t_{2}),\\ldots,T_{d}(t_{d}))\n        } \\leq \\sup_{\\boldsymbol{\\lambda} \\in \\Torus^{d}}\n        \\abs{\n            p(\\lambda_{1},\\lambda_{2},\\ldots,\\lambda_{d})\n        }\n    $$\n\n\\noindent\nholds for all $\\mathbf{t} = (t_{i})_{i=1}^{d} \\in \\realsNonNeg^{d}$,\nwhere $\\Torus$ is the unit circle in the complex plane.\nUsing these notions, a second characterisation is obtained\nin \\cite{Dahya2023dilation},\nwhich for the purposes of this paper, we restate as follows:\n\n\\begin{thm}[Characterisation via polynomial bounds]\n\\makelabel{thm:classification-bounded:poly:sig:article-stochastic-raj-dahya}\n    Let\n        $d\\in\\naturals$,\n        $\\HilbertRaum$ be a Hilbert space,\n        and\n        $\\{T_{i}\\}_{i=1}^{d}$\n        be a commuting family of $\\Cnought$-semigroups on $\\HilbertRaum$\n        with generators $\\{A_{i}\\}_{i=1}^{d}$.\n    If the semigroups have bounded generators,\n    then the following are equivalent:\n\n    \\begin{kompaktenum}{\\bfseries (a)}\n        \\item\\punktlabel{1}\n            The family $\\{T_{i}\\}_{i=1}^{d}$ has a simultaneous regular unitary dilation.\n        \\item\\punktlabel{2}\n            The family $\\{T_{i}\\}_{i=1}^{d}$ satisfies regular polynomial bounds.\n        \\item\\punktlabel{3}\n            For each ${K \\subseteq \\{1,2,\\ldots,d\\}}$\n            and all ${\\mathbf{t}=(t_{i})_{i=1}^{d} \\in \\realsNonNeg^{d}}$\n            the operator\n                $p_{K}(T_{1}(t_{1}),T_{2}(t_{2}),\\ldots,T_{d}(t_{d}))$\n            is positive, where\n\n            \\vspace{-3\\parskip}\n            \\begin{eqnarray*}\n                p_{K}\n                    \\colonequals\n                        \\displaystyle\n                        \\sum_{\\mathclap{\n                            \\isPartition{(C_{1},C_{2})}{K}\n                        }}\n                            \\displaystyle\n                            \\prod_{i \\in C_{1}}\n                                (1 - X_{i}^{-1})\n                            \\cdot\n                            \\displaystyle\n                            \\prod_{j \\in C_{2}}\n                                (1 - X_{j}).\\\\\n            \\end{eqnarray*}\n\n        \\item\\punktlabel{4}\n            The generators $\\{A_{i}\\}_{i=1}^{d}$ are completely dissipative.\n    \\end{kompaktenum}\n\n    \\noindent\n    Furthermore, the implications\n        \\punktcref{1}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}\n    hold without the boundedness assumption.\n\\end{thm}\n\nDue to the inclusion of the intermediate step \\punktcref{3},\nwe sketch the proof for the reader's convenience.\n\n    \\begin{proof}[of \\Cref{thm:classification-bounded:poly:sig:article-stochastic-raj-dahya}, sketch]\n        The equivalence of\n            \\punktcref{1}, \\punktcref{2}, and \\punktcref{4}\n        is proved directly in \\cite[Theorem~1.4]{Dahya2023dilation} in reliance upon \\Cref{thm:classification:dissipativity:sig:article-stochastic-raj-dahya}.\n        For \\punktcref{1}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{2}\n        the assumption of bounded generators is not required\n        (see \\cite[Remark~6.6]{Dahya2023dilation}).\n\n        Towards \\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}:\n        Suppose (without the boundedness assumption!)\n        that the family $\\{T_{i}\\}_{i=1}^{d}$ satisfies regular polynomial bounds.\n        Let\n            $K \\subseteq \\{1,2,\\ldots,d\\}$\n            and\n            $\\mathbf{t} = (t_{i})_{i=1}^{d} \\in \\realsNonNeg^{d}$\n        be arbitrary.\n        Using binomial expansions, one can see that the regular polynomial, $p_{K}$,\n        can be expressed as\n            $\n                p_{K}\n                = \\prod_{i=1}^{d}(2 - X_{i} - X_{i}^{-1})\n            $,\n        and thus\n            $\n                p_{K}(\\lambda_{1},\\lambda_{2},\\ldots,\\lambda_{d})\n                = \\prod_{i=1}^{d}(2 - \\lambda_{i} - \\lambda_{i}^{-1})\n                = \\prod_{i=1}^{d}(2 - 2\\mathop{\\mathfrak{R}\\mathrm{e}}\\lambda_{i})\n                \\in [0, 4^{d}]\n            $\n        for all $\\boldsymbol{\\lambda} = (\\lambda_{1},\\lambda_{2},\\ldots,\\lambda_{d}) \\in \\Torus^{d}$.\n        It follows that\n            $\n                \\sup_{\\boldsymbol{\\lambda}}\n                    \\abs{1 - \\alpha p_{K}(\\lambda_{1},\\lambda_{2},\\ldots,\\lambda_{d})}\n                \\leq 1\n            $\n        for sufficiently small $\\alpha\\in\\realsPos$.\n        Since the family of semigroups satisfies regular polynomial bounds,\n        it follows that\n            $\n                \\norm{\\onematrix - \\alpha p_{K}(T_{1}(t_{1}), T_{2}(t_{2}), \\ldots, T_{d}(t_{d}))}\n                \\leq 1\n            $\n        for sufficiently small $\\alpha\\in\\realsPos$.\n        As argued in the proof of \\cite[Theorem~1.4]{Dahya2023dilation}, this implies that\n            $p_{K}(T_{1}(t_{1}), T_{2}(t_{2}), \\ldots, T_{d}(t_{d}))$\n        is a positive operator.\n\n        Finally, the implication \\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4}\n        (under the assumption of bounded generators)\n        is proved exactly as in \\cite[Theorem~1.4]{Dahya2023dilation}.\n        By taking limits of these positive operators appropriately scaled,\n        one obtains that the generators are completely dissipative.\n    \\end{proof}\n\nThis result raises the question\n(\\cf \\cite[Remark~6.6]{Dahya2023dilation}),\nwhich along with \\Cref{qstn:classical-approximants:dilation:sig:article-stochastic-raj-dahya}\nalso motivated the research in the current paper.\n\n\\begin{qstn}\n\\makelabel{qstn:polynomial:sig:article-stochastic-raj-dahya}\n    Does the equivalence of\n        \\eqcref{it:1:thm:classification-bounded:poly:sig:article-stochastic-raj-dahya}\n        and\n        \\eqcref{it:2:thm:classification-bounded:poly:sig:article-stochastic-raj-dahya}\n        in\n        \\Cref{thm:classification-bounded:poly:sig:article-stochastic-raj-dahya}\n    continue to hold without the assumption of bounded generators?\n\\end{qstn}\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Characterisation via expectation-approximants]{Characterisation via expectation-approximants}\n\\label{sec:introduction:results:sig:article-stochastic-raj-dahya}\n\n\\noindent\nWe shall demonstrate a further characterisation related\nto the above two results\nwithout the assumption of bounded generators.\nThe key idea is to make the implication\n    \\eqcref{it:1:thm:classification:dissipativity:sig:article-stochastic-raj-dahya}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\eqcref{it:3:thm:classification:dissipativity:sig:article-stochastic-raj-dahya}\n    of\n    \\Cref{thm:classification:dissipativity:sig:article-stochastic-raj-dahya}\nwork by considering suitable canonical choices for approximants\n(\\cf \\Cref{qstn:classical-approximants:dilation:sig:article-stochastic-raj-dahya}).\n\nLet $T$ be an arbitrary $\\Cnought$-semigroup on a Banach space $\\BanachRaum$\nwith generator\n    ${A \\colon \\opDomain{A} \\subseteq \\BanachRaum \\to \\BanachRaum}$.\nWe now consider two concrete nets of approximants\nof the form\n    $(\n        T^{(\\lambda)}\n        = (\n            e^{tA^{(\\lambda)}}\n        )_{\\mathbf{t} \\in \\realsNonNeg^{d}}\n    )_{\\lambda \\in I}$\nfor some $I \\subseteq \\realsPos$ directed by increasing values of $\\lambda$.\n\nIf $\\omega_{0}(T)\\in[-\\infty,\\:\\infty)$ is the \\highlightTerm{growth bound} for $T$\n(\\cf \\cite[Proposition~I.5.5 and Definition~I.5.6]{EngelNagel2000semigroupTextBook},\n    \\cite[Lemma~I.2.12]{Goldstein1985semigroups}%\n),\nthe \\highlightTerm{$\\lambda$\\textsuperscript{th}-Yosida-approximant}\nis defined by\n    $\n        T^{(\\lambda)} = (e^{tA^{(\\lambda)}})_{t\\in\\realsNonNeg}\n    $\nfor each $\\lambda \\in (\\omega_{0}(T),\\infty)$,\nwhere\n\n    \\vspace{-3\\parskip}\n    \\begin{eqnarray}\n    \\label{eq:yosida-approx-generator:sig:article-stochastic-raj-dahya}\n        A^{(\\lambda)}\n        \\colonequals\n            \\lambda A \\opResolvent{A}{\\lambda}\n        = \\lambda^{2} \\opResolvent{A}{\\lambda}\n            - \\lambda\\onematrix\n    \\end{eqnarray}\n\n\\noindent\nis a bounded operator.\nThe Yosida-approximants satisfy\n\n    \\vspace{-3\\parskip}\n    $$\n        \\sup_{t \\in L}\\norm{(T^{(\\lambda)}(t) - T(t))\\xi} \\longrightarrow 0\n    $$\n\n\\noindent\nas ${(\\omega_{0}(T),\\infty) \\ni \\lambda \\longrightarrow \\infty}$\nfor all\n    $\\xi \\in \\BanachRaum$\n    and\n    compact $L \\subseteq \\realsNonNeg$.\nFurthermore, if $T$ is contractive, then each of the Yosida-approximants are contractive.\n(For a proof of these classical results, see \\exempli\n    \\cite[Theorem~G.4.3]{HytNeervanMarkLutz2016bookVol2},\n    \\cite[Theorem~II.3.5, pp.~73--74]{EngelNagel2000semigroupTextBook},\n    \\cite[(12.3.4), p.~361]{Hillephillips1957faAndSg}.%\n)\n\nNow, if a family, $\\{T_{i}\\}_{i=1}^{d}$, of commuting $\\Cnought$-semigroups\nhas a simultaneous regular unitary dilation,\nthen in particular each of the $T_{i}$ must be contractive,\nand thus the family, $\\{T^{(\\lambda_{i})}_{i}\\}_{i=1}^{d}$,\nof Yosida-approximants consists of contractive $\\Cnought$-semigroups\nfor each $\\boldsymbol{\\lambda} = (\\lambda_{i})_{i=1}^{d} \\in \\realsPos^{d}$\n(\\cf the subsequent paragraph below (3.8) in \\cite{EngelNagel2000semigroupTextBook}).\nFurthermore, the commutativity of the $T_{i}$ implies the commutativity of the resolvents,%\n\\footnote{\n    see \\exempli \\cite[Theorem~1]{Abdelaziz1983commutingSemigroups},\n    where this is proved under slightly more general assumptions.\n    One can also directly verify this\n    by relying on the Laplace integral representation of resolvents.\n}\nwhich by \\eqref{eq:yosida-approx-generator:sig:article-stochastic-raj-dahya}\nimplies the commutativity of the generators\n    $\\{A^{(\\lambda_{i})}_{i}\\}_{i=1}^{d}$,\nwhich in turn implies that\n    $\\{T^{(\\lambda_{i})}\\}_{i=1}^{d}$\nis a commuting family.\n\nAnother appropriate approximation is constructed in \\emph{Hille's first exponential formula}.\nFor $\\lambda\\in\\realsPos$ call $T^{(\\lambda)} = (e^{tA^{(\\lambda)}})_{t\\in\\realsNonNeg}$\nthe \\highlightTerm{$\\lambda$\\textsuperscript{th}-Hille-approximant} for $T$,\nwhere\n\n    \\vspace{-3\\parskip}\n    \\begin{eqnarray}\n    \\label{eq:hille-approx-generator:sig:article-stochastic-raj-dahya}\n        A^{(\\lambda)}\n            \\colonequals\n                \\lambda (T(\\tfrac{1}{\\lambda}) - \\onematrix).\n    \\end{eqnarray}\n\n\\noindent\nis a bounded operator.\nThen again it holds that\n    ${T^{(\\lambda)}(t) \\longrightarrow T(t)}$\nfor ${\\realsPos \\ni \\lambda \\longrightarrow \\infty}$\n    \\wrt the \\topSOT-topology\n    uniformly in $t$\n    on compact subsets of $\\realsNonNeg$\n(see \\cite[\\S{}1.2 and Theorem~1.2.2]{Butzer1967semiGrApproximationsBook}).\nIt is easy to verify that\n    $\\norm{T^{(\\lambda)}(t)} \\leq e^{-\\lambda t}e^{\\lambda t \\norm{T(\\tfrac{1}{\\lambda})}}$.\nThus the Hille-approximants of contractive $\\Cnought$-semigroups\nare themselves contractive $\\Cnought$-semigroups\nwith bounded generators.\nMoreover, if $\\{T_{i}\\}_{i=1}^{d}$ is a commuting family of (contractive) $\\Cnought$-semigroups,\nthen for each $\\boldsymbol{\\lambda} = (\\lambda_{i})_{i=1}^{d} \\in \\realsPos^{d}$\nby \\eqcref{eq:hille-approx-generator:sig:article-stochastic-raj-dahya}\nthe generators\n    $\\{A^{(\\lambda_{i})}\\}_{i=1}^{d}$\nclearly commute,\nso that the family of Hille-approximants\n    $\\{T^{(\\lambda_{i})}\\}_{i=1}^{d}$\nis a commuting family of (contractive) $\\Cnought$-semigroups.\n\nIt turns out that these classically defined approximants in semigroup theory\ncan be naturally generalised to a class of approximants,\nwhich we shall call \\highlightTerm{expectation-approximants}\n(see \\Cref{defn:expectation-approximants:sig:article-stochastic-raj-dahya} below).\nWe now state the first main result of this paper:\n\n\\begin{schattierteboxdunn}[backgroundcolor=leer,nobreak=false]\n\\begin{thm}\n\\makelabel{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n    Let\n        $d\\in\\naturals$,\n        $\\HilbertRaum$ be a Hilbert space,\n        and\n        $\\{T_{i}\\}_{i=1}^{d}$\n        be a commuting family of contractive $\\Cnought$-semigroups on $\\HilbertRaum$.\n    Further let\n        $(T^{(\\alpha)}_{i})_{\\alpha \\in \\Lambda_{i}}$\n    be a net of expectation-approximants for $T_{i}$\n    (\\exempli Hille- and Yosida-approximants)\n    for each $i\\in\\{1,2,\\ldots,d\\}$.\n    Then the following are equivalent:\n\n    \\begin{kompaktenum}{\\bfseries (a)}\n        \\item\\punktlabel{1}\n            The family $\\{T_{i}\\}_{i=1}^{d}$ has a simultaneous regular unitary dilation.\n        \\item\\punktlabel{2}\n            The family $\\{T_{i}\\}_{i=1}^{d}$ satisfies regular polynomial bounds.\n        \\item\\punktlabel{3}\n            For each ${K \\subseteq \\{1,2,\\ldots,d\\}}$\n            and all ${\\mathbf{t}=(t_{i})_{i=1}^{d} \\in \\realsNonNeg^{d}}$\n            the operator\n                $p_{K}(T_{1}(t_{1}),T_{2}(t_{2}),\\ldots,T_{d}(t_{d}))$\n            is positive, where\n\n            \\vspace{-3\\parskip}\n            \\begin{eqnarray*}\n                p_{K}(X_{1},X_{2},\\ldots,X_{d})\n                    \\colonequals\n                        \\displaystyle\n                        \\sum_{\\mathclap{\n                            \\isPartition{(C_{1},C_{2})}{K}\n                        }}\n                            \\displaystyle\n                            \\prod_{i \\in C_{1}}\n                                (1 - X_{i}^{-1})\n                            \\cdot\n                            \\displaystyle\n                            \\prod_{j \\in C_{2}}\n                                (1 - X_{j}).\\\\\n            \\end{eqnarray*}\n\n        \\item\\punktlabel{4}\n            The family,\n                $\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d}$,\n            of approximants\n            has a simultaneous regular unitary dilation\n            for each $\\boldsymbol{\\alpha} = (\\alpha_{i})_{i=1}^{d} \\in \\prod_{i=1}^{d}\\Lambda_{i}$.\n    \\end{kompaktenum}\n\n    \\@ifnextchar\\bgroup{\\nvraum@c}{\\nvraum@bes}{1}\n\n\\end{thm}\n\\end{schattierteboxdunn}\n\nBy \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya},\nwe shall be able to positively address\n    \\Cref{%\n        qstn:classical-approximants:dilation:sig:article-stochastic-raj-dahya,%\n        qstn:polynomial:sig:article-stochastic-raj-dahya%\n    }.\nNote that the main implication to prove is\n    \\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4},\nas the others have essentially been proved in\n\\cite{Dahya2023dilation}\n(\\cf the discussion in \\S{}\\ref{sec:introduction:dissipativity:sig:article-stochastic-raj-dahya}--\\ref{sec:introduction:polynomial:sig:article-stochastic-raj-dahya}).\nNow, in the case of bounded generators the spectral-theoretic concept of complete dissipativity\nwas crucial for the characterisations.\nIn the unbounded setting, it is unclear how to extend the notion\nof dissipation operators and thus of complete dissipativity.\nNonetheless, one possibility and its limitations shall be discussed\n(\\cf \\Cref{rem:complete-dissipativity:sig:article-stochastic-raj-dahya}).\nInstead, we make use of stochastic methods,\nbuilding on the results in \\cite{Chung1962exp}.\nUsing suitable stochastic processes\nand expectations computed strongly via Bochner-integrals,\nwhich we lay out in \\S{}\\ref{sec:stochastic:sig:article-stochastic-raj-dahya},\nwe show that expectation-approximants\ncan be expressed in terms of their original semigroups.\nThis provides a key ingredient to prove \\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4}.\nThis shall all be covered in\n    \\S{}\\ref{sec:stochastic:sig:article-stochastic-raj-dahya}--\\ref{sec:first-results:sig:article-stochastic-raj-dahya}.\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Special conditions on topological monoids]{Special conditions on topological monoids}\n\\label{sec:introduction:monoids:sig:article-stochastic-raj-dahya}\n\n\\noindent\nIn the second part of this paper, we concentrate on classical dynamical systems\ndefined more broadly over topological monoids.\nWe are particularly interested in topological monoids, $M$,\nwhich occur as (closed or at least measurable) submonoids of (ideally locally compact) topological groups $G$.\n\n\\begin{e.g.}\n    Simple examples of locally compact topological groups and closed submonoids thereof\n    include any discrete group and submonoid thereof,\n    \\exempli $(\\integers^{d},+,\\zerovector)$ and $\\naturalsZero^{d}\\subseteq\\integers^{d}$ for $d\\in\\naturalsPos$.\n    We also consider\n        $\\realsNonNeg^{d}$\n    as a closed submonoid of the locally compact topological group $(\\reals^{d},+,\\zerovector)$.\n\\end{e.g.}\n\n\\begin{e.g.}\n    As a non-discrete and non-commutative example,\n    consider the Heisenberg group\n    $\\Heisenberg_{d}$ of order $2d+1$, $d\\in\\naturalsPos$,\n    which can be represented topologically as\n    ${\\Heisenberg_{d}=\\reals^{d}\\times\\reals^{d}\\times\\reals}$\n    and algebraically via the operation%\n    \\footnote{\n        There are various different presentations of the Heisenberg group in the literature\n        (\\cf \\cite[\\S{}1]{Coburn1999artHeisneberg}, \\cite[\\S{}10.1]{Deitmar2014bookHarmonicAn}, \\cite[\\S{}6.7.4]{Folland2015bookHarmonicAnalysis}).\n        We choose this particular form for convenience.\n    }\n\n        \\vspace{-3\\parskip}\n        $$\n            (\\mathbf{x},\\mathbf{p},E)\n            (\\mathbf{x}^{\\prime},\\mathbf{p}^{\\prime},E^{\\prime})\n            = \\Big(\n                \\mathbf{x} + \\mathbf{x}^{\\prime},\n                \\mathbf{p} + \\mathbf{p}^{\\prime},\n                E + E^{\\prime}\n                + \\frac{1}{2}(\n                    \\brkt{\\mathbf{p}}{\\mathbf{x}^{\\prime}}\n                    - \\brkt{\\mathbf{p}^{\\prime}}{\\mathbf{x}}\n                )\n            \\Big)\n        $$\n\n    \\noindent\n    for\n        $\n            (\\mathbf{x},\\mathbf{p},E),\n            (\\mathbf{x}^{\\prime},\\mathbf{p}^{\\prime},E^{\\prime})\n            \\in \\Heisenberg_{d}\n        $.\n    The identity element of $\\Heisenberg_{d}$ is clearly $(\\zerovector,\\zerovector,0)$\n    and inverse of $g\\in\\Heisenberg_{d}$ is given by $g^{-1}=-g$,\n    if we view $g$ as a vector in $\\reals^{2d+1}$.\n    One can readily verify that\n\n        \\vspace{-3\\parskip}\n        $$\n            \\Heisenberg^{+}_{d}\n            \\colonequals\n            \\{\n                (\\mathbf{x},\\mathbf{p},E)\n                \\mid\n                \\mathbf{x},\\mathbf{p}\\in\\realsNonNeg^{d},\n                E\\in\\reals\n            \\}\n            \\subseteq \\Heisenberg_{d}\n        $$\n\n    \\noindent\n    is a closed submonoid.\n    As a further non-commutative example,\n    consider for some antisymmetric matrix $C \\in M_{d}(\\reals)$\n    the closed subgroup\n        $\n            \\Heisenberg_{d,C}\n            \\colonequals\n            \\{\n                (\\mathbf{x},C\\mathbf{x},E)\n                \\mid\n                \\mathbf{x}\\in\\reals^{d},\n                E\\in\\reals\n            \\}\n            \\subseteq \\Heisenberg_{d}\n        $\n    and the closed submonoid\n        $\n            \\Heisenberg^{+}_{d,C}\n            \\colonequals\n            \\{\n                (\\mathbf{x},C\\mathbf{x},E)\n                \\mid\n                \\mathbf{x}\\in\\realsNonNeg^{d},\n                E\\in\\reals\n            \\}\n            \\subseteq \\Heisenberg_{d,C}\n        $.\n    Without loss of generality, we can replace the above representations of\n        $\\Heisenberg_{d,C}$ and $\\Heisenberg^{+}_{d,C}$\n    by\n        $\\reals^{d}\\times\\reals$ and $\\realsNonNeg^{d}\\times\\reals$\n    respectively.\n    Observe that the group operation satisfies\n\n        \\vspace{-3\\parskip}\n        $$\n            (\\mathbf{x},E)\n            (\\mathbf{x}^{\\prime},E)\n            = \\Big(\n                \\mathbf{x} + \\mathbf{x}^{\\prime},\n                E + E^{\\prime}\n                + \\frac{1}{2}(\n                    \\brkt{C\\mathbf{x}}{\\mathbf{x}^{\\prime}}\n                    - \\brkt{C\\mathbf{x}^{\\prime}}{\\mathbf{x}}\n                )\n            \\Big)\n            = \\Big(\n                \\mathbf{x} + \\mathbf{x}^{\\prime},\n                E + E^{\\prime} + \\brkt{C\\mathbf{x}}{\\mathbf{x}^{\\prime}}\n            \\Big)\n        $$\n\n    \\noindent\n    for $(\\mathbf{x},E),(\\mathbf{x}^{\\prime},E^{\\prime})\\in\\Heisenberg_{d,C}$.\n    One can thus think of the $i$\\textsuperscript{th} and $j$\\textsuperscript{th}\n    $x$-co-ordinates of elements of $\\Heisenberg_{d,C}$\n    as being \\emph{correlated} with $C_{ij}$ encoding this correlation.\n\\end{e.g.}\n\n\\begin{e.g.}[Non-commuting families, Weyl Form of CCR]\n\\makelabel{e.g.:non-commuting-family-heisenberg-d-C:sig:article-stochastic-raj-dahya}\n    Let\n        $d\\in\\naturals$,\n        $\\HilbertRaum$ be a Hilbert space,\n        and\n        $C\\in M_{d}(\\reals)$ be an antisymmetric matrix,\n    \\idest\n        $C = D - D^{T}$, where $D \\in M_{d}(\\reals)$\n    is a strictly upper triangular matrix.\n    One use of classical dynamical systems defined over $\\Heisenberg^{+}_{d,C}$\n    is as follows is as follows:\n   \n    Consider an arbitrary \\topSOT-continuous homomorphism,\n        ${T \\colon \\Heisenberg^{+}_{d,C} \\to \\BoundedOps{\\HilbertRaum}}$.\n    Set\n        ${\n            T_{i} \\colonequals (T(t\\mathbf{e}_{i},0))_{t\\in\\realsNonNeg}\n        }$\n        for each $i\\in\\{1,2,\\ldots,d\\}$,\n        and\n        ${\n            U \\colonequals (T(\\zerovector, E))_{E\\in\\reals}\n        }$,\n    whereby each\n        $\\mathbf{e}_{i} \\colonequals (0,0,\\ldots,\\underset{i}{1},\\ldots,0)$\n    denotes the canonical $i$\\textsuperscript{th} basis vector of $\\reals^{d}$.\n    Then one can readily verify that\n        $\\{T_{i}\\}_{i=1}^{d}$ is a family of $\\Cnought$-semigroups on $\\HilbertRaum$\n        and\n        $U$ is an \\topSOT-continuous representation of $(\\reals^{d},+,\\zerovector)$ on $\\HilbertRaum$\n        which commutes with each of the $T_{i}$.\n    Furthermore the relations\n\n        \\vspace{-3\\parskip}\n        \\begin{eqnarray}\n        \\label{eq:commutation-relations:sig:article-stochastic-raj-dahya}\n            T_{j}(t) T_{i}(s)\n                &= &U(2stC_{ij}) T_{i}(s) T_{j}(t)\n        \\end{eqnarray}\n\n    \\noindent\n    hold for all $i,j\\in\\{1,2,\\ldots,d\\}$,\n    which is a slight generalisation of\n    the semigroup version of the \\emph{canonical commutation relations} (CCR)\n    in the Weyl form.\n    Such systems are of interest in quantum mechanics\n    (see \\exempli \\cite[\\S{}5.2.1.2]{Bratteli1997bookQSM}).\n   \n    Conversely, one may consider an arbitrary family,\n        $\\{T_{i}\\}_{i=1}^{d}$,\n    of $\\Cnought$-semigroups on $\\HilbertRaum$\n    and an arbitrary \\topSOT-continuous representation,\n        $U$,\n    of $(\\reals^{d},+,\\zerovector)$ on $\\HilbertRaum$,\n    which commutes with each of the $T_{i}$,\n    and such that the relations in\n    \\eqcref{eq:commutation-relations:sig:article-stochastic-raj-dahya} hold.%\n    \\footnote{\n        For example consider\n            $\\HilbertRaum = L^{2}(\\realsNonNeg^{m})$, $m\\in\\naturals$.\n        Let\n            $\\mathbf{u}^{(i)}\\in\\realsNonNeg^{m}$\n        and either\n            $\\lambda\\in\\iunit\\reals$\n            and\n            $\\boldsymbol{\\alpha}^{(i)}\\in\\reals^{m}$,\n        or\n            $\\lambda\\in\\complex$ with $\\mathop{\\mathfrak{R}\\mathrm{e}}\\lambda < 0$\n            and\n            $\\boldsymbol{\\alpha}^{(i)}\\in\\realsNonNeg^{m}$\n        for each $i\\in\\{1,2,\\ldots,d\\}$.\n        Set\n            $U \\colonequals (e^{\\lambda t}\\onematrix)_{t\\in\\reals}$\n        and\n            $\n                (T_{i}(t)f)(\\mathbf{x})\n                \\colonequals\n                e^{\n                    \\lambda t\n                    \\brkt{\\boldsymbol{\\alpha}^{(i)}}{\\mathbf{x}}\n                }\n                f(\\mathbf{x} + t\\mathbf{u}^{(i)})\n            $\n        for\n            $t\\in\\realsNonNeg$,\n            $f\\in L^{2}(\\realsNonNeg^{m})$,\n            $\\mathbf{x}\\in\\realsNonNeg^{m}$,\n            $i\\in\\{1,2,\\ldots,d\\}$.\n        Then \\eqcref{eq:commutation-relations:sig:article-stochastic-raj-dahya} holds with\n            $\n                C \\colonequals (\n                    \\frac{1}{2}(\n                        \\brkt{\\boldsymbol{\\alpha}^{(i)}}{\\mathbf{v}^{(j)}}\n                        -\n                        \\brkt{\\boldsymbol{\\alpha}^{(j)}}{\\mathbf{v}^{(i)}}\n                    )\n                )_{i,j=1}^{d}\n                \\in M_{d}(\\reals)\n            $.\n    }\n    Then, defining\n        ${T \\colon \\Heisenberg^{+}_{d,C} \\to \\BoundedOps{\\HilbertRaum}}$\n    via\n\n        \\vspace{-3\\parskip}\n        $$\n            T(\\mathbf{x},E)\n            \\colonequals\n            U(E + \\brkt{D\\mathbf{x}}{\\mathbf{x}})\n            \\prod_{i=1}^{d}\n                T_{i}(x_{i})\n        $$\n\n    \\noindent\n    for each $\\mathbf{x}\\in\\realsNonNeg^{d}$, $E\\in\\reals$,\n    one can verify that $T$ is an \\topSOT-continuous homomorphism.\n    These two constructions constitute a correspondence\n    (this is left as an exercise to the reader)\n    between families satisfying\n    \\eqcref{eq:commutation-relations:sig:article-stochastic-raj-dahya}\n    and \\topSOT-continuous homomorphisms defined over\n    $\\Heisenberg^{+}_{d,C}$.\n\\end{e.g.}\n\nFor submonoids of topological groups,\nthere are natural conditions which will aid us when investigating properties (\\viz dilation)\nof classical dynamical systems defined over them.\nThe following definition is due to Mueller \\cite[\\S{}2]{Mueller1965positiveInIdentity}:%\n\\footnote{\n    \\cf also \\cite[Appendix~A]{Dahya2022complmetrproblem},\n    where a similar, but stronger, property called \\emph{positivity in the identity} is defined.\n}\n\n\\begin{defn}\n\\makelabel{defn:monoids:e-jointedness:sig:article-stochastic-raj-dahya}\n    Let $G$ be a topological group and $M \\subseteq G$ be a measurable submonoid.\n    Say that $M$ is \\highlightTerm{$e$-joint}\n    if $\\lambda_{G}(U \\cap M) > 0$\n    for all open neighbourhoods $U \\subseteq G$ of the identity $e \\in G$.\n\\end{defn}\n\n\\begin{e.g.}\n\\makelabel{e.g.:monoids:e-joint:sig:article-stochastic-raj-dahya}\n    Let\n        $d\\in\\naturals$,\n        $p\\in\\mathbb{P}$ be a prime number,\n        $C\\in M_{d}(\\reals)$ be an antisymmetric matrix,\n        $G_{i}$ be a topological group,\n        and $M_{i} \\subseteq G_{i}$ an $e$-joint measurable submonoid\n    for $i\\in\\{1,2,\\ldots,d\\}$.\n\n    \\begin{kompaktenum}{(i)}\n    \\item\\punktlabel{1}\n        Consider $(G,M) = (\\reals^{d},\\realsNonNeg^{d})$.\n        For an open neighbourhood $U \\subseteq G$ of $\\zerovector$,\n        there exists $\\varepsilon>0$, such that $U \\supseteq (-\\varepsilon,\\:\\varepsilon)^{d}$.\n        Thus\n            $M \\cap U \\supseteq (0,\\:\\varepsilon)^{d}$,\n        which is a non-empty and thus non-null subset of $G$.\n        Hence $M$ is an $e$-joint submonoid.\n    \\item\\punktlabel{2}\n        Consider\n            $\n                (G,M)\n                = (\\Heisenberg_{d},\\Heisenberg^{+}_{d})\n                = (\n                    \\reals^{d}\\times\\reals^{d}\\times\\reals,\n                    \\realsNonNeg^{d}\\times\\realsNonNeg^{d}\\times\\reals\n                )\n            $.\n        As in \\punktcref{1},\n        for any open neighbourhood $U \\subseteq G$ of $(\\zerovector,\\zerovector,0)$,\n        one has\n            $M \\cap U \\supseteq (0,\\:\\varepsilon)^{d} \\times (0,\\:\\varepsilon)^{d} \\times (-\\varepsilon,\\:\\varepsilon)$\n        for some $\\varepsilon > 0$,\n        which is a non-empty and thus non-null subset of $G$.\n        Hence $M$ is an $e$-joint submonoid.\n    \\item\n        Similar to \\punktcref{2} one can show that\n        $\\Heisenberg^{+}_{d,C}$ is a (closed) $e$-joint submonoid\n        of $\\Heisenberg_{d,C}$.\n    \\item\n        The submonoid $\\integers_{p} \\mathbin{\\setminus} \\{0\\}$ of non-zero $p$-adic integers\n        within the locally compact multiplicative group,\n            $(\\rationals_{p} \\mathbin{\\setminus} \\{0\\},\\cdot,1)$,\n        of non-zero $p$-adic numbers is clearly $e$-joint,\n        since it is a clopen subset.\n    \\item\n        By simple computations with product measures\n        one can readily verify that\n            $\\prod_{i=1}^{d}M_{i}$ is a measurable $e$-joint submonoid in $\\prod_{i=1}^{d}G_{i}$\n        (\\cf \\cite[Proposition~A.7]{Dahya2022complmetrproblem}).\n    \\end{kompaktenum}\n\n    \\@ifnextchar\\bgroup{\\nvraum@c}{\\nvraum@bes}{1}\n\n\\end{e.g.}\n\n\\begin{defn}\n\\makelabel{defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya}\n    Let $(G,\\cdot,e)$ be a (not necessarly locally compact!) topological group\n    and $M \\subseteq G$ a submonoid.\n    If ${\\cdot^{+} \\colon G \\to M}$ is a continuous function which satisfies\n\n    \\begin{kompaktenum}{(i)}\n        \\item\\punktlabel{ax:identity}\n            $e^{+} = e$ for the identity element $e \\in G$;\n        \\item\\punktlabel{ax:idempotent}\n            $x^{++} = x^{+}$ for all $x\\in G$,\n            \\idest $\\cdot^{+}$ is idempotent;\n            and\n        \\item\\punktlabel{ax:representation}\n            $(x^{-})^{-1}x^{+} = x$\n            for all $x\\in G$,\n            where $x^{-} \\colonequals (x^{-1})^{+}$,\n    \\end{kompaktenum}\n\n    \\noindent\n    then we call $(G,M,\\cdot^{+})$\n    a \\highlightTerm{positivity structure}.\n\\end{defn}\n\n\\begin{e.g.}\n\\makelabel{e.g.:monoids:positivity:sig:article-stochastic-raj-dahya}\n    Let $d\\in\\naturals$ and $C \\in M_{d}(\\reals)$ be an antisymmetric matrix.\n    The pairs $(G,M)$ of topological groups and submonoids:\n        $(\\reals^{d},\\realsNonNeg^{d})$,\n        $(\\Heisenberg_{d},\\Heisenberg^{+}_{d})$,\n        $(\\Heisenberg_{d,C},\\Heisenberg^{+}_{d,C})$\n    admit natural positivity structures.\n    Furthermore, if\n        $(G_{i},M_{i},\\cdot^{+_{i}})$\n    are positivity structures,\n    then the product\n        $(\\prod_{i=1}^{d}G_{i},\\prod_{i=1}^{d}M_{i})$\n    admits a positivity via the pointwise definition.\n    The constructions are presented in\n        \\Cref{table:examples:monoids:positivity:sig:article-stochastic-raj-dahya}\n    and left to the reader to verify.\n\\end{e.g.}\n\n\\begin{table}[!htb]\n    \\begin{tabular}[t]{|p{0.15\\textwidth}|p{0.15\\textwidth}|p{0.4\\textwidth}|}\n        \\hline\n            Group $G$\n            &Submonoid $M$\n            &Description of ${\\cdot^{+} \\colon G\\to M}$\\\\\n        \\hline\n        \\hline\n            $\\prod_{i=1}^{d}G_{i}$\n                &$\\prod_{i=1}^{d}M_{i}$\n                &$\\mathbf{x} \\mapsto (x_{i}^{+_{i}})_{i=1}^{d}$\\\\\n            $(\\reals,+,0)$\n                &$\\realsNonNeg$\n                &$t \\mapsto \\max\\{t,0\\}$\\\\\n            $(\\reals^{d},+,\\zerovector)$\n                &$\\realsNonNeg^{d}$\n                    &$\\mathbf{t} \\mapsto (t_{i}^{+})_{i=1}^{d}$\\\\\n            $\\Heisenberg_{d}$\n                &$\\Heisenberg^{+}_{d}$\n                &$(\\mathbf{x},\\mathbf{p},E)\n                    \\mapsto\n                    (\n                        \\mathbf{x}^{+},\n                        \\mathbf{p}^{+},\n                        E^{+}\n                    )\n                $\\\\\n            $\\Heisenberg_{d,C}$\n                &$\\Heisenberg^{+}_{d}$\n                &$(\\mathbf{x},E)\n                    \\mapsto\n                    (\n                        \\mathbf{x}^{+},\n                        E^{+}\n                    )\n                $\\\\\n        \\hline\n    \\end{tabular}\n    \\caption{\n        Examples of positivity structures $(G,M,\\cdot^{+})$.\n        Here $d\\in\\naturals$, $C \\in M_{d}(\\reals)$ is an antisymmetric matrix,\n        and $(G_{i},M_{i},\\cdot^{+_{i}})$\n        are positivity structures for $i\\in\\{1,2,\\ldots,d\\}$.\n    }\n    \\label{table:examples:monoids:positivity:sig:article-stochastic-raj-dahya}\n\\end{table}\n\nNote that for the subset\n    $G^{+} \\colonequals \\{x^{+}\\mid g\\in G\\} \\subseteq M$,\nit is neither required that  $G^{+} = M$\nnor even that $G^{+}$ be closed under the group operation.\nIn the case of $\\reals^{d}$, this happens to be the case,\nbut in the case of the Heisenberg group,\nneither of these additional properties holds.\nWe now state some basic facts about positivity structures.\n\n\\begin{prop}\n    Let $(G,M,\\cdot^{+})$ be a positivity structure,\n    where $(G,\\cdot,e)$ is a topological group\n    and $M \\subseteq G$ is a submonoid.\n    Then\n        $(G^{-})^{-1} \\cap G^{+} = (G^{+})^{-1} \\cap G^{+} = \\{e\\}$.\n\\end{prop}\n\n    \\begin{proof}\n        First observe that\n            $(G^{-})^{-1} \\cap G^{+} = ((G^{-1})^{+})^{-1} \\cap G^{+} = (G^{+})^{-1} \\cap G^{+}$\n        Thus it suffices to prove that $(G^{+})^{-1} \\cap G^{+} = \\{e\\}$.\n        Since $e^{+} = e$, the $\\supseteq$-inclusion holds.\n        Towards the $\\subseteq$-inclusion,\n        let $x \\in (G^{+})^{-1} \\cap G^{+}$ be arbitrary.\n        Then $x = y^{+} = (z^{+})^{-1}$\n        for some $y, z \\in G$.\n        By the idempotence axiom,\n        one has that\n            $\n                x^{+}\n                = y^{++}\n                = y^{+}\n                = x\n            $\n        and\n            $\n                x^{-}\n                = (x^{-1})^{+}\n                = z^{++}\n                = z^{+}\n                = x^{-1}\n            $.\n        By axiom \\eqcref{it:ax:representation:defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya},\n        it follows that\n            $x = (x^{-})^{-1}(x^{+}) = x^{-1}x = e$.\n    \\end{proof}\n\n\\begin{prop}\n\\makelabel{prop:positivity-structure-basic:sig:article-stochastic-raj-dahya}\n    Let $(G,\\cdot,e)$ be a topological group\n    and $M \\subseteq G$ a submonoid.\n    Let ${\\cdot^{+} \\colon G \\to M}$\n    be a continuous map satisfying\n    axiom \\eqcref{it:ax:representation:defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya}\n    of \\eqcref{defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya}.\n    Then axiom\n        \\eqcref{it:ax:idempotent:defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya}\n    is equivalent to\n    the condition that $x^{+-}=e$ for all $x\\in G$.\n\\end{prop}\n\n    \\begin{proof}\n        Let $x \\in G$ be arbitrary.\n        Then by\n            axiom \\eqcref{it:ax:representation:defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya}\n        one has\n            $x^{+} = (x^{+-})^{-1}x^{++}$.\n        It follows that\n            $x^{++} = x^{+}$\n            if and only if\n            $x^{+-} = e$.\n    \\end{proof}\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Characterisation of unitary approximations]{Characterisation of unitary approximations}\n\\label{sec:introduction:applications:sig:article-stochastic-raj-dahya}\n\n\\noindent\nLetting $M$ be a topological monoid and $\\HilbertRaum$ a Hilbert space,\na classical dynamical system modelled by a homomorphism,%\n\\footnote{\n    \\idest $T(e) = \\onematrix$ and $T(xy) = T(x)T(y)$ for $x,y\\in M$.\n}\n    ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$,\ncan be thought of as \\emph{reversible}\nif it is unitary valued.\nAs this need not be the case, the question arises whether and in what sense\none can \\emph{approximate} $T$ via unitary systems.\n\nIn\n    \\Cref{thm:classification:dissipativity:sig:article-stochastic-raj-dahya}\n    and\n    \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\na strong notion of convergence is used for the approximants.\nIn the literature a similar weak notion of convergence for $\\Cnought$-semigroups is studied\n(\\cf \\exempli\n    \\cite{Krol2009,Eisner2008kato,Eisnersereny2009catThmStableSemigroups,Dahya2022weakproblem,Dahya2022complmetrproblem}\n    and\n    \\cite[\\S{}III.6]{Eisner2010buchStableOpAndSemigroups}\n    in the case of Hilbert spaces%\n).\nThese notions are defined as follows:\nLet\n    $d\\in\\naturals$,\n    $\\BanachRaum$ be a Banach space,\n    $\\{T_{i}\\}_{i=1}^{d}$\n    be a commuting family of $\\Cnought$-semigroups on $\\BanachRaum$,\n    and\n    $(\\{T^{(\\alpha)}_{i}\\}_{i=1}^{d})_{\\alpha\\in\\Lambda}$\n    be a net of commuting families of $\\Cnought$-semigroups on $\\BanachRaum$.\nWe say that\n    $(\\{T^{(\\alpha)}_{i}\\}_{i=1}^{d})_{\\alpha\\in\\Lambda}$\n    converges to $\\{T_{i}\\}_{i=1}^{d}$\n    \\highlightTerm{\\wrt the \\topSOT-toplogy}\n    (\\respectively \\highlightTerm{\\wrt the \\topWOT-toplogy})\n    \\highlightTerm{uniformly on compact subsets of $\\realsNonNeg^{d}$},\nif\n    for all $\\xi\\in\\BanachRaum$\n    (\\respectively for all $\\xi\\in\\BanachRaum$ and $\\eta\\in\\BanachRaum^{\\prime}$)\n    and all compact $L\\subseteq\\realsNonNeg^{d}$\nit holds that\n        ${\n            \\sup_{\\mathbf{t}\\in L}\n                \\norm{\n                    (\n                        \\prod_{i=1}^{d}\n                            T^{(\\alpha)}_{i}(t_{i})\n                        -\n                        \\prod_{i=1}^{d}\n                            T_{i}(t_{i})\n                    )\n                    \\:\\xi\n                }\n            \\underset{\\alpha}{\\longrightarrow}\n            0\n        }$\n(\\respectively\n    ${\n        \\sup_{\\mathbf{t}\\in L}\n            \\abs{\\brkt{\n                (\n                    \\prod_{i=1}^{d}\n                        T^{(\\alpha)}_{i}(t_{i})\n                    -\n                    \\prod_{i=1}^{d}\n                        T_{i}(t_{i})\n                )\n                \\:\\xi\n            }{\\eta}}\n        \\underset{\\alpha}{\\longrightarrow}\n        0\n    }$\n).\nFor classical systems on Hilbert spaces,\nusing the concepts in \\S{}\\ref{sec:introduction:monoids:sig:article-stochastic-raj-dahya}\nwe may generalise the weak notion of convergence to classical dynamical systems\nin general in the following natural manner:\n\n\\begin{defn}[Weak topologies]\n\\makelabel{defn:weak-convergence-over-exact-uniform-ptwise:sig:article-stochastic-raj-dahya}\n    Let\n        $G$ be a topological group\n        $M \\subseteq $ a submonoid,\n        $\\HilbertRaum$ a Hilbert space,\n        ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$\n        an \\topSOT-continuous homomorphism\n        and\n        ${(T^{(\\alpha)} \\colon M \\to \\BoundedOps{\\HilbertRaum})_{\\alpha\\in\\Lambda}}$\n        a net of \\topSOT-continuous homomorphisms.\n    We say that\n        $(T^{(\\alpha)})_{\\alpha\\in\\Lambda}$\n        converges to\n        $T$\n\n    \\begin{kompaktenum}{(i)}\n    \\item\\punktlabel{1}\n        \\highlightTerm{exactly weakly},\n        if for each\n            $\\xi,\\eta\\in\\HilbertRaum$\n        there exists an index $\\alpha_{0}\\in\\Lambda$\n        such that for all\n            $x \\in M$\n            and\n            $\\alpha \\geq \\alpha_{0}$\n\n            \\vspace{-3\\parskip}\n            $$\n                \\brkt{T^{(\\alpha)}(x)\\xi}{\\eta}\n                =\n                \\brkt{T(x)\\xi}{\\eta};\n            $$\n    \\item\\punktlabel{2}\n        \\highlightTerm{uniformly weakly},\n        if for all\n            $\\xi,\\eta\\in\\HilbertRaum$\n            and\n            compact $L \\subseteq M$\n\n            \\vspace{-3\\parskip}\n            $$\n                \\sup_{\\mathbf{t}\\in L}\n                    \\abs{\n                        \\brkt{T^{(\\alpha)}(x)\\xi}{\\eta}\n                        -\n                        \\brkt{T(x)\\xi}{\\eta}\n                    }\n                \\underset{\\alpha}{\\longrightarrow}\n                0;\n            $$\n    \\item\\punktlabel{3}\n        \\highlightTerm{pointwise weakly},\n        if for all\n            $\\xi,\\eta\\in\\HilbertRaum$\n            and\n            $x \\in M$\n\n            \\vspace{-3\\parskip}\n            $$\n                \\abs{\n                    \\brkt{T^{(\\alpha)}(x)\\xi}{\\eta}\n                    -\n                    \\brkt{T(x)\\xi}{\\eta}\n                }\n                \\underset{\\alpha}{\\longrightarrow}\n                0.\n            $$\n    \\end{kompaktenum}\n\n    \\noindent\n    If each $T^{(\\alpha)} = U^{(\\alpha)}\\restr{M}$,\n    where ${U^{(\\alpha)} \\colon G \\to \\BoundedOps{\\HilbertRaum}}$\n    is an \\topSOT-continuous unitary representation of $G$ on $\\HilbertRaum$,\n    we say that $T$ has an\n        \\highlightTerm{exact}\n        (%\n            \\respectively\n            \\highlightTerm{uniform}\n            \\respectively\n            \\highlightTerm{pointwise}%\n        )\n        \\highlightTerm{weak unitary approximation},\n    if \\punktcref{1}\n        (%\n            \\respectively\n            \\punktcref{2}\n            \\respectively\n            \\punktcref{3}%\n        )\n    holds.\n\\end{defn}\n\n\\begin{defn}[Regular weak topologies]\n\\makelabel{defn:regular-weak-convergence-over-exact-uniform-ptwise:sig:article-stochastic-raj-dahya}\n    Let\n        $(G,M,\\cdot^{+})$ be a positivity structure,\n        $\\HilbertRaum$ a Hilbert space,\n        ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$\n        an \\topSOT-continuous homomorphism\n        and\n        ${(T^{(\\alpha)} \\colon M \\to \\BoundedOps{\\HilbertRaum})_{\\alpha\\in\\Lambda}}$\n        a net of \\topSOT-continuous homomorphisms.\n    We say that\n        $(T^{(\\alpha)})_{\\alpha\\in\\Lambda}$\n        converges to\n        $T$\n\n    \\begin{kompaktenum}{(i)}\n    \\item\\punktlabel{1}\n        \\highlightTerm{exactly regularly weakly},\n        if for each\n            $\\xi,\\eta\\in\\HilbertRaum$\n        there exists an index $\\alpha_{0}\\in\\Lambda$\n        such that for all\n            $x \\in G$\n            and\n            $\\alpha \\geq \\alpha_{0}$\n\n            \\vspace{-3\\parskip}\n            $$\n                \\brkt{T^{(\\alpha)}(x^{-})^{\\ast}T^{(\\alpha)}(x^{+})\\xi}{\\eta}\n                =\n                \\brkt{T(x^{-})^{\\ast}T(x^{+})\\xi}{\\eta};\n            $$\n    \\item\\punktlabel{2}\n        \\highlightTerm{uniformly regularly weakly},\n        if for all\n            $\\xi,\\eta\\in\\HilbertRaum$\n            and\n            compact $L \\subseteq G$\n\n            \\vspace{-3\\parskip}\n            $$\n                \\sup_{x \\in L}\n                    \\abs{\n                        \\brkt{T^{(\\alpha)}(x^{-})^{\\ast}T^{(\\alpha)}(x^{+})\\xi}{\\eta}\n                        -\n                        \\brkt{T(x^{-})^{\\ast}T(x^{+})\\xi}{\\eta}\n                    }\n                \\underset{\\alpha}{\\longrightarrow}\n                0;\n            $$\n    \\item\\punktlabel{3}\n        \\highlightTerm{pointwise regularly weakly},\n        if for all\n            $\\xi,\\eta\\in\\HilbertRaum$\n            and\n            $x \\in G$\n\n            \\vspace{-3\\parskip}\n            $$\n                \\abs{\n                    \\brkt{T^{(\\alpha)}(x^{-})^{\\ast}T^{(\\alpha)}(x^{+})\\xi}{\\eta}\n                    -\n                    \\brkt{T(x^{-})^{\\ast}T(x^{+})\\xi}{\\eta}\n                }\n                \\underset{\\alpha}{\\longrightarrow}\n                0.\n            $$\n    \\end{kompaktenum}\n\n    \\noindent\n    If each $T^{(\\alpha)} = U^{(\\alpha)}\\restr{M}$,\n    where ${U^{(\\alpha)} \\colon G \\to \\BoundedOps{\\HilbertRaum}}$\n    is an \\topSOT-continuous unitary representation of $G$ on $\\HilbertRaum$,\n    we say that $T$ has an\n        \\highlightTerm{exact}\n        (%\n            \\respectively\n            \\highlightTerm{uniform}\n            \\respectively\n            \\highlightTerm{pointwise}%\n        )\n        \\highlightTerm{regular weak unitary approximation},\n    if \\punktcref{1}\n        (%\n            \\respectively\n            \\punktcref{2}\n            \\respectively\n            \\punktcref{3}%\n        )\n    holds.\n\\end{defn}\n\nClearly, exact (regular) weak approximations\nare uniform (regular) weak approximations,\nwhich in turn are pointwise (regular) weak approximations.\nIn the case of\n    $(G,M)=(\\reals^{d},\\realsNonNeg^{d})$,\nif $d=1$, then\neach \\emph{regular}-notion of convergence clearly coincides\nwith the corresponding notion without the \\emph{regular} prefix.\nIn this special case, the following result is known:\n\n\\begin{thm}[Kr\\'ol, 2009]\n\\makelabel{thm:unitary-approx:one-param:krol:sig:article-stochastic-raj-dahya}\n    Let $T$ be a contractive $\\Cnought$-semigroup\n    on an infinite dimensional Hilbert space $\\HilbertRaum$.%\n    \\footref{ft:approx-thm:hilbert-space-dim:sig:article-stochastic-raj-dahya}\n    Then $T$ has a uniform weak unitary approximation.\n\\end{thm}\n\nFor a proof see \\cite[Theorem~2.1 and Remark~2.3]{Krol2009},\nin which Kr\\'ol constructs the unitary approximants directly from a regular unitary dilation of $T$.\nIn this reference, it is questioned whether a proof is possible without reliance upon dilations\n(\\cf \\cite[Remark~2.2]{Krol2009}).\nOur results partially address this by showing\nthat the existence of simultaneous (regular) dilations is necessary.\nThus any dilation-free proof of\n    \\Cref{thm:unitary-approx:one-param:krol:sig:article-stochastic-raj-dahya}\nmight necessarily involve some characterisation of (regular) unitary dilations.\n\n\\begin{schattierteboxdunn}[backgroundcolor=leer,nobreak=true]\n\\begin{thm}[Characterisation of weak unitary approximations]\n\\makelabel{thm:unitary-approx:weak:sig:article-stochastic-raj-dahya}\n    Let\n        $G$ be a locally compact topological group\n        and\n        $M \\subseteq G$ be an $e$-joint closed submonoid.\n    Further let\n        $\\HilbertRaum$ be a Hilbert space\n        and\n        ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$\n        be an \\topSOT-continuous homomorphism.\n    If $G$ contains a dense subspace $D \\subseteq G$\n    with $\\card{D} \\leq \\mathop{\\textup{dim}}(\\HilbertRaum)$,%\n    \\footref{ft:approx-thm:hilbert-space-dim:sig:article-stochastic-raj-dahya}\n    then the following are equivalent:\n\n    \\begin{kompaktenum}{\\bfseries (a)}\n        \\item\\punktlabel{1}\n            The classical system $T$ has a unitary dilation.\n        \\item\\punktlabel{2}\n            The classical system $T$ has an exact weak unitary approximation.\n        \\item\\punktlabel{3}\n            The classical system $T$ has a uniform weak unitary approximation.\n    \\end{kompaktenum}\n\n    \\noindent\n    Without the above assumption on the dimension of $\\HilbertRaum$,\n        \\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1}\n    hold.\n\\end{thm}\n\\end{schattierteboxdunn}\n\n\\begin{schattierteboxdunn}[backgroundcolor=leer,nobreak=true]\n\\begin{thm}[Characterisation of regular weak unitary approximations]\n\\makelabel{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n    Let\n        $(G,M,\\cdot^{+})$ be a positivity structure,\n    where\n        $G$ is a topological group\n        and\n        $M \\subseteq G$ is a submonoid.%\n        \\footref{ft:approx-thm:G-not-locally-compact:sig:article-stochastic-raj-dahya}\n    Further let\n        $\\HilbertRaum$ be a Hilbert space\n        and\n        ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$\n        be an \\topSOT-continuous homomorphism.\n    If $\\HilbertRaum$ is infinite dimensional\n    and $G$ contains a dense subspace $D \\subseteq G$\n    with $\\card{D} \\leq \\mathop{\\textup{dim}}(\\HilbertRaum)$,%\n        \\footref{ft:approx-thm:hilbert-space-dim:sig:article-stochastic-raj-dahya}\n    then the following are equivalent:\n\n    \\begin{kompaktenum}{\\bfseries (a)}\n        \\item\\punktlabel{1}\n            The classical system $T$ has a regular unitary dilation.\n        \\item\\punktlabel{2}\n            The classical system $T$ has an exact regular weak unitary approximation.\n        \\item\\punktlabel{3}\n            The classical system $T$ has a uniform regular weak unitary approximation.\n        \\item\\punktlabel{4}\n            The classical system $T$ has a pointwise regular weak unitary approximation.\n    \\end{kompaktenum}\n\n    \\noindent\n    Without the above assumption on the dimension of $\\HilbertRaum$,\n        \\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1}\n    hold.\n\\end{thm}\n\\end{schattierteboxdunn}\n\n\\footnotetext[ft:approx-thm:hilbert-space-dim:sig:article-stochastic-raj-dahya]{\n    In the case of separable topological groups,\n        \\exempli $G=\\reals^{d}$, $d\\in\\naturals$,\n    this holds as soon as $\\HilbertRaum$ is infinite dimensional.\n}\n\n\\footnotetext[ft:approx-thm:G-not-locally-compact:sig:article-stochastic-raj-dahya]{\n    Note that we neither require $G$ to be locally compact\n    nor $M$ to be a measurable subset in this theorem!\n}\n\n\\Cref{%\n    thm:unitary-approx:weak:sig:article-stochastic-raj-dahya,%\n    thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya,%\n}\nprovide a sharp distinction between the two notions of dilation.\nBy\n    \\Cref{%\n        e.g.:monoids:e-joint:sig:article-stochastic-raj-dahya,%\n        e.g.:monoids:positivity:sig:article-stochastic-raj-dahya,%\n    }\nthese results are immediately applicable\nto commuting families of $\\Cnought$-semigroups\nas well as non-commuting families satisfying the \\emph{canonical commutation relations} (CCR) in the Weyl form\n(see \\Cref{e.g.:non-commuting-family-heisenberg-d-C:sig:article-stochastic-raj-dahya}).\n\nAs an example, applying these characterisations to\n    $(G,M) = (\\reals^{d},\\realsNonNeg^{d})$\nfor any $d \\geq 2$ and infinite dimensional Hilbert space $\\HilbertRaum$,\nwe shall construct commuting families of $d$ contractive $\\Cnought$-semigroups on $\\HilbertRaum$\nthat admit no unitary approximations\n(see \\Cref{cor:counter-examples-unitary-approx:sig:article-stochastic-raj-dahya}).\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Notation]{Notation}\n\\label{sec:introduction:notation:sig:article-stochastic-raj-dahya}\n\n\\noindent\nIn this paper we fix the following notation and conventions:\n\n\\begin{kompaktitem}\n    \\item\n        We write\n            $\\naturalsPos = \\{1,2,\\ldots\\}$,\n            $\\naturalsZero = \\{0,1,2,\\ldots\\}$,\n            $\\realsNonNeg = \\{r\\in\\reals \\mid r\\geq 0\\}$,\n            $\\realsPos = \\{r\\in\\reals \\mid r > 0\\}$,\n            and\n            $\\Torus = \\{z\\in\\complex \\mid \\abs{z} = 1\\}$\n            (unit circle in the complex plane).\n        To distinguish from indices $i$ we use $\\iunit$ for the imaginary unit $\\sqrt{-1}$.\n    \\item\n        We write elements of product spaces in bold\n        and denote their components in light face fonts with appropriate indices,\n        \\exempli the $i$\\textsuperscript{th} components of\n            $\\mathbf{t} \\in \\realsNonNeg^{n}$\n            and\n            $\\boldsymbol{\\alpha} \\in \\prod_{i=1}^{n}\\Lambda_{i}$\n            are denoted\n                $t_{i}$ and $\\alpha_{i}$\n            respectively.\n    \\item\n        In some instances we shall work with concrete constructions of approximants of\n        $\\Cnought$-semigroups or families thereof\n        (\\exempli the Hille- and Yosida-approximants).\n        In such cases we use\n            $\\lambda \\in \\realsPos$\n            and\n            $\\boldsymbol{\\lambda} \\in \\realsPos^{d}$\n        to index the approximants.\n        In others instances we work with the generalisation: \\highlightTerm{expectation-approximants}\n        (defined below in \\Cref{defn:expectation-approximants:sig:article-stochastic-raj-dahya}).\n        To indicate the abstract setting,\n            $\\alpha \\in \\Lambda$\n            and\n            $\\boldsymbol{\\alpha} \\in \\prod_{i=1}^{d}\\Lambda_{i}$\n        are used to index the approximants.\n    \\item\n        For bounded operators, $S$, over a Banach space,\n        $S^{\\prime}$ denotes the adjoint (dual) operator.\n        For bounded operators, $S$, over a Hilbert space,\n        $S^{\\ast}$ denotes the Hermitian adjoint.\n    \\item\n        For a measure (or probability) space $(X,\\Sigma,\\mu)$,\n        a measurable space $(Y,S)$,\n        and\n        a measurable function ${f \\colon X \\to Y}$,\n        the push-forward measure, $\\PushForward{f}{\\mu}$,\n        which we denote $\\mu_{f}$,\n        is the measure (\\respectively probability measure) on $(Y, S)$\n        defined by\n            $\\PushForward{f}{\\mu}[B] = \\mu[f^{-1}(B)]$\n        for all measurable $B \\subseteq Y$.\n    \\item\n        For a probability distribution, $\\Gamma$, over a set $X$\n        we write ${\\theta \\distributedAs \\Gamma}$\n        to denote that $\\theta$ is an $X$-valued random variable (\\randomvar)\n        with distribution $\\Gamma$.\n    \\item\n        For $t\\in\\reals$ the distribution $\\delta_{t}$ denotes the point distribution concentrated in $t$.\n    \\item\n        For $\\lambda\\in\\realsPos$\n        we denote with $\\DistExp{\\lambda}$\n        the exponential distribution with \\highlightTerm{rate} $\\lambda$.\n        For ${\\theta \\distributedAs \\DistExp{\\lambda}}$\n        it holds that\n            ${\n                \\Prob_{\\theta}[B] = \\int_{B}\\lambda e^{\\lambda s}\\:\\dee s\n            }$\n        for all measurable $B \\subseteq \\realsNonNeg$.\n    \\item\n        For $\\lambda\\in\\realsNonNeg$ and $c\\in\\realsPos$\n        we denote with $\\DistPoissScale{\\lambda}{c}$\n        the distribution of a Poisson distributed \\randomvar scaled by $c$.\n        For $\\theta \\distributedAs \\DistPoissScale{\\lambda}{c}$\n        it holds that\n            ${\n                \\Prob[\\theta=cn] = \\frac{\\lambda^{n}}{n!}e^{-\\lambda}\n            }$\n        for $n\\in\\naturals$ with the convention that $0^{0} \\colonequals 0$.\n        In particular, $\\DistPoissScale{\\lambda}{c}=\\delta_{0}$ if $\\lambda=0$.\n    \\item\n        For $\\lambda\\in\\realsPos$ and $t\\in\\realsNonNeg$,\n        we denote with $\\DistPoissAux{t}{\\lambda}$\n        an \\highlightTerm{auxiliary Poisson process}\n        with \\highlightTerm{rate} $\\lambda$\n        over a \\highlightTerm{time duration} $t$\n        (defined below in \\S{}\\ref{sec:stochastic:distributions:sig:article-stochastic-raj-dahya},\n        see also \\cite[\\S{}4]{Chung1962exp}).\n    \\item\n        For a (unital) {($C^{\\ast}$-)algebra} $\\mathcal{A}$,\n        $M_{n}(\\mathcal{A})$ denotes the (unital) matrix {($C^{\\ast}$-)algebra}\n        of $\\mathcal{A}$-valued $n \\times n$-matrices\n        for $n\\in\\naturals$.\n        If $\\mathcal{A}$ is an algebra of operators over some Hilbert space, $\\HilbertRaum$,\n        then the elements of $M_{n}(\\mathcal{A})$\n        are viewed as operators acting on $\\bigoplus_{i=1}^{n}\\HilbertRaum$,\n        and we have\n            $\n                \\brkt{(a_{ij})_{ij}\\oplus_{i=1}^{d}\\xi_{i}}{\\oplus_{i=1}^{d}\\eta_{i}}\n                = \\sum_{ij}\n                    \\brkt{a_{ij}\\xi_{j}}{\\eta_{i}}\n            $\n        for \\usesinglequotes{matrices} $(a_{ij})_{ij} \\in M_{n}(\\mathcal{A})$\n        and vectors\n            $\n                \\oplus_{i=1}^{n}\\xi_{i},\\oplus_{i=1}^{n}\\eta_{i}\n                \\in \\bigoplus_{i=1}^{n}\\HilbertRaum\n            $.\n    \\item\n        A map ${\\Psi \\colon \\mathcal{A} \\to \\mathcal{B}}$\n        between (subalgebras of) $C^{\\ast}$-algebras is called\n        \\emph{completely bounded} if\n            $\n                \\normCb{\\Psi}\n                \\colonequals\n                    \\sup_{n\\in\\naturalsPos}\\norm{\\Psi \\otimes \\mathrm{\\textit{id}}_{M_{n}}}\n                < \\infty\n            $,\n        and \\emph{completely positive} if\n            $\n                \\Psi \\otimes \\mathrm{\\textit{id}}_{M_{n}}\n            $\n            is positive for all $n\\in\\naturals$.\n        Here\n            ${\\Psi \\otimes \\mathrm{\\textit{id}}_{M_{n}} \\colon M_{n}(\\mathcal{A}) \\to M_{n}(\\BoundedOps{\\HilbertRaum})}$\n        is defined by\n            $(\\Psi \\otimes \\mathrm{\\textit{id}}_{M_{n}})((a_{ij})_{ij}) = (\\Psi(a_{ij}))_{ij} \\in M_{n}(\\BoundedOps{\\HilbertRaum})$\n        for each\n            $(a_{ij})_{ij} \\in M_{n}(\\mathcal{A})$\n            and\n            $n\\in\\naturals$\n        (%\n            see\n            \\cite[Chapter~1]{Paulsen1986bookCBmapsAndDilations},\n            \\cite[Chapter~3]{Pisier2001bookCBmaps}%\n        ).\n\\end{kompaktitem}\n\nFor a Banach space $\\BanachRaum$,\na measure space $(X,\\Sigma,\\mu)$\nand an operator-valued function ${T \\colon X \\to \\BoundedOps{\\BanachRaum}}$,\nfor which ${T(\\cdot)\\xi \\colon X \\to \\BanachRaum}$ is\n\\highlightTerm{strongly measurable} for each $\\xi\\in\\BanachRaum$,\nthe integral ${\\sotInt_{X} T\\:\\dee\\mu}$, when it exists,\ndenotes the unique bounded operator $\\tilde{T}\\in\\BoundedOps{\\BanachRaum}$\nthat satisfies\n    ${\\tilde{T}\\xi = \\int_{X} T(\\cdot)\\xi\\:\\dee\\mu}$\nfor all $\\xi\\in\\BanachRaum$,\nwhere $\\int_{X} T(\\cdot)\\xi\\:\\dee\\mu$\nis computed as a Bochner-integral.\nThis holds, for example, if $X$ is a locally compact Polish space\n(\\exempli $\\realsNonNeg^{d}$ for some $d\\in\\naturals$),\nand $T$ is contractive and \\topSOT-continuous\n(\\exempli a product of contractive $\\Cnought$-semigroups).\nIf $(\\Omega,\\Sigma,\\Prob)$ is a probability space\nand ${\\tau \\colon \\Omega \\to X}$ is an $X$-valued \\randomvar (\\idest a measurable function),\nwe refer to\n    $\n        \\Expected[T(\\theta)]\n        = \\sotInt_{\\omega\\in\\Omega} T(\\theta(\\omega))\\:\\Prob(\\dee\\omega)\n        = \\sotInt_{t \\in X} T(t)\\:\\Prob_{\\theta}(\\dee t)\n    $,\nwhen it exists,\nas the \\highlightTerm{expectation} (computed strongly via Bochner-integrals).\nIf $X$ is a locally compact Polish space\nand $T$ is a contractive \\topSOT-continuous function\nthen the expectation exists.\n\nThe existence and properties of Bochner-integrals\n(including linearity, convexity and triangle inequalities, Fubini's theorem for products, \\etcetera)\nas well as the validity of various computations with Bochner-integrals and expectations\nused in the rest of this paper\ncan be found in or readily derived from the literature.\nWe refer the reader in particular to\n    \\cite[\\S{}3.7, Theorems~3.7.4--6, and Theorems~3.7.12--13]{Hillephillips1957faAndSg},\n    \\cite[\\S{}II.2, Theorem~2, and Theorem~4]{DiestelUhl1977VectorMeasures},\n    \\cite[\\S{}C.1--4]{EngelNagel2000semigroupTextBook},\n    and\n    \\cite[\\S{}1.1.c--\\S{}1.2.a]{HytNeervanMarkLutz2016bookVol1}.\nFor example, using Fubini's theorem one can derive that\n    $\n        \\prod_{i=1}^{n}\\Expected[T_{i}(\\theta_{i})]\n        = \\Expected[\\prod_{i=1}^{n}T_{i}(\\theta_{i})]\n    $\nfor $n\\in\\naturals$,\nindependent $\\realsNonNeg$-valued \\randomvar's\n    $\\theta_{1},\\theta_{2},\\ldots,\\theta_{n}$\nand $\\Cnought$-semigroups $T_{1},T_{2},\\ldots,T_{n}$ on a Banach space $\\BanachRaum$\nwhich are uniformly bounded on the essential ranges of\n$\\theta_{1},\\theta_{2},\\ldots,\\theta_{n}$ respectively\n(\\exempli contractive semigroups).\nWe shall take advantage of this computation throughout.\nFurther fundamental applications of Bochner-integrals\nin the context of $\\Cnought$-semigroups\ncan be found \\exempli in \\cite{Chung1962exp,Dunfordschwartz1988BookLinOpI,Reissig2005abstractresolvent}.\n\n\n\n\n\\@startsection{section}{1}{\\z@}{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}{\\formatsection@text}[Stochastic presentation of classical approximants]{Stochastic presentation of classical approximants}\n\\label{sec:stochastic:sig:article-stochastic-raj-dahya}\n\n\\noindent\nIn this section we provide standalone results for $\\Cnought$-semigroups\nover Banach spaces and then for (commuting) families.\nWe assume basic knowledge of stochastic processes\nas well as Poisson and exponential distributions.\nWe shall also rely on the theory of Bochner-integrals,\nin particular those that occur in the integral representations of\npowers of resolvents of unbounded generators of $\\Cnought$-semigroups.\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Semigroups of distributions]{Semigroups of distributions}\n\\label{sec:stochastic:distributions:sig:article-stochastic-raj-dahya}\n\n\\noindent\nA family $(\\Gamma(t))_{t\\in\\realsNonNeg}$ of parameterised distributions\nof $\\reals$-valued random variables (\\randomvar's)\nshall be called a \\highlightTerm{continuous semigroup of distributions} if\n\n\\begin{kompaktenum}{(i)}\n    \\item\n        $\\Gamma(0)$ is the point distribution concentrated in $0$ (\\idest $\\delta_{0}$);\n    \\item\n        $\\theta_{1} + \\theta_{2} \\distributedAs \\Gamma(s + t)$\n        for $s,t\\in\\realsNonNeg$\n        and any independent \\randomvar's\n            $\\theta_{1} \\distributedAs \\Gamma(s)$\n            and\n            $\\theta_{2} \\distributedAs \\Gamma(t)$;\n        and\n    \\item\n        letting $\\theta_{t} \\distributedAs \\Gamma(t)$\n        and $\\mu_{t}$ be the associated probability distribution\n        for $t\\in\\realsNonNeg$,\n        we have\n            ${\\mu_{t} \\longrightarrow \\mu_{0} = \\delta_{0}}$\n        weakly for ${\\realsPos \\ni t \\longrightarrow 0}$,\n        \\idest\n            ${\\Expected[f(\\theta_{t})] \\longrightarrow \\Expected[f(\\theta_{0})] = f(0)}$\n        for all bounded continuous functions $f\\in\\@ifnextchar_{\\Cts@tief}{\\Cts@tief_{}}{\\realsNonNeg}$\n\\end{kompaktenum}\n\n\\noindent\n(\\cf\n    \\cite[Definition~14.46 and Example~17.7]{Klenke2008probTheory}\n    and\n    \\cite[\\S{}17.E]{Kechris1995BookDST}%\n).\nSimple examples of this include\n    $(\\delta_{t})_{t\\in\\realsNonNeg}$,\n    $(\\DistPoissScale{\\lambda t}{c})_{t\\in\\realsNonNeg}$\n    for $c\\in\\reals$, $\\lambda\\in\\realsPos$,\n    and\n    $(\\mathcal{N}(\\mu t,\\sigma^{2}t))_{t\\in\\realsNonNeg}$\n    (the normal distributions)\n    for $\\mu\\in\\reals$, $\\sigma\\in\\realsNonNeg$\n    with the convention that\n    $\\mathcal{N}(0,0)$ denotes the point distribution $\\delta_{0}$\n(\\cf \\cite[Corollary~15.13]{Klenke2008probTheory}).\n\nWe now construct a further family of parameterised distributions.\nFor $\\lambda \\in (0,\\:\\infty)$ we construct a random distribution\nvia two independent homogeneous point Poisson processes (PPP) as follows\n(depicted in \\Cref{fig:example-auxiliary-poisson:sig:article-stochastic-raj-dahya}):\n\n\\begin{kompaktenum}{\\bfseries 1.}\n    \\item\n        Let\n            $\n                \\tilde{\\tau}_{0}, \\tilde{\\tau}_{1}, \\tilde{\\tau}_{2},\\ldots,\n                \\tau_{0}, \\tau_{1}, \\tau_{2},\\ldots\n                \\distributedAs\n                \\DistExp{\\lambda}\n            $\n        be independent identically distributed (\\iid) \\randomvar's.\n    \\item\n        For each $n\\in\\naturalsZero$ set\n            $\n                \\tau_{<n} \\colonequals \\sum_{i=0}^{n-1}\\tau_{i}\n            $,\n        with the convention that the empty sum is just\n        the \\randomvar equal to $0$ \\almostSurely.\n        As a sum of $n$ exponentially distributed \\randomvar's,\n        we have\n            $\n                \\Prob[\\tau_{<n} \\in B]\n                = \\int_{s \\in B}\n                    \\lambda\\frac{(\\lambda s)^{n-1}}{(n-1)!}e^{-\\lambda s}\n                \\:\\dee s\n            $\n        for all measurable $B \\subseteq \\realsNonNeg$,\n        provided $n \\geq 1$\n        (see \\exempli \\cite[Theorem~15.12 and Corollary~15.13~(ii)]{Klenke2008probTheory}).\n    \\item\n        For each $t\\in\\realsNonNeg$, let\n            $\n                N_{t}\n                \\colonequals \\sup_{[0,\\:\\infty]}\\{\n                    n\\in\\naturalsZero \\mid \\sum_{i=0}^{n-1}\\tilde{\\tau}_{i} < t\n                \\}\n            $.\n        Then\n            $N_{t} < \\infty$ \\almostSurely\n        and\n            $N_{t} \\distributedAs \\DistPoissScale{\\lambda t}{1}$,\n        including when $t=0$,\n        since $\\DistPoissScale{0}{1}$ is just the point distribution $\\delta_{0}$.\n        Moreover, by construction each $N_{t}$ is independent of the $\\tau_{i}$.\n    \\item\n        Finally, set\n            $\n                \\theta_{t}\n                \\colonequals\n                    \\tau_{< N_{t}}\n                = \\sum_{i=0}^{N_{t} - 1}\\tau_{i}\n            $\n        for $t\\in\\realsNonNeg$.\n\\end{kompaktenum}\n\n\n\\begin{figure}[!ht]\n\n    \\begin{subfigure}[m]{\\textwidth}\n        \\centering\n        \\begin{tikzpicture}[node distance=1cm, thick]\n            \\pgfmathsetmacro\\hunit{3}\n            \\pgfmathsetmacro\\vunit{1}\n\n           \n            \\draw [draw=none, fill=none]\n                ({-0.1 * \\hunit}, {0.5 * \\vunit})\n                -- ({2.6 * \\hunit}, {0.5 * \\vunit})\n                -- ({2.6 * \\hunit}, {-0.5 * \\vunit})\n                -- ({-0.1 * \\hunit}, {-0.5 * \\vunit})\n                -- cycle;\n\n           \n            \\draw [draw=none, fill=drawing_light_grey]\n                ({0 * \\hunit}, {0 * \\vunit})\n                -- ({2.5 * \\hunit}, {0 * \\vunit})\n                -- ({2.5 * \\hunit}, {0.5 * \\vunit})\n                -- ({0 * \\hunit}, {0.5 * \\vunit})\n                -- cycle;\n\n           \n            \\draw [draw=black, line width=0.5pt, fill=none] ({0.16423594284219328 * \\hunit}, {0 * \\vunit}) -- ({0.16423594284219328 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=black, line width=0.5pt, fill=none] ({0.30852786449475916 * \\hunit}, {0 * \\vunit}) -- ({0.30852786449475916 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=black, line width=0.5pt, fill=none] ({0.4849472124336579 * \\hunit}, {0 * \\vunit}) -- ({0.4849472124336579 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=black, line width=0.5pt, fill=none] ({0.5758038383012576 * \\hunit}, {0 * \\vunit}) -- ({0.5758038383012576 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=black, line width=0.5pt, fill=none] ({0.6974385112680267 * \\hunit}, {0 * \\vunit}) -- ({0.6974385112680267 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=black, line width=0.5pt, fill=none] ({0.9444818148614873 * \\hunit}, {0 * \\vunit}) -- ({0.9444818148614873 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=black, line width=0.5pt, fill=none] ({1.1158979979012023 * \\hunit}, {0 * \\vunit}) -- ({1.1158979979012023 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=black, line width=0.5pt, fill=none] ({1.6668098229013812 * \\hunit}, {0 * \\vunit}) -- ({1.6668098229013812 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=black, line width=0.5pt, fill=none] ({1.7311098678037797 * \\hunit}, {0 * \\vunit}) -- ({1.7311098678037797 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=black, line width=0.5pt, fill=none] ({2.147702862254488 * \\hunit}, {0 * \\vunit}) -- ({2.147702862254488 * \\hunit}, {0.5 * \\vunit});\n\n           \n            \\draw [decorate, decoration={brace, amplitude=5pt}]\n                ({0 * \\hunit}, {0.55 * \\vunit}) -- ({0.9444818148614873 * \\hunit}, {0.55 * \\vunit})\n                node[midway, above=5pt, align=center]{\\footnotesize $N_{t}$};\n\n           \n            \\draw [draw=black, line width=2pt, fill=none]\n                ({0 * \\hunit}, {-0.05 * \\vunit})\n                node[label=below:{\\footnotesize $0$}]{}\n                -- ({0 * \\hunit}, {0.1 * \\vunit});\n\n           \n            \\draw [draw=black, line width=2pt, fill=none]\n                ({1 * \\hunit}, {-0.05 * \\vunit})\n                node[label=below:{\\footnotesize $t$}]{}\n                -- ({1 * \\hunit}, {0.1 * \\vunit});\n\n           \n            \\draw [->, draw=black, line width=0.5pt, fill=none]\n                ({-0.1 * \\hunit}, {0 * \\vunit}) -- ({2.6 * \\hunit}, {0 * \\vunit})\n                node [below, align=center]{\\footnotesize time};\n        \\end{tikzpicture}\n        \\caption{%\n            1\\textsuperscript{st} homogeneous PPP with rate $\\lambda$,\n            where $N_{t}$ counts the number of \\usesinglequotes{events} in $[0,\\:t)$.\n        }\n        \\label{fig:example-auxiliary-poisson:a:sig:article-stochastic-raj-dahya}\n    \\end{subfigure}\n\n    \\begin{subfigure}[m]{\\textwidth}\n        \\centering\n        \\begin{tikzpicture}[node distance=1cm, thick]\n            \\pgfmathsetmacro\\hunit{3}\n            \\pgfmathsetmacro\\vunit{1}\n\n           \n            \\draw [draw=none, fill=none]\n                ({-0.1 * \\hunit}, {0.5 * \\vunit})\n                -- ({2.6 * \\hunit}, {0.5 * \\vunit})\n                -- ({2.6 * \\hunit}, {-0.5 * \\vunit})\n                -- ({-0.1 * \\hunit}, {-0.5 * \\vunit})\n                -- cycle;\n\n           \n            \\draw [draw=none, fill=drawing_light_grey]\n                ({0 * \\hunit}, {0 * \\vunit})\n                -- ({2.5 * \\hunit}, {0 * \\vunit})\n                -- ({2.5 * \\hunit}, {0.5 * \\vunit})\n                -- ({0 * \\hunit}, {0.5 * \\vunit})\n                -- cycle;\n\n           \n            \\draw [draw=blue, line width=0.5pt, fill=none] ({0.1704947206981121 * \\hunit}, {0 * \\vunit}) -- ({0.1704947206981121 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=blue, line width=0.5pt, fill=none] ({0.986362751410455 * \\hunit}, {0 * \\vunit}) -- ({0.986362751410455 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=blue, line width=0.5pt, fill=none] ({1.1292144647056632 * \\hunit}, {0 * \\vunit}) -- ({1.1292144647056632 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=blue, line width=0.5pt, fill=none] ({1.2332491071729783 * \\hunit}, {0 * \\vunit}) -- ({1.2332491071729783 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=blue, line width=0.5pt, fill=none] ({1.4080395043214222 * \\hunit}, {0 * \\vunit}) -- ({1.4080395043214222 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=blue, line width=0.5pt, fill=none] ({1.5578881324962115 * \\hunit}, {0 * \\vunit}) -- ({1.5578881324962115 * \\hunit}, {0.5 * \\vunit});\n            \\draw [draw=black, line width=0.5pt, fill=none] ({2.0546540974785676 * \\hunit}, {0 * \\vunit}) -- ({2.0546540974785676 * \\hunit}, {0.5 * \\vunit});\n\n           \n            \\draw [decorate, decoration={brace, amplitude=5pt}]\n                ({0 * \\hunit}, {0.55 * \\vunit}) -- ({1.5578881324962115 * \\hunit}, {0.55 * \\vunit})\n                node[midway, above=5pt, align=center]{\\footnotesize $N_{t}$};\n\n           \n            \\draw [draw=black, line width=2pt, fill=none]\n                ({0 * \\hunit}, {-0.05 * \\vunit})\n                node [label=below:{\\footnotesize $0$}]{}\n                -- ({0 * \\hunit}, {0.1 * \\vunit});\n\n           \n            \\draw [draw=blue, line width=1pt, fill=none]\n                ({1.5578881324962115 * \\hunit}, {-0.05 * \\vunit})\n                node [label=below:{\\footnotesize $\\theta$}]{}\n                -- ({1.5578881324962115 * \\hunit}, {0.5 * \\vunit});\n\n           \n            \\draw [->, draw=black, line width=0.5pt, fill=none]\n                ({-0.1 * \\hunit}, {0 * \\vunit}) -- ({2.6 * \\hunit}, {0 * \\vunit})\n                node [below, align=center]{\\footnotesize time};\n        \\end{tikzpicture}\n        \\caption{%\n            2\\textsuperscript{nd} homogeneous PPP with rate $\\lambda$,\n            independent of 1\\textsuperscript{st} PPP,\n            but with $N_{t}$ determined by (a).\n        }\n        \\label{fig:example-auxiliary-poisson:b:sig:article-stochastic-raj-dahya}\n    \\end{subfigure}\n\n    \\caption{%\n        Visualisation of the construction of an \\highlightTerm{auxiliary Poisson process},\n        $\\theta \\distributedAs \\DistPoissAux{t}{\\lambda}$.\n    }\n    \\label{fig:example-auxiliary-poisson:sig:article-stochastic-raj-dahya}\n\\end{figure}\n\n\nFor each $\\lambda\\in\\realsPos$ and $t\\in\\realsNonNeg$\nlet $\\DistPoissAux{t}{\\lambda}$ denote the distribution of $\\theta_{t}$\nconstructed as above.\nWe refer to any \\randomvar distributed as $\\DistPoissAux{t}{\\lambda}$\nas an \\highlightTerm{auxiliary Poisson process}\nwith \\highlightTerm{rate} $\\lambda$\nover a \\highlightTerm{time duration} $t$.\nWe observe some basic properties of auxiliary Poisson processes.\n\n\\begin{prop}\n\\makelabel{prop:characteristic-function:aux-poisson-process:sig:article-stochastic-raj-dahya}\n    Let $\\lambda\\in\\realsPos$ and $t\\in\\realsNonNeg$.\n    The characteristic function\n    of a $\\DistPoissAux{t}{\\lambda}$-distributed \\randomvar%\n    \\footnote{\n        For an $\\reals$-valued \\randomvar $X$\n        the characteristic function is defined as the map\n        ${\\altvarphi \\colon \\reals \\ni \\omega \\mapsto \\Expected[e^{\\iunit \\omega X}}]$\n        (\\cf \\cite[Definition~15.7]{Klenke2008probTheory}).\n    }\n    is given by\n        $\n            \\altvarphi_{t, \\lambda}(\\omega)\n            = e^{\\frac{\\iunit\\omega}{\\lambda - \\iunit\\omega} \\lambda t}\n        $\n    for all $\\omega \\in \\reals$.\n    Moreover, the mean and variance of\n    $\\DistPoissAux{t}{\\lambda}$-distributed \\randomvar's\n    are $t$ and $\\frac{2t}{\\lambda}$ respectively.\n\\end{prop}\n\n    \\begin{proof}\n        Let $\\theta \\distributedAs \\DistPoissAux{t}{\\lambda}$.\n        Without loss of generality, we may assume that $\\theta$\n        is given by the above construction, \\idest\n            $\\theta = \\theta_{t} = \\tau_{<N_{t}}$.\n        For $n\\in\\naturalsPos$\n        we have\n        $\n            \\Expected[e^{\\iunit\\omega\\tau_{<n}}]\n            = (\\frac{\\lambda}{\\lambda - \\iunit\\omega})^{n}\n        $\n        (see \\exempli \\cite[Theorem~15.12]{Klenke2008probTheory})\n        and for $n=0$ clearly\n        $\n            \\Expected[e^{\\iunit\\omega\\tau_{<n}}]\n            = \\Expected[e^{\\iunit\\omega\\cdot 0}]\n            = 1\n            \\equalscolon (\\frac{\\lambda}{\\lambda - \\iunit\\omega})^{0}\n        $.\n        Since by construction, $N_{t}$ is independent of the $\\tau_{i}$,\n        one thus obtains\n\n            \\vspace{-3\\parskip}\n            \\begin{eqnarray*}\n                \\Expected[e^{\\iunit\\omega\\theta}]\n                &= &\\Expected[\\Expected[e^{\\iunit\\omega\\tau_{<N_{t}}}]|N_{t}]\\\\\n                &= &\\Expected[(\\tfrac{\\lambda}{\\lambda - \\iunit\\omega})^{N_{t}}|N_{t}]\\\\\n                &= &\\displaystyle\n                    \\sum_{n=0}^{\\infty}\n                        e^{-\\lambda t}\n                        \\frac{(\\lambda t)^{n}}{n!}\n                        \\,\\Big(\n                            \\frac{\\lambda}{\\lambda - \\iunit\\omega}\n                        \\Big)^{n}\\\\\n                &= &e^{-\\lambda t}e^{\\lambda t \\, \\frac{\\lambda}{\\lambda - \\iunit\\omega}}\\\\\n            \\end{eqnarray*}\n\n        \\noindent\n        for all $\\omega\\in\\reals$.\n        Hence the characteristic function is as claimed.\n\n        The mean can be computed as follows:\n        For $n\\in\\naturalsPos$\n        one has\n            $\n                \\Expected[\\tau_{<n}]\n                = \\sum_{i=1}^{n}\\Expected[\\tau_{i}]\n                = \\frac{n}{\\lambda}\n            $\n        and\n            $\n                \\Expected[\\tau_{<0}]\n                = \\Expected[0]\n                = 0\n            $,\n        whence\n            $\n                \\Expected[\\theta]\n                = \\Expected[\\Expected[\\tau_{<N_{t}} | N_{t}]]\n                = \\Expected[\\frac{N_{t}}{\\lambda}]\n                = \\frac{\\lambda t}{\\lambda}\n                = t\n            $.\n        To compute the variance, we first compute the $2$\\textsuperscript{nd} moment of $\\theta$.\n        Since the characteristic function, $\\altvarphi_{t,\\lambda}$, of $\\theta$\n        is $2$-times (in fact $\\infty$-times) continuously differentiable,\n        one may compute (\\cf \\cite[Theorem~15.34]{Klenke2008probTheory})\n\n            \\vspace{-3\\parskip}\n            $$\n                \\Expected[\\theta^{2}]\n                = \\frac{1}{\\iunit^{2}}\\altvarphi^{\\prime\\prime}_{t,\\lambda}(0)\n                = e^{-\\lambda t}e^{\\frac{\\lambda^{2} t}{\\lambda - \\iunit\\omega}}\n                    \\Big(\n                        \\Big(\n                            \\frac{\\lambda^{2} t}{(\\lambda - \\iunit\\omega)^{2}}\n                        \\Big)^{2}\n                        +\n                        2\\frac{\\lambda^{2} t}{(\\lambda - \\iunit\\omega)^{3}}\n                    \\Big)\n                    \\Big\\rvert_{\\:\\omega=0}\n                = t^{2} + \\frac{2t}{\\lambda},\n            $$\n\n        \\noindent\n        from which\n            $\n                \\Var(\\theta)\n                = \\Expected[\\theta^{2}] - \\Expected[\\theta]^{2}\n                = \\frac{2t}{\\lambda}\n            $\n        follows.\n    \\end{proof}\n\n\\begin{prop}\n\\makelabel{prop:app-semigroup-law:sig:article-stochastic-raj-dahya}\n    For each $\\lambda\\in\\realsPos$,\n    the family,\n        $(\\DistPoissAux{t}{\\lambda})_{t\\in\\realsNonNeg}$,\n    of distributions of auxiliary Poisson processes\n    constitutes a continuous semigroup of distributions.\n\\end{prop}\n\n    \\begin{proof}\n        If $\\theta \\distributedAs \\DistPoissAux{0}{\\lambda}$,\n        then by the above construction,\n            $\\theta = \\tau_{< N_{0}} = \\tau_{< 0} = 0$ \\almostSurely.\n        We now show for independent \\randomvar's\n            $t_{1},t_{2} \\in \\realsNonNeg$\n        with\n            ${\\theta_{1} \\distributedAs \\DistPoissAux{t_{1}}{\\lambda}}$\n            and\n            ${\\theta_{2} \\distributedAs \\DistPoissAux{t_{2}}{\\lambda}}$,\n        that\n            ${\\theta_{1} + \\theta_{2} \\distributedAs \\DistPoissAux{t_{1} + t_{2}}{\\lambda}}$.\n        To this end let\n            $\\altvarphi_{\\theta_{1}}$, $\\altvarphi_{\\theta_{2}}$, and $\\altvarphi_{\\theta_{1} + \\theta_{2}}$\n        denote the characteristic functions of\n            $\\theta_{1}$, $\\theta_{2}$, and  $\\theta_{1}+\\theta_{2}$\n        respectively.\n        By \\Cref{prop:characteristic-function:aux-poisson-process:sig:article-stochastic-raj-dahya},\n        it holds that\n            $\\altvarphi_{\\theta_{1}} = \\altvarphi_{t_{1},\\lambda}$\n            and\n            $\\altvarphi_{\\theta_{2}} = \\altvarphi_{t_{2},\\lambda}$\n        and by independence\n        (\\cf \\exempli \\cite[Lemma~15.11]{Klenke2008probTheory})\n            $\n                \\altvarphi_{\\theta_{1} + \\theta_{2}}(\\omega)\n                = \\altvarphi_{\\theta_{1}}(\\omega) \\cdot \\altvarphi_{\\theta_{2}}(\\omega)\n                = \\altvarphi_{t_{1},\\lambda}(\\omega) \\cdot \\altvarphi_{t_{2},\\lambda}(\\omega)\n                = e^{\\frac{\\iunit\\omega}{\\lambda - \\iunit\\omega} \\lambda t_{1}}\n                    e^{\\frac{\\iunit\\omega}{\\lambda - \\iunit\\omega} \\lambda t_{2}}\n                = \\altvarphi_{t_{1}+t_{2},\\lambda}(\\omega)\n            $\n        for all $\\omega\\in\\reals$.\n        Since the characteristic function uniquely determines the distribution of a \\randomvar\n        (\\cf \\cite[Theorem~15.8]{Klenke2008probTheory}),\n        it follows by \\Cref{prop:characteristic-function:aux-poisson-process:sig:article-stochastic-raj-dahya}\n        that $\\theta_{1} + \\theta_{2} \\distributedAs \\DistPoissAux{t_{1} + t_{2}}{\\lambda}$.\n\n        Towards continuity at $0$, let $f\\in\\@ifnextchar_{\\Cts@tief}{\\Cts@tief_{}}{\\realsNonNeg}$ be an arbitrary bounded function.\n        Let $t\\in\\realsNonNeg$ and consider ${\\theta_{t} \\distributedAs \\DistPoissAux{t}{\\lambda}}$.\n        Without loss of generality, assume that $\\theta_{t}$ is constructed as above,\n        \\idest $\\theta_{t} = \\tau_{<N_{t}}$.\n        We need to show that\n            ${\\Expected[f(\\theta_{t})] \\longrightarrow f(0)}$\n        for ${\\realsPos \\ni t \\longrightarrow 0}$.\n        For $n \\in \\naturalsPos$\n        one has\n            $\n                \\abs{\\Expected[f(\\tau_{n})]}\n                \\leq \\Expected[\\abs{f(\\tau_{n})}]\n                \\leq \\norm{f}_{\\infty}\n            $\n        and for $n=0$ one has\n            $\\Expected[f(\\tau_{0})] = \\Expected[f(0)] = f(0)$.\n        Thus\n\n            \\vspace{-3\\parskip}\n            \\begin{eqnarray*}\n                \\absLong{\\Expected[f(\\theta_{t})] - f(0)}\n                &= &\\absLong{\\Expected[\\Expected[f(\\theta_{t}) \\mid N_{t}]] - f(0)}\\\\\n                &= &\\absLong{\n                        \\displaystyle\n                        \\sum_{n=0}^{\\infty}\n                            \\Prob[N_{t}=n]\n                            \\,\\Expected[f(\\theta_{t}) \\mid N_{t}=n]\n                        -\n                        f(0)\n                    }\\\\\n                &= &\\absLong{\n                        \\displaystyle\n                        \\sum_{n=0}^{\\infty}\n                            \\Prob[N_{t}=n]\n                            \\underbrace{\n                                \\Expected[f(\\tau_{n})]\n                            }_{=f(0)~\\text{for $n=0$}}\n                        -\n                        f(0)\n                    }\\\\\n                &\\leq &(1 - \\Prob[N_{t}=0])\\abs{f(0)}\n                    + \\displaystyle\n                    \\sum_{n=1}^{\\infty}\n                        \\Prob[N_{t}=n]\n                        \\norm{f}_{\\infty}\\\\\n                &= &(1 - \\Prob[N_{t}=0])(\\abs{f(0)} + \\norm{f}_{\\infty})\\\\\n                &= &(1 - e^{-\\lambda t})(\\abs{f(0)} + \\norm{f}_{\\infty})\\\\\n                &\\longrightarrow &0\\\\\n            \\end{eqnarray*}\n\n        \\noindent\n        for ${\\realsPos \\ni t \\longrightarrow 0}$.\n    \\end{proof}\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Approximants as expectations]{Approximants as expectations}\n\\label{sec:stochastic:one-parameter:sig:article-stochastic-raj-dahya}\n\n\\noindent\nOur goal is to now associate continuous semigroups of distributions\nto $\\Cnought$-semigroups of operators via the machinery of expectations\ncomputed strongly via Bochner-integrals.\n\n\\begin{defn}[Expectation-approximants]\n\\makelabel{defn:expectation-approximants:sig:article-stochastic-raj-dahya}\n    Let $\\Lambda$ be a directed index set\n    and ${\\Gamma^{(\\alpha)} \\colonequals (\\Gamma^{(\\alpha)}(t))_{t \\in \\realsNonNeg}}$\n    be a continuous semigroup of distributions\n    for each $\\alpha\\in \\Lambda$.\n    Further let $T$ be a contractive $\\Cnought$-semigroup on a Banach space $\\BanachRaum$\n    and define\n\n        \\vspace{-3\\parskip}\n        $$\n            T^{(\\alpha)} \\colonequals (\\Expected[T(\\theta^{(\\alpha)}_{t})])_{t\\in\\realsNonNeg}\n        $$\n\n    \\noindent\n    for each $\\alpha \\in \\Lambda$,\n    where ${\\theta^{(\\alpha)}_{t} \\distributedAs \\Gamma^{(\\alpha)}(t)}$.\n    Furthermore, letting\n        $\\mu^{(\\alpha)}_{t}$ and $(\\sigma^{(\\alpha)}_{t})^{2}$\n    denote the mean and variance of $\\Gamma^{(\\alpha)}(t)$-distributed \\randomvar's\n    respectively, suppose that\n\n        \\vspace{-3\\parskip}\n        $$\n            \\mu^{(\\alpha)}_{t}\n                \\underset{\\alpha}{\\longrightarrow} t\n            \\quad\\text{and}\\quad\n            (\\sigma^{(\\alpha)}_{t})^{2}\n                \\underset{\\alpha}{\\longrightarrow} 0\n        $$\n\n    \\noindent\n    uniformly in $t$ on compact subsets of $\\realsNonNeg$.\n    In this case, we say that\n        $(T^{(\\alpha)})_{\\alpha\\in\\Lambda}$\n    is a net of \\highlightTerm{expectation-approximants}\n    with \\highlightTerm{associated distribution semigroups}\n        $(\\Gamma^{(\\alpha)})_{\\alpha\\in\\Lambda}$.\n    We furthermore refer to $T$ as the \\highlightTerm{original semigroup}.\n\\end{defn}\n\nOur motivation in using expectation-approximants\nis that they can be expressed in terms of the original $\\Cnought$-semigroup.\nBefore we proceed with using this broad definition,\nwe provide some concrete examples.\nIn particular, we show that the Hille- and Yosida-approximants\nsatisfy the above definition.\n\n\\begin{schattierteboxdunn}[backgroundcolor=leer,nobreak=true]\n\\begin{lemm}[Hille- and Yosida-approximants as expectations, Chung 1962]\n\\makelabel{lemm:classical-examples-expectation-approximants:sig:article-stochastic-raj-dahya}\n    Let $T$ be a contractive $\\Cnought$-semigroup on a Banach space $\\BanachRaum$\n    with generator $A$.\n    The Hille- and Yosida-approximants of $T$\n    constitute expectation-approximants.\n\\end{lemm}\n\\end{schattierteboxdunn}\n\nFor our purposes, it will suffice to assume that $T$ is contractive,\nalthough this result holds without this restriction.\nThis result can be attributed to Chung\n(see \\cite[Theorem~1 and Theorem~4]{Chung1962exp}).\nFor completeness and the reader's convenience, we provide a proof.\n\n    \\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{lemm:classical-examples-expectation-approximants:sig:article-stochastic-raj-dahya}\n    \\begin{proof}[of \\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}]\n        \\paragraph{Hille-approximants:}\n            The Hille-approximants are given by the net\n                $(T^{(\\lambda)})_{\\lambda\\in\\realsPos}$,\n            ordered by increasing values of $\\lambda$,\n            where\n                $T^{(\\lambda)} = (e^{t A^{(\\lambda)}})_{t\\in\\realsNonNeg}$\n                and\n                $A^{(\\lambda)} = \\lambda (T(\\tfrac{1}{\\lambda}) - \\onematrix)$\n            for each $\\lambda\\in\\realsPos$.\n            For $\\lambda\\in\\realsPos$ the parameterised distributions\n                ${\n                    \\Gamma^{(\\lambda)}\n                    \\colonequals\n                    (\\DistPoissScale{\\lambda t}{\\tfrac{1}{\\lambda}})_{t\\in\\realsNonNeg}\n                }$\n            form a continuous semigroup of distributions (see \\S{}\\ref{sec:stochastic:distributions:sig:article-stochastic-raj-dahya}),\n            with means\n                ${\\mu^{(\\lambda)}_{t} = \\frac{1}{\\lambda} \\cdot \\lambda t = t}$\n            and variances\n                ${\n                    (\\sigma^{(\\lambda)}_{t})^{2}\n                    = \\frac{1}{\\lambda^{2}} \\cdot \\lambda t\n                    = \\frac{t}{\\lambda}\n                    \\underset{\\lambda}{\\longrightarrow} 0\n                }$\n            for ${\\realsPos \\ni \\lambda \\longrightarrow \\infty}$,\n            and this convergence is clearly uniform in $t$\n            on compact subsets of $\\realsNonNeg$.\n            To satisfy the definition of a net of expectation-approximants,\n            it thus suffices to show for\n                $t\\in\\realsNonNeg$,\n                $\\lambda\\in\\realsPos$,\n                and\n                $\\theta \\distributedAs \\DistPoissScale{\\lambda t}{\\tfrac{1}{\\lambda}}$\n            that\n                $T^{(\\lambda)}(t) = \\Expected[T(\\theta)]$.\n            Indeed\n                $\n                    T^{(\\lambda)}(t)\n                    = e^{t A^{(\\lambda)}}\n                    = e^{\\lambda t (T(\\tfrac{1}{\\lambda}) - \\onematrix)}\n                    = e^{-\\lambda t}e^{\\lambda t \\, T(\\tfrac{1}{\\lambda})}\n                $\n            and\n\n                \\vspace{-3\\parskip}\n                $$\n                    e^{-\\lambda t}e^{\\lambda t \\, T(\\tfrac{1}{\\lambda})}\n                    = \\sum_{n=0}^{\\infty}\n                        \\frac{e^{-\\lambda t}(\\lambda t)^{n}}{n!}\n                        \\,T(\\tfrac{1}{\\lambda})^{n}\n                    = \\sum_{n=0}^{\\infty}\n                        \\Prob[\\theta=\\tfrac{1}{\\lambda}n]\\,T(\\tfrac{1}{\\lambda}n)\n                    = \\Expected[T(\\theta)].\n                $$\n\n            \\noindent\n            Hence the net of Hille-approximants for $T$\n            forms a net of expectation-approximants for $T$.\n\n        \\paragraph{Yosida-approximants:}\n            The Yosida-approximants are given by the net\n                $(T^{(\\lambda)})_{\\lambda\\in(\\omega_{0}(T)^{+},\\:\\infty)}$,\n            ordered by increasing values of $\\lambda$,\n            where\n                $T^{(\\lambda)} = (e^{t A^{(\\lambda)}})_{t\\in\\realsNonNeg}$\n                and\n                ${\n                    A^{(\\lambda)}\n                    = \\lambda \\cdot (\n                        \\lambda \\opResolvent{A}{\\lambda}\n                        - \\onematrix\n                    )\n                }$\n            for each $\\lambda\\in(\\omega_{0}(T)^{+},\\:\\infty)$.\n            For $\\lambda\\in(\\omega_{0}(T)^{+},\\:\\infty)$\n            the parameterised distributions\n                ${\n                    \\Gamma^{(\\lambda)}\n                    \\colonequals\n                    (\\DistPoissAux{t}{\\lambda})_{t\\in\\realsNonNeg}\n                }$\n            form a continuous semigroup of distributions\n            (see \\Cref{prop:app-semigroup-law:sig:article-stochastic-raj-dahya}),\n            with means\n                ${\\mu^{(\\lambda)}_{t} = \\frac{1}{\\lambda} \\cdot \\lambda t = t}$\n            and variances\n                ${\n                    (\\sigma^{(\\lambda)}_{t})^{2}\n                    = \\frac{2t}{\\lambda}\n                    \\underset{\\lambda}{\\longrightarrow} 0\n                }$\n            for ${(\\omega_{0}(T)^{+},\\:\\infty) \\ni \\lambda \\longrightarrow \\infty}$\n            (see \\Cref{prop:characteristic-function:aux-poisson-process:sig:article-stochastic-raj-dahya}),\n            and this convergence is clearly uniform in $t$\n            on compact subsets of $\\realsNonNeg$.\n            To demonstrate that\n                $(T^{(\\lambda)})_{\\lambda\\in(\\omega_{0}(T)^{+},\\:\\infty)}$\n            forms a net of expectation-approximants for $T$,\n            it thus suffices to show for\n                $t\\in\\realsNonNeg$,\n                $\\lambda\\in(\\omega_{0}(T)^{+},\\:\\infty)$,\n            and\n                ${\\theta \\distributedAs \\DistPoissAux{t}{\\lambda}}$\n            that\n                $T^{(\\lambda)}(t) = \\Expected[T(\\theta)]$.\n\n            By the construction of auxiliary Poisson processes\n            in \\S{}\\ref{sec:stochastic:distributions:sig:article-stochastic-raj-dahya},\n            it suffices to prove the claim concretely for\n                ${\\theta \\colonequals \\theta_{t} = \\tau_{<N_{t}}}$.\n            For $n\\in\\naturalsPos$, the \\emph{Phillips calculus} applied to $A$\n            (see \\exempli\n                \\cite[Lemma~VIII.1.12~{[$\\ast$]}]{Dunfordschwartz1988BookLinOpI},\n                \\cite[Proposition~3.3.5]{Reissig2005abstractresolvent}%\n            )\n            yields\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray*}\n                    (\\onematrix + \\lambda^{-1}A^{(\\lambda)})^{n}\n                        &= &(\\lambda \\opResolvent{A}{\\lambda})^{n}\\\\\n                        &= &\\lambda^{n}\n                            \\cdot\n                            \\displaystyle\n                            \\sotInt_{s=0}^{\\infty}\n                                \\frac{s^{n-1}}{(n-1)!}\n                                e^{-\\lambda s}\n                                T(s)\n                                \\:\\dee s\\\\\n                        &= &\\displaystyle\n                            \\sotInt_{s=0}^{\\infty}\n                                T(s)\n                                \\:\n                                \\underbrace{\n                                    \\lambda\n                                    \\frac{(\\lambda s)^{n-1}}{(n-1)!}\n                                    e^{-\\lambda s}\n                                    \\cdot\n                                    \\:\\dee s\n                                }_{=\\Prob_{\\tau_{<n}}(\\dee s)},\\\\\n                        &= &\\displaystyle\n                            \\sotInt_{s \\in \\realsNonNeg}\n                                T(s)\n                                \\:\\Prob_{\\tau_{<n}}(\\dee s)\\\\\n                        &\\textoverset{defn}{=}\n                            &\\Expected[T(\\tau_{<n})].\\\\\n                \\end{eqnarray*}\n\n            \\noindent\n            Moreover, since $\\tau_{<0} = 0$ \\almostSurely,\n            one has $T(\\tau_{<0}) = \\onematrix$ \\almostSurely\n            and thus\n                $\n                    \\Expected[T(\\tau_{<0})]\n                    = \\onematrix\n                    = (\\onematrix + \\lambda^{-1}A^{(\\lambda)})^{0}\n                $,\n            using the convention of $S^{0} \\colonequals \\onematrix$\n            for $S \\in \\BoundedOps{\\BanachRaum}$.\n            Thus\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray}\n                \\label{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n                    (\\onematrix + \\lambda^{-1}A^{(\\lambda)})^{n}\n                    = \\Expected[T(\\tau_{<n})]\n                \\end{eqnarray}\n\n            \\noindent\n            for all $n\\in\\naturalsZero$\n            (\\cf\n                \\cite[Lemma~2]{Chung1962exp}%\n            ).\n            Let $t\\in\\realsNonNeg$ be arbitrary.\n            By the boundedness of $A^{(\\lambda)}$\n            one has\n                $\n                    T^{(\\lambda)}(t)\n                    = e^{t A^{(\\lambda)}}\n                    = e^{\\lambda t (-\\onematrix + (\\onematrix + \\lambda^{-1} A^{(\\lambda)}))}\n                    = e^{-\\lambda t} e^{\\lambda t (\\onematrix + \\lambda^{-1} A^{(\\lambda)}))}\n                $\n            and\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray*}\n                    e^{-\\lambda t} e^{\\lambda t (\\onematrix + \\lambda^{-1} A^{(\\lambda)}))}\n                        &= &e^{-\\lambda t}\n                            \\displaystyle\n                            \\sum_{n=0}^{\\infty}\n                                \\frac{(\\lambda t)^{n}}{n!}\n                                \\underbrace{\n                                    (\\onematrix + \\lambda^{-1} A^{(\\lambda)})^{n}\n                                }_{\\text{%\n                                    $\\colonequals\\onematrix$ for $n=0$%\n                                }}\\\\\n                        &\\eqcrefoverset{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}{=}\n                        &\\displaystyle\n                            \\sum_{n=0}^{\\infty}\n                                \\underbrace{\n                                    e^{-\\lambda t}\n                                    \\frac{(\\lambda t)^{n}}{n!}\n                                }_{=\\Prob[N_{t}=n]}\n                                \\Expected[T(\\tau_{<n})]\\\\\n                        &= &\\Expected[T(\\tau_{<N_{t}})].\\\\\n                \\end{eqnarray*}\n\n            \\noindent\n            Hence\n                ${T^{(\\lambda)}(t) = \\Expected[T(\\tau_{<N_{t}})] = \\Expected[T(\\theta)]}$.\n    \\end{proof}\n\n\\begin{rem}\n    The setup of the auxiliary Poisson processes\n    used to demonstrate that Yosida-approximants are expectation-approximants\n    in \\Cref{lemm:classical-examples-expectation-approximants:sig:article-stochastic-raj-dahya}\n    bears some resemblance to \\emph{expectation semigroups} in \\emph{random evolutions}.\n    Note that a random evolution involves a \\emph{single Markov-process}:\n    both\n        for the number of events (referred to as \\usesinglequotes{jumps})\n        that occur in a time interval $[0,\\:t)$,\n    as well as\n        for the durations between each jump used with the semigroup(s),\n    which accumulate to a total duration\n    which necessarily lies in $[0,\\:t)$\n    (\\cf \\cite[\\S{}II.15.7]{Goldstein1985semigroups}).\n    By contrast, the construction in \\S{}\\ref{sec:stochastic:distributions:sig:article-stochastic-raj-dahya}\n    involves two independent processes:\n        $N_{t}$ counts up the number of events that occur in $[0,\\:t)$\n        for the first PPP,\n    whereas the total duration,\n        $\\theta_{t} = \\sum_{i=0}^{N_{t}}\\tau_{i}$,\n    used as inputs to the semigroup,\n    is determined by a second independent PPP\n    and may well lie outside $[0,\\:t)$.\n\\end{rem}\n\nWe now observe basic facts about expectation-approximants,\nwhich provide alternative unified proofs of facts\nwe already knew about Hille- and Yosida-approximants\n(\\cf \\S{}\\ref{sec:introduction:results:sig:article-stochastic-raj-dahya}).\nWe first demonstrate that expectation-approximants\nare themselves bona fide $\\Cnought$-semigroups\nand that the nets\napproximate the original $\\Cnought$-semigroup.\nThis result is likely well known (\\cf \\exempli \\cite[Lemma~1]{Chung1962exp}).\nFor the sake of completeness\nand the reader's convenience we present the proofs.\n\n\\begin{prop}\n\\makelabel{prop:expectation-approximants:basic:c0-semigroups-converege:sig:article-stochastic-raj-dahya}\n    Let $T$ be a contractive $\\Cnought$-semigroup on a Banach space $\\BanachRaum$\n    and\n        $(T^{(\\alpha)})_{\\alpha \\in \\Lambda}$\n    a net of expectation-approximants for $T$.\n    Then\n        $T^{(\\alpha)}$ is a contractive $\\Cnought$-semigroup\n    for each $\\alpha\\in\\Lambda$.\n    Moreover,\n        ${\n            T^{(\\alpha)}(t) \\underset{\\alpha}{\\longrightarrow} T(t)\n        }$\n    \\wrt the \\topSOT-topology\n    uniformly in $t$\n    on compact subsets of $\\realsNonNeg$.\n\\end{prop}\n\n    \\begin{proof}\n        Let $(\\Gamma^{(\\alpha)})_{\\alpha\\in\\Lambda}$\n        be the distribution semigroups associated with\n        the net of expectation-approximants.\n        Further let\n            ${\\theta^{(\\alpha)}_{t} \\distributedAs \\Gamma^{(\\alpha)}(t)}$\n        and let\n            $\\mu^{(\\alpha)}_{t}$ and $(\\sigma^{(\\alpha)}_{t})^{2}$\n            be the mean and variance of $\\theta^{(\\alpha)}_{t}$ respectively\n        for each $\\alpha \\in \\Lambda$ and $t\\in\\realsNonNeg$.\n\n        \\paragraph{$\\Cnought$-semigroup:}\n            Let $\\alpha\\in \\Lambda$ be arbitrary.\n            First observe that\n                $\n                    \\norm{T^{(\\alpha)}(t)}\n                    = \\norm{\\Expected[T(\\theta^{(\\alpha)}_{t})]}\n                    \\leq \\Expected[\\norm{T(\\theta^{(\\alpha)}_{t})}]\n                    \\leq 1\n                $\n            for each $t\\in\\realsNonNeg$,\n            whence each expectation-approximant is contractive.\n            Towards the semigroup law,\n            by definition of continuous semigroups of distributions,\n                $\n                    T^{(\\alpha)}(0)\n                    = \\Expected[T(\\theta^{(\\alpha)}_{0})]\n                    = \\Expected[T(0)]\n                    = \\onematrix\n                $\n            since $\\theta^{(\\alpha)}_{0} \\distributedAs \\delta_{0}$,\n            and for each $s,t\\in\\realsNonNeg$,\n            letting\n                $\\tau_{1} \\distributedAs \\theta^{(\\alpha)}_{s} \\distributedAs \\Gamma^{(\\alpha)}(s)$\n                and\n                $\\tau_{2} \\distributedAs \\theta^{(\\alpha)}_{t} \\distributedAs \\Gamma^{(\\alpha)}(t)$\n            be independent,\n            one has\n                $\\tau_{1} + \\tau_{2} \\distributedAs \\theta^{(\\alpha)}_{s+t} \\distributedAs \\Gamma^{(\\alpha)}(s+t)$,\n            whence\n                $\n                    T^{(\\alpha)}(s)T^{(\\alpha)}(t)\n                    = \\Expected[T(\\tau_{1})]\\Expected[T(\\tau_{2})]\n                    = \\Expected[T(\\tau_{1})T(\\tau_{2})]\n                    = \\Expected[T(\\tau_{1}+\\tau_{2})]\n                    = \\Expected[T(\\theta^{(\\alpha)}_{s+t})]\n                    = T^{(\\alpha)}(s+t)\n                $.\n            Towards \\topSOT-continuity of $T^{(\\alpha)}$,\n            it suffices to prove continuity in $0$.\n            To this end, let $\\xi\\in\\BanachRaum$ and $\\varepsilon > 0$ be arbitrary.\n            Since $T$ is \\topSOT-continuous,\n                ${\n                    \\sup_{s\\in[0,\\:\\delta]}\n                        \\norm{(T(s) - \\onematrix)\\xi}\n                    < \\varepsilon\n                }$\n            for some $\\delta > 0$.\n            Thus\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray*}\n                    \\norm{(T^{(\\alpha)}(t) - T^{(\\alpha)}(0))\\xi}\n                    &= &\\normLong{\\Big(\\Expected[T(\\theta^{(\\alpha)}_{t})] - \\onematrix\\Big)\\xi}\\\\\n                    &= &\\normLong{\n                            \\displaystyle\n                            \\int_{s\\in\\realsNonNeg}\n                                (T(s) - \\onematrix)\\xi\n                            \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\n                        }\\\\\n                    &\\leq &\\displaystyle\n                        \\int_{s\\in\\realsNonNeg}\n                            \\norm{(T(s) - \\onematrix)\\xi}\n                        \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\\\\\n                    &= &\\displaystyle\n                        \\int_{s\\in[0,\\:\\delta]}\n                            \\underbrace{\n                                \\normLong{(T(s) - \\onematrix)\\xi}\n                            }_{<\\varepsilon}\n                        \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\n                        + \\displaystyle\n                        \\int_{s\\in(\\delta,\\:\\infty)}\n                            \\underbrace{\n                                \\normLong{(T(s) - \\onematrix)\\xi}\n                            }_{\\leq 2\\norm{\\xi}}\n                        \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\\\\\n                    &\\leq &\\varepsilon\\,\\Prob_{\\theta^{(\\alpha)}_{t}}[[0,\\:\\delta]]\n                        + 2\\norm{\\xi}\\,\\Prob_{\\theta^{(\\alpha)}_{t}}[(\\delta,\\:\\infty)]\\\\\n                \\end{eqnarray*}\n\n            \\noindent\n            for all $t\\in\\realsNonNeg$.\n            By the Portmanteau theorem (see \\cite[Theorem~17.20~v)]{Kechris1995BookDST}) and since by definition\n                ${\\theta^{(\\alpha)}_{t} \\underset{t}{\\longrightarrow} \\delta_{0}}$\n            and\n                $\n                    \\delta_{0}(\\quer{(\\delta,\\:\\infty)} \\mathbin{\\setminus} \\topInterior{(\\delta,\\:\\infty)}\n                    = \\delta_{0}(\\{\\delta\\})\n                    = 0\n                $,\n            one has\n                ${\n                    \\Prob_{\\theta^{(\\alpha)}_{t}}[(\\delta,\\:\\infty)]\n                    \\longrightarrow\n                    \\Prob_{\\delta_{0}}[(\\delta,\\:\\infty)]\n                    = 0\n                }$\n            for ${\\realsPos \\ni t \\longrightarrow 0}$.\n            From the above inequality and since $\\varepsilon > 0$ was arbitrarily chosen,\n            it follows that\n                ${T^{(\\alpha)}(t)\\xi \\longrightarrow T^{(\\alpha)}(0)\\xi}$\n            for ${\\realsPos \\ni t \\longrightarrow 0}$.\n            So $T^{(\\alpha)}$ constitutes a contractive $\\Cnought$-semigroup.\n\n        \\paragraph{Approximation:}\n            Fix an arbitrary\n                $\\xi\\in\\BanachRaum$\n                and\n                compact subset $L \\subseteq \\realsNonNeg$.\n            Let $\\varepsilon > 0$ be arbitrary.\n            Without loss of generality, we can assume that $L=[0,\\:a]$\n            for some $a\\in\\realsPos$.\n            By the \\topSOT-continuity of $T$ and compactness of $[0,\\:a+1]$,\n            there exists $\\delta \\in (0,\\:1)$ such that\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray}\n                \\label{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n                    \\sup_{s \\in U}\n                        \\norm{(T(s) - T(t))\\xi}\n                    < \\varepsilon\n                \\end{eqnarray}\n\n            \\noindent\n            for all $t \\in L$,\n            where ${U \\colonequals \\realsNonNeg \\cap (t-\\delta,\\:t+\\delta)}$.\n            Since by definition of expectation-approximants,\n                ${\\mu^{(\\alpha)}_{t} \\underset{\\alpha}{\\longrightarrow} t}$\n            uniformly for $t \\in L$,\n            by \\eqcref{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya} it follows that\n                ${\\sup_{t \\in L}\\norm{(T(\\mu^{(\\alpha)}_{t}) - T(t))\\xi} \\leq \\varepsilon}$\n            for sufficiently large indices $\\alpha \\in \\Lambda$.\n            Furthermore, by the Chebyshev-inequality (see \\exempli \\cite[Theorem~5.11]{Klenke2008probTheory}), one has that\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray}\n                \\label{eq:chebyshev:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n                    \\Prob_{\\theta^{(\\alpha)}_{t}}[\\realsNonNeg \\mathbin{\\setminus} U]\n                        \\leq \\delta^{-2}(\\sigma^{(\\alpha)}_{t})^{2}\n                \\end{eqnarray}\n\n            \\noindent\n            for each $t \\in L$.\n            Since by definition of expectation-approximants,\n                ${(\\sigma^{(\\alpha)}_{t})^{2} \\underset{\\alpha}{\\longrightarrow} 0}$\n            uniformly for $t \\in L$,\n            by \\eqcref{eq:chebyshev:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya} one has\n                ${\\sup_{t \\in L}\\Prob_{\\theta^{(\\alpha)}_{t}}[\\realsNonNeg \\mathbin{\\setminus} U] \\leq \\varepsilon}$\n            for sufficiently large indices $\\alpha \\in \\Lambda$.\n            For sufficiently large indices\n                $\\alpha \\in \\Lambda$ and all $t \\in L$\n            it follows that\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray*}\n                    \\norm{(T^{(\\alpha)}(t) - T(t))\\xi}\n                        &\\leq\n                            &\\norm{(T(\\mu^{(\\alpha)}_{t}) - T(t))\\xi}\n                            + \\norm{(T^{(\\alpha)}(t) - T(\\mu^{(\\alpha)}_{t}))\\xi}\\\\\n                    &=\n                        &\\underbrace{\n                            \\norm{(T(\\mu^{(\\alpha)}_{t}) - T(t))\\xi}\n                        }_{\\leq\\varepsilon}\n                        + \\normLong{\n                            \\Big(\\Expected[T(\\theta^{(\\alpha)}_{t})] - T(\\mu^{(\\alpha)}_{t})\\Big)\\xi\n                        }\\\\\n                    &\\leq &\\varepsilon + \\normLong{\n                            \\displaystyle\n                            \\int_{s\\in\\realsNonNeg}\n                                (T(s) - T(\\mu^{(\\alpha)}_{t}))\\xi\n                            \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\n                        }\\\\\n                    &\\leq &\\varepsilon\n                        + \\displaystyle\n                        \\int_{s\\in\\realsNonNeg}\n                            \\norm{(T(s) - T(\\mu^{(\\alpha)}_{t}))\\xi}\n                        \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\\\\\n                    &\\leq &\\begin{array}[t]{0l}\n                            \\varepsilon + \\displaystyle\n                            \\int_{s \\in U}\n                                \\underbrace{\n                                    \\normLong{(T(s) - T(t))\\xi}\n                                    + \\normLong{(T(t) - T(\\mu^{(\\alpha)}_{t}))\\xi}\n                                }_{\\leq 2\\varepsilon}\n                            \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\\\\\n                            + \\displaystyle\n                            \\int_{s \\in \\realsNonNeg \\mathbin{\\setminus} U}\n                                \\underbrace{\n                                    \\normLong{(T(s) - T(\\mu^{(\\alpha)}_{t}))\\xi}\n                                }_{\\leq 2\\norm{\\xi}}\n                            \\:\\Prob_{\\theta^{(\\alpha)}_{t}}(\\dee s)\\\\\n                        \\end{array}\\\\\n                    &\\leq &\\varepsilon + 2\\varepsilon\\,\\Prob_{\\theta^{(\\alpha)}_{t}}[U]\n                        + 2\\norm{\\xi}\\,\\underbrace{\n                            \\Prob_{\\theta^{(\\alpha)}_{t}}[\\realsNonNeg \\mathbin{\\setminus} U]\n                        }_{\\leq\\varepsilon}\n                    \\leq 3\\varepsilon + 2\\norm{\\xi}\\varepsilon.\\\\\n                \\end{eqnarray*}\n\n            \\noindent\n            Since $\\xi$ and $\\varepsilon$ were arbitrarily chosen,\n            it follows that\n                ${T^{(\\alpha)}(t) \\longrightarrow T(t)}$\n                \\wrt the \\topSOT-topology\n                uniformly in $t$\n                on compact subsets of $\\realsNonNeg$.\n    \\end{proof}\n\n\\begin{prop}\n\\makelabel{prop:expectation-approximants:basic:commuting-contractive-families:sig:article-stochastic-raj-dahya}\n    Let\n        $d\\in\\naturals$,\n        $\\BanachRaum$ be a Banach space,\n        and\n        $\\{T_{i}\\}_{i=1}^{d}$ be a commuting family\n        of contractive $\\Cnought$-semigroups on $\\BanachRaum$.\n    Furthermore, let\n        $(T^{(\\alpha)}_{i})_{\\alpha\\in\\Lambda_{i}}$\n        be expectation-approximants for $T_{i}$\n        for each $i\\in\\{1,2,\\ldots,d\\}$.\n    Then for each\n        $\\boldsymbol{\\alpha} \\in \\prod_{i=1}^{d} \\Lambda_{i}$\n    the family of approximants\n        $\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d}$\n    is a commuting family of contractive $\\Cnought$-semigroups.\n    Moreover,\n        $(\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d})_{\\boldsymbol{\\alpha}\\in\\prod_{i=1}^{d} \\Lambda_{i}}$\n    converges to $\\{T_{i}\\}_{i=1}^{d}$\n    \\wrt the \\topSOT-topology\n    uniformly on compact subsets of $\\realsNonNeg^{d}$.\n\\end{prop}\n\n    \\begin{proof}\n        \\paragraph{Commuting family:}\n            By \\Cref{prop:expectation-approximants:basic:c0-semigroups-converege:sig:article-stochastic-raj-dahya},\n            the approximants $T^{(\\alpha_{i})}_{i}$\n            are contractive $\\Cnought$-semigroups\n            for each $i\\in\\{1,2,\\ldots,d\\}$.\n            Let $\\mathbf{t}\\in\\realsNonNeg^{d}$ be arbitrary.\n            Per definition of expectation-approximants\n            there exist $\\realsNonNeg$-valued \\randomvar's\n                $\\theta_{1},\\theta_{1},\\ldots,\\theta_{d}$\n            satisfying\n                ${T^{(\\alpha_{i})}_{i}(t_{i}) = \\Expected[T_{i}(\\theta_{i})]}$\n            for each $i\\in\\{1,2,\\ldots,d\\}$.\n            Without loss of generality, we may assume that the $\\theta_{i}$ are independent \\randomvar's.\n            By independence and commutativity\n                $\n                    \\Expected[T_{i}(\\theta_{i})]\n                    \\Expected[T_{j}(\\theta_{j})]\n                    = \\Expected[T_{i}(\\theta_{i})T_{j}(\\theta_{j})]\n                    = \\Expected[T_{j}(\\theta_{j})T_{i}(\\theta_{i})]\n                    = \\Expected[T_{j}(\\theta_{j})]\n                    \\Expected[T_{i}(\\theta_{i})]\n                $\n            for all $i,j\\in\\{1,2,\\ldots,d\\}$ with $i \\neq j$.\n            Thus $\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d}$\n            is a commuting family of contractive $\\Cnought$-semigroups.\n\n        \\paragraph{Approximation:}\n            Let $L \\subseteq \\realsNonNeg^{d}$ be an arbitrary compact subset.\n            Without loss of generality, one may assume $L = \\prod_{i=1}^{d}L_{i}$\n            for some compact subsets $L_{i} \\subseteq \\realsNonNeg$, $i\\in\\{1,2,\\ldots,d\\}$.\n            We prove by induction over $k\\in\\{1,2,\\ldots,d\\}$ that\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray}\n                \\label{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n                    \\sup_{\\mathbf{t} \\in \\prod_{i=1}^{k}L_{i}}\n                    \\normLong{\n                        \\Big(\n                            \\prod_{i=1}^{k}\n                                T^{(\\alpha_{i})}_{i}(t_{i})\n                            -\n                            \\prod_{i=1}^{k}\n                                T_{i}(t_{i})\n                        \\Big)\n                        \\xi\n                    }\n                    \\underset{\\boldsymbol{\\alpha}}{\\longrightarrow} 0\n                \\end{eqnarray}\n\n            \\noindent\n            for all $\\xi\\in\\BanachRaum$.\n            For $k=1$, this holds by \\Cref{prop:expectation-approximants:basic:c0-semigroups-converege:sig:article-stochastic-raj-dahya}.\n            Let $1 < k \\leq d$ and assume that \\eqcref{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya} holds for $k-1$.\n            Let $\\xi\\in\\BanachRaum$ and $\\varepsilon > 0$ be arbitrary.\n            Since $T_{k}$ is \\topSOT-continuous and $L_{k}$ is compact,\n            there is a finite subset $F \\subseteq L_{k}$ such that\n                $\\min_{t^{\\prime} \\in F}\\norm{(T_{k}(t) - T_{k}(t^{\\prime}))\\xi} < \\varepsilon$\n            for each $t \\in L_{k}$.\n            Let $\\mathbf{t} \\in L$ be arbitrary\n            and let $t^{\\prime} \\in F$ be such\n            that $\\norm{(T_{k}(t_{k}) - T_{k}(t^{\\prime}))\\xi} < \\varepsilon$.\n            Since the approximants are all contractive,\n            one obtains\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray*}\n                    \\normLong{\n                        \\Big(\n                            \\displaystyle\n                            \\prod_{i=1}^{k}\n                                T^{(\\alpha_{i})}_{i}(t_{i})\n                            -\n                            \\displaystyle\n                            \\prod_{i=1}^{k}\n                                T_{i}(t_{i})\n                        \\Big)\n                        \\xi\n                    }\n                        &\\leq\n                            &\\normLong{\n                                \\Big(\n                                    \\displaystyle\n                                    \\prod_{i=1}^{k-1}\n                                        T^{(\\alpha_{i})}_{i}(t_{i})\n                                    -\n                                    \\displaystyle\n                                    \\prod_{i=1}^{k-1}\n                                        T_{i}(t_{i})\n                                \\Big)\n                                T_{k}(t^{\\prime})\n                                \\xi\n                            }\\\\\n                            &&+ \\underbrace{\n                                \\normLong{\n                                    \\displaystyle\n                                    \\prod_{i=1}^{k-1}\n                                        T^{(\\alpha_{i})}_{i}(t_{i})\n                                    -\n                                    \\displaystyle\n                                    \\prod_{i=1}^{k-1}\n                                        T_{i}(t_{i})\n                                }\n                            }_{\\leq 2}\n                            \\underbrace{\n                                \\normLong{\n                                    \\Big(\n                                        T_{k}(t^{\\prime}) - T_{k}(t_{k})\n                                    \\Big)\n                                    \\xi\n                                }\n                            }_{< \\varepsilon}\\\\\n                            &&+ \\underbrace{\n                                \\normLong{\n                                    \\displaystyle\n                                    \\prod_{i=1}^{k-1}\n                                        T^{(\\alpha_{i})}_{i}(t_{i})\n                                }\n                            }_{\\leq 1}\n                            \\normLong{\n                                \\Big(\n                                    T^{(\\alpha_{k})}_{k}(t_{k}) - T_{k}(t_{k})\n                                \\Big)\n                                \\xi\n                            }\\\\\n                        &\\leq\n                            &\\displaystyle\n                            \\max_{t^{\\prime\\prime} \\in F}\n                                \\normLong{\n                                    \\Big(\n                                        \\displaystyle\n                                        \\prod_{i=1}^{k-1}\n                                            T^{(\\alpha_{i})}_{i}(t_{i})\n                                        -\n                                        \\displaystyle\n                                        \\prod_{i=1}^{k-1}\n                                            T_{i}(t_{i})\n                                    \\Big)\n                                    T_{k}(t^{\\prime\\prime})\n                                    \\xi\n                                }\\\\\n                        &&+ 2\\varepsilon + \\norm{\n                                (T^{(\\alpha_{k})}_{k}(t_{k}) - T_{k}(t_{k}))\n                                \\xi\n                            }\\\\\n                \\end{eqnarray*}\n\n            \\noindent\n            for each $\\boldsymbol{\\alpha} \\in \\prod_{i=1}^{k}\\Lambda_{i}$.\n            By\n                induction\n                (%\n                    \\eqcref{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya} applied to $k-1$\n                    and the finite set of vectors\n                    $\\{T_{k}(t^{\\prime\\prime})\\xi \\mid t^{\\prime\\prime} \\in F\\}$%\n                )\n                and\n                \\Cref{prop:expectation-approximants:basic:c0-semigroups-converege:sig:article-stochastic-raj-dahya}\n                (applied to $T_{k}$),\n            by taking $\\limsup$ over $\\boldsymbol{\\alpha}$,\n            the right-hand expression is bounded by $0 + 2\\varepsilon + 0$.\n            Since $\\varepsilon > 0$ was arbitrarily chosen, it follows that \\eqcref{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya} holds.\n            Hence the claim holds by induction.\n    \\end{proof}\n\nWe also obtain the following auxiliary results for simple modifications of $\\Cnought$-semigroups:\n\n\\begin{prop}\n\\makelabel{prop:basic-operations:expectation-approximations:sig:article-stochastic-raj-dahya}\n    Let\n        $\\BanachRaum$ be a Banach space\n        and\n        $T$ be a contractive $\\Cnought$-semigroup on $\\BanachRaum$.\n    Furthermore, let\n        $(T^{(\\alpha)})_{\\alpha \\in \\Lambda}$\n    be a net of expectation-approximants for $T$\n    with associated distribution semigroups\n        $(\\Gamma^{(\\alpha)})_{\\alpha \\in \\Lambda}$.\n    Then\n    for any (equivalently: every) \\randomvar\n        ${\\theta\\distributedAs\\Gamma^{(\\alpha)}(t)}$,\n        $t\\in\\realsNonNeg$,\n        and\n        $\\alpha \\in \\Lambda$\n\n    \\begin{kompaktenum}{\\bfseries (i)}\n        \\item\\punktlabel{1}\n            $(T^{(\\alpha)}(t))^{\\prime} = \\Expected[T(\\theta)^{\\prime}]$,\n            provided $\\BanachRaum$ is reflexive; and\n        \\item\\punktlabel{2}\n            $(T^{(\\alpha)}(t))^{\\ast} = \\Expected[T(\\theta)^{\\ast}]$,\n            if $\\BanachRaum$ is a Hilbert space.\n    \\end{kompaktenum}\n\n    \\noindent\n    Furthermore, in the case of Hille-(\\respectively Yosida-)approximants\n        $(T^{(\\lambda)})_{\\lambda \\in \\realsPos}$\n    it holds that\n\n    \\begin{kompaktenum}{\\bfseries (i)}\n    \\setcounternach{enumi}{3}\n        \\item\\punktlabel{3}\n            $T^{(\\lambda)}(rt) = \\Expected[T(r\\theta)]$;\n    \\end{kompaktenum}\n\n    \\noindent\n    for $r\\in\\realsPos$\n    and any (equivalently: every) \\randomvar\n        ${\\theta \\distributedAs \\DistPoissScale{r\\lambda t}{\\tfrac{1}{r\\lambda}}}$\n    (\\respectively ${\\theta \\distributedAs \\DistPoissAux{t}{r\\lambda}}$).\n\\end{prop}\n\n    \\begin{proof}\n        \\paragraph{\\punktcref{1}:}\n            First observe that\n                $(T^{(\\alpha)}(t)^{\\prime})_{t\\in\\realsNonNeg}$\n                and\n                $(T(t)^{\\prime})_{t\\in\\realsNonNeg}$\n            are $\\Cnought$-semigroups\n            (see \\exempli \\cite[Theorem~I.4.9]{Goldstein1985semigroups}).\n            Let $\\xi\\in\\BanachRaum^{\\prime\\prime} \\cong \\BanachRaum$\n            and $\\eta\\in\\BanachRaum^{\\prime}$ be arbitrary.\n            It holds that\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray*}\n                    \\brkt{\\xi}{T^{(\\alpha)}(t)^{\\prime}\\eta}\n                        &= &\\brkt{T^{(\\alpha)}(t)\\xi}{\\eta}\\\\\n                        &= &\\brkt{\\Expected[T(\\theta)]\\xi}{\\eta}\\\\\n                        &= &\\brktLong{\n                                \\Big(\n                                    \\displaystyle\n                                    \\sotInt_{s\\in\\realsNonNeg}\n                                        T(s)\n                                    \\:\\Prob_{\\theta}(\\dee s)\n                                \\Big)\n                                \\xi\n                            }{\\eta}\\\\\n                        &= &\\displaystyle\n                            \\int_{s\\in\\realsNonNeg}\n                                \\brkt{T(s)\\xi}{\\eta}\n                            \\:\\Prob_{\\theta}(\\dee s)\\\\\n                        &= &\\displaystyle\n                            \\int_{s\\in\\realsNonNeg}\n                                \\brkt{\\xi}{T(s)^{\\prime}\\eta}\n                            \\:\\Prob_{\\theta}(\\dee s)\\\\\n                        &= &\\brktLong{\\xi}{\n                                \\Big(\n                                    \\displaystyle\n                                    \\sotInt_{s\\in\\realsNonNeg}\n                                        T(s)^{\\prime}\n                                    \\:\\Prob_{\\theta}(\\dee s)\n                                \\Big)\n                                \\eta\n                            }\\\\\n                        &= &\\brkt{\\xi}{\\Expected[T(\\theta)^{\\prime}]\\eta}.\\\\\n                \\end{eqnarray*}\n\n            \\noindent\n            It follows that $T^{(\\alpha)}(t)^{\\prime} = \\Expected[T(\\theta)^{\\prime}]$.\n            The expression in \\punktcref{2} can be proved analogously.\n\n        \\paragraph{\\punktcref{3}:}\n            First observe that\n                ${T_{r} \\colonequals (T(rt^{\\prime}))_{t^{\\prime}\\in\\realsNonNeg}}$\n            is a $\\Cnought$-semigroup with generator $A_{r} = r A$.\n            Let $\\lambda\\in\\realsPos$.\n            In the case of Hille-approximants, one has\n\n                \\vspace{-3\\parskip}\n                $$\n                    A_{r}^{(r\\lambda)}\n                    = (r\\lambda) \\cdot (\n                        T_{r}(\\tfrac{1}{r\\lambda}) - \\onematrix\n                    )\n                    = (r\\lambda) \\cdot (\n                        T(\\tfrac{1}{\\lambda}) - \\onematrix\n                    )\n                    = r A^{(\\lambda)}\n                $$\n\n            \\noindent\n            and in the case of Yosida-approximants\n\n                \\vspace{-3\\parskip}\n                $$\n                    A_{r}^{(r\\lambda)}\n                    = (r\\lambda) \\cdot (\n                        (r\\lambda) \\opResolvent{A_{r}}{r\\lambda}\n                        - \\onematrix\n                    )\n                    = r\\lambda \\cdot (\n                        (r\\lambda) \\opResolvent{rA}{r\\lambda}\n                        - \\onematrix\n                    )\n                    = rA^{(\\lambda)}.\n                $$\n\n            \\noindent\n            Thus in both cases\n                $\n                    T^{(r\\lambda)}_{r}(t)\n                    = e^{t A_{r}^{(r\\lambda)}}\n                    = e^{t r A^{(\\lambda)}}\n                    = T^{(\\lambda)}(rt)\n                $\n            holds.\n            Applying the properties of the\n                $(r\\lambda)$\\textsuperscript{th}-Hille-approximant\n            (\\respectively $(r\\lambda)$\\textsuperscript{th}-Yosida-approximant)\n            of $T_{r}$ thus yields\n\n                \\vspace{-3\\parskip}\n                $$\n                    T^{(\\lambda)}(rt)\n                    = T^{(r\\lambda)}_{r}(t)\n                    = \\Expected[T_{r}(\\theta)]\n                    = \\Expected[T(r\\theta)]\n                $$\n\n            \\noindent\n            for any (equivalently: every) \\randomvar\n                ${\\theta \\distributedAs \\DistPoissScale{r\\lambda t}{\\tfrac{1}{r\\lambda}}}$\n            (\\respectively ${\\theta \\distributedAs \\DistPoissAux{t}{r\\lambda}}$).\n    \\end{proof}\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Families of approximants as expectations]{Families of approximants as expectations}\n\\label{sec:stochastic:multi-parameter:sig:article-stochastic-raj-dahya}\n\n\\noindent\nIn oder to prove\n    \\eqcref{it:3:thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\eqcref{it:4:thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n    of\n    \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya},\nwe shall rely on results about the expectations of products.\n\n\\begin{prop}\n\\makelabel{prop:expectation-approximants:product:sig:article-stochastic-raj-dahya}\n    Let\n        $d\\in\\naturals$,\n        $\\BanachRaum$ be a Banach space,\n        and\n        $\\{T_{i}\\}_{i=1}^{d}$ be a (not necessarily commuting) family of\n        contractive $\\Cnought$-semigroups on $\\BanachRaum$.\n    Further let\n        $(T^{(\\alpha)}_{i})_{\\alpha \\in \\Lambda_{i}}$\n    be a net of expectation-approximants for $T_{i}$\n    with associated distribution semigroups\n        $(\\Gamma^{(\\alpha)}_{i}(t))_{t\\in\\realsNonNeg,\\alpha \\in \\Lambda_{i}}$\n    for each $i \\in \\{1,2,\\ldots,d\\}$.\n    Then for\n        $\\boldsymbol{\\alpha}\\in\\prod_{i=1}^{d}\\Lambda_{i}$,\n        and\n        $\\boldsymbol{t} \\in \\realsNonNeg^{d}$\n    it holds that\n\n        \\vspace{-3\\parskip}\n        \\begin{eqnarray}\n        \\label{eq:yosida-stoch:prod:sig:article-stochastic-raj-dahya}\n            \\displaystyle\n            \\prod_{i=1}^{d}\n                T^{(\\alpha_{i})}_{i}(t_{i})\n            = \\Expected\\Big[\n                \\displaystyle\n                \\prod_{i=1}^{d}\n                    T_{i}(\\theta_{i})\n            \\Big],\n        \\end{eqnarray}\n\n    \\noindent\n    for any (equivalently: every) family of\n    independent \\randomvar's\n        ${\\theta_{i} \\distributedAs \\Gamma^{(\\alpha_{i})}_{i}(t)}$,\n        $i\\in\\{1,2,\\ldots,d\\}$.\n\\end{prop}\n\n    \\begin{proof}\n        By definition of expectation-approximants we have\n            $\n                T^{(\\alpha_{i})}_{i}(t_{i})\n                = \\Expected[\n                    T_{i}(\\theta_{i})\n                ]\n            $\n        for each $i\\in\\{1,2,\\ldots,d\\}$\n        and thus by independence\n            $\n                \\prod_{i=1}^{d}\n                    T^{(\\alpha_{i})}_{i}(t_{i})\n                = \\prod_{i=1}^{d}\n                    \\Expected[T_{i}(\\theta_{i})]\n                = \\Expected[\n                    \\prod_{i=1}^{d}\n                        T_{i}(\\theta_{i})\n                ]\n            $.\n    \\end{proof}\n\nRestricting to the context of semigroups over Hilbert spaces yields a result,\nwhich can be utilised with regular polynomial evaluations.\nFor $p \\in \\complex[X_{1},X_{1}^{-1},X_{2},X_{2}^{-1},\\ldots,X_{d},X_{d}^{-1}]$\nlet the \\highlightTerm{absolute degree of $p$}\ndenote the largest $n\\in\\naturalsZero$,\nsuch that $X_{i}^{n}$ or $X_{i}^{-n}$ occurs in some monomial in $p$,\nor else $0$ if $p=0$.\nIn particular, the absolute degree of $p$ is at most $1$\nif and only if\nthe only powers of the $X_{i}$ that occur in monomials in $p$ are $\\pm 1$.\n\n\\begin{prop\n\\makelabel{prop:expectation-approximants:poly:sig:article-stochastic-raj-dahya}\n    Let\n        $d\\in\\naturals$,\n        $\\HilbertRaum$ be a Hilbert space,\n        and\n        $\\{T_{i}\\}_{i=1}^{d}$\n        be a commuting family of contractive $\\Cnought$-semigroups on $\\HilbertRaum$.\n    Further let\n        $(T^{(\\alpha)}_{i})_{\\alpha\\in\\Lambda_{i}}$\n    be a net of expectation-approximants for $T_{i}$\n    with associated distribution semigroups\n        $(\\Gamma^{(\\alpha)}_{i})_{\\alpha\\in\\Lambda_{i}}$\n    for each $i \\in \\{1,2,\\ldots,d\\}$.\n    Then for\n        $\\boldsymbol{\\alpha}\\in\\prod_{i=1}^{d}\\Lambda_{i}$,\n        $\\boldsymbol{t} \\in \\realsNonNeg^{d}$\n    and any regular polynomial\n        $p \\in \\complex[X_{1},X_{1}^{-1},X_{2},X_{2}^{-1},\\ldots,X_{d},X_{d}^{-1}]$\n    with absolute degree at most $1$\n\n        \\vspace{-3\\parskip}\n        \\begin{eqnarray}\n        \\label{eq:yosida-stoch:poly:sig:article-stochastic-raj-dahya}\n            p(\n                T^{(\\lambda_{1})}_{1}(t_{1}),\n                T^{(\\lambda_{2})}_{2}(t_{2}),\n                \\ldots,\n                T^{(\\lambda_{d})}_{d}(t_{d})\n            )\n            = \\Expected[\n                p(\n                    T_{1}(\\theta_{1}),\n                    T_{2}(\\theta_{2}),\n                    \\ldots,\n                    T_{d}(\\theta_{d})\n                )\n            ]\n        \\end{eqnarray}\n\n    \\noindent\n    for any (equivalently: every) family of\n    independent \\randomvar's\n        ${\\theta_{i} \\distributedAs \\Gamma^{(\\alpha_{i})}_{i}(t_{i})}$,\n        $i\\in\\{1,2,\\ldots,d\\}$.\n\\end{prop}\n\n    \\begin{proof}\n        By linearity, it suffices to simply consider monomials of the form\n            $p=\\prod_{i=1}^{n}X_{i}^{n_{i}}$\n        where $\\mathbf{n} \\in \\{0, \\pm 1\\}^{d}$.\n        Let\n            $C_{1} \\colonequals \\mathop{\\textup{supp}}(\\mathbf{n}^{-})$,\n            $C_{2} \\colonequals \\mathop{\\textup{supp}}(\\mathbf{n}^{+})$,\n            and\n            $K \\colonequals \\mathop{\\textup{supp}}(\\mathbf{n}) \\subseteq \\{1,2,\\ldots,d\\}$,\n        where\n            $\\mathbf{n}^{-} \\colonequals (n_{i}^{-})_{i=1}^{d}$\n            and\n            $\\mathbf{n}^{+} \\colonequals (n_{i}^{+})_{i=1}^{d}$.\n        Then $\\isPartition{(C_{1},C_{2})}{K}$.\n        By taking regular polynomial evaluations, we thus have to prove that\n\n            \\vspace{-3\\parskip}\n            $$\n                \\prod_{i \\in C_{1}}T^{(\\lambda_{i})}_{i}(t_{i})^{\\ast}\n                \\cdot\n                \\prod_{j \\in C_{2}}T^{(\\lambda_{j})}_{j}(t_{j})\n                =\n                \\Expected\\Big[\n                    \\prod_{i \\in C_{1}}T_{i}(\\theta_{i})^{\\ast}\n                    \\cdot\n                    \\prod_{j \\in C_{2}}T_{j}(\\theta_{j})\n                \\Big]\n            $$\n\n        \\noindent\n        for any (equivalently: every) family of\n        independent \\randomvar's\n            ${\\theta_{i} \\distributedAs \\Gamma^{(\\alpha_{i})}_{i}(t)}$,\n            $i \\in K$.\n        This follows by applying\n            \\Cref{%\n                prop:basic-operations:expectation-approximations:sig:article-stochastic-raj-dahya,%\n                prop:expectation-approximants:product:sig:article-stochastic-raj-dahya,%\n            }.\n    \\end{proof}\n\n\\begin{rem}\n    For this paper we only need the statement in\n        \\Cref{prop:expectation-approximants:poly:sig:article-stochastic-raj-dahya}\n    to hold for regular polynomials of absolute degree at most $1$.\n    If we nonetheless sought to widen the scope of this result,\n    observe that limitations arise for powers $n$ of $X_{i}$ with $\\abs{n}\\geq 2$,\n    since then the application of\n        \\Crefit{prop:basic-operations:expectation-approximations:sig:article-stochastic-raj-dahya}{3}\n    leads to random variables with different parameterisations.\n\\end{rem}\n\nAs an immediate consequence of \\Cref{prop:expectation-approximants:poly:sig:article-stochastic-raj-dahya},\nwe obtain:\n\n\\begin{schattierteboxdunn}[backgroundcolor=leer,nobreak=true]\n\\begin{lemm}[Transfer result]\n\\makelabel{lemm:positivity-transfer:poly:sig:article-stochastic-raj-dahya}\n    Let\n        $d\\in\\naturals$,\n        $\\HilbertRaum$ be a Hilbert space,\n        and\n        $\\{T_{i}\\}_{i=1}^{d}$\n        be a commuting family of contractive $\\Cnought$-semigroups on $\\HilbertRaum$.\n    Further for each $i \\in \\{1,2,\\ldots,d\\}$ let\n        $(T^{(\\alpha)}_{i})_{\\alpha \\in \\Lambda_{i}}$\n    be a net of expectation-approximants for $T_{i}$.\n    Then for any regular polynomial\n        $p \\in \\complex[X_{1},X_{1}^{-1},X_{2},X_{2}^{-1},\\ldots,X_{d},X_{d}^{-1}]$\n    with absolute degree at most $1$,\n    if\n        $p(T_{1}(t_{1}),T_{1}(t_{2}),\\ldots,T_{d}(t_{d}))$\n        is a positive operator\n        for all $\\mathbf{t} \\in \\realsNonNeg^{d}$,\n    then\n        $p(T^{(\\alpha_{1})}_{1}(t_{1}),T^{(\\alpha_{1})}_{1}(t_{2}),\\ldots,T^{(\\alpha_{d})}_{d}(t_{d}))$\n        is positive\n    for all\n        $\\boldsymbol{\\alpha} \\in \\prod_{i=1}^{d}\\Lambda_{i}$\n        and\n        $\\mathbf{t} \\in \\realsNonNeg^{d}$.\n\\end{lemm}\n\\end{schattierteboxdunn}\n\n\n\n\n\\@startsection{section}{1}{\\z@}{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}{\\formatsection@text}[First main result]{First main result: Expectation-approximants and polynomial bounds}\n\\label{sec:first-results:sig:article-stochastic-raj-dahya}\n\n\n\\noindent\nWe can now prove \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}.\n\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n\n\\begin{proof}[of \\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}]\n    First observe by\n        \\Cref{prop:expectation-approximants:basic:commuting-contractive-families:sig:article-stochastic-raj-dahya}\n    that the expectation-approximants constitute commuting families of $\\Cnought$-semigroups,\n    which converge \\wrt the \\topSOT-topology uniformly on compact subsets of $\\realsNonNeg^{d}$\n    to the original family $\\{T_{i}\\}_{i=1}^{d}$.\n    In particular, the implication\n        \\punktcref{4}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1}\n    holds by \\Cref{thm:classification:dissipativity:sig:article-stochastic-raj-dahya}.\n    The implications\n        \\punktcref{1}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}\n    hold by \\Cref{thm:classification-bounded:poly:sig:article-stochastic-raj-dahya},\n    since these do not require the assumption of bounded generators.\n    It remains to prove \\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4}.\n    We first prove this under the assumption\n    that the expectation-approximants have bounded generators.\n\n    \\paragraph{\\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4}, under boundedness assumption on approximants:}\n        Let\n            $\n                \\boldsymbol{\\alpha}\n                \\in \\prod_{i=1}^{d}\\Lambda_{i}\n            $\n        be arbitrary.\n        Let $K \\subseteq \\{1,2,\\ldots,d\\}$ be arbitrary.\n        By assumption,\n\n            \\vspace{-3\\parskip}\n            \\begin{eqnarray*}\n                p_{K}(T_{1}(t_{1}),T_{2}(t_{2}),\\ldots,T_{d}(t_{d}))\n                = \\displaystyle\n                    \\sum_{\\mathclap{\n                        \\isPartition{(C_{1},C_{2})}{K}\n                    }}\n                        \\displaystyle\n                        \\prod_{i \\in C_{1}}T_{i}(t_{i})^{\\ast}\n                        \\displaystyle\n                        \\prod_{j \\in C_{2}}T_{j}(t_{j})\n                \\geq \\zeromatrix\n            \\end{eqnarray*}\n\n        \\noindent\n        for all\n            $\\mathbf{t} \\in \\realsNonNeg^{d}$.\n        By \\Cref{lemm:positivity-transfer:poly:sig:article-stochastic-raj-dahya},\n        this positivity can be transferred to the family of expectation-approximants,\n        \\idest\n            ${\n                p_{K}(T^{(\\alpha_{1})}_{1}(t_{1}),T^{(\\alpha_{2})}_{2}(t_{2}),\\ldots,T^{(\\alpha_{d})}_{d}(t_{d}))\n                \\geq \\zeromatrix\n            }$\n        for all\n            $\\mathbf{t} \\in \\realsNonNeg^{d}$.\n        Since this holds for all $K \\subseteq \\{1,2,\\ldots,n\\}$\n        and since by assumption the semigroups in\n            $\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d}$\n        have bounded generators,\n        by \\Cref{thm:classification-bounded:poly:sig:article-stochastic-raj-dahya}\n            $\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d}$\n        has a simultaneous regular unitary dilation.\n\n       \n        We have thus established the equivalence of\n            \\punktcref{1}, \\punktcref{2}, \\punktcref{3}, \\punktcref{4}\n        under the assumption that the expectation-approximants have bounded generators.\n        Since by \\Cref{lemm:classical-examples-expectation-approximants:sig:article-stochastic-raj-dahya}\n        one can always use the Hille- or Yosida-approximants (which by construction have bounded generators),\n        the equivalences\n            \\punktcref{1}{\\,}\\ensuremath{\\Leftrightarrow}{\\,}\\punktcref{2}{\\,}\\ensuremath{\\Leftrightarrow}{\\,}\\punktcref{3}\n        hold in general. (\\textdagger)\n\n        \\paragraph{\\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4}, without boundedness assumption on approximants:}\n        By the above,\n            \\punktcref{4}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}\n        continue to hold.\n        Exactly as argued above, \\punktcref{3} implies that\n            ${\n                p_{K}(T^{(\\alpha_{1})}_{1}(t_{1}),T^{(\\alpha_{2})}_{2}(t_{2}),\\ldots,T^{(\\alpha_{d})}_{d}(t_{d}))\n                \\geq \\zeromatrix\n            }$\n        for all\n            $\\mathbf{t} \\in \\realsNonNeg^{d}$,\n            $K\\subseteq\\{1,2,\\ldots,d\\}$,\n            and\n            $\\boldsymbol{\\alpha} \\in \\prod_{i=1}^{d}\\Lambda_{i}$.\n        Let $\\boldsymbol{\\alpha} \\in \\prod_{i=1}^{d}\\Lambda_{i}$ be arbitrary.\n        By the general validity of \\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1} (see (\\textdagger))\n        applied to $\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d}$,\n        it follows that\n            $\\{T^{(\\alpha_{i})}_{i}\\}_{i=1}^{d}$\n        has a simultaneous regular unitary dilation.\n        Returning to the current context, this means that \\punktcref{4} holds.\n\\end{proof}\n\n\n\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n\\begin{rem}\n\\label{rem:explicit-requirement-contractive-not-needed:sig:article-stochastic-raj-dahya}\n    In \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya},\n    since \\punktcref{1}, \\punktcref{2}, and \\punktcref{3}\n    each imply that the $T_{i}$ are contractive,\n    for the equivalences\n        \\punktcref{1}{\\,}\\ensuremath{\\Leftrightarrow}{\\,}\\punktcref{2}{\\,}\\ensuremath{\\Leftrightarrow}{\\,}\\punktcref{3}\n    it is not necessary to explicitly demand that the $T_{i}$ are contractive.\n    To see that \\punktcref{2} implies that the $T_{i}$ are contractive,\n    consider polynomial bounds applied to the regular polynomials\n    $X_{i}$ for each $i\\in\\{1,2,\\ldots,d\\}$.\n    To see that \\punktcref{3} implies that the $T_{i}$ are contractive,\n    let $i\\in\\{1,2,\\ldots,d\\}$ be arbitrary.\n    By considering the polynomial, $p_{\\{i\\}}$,\n    we have that\n        $(\\onematrix-T_{i}(t)^{\\ast}) + (\\onematrix-T_{i}(t)) \\geq \\zeromatrix$\n    for all $t\\in\\realsNonNeg$.\n    It readily follows that the generator, $A_{i}$, of $T_{i}$\n    is dissipative and thus that $T_{i}$ is contractive.\n\\end{rem}\n\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n\\begin{rem}\n\\label{rem:significance-of-result-other-dilation:sig:article-stochastic-raj-dahya}\n    If we replace \\emph{\\usesinglequotes{simultaneous regular unitary dilations}}\n    by \\emph{\\usesinglequotes{simultaneous unitary dilations}},\n    classifications via bounds of algebraic expressions\n    include \\cite[Theorem~2.2]{LeMerdy1996DilMultiParam} in the continuous setting,\n    and \\cite[Corollaries~4.9]{Pisier2001bookCBmaps} in the discrete setting (\\idest for tuples of commuting contractions).\n    In the discrete setting,\n    the existence of a simultaneous \\emph{power dilation}\n    is characterised by the \\emph{complete boundedness}\n    of polynomials defined on the operators.\n    In the continuous setting,\n    le~Merdy characterised the existence of a simultaneous unitary dilation\n    by the complete boundedness of a certain functional calculus map\n    generated by the resolvents of the semigroups\n    and defined on an algebra of holomorphic functions.\n    Adding to this picture, our result in\n    \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n    (see also \\Cref{rem:explicit-requirement-contractive-not-needed:sig:article-stochastic-raj-dahya})\n    answers \\Cref{qstn:polynomial:sig:article-stochastic-raj-dahya} positively\n    and we now know in full generality\n    that finite commuting families of $\\Cnought$-semigroups over Hilbert spaces\n    have a simultaneous regular unitary dilation if and only if\n    they satisfy regular polynomial bounds.\n    (In \\S{}\\ref{sec:functional-calculus:sig:article-stochastic-raj-dahya} a further characterisation\n    of regular unitary dilations via the complete positivity of a functional calculus shall be presented,\n    which is more general than the regular polynomial bounds and further adds to this picture.)\n\\end{rem}\n\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n\\begin{rem}\n\\label{rem:theorem-positivity-in-a-neighbourhood:sig:article-stochastic-raj-dahya}\n    Consider a neighbourhood $U \\subseteq \\realsNonNeg^{d}$ of $\\zerovector$\n    and let (\\punktref{3}') be the assertion of the positivity of the operators in \\punktcref{3}\n    for $\\mathbf{t} \\in U$ instead of for all $\\mathbf{t}\\in\\realsNonNeg^{d}$.\n    We show that \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n    holds with \\punktcref{3} replaced by (\\punktref{3}').\n    For this it suffices to show that (\\punktref{3}') implies \\punktcref{1}\n    (\\idest that $\\{T_{i}\\}_{i=1}^{d}$ has a simultaneous regular unitary dilation).\n\n    To this end, fix some ${a\\in\\realsPos}$ such that ${U \\supseteq [0,\\:a)^{d}}$\n    and consider the subnet of Hille-approximants:\n        $(T^{(\\lambda)}_{i})_{\\lambda\\in(a^{-1},\\:\\infty)}$ for $T_{i}$\n        for each ${i\\in\\{1,2,\\ldots,d\\}}$.\n    Clearly, taking subnets does not affect the fact that these are expectation-approximants.\n    Thus applying \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n    yields that $\\{T_{i}\\}_{i=1}^{d}$ has a simultaneous regular unitary dilation,\n    if and only if\n        $\\{T^{(\\lambda_{i})}_{i}\\}_{i=1}^{d}$\n    has a simultaneous regular unitary dilation\n    for each ${\\boldsymbol{\\lambda}\\in(a^{-1},\\:\\infty)^{d}}$.\n    Since the Hille-approximants have bounded generators,\n    by \\Cref{thm:classification-bounded:poly:sig:article-stochastic-raj-dahya}\n    this holds if and only if the generators\n        $\\{A^{(\\lambda_{i})}_{i}\\}_{i=1}^{d}$\n    of the Hille-approximants are completely dissipative\n    for each ${\\boldsymbol{\\lambda}\\in(a^{-1},\\:\\infty)^{d}}$.\n    By construction of the Hille-approximants, this holds if and only if\n\n        \\vspace{-3\\parskip}\n        $$\n            (-\\tfrac{1}{2})^{\\card{K}}\n            \\sum_{\\isPartition{(C_{1},C_{2})}{K}}\n                \\Big(\n                    \\prod_{i\\in C_{1}}\n                        \\Big(\n                            \\lambda_{i}\n                            (T_{i}(\\tfrac{1}{\\lambda_{i}})-\\onematrix)\n                        \\Big)\n                \\Big)^{\\ast}\n                \\prod_{j\\in C_{2}}\n                    \\Big(\n                        \\lambda_{j}\n                        (T_{j}(\\tfrac{1}{\\lambda_{j}})-\\onematrix)\n                    \\Big)\n            \\geq \\zeromatrix\n        $$\n\n    \\noindent\n    for all ${\\boldsymbol{\\lambda}\\in(a^{-1},\\:\\infty)^{d}}$\n    and all $K \\subseteq \\{1,2,\\ldots,d\\}$.\n    This holds if and only if\n        ${p_{K}(T_{1}(t_{1}), T_{1}(t_{2}),\\ldots,T_{d}(t_{d})) \\geq \\zeromatrix}$\n    for all ${\\mathbf{t}\\in(0,\\:a)^{d}}$\n    and all $K \\subseteq \\{1,2,\\ldots,d\\}$.\n    This in turn is clearly implied by  (\\punktref{3}').\n\\end{rem}\n\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n\\begin{rem}\n\\label{rem:other-approximants:sig:article-stochastic-raj-dahya}\n    By\n        \\punktcref{1}{\\,}\\ensuremath{\\Leftrightarrow}{\\,}\\punktcref{4}\n        of\n        \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n    as well as\n        \\Cref{lemm:classical-examples-expectation-approximants:sig:article-stochastic-raj-dahya},\n    we have answered\n        \\Cref{qstn:classical-approximants:dilation:sig:article-stochastic-raj-dahya}\n    positively\n    for a large class of naturally definable examples:\n    The simultaneous regular unitary dilatability\n        of a commuting contractive family\n    is characterised by\n    the simultaneous regular unitary dilatability\n        of families of semigroups\n    in any given net of expectation-approximants,\n    \\exempli the nets of Hille- and Yosida-approximants.\n    It would be interesting to know whether this characterisation holds\n    for other classically defined approximants,\n    such as the approximants that occur in Kendall's formula\n    and the semigroup version of the Post-Widder theorem\n    (\\cf\n        \\cite[Theorem~10.4.3~and~11.6.6]{Hillephillips1957faAndSg},\n        \\cite[Theorems~2--3]{Chung1962exp}%\n    ).\n    In these cases, in place of the stochastic methods used in the present paper,\n    other techniques such as product formulae used with path integrals,\n    \\exempli Chernoff approximations\n    (see \\cite{Chernoff1968article,Chernoff1974book,Butko2020chernoff}),\n    may be better suited.\n\\end{rem}\n\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n\\begin{rem}\n\\label{rem:complete-dissipativity:sig:article-stochastic-raj-dahya}\n    Let $\\{A_{i}\\}_{i=1}^{d}$ be the generators of a commuting family,\n    $\\{T_{i}\\}_{i=1}^{d}$, of $\\Cnought$-semigroups on a Hilbert space $\\HilbertRaum$.\n    Consider now the subset\n        $D \\subseteq \\HilbertRaum$,\n    of elements $\\xi$ which lie in the domain of\n        $A_{k_{n}} \\cdot \\ldots \\cdot A_{k_{2}} \\cdot A_{k_{1}}$\n    and such that the value of\n        ${(A_{k_{n}} \\cdot \\ldots \\cdot A_{k_{2}} \\cdot A_{k_{1}})\\xi}$\n    does not depend on the order of the $k_{i}$\n    for injective sequences\n        $(k_{i})_{i=1}^{n} \\subseteq \\{1,2,\\ldots,d\\}$,\n        $n\\in\\{1,2,\\ldots,d\\}$.\n    It can be shown that $D$ is a dense linear subspace of $\\HilbertRaum$\n    (see \\Cref{prop:dense-subspace-multi-param:sig:article-stochastic-raj-dahya}).\n    One may thus extend the notion of complete dissipativity to\n    families of generators $\\{A_{i}\\}_{i=1}^{d}$\n    (without the boundedness assumption),\n    by demanding that\n\n        \\vspace{-3\\parskip}\n        $$\n            (-\\tfrac{1}{2})^{\\card{K}}\n            \\sum_{\\isPartition{(C_{1},C_{2})}{K}}\n                \\brktLong{\n                    \\Big(\n                    \\prod_{j\\in C_{2}}\n                        A_{j}\n                    \\Big)\n                    \\xi\n                }{\n                    \\Big(\n                        \\prod_{i\\in C_{1}}\n                            A_{i}\n                    \\Big)\n                    \\xi\n                }\n            \\geq 0\n        $$\n\n    \\noindent\n    for all ${\\xi \\in D}$ and ${K \\subseteq \\{1,2,\\ldots,d\\}}$.\n\n    Appealing to condition \\eqcref{it:3:thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}\n    of \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya},\n    if $\\{T_{i}\\}_{i=1}^{d}$ has a simultaneous regular unitary dilation,\n    then by taking limits of the positive expressions\n        $\n            \\frac{1}{2^{K}\\prod_{i=1}^{d}t_{i}}\n            \\brktLong{\n                p_{K}(\n                    T_{1}(t_{1}),\n                    T_{2}(t_{2}),\n                    \\ldots,\n                    T_{d}(t_{d})\n                )\\xi\n            }{\\xi}\n        $\n    for ${\\realsPos \\ni t_{i} \\longrightarrow 0}$\n    successively for each ${i \\in \\{1,2,\\ldots,d\\}}$\n    and for each ${\\xi \\in D}$,\n    we obtain that $\\{A_{i}\\}_{i=1}^{d}$ is completely dissipative\n    by the above definition.\n    However, it is unclear whether the reverse implication holds.\n    The approach used in \\cite[Theorem~1.1]{Dahya2023dilation}\n    to link complete dissipativity to\n    a previously known condition%\n    \\footnote{\n        \\emph{Brehmer positivity},\n        see\n        \\cite[Theorem~3.2]{Ptak1985}.\n    }\n    which characterises the existence of simultaneous regular unitary dilations in general,\n    relies on asymptotic expressions,\n    which in turn rely on the boundedness of the generators.\n    Hence an alternative approach is needed for the unbounded setting.\n    It would thus be of interest to know whether the above (or an alternative)\n    definition of complete dissipativity can be shown to be equivalent\n    to any (and thus all) of the conditions in \\Cref{thm:classification-unbounded:poly:sig:article-stochastic-raj-dahya}.\n\\end{rem}\n\n\n\n\n\\@startsection{section}{1}{\\z@}{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}{\\formatsection@text}[Functional calculi associated with dilations]{Functional calculi associated with dilations}\n\\label{sec:functional-calculus:sig:article-stochastic-raj-dahya}\n\n\\noindent\nWe now leave the setting of commuting families and turn our attention\nto classical dynamical systems modelled by\n\\topSOT-continuous homomorphisms between topological monoids\nand bounded operators over a Hilbert space.\nIn this section we provide characterisations of unitary and regular unitary dilations\nvia \\emph{functional calculi} defined on (subalgebras of) certain $C^{\\ast}$-algebras\nrelated to topological groups.\nWe thus begin in \\S{}\\ref{sec:functional-calculus:harmonic-analysis:sig:article-stochastic-raj-dahya}\nby recalling the correspondence between ${}^{\\ast}$-representations of group $C^{\\ast}$-algebras\nand unitary representations of topological groups.\nThen in \\S{}\\ref{sec:functional-calculus:continuous:sig:article-stochastic-raj-dahya}\nwe shall use the Wittstock-Haagerup result,\nwhich involves extending and then dilating \\emph{completely bounded} maps.\nWe build on this to generalise le~Merdy's approach to characterise unitary dilatability.\nFinally, in \\S{}\\ref{sec:functional-calculus:discrete:sig:article-stochastic-raj-dahya}\nwe shall use Averson's result and Stinespring's theorem,\nwhich involves extending and then dilating \\emph{completely positive} maps.\nWe build on this to obtain a characterisation of regular unitary dilations\nsimilar to that of Sz.-Nagy and Foias\n(see \\cite[Theorem~7.1~b]{Nagy1970}).\nThe underlying algebras involved in the two functional calculi\ndemonstrate that unitary dilations are in some sense a \\emph{continuous} phenomenon,\nwhilst regular unitary dilations are in some sense a \\emph{discrete} phenomenon.\n\nThroughout this section, $(G,\\cdot,e)$ (or simply: $G$)\nshall denote a locally compact topological group%\n\\footnote{\n    Since we are only concerned with continuous maps between $G$\n    and other Hausdorff topological groups\n    (\\exempli the group of unitaries on a Hilbert space under the $\\topSOT$-topology),\n    it is not important to assume that $G$ be Hausdorff\n    (\\cf \\cite[\\S{}1.2]{Deitmar2014bookHarmonicAn}).\n    Nonetheless, all of our examples\n    (see \\Cref{sec:introduction:monoids:sig:article-stochastic-raj-dahya})\n    are Hausdorff.\n}\nand $M$ shall denote a (closed) submonoid of $G$\nso that\n    $(M,\\cdot,e)$ (or simply $M$),\nequipped with the relative topology,\ncomprises a (locally compact) topological monoid.\nWe furthermore let $\\lambda_{G}$ denote a left-invariant Haar-measure on $G$\nand express integrals of $G$ via ${\\int_{x \\in G}\\:\\cdot\\:\\dee x}$.\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[Abstract harmonic analysis]{Abstract harmonic analysis}\n\\label{sec:functional-calculus:harmonic-analysis:sig:article-stochastic-raj-dahya}\n\n\\noindent\nIn order to study dilations of classical dynamical systems on Hilbert spaces,\nwe shall make use of a fundamental relationship between unitary representations\nand ${}^{\\ast}$-representations of $C^{\\ast}$-algebras.\nWe recall these facts here.\nFor a detailed exposition,\nsee \\exempli\n    \\cite[\\S{}3.2 and \\S{}7.1]{Folland2015bookHarmonicAnalysis},\n    \\cite[\\S{}3.3]{Deitmar2014bookHarmonicAn}.\nLet $G$ be a locally compact group\n(for which we can fix a left-invariant Haar measure)\nand let ${\\Delta(\\cdot) \\colon G\\to(\\realsPos,\\cdot,1)}$\nbe the \\emph{modular function},\nwhich is a continuous homomorphism.\nThen $L^{1}(G)$ forms a Banach ${}^{\\ast}$-algebra\nunder the convolution operation (viewed as \\usesinglequotes{multiplication})\nand involution defined by\n\n    \\vspace{-3\\parskip}\n    $$\n        (f_{1}\\ast f_{2})(x)\n        \\colonequals\n        \\int_{y \\in G}\n            f_{1}(y)f_{2}(y^{-1}x)\n        \\:\\dee y\n    $$\n\n\\noindent\nand\n\n    \\vspace{-3\\parskip}\n    $$\n        f^{\\ast}(x)\n        \\colonequals\n            (f(x^{-1}))^{\\ast}\n            \\Delta(x^{-1})\n    $$\n\n\\noindent\nrespectively,\nfor $f, f_{1},f_{2}\\in L^{1}(G)$, $x\\in G$.\nIt can then be shown, for any Hilbert space $\\HilbertRaum$,\nthat there is a natural a correspondence between\n    non-degenerate%\n\\footnote{\n    \\idest there is no $\\xi\\in\\HilbertRaum\\mathbin{\\setminus}\\{\\zerovector\\}$\n    such that\n        $\\pi(f)\\xi=\\zerovector$\n    for all $f \\in L^{1}(G)$.\n    This is the case, \\exempli for irreducible representations.\n}\n    ${}^{\\ast}$-representations, $\\pi$, of $(L^{1}(G),\\ast,{}^{\\ast})$\n    on $\\HilbertRaum$\nand \\topSOT-continuous unitary representations, $U$, of $G$\n    on $\\HilbertRaum$\ngiven by\n\n    \\vspace{-3\\parskip}\n    \\begin{eqnarray}\n    \\label{eq:correspondence-unitary-to-star-repr:shift:sig:article-stochastic-raj-dahya}\n        U(x)\\pi(f) = \\pi(L_{x}f) = \\pi(f(x^{-1}\\cdot ))\n    \\end{eqnarray}\n\n\\noindent\nand\n\n    \\vspace{-3\\parskip}\n    \\begin{eqnarray}\n    \\label{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}\n        \\brkt{\\pi(f)\\xi}{\\eta}\n        = \\int_{x \\in G}\n            f(x)\\brkt{U(x)\\xi}{\\eta}\n        \\:\\dee g\n    \\end{eqnarray}\n\n\\noindent\nfor $f\\in L^{1}(G)$, $x\\in G$, $\\xi,\\eta\\in\\HilbertRaum$.\n\nLet $\\hat{G}$ denote the set of all\n    irreducible ${}^{\\ast}$-representations of $L^{1}(G)$\n    (equivalently: irreducible unitary representations of $G$),\n    up to unitary equivalence.\nOne can then equip\n    $(L^{1}(G),\\ast,{}^{\\ast})$\nwith the norm\n\n    \\vspace{-3\\parskip}\n    $$\n        \\norm{f}_{\\ast}\n        \\colonequals\n            \\sup_{[\\pi]\\in\\hat{G}}\n                \\norm{\\pi(f)}\n    $$\n\n\\noindent\nfor $f\\in L^{1}(G)$.\nThis renders $(L^{1}(G),\\ast,{}^{\\ast})$\na dense ${}^{\\ast}$-subalgebra of a $C^{\\ast}$-algebra,\nwhich is referred to as the\n\\highlightTerm{group $C^{\\ast}$-algebra} for $G$,\nand is denoted $C^{\\ast}(G)$.\nNote that in particular $\\norm{f}_{\\ast} \\leq \\norm{f}$\nholds for all $f \\in L^{1}(G)$.\n\n\\begin{conv}\n    For simplicity we shall view $L^{1}(G)$ as a subset of $C^{\\ast}(G)$.\n    We shall interchangeably denote the group $C^{\\ast}$-algebra for $G$\n    as $C^{\\ast}(G)$ and as $\\quer{L^{1}(G)}$.\n    Equipping $G$ with the discrete topology yields a locally compact group\n    with the counting measure as the Haar measure.\n    In this case, the $L^{1}$-space is simply $\\ell^{1}(G)$.\n    We shall thus denote the group $C^{\\ast}$-algebra associated\n    with the discretised $G$ via $\\quer{\\ell^{1}(G)}$.\n\\end{conv}\n\nAs a $C^{\\ast}$-algebra, $C^{\\ast}(G)$ has a unique unital extension.\nNote that $L^{1}(G)$ contains a unital element\nif and only if $G$ is discrete,\nin which case, the unit is $\\delta_{e}$.\nFor this reason,\nwe shall denote the unital extension of $C^{\\ast}(G)$ by\n\n    \\vspace{-3\\parskip}\n    $$\n        \\complex\\cdot\\delta_{e} + C^{\\ast}(G)\n    $$\n\n\\noindent\nin case $G$ is continuous.\nIf $G$ is discrete, we have that\n    $\\complex\\cdot\\delta_{e} + C^{\\ast}(G) = C^{\\ast}(G)$.\nOtherwise the above extension can be understood as\n    $\\complex\\cdot\\delta_{e} \\oplus C^{\\ast}(G)$.\n\nRelying on the density of $L^{1}(G)$ in $C^{\\ast}(G)$,\nthe above correspondence can be restated,\nthe proof of which is due to Folland.\n\n\\begin{prop}[Correspondence between ${}^{\\ast}$- and unitary representations]\n\\makelabel{prop:correspondence-unitary-to-star:sig:article-stochastic-raj-dahya}\n    Let\n        $G$ be a locally compact topological group\n        and\n        $\\HilbertRaum$ a Hilbert space.\n    Then for every\n        \\topSOT-continuous unitary representation, $U$, of\n        $G$ on $\\HilbertRaum$\n    there exists a\n        non-degenerate ${}^{\\ast}$-representation, $\\pi = \\pi_{U}$, of\n        $C^{\\ast}(G)$ on $\\HilbertRaum$,\n    such that\n        \\eqcref{eq:correspondence-unitary-to-star-repr:shift:sig:article-stochastic-raj-dahya}\n        and\n        \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}\n    hold.\n    And for every\n        non-degenerate ${}^{\\ast}$-representation, $\\pi$, of\n        $C^{\\ast}(G)$ on $\\HilbertRaum$,\n    there exists an\n        \\topSOT-continuous unitary representation, $U = U_{\\pi}$, of\n        $G$ on $\\HilbertRaum$,\n    such that\n        \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}\n    holds.\n    The constructions\n        ${U\\mapsto\\pi_{U}}$ and ${\\pi\\mapsto U_{\\pi}}$\n    establish a correspondence.\n    between \\topSOT-continuous unitary representations of $G$ on $\\HilbertRaum$\n    and non-degenerate ${}^{\\ast}$-representations of $C^{\\ast}(G)$ on $\\HilbertRaum$.\n\\end{prop}\n\n    \\begin{proof}\n        For the first claim, by\n            \\cite[Theorem~3.9]{Folland2015bookHarmonicAnalysis},\n        there exists a non-degenerate ${}^{\\ast}$-representation, $\\pi$, of\n            $L^{1}(G)$ on $\\HilbertRaum$\n        satisfying\n            \\eqcref{eq:correspondence-unitary-to-star-repr:shift:sig:article-stochastic-raj-dahya}\n            and\n            \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}.\n        Using Zorn's lemma, one can decompose\n            $\\HilbertRaum$\n        into closed $\\pi$-invariant subspaces with cyclic vectors,\n            $\\HilbertRaum=\\bigoplus_{i}\\HilbertRaum_{i}$,\n        thereby obtaining irreducible ${}^{\\ast}$-representations, $\\pi_{i}$,\n        of $L^{1}(G)$ on each $\\HilbertRaum_{i}$.\n        By construction of the $\\norm{\\cdot}_{\\ast}$-norm,\n        each $\\pi_{i}$ and thus $\\pi$ itself are contractive.\n        It follows that $\\pi$ can be extended\n        to a bounded linear operator between\n            $\\quer{L^{1}(G)} = C^{\\ast}(G)$\n        and $\\BoundedOps{\\HilbertRaum}$,\n        which we may also call $\\pi$.\n        Since the algebraic operations in $C^{\\ast}(G)$ are continuous,\n        $\\pi$ remains a ${}^{\\ast}$-representation.\n\n        Towards the second claim, by density, $\\pi$ is a non-degenerate ${}^{\\ast}$-representation\n        of $L^{1}(G)$ on $\\HilbertRaum$,\n        and thus by \\cite[Theorem~3.11]{Folland2015bookHarmonicAnalysis},\n        there exists an \\topSOT-continuous unitary representation, $U$, of\n            $G$ on $\\HilbertRaum$,\n        satisfying\n            \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}.\n\n        Towards the final claim, we first show that $\\pi_{U_{\\pi}} = \\pi$\n        for each ${}^{\\ast}$-representation, $\\pi$, of $C^{\\ast}(G)$ on $\\HilbertRaum$.\n        Using\n            \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}\n        yields\n            $\n                \\brkt{\\pi_{U_{\\pi}}(f)xi}{\\eta}\n                = \\int_{x \\in G}f(x)\\brkt{U_{\\pi}(x)\\xi}{\\eta}\\:\\dee x\n                = \\brkt{\\pi(f)\\xi}{\\eta}\n            $\n        for all $f\\in L^{1}(G)$ and $\\xi,\\eta\\in\\HilbertRaum$.\n        By the density of $L^{1}(G)$ in $C^{\\ast}(G)$\n        and continuity of $\\pi$, $\\pi_{U_{\\pi}}$,\n        it follows that $\\pi_{U_{\\pi}} = \\pi$.\n        To show that $U_{\\pi_{U}} = U$\n        for each \\topSOT-continuous unitary representation, $U$, of\n            $G$ on $\\HilbertRaum$,\n        using\n            \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}\n        yields\n            $\n                \\int_{K}\\brkt{U_{\\pi_{U}}(x)\\xi}{\\eta}\\:\\dee x\n                = \\brkt{\\pi_{U}(\\einser_{K})xi}{\\eta}\n                = \\int_{K}\\brkt{U(x)\\xi}{\\eta}\\:\\dee x\n            $\n        for all compact $K \\subseteq G$\n        and all $\\xi,\\eta\\in\\HilbertRaum$.\n        By the \\topWOT-continuity of $U$, $U_{\\pi_{U}}$,\n        it follows that $U_{\\pi_{U}} = U$.\n    \\end{proof}\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[The continuous functional calculus]{The continuous functional calculus}\n\\label{sec:functional-calculus:continuous:sig:article-stochastic-raj-dahya}\n\n\\noindent\nFor a locally compact topological group $G$ and closed submonoid $M \\subseteq G$,\nwe consider\n\n    \\vspace{-3\\parskip}\n    $$\n        \\LOneCPlus{M}{G}\n        \\colonequals\n        \\{\n            f \\in L^{1}(G)\n            \\mid\n            \\quer{\\mathop{\\textup{supp}}}(f) \\subseteq M,\n            ~\\text{compact}\n        \\},\n    $$\n\n\\noindent\nwhich is a subalgebra of the convolution algebra $(L^{1}(G),\\ast)$\nand thus of the unital $C^{\\ast}$-algebra ${\\complex\\cdot\\delta_{e} + C^{\\ast}(G)}$.\nFor an \\topSOT-continuous homomorphism,\n    ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$,\non $\\HilbertRaum$, consider the map\n    ${\\funcCalcCts \\colon \\complex\\cdot\\delta_{e} + \\LOneCPlus{M}{G} \\to \\BoundedOps{\\HilbertRaum}}$\ndefined by\n\n    \\vspace{-3\\parskip}\n    \\begin{eqnarray}\n    \\label{eq:functional-calc:cts:sig:article-stochastic-raj-dahya}\n        \\funcCalcCts(c\\delta_{e} + f)\n            = c\\onematrix\n            + \\sotInt_{x \\in \\quer{\\mathop{\\textup{supp}}}(f)}\n                f(x)T(x)\n            \\:\\dee x\n    \\end{eqnarray}\n\n\\noindent\nfor $f\\in\\LOneCPlus{M}{G}$, $c\\in\\complex$.\nIt is easy to verify that $\\funcCalcCts$ is a linear unital map\nwhich satisfies\n    $\n        \\funcCalcCts(f \\ast g)\n        = \\funcCalcCts(f)\n        \\funcCalcCts(g)\n    $\nfor $f,g\\in\\LOneCPlus{M}{G}$.\nWe shall refer to this unital homomorphism\nas the \\highlightTerm{continuous functional calculus}\nassociated with $T$.\n\n\\begin{rem}\n\\makelabel{rem:continuous-functional-calculus-le-merdy-phillips:sig:article-stochastic-raj-dahya}\n    Consider the case $(G,M)=(\\reals^{d},\\realsNonNeg^{d})$.\n    Let\n        $\\HilbertRaum$ be a Hilbert space\n        and\n        ${T \\colon \\realsNonNeg^{d} \\to \\BoundedOps{\\HilbertRaum}}$\n        be an \\topSOT-continuous contractive homomorphism on $\\HilbertRaum$.\n    Further let\n        $\\{T_{i}\\}_{i=1}^{d}$\n    be the corresponding commuting family of contractive $\\Cnought$-semigroups.\n    Then \\eqcref{eq:functional-calc:cts:sig:article-stochastic-raj-dahya}\n    can be naturally extended\n    to\n        ${\n            \\mathcal{B}\n            \\colonequals\n            \\complex\\cdot\\delta_{\\zerovector} \\oplus \\mathcal{B}_{0}\n        }$,\n    where\n        ${\n            \\mathcal{B}_{0}\n            \\colonequals\n            \\{f\\in L^{1}(\\reals^{d}) \\mid \\mathop{\\textup{supp}}(f) \\subseteq \\realsNonNeg^{d}\\}\n        }$.\n    This extension of $\\funcCalcCts$ to $\\mathcal{B}$\n    is essentially a restriction of the \\emph{Phillips calculus}\n    (see \\exempli\n        \\cite[Lemma~VIII.1.12~{[$\\ast$]}]{Dunfordschwartz1988BookLinOpI},\n        \\cite[Proposition~3.3.5]{Reissig2005abstractresolvent}%\n    )\n    applied to the generators $\\{A_{i}\\}_{i=1}^{d}$ of $\\{T_{i}\\}_{i=1}^{d}$.\n    Consider now arbitrary\n        $\\mathbf{n}\\in\\naturalsPos^{d}$\n    and\n        $\\boldsymbol{\\lambda}\\in\\realsPos^{d}$,\n    and let\n        $f\\in \\mathcal{B}_{0}$\n    be defined by\n    ${\n        f(\\mathbf{t})\n        \\colonequals\n        \\prod_{i=1}^{d}\n            \\lambda_{i}\\frac{(\\lambda_{i}t_{i})^{n_{i}-1}}{(n-1)!}\n            e^{-\\lambda_{i}t_{i}}\n    }$\n    for $\\mathbf{t}\\in\\realsNonNeg^{d}$.\n    One has\n        ${\n            (\\Fourier f)(\\boldsymbol{\\omega})\n            = \\prod_{i=1}^{d}\n                (\\frac{\\lambda_{i}}{\\lambda_{i} + \\iunit\\omega_{i}})^{n_{i}}\n        }$\n    for $\\boldsymbol{\\omega}\\in\\reals^{d}$,\n    where ${\\Fourier \\colon L^{1}(\\reals^{d}) \\to \\@ifnextchar_{\\Cts@tief}{\\Cts@tief_{}}_{0}{\\reals^{d}}}$\n    denotes the Fourier transform.\n    Then\n        ${\n            \\funcCalcCts(f)\n            = \\prod_{i=1}^{d}\n                \\sotInt_{s=0}^{\\infty}\n                    \\lambda_{i}\\frac{(\\lambda_{i}s)^{n_{i}-1}}{(n-1)!}\n                    e^{-\\lambda_{i}s}\n                    T_{i}(s)\n                    \\:\\dee s\n            = \\prod_{i=1}^{d}\n                (\\lambda_{i} \\opResolvent{A_{i}}{\\lambda_{i}})^{n_{i}}\n        }$.\n    The Phillips calculus is thus a common extension\n    of the continuous functional calculus\n    and (upon application of the Fourier transform)\n    the one defined by le~Merdy in \\cite[Definition~2.1]{LeMerdy1996DilMultiParam}.\n\\end{rem}\n\nThe main result in this subsection is thus inspired by le~Merdy's characterisation of simultaneous unitary dilations.\nBy simplifying his approach and utilising results from abstract harmonic analysis we obtain a generalisation.\nBefore presenting this, we need the following approximation.\n\n\\begin{prop}\n\\makelabel{prop:func-calc-cts:basic:shift-unit:sig:article-stochastic-raj-dahya}\n    Let $G$ be a locally compact topological group and\n    $M \\subseteq G$ a closed submonoid.\n    Further let\n        ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$\n    be an \\topSOT-continuous homomorphism.\n    If $M$ is $e$-joint,\n    then there exists a net\n        $(f_{i})_{i} \\subseteq \\LOneCPlus{M}{G}$\n    such that\n        $\\norm{f_{i}}_{\\ast} \\leq 1$ for all $i\\in I$,\n        ${\\norm{f_{i} \\ast g - g}_{\\ast} \\underset{i}{\\longrightarrow} 0}$\n    for all $g \\in L^{1}(G)$,\n    and\n        $\\funcCalcCts(f_{i}) \\underset{i}{\\longrightarrow} \\onematrix$\n    \\wrt the \\topSOT-topology.\n\\end{prop}\n\n    \\begin{proof}\n        Let $\\mathcal{N}$ be the filter of all compact neighbourhoods of\n        the group identity $e \\in G$.\n        By compactness and $e$-jointedness,\n            $0 < \\lambda_{G}(K \\cap M) \\leq \\lambda_{G}(K) < \\infty$\n        for all $K \\in \\mathcal{N}$.\n        Thus\n            $f_{K} \\colonequals \\frac{1}{\\lambda_{G}(K \\cap M)}\\einser_{K \\cap M}$\n        is a well-defined element of $L^{1}(G)$\n        with $\\quer{\\mathop{\\textup{supp}}}(f) = K \\cap M \\subseteq M$\n        for each $K\\in\\mathcal{N}$,\n        and moreover\n            $\n                \\norm{f}_{\\ast}\n                \\leq \\norm{f}_{1}\n                = 1\n            $.\n        Hence it suffices to consider the net\n            $(f_{K})_{K\\in\\mathcal{N}}$\n        directly ordered by reverse inclusion.\n\n        Let $g \\in L^{1}(G)$ be arbitrary.\n        Then\n            $\\norm{f_{K} \\ast g - g}_{\\ast} \\leq \\norm{f_{K} \\ast g - g}_{1}$\n        for each $h\\in\\realsPos$.\n        Since convolution is $L^{1}$-continuous\n        (see \\exempli \\cite[Proposition~2.40a)]{Folland2015bookHarmonicAnalysis}) and $\\@ifnextchar_{\\Cts@tief}{\\Cts@tief_{}}_{c}{G}$ is dense in $L^{1}(G)$,\n        it suffices to prove that\n            ${\\norm{f_{K} \\ast g - g}_{1} \\underset{K}{\\longrightarrow}  0}$\n        for each $g\\in\\@ifnextchar_{\\Cts@tief}{\\Cts@tief_{}}_{c}{G}$.\n        This is a straightforward matter that can be derived\n        using uniform continuity arguments.\n\n        Towards the final claim, one has\n            $\n                \\norm{\\funcCalcCts(f_{K})\\xi - \\xi}\n                = \\normLong{\n                    \\Big(\n                        \\frac{1}{\\lambda_{G}(K \\cap M)}\n                        \\sotInt_{x \\in K \\cap M}\n                            T(x)\n                        \\:\\dee x\n                    \\Big)\n                    \\xi\n                    -\n                    \\xi\n                }\n                \\leq\n                    \\frac{1}{\\lambda_{G}(K \\cap M)}\n                    \\int_{x \\in K \\cap M}\n                        \\norm{T(x)\\xi - T(e)\\xi}\n                    \\:\\dee x\n                \\leq\n                    \\sup_{x \\in K \\cap M}\n                        \\norm{(T(x) - T(e))\\xi}\n            $\n        for each $\\xi\\in\\HilbertRaum$,\n        whereby the latter expression\n        converges to $0$ since $T$ is \\topSOT-continuous.\n    \\end{proof}\n\n\\begin{schattierteboxdunn}[backgroundcolor=leer,nobreak=true]\n\\begin{lemm}[Generalisation of le~Merdy's characterisation of dilations]\n\\makelabel{lemm:cts-functional-calculus:sig:article-stochastic-raj-dahya}\n    Let $G$ be a locally compact topological group and\n    $M \\subseteq G$ an $e$-joint closed submonoid.\n    Further let\n        ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$\n    be an \\topSOT-continuous homomorphism.\n    Then $T$ has a unitary dilation if and only if $\\funcCalcCts$ is completely bounded\n    with $\\normCb{\\funcCalcCts} \\leq 1$.\n\\end{lemm}\n\\end{schattierteboxdunn}\n\n    \\begin{proof}\n        Via the GNS-construction, we may view\n        the unital $C^{\\ast}$-algebra,\n            ${\\mathcal{A} \\colonequals \\complex\\cdot\\delta_{e} + C^{\\ast}(G)}$\n        as a unital $C^{\\ast}$-subalgebra of $\\BoundedOps{\\HilbertRaum_{0}}$\n        for some Hilbert space $\\HilbertRaum_{0}$.\n\n        \\paragraph{Necessity:}\n            Suppose that $(\\HilbertRaum_{1},U,r)$ is a unitary dilation of $T$.\n            By the correspondence between unitary representations and non-degenerate ${}^{\\ast}$-representations\n            in abstract harmonic analysis\n            (see \\Cref{prop:correspondence-unitary-to-star:sig:article-stochastic-raj-dahya}),\n            there exists a non-degenerate ${}^{\\ast}$-representation\n                ${\n                    \\pi \\colon\n                    C^{\\ast}(G)\n                    \\to\n                    \\BoundedOps{\\HilbertRaum_{1}}\n                }$\n            such that \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya} holds.\n            Now, $\\pi$ can be extended to\n                ${\n                    \\tilde{\\pi} \\colon\n                    \\complex\\cdot\\delta_{e} + C^{\\ast}(G)\n                    \\to\n                    \\BoundedOps{\\HilbertRaum_{1}}\n                }$\n            defined by\n                ${\\tilde{\\pi}(c\\delta_{e} + f) \\colonequals c\\onematrix + \\pi(f)}$\n                for $f\\in C^{\\ast}(G)$ and $c\\in\\complex$.\n            By non-degeneracy, $\\tilde{\\pi}$ is a well-defined%\n            \\footnote{\n                in particular, if $C^{\\ast}(G)$ already contains the $\\delta_{e}$,\n                then by non-degeneracy $\\pi(\\delta_{e})=\\onematrix$\n                must hold.\n            }\n            unital ${}^{\\ast}$-algebra representation.\n            Using the unitary dilation and the ${}^{\\ast}$-representation,\n            one obtains\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray*}\n                    \\brkt{\\funcCalcCts(c\\delta_{e} + f)\\xi}{\\eta}\n                    &= &\\brktLong{\n                            \\Big(\n                                c\\onematrix\n                                +\n                                \\displaystyle\n                                \\sotInt_{x \\in \\quer{\\mathop{\\textup{supp}}}(f)}\n                                    f(x)T(x)\n                                \\:\\dee x\n                            \\Big)\n                            \\xi\n                        }{\\eta}\\\\\n                    &= &c\\brkt{\\xi}{\\eta}\n                        +\n                        \\displaystyle\n                        \\int_{x\\in\\quer{\\mathop{\\textup{supp}}}(f)}\n                            f(x)\n                            \\underbrace{\n                                \\brkt{T(x)\\xi}{\\eta}\n                            }_{=\\brkt{r^{\\ast}U(x)r\\xi}{\\eta}}\n                        \\:\\dee x\\\\\n                    &= &c\\brkt{r\\xi}{r\\eta}\n                        +\n                        \\displaystyle\n                        \\int_{x\\in\\quer{\\mathop{\\textup{supp}}}(f)}\n                            f(x)\n                            \\brkt{U(x)r\\xi}{r\\eta}\n                        \\:\\dee x\\\\\n                    &\\eqcrefoverset{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}{=}\n                        &\\brkt{c\\onematrix\\,r\\xi}{r\\eta}\n                        +\n                        \\brkt{\\pi(f)r\\xi}{r\\eta}\\\\\n                    &= &\\brkt{r^{\\ast}\\tilde{\\pi}(c\\cdot\\delta_{e} + f)r\\xi}{\\eta}\n                \\end{eqnarray*}\n\n            \\noindent\n            for all\n                $f \\in \\LOneCPlus{M}{G} \\subseteq L^{1}(G)$,\n                $c\\in\\complex$,\n                $\\xi,\\eta\\in\\HilbertRaum$.\n            Thus $\\funcCalcCts(a) = r^{\\ast}\\tilde{\\pi}(a)r$\n            for all $a\\in \\complex\\cdot\\delta_{e} + \\LOneCPlus{M}{G}$.\n            Let $n\\in\\naturals$ and $\\mathbf{a} = (a_{ij})_{ij} \\in M_{n}(\\complex\\cdot\\delta_{e} + \\LOneCPlus{M}{G})$.\n            Then\n                $\n                    (\\funcCalcCts\\otimes\\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n                    = (\\funcCalcCts(a_{ij}))_{ij}\n                    = (r^{\\ast}\\tilde{\\pi}(a_{ij})r)_{ij}\n                    = (r \\otimes \\onematrix_{M_{n}})^{\\ast}\n                        (\\tilde{\\pi} \\otimes \\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n                    (r \\otimes \\onematrix_{M_{n}})\n                $.\n            Now, since $\\tilde{\\pi} \\otimes \\mathrm{\\textit{id}}_{M_{n}}$\n            is a unital ${}^{\\ast}$-algebra representation,\n            it is necessarily contractive.\n            It follows that $\\funcCalcCts\\otimes\\mathrm{\\textit{id}}_{M_{n}}$ is contractive for each $n\\in\\naturals$.\n            Thus $\\normCb{\\funcCalcCts} \\leq 1$.\n\n        \\paragraph{Sufficiency:}\n            If $\\funcCalcCts$ is completely bounded with $\\normCb{\\funcCalcCts} \\leq 1$,\n            then, since $\\funcCalcCts$ is a (contractive!) unital homomorphism\n            defined on the unital subalgebra\n                ${\n                    \\complex\\cdot\\delta_{e} + \\LOneCPlus{M}{G}\n                    \\subseteq \\mathcal{A}\n                    \\subseteq \\BoundedOps{\\HilbertRaum_{0}}\n                }$,\n            one may apply the dilation theorem in \\cite[Theorem~4.8]{Pisier2001bookCBmaps}%\n            \\footnote{\n                This result is based on the factorisation of completely bounded maps\n                (see \\cite[Theorem~4.8]{Pisier2001bookCBmaps}),\n                which builds on Stinespring's theorem\n                and can be attributed to\n                    Wittstock \\cite{Wittstock1981ArticleDilationDe,Wittstock1984inCollDilationEn},\n                    Haagerup \\cite{Haagerup1985inCollCbMaps},\n                as well as Paulsen.\n            }\n            and obtain\n                a Hilbert space $\\HilbertRaum_{1}$,\n                an isometry $r\\in\\BoundedOps{\\HilbertRaum}{\\HilbertRaum_{1}}$,\n                and\n                a ${}^{\\ast}$-algebra representation\n                ${\\pi \\colon \\BoundedOps{\\HilbertRaum_{0}} \\to \\BoundedOps{\\HilbertRaum_{1}}}$\n            such that\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray}\n                \\label{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n                    \\funcCalcCts(a) = r^{\\ast}\\,\\pi(a)\\,r\n                \\end{eqnarray}\n\n            \\noindent\n            for all $a\\in\\complex\\cdot\\delta_{e} + \\LOneCPlus{M}{G}$.\n            Since $\\mathcal{A}$ is unital,\n            we can replace $\\HilbertRaum_{1}$\n            with the $\\pi$-invariant subspace $\\quer{\\pi(\\mathcal{A})r\\HilbertRaum}$,\n            which contains $r\\HilbertRaum$.\n            In particular, one can assume that $\\pi$ is a unital\n            (and thus non-degenerate) ${}^{\\ast}$-representation\n            of $\\mathcal{A}$ on $\\HilbertRaum_{1}$\n            and that $\\pi(\\mathcal{A})r\\HilbertRaum$ is dense in $\\HilbertRaum_{1}$.\n            Since $L^{1}(G)$ is $\\norm{\\cdot}_{\\ast}$-dense in $C^{\\ast}(G)$,\n            it follows that\n                $(\\complex\\cdot\\onematrix + \\pi(L^{1}(G)))r\\HilbertRaum$\n            is dense in $\\HilbertRaum_{1}$.\n\n            By the correspondence between unitary representations and non-degenerate ${}^{\\ast}$-representations\n            in abstract harmonic analysis\n            (see \\Cref{prop:correspondence-unitary-to-star:sig:article-stochastic-raj-dahya}),\n            there exists a (unique) \\topSOT-continuous unitary representation\n                ${U:G \\to \\BoundedOps{\\HilbertRaum_{1}}}$\n            such that\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray}\n                \\label{eq:2:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n                    U(x)\\pi(f) = \\pi(L_{x}f) = \\pi(f(x^{-1}\\cdot))\n                \\end{eqnarray}\n\n            \\noindent\n            for all $x \\in G$ and all $f \\in L^{1}(G)$.\n            Our goal is to show that $(\\HilbertRaum_{1},U,r)$ is a unitary dilation of $T$.\n\n            Let $x \\in M$ and $f \\in \\LOneCPlus{M}{G}$ be arbitrary.\n            Then\n                $L_{x}f = f(x^{-1}\\cdot) \\in \\LOneCPlus{M}{G}$,\n            since\n                $\\quer{\\mathop{\\textup{supp}}}(f(x^{-1}\\cdot)) = x \\cdot \\quer{\\mathop{\\textup{supp}}}(f) \\subseteq M$.\n            Applying the construction of $\\funcCalcCts$ yields\n\n            \\vspace{-3\\parskip}\n            \\begin{eqnarray*}\n                r^{\\ast}\\,U(x)\\,\\pi(f)\\,r\n                    &\\eqcrefoverset{eq:2:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}{=}\n                        &r^{\\ast}\\,\\pi(\\Fourier (f(x^{-1}\\cdot)))\\,r\\\\\n                    &\\eqcrefoverset{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}{=}\n                        &\\funcCalcCts(f(x^{-1}\\cdot))\\\\\n                    &= &\\displaystyle\n                        \\sotInt_{y \\in \\quer{\\mathop{\\textup{supp}}}(f(x^{-1}\\cdot))}\n                            f(x^{-1}y) T(y)\n                            \\:\\dee y\\\\\n                    &= &\\displaystyle\n                        \\sotInt_{y \\in \\quer{\\mathop{\\textup{supp}}}(f)}\n                            f(y) T(xy)\n                            \\:\\dee y\\\\\n                    &= &T(x)\n                        \\cdot\n                        \\displaystyle\n                        \\sotInt_{y \\in \\quer{\\mathop{\\textup{supp}}}(f)}\n                            f(y) T(y)\n                            \\:\\dee y\\\\\n                    &= &T(x)\\funcCalcCts(f).\\\\\n            \\end{eqnarray*}\n\n            Consider now the net\n                $(f_{i})_{i \\in I} \\subseteq \\LOneCPlus{M}{G}$\n            constructed\n            in \\Cref{prop:func-calc-cts:basic:shift-unit:sig:article-stochastic-raj-dahya},\n            for which it holds that\n                ${\\funcCalcCts(f_{i}) \\underset{i}{\\longrightarrow} \\onematrix}$\n            \\wrt the \\topSOT-topology.\n            By construction,\n                $\\norm{f_{i}}_{\\ast} \\leq 1$\n            for each $i\\in I$,\n            and since $\\pi$ is a ${}^{\\ast}$-algebra representation,\n            it follows that\n                $\\norm{\\pi(f_{i})} \\leq \\norm{f_{i}}_{\\ast} \\leq 1$\n            for all $i\\in I$.\n            We now claim that\n                ${\\pi(f_{i})r \\underset{i}{\\longrightarrow} r}$\n            \\wrt the \\topWOT-topology.\n            Since $(\\pi(f_{i}))_{i \\in I}$ is uniformly bounded\n            and ${(\\complex\\cdot\\onematrix + \\pi(L^{1}(G)))r\\HilbertRaum}$\n            is dense in $\\HilbertRaum_{1}$,\n            it suffices to show that\n                ${\n                    \\brkt{\\pi(f_{i})r\\xi}{ar\\eta}\n                    \\underset{i}{\\longrightarrow}\n                    \\brkt{r\\xi}{ar\\eta}\n                }$\n            for\n                $a\\in\\complex\\cdot\\onematrix + \\pi(L^{1}(G))$\n                and\n                $\\xi,\\eta\\in\\HilbertRaum$.\n            To this end, consider\n                an arbitrary $\\eta\\in\\HilbertRaum$\n            and\n                $a = c\\cdot\\onematrix + \\pi(g)$\n            for arbitrary $c\\in\\complex$, $g\\in L^{1}(G)$.\n            By the properties of the construction in\n                \\Cref{prop:func-calc-cts:basic:shift-unit:sig:article-stochastic-raj-dahya}\n            one has\n                ${\\funcCalcCts(f_{i}) \\underset{i}{\\longrightarrow} \\onematrix}$\n                \\wrt the \\topSOT-topology\n            as well as\n                ${\\norm{g^{\\ast} \\ast f_{i} - g^{\\ast}}_{\\ast} \\underset{i}{\\longrightarrow} 0}$\n            and thus\n                ${\\pi(g^{\\ast} \\ast f_{i}) \\underset{i}{\\longrightarrow} \\pi(g^{\\ast}) = \\pi(g)^{\\ast}}$\n            in norm.\n            Hence\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray*}\n                    \\brkt{\\pi(f_{i})r\\xi}{ar\\eta}\n                        &= &c^{\\ast}\\brkt{\\pi(f_{i})r\\xi}{r\\eta}\n                            + \\brkt{\\pi(f_{i})r\\xi}{\\pi(g)r\\eta}\\\\\n                        &= &c^{\\ast}\\brkt{r^{\\ast}\\pi(f_{i})r\\xi}{\\eta}\n                            + \\brkt{\\pi(g)^{\\ast}\\pi(f_{i})r\\xi}{r\\eta}\\\\\n                        &\\eqcrefoverset{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}{=}\n                            &c^{\\ast}\\brkt{\\funcCalcCts(f_{i})\\xi}{\\eta}\n                            + \\brkt{\\pi(g^{\\ast} \\ast f_{i})r\\xi}{r\\eta}\\\\\n                        &\\underset{i}{\\longrightarrow}\n                            &c^{\\ast}\\brkt{\\onematrix\\,\\xi}{\\eta}\n                            + \\brkt{\\pi(g)^{\\ast}r\\xi}{r\\eta}\n                        = \\brkt{r\\xi}{ar\\eta},\\\\\n                \\end{eqnarray*}\n\n            \\noindent\n            from which the claim follows.\n            Taking weak limits in the above computation\n            applied to the $f_{i}$ thus yields\n                $\n                    r^{\\ast}\\,U(x)\\,r = T(x)\\cdot\\onematrix\n                $\n            for all $x \\in M$.\n            Hence $(\\HilbertRaum_{1},U,r)$ is a unitary dilation of $T$.\n    \\end{proof}\n\n\n\n\\@startsection{subsection}{2}{\\z@}{\\z@}{\\z@\\hspace{1em}}{\\formatsubsection@text}[The discrete functional calculus]{The discrete functional calculus}\n\\label{sec:functional-calculus:discrete:sig:article-stochastic-raj-dahya}\n\n\\noindent\nFor a (not necessarily locally compact!) topological group $G$, we consider\n\n    \\vspace{-3\\parskip}\n    $$\n        c_{00}(G)\n        =\n        \\{\n            f \\in \\ell^{1}(G)\n            \\mid\n            \\mathop{\\textup{supp}}(f)~\\text{finite}\n        \\},\n    $$\n\n\\noindent\nwhich is a ${}^{\\ast}$-subalgebra of the convolution algebra $(\\ell^{1}(G),\\ast)$\nand thus of the unital $C^{\\ast}$-algebra $\\quer{\\ell^{1}(G)}$.\nLet $M \\subseteq$ be an arbitrary submonoid\nand suppose that $(G,M,\\cdot^{+})$ is a positivity structure\n(see \\Cref{defn:positivity-structure-monoids:sig:article-stochastic-raj-dahya}).\nFor a (not necessarily \\topSOT-continuous) homomorphism,\n    ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$,\non $\\HilbertRaum$, consider the map\n    ${\\funcCalcDiscr \\colon c_{00}(G) \\to \\BoundedOps{\\HilbertRaum}}$\ndefined by\n\n    \\vspace{-3\\parskip}\n    \\begin{eqnarray}\n    \\label{eq:functional-calc:discr:sig:article-stochastic-raj-dahya}\n        \\funcCalcDiscr(f)\n            = \\sum_{x \\in \\mathop{\\textup{supp}}(f)}\n                f(x)T(x^{-})^{\\ast}T(x^{+})\n    \\end{eqnarray}\n\n\\noindent\nfor $f\\in c_{00}(G)$.\nOne can readily check that $\\funcCalcDiscr$ is a\nlinear, self-adjoint, unital map\nwhich satisfies\n    $\n        \\funcCalcDiscr(f \\ast g)\n        = \\funcCalcDiscr(f)\n        \\funcCalcDiscr(g)\n    $\nfor $f,g\\in c_{00}(G)$.\nWe shall refer to this unital ${}^{\\ast}$-algebraic homomorphism\nas the \\highlightTerm{discrete functional calculus}\nassociated with $T$.\n\n\\begin{schattierteboxdunn}[backgroundcolor=leer,nobreak=true]\n\\begin{lemm}[Characterisation of regular dilations \\`{a} la Sz.-Nagy]\n\\makelabel{lemm:discr-functional-calculus:sig:article-stochastic-raj-dahya}\n    Let $(G,M,\\cdot^{+})$ be a positivity structure\n    where $G$ is a topological group\n    and $M \\subseteq G$ is a submonoid.%\n    \\footref{ft:disc-func-calc:G-not-locally-compact:sig:article-stochastic-raj-dahya}\n    Further let\n        ${T \\colon M \\to \\BoundedOps{\\HilbertRaum}}$\n    be an \\topSOT-continuous homomorphism.\n    Then $T$ has a regular unitary dilation if and only if $\\funcCalcDiscr$ is completely positive.\n\\end{lemm}\n\\end{schattierteboxdunn}\n\n\\footnotetext[ft:disc-func-calc:G-not-locally-compact:sig:article-stochastic-raj-dahya]{\n    Note that we neither require $G$ to be locally compact\n    nor $M$ to be a measurable subset in this theorem!\n}\n\nParts of the proof of \\Cref{lemm:discr-functional-calculus:sig:article-stochastic-raj-dahya}\nare similar to \\cite[Theorem~7.1~b)]{Nagy1970}.\nHowever, there are two main differences.\nFirstly, Sz.-Nagy works with extensions of $T$ to all of $G$,\nwithout explicitly defining this\n(except in the special cases of $G=\\reals^{d}$ and $G=\\integers^{d}$).\nSecondly, our approach relies on Stinespring's dilation theorem for $C^{\\ast}$-algebras,\nwhilst Sz.-Nagy's approach is more directly connected the theory of unitary representations.\n\n    \\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{lemm:discr-functional-calculus:sig:article-stochastic-raj-dahya}\n    \\begin{proof}[of \\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}]\n        Let $\\mathcal{A} \\colonequals \\quer{\\ell^{1}(G)}$\n        be the (unital) group $C^{\\ast}$-algebra\n        for the discretised version of $G$.\n        Note that $c_{00}(G)$ is unital and self-adjoint,\n        and thus constitutes an \\emph{operator system}\n        (\\cf \\cite[Chapter~2, p.~9]{Paulsen1986bookCBmapsAndDilations}).\n\n        \\paragraph{Necessity:}\n            Suppose that $(\\HilbertRaum_{1},U,r)$ is a regular unitary dilation of $T$.\n            By the correspondence between unitary representations and non-degenerate ${}^{\\ast}$-representations\n            in abstract harmonic analysis\n            (see \\Cref{prop:correspondence-unitary-to-star:sig:article-stochastic-raj-dahya}),\n            there exists a non-degenerate ${}^{\\ast}$-representation\n                ${\n                    \\pi \\colon\n                    \\mathcal{A}\n                    \\to\n                    \\BoundedOps{\\HilbertRaum_{1}}\n                }$\n            such that \\eqcref{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya} holds.\n            Using the regular unitary dilation and the ${}^{\\ast}$-representation,\n            one obtains\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray*}\n                    \\brkt{\\funcCalcDiscr(f)\\xi}{\\eta}\n                    &= &\\brktLong{\n                            \\Big(\n                                \\displaystyle\n                                \\sum_{x \\in \\mathop{\\textup{supp}}(f)}\n                                    f(x)\n                                    \\underbrace{\n                                        T(x^{-})^{\\ast}T(x^{+})\n                                    }_{=r^{\\ast}U(x)r}\n                            \\Big)\n                            \\xi\n                        }{\\eta}\\\\\n                    &= &\\displaystyle\n                        \\sum_{x \\in \\mathop{\\textup{supp}}(f)}\n                            f(x)\n                            \\brkt{U(x)r\\xi}{r\\eta}\\\\\n                    &\\eqcrefoverset{eq:correspondence-unitary-to-star-repr:inner-product:sig:article-stochastic-raj-dahya}{=}\n                        &\\brkt{\\pi(f)r\\xi}{r\\eta}\\\\\n                    &= &\\brkt{r^{\\ast}\\,\\pi(f)\\,r\\xi}{\\eta}\n                \\end{eqnarray*}\n\n            \\noindent\n            for all\n                $f \\in c_{00}(G)$,\n                $\\xi,\\eta\\in\\HilbertRaum$.\n            Thus $\\funcCalcDiscr(a) = r^{\\ast}\\,\\pi(a)\\,r$\n            for all $a\\in c_{00}(G)$.\n            For $n\\in\\naturals$ and positive matrices\n                $\\mathbf{a} = (a_{ij})_{ij} \\in M_{n}(c_{00}(G))$\n            it follows that\n                $\n                    (\\funcCalcDiscr \\otimes \\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n                        = (\\funcCalcDiscr(a_{ij}))_{ij}\n                        = \\Big(\n                            r^{\\ast}\\,\\pi(a_{ij})\\,r\n                        \\Big)_{ij}\n                        = (r \\otimes \\onematrix_{M_{n}})^{\\ast}\n                            (\\pi \\otimes \\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n                            (r \\otimes \\onematrix_{M_{n}})\n                $,\n            which is positive,\n            since\n                ${\\pi \\otimes \\mathrm{\\textit{id}}_{M_{n}}}$\n            is a ${}^{\\ast}$-representation of\n            the $C^{\\ast}$-algebra $M_{n}(\\mathcal{A})$\n            and thus positive.\n            Thus $\\funcCalcDiscr$ is completely positive.\n\n        \\paragraph{Sufficiency:}\n            Since $c_{00}(G) \\subseteq \\mathcal{A}$ is an operator system,\n            Averson's extension theorem\n            (see \\cite[Theorem~7.5]{Paulsen1986bookCBmapsAndDilations})\n            yields an extension of $\\funcCalcDiscr$ to a completely positive map\n            between $\\mathcal{A}$ and $\\BoundedOps{\\HilbertRaum}$.\n            Stinespring's dilation theorem\n            (see \\cite[Theorem~1 and \\S{}3.~Remarks]{Stinespring1955dilation})\n            applied to this yields\n                a Hilbert space $\\HilbertRaum_{1}$,\n                an isometry $r\\in\\BoundedOps{\\HilbertRaum}{\\HilbertRaum_{1}}$,\n                and\n                a unital ${}^{\\ast}$-representation ${\\pi \\colon \\mathcal{A}\\to\\BoundedOps{\\HilbertRaum_{1}}}$,\n            such that\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray}\n                \\label{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n                    \\funcCalcDiscr(a) = r^{\\ast}\\,\\pi(a)\\,r\n                \\end{eqnarray}\n\n            \\noindent\n            holds for all $a\\in c_{00}(G)$.\n            Since $c_{00}(G)$ is a dense, unital ${}^{\\ast}$-subalgebra of $\\mathcal{A}$,\n            we can replace $\\HilbertRaum_{1}$ by the $\\pi$-invariant closed subspace,\n            $\\quer{c_{00}(G)r\\HilbertRaum}$, which contains $r\\HilbertRaum$.\n\n            Since for all $x,y \\in G$\n            one has\n                $\\delta_{x}\\ast\\delta_{y} = \\delta_{xy}$\n            and\n                $\\delta_{x}\\in c_{00}(G)\\subseteq\\mathcal{A}$\n            are unitary,%\n            \\footnote{\n                since\n                    $\\delta_{x}^{\\ast}\\ast\\delta_{x} = \\delta_{x^{-1}x} = \\delta_{e}$\n                and\n                    $\\delta_{x}\\ast\\delta_{x}^{\\ast} = \\delta_{xx^{-1}} = \\delta_{e}$.\n            }\n            and since $\\pi$ is a unital ${}^{\\ast}$-representation,\n            it follows that ${U \\colon G \\to \\BoundedOps{\\HilbertRaum}}$\n            defined by $U(x) \\colonequals \\pi(\\delta_{x})$\n            is a unitary homomorphism of $G$ on $\\HilbertRaum$.\n            Moreover by \\eqcref{eq:1:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n\n                \\vspace{-3\\parskip}\n                \\begin{eqnarray}\n                \\label{eq:2:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n                    T(x^{-})^{\\ast}T(x^{+})\n                    = \\funcCalcDiscr(\\delta_{x})\n                    = r^{\\ast}\\,\\pi(\\delta_{x})\\,r\n                    = r^{\\ast}\\,U(x)\\,r\n                \\end{eqnarray}\n\n            \\noindent\n            for all $x \\in G$.\n            To show that $(\\HilbertRaum_{1},U,r)$ is a regular unitary dilation of $T$,\n            it thus remains to show that $U$ is $\\topSOT$-continuous.\n\n            To this end, first note that\n                ${\n                    (\\BoundedOps{\\HilbertRaum},\\topSOT)\n                    \\times (\\BoundedOps{\\HilbertRaum},\\topSOT)\n                    \\ni (R, S)\n                    \\mapsto\n                    S^{\\ast}R \\in (\\BoundedOps{\\HilbertRaum},\\topWOT)\n                }$\n            is continuous.%\n            \\footnote{\n                This follows from the estimate\n                    $\n                        \\abs{\\brkt{\n                            (S_{2}^{\\ast}R_{2}-S_{1}^{\\ast}R_{1})\\xi\n                        }{\\eta}}\n                        =\n                            \\abs{\\brkt{\n                                (\n                                    S_{1}^{\\ast}\n                                    (R_{2}-R_{1})\n                                    +\n                                    (S_{2}-S_{1})^{\\ast}\n                                    R_{1}\n                                    +\n                                    (S_{2}-S_{1})^{\\ast}\n                                    (R_{2}-R_{1})\n                                )\n                                \\xi\n                            }{\n                                \\eta\n                            }}\n                        \\leq\n                            \\norm{(R_{2}-R_{1})\\xi}\n                            \\norm{S_{1}\\eta}\n                            +\n                            \\norm{R_{1}\\xi}\n                            \\norm{(S_{2}-S_{1})\\eta}\n                            +\n                            \\norm{(R_{2} - R_{1})\\xi}\n                            \\norm{(S_{2}-S_{1})\\eta}\n                    $\n                for $R_{1},R_{2},S_{1},S_{2}\\in\\BoundedOps{\\HilbertRaum}$\n                and $\\xi,\\eta\\in\\HilbertRaum$.\n                By the form of this estimate,\n                we do not need to restrict the operators to bounded subsets\n                (\\cf \\cite[Lemma~3.1]{Ptak1985}).\n            }\n            So since ${T \\colon M \\to (\\BoundedOps{\\HilbertRaum},\\topSOT)}$\n            and\n            $\\cdot^{+}$ (and thus $\\cdot^{-}$)\n            are continuous,\n            it follows that\n                ${\n                    G \\ni x\n                    \\mapsto\n                    r^{\\ast}U(x)r\n                    \\eqcrefoverset{eq:2:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}{=}\n                    T(x^{-})^{\\ast}T(x^{+})\n                    \\in \\BoundedOps{\\HilbertRaum}\n                }$\n            is \\topWOT-continuous.\n            This implies that\n\n                \\vspace{-3\\parskip}\n                $$\n                    G \\ni x\n                    \\mapsto\n                    \\begin{array}[t]{0l}\n                        \\brkt{U(x)\\pi(\\delta_{y})r\\xi}{\\pi(\\delta_{z})r\\eta}\\\\\n                        = \\brkt{U(x)U(y)r\\xi}{U(z)r\\eta}\\\\\n                        = \\brkt{r^{\\ast}U(z^{-1}xy)r\\xi}{r\\eta}\\\\\n                    \\end{array}\n                $$\n\n            \\noindent\n            is continuous\n            for all $\\xi,\\eta\\in\\BoundedOps{\\HilbertRaum}$, $y,z\\in G$,\n            which in turn entails the continuity of\n                ${\n                    G \\ni x \\mapsto \\brkt{U(x)\\pi(f)r\\xi}{\\pi(g)r\\eta}\n                }$\n            for all $\\xi,\\eta\\in\\BoundedOps{\\HilbertRaum}$, $f,g \\in c_{00}(G)$.\n            Since $U$ is unitary-valued and $\\pi(c_{00}(G))r\\HilbertRaum$ is dense in $\\HilbertRaum_{1}$,\n            it follows that $U$ is a \\topWOT- and thus indeed an \\topSOT-continuous\n            unitary representation of $G$ on $\\HilbertRaum_{1}$.\n    \\end{proof}\n\n\\begin{rem}\n    The \\topSOT-continuity of $T$ was only used\n    to prove the \\topSOT-continuity of the unitary representation.\n    Without this assumption, the above proof shows that\n    $T$ has a (not necessarily \\topSOT-continuous) regular unitary dilation\n    if and only if $\\funcCalcDiscr$ is completely positive.\n\\end{rem}\n\n\n\n\n\\@startsection{section}{1}{\\z@}{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}{\\formatsection@text}[Second main result]{Second main results: Unitary approximants}\n\\label{sec:second-results:sig:article-stochastic-raj-dahya}\n\n\\noindent\nThe functional calculi presented in \\S{}\\ref{sec:functional-calculus:sig:article-stochastic-raj-dahya}\nto characterise unitary and regular unitary dilations respectively,\nprovide us the means to study topological approximations of classical dynamical systems.\nWe exploit these results to prove\n\\Cref{%\n    thm:unitary-approx:weak:sig:article-stochastic-raj-dahya,%\n    thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya,%\n}.\n\n\n\\noindent\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:unitary-approx:weak:sig:article-stochastic-raj-dahya}\n\\begin{proof}[of \\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}]\n    The implications \\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3} are clear.\n\n    \\paragraph{\\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1}:}\n        Let\n            $(U^{(\\alpha)})_{\\alpha\\in\\Lambda}$,\n        be a net of \\topSOT-continuous unitary representations of $G$ on $\\HilbertRaum$.\n        Suppose that $(U^{(\\alpha)}\\restr{M})_{\\alpha\\in\\Lambda}$\n        approximates $T$ in the \\emph{uniform weak} sense.\n        We make use of the \\emph{continuous functional calculi}\n            ${\n                \\funcCalcCts,\\funcCalcCts^{(\\alpha)}\n                \\colon\n                \\LOneCPlus{M}{G} \\to \\BoundedOps{\\HilbertRaum}\n            }$\n        associated with $T$\n        and each $U^{(\\alpha)}\\restr{M}$ respectively\n        (see \\S{}\\ref{sec:functional-calculus:continuous:sig:article-stochastic-raj-dahya}).\n        By the characterisation in\n            \\Cref{lemm:cts-functional-calculus:sig:article-stochastic-raj-dahya},\n            $\\normCb{\\funcCalcCts^{(\\alpha)}} \\leq 1$\n        for each $\\alpha\\in\\Lambda$,\n        and, in order to show that $T$ has a unitary dilation,\n        it suffices to show that\n            $\\funcCalcCts$\n        is completely bounded\n        with $\\normCb{\\funcCalcCts} \\leq 1$.\n        To this end, first observe that for each\n            $\n                a = c\\cdot\\delta_{e} + f\n                \\in \\complex\\cdot\\delta_{e} + \\LOneCPlus{M}{G}\n                \\equalscolon \\mathcal{A}\n            $,\n        uniform weak convergence yields\n\n            \\vspace{-3\\parskip}\n            \\begin{eqnarray*}\n                \\brkt{\n                    \\funcCalcCts^{(\\alpha)}(a)\n                    \\xi\n                }{\\eta}\n                &= &\\brktLong{\n                        \\Big(\n                            c\\onematrix\n                            +\n                            \\displaystyle\n                            \\sotInt_{x \\in \\quer{\\mathop{\\textup{supp}}}(f)}\n                                f(x)\n                                U^{(\\alpha)}(x)\n                            \\:\\dee x\n                        \\Big)\n                        \\xi\n                    }{\\eta}\\\\\n                &= &c\\brkt{\\xi}{\\eta}\n                    +\n                    \\displaystyle\n                    \\int_{x \\in \\quer{\\mathop{\\textup{supp}}}(f)}\n                        f(x)\n                        \\brkt{U^{(\\alpha)}(x)\\xi}{\\eta}\n                    \\:\\dee x\\\\\n                &\\underset{\\alpha}{\\longrightarrow}\n                    &c\\brkt{\\xi}{\\eta}\n                    +\n                    \\displaystyle\n                    \\int_{x \\in \\quer{\\mathop{\\textup{supp}}}(f)}\n                        f(x)\n                        \\brkt{T(x)\\xi}{\\eta}\n                    \\:\\dee x\\\\\n                &= &\\brkt{\n                    \\funcCalcCts(a)\n                    \\xi\n                }{\\eta}\\\\\n            \\end{eqnarray*}\n\n        \\noindent\n        for all $\\xi,\\eta\\in\\HilbertRaum$.\n        Thus\n            ${\n                \\funcCalcCts^{(\\alpha)}(a)\n                \\underset{\\alpha}{\\longrightarrow}\n                \\funcCalcCts(a)\n            }$\n        \\wrt the $\\topWOT$-topology\n        for each $a\\in \\mathcal{A}$.\n        It follows that\n            ${\n                (\\funcCalcCts^{(\\alpha)} \\otimes \\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n                \\underset{\\alpha}{\\longrightarrow}\n                (\\funcCalcCts \\otimes \\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n            }$\n        \\wrt the $\\topWOT$-topology\n        for\n            $n\\in\\naturals$\n            and matrices\n            $\\mathbf{a} = (a_{ij})_{ij} \\in M_{n}(\\mathcal{A})$.\n        Since\n            $\\funcCalcCts^{(\\alpha)} \\otimes \\mathrm{\\textit{id}}_{M_{n}}$\n        is a contraction for each $\\alpha\\in\\Lambda$ and each $n\\in\\naturals$,\n        it follows that\n            $\\funcCalcCts \\otimes \\mathrm{\\textit{id}}_{M_{n}}$\n        is a contraction for each $n\\in\\naturals$.\n        Thus $\\funcCalcCts$ is completely bounded\n        with $\\normCb{\\funcCalcCts} \\leq 1$.\n\n    \\paragraph{\\punktcref{1}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{2}, under cardinality assumption:}\n        By assumption, $G$ contains a dense subspace $D \\subseteq G$\n        and $\\mathop{\\textup{dim}}(\\HilbertRaum) \\geq \\max\\{\\aleph_{0},\\card{D}\\}$.\n        Without loss of generality, one can replace $D$ by a dense subgroup of $G$.\n        Let\n            $(\\HilbertRaum_{1},U,r)$\n        be a regular unitary dilation of $T$.\n        Let $B \\subseteq \\HilbertRaum$ be an orthonormal basis (ONB) for $\\HilbertRaum$\n        and\n            $\\kappa \\colonequals \\card{B} = \\mathop{\\textup{dim}}(\\HilbertRaum) \\geq \\max\\{\\aleph_{0},\\card{D}\\}$\n        and consider\n\n            \\vspace{-3\\parskip}\n            $$\n                \\HilbertRaum_{0} \\colonequals \\quer{\\mathop{\\textup{lin}}}\\{\n                    U(x)r\\xi\n                    \\mid\n                    x \\in D,\n                    \\xi \\in B\n                \\},\n            $$\n\n        \\noindent\n        which is a $U$-invariant subspace.\n        By the above cardinality assumptions\n        and elementary computations with infinite cardinals\n        (see \\exempli \\cite[\\S{}I.10]{Kunen2011BookSetTh}),\n        one has cardinality\n            $\\card{\\HilbertRaum_{0}} = \\card{B} = \\kappa$.\n        It follows that\n            $(\\HilbertRaum_{0},U\\restr{\\HilbertRaum_{0}},r)$\n        is a regular unitary dilation of $T$.\n        Furthermore, since $r$ is isometric, we have\n            $\n                \\kappa\n                = \\mathop{\\textup{dim}}(\\HilbertRaum)\n                = \\mathop{\\textup{dim}}(\\mathop{\\textup{ran}}(r))\n                \\leq \\mathop{\\textup{dim}}(\\HilbertRaum_{0})\n                = \\kappa\n            $.\n        Thus $\\mathop{\\textup{dim}}(\\HilbertRaum_{0}) = \\kappa = \\mathop{\\textup{dim}}(\\HilbertRaum)$.\n        So without loss of generality, one may assume that $\\HilbertRaum_{0} = \\HilbertRaum$.\n        It follows that there exists\n            an isometry ${r\\in\\BoundedOps{\\HilbertRaum}}$\n            and\n            an $\\topSOT$-continuous\n            unitary representation $U$ of $G$ on $\\HilbertRaum$\n        such that\n\n            \\vspace{-3\\parskip}\n            \\begin{eqnarray}\n            \\label{eq:dil:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n                T(x)\n                =\n                r^{\\ast}\\,U(x)\\,r\n            \\end{eqnarray}\n\n        \\noindent\n        for all $x \\in M$.\n\n        Now let\n            $P \\subseteq \\BoundedOps{\\HilbertRaum}$\n        be the index set consisting of finite projections on $\\HilbertRaum$,\n        directly ordered by $p \\succeq q$ $:\\Leftrightarrow$ $\\mathop{\\textup{ran}}(p) \\supseteq \\mathop{\\textup{ran}}(q)$.\n        Let $p \\in P$ be arbitrary\n        and let $F_{p} \\subseteq \\HilbertRaum$ be a finite ONB for $\\mathop{\\textup{ran}}(p)$.\n        Since $r$ is an isometry,\n            $\\tilde{F}_{p} \\colonequals \\{r\\,e \\mid e\\in F_{p}\\}$\n        is also a finite orthonormal family of vectors.\n        Let\n            $B_{p},\\tilde{B}_{p}^{\\prime} \\subseteq \\HilbertRaum$\n        be ONBs extending $F_{p},\\tilde{F}_{p}$ respectively.\n        Since $\\HilbertRaum$ is infinite dimensional\n        and $F_{p},\\tilde{F}_{p}$ are finite,\n        one has\n            $\n                \\card{B_{p} \\setminus F_{p}}\n                = \\mathop{\\textup{dim}}(\\HilbertRaum)\n                = \\card{\\tilde{B}_{p} \\setminus \\tilde{F}_{p}}\n            $.\n        Thus there exists a bijection\n            ${f \\colon B_{p} \\setminus F_{p} \\to \\tilde{B}_{p} \\setminus \\tilde{F}_{p}}$.\n        Thus\n            $g \\colonequals r\\restr{F_{p}} \\cup f$\n        is a bijection between $B_{p}$ and $\\tilde{B}_{p}$.\n        This extends uniquely to a unitary operator\n            $w_{p} \\in \\BoundedOps{\\HilbertRaum}$.\n        By construction,\n            ${w_{p}\\restr{F_{p}} = r\\restr{F_{p}}}$\n        and thus by linearity\n\n            \\vspace{-3\\parskip}\n            \\begin{eqnarray}\n            \\label{eq:unitary-squash:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n                w_{p}p = rp\n            \\end{eqnarray}\n\n        \\noindent\n        for each $p \\in P$.\n        Finally, set\n\n            \\vspace{-3\\parskip}\n            $$\n                U^{(p)} \\colonequals w_{p}^{\\ast}\\,U(\\cdot)\\,w_{p}\n            $$\n\n        \\noindent\n        for each $p \\in P$,\n        which are clearly \\topSOT-continuous unitary representations of $G$ on $\\HilbertRaum$.\n        We demonstrate that the net,\n            $(U^{(p)}\\restr{M})_{p \\in P}$,\n        of \\topSOT-continuous homomorphisms,\n        is an \\emph{exact weak approximation} of $T$.\n        To this end, let\n            $\\xi,\\eta\\in\\HilbertRaum$ be arbitrary.\n        Let $p_{0}\\in\\BoundedOps{\\HilbertRaum}$\n        be the projection onto $\\mathop{\\textup{lin}}\\{\\xi,\\eta\\}$.\n        For each $p \\in P$ with $p \\succeq p_{0}$ one has\n        that\n            $p\\xi = \\xi$\n            and\n            $p\\eta = \\eta$.\n        By \\eqcref{eq:unitary-squash:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya},\n            $\n                w_{p}^{\\ast}r \\xi\n                = w_{p}^{\\ast} r p \\xi\n                = w_{p}^{\\ast} w_{p} p \\xi\n                = p \\xi\n                = \\xi\n            $\n        and similarly\n            $\n                w_{p}^{\\ast}r \\eta\n                = \\eta\n            $.\n        The dilation yields\n\n            \\vspace{-3\\parskip}\n            \\begin{eqnarray*}\n                \\brkt{\n                    T(x)\n                    \\xi\n                }{\\eta}\n                    &\\eqcrefoverset{eq:dil:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}{=} &\\brkt{\n                            r^{\\ast}\\,U(x)\\,r\n                            \\xi\n                        }{\\eta}\\\\\n                    &= &\\brkt{\n                            r^{\\ast}\n                            w_{p}\n                            U^{(p)}(x)\n                            w_{p}^{\\ast}\n                            r\n                            \\xi\n                        }{\\eta}\\\\\n                    &= &\\brkt{\n                            U^{(p)}(x)\n                            w_{p}^{\\ast}\n                            r\n                            \\xi\n                        }{w_{p}^{\\ast}r\\eta}\\\\\n                    &= &\\brkt{\n                            U^{(p)}(x)\n                            \\xi\n                        }{\\eta}\\\\\n            \\end{eqnarray*}\n\n        \\noindent\n        for all $x \\in M$\n        and $p \\succeq p_{0}$.\n        Hence\n            $(U^{(p)}\\restr{M})_{p \\in P}$\n        is an exact weak approximation of $T$.\n\\end{proof}\n\n\\defthm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n\\begin{proof}[of \\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}]\n    The implications \\punktcref{2}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{3}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{4} are clear.\n\n    \\paragraph{\\punktcref{4}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{1}:}\n        Let\n            $(U^{(\\alpha)})_{\\alpha\\in\\Lambda}$,\n        be a net of \\topSOT-continuous unitary representations of $G$ on $\\HilbertRaum$.\n        Suppose that $(U^{(\\alpha)}\\restr{M})_{\\alpha\\in\\Lambda}$\n        approximates $T$ in the \\emph{pointwise regular weak} sense.\n        We now make use of the \\emph{discrete functional calculi}\n            ${\n                \\funcCalcDiscr,\\funcCalcDiscr^{(\\alpha)}\n                \\colon\n                c_{00}(G) \\to \\BoundedOps{\\HilbertRaum}\n            }$\n        associated with $T$\n        and each $U^{(\\alpha)}\\restr{M}$ respectively\n        (see \\S{}\\ref{sec:functional-calculus:discrete:sig:article-stochastic-raj-dahya}).\n        By the characterisation in\n            \\Cref{lemm:discr-functional-calculus:sig:article-stochastic-raj-dahya},\n        each\n            $\\funcCalcDiscr^{(\\alpha)}$\n        is completely positive\n        and, in order to show that $T$ has a regular unitary dilation,\n        it suffices to show that\n            $\\funcCalcDiscr$\n        is completely positive.\n        To this end, first observe for $f\\in c_{00}(G)$,\n        that pointwise regular weak convergence yields\n\n            \\vspace{-3\\parskip}\n            \\begin{eqnarray*}\n                \\brkt{\n                    \\funcCalcDiscr^{(\\alpha)}(f)\n                    \\xi\n                }{\\eta}\n                &= &\\displaystyle\n                    \\sum_{x \\in \\mathop{\\textup{supp}}(f)}\n                        f(x)\n                        \\brkt{\n                            U^{(\\alpha)}(x^{-})^{\\ast}U^{(\\alpha)}(x^{+})\n                            \\xi\n                        }{\\eta}\\\\\n                &\\underset{\\alpha}{\\longrightarrow}\n                    &\\displaystyle\n                    \\sum_{x \\in \\mathop{\\textup{supp}}(f)}\n                        f(x)\n                        \\brkt{\n                            T(x^{-})^{\\ast}T(x^{+})\n                            \\xi\n                        }{\\eta}\\\\\n                &= &\\brkt{\n                    \\funcCalcDiscr(f)\n                    \\xi\n                }{\\eta}\\\\\n            \\end{eqnarray*}\n\n        \\noindent\n        for all $\\xi,\\eta\\in\\HilbertRaum$.\n        Thus\n            ${\n                \\funcCalcDiscr^{(\\alpha)}(f)\n                \\underset{\\alpha}{\\longrightarrow}\n                \\funcCalcDiscr(f)\n            }$\n        \\wrt the $\\topWOT$-topology\n        for each $f\\in c_{00}(G)$.\n        It follows that\n            ${\n                (\\funcCalcDiscr^{(\\alpha)} \\otimes \\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n                \\underset{\\alpha}{\\longrightarrow}\n                (\\funcCalcDiscr \\otimes \\mathrm{\\textit{id}}_{M_{n}})(\\mathbf{a})\n            }$\n        \\wrt the $\\topWOT$-topology\n        for\n            $n\\in\\naturals$\n            and matrices\n            $\\mathbf{a} = (a_{ij})_{ij} \\in M_{n}(c_{00}(G))$.\n        Since\n            $\\funcCalcDiscr^{(\\alpha)} \\otimes \\mathrm{\\textit{id}}_{M_{n}}$\n        is positive for each $\\alpha$ and each $n\\in\\naturals$,\n        it follows that\n            $\\funcCalcDiscr \\otimes \\mathrm{\\textit{id}}_{M_{n}}$\n        is positive for each $n\\in\\naturals$.\n        Thus $\\funcCalcDiscr$ is completely positive.\n\n    \\paragraph{\\punktcref{1}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\punktcref{2}, under cardinality assumption:}\n        The proof is analogous to proof of\n            \\eqcref{it:1:thm:unitary-approx:weak:sig:article-stochastic-raj-dahya}{\\,}\\ensuremath{\\Rightarrow}{\\,}\\eqcref{it:2:thm:unitary-approx:weak:sig:article-stochastic-raj-dahya}\n            of\n            \\Cref{thm:unitary-approx:weak:sig:article-stochastic-raj-dahya}.\n        Relying on the cardinality assumptions, the same arguments as above\n        yield a regular unitary dilation of $T$ of the form $(\\HilbertRaum,U,r)$,\n        \\idest\n\n            \\vspace{-3\\parskip}\n            \\begin{eqnarray}\n            \\label{eq:dil:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\n                T(x^{-})^{\\ast}T(x^{+})\n                = r^{\\ast}\\,U(x)\\,r\n            \\end{eqnarray}\n\n        \\noindent\n        for all $x \\in G$.\n        The nets of unitary operators,\n            $(w_{p})_{p \\in P}$,\n        and of \\topSOT-continuous unitary representations of $G$ on $\\HilbertRaum$,\n            $(U^{(p)} \\colonequals w_{p}^{\\ast}U(\\cdot)w_{p})_{p \\in P}$,\n        are constructed as above.\n        For $\\xi,\\eta\\in\\HilbertRaum$,\n        letting $p_{0}$ be the projection onto $\\mathop{\\textup{lin}}\\{\\xi,\\eta\\}$,\n        one has again\n            $w_{p}^{\\ast}r\\xi=\\xi$\n            and\n            $w_{p}^{\\ast}r\\eta=\\eta$\n        for $p \\succeq p_{0}$.\n        The regular dilation yields\n\n            \\vspace{-3\\parskip}\n            \\begin{eqnarray*}\n                \\brkt{\n                    T(x^{-})^{\\ast}\n                    T(x^{+})\n                    \\xi\n                }{\\eta}\n                    &\\eqcrefoverset{eq:dil:thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}{=}\n                        &\\brkt{\n                            r^{\\ast}\\,U(x)\\,r\n                            \\xi\n                        }{\\eta}\\\\\n                    &= &\\brkt{\n                            r^{\\ast}\n                            w_{p}\n                            U^{(p)}(x)\n                            w_{p}^{\\ast}\n                            r\n                            \\xi\n                        }{\\eta}\\\\\n                    &= &\\brkt{\n                            U^{(p)}(x)\n                            w_{p}^{\\ast}\n                            r\n                            \\xi\n                        }{w_{p}^{\\ast}r\\eta}\\\\\n                    &= &\\brkt{\n                            U^{(p)}(x)\n                            \\xi\n                        }{\\eta}\\\\\n            \\end{eqnarray*}\n\n        \\noindent\n        for all $x \\in G$\n        and $p \\succeq p_{0}$.\n        Hence\n            $(U^{(p)}\\restr{M})_{p \\in P}$\n        is an \\emph{exact regular weak approximation} of $T$.\n\\end{proof}\n\n\n\nAs an immediate application of \\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya},\nwe demonstrate an infinite class of commuting systems\nwhich admit no regular weak unitary approximations.\nThe following examples demonstrate in particular,\nthat the problem of unitary approximability (in the \\emph{regular} case)\nof a commuting system cannot be reduced to the unitary approximability of strict subsystems.\n\n\\begin{cor}\n\\makelabel{cor:counter-examples-unitary-approx:sig:article-stochastic-raj-dahya}\n    Let\n        $d\\in\\naturals$ with $d \\geq 2$\n        and\n        $\\HilbertRaum$ be an infinite dimensional Hilbert space.\n    Then there exists an infinite class\n    of commuting families, $\\{T_{i}\\}_{i=1}^{d}$,\n    of contractive $\\Cnought$-semigroups on $\\HilbertRaum$\n    whose generators have strictly negative spectral bounds,%\n    \\footnote{\n        The \\highlightTerm{spectral bound} of a linear operator\n            ${A \\colon \\mathop{\\textup{dom}}(A)\\subseteq\\HilbertRaum\\to\\HilbertRaum}$\n        is given by\n            $\\sup\\{\\mathop{\\mathfrak{R}\\mathrm{e}} \\lambda \\mid \\lambda\\in\\opSpectrum{A}\\}$\n        (\\cf \\cite[Definition~1.12]{EngelNagel2000semigroupTextBook}).\n    }\n    such that\n        $\\{T_{i}\\}_{i \\in C}$\n    has an exact regular weak unitary approximation\n    for each $C \\subsetneq \\{1,2,\\ldots,d\\}$,\n    whilst\n        $\\{T_{i}\\}_{i=1}^{d}$\n    has no pointwise regular weak unitary approximation.\n\\end{cor}\n\nThe class of semigroups can be constructed as in\n\\cite[Proposition~5.3]{Dahya2023dilation}.\nFor the reader's convenience, we sketch the construction.\nWe apply the characterisation in\n\\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya}\nto $(G,M) = (\\reals^{d},\\realsNonNeg^{d})$\n(see \\Cref{e.g.:monoids:positivity:sig:article-stochastic-raj-dahya})\nas well as the correspondence between\nbetween \\topSOT-continuous homomorphisms defined over $\\realsNonNeg^{d}$\nand commuting families of $\\Cnought$-semigroups\ndiscussed in \\S{}\\ref{sec:introduction:sig:article-stochastic-raj-dahya}.\n\n    \\begin{proof}[of \\Cref{cor:counter-examples-unitary-approx:sig:article-stochastic-raj-dahya}]\n        By assumption, one can find orthonormal closed subspaces\n            $\\HilbertRaum_{1},\\HilbertRaum_{2}\\subseteq\\HilbertRaum$\n        with\n            $0 < \\mathop{\\textup{dim}}(\\HilbertRaum_{2}) \\leq \\mathop{\\textup{dim}}(\\HilbertRaum_{1})$\n        such that\n            $\\HilbertRaum=\\HilbertRaum_{1}\\bigoplus\\HilbertRaum_{2}$.\n        Working \\wrt this partition,\n        and letting\n            $\\alpha \\in (\\frac{1}{\\sqrt{d}},\\:\\frac{1}{\\sqrt{d-1}})$\n        be arbitrary,\n        we consider bounded operators of the form\n\n            \\vspace{-3\\parskip}\n            $$\n                A_{i} = -\\onematrix\n                +\n                \\begin{smatrix}\n                    \\zeromatrix &2\\alpha V_{i}\\\\\n                    \\zeromatrix &\\zeromatrix\\\\\n                \\end{smatrix}\n            $$\n\n        \\noindent\n        where\n            $V_{i} \\in \\BoundedOps{\\HilbertRaum_{2}}{\\HilbertRaum_{1}}$\n        can be chosen to be any isometry.\n        One can show that $\\{A_{i}\\}_{i=1}^{d}$ is a commuting family of\n        dissipative operators whose spectra are each given by $\\{-1\\}$.\n        Thus, $\\{T_{i} \\colonequals (e^{t A_{i}})_{t\\in\\realsNonNeg}\\}_{i=1}^{d}$\n        is a commuting family of contractive $\\Cnought$-semigroups,\n        whose (bounded) generators have strictly negative spectral bounds.\n        As in \\cite[Proposition~5.3]{Dahya2023dilation},\n        it can be shown that\n            $\\{A_{i}\\}_{i \\in C}$ is completely dissipative\n            for each $C \\subsetneq \\{1,2,\\ldots,d\\}$,\n            whilst\n            $\\{A_{i}\\}_{i=1}^{d}$ is not completely dissipative.\n        By the characterisation for semigroups with bounded generators\n            (\\Cref{thm:classification:dissipativity:sig:article-stochastic-raj-dahya}),\n        it follows that\n            $\\{T_{i}\\}_{i \\in C}$ has a simultaneous regular unitary dilation\n            for each $C \\subsetneq \\{1,2,\\ldots,d\\}$,\n            whilst\n            $\\{T_{i}\\}_{i=1}^{d}$ does not.\n        By \\Cref{thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya},\n        the claim follows.\n    \\end{proof}\n\n\\begin{rem}\n\\label{rem:conjecture-ordinary-unitary-dilation:sig:article-stochastic-raj-dahya}\n    It is well known that commuting systems of $d=2$ contractive $\\Cnought$-semigroups\n    have simultaneous unitary dilations\n    (see\n        \\cite{Slocinski1974},\n        \\cite[Theorem~2]{Slocinski1982},\n        and\n        \\cite[Theorem~2.3]{Ptak1985}%\n    ).\n    Hence by\n        \\Cref{%\n            thm:unitary-approx:weak:sig:article-stochastic-raj-dahya,%\n            thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya,%\n        }\n    and \\Cref{cor:counter-examples-unitary-approx:sig:article-stochastic-raj-dahya},\n    the topologies defined by\n        exact weak\n        (\\respectively uniform weak)\n        convergence\n    are in general%\n    \\footnote{\n        \\viz for commuting families of $d \\geq 2$ semigroups\n        over infinite dimensional Hilbert spaces.\n    }\n    strictly weaker than their \\emph{regular} counterparts.\n    In particular,\n    the characterisations in\n        \\Cref{%\n            thm:unitary-approx:weak:sig:article-stochastic-raj-dahya,%\n            thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya,%\n        }\n    now provide a sharp topological distinction\n    between \\emph{unitary dilations}\n    and \\emph{regular unitary dilations}.\n\\end{rem}\n\nAt the start of this paper we mentioned two different ways to treat irreversible systems:\nvia embeddings into or approximations by reversible systems.\nThe characterisations in\n    \\Cref{%\n        thm:unitary-approx:weak:sig:article-stochastic-raj-dahya,%\n        thm:unitary-approx:regular-weak:sig:article-stochastic-raj-dahya,%\n    },\ndemonstrate that, unter moder conditions,\nthese are in fact equivalent\nfor respective choices of dilations and approximations.\nThis holds for the commutative setting\nand for a large class of classical dynamical systems,\nincluding dynamical systems consisting of\nfamilies of $\\Cnought$-semigroups on Hilbert spaces\nsatisfying a semigroup version of the CCR in Weyl-form\n(%\n    see\n    \\Cref{e.g.:non-commuting-family-heisenberg-d-C:sig:article-stochastic-raj-dahya},\n    \\Cref{e.g.:monoids:e-joint:sig:article-stochastic-raj-dahya},\n    and\n    \\Cref{e.g.:monoids:positivity:sig:article-stochastic-raj-dahya}%\n).\n\n\n\n\n\\setcounternach{section}{1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\lb{intro}\nIn its weak-field and slow-motion approximation, general relativity predicts that, in addition to the time-honored post-Newtonian (pN) gravitoelectric and gravitomagnetic   precessions induced by the mass $M$ (Schwarzschild) and the spin angular momentum $\\bds S$ (Lense-Thirring)  of the central body acting as source of the gravitational field, other  pN gravitoelectric and gravitomagnetic orbital effects related to its oblateness  arise as well \\citep{Sof89,1988CeMec..42...81S,1990CeMDA..47..205H,1991ercm.book.....B,2014PhRvD..89d4043W,2014CQGra..31x5012P,2015IJMPD..2450067I,2015CeMDA.123....1M,2016JGeod..90.1345S,2018RSOS....580640F,2018CeMDA.130...40S}. So far, they have never been put to the test in any astronomical and astrophysical scenarios, despite some recent preliminary investigations pertaining the planet Jupiter in our solar system \\citep{2013CQGra..30s5011I,2019MNRAS.484.4811I}; for some embryonic thoughts about an Earth-spacecraft scenario, see \\citet{2013CQGra..30s5011I,2015IJMPD..2450067I}. To the pN level, the oblateness of astronomical bodies modifies also the propagation of the electromagnetic waves in their deformed spacetime. About the perspectives of measuring the resulting deflection due to Jupiter with astrometric techniques, see, e.g.,\n\\citet{2006CQGra..23.4853C},\\,\\citet{2007PhRvD..75f2002K},\\,\\citet{2008PhRvD..77d4029L},\\,\\citet{2019MNRAS.485.1147A}, and references therein.\n\nAn analysis of the analytical expressions of the pN gravitoelectric and gravitomagnetic orbital precessions due to the asphericity of the primary \\citep{2015IJMPD..2450067I,2019MNRAS.484.4811I} shows that the key ingredients  needed to enhance their magnitude are a strongly distorted central body, and a highly eccentric and close orbit of the moving particle.\n\nBinaries composed by a pulsar and a main-sequence star \\citep{1998MNRAS.298...67W} may offer, in principle, interesting natural laboratories to try to investigate such little known pN effects.\nIndeed, they are systems composed of a neutron star regularly emitting electromagnetic radio pulses orbiting an usually more massive main sequence star, which, in most cases, is highly distorted due to its fast rotation, along a generally elliptical orbit.\nA very accurate observable quantity in binary pulsars is represented by the measurement of the times of arrival (TOAs) $\\tau$ of the pulses emitted by the neutron star which, in case it has a gravitationally bound companion, exhibit a regular variation $\\delta\\tau$ due to, among other things, the Keplerian motion about the common centre of mass: it is the R{\\o}mer-like time delay. The full variation of the pulses' times of arrival is due to several other effects connected, e.g., with the propagation of the electromagnetic waves through the deformed spacetime of the system \\citep{1998MNRAS.298...67W}.\n\nThe first binary pulsar hosting a main sequence star to be discovered was PSR B1259-63 \\citep{1992ApJ...387L..37J,2014MNRAS.437.3255S}; it is characterized by a highly eccentric orbit ($e=0.870$) with an orbital period of $P_{\\rm b}=1237~\\textrm{d}=3.38~\\textrm{yr}$. The pulsar's non-degenerate companion is the fast spinning Be star LS 2883, whose equatorial velocity $V_\\textrm{e}$ is about $280~\\textrm{km~s}^{-1}$ corresponding to $\\sim 70\\%$ of its break-up velocity \\citep{1996MNRAS.280L..31P}. Its mass $M$ and equatorial radius $R_\\textrm{e}$ amounts to about 30 Solar masses $(\\textrm{M}_\\odot)$ and $9.7$ Solar radii $(\\textrm{R}_\\odot)$ \\citep{negueruela+11}.\nLater, the eccentric ($e=0.808$) binary PSR J0045-7319 was discovered \\citep{1994ApJ...423L..43K}. To date, it is the fastest orbiting system since it is $P_{\\rm b}=51.17~\\textrm{d}$. Its primary is a  main sequence B-star spinning close to its break-up velocity \\citep{1995ApJ...452..819L,1996Natur.381..584K}. PSR J1638-4725, having an orbital period of $P_{\\rm b}=1940~\\textrm{d}=5.3~\\textrm{yr}$ and $e=0.95$, was found by \\citet{2006MNRAS.372..777L}. Its stellar companion should be a rapidly rotating Be star. PSR J1740-3052 \\citep{2001MNRAS.325..979S}, with $P_{\\rm b}=231~\\textrm{d}$ and $e=0.578$, hosts most likely a B-type main-sequence star \\citep{2010MNRAS.406.1848T,2012MNRAS.425.2378M}. The most recently discovered main-sequence-star binary pulsar is the highly eccentric ($e=0.93$) PSR J2032+4127 \\citep{2015MNRAS.451..581L} characterized by $P_{\\rm b}=8578~\\textrm{d}=23.5~\\textrm{yr}$. The companion of the neutron star is the massive Be star MT91 213 with $M\\simeq 15\\,\\textrm{M}_\\odot$.\n\nFor the sake of completeness, we mention also a few other  binary pulsars hosting a non-degenerate star, although they are not relevant for our purposes in view of the nature of their non massive and fast-rotating partners. They are PSR J1903+0327 \\textcolor{black}{\\citep{2018ApJS..235...37A}}, whose companion is a F5V-GOV $\\sim 1~\\textrm{M}_\\odot$ star moving in $P_{\\rm b}=95~\\textrm{d}$ along a rather eccentric orbit with $e=0.44$, the transitional millisecond pulsar PSR J1023+0038 \\citep{2009Sci...324.1411A} orbiting a low-mass ($0.2~\\textrm{M}_\\odot$) companion star in a circular path with $P_{\\rm b}=4.75~\\textrm{hr}$.\nThe rms timing residuals of the aforementioned binary pulsars are all of the order of $\\lesssim \\textrm{ms}$, apart from PSR J1903+0327\nwhich \\textcolor{black}{is} at the $\\simeq \\textcolor{black}{1}\\,\\upmu\\textrm{s}$ level; more specifically, they are $\\simeq 0.46~\\textrm{ms}$  over $13~\\textrm{yr}$ for PSR B1259-63 \\citep{2004MNRAS.351..599W}, $7.4~\\textrm{ms}$ over $2~\\textrm{yr}$ for PSR J0045-7319 \\citep{1994ApJ...423L..43K}, $\\simeq 5.3~\\textrm{ms}$  over $4.35~\\textrm{yr}$ for PSR J1638-4725 \\citep{2006MNRAS.372..777L}, $\\simeq 0.8~\\textrm{ms}$  over $2.29~\\textrm{yr}$ for PSR J1740-3052 \\citep{2001MNRAS.325..979S}, $\\simeq 0.5-1~\\textrm{ms}$  over about $6~\\textrm{yr}$ for PSR J2032+4127 \\citep{2015MNRAS.451..581L}, $\\simeq 1~\\textrm{\\upmu s}$  over about $3~\\textrm{yr}$ for PSR J1903+0327 \\textcolor{black}{\\citep{2018ApJS..235...37A}},  $0.1~\\textrm{ms}$  over about $4~\\textrm{yr}$ for PSR J1023+0038 \\citep{2013arXiv1311.5161A}.\nTable~\\ref{tavola1} summarizes the key data for the binary pulsars hosting massive, fast rotating main sequence stars.\n\\begin{sidewaystable}[ht]\n\\begin{center}\n\\small{\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}\n  \\hline\nPulsar  &  Companion  & Distance (kpc) &  $M~(\\textrm{M}_\\odot)$ & $R_\\textrm{e}~(\\textrm{R}_\\odot)$ & $R_\\textrm{p}~(\\textrm{R}_\\odot)$ & $P_\\textrm{b}$ (d) &  $e$ & $P$ (ms)  & $\\upsigma_\\tau$ (ms) &  $\\Delta T$ (yr) \\\\\n\\hline\nPSR J0045-7319 & B1V star & In SMC & $8.8\\pm 1.8$ & $6.4\\pm 0.7$ & $-$ &  $51.17$ & $0.808$ & $930$ & $7.4$ & $2$ \\\\\nPSR J1740-3052 & Main sequence star & $-$ & $> 11$ & $-$ & $-$ & $231$ & $0.578$ & $570$ & $0.8$ & $2.29$ \\\\\nPSR B1259-63 & B2e star LS 2883 & $2.75$ & $\\simeq 30$ & $\\simeq 9.7$ & $\\simeq 8.1$ &  $1237$ & $0.870$ & $48$ & $0.46$ & $13$ \\\\\nPSR J1638-4725 & Be star & $-$ &  $>4$ & $-$ & $-$ & $1940$ & $0.95$ & $764$ & $5.3$ & $4.35$ \\\\\nPSR J2032+4127 & Be star MT91 213 & $1.7$ & $\\simeq 15$ & $-$  & $-$  & $8578$ & $0.93$ & $143$ & $0.5-1$ & $6$ \\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\caption{Binary pulsars with a massive, fast rotating main sequence star companion discovered so far. For each of them, we list the companion, the distance (when available), the mass $M$, the equatorial and polar radii $R_\\textrm{e},\\,R_\\textrm{p}$ (when available), the orbital period $P_\\textrm{b}$, the orbital eccentricity $e$, the spin period $P$, the rms timing residuals $\\upsigma_\\tau$, the data analysis time span $\\Delta T$. For the values of the listed parameters, see the references cited in the text and the online pulsar catalog at http:\/\/www.atnf.csiro.au\/research\/pulsar\/psrcat\/. For PSR B1256-63 we give the equatorial $R_\\textrm{e}$ and polar $R_\\textrm{p}$ radii as estimated by \\citet{negueruela+11}.}\\label{tavola1}\n\\end{sidewaystable}\n\nAbout the achievable accuracy level in timing residuals, in the case of the pulsars orbiting a main sequence star, their timing seems doomed to stay at the $\\simeq\\textrm{ms}$ level. It is so because, for evolutionary reasons, they are not fully recycled \\citep{2010NewAR..54...93S}. Thus, their spinning periods $P$ are not at the millisecond level, and their TOAs are not measured with a precision of the order of $\\simeq \\upmu\\textrm{s}$\\textcolor{black}{, and their timing is often contaminated by timing noise \\citep{Hobbsetal010}}. The timing of the non-recycled pulsars is almost always less accurate than for the fully recycled pulsars. Indeed, PSR J1903+0327 rotates with a period $P=2.5\\,\\textrm{ms}$, and its rms timing residuals are as little as $\\simeq 1\\,\\upmu\\textrm{s}$. The spin period of PSR J1023+0038 is $P=1.7\\,\\textrm{ms}$, and its rms timing residuals are  $0.1\\,\\textrm{ms}$.\nIncidentally, we mention the fact that, according to Table\\,2  of\n\\citet{2018ApJS..235...37A}, the rms timing accuracy of some fully recycled pulsars with $P=1.5-10\\,\\textrm{ms}$, isolated or with a white dwarf as companion, is of the order of $0.1-0.2\\,\\upmu\\textrm{s}$. It is expected that future instrumental improvements may push the rms timing accuracy of some of them to the\\footnote{A. Possenti, personal communication to L.I., April 2019.} $\\simeq 10\\,\\textrm{ns}$ level over time spans some yr long. For binary pulsars hosting another neutron star, the rms timing accuracy is of the order of $1-100\\,\\upmu\\textrm{s}$.\n\nHere, we will preliminarily investigate the size and the temporal patterns of the perturbations $\\Delta\\ton{\\delta\\tau}$ induced on the R{\\o}mer-like orbital part of the pulsar's time delay $\\delta\\tau$ by both the standard (Schwarzschild and Lense-Thirring) and the oblateness-driven pN accelerations felt by a fictitious neutron star orbiting a highly distorted, fast rotating B-type main sequence star in view of a possible detection in new binaries that may eventually be discovered in the future.\nHowever, caution is in order before inferring too optimistic conclusions from a  straightforward comparison of our simulated time series with the rms timing residuals listed in Table\\,\\ref{tavola1}. Even if the size of some pK signatures were to be larger than the $\\simeq\\textrm{ms}$ level, it does not necessarily mean that such  effects will be measurable in the actual processing of the real observations. Indeed, careful, dedicated simulated data reductions and covariance analyses should be performed by explicitly modeling the signals of interest, estimating its characteristic parameters and inspecting the resulting correlations with the other parameters usually estimated. It should be kept in mind that, in principle, an unmodeled effect may be removed from the post-fit residuals, at least to a certain extent, being \\virg{absorbed} in the estimated values of the other parameters determined in the data reduction.\nThus, our investigation should be deemed just as a sensitivity analysis able to preliminarily explore the potential of the scenarios considered.\nWe will assume the validity of general relativity throughout the paper, which is organized as follows.\n\nIn Section~\\ref{J2S}, we discuss the magnitude of the angular momentum $S$ and the dimensionless quadrupole mass  moment $J_2$ of typical fast rotating massive B-type stars. Section~\\ref{GR} is devoted to the numerical calculation of the  perturbations $\\Delta\\ton{\\delta\\tau}$ induced on the pulsar's R{\\o}mer time delay by some post-Keplerian (pK) classical and pN  accelerations. We do not calculate the propagation delays accounting, e.g., for the effects on the pulsar's travelling electromagnetic waves through the deformed spacetime of the B-type star. Section~\\ref{conclu} summarizes our findings and contains our conclusions. Once again, we stress the preliminary nature of our sensitivity investigation; we do not perform a full covariance analysis implying, e.g., the simulation of the pulsar's TOAs and their reduction along with parameter estimation.\n\\section{Quadrupole mass moment, flattening, and angular momentum of fast rotating main sequence B-type massive stars}\\lb{J2S}\n\nThe binary pulsars on which we shall focus presumably own a main sequence B-type\nmassive star. Such stars are mostly fast rotators \\cite[][]{levato+13} and\nare hence distorted by the centrifugal acceleration. Such a\ndistortion takes the mass distribution away from spherical symmetry\nand endows these stars with a quadrupolar and higher gravitational\nmoments. In the case of Be stars (namely B stars with emission lines),\nrotation is believed to be almost critical, namely\nthe rotational velocity is taken close ($> 70\\%$) to the Keplerian velocity at equator \\cite[e.g.][]{Rivinius2013_v21p69}. When\ncritical rotation is reached, the surface distortion is maximum and\nthe flattening $\\nu\\doteq (R_\\textrm{e}-R_\\textrm{p})\/R_\\textrm{e}$\\textcolor{black}{, expressed in terms of the equatorial and polar radii $R_\\mathrm{e},\\,R_\\mathrm{p}$, respectively,} is close to one third. For such\nstars the computation of the gravitational quadrupolar moment cannot\nbe done perturbatively as it is the case for the Sun, which is a slow\nrotator \\cite[e.g.][]{rozelot_etal01}. Modelling these stars requires\ntwo-dimensional models. Fortunately, self-consistent 2D-models have recently been\nachieved with the ESTER code \\cite[][]{ELR13,RELP16}. Compared to previous\n2D-models, ESTER models include self-consistently the baroclinicity of\nthe stellar envelopes and can thus predict the associated differential\nrotation. They therefore provide unambiguously the total angular momentum\nof the star given, for instance, its equatorial velocity.\n\nWith the ESTER code, we computed the parameters of three stellar models of 10, 15, and 30\\,$\\textrm{M}_\\odot$ as they can represent the companions of PSR J0045-7319, PSR J2032+4127 and PSR B1256-63 respectively. Since the evolutionary status of the stars is unknown but presumably on or close to the main sequence, we computed their steady state at ZAMS (Zero-Age Main Sequence) and at half-main sequence to get an idea of the effects of evolution. We use standard galactic metallicity $Z=0.02$ with the solar mixture (which may be approximate for PSR J0045-7319 which is in the SMC, known to be less metallic than the Galaxy). Results of ZAMS and evolved models are displayed in Tables~\\ref{Ester_zams}\\,to\\,\\ref{Ester_ev} respectively. There we give the total spin angular momentum $S$ and the dimensionless quadrupole mass moment $J_2$ along with other bulk parameters of the models. We recall that multipole gravitational moments $J_\\ell$ are defined by the multipole expansion of the gravitational potential of a mass $M$, namely\n\\begin{equation}\nU\\ton{\\bds r} = -\\frac{\\mu}{r}\\left[1 - \\sum_\\ell J_\\ell\\left(\\frac{R_\\textrm{e}}{r}\\right)^\\ell \\mathcal{P}_\\ell\\ton{\\xi}\\right],\\lb{potenz}\n\\end{equation}\nwhere $\\mu\\doteq GM$ is the star's gravitational parameter, $G$ is the Newtonian constant of gravitation, $\\xi\\doteq \\bds{\\hat{S}}\\bds\\cdot\\bds{\\hat{r}}$ is the cosine of the angle $\\theta$ between the directions of the body's spin axis and of an external point at $\\bds r$,  $\\mathcal{P}_\\ell\\ton{\\xi}$ is the Legendre polynomial of degree $\\ell$.\nWe consider the mass distribution of the stellar models to be symmetric with respect to equator thus making the $J_\\ell$ of odd order all vanish. The remaining $J_{2p}$ can be computed with the integral expression\n\\begin{equation}\nJ_{2p} = -\\frac{1}{MR^{2p}_\\textrm{e}}\\int_{\\mathcal{V}} r^{2p}\\,\\mathcal{P}_{2p}\\ton{\\xi}\\,\\rho\\ton{\\bds r}\\,d^3\\bds r\n\\end{equation}\nwhere the integration is over the volume $\\mathcal{V}$ of the star. The same expression is given\nin, for instance, \\cite{helled+11}. Here we are especially interested in $J_2$ and $S$, namely in\n\\begin{equation}\nJ_2 = -\\frac{1}{MR^2_\\textrm{e}}\\int_{\\mathcal{V}} r^2\\,\\mathcal{P}_2\\ton{\\xi}\\,\\rho\\ton{\\bds r}\\,d^3\\bds r\\quad {\\rm and}\\quad S =  \\int_{\\mathcal{V}}r^2(1-\\xi^2)\\,\\Xi\\ton{\\bds r}\\rho\\ton{\\bds r}\\,d^3\\bds r\n\\end{equation}\nwhich are directly computed from the ESTER models; $\\Xi\\ton{\\bds r}$ is the local angular speed.\n\nWe choose an angular rotation rate of 70\\% of the actual critical (Keplerian) angular velocity of the star. Such a rotation rate is typical of the nearby fast rotating stars that have been measured by interferometry. Their flattening is typically $\\sim0.2$  \\cite[e.g.][]{domiciano_etal14}, as our models.\n\\begin{table}[ht]\n\\caption{ESTER models for Zero-Age Main Sequence (ZAMS) stars with an equatorial angular velocity at\n70\\% of the critical angular velocity. $R_\\textrm{e}$ and $V_\\textrm{e}$ are the equatorial radius and\nvelocity, respectively, $S$ is the total spin angular momentum, $\\nu$ is the\nflattening, and $J_2$ is the dimensionless mass quadrupole  moment. The metallicity is Z=0.02 and the hydrogen mass fraction is $X=0.7$. In our simulations, we use the parameters of the star with $15\\,\\textrm{M}_\\odot$ listed here.}\n\\label{Ester_zams}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|}\n\\hline\n$M~(\\textrm{M}_\\odot)$ & $R_\\textrm{e}~(\\textrm{R}_\\odot)$ & $V_\\textrm{e}~(\\textrm{km}~\\textrm{s}^{-1})$ & $S~(\\times 10^{44}\\textrm{J}~\\textrm{s})$ & $\\nu$ & $J_2~(\\times 10^{-3})$\\\\\n\\hline\n  10    &    4.74        &  444                &  15.3               & 0.201      & 1.63 \\\\\n  15    &    5.96        &  485                &  34.1               & 0.203      & 1.92 \\\\\n  30    &    8.89        &  562                &  125.               & 0.210      & 2.17 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\begin{table}[ht]\n\\caption{Same as in table~\\ref{Ester_zams} but for ESTER models of\nstars at mid-main-sequence, namely when the hydrogen mass fraction in the convective core is half of the initial one.}\n\\label{Ester_ev}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|}\n\\hline\n$M~(\\textrm{M}_\\odot)$ & $R_\\textrm{e}~(\\textrm{R}_\\odot)$ & $V_\\textrm{e}~(\\textrm{km}~\\textrm{s}^{-1})$ & $S~(\\times 10^{44}~\\textrm{J}~\\textrm{s})$ & $\\nu$ & $J_2~(\\times 10^{-3})$\\\\\n\\hline\n  10  &    6.93                &  367                &  13.3               & 0.203      & 0.816\\\\\n  15  &    9.07                &  393                &  28.0               & 0.207      & 0.788\\\\\n  30  &    14.8                &  435                &  870.               & 0.227      & 0.495\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nFrom Tables~\\ref{Ester_zams}~to~\\ref{Ester_ev}, we clearly see that as evolution proceeds, namely as the hydrogen content of the core decreases, $J_2$ decreases as expected from the resulting contraction of the convective core. From the work of \\cite{james64} we can compute $J_2$ for a polytrope of index $n=3$  with a similar flattening as the ESTER models. We find that $J_2\\simeq 2.1\\times10^{-3}$ which is quite similar to the ZAMS models. For evolved models one should use polytropes with a higher polytropic index, as they are more centrally condensed, and typically $n=3.43$ matches the ESTER evolved models as far as $J_2$ is concerned. This result may be useful for simulating the orbital evolution of binary pulsars since polytropic models are much easier to compute.\n\\section{The perturbations of the R{\\o}mer-type pulsar's time delay due to some pK Newtonian and pN accelerations}\\lb{GR}\nHere, we will assume a coordinate system centered in the binary's barycenter whose reference $z$-axis is directed along the line of sight from the binary to the observer, while the reference $\\grf{x,~y}$ plane spans the plane of the sky.\n\\subsection{The pulsar as a structureless, pointlike particle}\\lb{punto}\nWe will, first, consider the following pK accelerations experienced by a test particle moving with velocity $\\textcolor{black}{\\bds{v}}$ in the external field of an oblate body of mass $M$, equatorial and polar  radii $R_\\textrm{e},~R_\\textrm{p}$, ellipticity $\\varepsilon \\doteq \\sqrt{1- R_\\textrm{p}^2\/R_\\textrm{e}^2}$, angular momentum $\\bds S$ and dimensionless quadrupole mass moment $J_2$. In Section\\,\\ref{esteso}, we will discuss the limits of validity of the point-particle approximation.\n\nTo the Newtonian level, the external potential of the distorted star at the position $\\bds r$ is, from \\rfr{potenz},\n\\begin{equation}\nU\\ton{\\bds r} = U_0 + \\Delta U_2 = -\\frac{\\mu}{r}\\qua{1-\\ton{\\frac{R_\\textrm{e}}{r}}^2 J_2 \\mathcal{P}_2\\ton{\\xi}},\n\\end{equation}\nwhere  $\\mathcal{P}_2\\ton{\\xi} = \\ton{3\\xi^2 - 1}\/2$ is the Legendre polynomial of degree 2.\nThe classical acceleration due to $J_2$ is\n\\begin{equation}\n{\\bds A}^{\\textrm{N}J_2} = -\\bds\\nabla \\Delta U_{J_2}= \\frac{3\\mu J_2 R_\\textrm{e}^2}{2r^4}\\qua{\\ton{5\\xi^2 - 1}\\bds{\\hat{r}} - 2\\xi\\bds{\\hat{S}}}.\\lb{NJ2}\n\\end{equation}\n\nThe 1pN gravitoelectric Schwarzschild-like acceleration affecting the motion of a test particle in the static field of a nonrotating, spherically symmetric body is \\citep{2010ITN....36....1P}\n\\begin{equation}\n{\\bds A}^{\\textrm{1pN}M} = \\frac{\\mu}{c^2 r^2}\\qua{\\ton{\\frac{4\\mu}{r}-{\\mathrm{v}}^2}\\bds{\\hat{r}} + 4{\\mathrm{v}}_r\\textcolor{black}{\\bds{v}}},\\lb{1pNM}\n\\end{equation}\nwhere $c$ is the speed of light in vacuum, and ${\\mathrm{v}}_r \\doteq \\textcolor{black}{\\bds{v}}\\bds\\cdot\\bds{\\hat{r}}$ is the radial velocity of the test particle. \\Rfr{1pNM} is responsible for the formerly anomalous perihelion precession of Mercury whose explanation was the first empirical confirmation of general relativity \\citep{1915SPAW...47..831E}.\n\nThe 1pN gravitomagnetic Lense-Thirring acceleration in the stationary field due to the rotating primary is \\citep{2010ITN....36....1P}\n\\begin{equation}\n{\\bds A}^{\\textrm{1pN}S} \\lb{1pNS}  = \\frac{2GS}{c^2 r^3}\\qua{3\\xi\\bds{\\hat{r}}\\bds\\times\\textcolor{black}{\\bds{v}} + \\textcolor{black}{\\bds{v}}\\bds\\times\\bds{\\hat{S}}}.\n\\end{equation}\nThe gravitomagnetic field of the Earth was unambiguously measured for the first time by the Gravity Probe B (GP-B) mission \\citep{2011PhRvL.106v1101E}. Tests of the Lense-Thirring orbital precessions with some terrestrial geodetic satellites are ongoing; see, e.g. \\citet{2013CEJPh..11..531R}, and \\citet{2015CQGra..32o5012L} and references therein for comprehensive reviews.\n\nThe 1pN gravitoelectric acceleration felt by a test particle in the field of an oblate body  is \\citep{1988CeMec..42...81S, Sof89, 1991ercm.book.....B, 2014PhRvD..89d4043W,2015IJMPD..2450067I}\n\\begin{equation}\n{\\bds A}^{\\textrm{1pN}M J_2} \\lb{1pNMJ2} =  \\frac{\\mu J_2 R_\\textrm{e}^2}{c^2 r^4}\\grf{\\frac{3}{2}\\qua{\\ton{5\\xi^2 - 1}\\bds{\\hat{r}} - 2\\xi\\bds{\\hat{S}}  }\\ton{{\\mathrm{v}}^2 -\\frac{4\\mu}{r} }- 6\\qua{\\ton{5\\xi^2 - 1}{\\mathrm{v}}_r  - 2\\xi{\\mathrm{v}}_S   }\\textcolor{black}{\\bds{v}}\n- \\frac{2\\mu}{r}\\ton{3\\xi^2 - 1}\\bds{\\hat{r}}},\n\\end{equation}\nwhere ${\\mathrm{v}}_S \\doteq \\textcolor{black}{\\bds{v}}\\bds\\cdot\\bds{\\hat{S}}$ is the component of the particle's velocity along the direction of the primary's spin.\nNote that the parameter $J_2$ in \\rfr{1pNMJ2} is the same entering \\rfr{NJ2}, as per Equations 1 to 2 of \\citet{1988CeMec..42...81S}.\n\nThe 1pN gravitomagnetic acceleration imparted to a test particle by the spin octupole moment\\textcolor{black}{\\footnote{\\textcolor{black}{It will be shown that its effects are small enough to justify order-of-magnitude calculations, without need of detailed stellar models.}}} of a uniformly rotating homogenous oblate spheroid  \\citep{2014CQGra..31x5012P,2015CeMDA.123....1M,2019MNRAS.484.4811I} can be cast into the compact form\n\\begin{equation}\n\\bds A^{\\textrm{1pN}SJ_2} = \\frac{3GSR_\\textrm{e}^2\\varepsilon^2 }{7c^2r^5}\\textcolor{black}{\\bds{v}}\\bds\\times\\grf{5\\xi\\qua{7\\xi^2 - 3}\\bds{\\hat{r}} + 3\\qua{1 - 5\\xi^2}\\bds{\\hat{S}}}.\\lb{1pNSJ2}\n\\end{equation}\n\nThe pK accelerations of \\rfrs{NJ2}{1pNSJ2} perturb the otherwise Keplerian motion of the binary causing a change $\\Delta\\ton{\\delta\\tau}$ of the regular variation $\\delta\\tau$ of the TOAs due to the relative orbital motion of the pulsar and the massive companion. It can be modeled as the\nratio of the projection of the barycentric orbit of the pulsar onto the line of sight to $c$ \\citep{1991PhRvL..66.2549D,2000ApJ...544..921K}. Thus, $\\Delta\\ton{\\delta\\tau}$ can be calculated by looking at the perturbations $\\Delta z$ induced by \\rfrs{NJ2}{1pNSJ2} on the $z$-component of the pulsar's barycentric orbital motion. We do that by numerically integrating the equations of motion of a fictitious pulsar \\textcolor{black}{with mass $M_\\mathrm{p}=1.4\\,\\mathrm{M}_\\odot$} having as a companion a Be-type star with $M = 15~\\textrm{M}_\\odot,~R_\\textrm{e} = 5.96~\\textrm{R}_\\odot,~\\nu=0.203,~S = 3.41\\times 10^{45}~\\textrm{J}~\\textrm{s},\\,J_2 = 1.92\\times 10^{-3}$, as per Table~\\ref{Ester_zams}, for different values of its orbital configuration, determined by the initial values of the semimajor axis $a$, the eccentricity $e$, the orbital inclination $I$ to the plane of the sky, the longitude of the ascending node $\\Omega$, the argument of periastron $\\omega$, the true anomaly \\textcolor{black}{at epoch} $f_0$, and of the stellar spin axis characterized by its inclination $i$ to the line of sight, and the longitude $\\phi$ of the projection of the stellar spin  onto the plane of the sky. For a chosen pK acceleration ${\\bds A}^\\textrm{pK}$, in order to compute its perturbation $\\Delta\\ton{\\delta\\tau}=\\Delta z\\ton{t}\/c$ over, say, 10 yrs, we perform two runs sharing the same initial conditions with and without ${\\bds A}^\\textrm{pK}$, calculate the resulting time series of $z\\ton{t}\/c$ and take their difference. Figures~\\ref{fig_NJ2}~to~\\ref{fig_1pNSJ2} depict our results for a given orbital configuration and the aforementioned Be-type main sequence star; in the panels of each Figure, we vary the Keplerian orbital elements and the orientation of $\\bds{\\hat{S}}$ in order to investigate the sensitivity to the parameter space of the adopted binary system. \\textcolor{black}{For the sake of a comparison, the Keplerian delay for the adopted reference orbital configuration lies in the range $-250\\,\\mathrm{s}\\lesssim\\delta\\tau\\lesssim 50\\,\\mathrm{s}$ over one orbital revolution.}\n\nFigure\\,\\ref{fig_NJ2} displays the Newtonian signatures due to the star's $J_2$. It can be noted that the resulting signal is strongly dependent on the initial value of the true anomaly, spanning the range from  $50\\,\\textrm{s}$ to $-150\\,\\textrm{s}$. Instead, for the given value $f_0 = 228\\,\\textrm{deg}$, the sensitivity to the other orbital parameters is rather modest, amounting to about $10\\,\\textrm{s}$.\n\\begin{figure}[ht]\n\\begin{center}\n\\centerline{\n\\vbox{\n\\begin{tabular}{cc}\n\\epsfysize= 4.4 cm\\epsfbox{Be_J2_a.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_J2_e.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_J2_I.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_J2_N.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_J2_p.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_J2_f.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_J2_LONG.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_J2_COLAT.pdf}\\\\\n\\end{tabular}\n}\n}\n\\caption{Numerically integrated time series of the timing shift $\\Delta\\ton{\\delta\\uptau}$ due to \\rfr{NJ2}, in $\\textrm{s}$, for variations of the parameter space of a fictitious binary pulsar, characterized by\n$P_{\\rm b} = 50~\\textrm{d},~e=0.8,~I=50~\\textrm{deg},~\\Omega=140~\\textrm{deg},~\\omega=149~\\textrm{deg},~f_0=228~\\textrm{deg}$, orbiting a Be-type star with $M=15~\\textrm{M}_\\odot,~R_\\textrm{e} = 5.96~\\textrm{R}_\\odot,~J_2 = 1.92\\times 10^{-3},~i=60~\\textrm{deg},~\\phi=217~\\textrm{deg}$.}\\label{fig_NJ2}\n\\end{center}\n\\end{figure}\n\nThe 1pN gravitoelectric Schwarzschild-like signatures due to the stellar mass monopole are reproduced in Figure\\,\\ref{fig_1pNM}.\nAlso in this case, the initial value of the true anomaly induces a marked variability in the decadal time series which ranges from to $-5\\,\\textrm{s}$ to $25\\,\\textrm{s}$. The other orbital parameters have less impact since the resulting variation of the signals is of the order of about $2\\,\\textrm{s}$.\n\\begin{figure}[ht]\n\\begin{center}\n\\centerline{\n\\vbox{\n\\begin{tabular}{cc}\n\\epsfysize= 4.4 cm\\epsfbox{Be_M_a.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_M_e.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_M_I.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_M_N.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_M_p.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_M_f.pdf}\\\\\n\\end{tabular}\n}\n}\n\\caption{Numerically integrated time series of the timing shift $\\Delta\\ton{\\delta\\uptau}$ due to \\rfr{1pNM}, in $\\textrm{s}$, for variations of the parameter space of a fictitious binary pulsar, characterized by\n$P_{\\rm b} = 50~\\textrm{d},~e=0.8,~I=50~\\textrm{deg},~\\Omega=140~\\textrm{deg},~\\omega=149~\\textrm{deg},~f_0=228~\\textrm{deg}$, orbiting a Be-type star with $M=15~\\textrm{M}_\\odot$.}\\label{fig_1pNM}\n\\end{center}\n\\end{figure}\n\nThe 1pN gravitoelectric time series induced by the quadrupole mass moment $J_2$ of the star are the subject of Figure\\,\\ref{fig_1pNMJ2}.\nIn this case, the ranges of variation due to all the orbital parameters are rather similar, amounting to about $20-50\\,\\textrm{ms}$.\n\\begin{figure}[ht]\n\\begin{center}\n\\centerline{\n\\vbox{\n\\begin{tabular}{cc}\n\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_a.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_e.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_I.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_N.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_p.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_f.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_LONG.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_MJ2_COLAT.pdf}\\\\\n\\end{tabular}\n}\n}\n\\caption{Numerically integrated time series of the timing shift $\\Delta\\ton{\\delta\\uptau}$ due to \\rfr{1pNMJ2}, in $\\textrm{ms}$, for variations of the parameter space of a fictitious binary pulsar, characterized by\n$P_{\\rm b} = 50~\\textrm{d},~e=0.8,~I=50~\\textrm{deg},~\\Omega=140~\\textrm{deg},~\\omega=149~\\textrm{deg},~f_0=228~\\textrm{deg}$, orbiting a Be-type star with $M = 15~\\textrm{M}_\\odot,~R_\\textrm{e} = 5.96~\\textrm{R}_\\odot,~J_2 = 1.92\\times 10^{-3},~i=60~\\textrm{deg},~\\phi=217~\\textrm{deg}$.}\\label{fig_1pNMJ2}\n\\end{center}\n\\end{figure}\n\nFigure\\,\\ref{fig_1pNS} shows the 1pN gravitomagnetic Lense-Thirring signatures due to the spin dipole moment of the star. They are mainly sensitive to the orbital inclination $I$ and to the spin's inclination $i$ to the line of sight which induces a variability as large as $10-12\\,\\textrm{ms}$. The ranges of variation induced by the other parameters are, instead, of the order of $1-5\\,\\textrm{ms}$.\n\\begin{figure}[ht]\n\\begin{center}\n\\centerline{\n\\vbox{\n\\begin{tabular}{cc}\n\\epsfysize= 4.4 cm\\epsfbox{Be_S_a.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_S_e.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_S_I.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_S_N.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_S_p.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_S_f.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_S_LONG.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_S_COLAT.pdf}\\\\\n\\end{tabular}\n}\n}\n\\caption{Numerically integrated time series of the timing shift $\\Delta\\ton{\\delta\\uptau}$ due to \\rfr{1pNS}, in $\\textrm{ms}$, for variations of the parameter space of a fictitious binary pulsar, characterized by\n$P_{\\rm b} = 50~\\textrm{d},~e=0.8,~I=50~\\textrm{deg},~\\Omega=140~\\textrm{deg},~\\omega=149~\\textrm{deg},~f_0=228~\\textrm{deg}$, orbiting a Be-type star with $M = 15~\\textrm{M}_\\odot,~S = 3.41\\times 10^{45}~\\textrm{J}~\\textrm{s},~i=60~\\textrm{deg},~\\phi=217~\\textrm{deg}$.}\\label{fig_1pNS}\n\\end{center}\n\\end{figure}\n\nThe 1pN gravitomagnetic time series caused by the spin octupole moment of the star are depicted in Figure\\,\\ref{fig_1pNSJ2}.\nThe widest range of variability, $30-50\\,\\upmu\\textrm{s}$, is due to the inclination $I$, the node $\\Omega$, and the longitude $\\phi$ of the spin's projection onto the plane of the sky. We also checked that, for a pulsar orbiting in $\\simeq 20-30\\,\\textrm{d}$ a star as massive as those in the last lines of Tables\\,\\ref{Ester_zams}\\,to\\,\\ref{Ester_ev} with periastron distances of $r_\\textrm{min}\\simeq 1.2-1.1\\,R_\\textrm{eq}$, the magnitude of the timing signatures would reach the $\\simeq 1-10\\,\\textrm{ms}$ level.\n\\begin{figure}[ht]\n\\begin{center}\n\\centerline{\n\\vbox{\n\\begin{tabular}{cc}\n\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_a.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_e.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_I.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_N.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_p.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_f.pdf}\\\\\n\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_LONG.pdf}&\\epsfysize= 4.4 cm\\epsfbox{Be_SJ2_COLAT.pdf}\\\\\n\\end{tabular}\n}\n}\n\\caption{Numerically integrated time series of the timing shift $\\Delta\\ton{\\delta\\uptau}$ due to \\rfr{1pNSJ2}, in $\\upmu\\textrm{s}$, for variations of the parameter space of a fictitious binary pulsar, characterized by\n$P_{\\rm b} = 50~\\textrm{d},~e=0.8,~I=50~\\textrm{deg},~\\Omega=140~\\textrm{deg},~\\omega=149~\\textrm{deg},~f_0=228~\\textrm{deg}$, orbiting a Be-type star with $M = 15~\\textrm{M}_\\odot,~R_\\textrm{e} = 5.96~\\textrm{R}_\\odot,~\\nu=0.203,~S = 3.41\\times 10^{45}~\\textrm{J}~\\textrm{s},~i=60~\\textrm{deg},~\\phi=217~\\textrm{deg}$.}\\label{fig_1pNSJ2}\n\\end{center}\n\\end{figure}\n\\subsection{The effects of the mass and spin multipoles of the pulsar}\\lb{esteso}\nUntil now, we have considered the neutron star as a structureless, point particle moving around its more massive companion.\nIn fact, a pulsar is an extended body with its own mass and spin multipole moments which, at least in principle, may have an impact on its orbital motion in a full two-body framework.\n\nThe  modification of \\rfr{1pNM} for two finite bodies of masses $M_\\textrm{A},\\,M_\\textrm{B}$\nis \\citep{Sof89}\n\\begin{align}\n{\\bds A}^{\\textrm{1pN}M} = \\frac{\\mu_\\textrm{tot}}{c^2 r^2}\\grf{\\qua{\\ton{4+2\\zeta}\\frac{\\mu_\\textrm{tot}}{r}-\\ton{1+3\\zeta}{\\mathrm{v}}^2 +\\frac{3}{2}\\zeta{\\mathrm{v}}_r^2}\\bds{\\hat{r}} + \\ton{4-2\\zeta}{\\mathrm{v}}_r\\textcolor{black}{\\bds{v}}},\\lb{megapN}\n\\end{align}\nwhere $\\mu_\\textrm{tot}=GM_\\textrm{tot}=G\\ton{M_\\textrm{A}+M_\\textrm{B}}$, and $\\zeta\\doteq M_\\textrm{A}M_\\textrm{B}\/M_\\textrm{tot}^2$.\nBy taking the standard value $M_\\textrm{p} = 1.4\\,\\textrm{M}_\\odot$ for the mass of the pulsar, it is $\\zeta = 0.078$ for the Be-star assumed in Section\\,\\ref{punto}. It turns out that the introduction of $\\zeta$ in our numerical code changes the size of the time series of Figure\\,\\ref{fig_1pNM} by  $\\simeq 0.1-0.2\\,\\textrm{s}$, while  their temporal patterns remain essentially unchanged. Given the current level of accuracy in the timing residuals, such a discrepancy might be significative, and \\rfr{megapN} should be used instead of \\rfr{1pNM}.\n\nIn regard to the angular momentum, the spin $\\bds S$ of the Be-star in \\rfr{1pNS}  should be replaced by the sum \\citep{1975PhRvD..12..329B}\n\\begin{equation}\n\\bds S\\rightarrow \\ton{1 + \\frac{3}{4}\\frac{M_\\textrm{p}}{M}}\\bds S +\\ton{1 + \\frac{3}{4}\\frac{M}{M_\\textrm{p}}}{\\bds S}_\\textrm{p},\\lb{pulsarS}\n\\end{equation}\nwhere ${\\bds S}_\\textrm{p}$ is the angular momentum of the pulsar.\nBy assuming, e.g., $S_\\textrm{p}= 3\\times 10^{40}\\,\\textrm{J}\\,\\textrm{s}$ as for PSR J0737-3039A \\citep{2003Natur.426..531B,2004Sci...303.1153L,Kramer2012,\nKehletal017,2018IAUS..337..128K}, it turns out that the magnitude of the second term in \\rfr{pulsarS} amounts to about $\\simeq 7\\times 10^{-5}$ of that of the first one. Moreover, it is  $\\ton{3\/4}\\ton{M_\\textrm{p}\/M}=0.07$. Thus, as far as the 1pN gravitomagnetic Lense-Thirring effect is concerned, our scenario can be well approximated by a restricted two-body system with a spinning primary, and  \\rfr{1pNS} is substantially adequate. Indeed, it turns out that rescaling the star's angular momentum in our numerical simulations as dictated by \\rfr{pulsarS} slightly modifies the size of the time series of Figure\\,\\ref{fig_1pNS} by just $\\simeq 1-1.5\\,\\textrm{ms}$.\n\nThe effect of the pulsar's quadrupole mass moment $J_2^\\textrm{p}$  can be accounted for by the replacement $M\\rightarrow M_\\textrm{tot}$ in \\rfr{NJ2} and by writing in its right-hand-side another term for $J_2^\\textrm{p}$ analogous to that for the stellar oblateness $J_2$. As a result, by introducing the dimensional quadrupole mass moment\n$Q_2 \\doteq -J_2 M R_\\textrm{e}^2$, the two terms in the right-hand-side of the resulting modified version of \\rfr{NJ2} are weigthed by \\citep{1975PhRvD..12..329B}\n\\begin{align}\n\\mathcal{Q}_2 \\lb{Qu2} & = \\ton{1+\\frac{M_\\textrm{p}}{M}}Q_2, \\\\ \\nonumber \\\\\n\\mathcal{Q}_2^\\textrm{p} & = \\ton{1+\\frac{M}{M_\\textrm{p}}}Q^\\textrm{p}_2.\n\\end{align}\nFor a neutron star, it is \\citep{1999ApJ...512..282L,2004MNRAS.350.1416B,2012PhRvL.108w1104P,2013ApJ...777...68B}\n\\begin{equation}\nQ_2^\\textrm{p} =  -q\\frac{M_\\textrm{p}^3 G^2}{c^4},\n\\end{equation}\nwith $0.07\\lesssim q \\lesssim 3.507$ for a variety of Equations of State (EOSs).\nThus, we have\n\\begin{align}\nQ_2 & = -9.8\\times 10^{47}\\,\\textrm{kg}\\,\\textrm{m}^2, \\\\ \\nonumber \\\\\nQ_2^\\textrm{p} & = -q\\,1.2\\times 10^{37}\\,\\textrm{kg}\\,\\textrm{m}^2,\n\\end{align}\nso that $\\mathcal{Q}_2^\\textrm{p}\\simeq 10^{-10}\\,\\mathcal{Q}_2$. On the other hand, $M_\\textrm{p}\/M = 0.09$ in \\rfr{Qu2}.\nThus, for the Newtonian signature of the quadrupole mass moment, the restricted two-body scenario with an oblate primary is adequate in the present case, and the use of \\rfr{NJ2} is justified provided that the stellar quadrupole moment $Q_2$ is replaced by \\rfr{Qu2}. Indeed, it turns out that the introduction of $\\mathcal{Q}_2$ in the numerical integration changes the size of the time series in Figure\\,\\ref{fig_NJ2} by $\\simeq 1\\,\\textrm{s}$ which may not be neglected, given the current level in $\\upsigma_{\\tau}$.\n\nDespite \\rfr{1pNMJ2} and \\rfr{1pNSJ2} were derived  so far only for the motion of a test particle around a spinning, oblate mass, there are no doubts that they are adequate to the scenario considered here.\n\nActually, general relativity predicts that, in general, there is also a self-force due to the spin angular momentum of an extended rotating body in motion in an external gravitational field  which acts on it modifying its orbit through a spin-orbit coupling \\citep{1951RSPSA.209..248P,1974RSPTA.277..59D,1979GReGr..11..149B,2006PhRvD..74l4006M,Mathis010,2012GReGr..44..719I}.\nIn order to quickly evaluate the possible impact of such an effect in our case, let us proceed as follows. To the 1pN level, the precession $\\dot\\Omega_{S_\\textrm{p}}$ of, say, the node $\\Omega$, averaged over one orbital revolution of the spinning pulsar  in its motion around the massive Be-type star, assumed nonrotating, is \\citep{2012GReGr..44..719I}\n\\begin{equation}\n\\dot\\Omega_{S_\\textrm{p}} = \\frac{3\\mu \\sigma_\\textrm{p}\\csc I\\ton{{\\bds{\\hat{\\sigma}}}_\\textrm{p}\\bds\\cdot\\bds{\\hat{m}}}}{2c^2a^3\\ton{1-e^2}^{3\/2}}.\\lb{spinor}\n\\end{equation}\nIn this expression, ${\\bds\\sigma}_\\textrm{p} = {\\bds S}_\\textrm{p}\/M_\\textrm{p}$ is the pulsar's spin angular momentum per unit mass, while $\\bds{\\hat{m}} =\\grf{-\\cos I\\sin\\Omega,\\,\\cos I\\cos\\Omega,\\,\\sin I}$ is a unit vector in the orbital plane perpendicular to the line of the nodes.\nLet us compare \\rfr{spinor}  to  the analogous precession induced by some of the effects previously considered in which the pulsar was treated as a point particle.  The Lense-Thirring node rate, induced by \\rfr{1pNS} which is responsible for the $\\simeq\\textrm{ms}$ time series of Figure\\,\\ref{fig_1pNS},\nis \\citep{2012GReGr..44..719I}\n\\begin{equation}\n\\dot\\Omega_\\textrm{LT} = \\frac{2GS\\csc I\n\\ton{\\bds{\\hat{S}}\\bds\\cdot\\bds{\\hat{m}}}}{c^2 a^3\\ton{1-e^2}^{3\/2}}.\n\\end{equation}\nIt turns out that, in the case of a pulsar like, e.g., PSR J0737-3039A orbiting the Be-type star of the Figures\\,\\ref{fig_NJ2}\\,to\\,\\ref{fig_1pNSJ2}, it is $\\left|\\dot\\Omega_{S_\\textrm{p}}\/\\dot\\Omega_\\textrm{LT}\\right|\\simeq 10^{-5}$. Thus, we conclude that the spin-orbit self-force experienced by the pulsar is completely negligible in our scenario.\nIn fact, there is also a further self-force acting on an extended rotating body moving in an external gravitational field due to the spin-spin coupling between the angular momenta of the source and of the orbiter itself \\citep{1979GReGr..11..149B,2012GReGr..44..719I}. By following the same strategy for the spin-orbit coupling, the results in \\citet{2012GReGr..44..719I} allow to conclude that, in our case, such an effect is even smaller than the previous one, being of the order of $\\simeq 10^{-7}$ of, say, the $\\simeq \\textrm{ms}$-level Lense-Thirring signal.\n\n\\section{Summary and conclusions}\\lb{conclu}\nWe preliminarily  explored the possibility of putting to the test several pK features of motion of Newtonian and pN origin in binaries hosting a pulsar and a massive, fast rotating, highly distorted main sequence star characterized by mass $M$, angular momentum $\\bds S$, equatorial and polar radii $R_\\textrm{e},~R_\\textrm{p}$, flattening $\\nu$, ellipticity $\\varepsilon$ and dimensionless quadrupole moment $J_2$. Indeed, in addition to the usual 1pN Schwarzschild and Lense-Thirring effects due to the mass monopole $\\ton{\\propto GM c^{-2}}$ and spin dipole $\\ton{\\propto GSc^{-2}}$ moments, respectively, of the distorted stellar field, there are also other 1pN orbital effects, induced by the mass quadrupole $\\ton{\\propto GMR^2_\\textrm{e}J_2 c^{-2}}$ and spin octupole $\\ton{\\propto GSR^2_\\textrm{e}\\varepsilon^2c^{-2}}$ moments, whose magnitudes may, perhaps, lie above the sensitivity threshold of the pulsar timing residuals in yet-to-be-discovered close binaries. However, the Newtonian perturbations due to $J_2$ are larger than the pN ones.\n\nIn order to perform a preliminary sensitivity analysis, we numerically investigated the orbital shifts $\\Delta\\ton{\\delta\\tau}$ induced over 10 yr by all of such pK perturbations on the otherwise Keplerian R{\\o}mer-type delay $\\delta\\tau$ in the pulsar's TOAs for a Be-type main sequence star characterized by  $M = 15~\\textrm{M}_\\odot,~R_\\textrm{e} = 5.96~\\textrm{R}_\\odot,~\\nu=0.203,~S = 3.41\\times 10^{45}~\\textrm{J}~\\textrm{s},\\,J_2 = 1.92\\times 10^{-3}$ orbited by a pulsar with an orbital geometry compatible $\\ton{P_{\\rm b}\\simeq 40-70~\\textrm{d}}$ with some of the tightest binaries of this kind out of those discovered so far. We also investigated the sensitivity of the pK timing shifts to the whole system's parameter space by varying both the orientation of the stellar spin axis and the orbital elements of the pulsar's orbit.\nIt turns out that the magnitude of the Newtonian signature due to $J_2$ can be as large as $\\lesssim 4-150~\\textrm{s}$,\nwhile the pN gravitoelectric quadrupolar one is $\\lesssim 10-40~\\textrm{ms}$.\nThe pN gravitoelectric (Schwarzschild-like) signal due to the stellar mass monopole can be as large as $\\lesssim 1.5-20~\\textrm{s}$ level.\nThe pN gravitomagnetic shifts due to the spin dipole (Lense-Thirring) and octupole moments, for the evaluation of whose size the knowledge of the stellar spin angular momentum $S$ and ellipticity $\\varepsilon$ is crucial, are of the order of $\\lesssim 0.5-6~\\textrm{ms}$, and $\\lesssim 5-20~\\upmu\\textrm{s}$, respectively.\nThe rms of the timing residuals of all the non-recycled, non-millisecond pulsars like those having a fast rotating main sequence star companion discovered so far are $\\lesssim \\textrm{ms}$ over $2-13~\\textrm{yr}$. It seems difficult that they can be substantially improved in the future. This implies that, in principle, all the pK effects considered fall within the potential measurability domain, except the pN  spin octupole which is $\\simeq 1-2$ orders of magnitude weaker, at least for the orbital configurations and the star considered in this study. The pN Lense-Thirring signatures are just at the $\\simeq\\textrm{ms}$ level.\nThe different temporal patterns characteristic of the signals investigated may be helpful in separating them. It turns out that the tiniest pN effect may reach the $\\simeq 1-10\\,\\textrm{ms}$ level only for a very tight, eccentric binary $(P_\\textrm{b}\\simeq 20-30\\,\\textrm{d},\\,r_\\textrm{min}\\simeq 1.1-1.2\\,R_\\textrm{e})$ hosting a Be-star with $30\\,\\textrm{M}_\\odot,\\,S=125-870\\times 10^{44}\\,\\textrm{J\\,s}$.\n\nFinally, we stress the preliminary nature of our sensitivity analysis. To this aim, we remark that we did not compute the other kinds of time delay connected, e.g., with the propagation of the electromagnetic waves in the deformed spacetime.  Moreover, we did not perform a full covariance analysis implying a simulation of the pulsar's TOAs, their reduction, and parameter estimation.\n\\section*{Acknowledgements}\nWe are grateful to A. Possenti for useful information. MR and ADS also acknowledge the strong support of the French Agence Nationale de la Recherche (ANR), under grant ESRR (ANR-16-CE31-0007-01).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\nSelf-assembly is a ubiquitous phenomenon in nature, producing complex structures in biology, chemistry and physics.\nExamples include DNA \\cite{winfree1998design,goodman2005rapid,rothemund2006folding,ke2012three}, protein quaternary structure \\cite{Levy:2006ez,villar2009self}, protein aggregation \\cite{Cohen:2013fd}, viruses \\cite{lavelle2009phase}, micelles \\cite{israelachvili1994self}, and thin films \\cite{krausch2002nanostructured}.\n\nTwo fundamental questions about a self-assembling system are: do the building blocks always form the same structure and does the assembly grow indefinitely.\nThe former property is referred to as the {\\em determinism} and the latter as {\\em boundedness} \\cite{Evo,ARXIV2015}.\nTogether they form the assembly classification.\n\nProtein complexes are a prominent example of deterministic, bound self-assembly.\nMisfolding or erroneous binding of mutated versions of such proteins can cause the self-assembly of a protein complex to become nondeterministic, unbound protein aggregation.\nThis in turn is the hallmark of a number of severe diseases, such as sickle-cell anemia and Alzheimer's \\cite{Cohen:2013fd,Bunn:1997de}.\n\nIn this paper we introduce a framework for establishing the determinism and boundedness of a given set of self-assembling building blocks.\nWe apply this approach to a previously studied lattice self-assembly model introduced in \\cite{Modularity} and studied further in \\cite{Evo,green,greengen,ARXIV2015,TES2016}.\nIn this model square tiles have attractive interfaces of different types, with interactions governed by a simple set of rules.\nThe final structures assembled are sets of connected lattice sites known as polyominoes.\n\nSuch tile self-assembly or polyomino models have been useful for the study of genotype-phenotype maps, where the specification of the building blocks can be viewed as a genotype and the resulting structure as the phenotype \\cite{Evo,green,greengen}.\nThis approach can be combined with genetic algorithms to model evolutionary processes \\cite{Evo}.\n\nDespite being an abstract model, this polyomino model can directly and meaningfully map to real biological self-assembly phenomena.\nFor example, the sickle-cell mutation of hemoglobin that leads to unbound protein aggregation can be modelled using polyominoes \\cite{green}.\nFurthermore, experimental implementations of the polyomino model have been realized using DNA tiles \\cite{TES2016}. \n\nThe assembly process previously used is fully stochastic, and may be sketched out as: (a) the structure is seeded with a randomly selected tile, (b) a random face on the structure is chosen, (c) a random tile is drawn with a random orientation, and (d) the drawn tile binds to the structure if the interfaces are interacting.\nSteps (b-d) are then repeated until no further attachments are possible and assembly terminates.\n\nIn this paper, we introduce a graph based approach to replace stochastic assembly as the primary method for identifying the determinism and boundedness of a given set of self-assembly building blocks.\nThe stochastic approach is inelegant and suffers from a compromise between accuracy and speed, whereas the graph theoretical approach offers a robust methodology that improves accuracy and speed.\n\nThis paper proceeds by discussing self-assembly basics in Section \\ref{sec:Lsa}. Assembly graphs and their construction from tile sets are introduced in Section \\ref{sec:AG}.\nClassification preserving transformations of assembly graphs which simplify graph features into deterministic motifs are discussed in Section \\ref{sec:ATAP}.\nAfter classifying the assembly graph, the structure is validated under steric constraints in Section \\ref{sec:CP}.\nExtensions to the method and general discussion are in Sections \\ref{sec:FE} and \\ref{sec:D} respectively.\n\nWith generalizations to other geometries and dimensions, there are potential applications in the study of protein complexes and protein aggregation, as well as bioengineering and nanotechnology.\n\n\\section{Lattice self-assembly}\\label{sec:Lsa}\nFollowing the model introduced in \\cite{Modularity}, a lattice self-assembly tile set consists of one or more tile types.\nEach side of a tile (the geometric face) has an interface type, with every tile type in the set exhibiting unique configurations of these interfaces.\nConventionally, interactions are defined with {\\bf 0} as non-interacting and {\\bf 1} $\\leftrightarrow$ {\\bf 2}, {\\bf 3} $\\leftrightarrow$ {\\bf 4}, etc. as interacting pairs.\nHowever, these interaction rules can take any arbitrarily complex form provided interactions are bidirectional.\nInteractions are infinite in strength, meaning two tiles bind irreversibly if the adjoined interfaces are interacting.\nThere is an infinite population of each tile type, precluding any stoichiometric limitations.\n\n\n\\subsection{Definitions of structure and determinism}\n\nA structure is defined as a set of connected tiles, each with an associated tile type, orientation, and unique lattice coordinates.\nStructure determinism can be defined in three distinct categories, listed in increasing order of strictness: shape, tile, and orientation determinism.\nStructures are defined independently of absolute position and rotation, and so lattice coordinate translations or rotations of entire structures are considered indistinguishable (structures are {\\em one-sided} polyominoes).\nAn example of a tile set and its assembled polymino is shown in Figure \\ref{determinism}.\n\n{\\bf Shape determinism} requires all produced structures to be the same one-sided polyomino.\n\n{\\bf Tile determinism} additionally requires corresponding coordinates between two structures to have matching tile types.\n\n{\\bf Orientation determinism} further requires matching relative inter-tile orientations.\n\nOrientation determinism is standard choice due to the impact that tile type and relative orientation could have on the function of the assembled structure. \n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{determinism}\n  \\caption{A tile set (left) and its assembled polyomino (right).\n    Each tile type in the set can be assigned a number, which is used to label the tile type in the polyomino.\n    Rotating the label indicates the corresponding rotation of the tile during assembly.}\\label{determinism}\n\\end{figure}\n\n\\section{Assembly graphs}\\label{sec:AG}\nAssembly tile sets are represented using the notation $\\{\\text{tile type 1}, \\text{tile type 2}, \\ldots \\}$, where tile type ordering is irrelevant.\nTile types have cyclic symmetry and have a length fixed by the geometry, so square tiles have four faces and can be represented as $\\left( F_1, F_2,F_3,F_4 \\right)$, where $F_i$ indicates the interface type of face $i$.\nWe use the convention that faces are encoded clockwise starting from the top.\nAn example of this notation can be found in the caption of Figure \\ref{assemblygraphdemo}.\n\nThe {\\bf assembly graph} of a tile set is constructed with nodes for each face on every tile type.\nThe faces within each tile type form cliques with edges labeled as internal, while interactions between interfaces (both inter- and intra-tile) are encoded with edges labeled as external. \nHence an assembly graph can be represented using an edge-labeled pseudograph (multigraph with loops).\nA detailed assembly graph example is shown in Figure \\ref{assemblygraphdemo}.\nInternal edges only depend on geometry, and are not displayed in future examples for clarity.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.475\\textwidth]{graph_construction}\n  \\caption{(a) The building blocks for the tile set $\\{\\left(1,3,0,0\\right),\\left(2,5,0,4\\right),\\left(0,0,0,6\\right)\\}$ explicitly labeled with interfaces and the assembled structure.\n  (b) The corresponding complete assembly graph with black (dashed) internal edges, and gray (solid) external edges.}\\label{assemblygraphdemo}\n\\end{figure}\n\n\\subsection{Assembly graph terminology}\nAssembly graphs contain three features of interest: single interacting faces, branching points, and cycles.\nFigure \\ref{assemblygraph} shows examples of partial assembly graphs with such features.\n\n{\\bf Single interacting face} (SIF) tiles are tile type which have only a single face with one or more external edges.\n\n{\\bf Branching points} occur when a given face has multiple external edges.\nNon-SIF branching points occur if a tile type has a branching point and at least one other face with external edges.\nNon-SIF branching points frequently cause nondeterminism due to diverging assembly pathways.\n\n{\\bf Cycles} have two varieties with identical impact on assembly classification: inter- and intra-tile.\nInter-tile cycles are walks on the assembly graph which alternate stepping on external and internal edges.\nIntra-tile cycles are walks of only a single step on an external edge connecting to the same tile.\nCycles are the primary source of unboundedness, due to the potential to endlessly traverse (and thus assemble) around the cycle.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.475\\textwidth]{graph_features}\n  \\caption{Five partial assembly graphs demonstrating various features.\n  (a) SIF tile (b) SIF branching point (c) non-SIF branching point (d) two intra-tile cycles (e) inter-tile cycle.}\\label{assemblygraph}\n\\end{figure}\n\n\\subsection{Graph connectivity}\nIf an assembly graph contains disconnected components, it is trivially nondeterministic due to seed dependence.\nThis is due to the inability to assemble between disconnected components (by definition), and hence the structures formed by seeds in different components are nondeterministic.\n\nHowever, each connected component can be assessed independently for determinism.\nBiologically, this can be related to genotypes encoding multiple independent phenotypes.\n\n\\subsection{Fundamental Deterministic Graph}\\label{subsec:FDG}\n\nA {\\bf treelike} assembly graph is one without any cycles or non-SIF branching points.\nThe simplest treelike graph is a single tile with no interactions, and is evidently bound and deterministic.\nAdding a SIF tile (potentially SIF branching point) with a new interaction pair to this assembly graph cannot introduce a non-SIF branching point or cycle by definition.\nSince these are the only sources of nondeterminism and unboundedness, the assembly classification consequently cannot be altered.\nBy induction, any treelike graph is therefore bound and deterministic.\n\nAs such, any assembly graph that can be transformed into a treelike graph will also be bound and deterministic.\nWe now focus on the procedures of reducing an arbitrarily complex assembly graph to this deterministic state, or if this is not possible, identifying the source of nondeterminism or unboundedness.\n\nSequentially, we prune SIF tiles and check if the assembly graph is treelike.\nIf we cannot classify the assembly graph at this stage, we analyze the nature of any cycles present and remove trivial cycles, and examine the remaining cycles for unbound behavior.\nAlthough the graph approach identifies sources of nondeterminism and unboundedness, it is unable to predict spatial conflicts that lead to steric nondeterminism, a limitation discussed later.\n\n\\section{Alternative treelike assembly procedures}\\label{sec:ATAP}\n\nExplicitly transforming assembly graphs to be treelike is unnecessary; the steps of SIF tile pruning to remove trivial branching points and checking for infinite cycles is sufficient to determine the assembly classification.\n\nAssembly graphs which cannot undergo or fail the above procedures are necessarily nondeterministic or unbound.\nFor instance, any assembly graph with multiple of each complementary interface, i.e.\\ branching points connected to branching points, cannot be made treelike and thus is nondeterministic.\n\n\n\\subsection{SIF tiles elimination procedure} \\label{subsec:SEP}\nSIF tiles, by converse reasoning to that of the fundamental deterministic graph in Section \\ref{subsec:FDG}, can be `pruned' from an assembly graph without altering its assembly classification.\nThis is done by removing a SIF tile from the assembly graph, and neutralizing its complementary interface (or interfaces if it's a SIF branching point).\nSIF tiles whose complementary interface is a branching point are nondeterministic and may be classified without pruning.\n\nThis procedure is applied iteratively, and if a treelike graph is obtained after a removal, the assembly graph is known to be bound and deterministic.\nExamples of the procedure is shown in Figure \\ref{pruning}.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.425\\textwidth]{SIF_elimination}\n  \\caption{The SIF elimination procedure applied iteratively on two assembly graphs.\n    The `1' SIF tile may be pruned in both (a) and (b), but only in (b) does the branching point become a SIF and allow further pruning.\n    Note that the second stage of (b) is already treelike (no cycles or non-SIF branching points) and thus classified.\n    Since (a) cannot be reduced to treelike, the assembly graph cannot be classified yet.}\\label{pruning}\n\\end{figure}\n\n\n\\subsection{Cycle rank classification}\n\nCycles can be categorized by the number of times the cycle pattern is repeated during the assembly, a quantity known as the rank.\nThe rank of the cycle can be determined from a single traversal of the cycle and noting the net rotation accumulated at the end of the walk.\nFigure \\ref{cycle_rank} shows several examples of cycles and their rank classification.\n\nFor square geometry, there are 4 periodic net rotations to consider, $\\theta=n \\pi \/ 2$ with $n \\in \\{-1,0,1,2\\}$. \nRank 4 cycles occur from $\\lvert n \\rvert=1$, while $n=2$ produces rank 2 cycles.\nThere are two subcategories for $n=0$, rank $\\infty$ and rank 1, based on spatial considerations.\n\nIf after traversing the cycle once, there is a zero net spatial translation, i.e.\\ the final tile placed reattaches in real-space to the first tile, then the cycle is rank 1.\nBy contrast, if there is a nonzero net spatial translation and the final tile placed leaves the final interface exposed, then the cycle will grow ad infinitum and is rank $\\infty$.\n\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.475\\textwidth]{cycle_ranks}\n  \\caption{The four cycle ranks possible with square geometry with assembly graphs on the left and assembled structures on the right in each example.\n    (a) rank 2 intra-tile cycle ($\\Delta \\theta=\\pi$) (b) rank 4 inter-tile cycle ($\\Delta \\theta=-\\pi\/2$) (c) rank $\\infty$ inter-tile cycle ($\\Delta \\theta=\\infty, \\Delta_{XY} \\neq 0$) (d) rank 1 inter-tile cycle ($\\Delta \\theta=\\infty, \\Delta_{XY} = 0$).\n    As soon as the initially placed tile is reused, the cycle rank may be classified, but for completeness the remaining assembly is shown in gray.\n}\\label{cycle_rank}\n\\end{figure}\n\n\n\\subsection{Establishing boundedness}\n\nCycles of rank 1, like SIFs, contribute trivial assembly behavior, and can be simplified without interfering with assembly classification.\nAny edge in the cycle may be removed, even if an edge is shared with another cycle.\nRegardless of the choice of cut, the resulting assemblies graphs will have the same quantity and rank of surviving cycles.\n\nThis procedure is likewise applied iteratively along with SIF elimination until the assembly graph is maximally simplified.\nMore details are given in Appendix \\ref{ap:cycle}.\nIf at any stage a rank $\\infty$ cycle is discovered, growth is immediately known to be unbound.\nMoreover, multiple surviving cycles ordinarily exhibit unboundness due to their amalgamated assembly pattern.\n\nSimplistically, this can be understood as the regeneration of cycle interfaces.\nWhen one cycle completes assembling, interfaces that assemble the other cycles are exposed and available for further growth.\nThis is demonstrated in Figure \\ref{pattern}.\n\nMultiple surviving cycles can have rank 1 behavior rather than rank $\\infty$, although it's rare due to the specific spatial constraints required.\nThis is detailed in Appendix \\ref{ap:cycle} for the specific case of Figure \\ref{pattern}.\n\nEstablishing the (in)finite behavior of the surviving cycles is hence nuanced, and proper care must be taken to identify infinite cycle behavior.\nFundamentally, infinite behavior can be identified if a tile in a cycle of finite rank $R$ is ever used in assembly more than $R$ times. \n\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.375\\textwidth]{cycle_regen}\n  \\caption{A section of an unbound structure resulting from an assembly graph with two rank 4 cycles.\nEvery time the blue (plain) cycle is completed, faces which interact with the red-hued (hatched) cycle are left exposed.\nA completion of the red-hued cycle likewise allows new copies of the blue cycle to start assembling, leading to unbound growth.}\\label{pattern}\n\\end{figure}\n\n\n\\section{Complete Procedure}\\label{sec:CP}\n\n\\subsection{Steric effects}\nIn the preceding sections we have addressed whether the assembly interactions alone make a tile set (non)deterministic or (un)bound.\nHowever, steric effects can also impact the growth of structures and cause nondeterminism and boundedness in assembly graphs that are otherwise deterministic or unbound.\n\nFor this reason a steric validation must be performed by growing the structure on the real-space lattice once.\nIf any lattice coordinate is used in assembly multiple times, then the assembly is sterically nondeterministic.\nSuch steric nondeterminism is clearly illustrated in Figure \\ref{mismatched}.\n\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.35\\textwidth]{steric}\n  \\caption{This assembly graph is rule-deterministic, but fails steric validation.\n    Growing the structure on the real-space lattice reveals steric nondeterminism as the hatched lattice site does not have a unique occupant.\n    This leads to nondeterminism as the final structure would depend on which occupant was built first.}\\label{mismatched}\n\\end{figure}\n\nSteric nondeterminism can prompt novel behavior in self-limiting cluster growth.\nIn previous experimental work \\cite{TES2016}, single-seed and multi-seed self-assembly were compared, observing that complementary pair interactions can limit growth in multi-seed assembly through local steric effects.\nSuch phenomenon is beyond the scope of this framework currently, but remains an topic for further analysis.  \n\n\\subsection{Geometric symmetry}\nThe presence of symmetric tiles will never impact classifying a bound and deterministic assembly, but may mistake unbound growth for nondeterminism due to the symmetry-induced branching points.\nThese spurious nondeterministic branching points can be replaced with alternative interfaces that preserve assembly classification through a desymmetrization procedure (see Appendix \\ref{ap:sym} for details).\n\n\n\n\\subsection{Analysis Sequence}\n\nThe flowchart in Figure \\ref{flow} illustrates the steps in determining the (un)bound and (non)deterministic nature of a self-assembling tile set. \nAn open-source implementation of the assembly graph and steric algorithms is available online\\cite{LeonardGithub}. \n\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{flowchart}\n  \\caption{The sequential analysis of an assembly graph.\nClassifications can be bound and rule-deterministic (BD), unbound rule-deterministic (UD), or nondeterministic (ND).\nSteric validation must still be performed to ensure determinism.\nNote disconnected graphs are trivially seed-dependent, but each connected component may undergo this analysis.}\\label{flow}\n\\end{figure}\n\n\n\\section{Framework extensions}\\label{sec:FE}\nThe modular nature of the assembly graph algorithms allows the pinpointing of unbounded and nondeterministic behavior.\nRelaxing the conditions which detect the above classifications allows examining a wider range of assembly conditions and constraints.\n\n\\subsection{Other dimensions and geometries}\nThe procedures introduced here directly extend to regular triangular and hexagonal geometries, which allow regular tilings of the plane\\cite{1977}. \nThe only conceptual modification is the $\\theta$-shifts in cycle rank classification.\nTriangular geometry allows finite cycles of rank 1, 2, and 6, while hexagonal geometry allow ranks of 1, 2, 3, 4, and 6. \nSome cycles examples for regular hexagons are presented in Figure \\ref{exo}.\n\nThe procedures can also be extended to other dimensions, again only modifying cycle rank classification.\nFor instance, cubic geometry still only supports finite cycles ranks of 1, 2, and 4, while in one dimension only rank 2 finite cycles are possible.\nAlthough identifying infinite cycle behavior, steric validation, and other mentioned procedures are more complicated to implement, the concepts are unaltered.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.35\\textwidth]{hexagonal}\n  \\caption{Hexagonal geometry allows for additional cycle ranks.\nA rank 3 and rank 2 cycle and the resulting structures are shown on the left and right respectively.}\\label{exo}\n\\end{figure}\n\n\\subsection{Nonstatic interaction rules}\nVarious biological assemblies do not the follow the earlier definition of determinism, due to the ability to form different final structures depending on external conditions, known as {\\em phenotype plasticity}.\nSuch external conditions can considerably impact self-assembly of biomolecules\\cite{deamer2006self,lavelle2009phase}.\nThis framework can encapsulate plasticity by considering e.g.\\ high pH interaction rules and low pH interaction rules, where the choice of rules determines which interactions are active.\nThis in turn can yield different deterministic structures.\n\nLikewise, it is possible to consider sequentially active interaction rules in a single assembly.\nAn assembly graph constructed under each set of interaction rules can be analyzed for (un)boundness and (non)determinism.\nThe assembly is then validated under steric constraints by assembling under the first set of interaction rules, followed by using the next set of rules on the existing structure etc.\nAs such, this approach can generalize to more complex assembly environments.\n\n\\subsection{Seed dependence}\nAlthough seed independence is ingrained in the method as outlined, incorporating fixed seed assembly allows for greater diversity in assembled structures.\nSeed dependence primarily results from branching points, as the behavior of SIF tiles and cycles are independent of the order of assembly.\n\nOne possible realization for modifying seed dependence is to walk on the assembly graph depth-first away from the seed.\nBranching points where a ``branched to'' face (i.e.\\ multiple faces with this interface type exist in the set) is discovered before the complementary ``branching'' face (i.e.\\ only face with this interface type in the set) can be ignored.\nIntuitively this removes the nondeterminism, as the assembly never encounters the multiple pathways when growing from those seeds.\nAn illustration of this extension is shown in Figure \\ref{seeded}.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{seed_dependence}\n  \\caption{An assembly graph that is classified as nondeterministic.\n    However, if the assembly is seeded with tile type `2' or `3'\/`4', the structures are deterministic.\n    The nondeterministic behavior of the branching point is only encountered walking from the `1' seed.}\\label{seeded}\n\\end{figure}\n\n\n\\section{Discussion}\\label{sec:D}\n\n\n\\subsection{Advantages of the assembly graph framework}\nThe stochastic assembly method suffers from the probabilistic detection of unboundness and nondeterminism.\nClassification accuracy is directly related to the number of repeated assemblies, $K$, which compromises speed.\nFor example, stochastic assembly can miss the unbounded and nondeterministic nature of the tile set $\\{\\left(1,0,2,0\\right), \\left(1,2,0,0\\right)\\}$ even with seed independence.\nThe misclassification probability for this tile set has an analytic form for $K$ repeated assemblies given by\n\\[\n\\Pr = \\sum\\limits_{N=1}^{\\infty}[N (1\/2)^{3+2N}]^K\n\\]\nwhere $N$ is the number of pairs of the infinite cycle $\\left(1,0,2,0\\right)$ tiles in the assembly.\nFor $K=10$, the probability of misclassification is inconsequential ($10^{-15}$).\nHowever, more complex tile sets or weaker determinism definitions can lead to non-negligible loss of accuracy.\nAssembling $\\{\\left(1,2,0,0\\right), \\left(1,0,0,0\\right)\\}$ with the less restrictive shape determinism has a misclassification rate of $\\left(7\/8\\right)^{K\/2}$, over 50\\% for $K=10$.\n\nThe form of stochastic assembly algorithm (an example of which is sketched out in Section \\ref{intro}) also impacts probabilistic detection.\nIn the immediately preceding tile set example, randomly drawing a new tile gives equal probability to the two tile types.\nHowever, the $\\left(1,2,0,0\\right)$ tile type has two potential bindings, and so could also be considered to have twice the probability to be selected over the other tile type.\nDepending on the implementation choice, the misclassification rate varies between $\\left(19\/27\\right)^{K\/2}$ and $\\left(7\/8\\right)^{K\/2}$.\nThe graph method removes the need for external parameters and model specifications inherent in stochastic assembly.\n\nDue to the exponential growth of configuration space, only one, two, and three tile systems have been exhaustively searched.\nHowever, assembly graphs up to 20 tiles were tested with $10^{11}$ samples per graph size ($N_T$).\nAll sampled tile sets exhibited no misclassifications when compared with the ``truth'' outcomes of sufficiently reliable ($K=50 N_T$) stochastic assemblies.\nA direct comparison of the two methods' speeds is shown in Figure \\ref{fig:spm} for connected assembly graphs.\nDetails on the simulations used can be found in Appendix \\ref{ap:speed}.\n\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.475\\textwidth]{speed_comparison}\n\n  \\caption{Elapsed CPU time (s) per million connected assembly graphs classified for the graph approach and stochastic assembly with $K=10$.\nBoth methods initially scale with $N_T$ similarly, although with the graph approach approximately 3 times faster. \nHowever, for larger $N_T$ the stochastic method scales superlinearly while the graph framework scaling remains approximately linear.}\\label{fig:spm}\n\n\\end{figure}\n\n\n\\subsection{Nondeterministic assembly dynamics}\nIn \\cite{ARXIV2015}, nondeterministic self-assembly of single lattice tiles and mixtures of two tiles at varying relative concentrations were studied in detail.\nIn addition to using the conventional interaction rules introduced initially, assembly was also examined using {\\bf 1}s and {\\bf 2}s as self-interacting.\nTile sets were classified in terms of their behavior upon variation of tile density, recognizing critical transitions from bound to unbound growth and noncritical density transitions.\n\nThese dynamics can be analyzed from a different perspective, examining the 106 topologically distinct tile sets with the assembly graph framework.\nWhile identifying deterministic assemblies is relatively unchanged, the ability to robustly distinguish unbound from nondeterministic behavior is greatly improved using the assembly graph framework.\n  \n\n\\subsection{Genotype-phenotype maps}\n\nThe lattice self-assembly model described here has been used as an abstract model of the genotype-phenotype (GP) map of protein quaternary structure \\cite{green}.\nIn these models, tile set is the genotype and the assembled structure represents the phenotype.\nThis work focused on bound deterministic assembly, while unbound or nondeterministic building block sets were regarded as a single unfavorable phenotype, and largely ignored.\n\nThe assembly graph framework allows assembly graphs to be used as an intermediate link between genotypes and phenotypes, and thus extend such work to examine the GP map of nondeterministic tile sets.\n\n\n\\subsection{Application to real proteins}\nThe study of such nondeterministic GP maps is essential, as nondeterministic self-assembly in the form of protein aggregation is a hallmark of numerous diseases.\nA classic example is that of hemoglobin and the sickle cell mutant.\n\nIn this representation, the $\\alpha$ and $\\beta$ chains of wild-type hemoglobin Hb A can be mapped to the $\\{ \\left(1, 3, 0, 0\\right), \\left(2, 0, 0, 4\\right)\\}$ tile set, and the sickle-cell mutant Hb S can be mapped to $\\{\\left(1, 3, 0, 0\\right),\\left(2, 5, 6, 4\\right)\\}$, displayed in Figure \\ref{GGP}.\nThe sickle-cell point mutation is sufficient to introduce an interaction labeled here with 5 and 6.\nThe wild-type assembly graph possesses a cycle of rank 2, and the mutation in the second tile introduces a new cycle of rank 4.\nFollowing the methodology we have outlined, this is sufficient to identify the unbound deterministic growth giving rise to the sickle cell anemia disease.\n \nBiological structures that are assembled through cycles with rank greater than 1 are prone to unbound growth via mutations which introduce additional cycles. \nProtein misfolding and resulting changes of interface angles in quaternary structures can be recast as introductions of cycles or branching points.\n\nWhile many proteins obviously exhibit more complex geometries than this framework is suited to, examining coarse-grained quaternary structure has already revealed insights into assembly properties and evolution history of proteins \\cite{Levy:2006ez,ahnert2015principles}.\nCombining these protein ``motifs'' with the introduced assembly graph formalisms is a rich vein for future exploration.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.4\\textwidth]{sickle_cell}\n  \\caption{The wild-type (left) and sickle-cell mutant (right) of the hemoglobin complex has been modelled previously using polyominoes\\cite{green}.\n    The unbound growth denoted by arrows resulting from the mutation can now succinctly be pinpointed to the addition of a higher order cycle to the assembly graph.}\\label{GGP}\n\\end{figure}\n\n\\subsection{Universal computing}\nThe motivations behind this method are unassociated with those of universal computing and Turing machines, but there is conceptual overlap with other tile assembly models (TAMs).\nThe interaction rules used here define {\\em noncooperative} bindings, meaning the interaction is independent of factors beyond the two adjoined faces.\nSuch noncooperative models have been demonstrated to not intrinsically allow universal computing\\cite{meunier2017non}, requiring significant generalizations to the TAMs to increase their computational ability\\cite{hendricks2016doubles}.\n\nAs such, the framework outlined here for determining boundedness and determinism of self-assembling tile sets has little bearing on universal computation and the infamous halting problem.\n  \n\\section{conclusion}\n\nWith the assembly graph framework we have introduced a new approach that can determine the (un)boundedness and (non)determinism of self-assembling tile sets.\nWhile the results presented have focused on 2D square geometry, the assembly framework readily extends to other dimensionalities (1D or 3D) and other geometries which regularly tile the plane. \n\nThe graph based approach outperforms the existing stochastic assembly approach in both speed and accuracy, facilitating the study of significantly larger GP maps and evolutionary dynamics.\nExternal parameter dependencies, like the $K$ repeated assemblies, are also eliminated.\n\nThe topological nature of the assembly graphs also opens new options for describing complexity and other properties relevant to evolutionary dynamics.\nIn addition, this methodology is a powerful tool for generalizing the study of GP maps by extending to the nondeterministic realm, which yields a coarse-grained model relevant to dysfunction and disease in the context of biological self-assembly.\n\n\\begin{acknowledgments}\nThis work was supported by the Engineering and Physical Sciences Research Council (ST and ASL), the Gatsby Foundation (ASL and SEA), and the Royal Society (SEA). \nThe authors would like to thank the referees for extensive feedback and suggestions on this work.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nUtterance and document level emotion recognition has received significant attention from the research community \\cite{SemEval2018Task1,poria2020beneath}.  Given the utterance \\emph{Sudan protests: Outrage as troops open fire on protestors} an emotion recognition system will be able to detect that \\emph{anger} is the main expressed emotion, signaled by the word \"outrage\".  However, the semantic information associated with expressions of emotion, such as the cause (the thing that triggers the emotion) or the target (the thing toward which the emotion is directed), is important to provide a finer-grained understanding of the text that might be needed in real-world applications. In the above utterance, the cause of the anger emotion is the event ``troops open fire on protestors'', while the target is the entity \"troops\" (see \\autoref{fig:goodnewseveryone}). \n\nResearch on finer-grained emotion analysis such as detecting the cause for an emotion expressed in text is in its infancy. Most work on emotion-cause detection has utilized a Chinese dataset where the cause is always syntactically realized as a clause and thus was modeled as a classification task \\citep{gui-etal-2016-event}. However, recently \\citet{DBLP:conf\/lrec\/BostanKK20} and \\citet{oberlander-klinger-2020-token} argued that in English, an emotion cause can be expressed syntactically as a clause (\\textit{as troops open fire on protestors}), noun phrase (\\textit{1,000 non-perishable food donations}) or verb phrase (\\textit{jumped into an ice-cold river}), and thus we follow their approach of framing emotion cause detection as a sequence tagging task. \n\nWe propose several ways in which to approach the tasks of emotion recognition and emotion cause tagging. First, these two tasks should not be independent; because the cause is the trigger for the emotion, knowledge about what the cause is should narrow down what emotion may be expressed, and vice versa. Therefore, we present a multi-task learning framework to model them jointly. Second, considering that common-sense knowledge plays an important role in understanding implicitly expressed emotions and the reasons for those emotions, we explore the use of common-sense knowledge via adapted knowledge models  (COMET, \\citet{DBLP:conf\/acl\/BosselutRSMCC19}) for both tasks. A key feature of our approach is to combine these adapted knowledge models (i.e., COMET), which are specifically trained to use and express common-sense knowledge, with pre-trained language models such as BERT, \\citep{DBLP:conf\/naacl\/DevlinCLT19}. \n\nOur primary contributions are three-fold: (i) an under-studied formulation of the emotion cause detection problem as a sequence tagging problem; (ii) a set of models that perform the emotion classification and emotion cause tagging tasks jointly while using common-sense knowledge (\\autoref{sec:multitask}) with improved performance (\\autoref{sec:results}); and (iii) analysis to gain insight into both model performance and the GoodNewsEveryone dataset that we use \\citep{DBLP:conf\/lrec\/BostanKK20} (\\autoref{sec:analysis}).\n\n\\section{Related Work}\n\nEmotion detection is a widely studied subfield of natural language processing \\citep{SemEval2018Task1,poria2020beneath}, and has been applied to a variety of text genres such as fictional stories \\citep{10.3115\/1220575.1220648}, news headlines \\citep{DBLP:series\/sci\/StrapparavaM10}, and social media, especially microblogs such as Twitter \\citep{abdul-mageed-ungar-2017-emonet,10.5555\/2693068.2693087,DBLP:journals\/corr\/abs-1912-02387,SemEval2018Task1}. Earlier work, including some of the above, focused on feature-based machine learning models that could leverage emotion lexicons \\citep{DBLP:journals\/corr\/MohammadT13}), while recent work explores deep learning models (e.g., Bi-LSTM and BERT) and multi-task learning \\citep{DBLP:journals\/corr\/abs-1809-04505,demszky-etal-2020-goemotions}.\n\n\n\nHowever, comparatively few researchers have looked at the semantic roles related to emotion such as the cause, the target or the experiencer, with few exceptions for Chinese \\cite{gui-etal-2016-event,chen-etal-2018-joint,DBLP:journals\/corr\/abs-1906-01267,DBLP:journals\/corr\/abs-1906-01236,fan-etal-2020-transition,wei-etal-2020-effective,ding-etal-2020-ecpe}, English \\cite{mohammad-etal-2014-semantic,Ghazi2015DetectingES,kim-klinger-2018-feels,DBLP:conf\/lrec\/BostanKK20, oberlander-etal-2020-experiencers,oberlander-klinger-2020-token} and Italian \\cite{russo-etal-2011-emocause}. \nWe highlight some of these works here and draw connection to our work. Most recent work on emotion-cause detection has been carried out on a Chinese dataset compiled by \\citet{gui-etal-2016-event}. This dataset characterizes the emotion and cause detection problems as clause-level pair extraction problem -- i.e., of all the clauses in the input, one is selected to contain the expression of an emotion, and one or more (usually one) are selected to contain the cause of that emotion. Many publications have used this corpus to develop novel and effective model architectures for the clause-level classification problem \\citep{chen-etal-2018-joint,DBLP:journals\/corr\/abs-1906-01267,DBLP:journals\/corr\/abs-1906-01236,fan-etal-2020-transition,wei-etal-2020-effective,ding-etal-2020-ecpe}. The key difference between this work and ours is that we perform cause detection as a sequence-tagging problem: the cause may appear anywhere in the input, and may be expressed as any grammatical construction (a noun phrase, a verb phrase, or a clause). Moreover, we use common sense knowledge for both tasks (emotion and cause tagging), through the use of adapted language models such as COMET. \n\n\nFor English, several datasets have been introduced \\cite{mohammad-etal-2014-semantic, kim-klinger-2018-feels,Ghazi2015DetectingES,DBLP:conf\/lrec\/BostanKK20,poria2020recognizing}, and emotion cause detection has been tackled either as a classification problem \\cite{mohammad-etal-2014-semantic}, or as a sequence tagging or span detection problem \\cite{kim-klinger-2018-feels,Ghazi2015DetectingES,oberlander-klinger-2020-token,poria2020recognizing}. We particularly note the work of \\citet{oberlander-klinger-2020-token}, who argue for our problem formulation of cause detection as sequence tagging rather than as a classification task supported by empirical evidence on several datasets including the GoodNewsEveryone dataset \\cite{DBLP:conf\/lrec\/BostanKK20} we use in this paper. One contribution we bring compared to these models is that we formulate a multi-task learning framework to jointly learn the emotion and the cause span. Another contribution is the use of common-sense knowledge through the use of adapted knowledge models such as COMET (both in the single models and the multi-task models). \\newcite{ghosal-etal-2020-cosmic} have very recently shown the usefulness of common-sense reasoning to the task of conversational emotion detection. \n\n\\section{Data}\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.5]{annotation_example_ours}\n\t\\end{center}\n\t\\caption{An example of the semantic roles annotated by \\citet{DBLP:conf\/lrec\/BostanKK20}}\n\t\\label{fig:goodnewseveryone}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\includegraphics[scale=0.5]{label_distribution}\n\t\\caption{Distribution of adjudicated emotion labels in the GoodNewsEveryone train data, as a percentage of the data points. ``Positive'' and ``Negative'' are abbreviated as + and -.}\n\t\\label{fig:label-distribution}\n\\end{figure}\n\n\\begin{figure*}\n    \\begin{subfigure}[t]{0.5\\textwidth}\n        \\centering\n        \\includegraphics[scale=0.35]{causefirst_nolstm_new.PNG}\n        \\caption{The $\\text{Multi}_{C \\shortrightarrow E}$ model.} \\label{fig:causefirst-arch}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.5\\textwidth}\n        \\centering\n         \\includegraphics[scale=0.36]{emofirst_nolstm_new.PNG}\n        \\caption{The $\\text{Multi}_{E \\shortrightarrow C}$ model.} \\label{fig:emofirst-arch}\n    \\end{subfigure}\n    \\caption{Our multi-task models.}\n\\end{figure*}\n\n\nFor our experiments, we use the GoodNewsEveryone corpus \\citep{DBLP:conf\/lrec\/BostanKK20}, which contains 5,000 news headlines labeled with emotions and semantic roles such as the target, experiencer, and cause of the emotion, as shown in \\autoref{fig:goodnewseveryone}.\\footnote{While the dataset labels both the most dominant emotion expressed in text and the reader's emotion, for this paper we only focus on the former.}  \nWe focus on the emotion detection and cause tagging tasks in this work. To our knowledge, GoodNewsEveryone is the largest English dataset labeled for both of these tasks.\n\nIn our experiments, we limit ourselves to the data points for which a cause span was annotated (4,798). We also note that this dataset uses a 15-way emotion classification scheme, an extended set including the eight basic Plutchik emotions as well as additional emotions like \\textit{shame} and \\textit{optimism}. While a more fine-grained label set is useful for capturing subtle nuances of emotion, many external resources focus on a smaller set of emotions. We also note that the label distribution of this dataset heavily favors the more basic emotions, as shown in \\autoref{fig:label-distribution}. Therefore, for our work, we choose to limit ourselves to the six Ekman emotions (\\textit{anger}, \\textit{fear}, \\textit{disgust}, \\textit{joy}, \\textit{surprise}, and \\textit{sadness}). We also choose to keep \\textit{positive surprise} and \\textit{negative surprise} separated, to avoid severely unbalancing the label distribution for our experiments. We randomly split the remaining data (2,503 data points) into 80\\% train, 10\\% development, and 10\\% test.\n\n\\section{Models} \\label{sec:models}\n\nAn important feature showcased by the GoodNewsEveryone dataset is that causes of emotions can be expressed through different syntactic constituents such as clauses, verb phrases, or noun-phrases. Thus, we approach the cause detection problem as a sequence tagging problem using the IOB scheme \\citep{DBLP:journals\/corr\/cmp-lg-9505040}:\n$\\mathcal{C} = \\{\\text{I-cause}, \\text{O}, \\text{B-cause}\\}$. Our approach is supported by very recent results by \\citet{oberlander-klinger-2020-token} and \\citet{yuan-etal-2020-emotion} who show that modeling emotion cause detection as a sequence tagging problem is better suited than a clause classification problem, although not much current work has yet adopted this formulation.  We tackle the emotion detection task as a seven-way classification task with $\\mathcal{E} = \\{\\text{anger}, \\text{disgust}, \\text{fear}, \\text{joy}, \\text{sadness}, \\text{negative}$ $\\text{surprise}, \\text{positive surprise}\\}$. \n\n\\subsection{Single-Task Models} \\label{sec:singletask}\n\nAs a baseline, we train single-task models for each of emotion classification and cause span tagging. We use a pre-trained BERT language model\\footnote{We use \\textsc{BERT-base-uncased}. We experimented with \\textsc{BERT-base-cased}, but it underperformed as the headlines incorporated into GoodNewsEveryone come from different news sources and have different capitalization styles.}  \\citep{DBLP:conf\/naacl\/DevlinCLT19}, which we fine-tune on our data, as the basis of this model. Our preprocessing strategy for all of our models consists of the pretrained BERT vocabulary and WordPiece tokenizer\\footnote{In the tagging setting, we ignore all tags predicted for subword tokens and use only the tag of the first subword.} \\citep{DBLP:journals\/corr\/WuSCLNMKCGMKSJL16} from Huggingface \\citep{wolf-etal-2020-transformers}. Therefore, for a sequence of $n$ WordPiece tokens, our input to the BERT model is a sequence of $n + 2$ tokens, $X = [[\\text{CLS}], x_1, x_2, ... x_n, [\\text{SEP}]]$, where each $x_i$ is from a finite WordPiece vocabulary and [CLS] and [SEP] are BERT's begin and end tokens. Passing $X$ through BERT yields a sequence of vector hidden states $H = [h_{[CLS]}, h_1, h_2, ..., h_n, h_{[SEP]}]$ with dimension $d_{BERT} = 768$. For emotion classification, we pool these hidden states and allow hyperparameter tuning to select the best type: selecting the [CLS] token ($h_f = h_{[CLS]}$), mean pooling ($h_f = \\frac{\\sum_{i=1}^n h_i}{n}$), max pooling ($h_{f,j} = \\max{h_{i,j}}$), or attention as formulated by \\citet{DBLP:journals\/corr\/BahdanauCB14}:\n\n\\begin{equation} \\label{eqn:bahdanau-attn}\n\th_f = \\sum^{n}_{i=1} \\alpha_i h_i\n\\end{equation}\n\nwhere $\\alpha_i = \\frac{\\exp{(W_ah_i + b_a)}}{\\sum_{j=1}^n \\exp{(W_ah_j + b_a)}}$ for trainable weights $W_a \\in \\mathbb{R}^{1 \\times d_{BERT}}$ and $b_a \\in \\mathbb{R}^{1}$. Then, the final distribution of emotion scores is calculated by a single dense layer and a softmax: \n\n\\begin{equation} \\label{eqn:emotion}\n\te = \\text{softmax}(W_eh_f + b_e)\n\\end{equation}\n\nwith $e \\in \\mathbb{R}^{|\\mathcal{E}|}$ and for trainable parameters $W_e \\in \\mathbb{R}^{|\\mathcal{E}| \\times d_{BERT}}$ and $b_e \\in \\mathbb{R}^{|\\mathcal{E}|}$. For cause tagging, a tag probability distribution is calculated directly on each hidden state: \n\n\\begin{equation} \\label{eqn:cause}\n\tc_i = \\text{softmax} (W_ch_i + b_c)\n\\end{equation}\n\nwith $c_i \\in \\mathbb{R}^{|\\mathcal{C}|}$ and for trainable parameters $W_c \\in \\mathbb{R}^{|\\mathcal{C}| \\times d_{BERT}}$ and $b_c \\in \\mathbb{R}^{|\\mathcal{C}|}$. We refer to both of these single-task models as BERT; if the task is not clear from the context, we will refer to the emotion detection model as $\\text{BERT}_E$ and the cause tagging model as $\\text{BERT}_C$. Our training loss for emotion classification as well as emotion cause tagging is the mean negative log-likelihood (NLL) loss per minibatch of size \\textit{b}:\n\n\\begin{equation} \\label{eqn:emo-loss}\n\t\\text{NLL}_{\\text{emo}} = - \\frac{1}{b} \\sum_j \\sum_k y_{jk} \\log e_{jk}\n\\end{equation}\n\n\\begin{equation} \\label{eqn:cause-loss}\n\t\\text{NLL}_{\\text{cause}} = - \\frac{1}{b} \\sum_i \\sum_j \\sum_k y_{ijk} \\log c_{ijk}\n\\end{equation}\n\nwhere $j$ is the index of the sentence in the minibatch, $k$ is the index of the label being considered (emotion labels for $\\text{NLL}_{\\text{emo}}$ and IOB tags for $\\text{NLL}_{\\text{cause}}$), $i$ is the index of the $i^{th}$ token in the $j^{th}$ sentence in the minibatch, $y_{jk} \\in \\{0, 1\\}$ is the gold probability of the $k^{th}$ emotion label for the $j^{th}$ sentence, $y_{ijk} \\in \\{0, 1\\}$ is the gold probability of the $k^{th}$ cause tag for the $i^{th}$ token in the $j^{th}$ sentence, and $e_{jk}$ and $c_{ijk}$ are the output probabilities of the ${k^{th}}$ emotion label and of the $k^{th}$ cause label for the $i^{th}$ token, both for the $j^{th}$ sentence.\n\n\\subsection{Multi-Task Models} \\label{sec:multitask}\n\nOur hypothesis is that the emotion detection and cause tagging tasks are closely related and can inform each other; therefore we propose three multi-task learning models to test this hypothesis. For all multi-task models, we use the same base architecture (BERT) as the single models. Additionally, for these models, we combine the losses of both tasks and weight them with a tunable lambda parameter: $\\lambda \\text{NLL}_\\text{emo} + (1 - \\lambda) \\text{NLL}_\\text{cause}$, using $\\text{NLL}_\\text{emo}$ and $\\text{NLL}_\\text{cause}$ from \\autoref{eqn:emo-loss} and \\autoref{eqn:cause-loss}.\n\n\\paragraph{Multi.} The first model, $\\text{Multi}$, is the classical multi-task learning framework with hard parameter sharing, where both tasks share the same BERT layers. Two dense layers for emotion classification and cause tagging operate at the same time from the same BERT layers, and we train both of the tasks simultaneously. That is, we simply calculate our emotion scores $e$ and cause tag scores $c$ from the same set of hidden states $H$.\n\nWe further develop two additional multi-task models with the intuition that we can design more explicit and concrete task dependencies than simple parameter sharing in the representation layer.\n\n\\paragraph{$\\text{Multi}_{C \\shortrightarrow E}$.} We assume that if a certain text span is given as the cause of an emotion, it should be possible to classify that emotion correctly while looking only at the words of the cause span. Therefore, we propose the $\\text{Multi}_{C \\shortrightarrow E}$ model, the architecture of which is illustrated in \\autoref{fig:causefirst-arch}. This model begins with the single-task cause detection model, $BERT_C$, which produces a probability distribution  $P(y_i | x_i)$ over IOB tags for each token $x_i$, where $P(y_i | x_i) = c_i$ from \\autoref{eqn:cause}. Then, for each token, we calculate the probability that it is part of the cause as $P(\\text{Cause} | x_i) = P(B | x_i) + P(I | x_i) = 1 - P(O | x_i)$. We feed the resulting probabilities through a softmax over the sequence and use them as an attention distribution over the input tokens in order to pool the hidden representations and perform emotion classification: attention is computed as in \\autoref{eqn:bahdanau-attn}, where $\\alpha_i = \\frac{\\exp{P(\\text{Cause} | x_i)}}{\\sum_{j=1}^n \\exp{P(\\text{Cause} | x_i)}}$, and emotion classification as in \\autoref{eqn:emotion}. For the $\\text{Multi}_{C \\shortrightarrow E}$ model, we apply teacher forcing at training time, and the gold cause spans are used to create the attention weights before emotion classification (which means that $P(\\text{Cause} | x_i) \\in \\{0, 1\\}$). At inference time, the model uses the predicted cause span instead. \n\n\\paragraph{$\\text{Multi}_{E \\shortrightarrow C}$.} Next, we hypothesize that knowledge of the predicted emotion should help us identify salient cause words. The $\\text{Multi}_{E \\shortrightarrow C}$ model first performs emotion classification, which results in a probability distribution over predicted emotion labels, as in the $\\text{BERT}_E$ model and \\autoref{eqn:emotion}. We additionally keep an emotion embedding matrix $E$, where $E[i]$ is a learnable representation of the $i$-th emotion label (see \\autoref{fig:emofirst-arch}) with dimension $d_e$ (in our experiments, we set $d_e = 300$). We use the predicted label probabilities $e$ to calculate a weighted sum of the emotion embeddings, i.e., $M = \\sum_i e_i \\cdot E[i]$. We then concatenate $M$ to the hidden representation of each token and perform emotion cause tagging with a final dense layer, i.e., $c_i = \\text{softmax} (W_{c'}[h_i ; M] + b_{c'})$, where $;$ is the concatenation operator and $W_{c'} \\in \\mathbb{R}^{|\\mathcal{C}| \\times (d_{BERT} + d_e)}$ and $b_{c'} \\in \\mathbb{R}^{|\\mathcal{C}|}$ are trainable parameters. In the $\\text{Multi}_{E \\shortrightarrow C}$ model, we again do teacher forcing and use the gold emotion labels before doing the sequence tagging for cause detection (i.e., $e$ is a one-hot vector where the gold emotion label has probability 1 and all other emotion labels have probability 0). At inference time, the model will use the predicted emotion distribution instead.\n\n\\subsection{Adapted Knowledge Models}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[scale=0.35]{comet_causefirst_nolstm_new.PNG}\n\t\\caption{The architecture of our proposed $\\text{Multi}^{COMET}_{C \\shortrightarrow E}$ model.}\n\t\\label{fig:comet-causefirst-architecture}\n\\end{figure}\n\n\\begin{table*}[t!]\n\t\\centering\n\t\\begin{tabular}{l|r|r|r}\n\t\t& \\textbf{Emotion Macro F1} & \\textbf{Emotion Accuracy} & \\textbf{Cause Span F1} \\\\ \\hline\n\t\tBERT & 37.25 $\\pm$ 1.30 & 38.50 $\\pm$ 0.84 & 37.49 $\\pm$ 1.94 \\\\ \\hline \n\t\t$\\text{BERT}^{COMET}$ & 37.74 $\\pm$ 0.84 & 38.50 $\\pm$ 1.14 & 39.27 $\\pm$ 1.85 \\\\ \\hline\n\t\tMulti & 36.91 $\\pm$ 1.48 & 38.34 $\\pm$ 1.94 & 38.35 $\\pm$ 3.89 \\\\ \\hline\n\t\t$\\text{Multi}_{C \\shortrightarrow E}$ & 37.74 $\\pm$ 2.12 & 38.74 $\\pm$ 2.07 & 39.08 $\\pm$ 3.73 \\\\ \\hline\n\t\t$\\text{Multi}_{E \\shortrightarrow C}$ & 38.26 $\\pm$ 3.28 & 39.69 $\\pm$ 3.41 & 38.83 $\\pm$ 1.60 \\\\ \\hline\n\t\t$\\text{Multi}^{COMET}$ & 37.06 $\\pm$ 2.04 & 39.05 $\\pm$ 0.98 & \\textbf{39.50} $\\pm$ 2.25 \\\\ \\hline\n\t\t$\\text{Multi}_{C \\shortrightarrow E}^{COMET}$ & \\textbf{39.26}* $\\pm$ 1.13 & \\textbf{40.79} $\\pm$ 2.17 & 38.68 $\\pm$ 1.36 \\\\ \\hline\n\t\t$\\text{Multi}_{E \\shortrightarrow C}^{COMET}$ & 37.44 $\\pm$ 1.37 & 38.58 $\\pm$ 1.44 & 36.27 $\\pm$ 1.31 \\\\\n\t\\end{tabular}\n\t\\caption{The results of our models, averaged over five runs with the same five distinct random seeds. The model with the highest mean performance under each metric is bolded. Results marked with a * are statistically significant above the single-task BERT baseline by the paired t-test (p $<$ 0.05).}\n\t\\label{tab:results}\n\\end{table*}\n\nRecent work has shown that fine-tuning pre-trained language models such as GPT-2 on \\emph{knowledge graph tuples} such as ConceptNet~\\cite{li-etal-2016-commonsense} or ATOMIC \\citep{DBLP:journals\/corr\/abs-1811-00146} allows these models to express their implicit knowledge directly \\citep{DBLP:conf\/acl\/BosselutRSMCC19}. These adapted \\emph{knowledge models} (e.g., COMET \\citep{DBLP:conf\/acl\/BosselutRSMCC19}) can produce common-sense knowledge on-demand for any entity, relation or event.\nConsidering that common-sense knowledge plays an important role in understanding implicitly expressed emotions and the reasons for those emotions, we explore the use of common-sense knowledge for our tasks, in particular the use of COMET adaptively pre-trained on the ATOMIC event-centric knowledge base. ATOMIC's event relations include ``xReact'' and ``oReact'', which describe the feelings of certain entities after the input event occurs. For example, ATOMIC's authors present the example of $<$PersonX pays PersonY a compliment, xReact, PersonX will feel good$>$. xReact refers to the feelings of the primary entity in the event, and oReact refers to the feelings of others (in this instance, oReact yields ``PersonY will feel flattered''). For example, using the headline ``Sudan protests: Outrage as troops open fire on protestors\", COMET-ATOMIC outputs that PersonX feels justified, PersonX feels angry, Others feel angry, and so on (\\autoref{fig:comet-causefirst-architecture}). To use this knowledge model in our task, we modify our approach by reframing our single-sequence classification task as a sequence-pair classification task (for which BERT can be used directly). We feed our input headlines into COMET-ATOMIC (using the model weights released by the authors), collect the top two outputs for xReact and oReact using beam search decoding, and then feed them into BERT alongside the input headlines, as a second sequence using the SEP token. That is, our input to BERT is now $X = [[\\text{CLS}], x_1, x_2, ..., x_n, [\\text{SEP}], z_1, z_2, ..., z_m, [\\text{SEP}]]$, where $z_i$ are the $m$ WordPiece tokens of our COMET output and are preprocessed in the same way as $x_i$. We hypothesize that, since pre-trained BERT is trained with a next sentence prediction objective, expressing the COMET outputs as a grammatical sentence will help BERT make better use of them, so we formulate this second sequence as complete sentences (e.g., ``This person feels... Others feel...'') (\\autoref{fig:comet-causefirst-architecture}). \n \nThis approach allows us incorporate information from  COMET into all our single- and multi-task BERT-based models; the example shown in  \\autoref{fig:comet-causefirst-architecture} is our  $\\text{Multi}_{C \\shortrightarrow E}$ model. We refer to the COMET variants of these models as: $\\text{BERT}^{COMET}$ (single-task models) and $\\text{Multi}^{COMET}$, $\\text{Multi}^{COMET}_{C \\shortrightarrow E}$, $\\text{Multi}^{COMET}_{E \\shortrightarrow C}$ for the three multi-task models.\n\n\\section{Experimental Setup}\n\n\\paragraph{Evaluation Metrics}\nFor emotion classification, we report macro-averaged F1 and accuracy. For cause tagging, we report exact span-level F1 (which we refer to as \\textit{span F1}), as developed for named entity recognition (e.g., \\citet{tjong-kim-sang-de-meulder-2003-introduction}), where a span is marked as correct if and only if its type and span boundaries match the gold exactly\\footnote{Our cause tagging task has only one type, ``cause'', as GoodNewsEveryone is aggregated such that each data point has exactly one emotion-cause pair. We note that this problem formulation leaves open the possibility of multiple emotion-cause pairs.}.\n\n\\paragraph{Training and Hyperparameter Selection} The classification layers are initialized randomly from a uniform distribution over $[-0.07,0.07]$\\footnote{The default initialization from the \\texttt{gluon} package: \\url{https:\/\/mxnet.apache.org\/versions\/1.7.0\/api\/python\/docs\/api\/gluon\/index.html}}, and all the parameters are trained on our dataset for up to 20 epochs, with early stopping based on the performance on the validation data (macro F1 for emotion, span F1 for cause). All models are trained with the Adam optimizer \\citep{DBLP:journals\/corr\/KingmaB14}. We highlight again that for our $\\text{Multi}_{C \\shortrightarrow E}$ and $\\text{Multi}_{E \\shortrightarrow C}$  models, we use teacher forced during training to avoid cascading training error. Because the subset of the data we use is relatively small, we follow current best practices for dealing with neural models on small data and select hyperparameters and models using the average performance of five models with different fixed random seeds on the development set. We then base our models' performance on the average of the results from these five runs (e.g., reported emotion F1 is the average of the emotion F1 scores for each of our five runs). For our joint models, since our novel models revolve around using one task as input for the other, we separately tune two sets of hyperparameters for each model, one based on each of the single-task metrics, yielding, for example, one Multi model optimized for predicting emotion and one optimized for predicting cause. The hyperparameters we tune are dropout in our linear layers, initial learning rate of the optimizer, COMET relation type, lambda weight for our multi-task models, and the type of pooler for emotion classification (enumerated in \\autoref{sec:singletask}).\n\n\\section{Results} \\label{sec:results}\n\nWe present the results of our models in \\autoref{tab:results}\\footnote{\\citet{oberlander-klinger-2020-token} report an F1 score of 34 in this problem setting on this dataset, but on a larger subset of the data (as they do not limit themselves to the Ekman emotions) and so we cannot directly compare our work to theirs.}. We see that the overall best model for each task is a multi-task adapted knowledge model, with $\\text{Multi}_{C \\shortrightarrow E}^{COMET}$ performing best for emotion (which is a statistically significant improvement over BERT by the paired t-test, $p<0.05$) and $\\text{Multi}^{COMET}$ performing best for cause. These results seem to support our two hypotheses: 1) emotion recognition and emotion cause detection can inform each other and 2) common-sense knowledge is helpful to infer the emotion and the cause for that emotion expressed in text.\nSpecifically, we notice that $\\text{Multi}_{C \\shortrightarrow E}$ alone does not outperform BERT on either cause or emotion, but $\\text{Multi}_{C \\shortrightarrow E}^{COMET}$ outperforms both BERT and $\\text{Multi}_{C \\shortrightarrow E}$ on both tasks. For cause, we also see additional benefits of common-sense reasoning alone: $\\text{BERT}^{COMET}$ outperforms BERT (multi-task modeling alone, Multi, also outperforms BERT for this task) and $\\text{Multi}^{COMET}$ outperforms Multi. These results speak to the differences between the two tasks, suggesting that common-sense reasoning, which aims to generates implicit emotions, and cause information may be complementary for emotion detection, but that for cause tagging, common-sense reasoning and given emotion information may overlap. \nThe common-sense reasoning we have used in this task (xReact and oReact from ATOMIC) is expressed as possible emotional reactions to an input situation, so this makes intuitive sense.\n\n\\begin{figure}[t]\n    \\includegraphics[scale=0.55]{emotion_comparison_new.png}\n    \\caption{Performance of the BERT and $\\text{Multi}_{C \\shortrightarrow E}^{COMET}$ models on emotion classification.}\n    \\label{fig:emotion_breakdown}\n\\end{figure}\n\nFinally, we also present per-emotion results for our best model for each task ($\\text{Multi}_{C \\shortrightarrow E}^{COMET}$ for emotion and $\\text{Multi}^{COMET}$ for cause) against the single-task BERT baselines in \\autoref{fig:emotion_breakdown} and \\autoref{fig:cause_breakdown}; these per-emotion scores are again the average performance of models trained with each of our five random seeds. We see that each task improves on a different set of emotions: for emotion classification $\\text{Multi}_{C \\shortrightarrow E}^{COMET}$ consistently improves over BERT by a significant margin on joy and to a lesser extent on anger and sadness. Meanwhile, for cause tagging, $\\text{Multi}^{COMET}$ improves over BERT on anger, disgust, and fear, while yielding very similar performance on the rest of the emotions.\n\n\\section{Analysis and Discussion} \\label{sec:analysis}\n\nIn order to understand the impact of common-sense reasoning and multi-task modeling for the two tasks, we provide several types of analysis in addition to our results in \\autoref{sec:results}. First, we include examples of our various models' outputs showcasing the impact of our methods (\\autoref{sec:examples}). Second, we carry out an analysis  of the dataset, focusing on the impact of label variation among multiple annotators  on the models' performance (\\autoref{sec:label-analysis}).\n\n\\subsection{Example Outputs} \\label{sec:examples}\n\\begin{table*}[t]\n\t\\centering\n\t\\begin{tabular}{|c|c|c|c|}\n\t\t\\hline\n\t\t\\textbf{BERT}                                                                      & \\textbf{Multitask}                         & \\textbf{$\\text{BERT}^{COMET}$}                            & \\textbf{$\\text{Multitask}^{COMET}$}                         \\\\ \\hline\n\t\t\\multicolumn{4}{|c|}{\\begin{tabular}[c]{@{}c@{}}Mexico reels from \\hlcolor{shooting attack in El Paso}\\\\  \\textbf{\\textit{fear}}\\end{tabular}}                                                                                                                \\\\ \\hline\n\t\tnegative surprise                                                                  & negative surprise                          & fear                                      & fear                                             \\\\ \\hline\n\t\t\\multicolumn{4}{|c|}{\\begin{tabular}[c]{@{}c@{}}Insane video shows Viking Sky cruise \\hlcolor{ship thrown into chaos at sea}\\\\ \\textbf{\\textit{fear}}\\end{tabular}}                                                                                          \\\\ \\hline\n\t\tnegative surprise                                                                  & fear                                       & negative surprise                         & fear                                             \\\\ \\hline\n\t\t\\multicolumn{4}{|c|}{\\begin{tabular}[c]{@{}c@{}}Durant \\hlcolor{could return for Game 3}\\\\ \\textbf{\\textit{positive surprise}}\\end{tabular}}                                                                                                                           \\\\ \\hline\n\t\tfor game                                                                           & \\multicolumn{3}{|c|}{could return for game}                                                                                                 \\\\ \\hline\n\t\t\\multicolumn{4}{|c|}{\\begin{tabular}[c]{@{}c@{}}Dan Fagan: \\hlcolor{Triple shooting near New Orleans School yet another sign of city's crime problem}\\\\ \\textbf{\\textit{negative surprise}}\\end{tabular}}                                   \\\\ \\hline\n\t\t\\begin{tabular}[c]{@{}c@{}}school yet another sign \\\\ of city's crime\\end{tabular} & \\multicolumn{3}{|c|}{\\begin{tabular}[c]{@{}c@{}}: triple shooting near new orleans school yet another sign of city's \\\\ crime\\end{tabular}} \\\\ \\hline\n\t\\end{tabular}\n\t\\caption{Example outputs from our systems. For each example, the gold cause is highlighted in yellow and the gold emotion is given under the text; the first two examples give our models' emotion outputs; the latter two, their causes. Joined cells show that multiple models produced the same output. To make this table easier to read, ``Multitask'' here may refer to Multi, $\\text{Multi}_{E \\shortrightarrow C}$, or $\\text{Multi}_{C \\shortrightarrow E}$ (details on selection and results for each individual model available in appendix; most multi-task models gave similar outputs).}\n\t\\label{tab:examples}\n\\end{table*}\n\n\\begin{table*}[t]\n\t\\begin{adjustbox}{max width=\\textwidth}\n\t\t\\begin{tabular}{c|l|l|l|l|l|l|l|l}\n\t\t\t\\textbf{Metric} & \\textbf{BERT} & \\textbf{$\\text{BERT}^{COMET}$} & \\textbf{Multi} & \\textbf{$\\text{Multi}_{E \\shortrightarrow C}$} & \\textbf{$\\text{Multi}_{C \\shortrightarrow E}$} & \\textbf{$\\text{Multi}^{COM}$} & \\textbf{$\\text{Multi}_{E \\shortrightarrow C}^{COMET}$} & \\textbf{$\\text{Multi}_{C \\shortrightarrow E}^{COMET}$} \\\\ \\hline\n\t\t\t\\begin{tabular}[c]{@{}l@{}}\\textbf{Acc.}\\\\\\textbf{(Gold)}\\end{tabular} & 38.50 & 38.50 & 38.34 & 39.68 & 38.74 & 39.05 & 38.58 & 40.79 \\\\ \\hline\n\t\t\t\\begin{tabular}[c]{@{}l@{}}\\textbf{Acc.}\\\\\\textbf{($\\neg$Gold)}\\end{tabular}  & 23.48 & 23.24 & 22.37 & 21.11 & 22.85 & 21.26 & 22.45 & 20.08\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\caption{Comparison of gold accuracy and non-gold ($\\neg$gold) accuracy for our emotion classification models.}\n\t\\label{tab:annotator-comparison}\n\\end{table*}\n\nWe provide some example outputs from our systems for both cause and emotion in \\autoref{tab:examples}; the various Multi models have been grouped together for readability and because they often produce similar outputs, but the outputs for every model are available in the appendix. In the first example, the addition of COMET to BERT informs the model enough to choose the gold emotion label; in the third and fourth, either COMET or multi-task learning is enough to help the model select key words that should be included in the cause (\\textit{return} and \\textit{triple shooting}). We also particularly note the second example, in which multi-task learning is needed both for the BERT and $\\text{BERT}^{COMET}$ models to be able to correctly predict the gold emotion. This suggests that for cause, both common-sense reasoning and emotion classification may carry overlapping useful information for cause tagging, while for emotion, different instances may be helped more by different aspects of our models.\n\n\n\\subsection{Label Agreement} \\label{sec:label-analysis}\n\n\\begin{figure}[t]\n\t\\includegraphics[scale=0.55]{cause_comparison_new.png}\n\t\\caption{Performance of the BERT and $\\text{Multi}^{COMET}$ models on cause tagging, broken down by emotion.}\n\t\\label{fig:cause_breakdown}\n\\end{figure}\n\nUpon inspection of the GoodNewsEveryone data, we discover significant variation in the emotion labels produced by annotators as cautioned by the authors in their original publication\\footnote{While the authors selected data according to agreement on the emotion labeling task, they found that in only 75\\% of cases do at least 3 annotators agree, with diminishing rates of agreement for more annotators.}. \nFrom our inspection of the development data, we see recurring cases where different annotators give directly opposing labels for the same input, depending on how they interpret the headline and whose emotions they choose to focus on. For example, our development set includes the following example: \\textit{Simona Stuns Serena at Wimbledon: Game, Set and ``Best Match'' for Halep}. The gold adjudicated emotion label for this example is \\textit{negative surprise}, but annotators actually included multiple primary and secondary emotion labels including \\textit{joy}, \\textit{negative surprise}, \\textit{positive surprise}, \\textit{pride}, and \\textit{shame}, which can be understood as various emotions felt by the two entities participant in the event (Simona Halep and Serena Williams). For this input, COMET suggests xReact may be \\textit{happy} or \\textit{proud} and oReact may be \\textit{happy} --- these reactions are likely most appropriate for tennis player Simona Halep, but not the only possible emotion that can be inferred from the headline.\n\nInspired by the variation in the data, we compute also models' accuracy using the human annotations that did not agree with the gold (i.e., a predicted emotion label is correct if it was  suggested by a human annotator but was not part of a majority vote to be included in the gold). We denote this $\\neg$Gold, and we compare the performance of our models with respect to Gold and $\\neg$Gold. We present the results of this analysis in \\autoref{tab:annotator-comparison}\\footnote{Note that we perform this analysis on just one of our five runs of the model, so the accuracy numbers do not exactly correspond to those in \\autoref{tab:results}.}. \nIn this table, a higher $\\neg$Gold accuracy means that the model is more likely to produce emotion labels that were not the gold but were suggested by some annotator.  First of all, we note that all models have a relatively high $\\neg$Gold accuracy (about half the magnitude of their gold accuracy); we believe this reflects the wide variety of annotations given by the annotators. We see a tradeoff between the Gold and $\\neg$Gold accuracy, and we note that generally the single-task models have higher $\\neg$Gold accuracy and the COMET-enhanced multi-task models have higher Gold accuracy. This suggests that our language models have general knowledge about emotion already, but that applying common-sense knowledge helps pare down the space of plausible outputs to those that are most commonly selected by human annotators. Recall that this dataset was annotated by taking the most frequent of the annotator-provided emotion labels.\nFurther, since the multi-task models have higher Gold accuracy and lower $\\neg$Gold accuracy than the single-task models, this suggests that also predicting the cause of an emotion causes the model to narrow down the space of possible emotion labels to only those that are most common.\n\n\\section{Conclusions and Future Work}\n\nWe present a common-sense knowledge-enhanced multi-task framework for joint emotion detection and emotion cause tagging. Our inclusion of common-sense reasoning through COMET, combined with multi-task learning, yields performance gains on both tasks including significant gains on emotion classification.\nWe highlight the fact that this work frames the cause extraction task as a span tagging task, allowing for the future possibility of including multiple emotion-cause pairs per input or multiple causes per emotion and allowing the cause to take on any grammatical role. Finally, we present an analysis of our dataset and models, showing that labeling emotion and its semantic roles is a hard task with annotator variability,\nbut that common-sense knowledge helps language models focus on the most prominent emotions according to human annotators. In future work, we hope to explore ways to integrate common-sense knowledge more innately into our classifiers and ways to apply these models to other fine-grained emotion tasks such as detecting the experiencer or the target of an emotion.\n\n\\section*{Acknowledgements}\n\nWe would like to thank our reviewers as well as the members of Amazon AI team for their constructive and insightful feedback.\n\n\n\\section*{Ethical Considerations}\n\nOur intended use for this work is as a tool to help understand emotions expressed in text. We propose that it may be useful for things like product reviews (where producers and consumers can rapidly assess reviews for aspects of their products to improve or expand), disaster relief (where those in need of help from any type of disaster can benefit if relief agents can understand what events are causing negative emotions, during and after the initial disaster), and policymaking (where constituents can benefit if policymakers can see real data about what policies are helpful or not and act in their interests). These applications do depend on the intentions of the user, and a malicious actor may certainly misuse the ability to (accurately or inaccurately) detect emotions and their causes. We do not feel it responsible to publicly list the ways in which this may happen in this paper. We also believe that regulators and operators of this technology should be aware that it is still in its nascent stages and does not represent an infallible oracle --- the predictions of this and any model should be reviewed by humans in the loop, and we feel that general public awareness of the limitations and mistakes of these models may help mitigate any possible harm. If these models are inaccurate, they will output either the incorrect emotion or the incorrect cause; blindly trusting the model's predictions without examining them may lead to unfair consequences in any of the above applications (e.g., failure to help someone whose text is misclassified as positive surprise during a natural disaster or a worsened product or policy if causes are incorrectly predicted). We additionally note that in its current form, this work is intended to detect the emotions that are expressed in text (headlines), and not those of the reader.\n\n\nWe concede that the data used in this work consists of news headlines and may not be the most adaptable to the use cases we describe above; we caution that models trained on these data will likely require domain adaptation to perform well in other settings. \\citet{DBLP:conf\/lrec\/BostanKK20} report that their data comes from the Media Bias Chart\\footnote{\\url{https:\/\/www.adfontesmedia.com\/about-the-interactive-media-bias-chart\/}}, which reports that their news sources contain a mix of political views, rated by annotators who also self-reported a mix of political views. We note that these data are all United States-based and in English. \\citet{DBLP:conf\/lrec\/BostanKK20} do sub-select the news articles according to impact on Twitter and Reddit, which have their own user-base biases\\footnote{\\url{https:\/\/www.pewresearch.org\/internet\/fact-sheet\/social-media\/}}, typically towards young, white American men; therefore, the data is more likely to be relevant to these demographics. The language used in headlines will likely most resemble Standard American English as well, and therefore our models will be difficult to use directly on other dialects and vernaculars.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"Criminal or civil actions, where the adviser or a management person of the adviser was convicted, pleaded guilty or 'no contest', or was subject to certain disciplinary actions with respect to conduct involving investment-related businesses, statutes, regulations, or activities; fraud, false statements, or omissions; wrongful taking of property; or bribery, forgery, counterfeiting, or extortion.\nSelf-regulatory organization (SRO) proceedings in which the adviser or a management person was found to have caused and investment-related business to lose its authorization to do business; or was found to have been involved in a violation of the SRO's rules and was the subject of specific disciplinary actions taken by the organization.\nCourt judgments pertaining to investment-related statutory and regulatory violations by the firm and its affiliated persons, for the last ten (10) years.\nRIA's registered with the SEC must disclose all of the above items on their annual report to the SEC, consisting of an updated Form ADV Part 1A.\nIn addition to the above items, RIAs registered with the state of their principal place of business must disclose information about surety bonds if required by the RIA home state, unsatisfied judgments and liens, investment-related judicial actions (court lawsuits) AND arbitrations filed against the RIA or its representatives. These additional disclosures are unique to state-registered RIA's; SEC-registered RIA's need not report these items on Form ADV, Part 1A. Those RIA's registered with a state must disclose the criminal and regulatory items described in the previous paragraph on their Form ADV Part 1A, and the items in this section (surety bonds, unsatisfied judgments and liens, judicial actions, and arbitrations) on Form ADV Part 1B.\nFor more information about investor education, visit the SEC's retail investor information page at http:\/\/investor.gov\/.\n1 Ms. Toleno is a partner in Shustak & Partner's San Diego office specializing in securities litigations and arbitrations.\n3 Pursuant to SEC rule adopted in January, 2011, that amount is going to increase to $100 million but the effective date of the implementation of the rule has been delayed until July, 2011 and may be delayed even further. This is the current rule as of the time this article was written.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gently stir the strawberries and sugar together in a bowl. Cover and let sit at room temperature until juices release, about 1 hour.\nDivide fruit between 2 small, clear serving bowls or stemmed cocktail glasses.\nPlace egg yolks, sugar, and marsala into metal mixing bowl. Set over burner on low heat. Hold rim of bowl with a potholder. Whisk until mixture forms loose peaks, 7 or 8 minutes. Mixture should feel warm to the touch.\nSpoon custard over the strawberries. Serve while custard is still warm.\nThe most tempting of romantic desserts.\nInstead of placing the metal mixing bowl over low heat to whisk the custard, you can set the bowl over a pot of slowly simmering water.\nI have never made a custard before, I didn't have a metal bowl so I just used a small sauce pan. It turned out amazing!\nCustard tasted eggy; not sweet enough. Not very good.\nThis is the first time I have ever made a homemade custard. I followed the directions exactly and it was delicious. I will definitely make this again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This strong woman reaches for weights. Time to workout. Photo via Pexels.\nNot so long ago, there was a lot of pressure on women to be skinny. But there is a new voice now, our voice, one that says we want to be strong, not skinny, and it is a mentality that seems to be here to stay. Women are embracing their muscles and toning their minds at the same time. However, as with any exercise where the efforts are high, so are the risks, and there is nothing more effective at stopping your progress than a severe injury.\nUnfortunately, there are no shortcuts when it comes to getting off the sofa and back into the gym. Recovering is just one of those things that have to be done right. If not, you risk making your temporary injury a chronic issue. We know it's frustrating, but we have laid out some steps for you to follow and get you back to fighting fit as soon as possible.\nIce the injury, ladies. Photo via Pexels.\nThe first thing you should do after sustaining an injury is pack on the ice on. Ice-pack, ice cubes wrapped in a towel, frozen peas, it doesn't matter what, it just matters that you do. Do a minute on and ten minutes off three times. What this will do is slow down any bleeding and speed up your recovery further down the line.\nIf you don't understand your injury then you can't hope to recover properly. Take it seriously. Yes, there are some injuries where you just have to woman up and walk them off, but serious ones don't fall into the category. That's why it is important you see a doctor and get a professional and full diagnosis. This way you will know what sort of recovery you are facing and you can plan your next moves from there.\nEat healthy, during recovery and beyond. Photo via Pexels.\nOne of the secrets to a speedy recovery is your diet. Put plenty of nutrients in your diet, drink plenty of water, consume plenty of vitamins and eat plenty of greens. Food is what fuels your body and it is also what repairs it. This is your chance to try out some of those gorgeous recipes you've seen on Tasty, while also chipping away at your boredom. Oh, no alcohol okay. Want to get back to pumping iron, then no alcohol during recovery.\nThe fastest way to get back onto your feet is to have a proper rehabilitation plan in place. Soft tissue massages, physical therapy, chiropractic care, sciatic nerve treatment, hydrotherapy; if it will help you in the long run then it should be considered. You don't want the future you to pay for your mistakes now. Trust us on that.\nGetting back to it with weights, after recovering from the injury. Always be safe and consult with a professional. Photo via Pexels.\nThe worst things about recovering from injury are the boredom and the constant tussling with your eagerness to get back out there, after all, there are only so many TED Talks you can watch. If it is a leg injury you are recovering from then why not ease into some upper body exercises, or if it is your upper body that suffered the injury then try some squats. Just remember to take it steady. You are in recovery and the last thing you want to do is make this period last longer than it has to.\nGreat article as always and agree it is time the girls add a little weight and adding muscle is a great way through workouts.\nYou mentioned ice if an injury occurs. Several years ago I discovered a great ice pack that wraps nicely around an injury, better than any ice pack, very simple and inexpensive.\nAdd a cup of rubbing alcohol and two cups of water to a large zip lock making certain it is sealed tightly getting as much air out as you can. If in doubt add another zip lock baggie and toss in the freezer. I few hours later you have this amazing ice pack that retains the cold for very long periods of time. Just refreeze it and use it over and over.\nI am honoured by your words!\nI like this series of life and health tips you're onto! Hugs!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Consuming protein in the morning will stabilize blood sugar, Sharon Collison, RD, sports dietitian and clinical instructor at the University of Delaware says. \"Make sure to have an excellent source of protein with every breakfast such as Greek yogurt, cottage cheese, or two to three eggs,\" she says. Eating more protein earlier in the day could also help reduce sugar cravings in the late afternoon, Collison adds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your little explorer will delight in the Animal Kingdom Collection from Just Born. The adorable plush animals of the Musical Mobile sway gently to Brahms' Lullaby. Perfect for soothing your baby to sleep.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Get ready for an exciting episode! The Handmaid's Tale airs its penultimate episode before the Season Finale! So many questions left unanswered. What will become of Emily? Having her Commander die while trying to impregnate her carries a great deal of notoriety. And what of June, her baby taken away from her? How will she cope with Holly's absence? What becomes of the Waterfords? Are Fred and Serena now Parents of the Year in Gilead soceity or struggling to adapt to the changes that a newborn brings into the house?\nWhat will become of Eden, our innocent child bride, sworn to protect all that Gilead stands for? With her relationship with Isaac, are they doomed to be tragic lovers or somehow receive God's forgiveness and survive another day? June and Nick may have plenty to think about in the aftermath of June's abandonment and Nick's part in Fred's little \"gift\" to June. Is Fred secretly in love with June, or are his feelings based in his desire to control her destiny and have her at his mercy? And what will become of baby Holly? Is she going to be well cared for in a house filled with hostility and violence or feel the loss of her dear mother?\nWhatever fates are awaiting our characters, there is a sense of gloom and unabashed terror growing in Gilead now. With June clearly traumatized and Emily serving a new Commander, it's going to be a hell of a ride!\nA Discovery of Witches - TrailerSan Diego Comic Con Exclusives Portal Now Open!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fanwood, NJ, March 6, 2018 \u2013 Symphony Solution is proud to announce successful integration of Voice Search feature in My AdultDayCare mobile application with the help of Artificial Intelligence.\nMy AdultDayCare is a cloud based Practice management software with features such as Electronic Medical Record (EMR), Patient Care Plan, Fleet Management etc. with AI capabilities. With the new Voice Search feature, the users can navigate through the My Adult DayCare app by sending voice commands.\nCEO of Symphony Solution, Mr. Anish Desai, explained how crucial AI integration was by saying, \"We wanted to create a feature which easily identifies voice commands and has a quick response rate. This feature will improve productivity whilst solving some of the commonly faced logistical barriers faced by an adult day care\".\nUsers can search for patients and view key information about them through the Voice Search feature in few seconds. Apart from increasing the productivity, this feature will also help attendees and fleet managers to quickly access emergency contact information of the people travelling in their vehicles.\n\"AI integration through Voice Search is vital for an app like My AdultDayCare where time and productivity are of key importance. We aimed at providing a completely new easy-to-navigate option and we have successfully integrated it,\" said a product developer working on AI integration.\nWith effective use of artificial intelligence Symphony Solution has made it safer for the driver to navigate through the app without putting any passengers at risk. Through the new feature Symphony Solution has raised the bar for custom built mobile application offering IT based solution. My AdultDayCare has features which can solve different problems faced by caretakers, dieticians, fleet managers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Education Department at the Ohr-O'Keefe Museum of Art offers fun projects and arts integration opportunities to our community in a variety of outlets. We feel that the creative spirit is vital to development and growth for kids and adults, which is why we offer free resources for parents, teachers, and kids.\nHere's how it works: Your group arrives and sets up any food or beverages you've brought along. Wine and beer are fine to bring! We'll have everything creative ready to go. When everyone arrives, we'll walk you through the studio and then jump right into our clay play, helping and guiding you along the way. No clay experience is necessary!\nPlease keep in mind this is a \"wet clay\" studio, and so the work created at the party must be carefully dried and fired in our kilns. Your work will be ready for pickup a few weeks after your class. Read the \"fine print\" below, and feel free to call us with any questions.\n*Hand-building (by the way) means sitting at a table with some clay and hand tools to make things, and it is generally easier and more social, so we recommend it for parties and team building events.\nSculptural Art Birthday Party: Each participant will make one piece of sculptural art. When you register, we will provide a list of available projects, and you may select one to create. Pricing is $15 \u2013 $25 per child or participating adult (with an 8 child minimum) dependent upon the chosen art project. There is no maximum number allowed, so if you have a large group of children or adults, this option is a good choice. Your registration fee covers 2 hours of studio rental, which includes time to eat and open gifts. Work is generally ready to pick up in two weeks (often sooner!), and should be picked up by ONE person.\nItems to bring: A tablecloth to cover one of our tables for food service, refreshments, napkins and cups, plates, etc. You are welcome to bring decorations or a pi\u00f1ata to hang outside in our tree.\nArrival and departure: Please arrive no more than 10 minutes before your party is scheduled to begin. Parents will be able to set up food tables while children are at work on their clay projects. Also, please remember that your rental time is for two hours.\nSafety in the studio: Our ceramics studio is a working studio with ongoing classes and students conducting independent study. We want you to have a safe and fun birthday experience, and therefore children must be aware of boundaries within the studio space. No climbing on our beautiful studio tree, the external walls, or the stairs is allowed. Children should refrain from touching studio artwork and tools. We have a designated area for birthday parties, and parents are asked to help our staff keep children safely in this area.\nDiscounts\/Deposits\/ Registration: Museum members hosting an entire class receive 10% off the total class fee. You must register your group at least 10 days in advance, and a 50% non refundable deposit is required at the time of booking. Balance is due on the day of your class.\nOhr-O'Keefe Museum of Art\/ 386 Beach Blvd\/ Biloxi\/ MS\/ 39530\/www.georgeohr.org.\nWe can accommodate all types of groups, including classes for team building, creativity workshops, and adult gatherings. Whether you want to gather a group of coworkers or just have a fun night out with family or friends, our two hour clay workshops let you relax and be creative in a very low-stress environment. You can choose from a list of fun projects and let our instructors guide you through the steps of creation, or let us know if you have something special in mind, and we will try to make it happen!\nRegistration for a hand building workshop is $15 per participant with a $150 minimum registration fee (8 participants). Your fee includes all instruction, materials, and firing costs. Registration for wheel throwing workshops is $25 and includes the same. Grab a bottle of your favorite beverage, bring some snacks, and come create something original in a fun environment!\nThe fine print for all creative events: Reservations for private studio bookings are required at least two weeks in advance and are based on studio and instructor availability. We require a $75 non-refundable deposit and completed reservation paperwork to reserve the studio for events. The Museum has an alcohol permit for beer only. If you would like to bring wine or liquor to your event, there is a $30 permit charge and we must have 30 days advance notice. Museum members hosting an entire class receive 10% off the total class fee, excluding alcohol permit. Balance is due on the day of your class after a head count is taken of those creating projects. Projects will be available for pickup 2 weeks after your class, potentially 3 weeks if your choice is a wheel class. We are available to mail finished work to participants for a $10 per person charge for packing and postage. Minimum number of students required for booking a party is eight. The Museum does not provide linens, silverware, food, or beverages.\nFor questions, please email education@georgeohr.org or call 228.374.5547.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chris Orme survived a strong, one-loss side bid by Chance Chin to complete an undefeated weekend at the Joss Northeast 9-Ball Tour the weekend of March 28-29. The $2000-added purse event drew 39 entrants to Classic Billiards in Rochester, NY.\nOrme gained the hot seat after a 9-6 win over Bucky Souvanthong and a 9-7 victory over John Morra. Morra had sent Dave Grau west 9-5.\nOn the one-loss side, it was Chin and Shane Winters from among the final 12 who emerged to face Grau and Souvanthong. Winters dropped Tim Wendt 9-3 for the right to meet Grau as Chin dispatched tour director Mike Zuglan to the tie for seventh place with his first of three hill-hill wins that put him into the finals. It was Grau and Chin advancing to the quarterfinals; Grau with a 9-4 win over Winters and Chin with his second hill-hill victory, this time over Souvanthong.\nChin got a reprieve from his double-hill struggles with a 9-7 win over Grau in the quarterfinals but stepped right back into the fray with a hill-hill battle against Morra in the semifinals. He survived that third straight double-hill struggle and turned to face Orme in the finals.\nSeemingly tired of matches that kept coming down to the wire, Chin opened the finals by taking a commanding 6-1 lead. He watched those as Orme battled back to tie it at 6 and move out in front by a single rack at 7-6. Chin tied it at 7, and then Orme reached the hill. It was Orme's break on the last rack, and he sunk the 9 ball to win it.\nChris Orme survived a strong, one-loss side bid by Chance Chin to complete an undefeated weekend at the Joss Northeast 9-Ball Tour the weekend of March 28-29.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The legend says that Bangalore's history dates back to 1120AD. From Benda Kalooro to Bangalore, the city as evolved so much but the rich cultural inheritance that it is owning is both magnificent and eye catching.\nThere are temples that are centuries old, mosques and churches that are visited by a large number of devotees across the country. The oldest of all is Bull Temple that was built by Kempe Gowda, the founder of the city. There are cave temples that are interesting to explore. Temples from the period of chola dynasty still attracts hordes of people. Hinduism being the main religion followed here, most of the Hindu Shrines can be found here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztjoz b/data_all_eng_slimpj/shuffled/split2/finalzztjoz
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@@ -0,0 +1,5 @@
+{"text":"The Geothermal Research Dept. at Sandia Natl. Laboratories, in collaboration with Drill Cool Systems Inc., has worked to develop and test insulated drillpipe (IDP). IDP will allow much cooler drilling fluid to reach the bottom of the hole, making possible the use of downhole motors, electronics, and steering tools that are now useless in high-temperature formations. Other advantages of cooler fluid include reduced degradation of drilling fluid, longer bit life, and reduced corrosion rates.\nThis article describes the theoretical background, laboratory testing, and field testing of IDP, including structural and thermal laboratory testing procedures and results. We also give results for a field test in a geothermal well in which circulating temperatures in IDP are compared with those in conventional drillpipe (CDP) at different flow rates. A brief description of the software used to model wellbore temperature and to calculate sensitivity in IDP design differences is included, along with a comparison of calculated and measured wellbore temperatures in the field test. There is also analysis of mixed (IDP and CDP) drillstrings and discussion of where IDP should be placed in a mixed string.\nSandia first considered IDP during an investigation of energy extraction from magma, which would have involved drilling into extremely high temperature (>1600\u00b0F) molten rock. This presented a fundamental problem - the damaging effect of high formation temperature on drilling tools, materials, and processes.1 When drilling fluid gets too hot, it can irreversibly deteriorate in its ability to carry the drilled cuttings, increase drillpipe corrosion by orders of magnitude, shorten the life of bits and other downhole tools, prevent the use of downhole motors and electronic instrumentation\/ tools, and even affect borehole stability and well control. Each of these problems can be attacked by individual technology developments, but all can be solved or greatly mitigated by simply cooling the downhole environment with a lowertemperature drilling fluid. IDP delivers drilling fluid to the bottom of the hole at a much lower temperature than CDP.\nMore detailed calculations and measurements are presented later, but an example of IDP's effect in a geothermal well is shown in Fig. 1. In this and other fluid-temperature plots, the right side of the curve shows the temperature in the annulus, and the left side shows the temperature in the drillpipe. Several important points are illustrated by the following calculation.\nBottomhole fluid temperature is reduced from 401\u00b0F with CDP to 166\u00b0F with IDP. This is the temperature that steering tools, mud motors, and other electronics must survive.\nMaximum fluid temperature (not at the bottom of the hole) is reduced from 406 to 214\u00b0F, the maximum temperature seen by drilling-fluid additives.\nThe mud return temperature is higher with IDP, 196\u00b0F vs. 163\u00b0F. This may mean that mud coolers are necessary, but it also means that more heat is being removed from the hole, which will be beneficial for logging and cementing after drilling.\nThese curves describe temperatures in holes that have been drilled from surface with either CDP or IDP - they do not reflect conditions that would exist while running measurement-whiledrilling (MWD) tools into a hot hole already drilled and shut in, for example. This kind of operation has been successfully performed in exploratory geothermal drilling at temperatures greater than 660\u00b0F by staging into the hole with a top drive and circulating cooler fluid after adding each stand of pipe,2 but IDP would make the process quicker and allow operation at even higher temperatures. (In the hole described in the Ref. 3, it required 14.5 hours to run in the hole to 8,500 ft.) In this kind of operation, it is essential to know the downhole temperature, so having an MWD tool in the drilling assembly is an important advantage.\nAn alternative approach to this scenario is described in the Wellbore Cooling section.\nThe IDP described in this paper was constructed by welding a 3.5-in. outer diameter (OD) by 3.068-in. inside diameter (ID) liner tube inside conventional 5-in. drillpipe (Fig. 2) and filling the annulus between the tubulars with an insulating material. With the compressive strength of the insulation to support the liner tube against internal pressure, it can be made of a thinner material, saving cost and weight and preserving the ID as large as possible.\nTwo criteria, compressive strength and conductivity, drive the choice of an insulating material. Required compressive strength can be estimated from the diameters of the drillpipe and liner, along with the expected internal pressure, but typical values are less than 8,000 psi.\nInsulation conductivity does not need to be exceptionally low because there is an alternate heat-flow path through the uninsulated tool joints. That is, at less than a nominal value of conductivity, further improvement in the insulation has little effect on the total heat flow because of the heat flow through the tool joints. Drilling-fluid temperatures are calculated for three values (k=0.05, 0.3, and 1.0 B\/hr-ft-F) of insulation conductivity (Fig. 3), and it can be seen that the variation in insulation properties has little effect relative to the difference between even the poorest insulation and conventional pipe. A number of materials have conductivity in the range of 0.2 to 0.8 B\/hr-ft-F, and a selected fracture proppant was chosen as the insulation for the IDP.\nBecause IDP is heavier (approximately 33 lb\/ft) than CDP (approximately 19.5 lb\/ft), drillstring weight could become a problem in deeper wells. Tensile stress induced by string weight is highest at the top of the drillstring. The other controlling stress is the hoop stress in the liner tube caused by mud weight and pump pressure; the liner stress is partially supported by the insulation and is highest at the bottom of the drillstring.\nIf principal stresses in the liner tube are combined and tested by the von Mises criterion, we can identify failure conditions. A plot of combined stress (Fig. 4) for various conditions shows that in a 20,000-ft well, failure (i.e., stress greater than the assumed 8,000- psi yield limit) may occur at both the top and bottom of a full string of IDP (Curve A) if the liner tube does not get full support from the insulation. On the other hand, both a 15,000-ft string of IDP (Curve B) and a mixed 20,000-ft string with 10,000 ft of IDP at the top and 10,000 ft of CDP below (Curve C) have liner stresses well within the assumed yield limit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SKU: N\/A. Categories: Leather Pants, Special Categories.\nGet this crop style flared women leather pants which is made from lamb skin leather and featured with high-waist and has pull-on style. This pant has flared hem and cropped length. This leather pants has side zipper closure and it is fully lined. Also it is available in many colors and all sizes. So women what are you waiting for?? Hurry up!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A man and woman have been arrested on suspicion of criminal damage after allegedly trying to cut down a tree with a petrol chainsaw in the early hours of the morning.\nOfficers were called shortly after 1am on Tuesday following reports of a small group of people attempting to fell the tree outside Manjaros in Linthorpe Road, Middlesbrough.\nThe road, which is a major thoroughfare in the town, was closed in both directions at 1.25am after the half-cut tree was left with the chainsaw still inside its trunk.\nThe tree, which was opposite Psyche and KFC, was assessed to be unsafe and was felled by a tree surgeon at around 3am in the interests of public safety.\nA large stump was left in place to avoid creating a trip hazard.\nThe road was reopened by 5am.\nCouncillor Denise Rooney, Middlesbrough Council's executive member for environment and commercial services, said: \"Our trees are a precious natural resource that add life and colour to our streets.\n\"To attack them in this way is utterly inexcusable, and those responsible must be held to account for their actions.\nA spokeswoman for the Force added: \"Police received numerous calls from the public and nearby just after 1am today reporting a small group of people damaging a tree with a petrol chain saw on Linthorpe Road.\n\"A 31-year-old woman and a 46-year-old man have been arrested on suspicion of criminal damage and are currently in custody where they will be questioned.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Areteem's Summer Camps are offered at five locations during the summer of 2019.\nAreteem's traditional Math Zoom program will be offered in Chicago, at Georgetown University, in Milwaukee, in San Jose, and at UC San Diego.\nAdvanced students are invited to apply to the Residential\/Day Camp at UC San Diego, which will feature a special group dedicated to advanced high school math, including material at the USAMO level.\nYounger students who do not yet qualify for the full Math Zoom program can attend the STEM Day Camp at UC San Diego which will feature math problem solving in the morning and STEM topics in the afternoon.\nThe 2019 International ZIML Convention will take place as part of the Residential\/Day Camp at UC San Diego, and all camp students will automatically participate. Students from the other 2019 Summer Camps also receive free tickets to the convention (but travel and housing costs are not included). Click here for more information.\nUse the links below to pay and reserve a spot at the camp.\nMore information about each of the camps can be found below or by clicking here.\nMake payment to reserve your spot at the camp.\nLogin on the camp page and complete the Student Questionnaire to provide the student's academic background needed for acceptance to the camp.\nAfter receiving your acceptance letter, complete consent forms.\nOne month prior to the camp start, complete the arrival and departure forms.\nDue to conflicts with local school schedules, a special option is available for students to register for the 2nd Week Only for the Chicago Day Camp.\nOn the payment page, be sure to choose the \"(2nd Week Only)\" option for the prorated price (currently \\$980).\nClick Here for the Payment Page for the Chicago Day Camp!\nThere is a \\$200 administrative fee for international residential students.\nMake payment to reserve your spot at the camp. Registration is available as either a Residential Student (first option), Day Camp Student (second option) or International Residential Student (third option).\nClick Here for the Payment Page for the East Coast Camp!\nRegister before April 30th to receive the discounted price of \\$900!\nClick Here for the Payment Page for the Milwaukee Day Camp!\nClick Here for the Payment Page for the San Jose Day Camp!\nClick Here for the Payment Page for the West Coast Camp!\nMornings will cover fundamental math at either upper elementary school level (our Fun Math curriculum) or a middle school level (our Math Challenge I-AB curriculum). Afternoons will cover the various STEM topics listed below.\nThe full schedule, divided into 3 weeks is found below.\nTo register, pay the tuition for the corresponding number of weeks. Once payment is complete, fill in the questionnaire to pick the topics\/dates you wish to attend.\nClick Here for the Payment Page for the 2019 West Coast Summer STEM Day Camp!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"White plastic cap for our one gallon container.\nDispensing spigot for our one gallon container.\nYorker spout for our one gallon container.\nCap plug for our one gallon container.\n1 gallon plastic container perfect for dispensing most liquids. Stands up to most chemicals and accepts a 38 mm cap or dispensing spigot,which are sold seperately.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxeyp b/data_all_eng_slimpj/shuffled/split2/finalzzxeyp
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@@ -0,0 +1,5 @@
+{"text":"Full Pan of Pulled Pork, 50 Pieces of BBQ Chicken, Full Pan of Mac and Cheese, Full Pan of Vinegar Slaw, Full Pan of Garden Salad with Ranch Dressing, Half Pan of Pit-Smoked Beans, 50 Pieces of Fresh Corn on the Cob.\nDINE IN: All packages include pop, coffee, & tea. All packages served with dinner salad (family style), pretzel crusted mac & cheese, and veggie.\nBurnt Ends, Pretzels, Baked Brie, Cheese Curds, Fried Pickles, and Wings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bummis gifted me a DuoBrite to thank me for my work on the GCDC, and I have been meaning to do a review! I have the same pros & cons as you do! However, they sent me an insert from a newer batch that was heat-treated, and it barely wrinkled at all at the edges compared to the first inserts I received.\nThis is such a great brand that you can feel good about purchasing! It's interesting to compare this to the Super Brite, as I used those covers on my newborns, but did not care for how thick and stiff the PUL was. Is it the same with this shell?\nIt's equally thick. The heavy grade of PUL is part of why they hold up so well. I've had some donated to my chapter of Cloth for a Cause that are 6-8 years old and are still in fabulous shape.\nThank you for the review and the giveaway. I love that these diapers are north american made! I have never tried one and they are kind of out of my price range. From the pictures they do look like they are comfortable on your child.\nThanks for the review! The more I read about these, the more I want one. I'm pretty my next diaper will be one of these because they are cute and have great reviews. This would be my first ai2 and I'm very excited about that!\nWe plan on cloth diapering when baby gets here and want to use covers with pre-folds. I would like to try the Duo-Brite cover. I also like that the diaper is made in Canada from materials made in the US\u2026aka not from China (Winner!).\nI am a bummis fan! These are on my list to try. Thanks for the review. I usually use prefolds and covers so this is awesome.\nThis is good to learn about! I haven't heard much about this system at all. I love that they're made in North America.\nI like the insert idea! Would love to try this on my newborn and toddler. \ud83d\ude42 I'll buy more if I like it as our stash needs upgrading.\nI don't have any bummis at all in my stash yet, so it doesn't bother me if the prints are outdated. I really like the design!\nI love the Duo Brite! The inserts are so soft and padded. Would love to have more!\nI love that you not only explained how to tell the difference in insert sides but also added a picture! I don't have any bummis since I'm just beginning this journey but they sound wonderful and definite must-add!\nWe have several Bummis cover and we love them. It sure would be super to give their diapers a try!\nI just bought the limited edition elephant print and love it! I also had the same problem telling the difference for which side goes up.\nI love the double gussets but at that price I really don't know I can justify buying one! I would be afraid to use it and that just defeats the purpose! OR I would absolutely love it and it would break my heart that to buy more would break the bank!!\nI am clueless about cloth diapers but really want to try them. These look like they could work for us! Love the patterns!\nI too wish they had new prints.\nI'm just wondering. Made in Canada from products from the US. Are the materials not available in Canada to be 100% Canadian made or are the materials in the US of a higher quality or more inexpensive even with the currency conversion to purchase? I'm quite clueless when it comes to diapers that are not disposable. To be honest at $59.95 that is pricey and I would think out of a lot of folks price range. I could see purchasing one but that would be about the extent of it. Love the colours and patterns that I have seen and the diaper itself looks adorable.\nIt is quite expensive, but since it's for 3 changes, it at least brings the price down to something more in line with other brands. Plus, they hold up so, so very well. I've received Super Brite (which the Duo-Brite replaced) as donations for Cloth for a Cause\u2026 and found that they were several years old. No repair needed. Bummis uses extremely high quality materials.\nI love Ai2's and have not tried Bummis yet! Love that they are made in North America and definitely want to give them a try!\nI hope to win\u2026I'd love to give these a try!\nI like that they are made in North America. Lots of fun prints to chose from too.\nThese are such cute prints! I don't have any bummis and have never tried them! I love that they are made here!!!\nWow that is a BIG DIAPER!\nVery thorough review, which I always appreciate! My issue is that they don't offer solid color options and only prints.\nI'm just now getting into covers, I would love to try these. And I actually like the prints.\nI love the snap in soaker and the double gussets.\nI wish they would come in velcro as well. Dad and grandma find those easier to use.\nI would love to try this diaper, we are Bummis fans!\nBummis duo Brite covers are so cute and colorful! Sounds like the insert is quite nice as well. I would love to try these!\nI like the choice of Stay-Dry or Cotton all in one insert. It took me a little while to get used to the way it snapped in, but I like that with that design you never have a snap exposed if you chose to use the shell as a cover rather than as an AI2. I do wish they would come out with some more exciting print choices.\nI like that the diaper has the double gussets for extra protection and that it is will fit two sizes. I like that the insert is either cotton or stay dry; options are always nice! Thanks!\nI've never tried this brand but the review really makes me want one. I love how u can pick which side I want with just one insert.\nWOW! It sounds like these are great! The snaps and the extra absorbancy are great!\nI recently bought one of these to help start my stash. Glad to see that it got good reviews.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Holy Dress is a lie-detecting garment. It monitors speech, giving live feedback through on-board lights and electric shocks. The dress is a dystopian design imagining electronically enhanced clothing as a way to help people become the best version of themselves. The Holy Dress is technology taking over the function of religion in guiding us towards living a good life.\nSociety treats technology as the new religion. We want it to save us from our own bad decisions. We trust it through the magic of Big Data to make better decisions for us than we would alone. Technology promises to solve all our problems for us, making a complex world simple again. The dress is made using a hacked commercial lie detector.\nIt determines if you're lying through voice stress analysis, since vocal stress can be an indicator of untruthfulness. When you speak, the dress lights up, growing with intensity as the likelihood of a lie increases. When the dress detects a lie it will fully light up and flicker like lightning, while punishing you with an electric shock.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Five words: A Schlock and Roll Christmas.\nAs for the running order, there would only have to be two specifications: \"Happy Christmas (War Is Over)\" would have to be the last song. And \"Do They Know It's Christmas?\" would have to be the first.\nNow, I love \"Do They Know It's Christmas?\" and was touched to see people rush to its defense when I included it on my list of the worst Christmas songs of all-time. The song has always been a guilty pleasure for me, somewhere between my Monkees CDs and Doctor Demento singles, and I included it on the worst list because it felt like the better fit \u2013 after all, it was the things that were so wrong about the song that made me love it all the more.\nIt's like, Bing Crosby's \"White Christmas\" is like George Washington: Perfect, regal, and distant, something that can only be marveled at from afar. But \"Do They Know It's Christmas?\" is like John Adams: Flawed, imperfect, and human, something that tells us far more about ourselves. I mean, who would you rather have a beer with: George Washington or John Adams? Everyone knows that George Washington would be all intimidating and quiet right off the bat, while John Adams would make fun of the ugly waitress before she was even out of earshot.\nAlthough I do still stand by statement that \"Well, tonight thank God it's them instead of you\" is the cruelest line I've ever heard in a Christmas song.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join us at O'Niell's Pub in Nob Hill for April Pint Night! $1 per pint of Santa Fe beer will benefit NM BHA. If you are interested in learning more about BHA and how we are fighting for your public lands in NM and across North America we welcome you to stop by. We will have BHA merch for sale and there's still time to sign up for our spring raffle that gives you a 1\/100 chance to win a YETI hopper with the purchase of any membership!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxjdf b/data_all_eng_slimpj/shuffled/split2/finalzzxjdf
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index 0000000000000000000000000000000000000000..22c4e33be1ab9015b5ca247ef2baeea1ac4a98fb
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Great Britain, 1838 Waghorn & Co. Overland Route Notice, printed notice datelined \"Cario 26th Dec. 1838\" regarding the forwarding of letters, parcels and packages between Great Britain, the Continent of Europe, India and China, signed by agent G. Gliddon, Very Fine.\nGreat Britain, 1839, Treasury essays by Charles Whiting, two in different shades, each large ornately embossed sheet, about 8\" x 10\u00bc\", with four different essays, on the reverse is Whiting's intricate \"Printed for protection against forgery\" imprint in red & green; each slight edge faults, darker shade with foxing, F.-V.F. and scarce.\nGreat Britain, 1840, 1d black, plate 7, lettered CC, red Maltese cross cancel, ample to mostly large margins, Extremely Fine. Scott No. 1.\nSG No. 2 \u00a3400 ($530).\nGreat Britain, 1840, 1d black, lettered AD, full balanced margins, tied by red mute cancel on 1840 folded letter from Scotland to North Shields, reverse with \"Aberdeen, Aug 6, 1840\" origin cds and red receiving postmark, Very Fine. Scott No. 1.\nSG No. 2 Estimate $200 - 300.\nGreat Britain, 1870, 1\u00bdd rose red, error of lettering, OP-PC for CP-PC, duplex cancel, strong color, usual centering for this, Fine, a very scarce error. Scott No. 32d.\nSG No. 53 \u00a32,000 ($2,630).\nGreat Britain, 1867, Queen Victoria, 2s dull blue, Spray of Rose watermark, without gum, rich bright color, short perf at bottom, otherwise Fine. Scott No. 55.\nSG No. 118 \u00a34,000 ($5,260).\nGreat Britain, 1878, Queen Victoria, 10s greenish gray, Maltese Cross watermark, lettered G-G, neat central Glasgow town cancel, rich color and attractive centering, couple trivial tiny marks above \"Shillings\", otherwise Very Fine; with 2019 B.P.A. certificate. Scott No. 74.\nSG No. 128 \u00a33,200 ($4,200).\nGreat Britain, 1878, Queen Victoria, \u00a31 brown lilac, Maltese Cross watermark, town cancels, rich color, Fine. Scott No. 75.\nSG No. 129 \u00a34,500 ($5,910).\nGreat Britain, 1882, Queen Victoria, 5s rose on white paper, Anchor watermark, London town cancel, bright pastel color, corner repaired at upper left, Very Fine appearance. Scott No. 90a.\nSG No. 134 \u00a34,200 ($5,520).\nGreat Britain, 1883, Queen Victoria, 10s greenish gray on white paper, Anchor watermark, 1883 town cancel, rich bold color, well centered, tiny pinhole at upper left, Very Fine appearance. Scott No. 91a.\nSG No. 135 \u00a34,500 ($5,910).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After BlackBerry had reached its long-term brand licensing deal with TCL last month, the Chinese firm teased its first true BlackBerry handset, codenamed Mercury, at CES 2017. There weren't many details revealed at the time, but we now know the date of its official unveiling: February 25, the eve of Mobile World Congress.\nBlackBerry Mobile sent out a tweet yesterday confirming the date, complete with a GIF of the handset showing the distinctive QWERTY physical keyboard that the brand is known for \u2013 though this was absent from the TCL-manufactured DTEK50 and DTEK60, which were modified and rebranded versions of the company's own Alcatel phones.\nInformation on the Mercury remains thin on the ground. It will run Android 7.0 Nougat, features a USB Type-C port, and has a fingerprint sensor built into the space bar - much like the BlackBerry Passport. Additionally, the touch-enabled keyboard lets you perform functions such as swiping to scroll and flick to type.\nThe Mercury's other specifications are all rumors: a 2.02GHz Snapdragon 625 processor, 3GB of RAM, 32GB internal storage, a 4.5-inch display (due to the size of the keyboard), a 1080p resolution, a 3,400 mAh battery, and an 18MP rear camera along with an 8MP front-facing shooter. It seems TCL is aiming for the mid-range market with this one.\nAs with other BlackBerry phones, the Mercury is aimed primarily at enterprise customers, and as such it comes with extra security features. The handset will ship with business software created by BlackBerry the company, which has now moved away from the hardware side of things.\nExactly how much the Mercury will cost remains a mystery, though Engadget believes it will be close to the DTEK50's $300 price. We'll find out more at Barcelona.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pradhan Mantri Gramin Awaas Yojana (PMGAY), previously Indira Awaas Yojana (IAY), is a social welfare flagship programme, created by the Indian Government, to provide housing for the rural poor in India. A similar scheme for urban poor was launched in 2015 as Housing for all by 2022. Indira Awaas Yojana was launched in 1985 by Rajiv Gandhi, the then Prime Minister of India, as one of the major flagship programs of the Minisrty of Rural Development to construct houses for BPL population in the villages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your Guide to getting the most from our books, audiobooks, DVDs, and digital downloads including new arrivals and staff picks at . . .\nLearn how to download free audiobooks on May 2.\nCheck out the latest publishers' new releases!\nIn the Gallery: Landscapes by Kent Stucky.\nThe McPherson Public Library strives to enrich, inspire, and enlighten our community through exceptional collections, services, and programs.\nVisit our NEWS ROOM . . .\nfor detailed information about the latest library news . . .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Betta fish food \u2013 why is this solution increasingly often recommended by rising number of specialists, who have their aquariums?\nHaving an own aquarium is something plenty customers are nowadays interested in. It is indicated by the fact that owing to it we can realize our dream of having pets in our house. Hence, we should also be aware of the fact that in order to care about our fishes sufficiently, it is inevitable to think about services such as Betta fish food.\nAlthough the food is not the only factor that matters in terms of the way fishes are treated, in such case we are recommended to also be aware of the fact that the better food we offer our fishes, the more they are likely to live significantly longer. This implies that this kind solutions like inter alia those mentioned above is quite worth our attention if we would like to assure our fishes very good conditions for further growth.\nStart new hobby \u2013 be an aquarium lover!\nPossessing an aquarium indicates taking care of fish which do not stay in cold climate. The aquarium enthusiast must have the proper information of the sea fish when the individual makes a choice to raise such sorts of fish. The sea fish are one of the most beautiful fish in the worldwide. Many of them are very effortless to look after \u2013 they desire only clean h2o and appropriate flowers to eat. On the another hand, there are fish which need specialized therapy and commitment of plenty of time to guarantee that the fish will survive in our aquarium.\nThe reason why food is pretty crucial is that without offering our fishes with a variety of meaningful elements, they are not quite likely to grow appropriately as well as have the energy to swim rapidly and move a lot, which makes each person watching their behavior become relaxed and delighted at the same time. Such options like inter alia Betta fish food is , thus, something that can support us have the pleasure to recognize the life of our fishes for a longer period of time.\nEven though there are also several other factors that contribute to our satisfaction in the above mentioned topic, we ought to be aware of the fact that food belongs to the most influential elements.\nTo sum up, in order to make or aquarium a place each fish would dream to be in, investing in Betta fish food we are able to be ascertained that we will make a significant step towards realizing this aim. Therefore, if we would like our fishes to grow instantly and feel really good, we ought to think about giving them appropriate food that would allow us to be assured that the fishes would grow quite rapidly and look very good making each moment of relax we spend on staring at them be even more attractive experience.\nTrade \u2013 what are the most popular commodities that are sent between various states on the globe?\nTrade is a field that has been systematically developing since thousands of years. It is connected with the fact that always there is a situation, in which we demand something as well as we are likely to also offer some products we have in too big amount to another person.\nPregnancy is a quite crucial and difficult period of time for both parents. With no doubt it is more stressful for a woman, but in fact also every partner has significant amount of stress in doing his best to guarantee his woman good development conditions. An example of alternative that is likely to help here a lot to offer the woman significantly more attractive mood during this pretty stressful time is related to pregnancy diet. Owing to it a woman can not only feel better, but also care better about the health of the life that is growing inside her.\nImproving number of people at present tend to invest more and more money in diverse types of options that aim is to make the shopping online a process that is considerably quicker and less expensive. One of examples how to buy diverse products considerably less expensive is connected with so-called Harrods promo codes. Due to them each time we visit this shop we are provided with an occasion to acquire discount on different sorts of commodities that are available in the above analyzed store.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxzlr b/data_all_eng_slimpj/shuffled/split2/finalzzxzlr
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index 0000000000000000000000000000000000000000..65a0da3885da1d6cd1f4bd19c17083d90831df12
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzxzlr
@@ -0,0 +1,5 @@
+{"text":"Informal business proposal memo example examples of proposalinformal letter samplesample template. Informalsiness proposal examples memo templateinformal sample of memoinformal. Informal business proposal examplesinformalo templateinformal for social media protocol sample of.\nHow to write essay proposal sample examples of a informalness examplesinformal memo templateinformal for social media. 55 best of proposal memo document template ideas informal business photo examples proposalinformal templateinformal for. Template informal businessosal how to write a letter company image collections photo sample of memoinformal. Informal business proposal photomo templateinformal sample.\nExamples of informal business proposalinformal proposal letterlesample memo for social mediainformal template. Informals proposal sample free invitation templates for word photo memo examples of proposalinformal. Informal business proposal photo template word example mughals memo sample.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. Focus on the view - When the view is as spectacular as this keep furnishings simple and unassuming so the view become the main attraction.\n2. Pay attention to comfort - Patios and porches are the perfect place to enjoy the last days of summer and comfortable seating provides the perfect place to read a book, chat with friends and just relax.\n3. Introduce greenery - Not every outdoor space may have a view like those above. Bring in lots of plants to make the space feel more tropical and welcoming.\n4. Open up the space - Use louvered shutters to promote circulation and provide privacy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CityGratis.com Free Classifieds by category in Ireland Commercial and offices - GET YOUR DREAM JOB NOW !\nQuikr is a city based free classifieds website that helps people buy, sell, rent, find in categories like furniture, mobile phones, cars & bikes, services, pets, real estate, jobs, education, entertainment and matrimonial. You can post free ads and reply to ads on Quikr in the city you live in. Quikr's classifieds websites allow you to post free ads, find what you are looking for and address your regular needs by connecting people in a faster, simpler and easier way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Melissa studied wallaby prostate development in graduate school with Professor Marilyn Renfree and Professor Geoffrey Shaw at the University of Melbourne. In 2012 Melissa moved to Madison to do postdoctoral training with Associate Professor Paul Marker, working on mouse prostate development and BPH. She joined the McNeel lab in August 2016 as an assistant scientist. Her research is looking at the androgen receptor as a target for immunotherapy in prostate cancer, and to improve the efficacy of AR DNA vaccines by combining the vaccine with androgen deprivation therapy, particularly ADTs which have AR degrading properties such as galeterone or a newly identified compound DipG and immune checkpoint inhibition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TakeOffTalk is making travel personal again. Taking you on an inspiring journey through visual stories told by fascinating people from exciting cities in Europe, it is a platform to share the knowledge that only the locals will have. We want to inspire you to travel because of the people and not only the city itself. TakeOffTalk met with locals in the trendy travel destinations Berlin, Copenhagen, Dublin to hear what they like about the city and what they don't. Through their subjective and honest stories, TakeOffTalk users will get an authentic impression of the city and hopefully get inspired to visit the place. And yes \u2013 we asked them for all their insider tips, just one of the many reasons why you should check out takeofftalk.org!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyose b/data_all_eng_slimpj/shuffled/split2/finalzzyose
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@@ -0,0 +1,5 @@
+{"text":"Iranian troops crossed the border into Iraq today and attacked a group of Kurdish militants in the mountainous border region of Iraqi Kurdistan, killing at least 30, according to Iran's state media.\nIranian Revolutionary Guards General Mahmoudabadi says that those killed in the clash were responsible for last week's attack on the military parade in the city of Mahabad, which killed 12 people.\nBoth Iranian and Turkish troops have repeatedly launched strikes into Iraqi Kurdistan over the past several years, claiming that the region is being used as a refuge for separatist movements from their respective countries.\nThough Iran is on extremely good terms with the Maliki government, border tensions have occasionally flared, and Iraq has positioned a considerable number of troops along the border in places where the border is ill-defined. This generally does not apply in semi-autonomous Kurdistan, however, as that region has its own security forces, the Peshmerga.\nuh\u2026wasn't there someone on this blog , who in defense of Iran said \" the Iranians have killed no one \" ?\nBe prepared for UN (USA) to declare bogus NO FLY zone and bomb cr@p out of Iran.Like they did Saddam's military.\nBe prepared for Iranian air defenses to shoot them down or maybe launch their own retaliatory air or missile strikes. Iran aint Iraq!\nUS foreign policy is setting the leading example, and the new world disorder will follow: War Without Borders.\nDo not believe this bullchit. It is all a lie to attack Iran.\nFalse flag violations are going to increase. Forces in Israel and USA are staging BS to \"justify\" attack on Iran.\nDo not believe it, people!\nI do NOT believe this. Just another propaganda item to attack Iran.\nGood for Iran! They're going after terrorists wherever they're found, regardless of borders or nationality- just like the US. I'd like to see how this is spun as 'Iranian aggression' when we do it every.day.of.the.week.24\/7\/365. Will Iran be hailed as a partner in combating terrorism for this incident? I doubt it- at this point Jesus himself could decree Iran was blameless and certain parties in the US and Israeli governments would ignore him.\nwhat are the names of those killed?\nNext Next post: Settlers Celebrate End of Construction Freeze, But Will Peace Talks Survive?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you looking to start a Foreign Limited Company in Kenya? Well, you need to be aware that you are required to have Kenya Revenue Authority (KRA) PIN for at least two (2) company directors in order to avoid problems when applying for your foreign company's KRA PIN.\nKRA PIN is very crucial document that allows your company to open bank account, make tax returns, apply for government tenders, tax compliance purposes and many others. As you can see it is a very important document that can't be ignored.\nNow, if you are planning to register your foreign company, then it is advisable to include at least two Kenyan citizen who need to both have KRA PIN individually. This will make it easier to apply for the KRA PIN for the foreign company and also the tax compliance certificate if you intend to apply for Government tenders.\nHowever, if that's not possible, then you will need to make sure that at least two of the non-resident directors apply for KRA PIN individually.\n5. Acknowledgement Receipt (from KRA) \u2013 Once you apply for KRA PIN using iTax system, you will be issued with an acknowledgement receipt from KRA.\nOnce you have all the documents and have applied through the iTax system, you will be able to get your KRA PIN through your email within 10 working days or less. However, if you encounter any challenges with getting your KRA PIN, you can visit your tax station indicated on the Acknowledgement letter from KRA.\nThanks for this info. I am not a resident or a Kenyan national and I want to set up a trading company with import licencing in Kenya. My partner is a national but he told me as a non resident I CANNOT obtaim a pin number so as to be added to the company as a Director. Is this true?\nHe says I need to be added as an investor.\nSimon, you can register a limited company as a single director if you have Alien ID Card, which is equivalent to a Kenyan ID. Then you can use the alien ID to apply for your personal KRA PIN which will enable you to register a limited company as the only director.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We can all become better listeners and learn to facilitate more productive conversations. Watch our 5 tips for coaching employees and providing feedback.\nBlog: When Does Coaching Work?\nDan is a Senior Consultant and Executive Coach at DecisionWise. He is responsible for partnering with individuals, teams, and organizations to help them design experiences that illuminate leadership potential and improve employee engagement. Dan's love for inspired design evolved from art and film, to business and human relations as he pursued and received his B.A. from Loyola Marymount University and his M.B.A. from San Diego State University. Dan is also a Certified Executive Coach from Georgetown University. His 20 year career in Human Resources spans the field from employee relations, performance management, organizational design, change management, employee engagement to leadership and executive development. These experiences have been cultivated from positions with Packard Bell, Wells Fargo, Capital One, RealNetworks, Micron Technologies, and most recently as the Manager of Organizational Development for a global non-profit. Dan is a surfer from Hawaii and enjoys \"riding the wave\" with his team, clients and family.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you ever feel that you have unhealthy relationships with others? That you just can't seem to let go of codependent tendencies, or the opposite \u2013 feeling totally numb to your emotions and distant from others. You also feel like your creativity can't blossom as it should, and that you feel stuck in one particular feeling or emotion.\nYou don't feel that you relate to yourself well, and therefore, you can't possibly relate to the world around you. If this sounds like you, you might have a blocked sacral chakra. The sacral chakra, or the second chakra, is located three inches below the navel, right in the center of your lower abdomen. This chakra governs the emotional body, creativity, sensuality, and pleasure.\nBelow, let's talk a bit more about the sacral chakra, and what you can do to get it unblocked.\nThe Sanskrit name for the Sacral chakra is \"Svadhisthana\", meaning \"your own place\". We can infer from this that by balancing your Sacral chakra, you will feel more at home, and know your place in the world. This chakra is represented by the color orange most commonly.\nUltimately, the sacral chakra represents our ability to relate to ourselves and the world around us. It helps us to express and understand our emotions, when properly balanced. It's the chakra of pleasure, allowing us to enjoy life through our senses, and encourages us to remain flexible in our feelings and emotions.\nA second function of the throat chakra deals with expanding our creativity, and really diving into our innermost worlds to understand what brings us passion in life. The sacral chakra also helps us to express our sexuality, and not feel ashamed or embarrassed by it. A balanced sacral chakra will allow you to feel comfortable with your emotions, sexuality, and creativity. Opening the sacral chakra means letting go of negativity surrounding your feelings, expression of sexuality, and the way you spread your creative gifts with the world.\nA blocked sacral chakra can make you feel out of touch with yourself and those around you. It can also make you feel trapped in your emotions, and unable to express them clearly. You also will likely feel insecure about your sexuality. However, an overactive sacral chakra can lead to sexual addiction, overreacting, emotional attachment and neediness in relationships, as well as stomach pains and muscle aches.\nA blocked sacral chakra can affect your relationships with others, as well as your zest for life. A sacral chakra blockage can also make it difficult to express your emotions clearly, and can make you prone to either overreacting or stifling your feelings. Upon opening this chakra, you will find it easy to express yourself sexually and creatively, and will feel close and connected to others.\nAn imbalanced sacral chakra can lead to two extremes; either total detachment from others, or total dependency on others. You might feel as though your emotions have a mind of their own, and you cannot control them. If you feel that you have a blocked sacral chakra, read on to find out how to clear this blockage so you can live authentically and freely.\nClearing the sacral chakra involves a lot of tuning into your higher self, and giving your mind and body what they need to thrive.\nLetting go of past emotional pain. One reason why you might feel either unable to open up, or trapped by your emotions, is that you carry so much of your past around with you. This can greatly dampen our emotional spirit, so make sure you talk with someone regularly about your emotions, journal, or even sing songs that reflect your feelings.\nIncorporate orange colors into your life to help boost this chakra's energy. Wear orange clothing, eat more orange foods, such as oranges, orange bell peppers, carrots, cantaloupe, and even cinnamon, and place orange items around your home.\nMeditate on this chakra. Imagine the orange lotus flower near your belly button, and hold this image in your mind. Remember to breathe deeply, and practice positive affirmations as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"'I was involved in a serious car accident which left me seriously injured to the point where I could no longer work. I had to use crutches wherever I went. The doctors said that I would never be able to walk properly again, and that I would be in a wheel chair in 5 years. This was not looking good for me. My business was going to collapse, followed by my life. I heard of a healer that was performing miracles of healing in a nearby town. I don't believe in all that rubbish and spooky stuff, but I was left with no other choice. The pain was so bad that I had to take many painkillers each day and night. It was starting to give me stomach ulcers. I had no choice but to call this healer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaaker b/data_all_eng_slimpj/shuffled/split2/finalzzzaaker
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index 0000000000000000000000000000000000000000..fd2ae634b7d015cfdff023be9f45b17af791ab39
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The following are the stone colors available in large slabs in our warehouse. The availability of our inventory is subject to change without notice. Please contact us if you are looking for materials that are not available in our warehouse.\n\u200bGranite slabs are products of nature. There are variations in color and pattern in each slab. The variations are not defect; they are the characteristics of natural stone that give the material its beauty and uniqueness. Actual granite colors may vary. We cannot guarantee that the color you see accurately portrays the true color of the granite.\n\u200b\u200bCopyright \u00a9 2019 Metro Granite LLC. All Rights Reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Read More Vote for THE SHOP in the statewide tattoo shop competition!\nRead More Last Day! One More Chance To Vote For THE SHOP in Fowlerville!\nRead More Small Towns Rock -Now, Let's Win This Contest Together!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discover the outstanding cuisine at the Kyriad Meaux restaurant at breakfast, lunch and evening meals.\nA full breakfast buffet is available from 7 a.m. to 9:30 a.m., for a tasty and energizing start to the day. Rendez-vous at the Karousel restaurant, from 12 p.m. to 2 p.m. and from 7 p.m. to 10 p.m, except on Sundays. Sometimes spicy, often colorful, always tempting, discover a cuisine which dares some astonishing combinations. Let yourself be amazed. The restaurant seats up to 80 people. Relax and mingle in the hotel bar.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a sentiment we often hear repeated by scientifically-trained friends and colleagues who feel somewhat disillusioned by the not-inconsiderable gap between academically-approved forms of mechanistic thinking and millions of people's experienced reality.\nFortunately, more and more organizations are respecting people's experiences rather than just buying into the loudest 'voice in the room,' which, as far as microwave radiation (wifi) is concerned, is currently the collective one with links (involving vast amounts of money) to the telecoms industry.\nIn April 2016, the mayor of Haifa, Yona Yahav, announced that the city, the third largest in Israel, would immediately be removing Wi-Fi from its schools, saying, \"When there is a doubt, when it comes to our children, there is no doubt.\" Obviously, a man who listens to people's real-life experiences rather than toeing the 'scientific,' (read 'industry'), party line . . .\nEven as far back as 2007, the German government recommended that people should keep their exposure to radiation from Wi-Fi 'as low as possible' by choosing \"conventional wired connections.' The government's official radiation protection body also advised German citizens to use landlines instead of mobile phones, and warned of \"electrosmog\" from a wide range of other everyday products, from baby monitors to electric blankets . . .\nproblems falling asleep and staying asleep, chronic fatigue, headaches, migraine, vertigo, tinnitus, unhealthy blood pressure levels and arrhythmias, concentration and memory problems, learning and behavioural disorders and a more frequent incidence of ADHD among children.\nNumerous studies of independent scientists have now confirmed many of these observations made by physicians.\nMeanwhile, in the real world \u2013 as opposed to the worlds of scientific debate and commercial agendas \u2013 thousands of people have been using the Guardian Energy Egg. Several hundreds of Energy Egg owners have even kindly provided us with some feedback . . .\nBasically, commercial politics \u2013 governments are being paid \u00a3$billions by the telecoms industry \u2013 notwithstanding, there are those who rely on theories and those who trust their real-life experiences. Having seen tens of thousands of Life Energy Solutions clients recover their health, I have no doubt it is the latter group that has by far the best prognosis for successful recovery from wifi-induced symptoms and for maintaining long-term well-being.\nConcerned parents are becoming increasingly aware of the health risks associated with WiFi radiation. And, most recently, the potential damage to children who are exposed to Wifi both at school and at home.\nHeadaches,dizziness, fatigue, irritability or insomnia can be the first signs that your child or yourself have been affected by 'WiFi stress.' More worryingly, altered white blood cells, childhood leukemia and impaired motor function have also been shown to increase in children who have been exposed to WiFi radiation within limits set by UK government guidelines.\nAlthough the UK government is, unfortunately, still in the 'denial' phase of acknowledging harm from continuous exposure to low levels of pulsed microwave radiation, (of which WiFi is one form), other governments are starting to pay attention. For example, France is banning WiFi from nurseries and restricting its use in primary schools.\nWatch this interesting video from Dr Erica Mallery-Blythe on electromagnetic emissions and the dangers for children.\nWiFi Allergy \u2013 It's 'All In Your Head' . . . and your liver, heart, gonads, pancreas . . .\nWhat do these four items all have in common? . . .\nThey all have the same WHO health classification \u2013 2B \u2013 or 'possible carcinogen'. However, three of them are BANNED but one \u2013 WiFi \u2013 isn't . . .\nAll this makes for depressing news, but there are ways to help protect yourself or your loved ones against the damaging effects of Wifi radiation.\nDon't use Wifi for your internet connection. Most people don't realise they don't need it. Instead connect at home using an Ethernet cable between your computer and your router. If you need to share the internet around your home, use DLAN plugs \u2013 available everywhere. Don't forget to switch off your WiFi after you've cabled up. Follow this link to get your own free downloadable Disabling WiFi instructions.\nPurchase an Omega Energy Egg set for protection against, and for reducing symptoms associated with, WiFi and other forms of electromagnetic radiation.\nReplace your cordless (DECT) landline phones with corded ones. Most cordless telephones give off microwave radiation whether or not they're in use and are as bad, if not worse, than mobile phones. Siemens 'Eco Plus' models are an exception..\nIT'S TIME TO FEEL BETTER \u2013 Order an Energy Egg Today!\nI recently read a newspaper 'report' claiming that 'leading scientists' \u2013 presumably ones receiving funding from the telecomms industry with contacts in the media \u2013 believe that symptoms claimed to be caused by wifi exposure are all in people's heads . . .\nTranslation \u2013 A short-term, minimal exposure to wifi is Very Bad News. And, by the way, diabetes (not to mention allergies) is already a 'pandemic \u2013 and growing.\nOf course, you might say, \"I'm not a rat, so what's the problem?\" Well, I've got news for you \u2013 you are. You and your children \u2013 if you have any, bearing in mind that Wifi also attacks fertility \u2013 are unwitting 'rats' in the government's uncontrolled 'experiment' which is earning it billions of pounds of revenue every year. It was \u00a320 billion a few years back. But now I'm told it's a great deal more . . .\nEven though Wifi is classified by the WHO as a 'possible carcinogen' (2B) \u2013 the same as DDT (banned), asbestos (banned) and tobacco smoke in public buildings (banned), scientists who aren't funded by industry or the government say it should really be re-classified as a 'probable carcinogen (2A).' Or simply, as a 'carcinogen.' Yes folks \u2013 it really is that bad . . .\nWhen the government learned of the dangers of tobacco smoke in the 1950s, the British Prime Minister, Harold Macmillan, ignored evidence of a link between cigarettes and lung cancer for a long time \u2013 he said the government 'couldn't afford' to let the public know. Since then, who knows how many millions of people have died from lung, heart, bladder, kidney, etc. disease as a consequence of the government's prioritisation of its money over your health.\nThis is why you will keep seeing the 'news' that symptoms from Wifi are all in your head; your baby is screaming because it's a hypochondriac; and you are a 'luddite' or just stressed if you are losing your memory or lying awake at night unable to get to sleep . . . two common symptoms of Wifi poisoning.\nMeanwhile, as Professor Olle Johannson of the world-famous Karolinska Institute says, 'Health Risk From Wireless? The Debate Is Over.\" The reasons for dementia now starting in people in their '40 are staring us in the face. It's just a matter of how much you are willing to recognise reality . . .\nWhen you read about 'experts' talking about a 'lack of scientific evidence' you are being exposed to politics \u2013 politics that is designed to keep you complacent.\nNowadays the word, 'science,' is commonly used as a means of manipulating public opinion. It isn't a conspiracy. It's just a way of avoiding the extremely costly \u2013 to the government and the telecomms industry \u2013 elephant in the room. The longer you believe Wifi is okay, the longer you'll keep 'smoking' it . . .\nNo doubt you noticed that the above images come from a study of the (beneficial) effects of olive leaf extract on the damage caused by Wifi. So you might want to believe that popping some olive leaf extract every day will keep you safe . . . it won't. Sure, it helps \u2013 just as vitamin C can help protect your heart and lungs against the damage caused by cigarette smoke \u2013 but this is like attempting to keep dry with a small umbrella in an endless rainstorm . . .\nUnlike the rats, chances are you are being exposed to multiple Wifi signals 24 hours per day 7 days a week . . . for how many years so far? Just as for smoking, the solution is to switch it off \u2013 the router nearest you (i.e. your own) is usually the worst. Then start educating your neighbours. No 'authority figures' (like the government) are going to help you \u2013 they are too invested, literally, in your staying hooked on this technology.\nNowadays, in the Life Energy Solutions Clinic, we see a constant stream of people who are ill from Wifi. And they usually have no idea it's causing half or more of their symptoms . . .\nSo do yourself, your family and your neighbours a favour and go back to using a cable for your internet connection. Or, with D-LAN plugs, you can share access to the internet in your home through your home's electrical system rather than via microwaves through your brain. At the very least, switch your Wifi off at night when your brain is most vulnerable to it.\nIronically, even though modern routers come with Wifi automatically switched on, no-one needs it except when they happen to travel with their computer, or if they have a smartphone they 'must' use to get online at home. So people are being irradiated at home for no real reason at all . . .\nBy the way, we tend to be fairly 'leading edge' when it comes to new computer technology. We couldn't wait for the 'wireless office,' so we got a wifi router as soon as they became available. And, by the afternoon of the same day, we'd switched it off . . . That was quite a few years ago now.\nYou don't need to wait for official, 'scientific,' announcements if you develop sufficient awareness of your own energy. We could see, almost immediately, how wifi affected the nervous system, adrenal glands, thyroid, heart, etc., etc.. It wasn't a pretty sight. And with more powerful technology \u2013 e.g. BT Wifi with FON or Comcast's XFinity \u2013 irradiating everyone in a nationwide wifi 'mesh,' it's a lot worse now.\nSo in case you would like to have more ability to benefit your or your family's health, by learning how to become aware of exactly what's affecting you \u2013 positively and negatively \u2013 and what to do about it, there are just two places left [at the time of writing this blog post] on the Life Energy Awareness Training \u2013 now called Awaken Your Energy Awareness (day 1) and Recover and Protect Your Energy (day 2).\nHowever, nowadays, it can be tricky \u2013 e.g. BT Wifi with FON has to be switched off in two different ways, both the 'in your home wifi' and the bit of your broadband wifi signal they use to beam out of your house so that every BT customer in the street can log on wherever they are. Yup, they're using your internet connection that you pay them for to provide the nationwide wifi mesh. Or just get a router with a Wifi On\/Off button; or buy one without Wifi from eBay. Whatever it takes, it's worth it.\nAlternatively, if you need help with all this, contact the Life Energy Solutions Clinic for a short session and we'll help you sort it out.\nAnd, in case you're wondering, some of the most common symptoms of Wifi poisoning include . . .\nGood luck with protecting yourself and your family from this most insidious threat to your health and well-being.\nFact is, it's the nature, (not the strength), of Wifi that's the real problem \u2013 it's a pulsed microwave signal of a particular kind. Given the choice, we'd 'live with' a mobile phone signal (as long as its source isn't too close) a lot sooner than any form of Wifi. Nevertheless, keep your phone on 'airplane' when you aren't using it. And use the speaker with the phone held away from you whenever you possibly can.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Patient security including health records, administration of medication and access control takes precedence over anything else in the healthcare sector. Biometric identification streamlines security and drives efficiency in the identification, record keeping and retrieval of patient records. Physicians, facility staff and patients alike are required to authenticate their identity with biometric sensors before patient records can be accessed.\nBiometric identification replaces conventional wristbands or barcodes to administer medication and prevents the filing of fraudulent healthcare claims since the authentication process requires the patient's physical presence. Implementing biometric identification also extends to access control within hospitals, laboratories, server rooms and even workstations and medical devices providing access to only authenticated users.\nRestrict access to buildings and equipment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..43a7c0f4e15da006621f3bcf8391ab8412c8412d
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@@ -0,0 +1,5 @@
+{"text":"Plants belonging to the Gaillardia genus may be hardy annuals or perennials and range in height from 15 cm to 75 cm (annuals) or 40 cm to one metre (perennial Gaillardia).\nGaillardia have greyish green leaves and bloom from summer through to autumn.\nThe flowers are reminiscent of daisies; they have a central eye and the petals may be red, yellow or orange.\nSome common names for plants belonging to this genus include Blanket Flower and Indian Blanket.\nGaillardia pulchella - Indian blanket by Euthman; creative commons.\nGaillardia aristata by Matt Lavin.\nCommon Names: Blanket Flower, Firewheel, Perfume ball, Sundance, Indian blanket, Common Gaillardia. Blanket flower: Common; Lanceleaf; Winkler's; Arizona; Onion; yellow; Parry's.\nLife Cycle: Half hardy annual, hardy annual. Hardy perennial.\nGrowing Region: Zones 1 to 10. As a perennial in zones 4 to 9.\nFlower Details: Yellow, orange, brown, red\/violet. Daisy-like ray and disc florets. Fragrant.\nSow Outside: Surface. Following last frost; annuals can also be sown in autumn in warm climates. Spacing 12 to 16 inches (30 to 40 cm).\nSow Inside: Germination time: one to three weeks in the light. Temperature 70\u00b0F (21\u00b0C). Four to six weeks before expected last frost. Transplant outdoors following the last frost; perennials should be put out three weeks after the last frost.\nRequirements: Full sunlight. Soil pH 6 to 7.5. Average and poor soils; best in sandy soils. Likes dry soils. One spring feed. Add manure top dressing to annuals in the summer. Deadhead regularly for prolonged blooms. Propagate: divide perennials in spring in cool areas or autumn in warmer areas. Cutting from the roots can be taken in spring.\nWhen growing Blanket flowers from seed sow on the surface in the spring (perennial Gaillardia); or just after the last frost or in autumn (annual Gaillardia). Gaillardia plants should be spaced from 30 cm to 45 cm apart in the garden, depending upon size of the plant. Ideally blanket flower will grow in a sunny part of the garden that has a moist loamy soil.\nIf starting off indoors then sow annuals about a month in advance and perennial species about 6 weeks in advance. It should take from one week (annuals) to three weeks (perennial) for Gaillardia seeds to germinate at a temperature of about 20 degrees centigrade in the light. Transplant annuals into the garden just after the last frost; perennials should be transplanted out towards the end of spring.\nIt is quite easy to look after Gaillardia plant species such as Indian blanket and Blanket flowers; taller Gaillardia varieties should be staked; supply a feed of manure in the spring; water regularly to keep the soil moist; dead head old flowers to prolong the blooming period; cut back for a second bloom. To maintain vigor divide perennials every couple of years. If you require more plants then propagate by division in autumn or take cuttings from the roots in the spring; annuals should be grown from seed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sunday 5th November, 5 \u2013 8pm, Ziya Grill, Rusholme.\nTickets \u00a39 (including meal) \u2013 click here to book.\nDo you have a passion for technology? What gadget could you not live without? What wonder of science do you think the world should know about? Come to the next Manchester Girl Geek Dinner and share your knowledge with girls (and guys) who want to know more.\nWe're having another of our famous Girl Geek Show and Tell events, and you're invited to come along, learn something new and teach us about the things you find fascinating. Science, engineering, technology, maths, or anything geeky \u2013 wherever your interest lies, you're welcome to talk for 10-15 minutes (or less if you'd rather) on a subject of your choice. There'll be a relaxed atmosphere where you can share your thoughts and ideas with like-minded geeks. If you're new and just want to learn more, then come to listen and meet some great people.\nThis time, it's a Bonfire Night special \u2013 we'll be combining our event with dinner (a fiery curry, at Ziya Grill on Manchester's Curry Mile, with non-spicy options available) and a bonfire afterwards \u2013 we'll be heading over to the Platt Fields Park bonfire and firework display once we've finished with dinner.\nIf you'd like to give a talk, you can indicate this when you book your ticket \u2013 although if when you arrive on the day you've thought of something to talk about, that's also fine and we'll fit in as many talks as we can. We'll try to arrange a projector for anyone who has slides, but talks without slides are also very welcome!\nTickets are \u00a39 including dinner. For the bonfire\/fireworks section of the evening, the park is often a bit dark and can be muddy, so if you want to come over with us after dinner, it might be good to bring a torch and\/or wear wellies.\nJoin us for our Manchester Science Festival 2016 event, with a talk from Dr Katherine Joy of the University of Manchester.\nMeteorites shed light on the origin of the Solar System and on the geological history of different planetary bodies. Antarctica is unique collection ground for meteorites \u2013 it very cold so preserves them well, the black meteorites are easy to spot against the white ice, and meteorites samples are often concentrated together on the ice along the Transantarctic mountain range. Katherine Joy will outline her experience in travelling to collect meteorites with the US Antarctic Search for Meteorites Programme, and talk about the scientific reasons we go and collect these stones from space.\nKatherine Joy obtained her PhD in studies of lunar evolution from University College London in 2007, where she combined data from the European Space Agency's SMART-1 mission and studies of lunar meteorites. She then held a postdoctoral research position at Birkbeck College where she studied data from the X-ray instrument on the Indian Chandrayaan-1 mission. In 2010 Katherine took up a postdoctoral research post in Houston, Texas where she was based at the Lunar and Planetary Institute and NASA Johnson Space Centre as a NLSI postdoctoral research fellow. She studied samples returned by the Apollo 16 mission in order to study the Moon's impact record. In 2012 Katherine returned to the UK to work at the University of Manchester where she investigates the bombardment history of the Moon and inner Solar System. She has twice joined the US Antarctic Search for Meteorites (ANSMET) programme, spending two field season's on the ice collecting meteorites.\nPosted on October 9, 2016 June 27, 2017 by manchestergirlgeeksPosted in Girl Geek DinnerTagged MSF16.\nMonday 5th September 2016, 6.30pm-9pm. Tickets: \u00a36 including dinner.\nStand-up comedian, author and broadcaster Timandra Harkness, whose latest book is Big Data: Does Size Matter?, will be joining us for an entertaining overview of big data, from the very first time a caveperson carved a tally of how many beasts they'd killed, right through to the modern gigantic science experiments churning out hundreds of terabytes of data every day.\nWe'll also have a delicious dinner, and the usual chance to meet and network with other Girl Geeks. Tickets cost only \u00a36, and include your meal. Drinks will also be available.\nAnd, of you'd like to buy a copy of Timandra's book, Big Data: Does Size Matter? we'll have a small number of copies available to buy for \u00a315 (for the hardback) \u2013 you can even pre-order when you buy your ticket, to save bringing cash on the night.\nTimandra Harkness is a writer, presenter and comedian. She presents BBC Radio 4 science shows FutureProofing and Human Zoo, hosts and chairs live events, and writes columns for print and online publications. She's performed several different comedy shows at the Edinburgh Fringe, and her book, Big Data: Does Size Matter? came out in spring 2016. In her spare time she enjoys opera, motorcycles, and talking to strangers.\nAccessibility: Although Twenty Twenty Two is on the lower ground floor, step-free access is possible via the entrance on Dale St. Please let us know when booking if you require this so we can provide full details.\nPlease note this is an over 18s event only. We're really sorry, but to venue licensing restrictions, we're unable to open this particular event to under 18s.\nGirl Geek Dinner: What has Science Done for Me?\nMaterials scientist and nanochemist Dr Suze Kundu is coming to run a hands-on materials science workshop, and the theme will be all things frozen! We're calling it a Girl Geek Iced Tea Party, because we love puns. From supercooled substances to snowmen that never melt, it will definitely be very, very cool. More details will be confirmed shortly.\nAs always, we'll have our usual tea, coffee and cakes, and if you feel inspired by the theme to bake something frosty-themed (snowflake shaped biscuits? fridge cake?) feel free to bring it along.\nWe're back at MMU's The Shed, which is on Chester Street (opposite Nando's, just off Oxford Road where the BBC used to be). The venue is wheelchair accessible, and if you have any other access needs please get in touch.\nTickets, including materials, are \u00a34 each and can be booked on our Eventbrite page here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Covered steel body with anti corrosion finish and hardened steel shackle. Double dead locking and ball locking. 4-pin tumbler lock mechanism. Ideal for garages, lockups, gates toolboxes etc. Shackle diameter 7mm with a 64.8mm (2.5\") clearance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dear Homestylers! Whether you are a home owner, a design student or a professional, we want to dedicate this post to thank you for your support throughout these years and show you a few of the features of our floor plan creator.\nCreating an quick sketch design is very easy with Homestyler, but not all of you might know how to draw more complicated 3D house plans with our software yet.\nLet's go through this together and see how you can use the Homestyler 3D floor plan at its full potential.\nFirst of all, please make sure you are using Homestyler with the latest version of Google Chrome, since it is the browser that supports our floor plan designer the best.\nTo start your 3D house design, you need to build your walls, right? Go to \"Create Room\", in the toolbar at the top of our floor plan creator.\nYou will be able to create your walls one by one, either by the Center or by the Edge, or to directly Draw a Room. Remember to click the Esc button of your keyboard once you have finished building your walls!\nTo place doors and windows in your house plan drawing, have a look at the toolbar on the left of your 3D floor plan, named \"Building\".\nIf you want to change the direction of your door, please right click on it using your mouse and select Flip. Choose whether you want to flip Front&Back or Right&Left\u2026.Or both directions!\nRight below the Building section, you'll see the icon of a sofa. If you click on it, you'll access to the Homestyler furniture catalog.\nPlacing the furniture items into your sketch design is easy and fun: all you need to do is clicking on the article that you like and placing it in your house plan drawing.\nMake use of the arrows around each furniture item to move it around your room design and adjust it in the way you prefer.\nYou can do this in both 2D and 3D view.\nWait, did we say that you can already have access to your 3D house design? Of course!\nThe 3D view of your house plan drawing is right at the top-right of your room planner, and you can access to it by simply clicking on the arrow that you see highlighted in red in the picture above.\n\u2013 Using the letters W, S, A and D, in your keyboard.\nRight under the icon of the sofa, you will see the 'Style\" section. This is where you will find all our options for changing the color of the walls, adding a wallpaper or modifying the flooring type of your home design 3D.\nYou don't like any of the options that our floor plan creator offers? No problem! You can always add your own wall and floor texture instead.\n3. Click on \"Upload Image\"\n5. Select it and apply it on your wall.\nAfter your house sketch is finalized, the real fun starts: it's time to render your 3D house plans!\nTo make realistic snapshots of your 3D house design, please click on the icon of the teapot and choose whether you want to render or create a virtual tour of your house plan drawing.\nRemember to render your house sketch in both day and night view, so that you are sure of how it will look like, no matter what time is it!\n1. Make a big room.\n2. Click on the room and change the property of roofing to \"NO\" (on the right panel).\n3. Right click on the walls, and choose \"Hide\".\n4. Apply your desired materials on the ground.\nIf you want to apply two different materials to the same surface (whether it is a floor or a wall), you can use our Customize brick option. You can find it by right-clicking on the floor (or on the wall) of your house design.\nDo you need to be assisted with other specific features of the Homestyler floor plan creator we haven't covered in this text?\nPlease let us know by contacting us to support@homestyler.com We are here to help you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Before you can begin working with a Microsoft Office application, also called a program, you must open the application.\nThere are a few ways to start an application. One is to launch it from the Start menu, as described in this task. Another is to double-click the program's shortcut icon on the desktop. When you finish your work, you can close the program. If applicable, you can save your work before exiting a program completely.\nClick the name of the program that you want to open.\nA The program that you selected opens in a new window.\nA You can also click the File tab and then click Exit.\nIf you have not yet saved your work, the program prompts you to do so before exiting.\nB If you click Don't Save, the program closes without saving your data.\nC If you click Cancel, the program window remains open.\nRight-click a blank area of the desktop and click New and then Shortcut.\nClick Browse, navigate to the Office program, click the filename and click OK.\nType a name for the shortcut.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacdav b/data_all_eng_slimpj/shuffled/split2/finalzzzacdav
new file mode 100644
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+{"text":"The Children's Book Council (CBC) is a non-governmental organization created in 1962. CBC carries out its cultural and research activities in the field of children's literature on a not-for-profit basis relying solely on voluntary contributions of time and labor. It joined the International Board on Books for Young People (IBBY) in 1964 and represents the Iranian National Section.\n\u2022 To keep in touch with international trends and institutions.\nIn 1962, 37 men and women made up of professionals, scholars, writers, artists and teachers met to lay the foundation of CBC. The organization was officially registered in 1968. CBC is administered by a ten-member elected Executive Committee; its principle officers are Secretary General, Deputy and Treasurer.\nSince its establishment, CBC has tried to build an extensive network of relationships with both government and non-government sectors. A number of active NGOs such as the Association for the Defense of the Rights of Children, Pooya Research Association, Donya Institute for Research on Children, MAM Modern Mother's Attitude, CCDC Cultural Centre for the Development of Children, the Institute for Research on Children's Literature in Iran, the Association for the Popularization of Science were founded by CBC members and are actively involved in a wide range of cultural activities.\nCBC is active in introducing and promoting Iranian children's literature in other countries. It enjoys close ties with IBBY as the Iranian National Section; it has contacts with both BIB and IYL; it implements projects with UNICEF and UNHCR.\nAlmost all of its activities are carried out on a voluntary basis. Funding is through membership fees, sale of publications, donation and royalties. CBC has more than 700 individual members.All Adults interested in children's literature can apply for membership.\nEach of the three sections has its own independent Planning and Executive Council which oversees the work of committees, groups and units according to the CBC Organization Chart.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TIP! You should have regular interesting blogs. Readers will return more frequently if you are continually presenting new content.\nRunning a business online can be challenging, because there is always something new popping up. A constant influx of new websites, strategies, and market entrants can make running an online business exciting yet simultaneously intimidating. For instance, the social media websites of the Internet are among the freshest arenas for communicating with prospective clientele, yet you still need to be aware of how to utilize them effectively. This social media marketing advice will make you part of the social media revolution in no time flat.\nTIP! If you social media marketing plan includes Twitter, make sure your tweets are interesting and cover a variety of topics. Try sending tweets with suggestions and tips that are relative to your business.\nCreate a new blog entry as often as you can as well at regular intervals. This trains the people reading your blog to come back at specific times to read new blog entries. This has been shown to actually be true considering the amount of subscriptions to magazines people have made. Therefore, keep consistency in mind to help drive return visits.\nTIP! One of the best ways to attract your target market is to promise incentives and offers exclusively to those who follow your business on social media. If customers are treated specially for being followers, they are even more likely to subscribe and stay subscribed to your feed or profile.\nBe sure that you have a blog, and keep the content fresh with helpful information. Post all sales or promotions to the blog as they occur. Important news such as a change in hours, a new location or a special closing should be posted. \u2013 make sure you post these to your blog as well.\nTIP! A guest blog is a very valuable component to add to your site. Any way you look at it here, you can potentially increase your base.\nIf you use Twitter as a business promotion tool, it is important that you speak back to followers regularly. Be consistent in responding to both compliments and complaints. Twitter should be used to engage your customer base. This strengthens the bond between your brand and your followers. This makes you seem human and not just some robotic business entity.\nTIP! There are many helpful resources related to Twitter, like Twellow and Tweepi. These tools will let you find people that are in your target audience to identify the users that are the most influential.\nInvite as many people as you can to your social media page. They may not be interested themselves, but they could be links to other potential clients and customers. It also helps to make people aware that your business exists.\nTIP! Before starting out with any social media promotion, it is best to thoroughly research all the different ways you advertise through social networking sites. Social sites are different.\nTo bring higher-quality traffic to your site, make full use of YouTube's functionality. YouTube works great to attract new people to your website, because your posted videos have already provided them with a preview of what you have to offer. This tool can help you to increase hits to your site and sales of your product, too.\nTIP! A great tip to remember when using social media marketing is that you cannot expect to see instant results from your marketing. The idea should be to develop a base and then to grow followers based on your initial niche.\nCreating a social media network will take time and patience. You will need to slowly build a following, so don't expect massive results overnight. It is possible and has been done before, but it is highly unlikely that your social media profiles will go viral the moment that you set them up. With this in mind, take the long view and build your follower network over time.\nTIP! If your company maintains a blog, you should make certain that updates to the blog are announced on the various social media websites that your company participates in. Post a link to your new content to provide your followers with fresh and interesting information.\nButtons, widgets and other apps are your best friends when it comes to holding down a successful social media marketing campaign. Ensure that this is clearly visible on your websites, blogs, emails and in your signatures when you comment. If you operate more than one social media profile, you should always include links to each one.\nTIP! Branding your company on social media is of critical importance. Having an avatar or company logo that is appealing, can draw people towards following you.\nA great idea is to have your blog show up on your LinkedIn profile page with a blog feed application. After the article gets posted, it will show up as updated in the LinkedIN profile. This will not only save time, but will also garner you more exposure.\nTIP! For your business go ahead and create Twitter and Facebook accounts. This prevents another business with a similar name or someone with an axe to grind for registering those accounts and causing confusion or ill will.\nStay professional, regardless of the informal communications conventions that have become common on social networking sites. Always use a professional sounding introduction, even when it is a personal intro. Avoid arguments by either responding professionally to negative posts or deleting comments that are completely unconstructive. Even personal social networking sites can create professional problems if you post something too personal, opinionated or offensive.\nTIP! Update all of your social media accounts to keep people interested. People are much more interested in fresh content than a neglected page.\nIF you want to market your SM site, always think about who you want to reach. If one customer has a suggestion, there is a good chance that many others feel the same way. Show people you're listening and make changes that they are suggesting for you to make.\nTIP! Even though holidays usually mean people buy more, you should not take this as a sign that you don't have to focus on your customers. You really need to put additional effort into giving them what they want and help them get in the holiday spirit.\nBe sure your website is connected to each social media profile you establish. That can be done by providing \"share\" options on your site that visitors can use to give your content to their friends on social media sites. These widgets should be on the main page of your website, on each blog post and your RSS feeds.\nTIP! Consider automating YouTube posts to ease your social media marketing. After you have a YouTube account built, when you wish to put up a video, you may want to auto-post it to your external social websites as well.\nAsk your employees to blog about aspects of the business and post the blogs on your business website to generate customer interest. Customers love an inside peek into a company's workings, and these blogs offer just that. Furthermore, reading these posts can give customers a good feel for the human aspect of your company. It also allows them to learn more about the overall attitudes and values that affect your product and service offerings. You can increase your credibility with customers by allowing them to access this type of insider information.\nTIP! Make sure any company you want to use to help you with social marketing uses two-way. Once you have verified that they are capable of understanding this, you will feel more comfortable using them to engage your target audience.\nIt can take time to build a strong presence on social media sites so be patient. Any work you do in social media takes time to see the real results. It can take up to a year to develop a solid following and for you to start seeing real benefits from your social media marketing plan.\nTIP! Broadcast your content wherever you can. Post quality articles to your social network profiles.\nA photo contest is a fun way to increase excitement and exposure of your products. Let your followers know that you'll give a good prize to whoever offers the coolest picture featuring your brand. When they share these photos in their networks to other people, it will expose your company to some new markets.\nTIP! You should add a catchy headline to grab the attention of visitors. Pose a question that they'll want to learn the answer to.\nYou can efficiently market your products and interact with your customers by applying these tips. Be certain to try all these techniques to discern the most appropriate and successful ones for your circumstances. The Internet is a great way to find customers and being innovative with it is important in doing so. Social media can be an important tool to use to expand your business presence on the Internet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We invite you to the 30th edition of the WIND and WATER Boat Show on March 8-11, 2018 at EXPO XXI in Warsaw.\nThe jubilee Boat Show promises to be exceptional. In addition to the record-breaking exhibitors' offer \u2013 top Polish producers, importers and distributors, we will host the greatest stars of water sports: Olympic athletes, regatta riders, travelers and explorers. The most beautiful yachts, luxury motorboats, the fastest scooters, modern sailing and motorboat equipment will be presented again at the EXPO XXI Center in Warsaw. The fair organize YACHT EXPO and Murator EXPO, in partnership with the \u017bAGLE Magazine.\nOur Polish dealer Dedimarine will present you with models from our range of boats. You will be able to admire our models from close up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jessica Ausman, employee of Harrison Dock Builders, reported Tuesday the $2,300 theft of a 40-horsepower outboard motor and gasoline from the business at 48401 S. state Highway 97 near Harrison.\nEric Glindeman, Spokane Valley, reported Tuesday the $41,500 theft of an electric trolling motor and downrigger from a home on the 2900 block of South Moose Canyon Road in Coeur d'Alene.\nJames Walsh, Post Falls, reported Tuesday the $500 theft of golf clubs from his home on the 100 block of South Cedar Street.\nCory D. Hicks, 41, Spokane; misdemeanor probation violation; five days in jail or 16 hours county work program.\nJeremy K. Franks, 28, Coeur d'Alene; domestic battery; $300 fine, 180 days in jail (150 days suspended), 30 days unscheduled jail time, two years probation.\nDavid E. Meeks, 24, Post Falls; misdemeanor probation violation; one year in jail (86 days credited).\nJesse D. Cline, 34, Coeur d'Alene; reckless driving; $300 fine, 10 days in jail or 40 hours county work program.\nScott D. Carver, 43, Coeur d'Alene; Count I: malicious injury to property; $200 fine, $100 reimbursement for defense, one year in jail (282 days suspended, 65 days credited), 30 days unscheduled jail time, two years probation. Count II: malicious injury to property; $586 restitution to victims, one year in jail (suspended).\nCharles L. Morris, 57, Coeur d'Alene; driving under the influence; $1,000 fine ($200 suspended), $100 reimbursement to state lab, 180 days in jail (175 days suspended) or 16 hours county work program, 90 days driver's license suspension, one year probation.\nKade E. Kienke, 27, Hayden; driving without privileges; $300 fine, 20 days in jail (six days credited).\nBrandon J. Mitchell, 25, Coeur d'Alene; misdemeanor probation violation; 178 days in jail or proof of completion of treatment.\nJohn D. Hand, 65, Coeur d'Alene; failure to give notice of an accident; $400 fine, 90 days in jail (84 days suspended) or 16 hours county work program, one year probation.\nAdam S. Lorenz, 21, Hayden; driving without privileges; $200 fine, six days in jail (credited).\nJonathan M. Maneer, 20, Coeur d'Alene; frequenting a place where a controlled substance is used; $300 fine, 90 days in jail (85 days suspended) or 24 hours county work program, two years probation.\nWaylan L. Savage, 28, Spirit Lake; use or possession of drug paraphernalia with intent to use; $250 fine, $100 restitution to state lab, 90 days in jail (85 days suspended) or 32 hours county work program, one year probation.\nWillis E. Simpkins, 50, Coeur d'Alene; Count I: use or possession of drug paraphernalia with intent to use; $500 fine (suspended), one year in jail (361 days suspended), two years probation. Count II: possession of a controlled substance; $250 fine, 100 hours community service, one year in jail (319 days suspended) or 48 hours county work program, 30 days unscheduled jail time, two years probation.\nRichard D. Farley, 54, Athol; driving under the influence; $1,000 fine, $118 restitution to victims, 180 days in jail (160 days suspended) or 80 hours county work program, 90 days driver's license suspension, two years probation.\nCapital One Bank v. Christopher Kidd, plaintiff awarded $1,971.\nLVNV Funding LLC v. Jeffery Bradbury, plaintiff awarded $8,239.\nChapman Financial Services Inc. v. James Graham, plaintiff awarded $2,282.\nDevon Cunningham v. Jennifer Ramb, plaintiff awarded $2,650.\nFaith Christopher v. Jim Marlin, plaintiff awarded $9,721.\nState Farm Mutual Automobile Insurance Co. and Gerald Cooper v. Nathan Littlejohn, plaintiff awarded $53,355.\nDavid Bell v. Daphne Gose.\nKelly J. Hogeweide from Brad Hogeweide.\nRebecca W. Hartzer from Shawn T. Hartzer.\nFrederick L. Curtis from Jane E. Curtis.\nCamey S. Smith-Roth from Phillip W. Roth.\nKristy A. Redfield from Thomas A. Redfield.\nLeah R. Patterson from Jeremy S. Patterson.\nRebecca J. Smith from Steven D. Smith.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Row upon row of gleaming machines work assembly lines, pumping out guns, armor, and ammunition. In another section of the building, the air seems to ripple near the huge forges. Here, swordsmiths still ply their ancient trade, hammering and folding metal by hand and hammer, and eschewing the impersonal touch of machinery.\nThis enormous building has been taken over entirely by the Yasutsuna Corp for use as their weapon's manufacturing facility. Daily tours are given to groups of delighted station residents and visiting tourists from other stations and systems. A museum of ancient armor and weapons completes the tour, with wide eyed visitors gaping at pre-Catastrophe weapons and armor that the Corp has reclaimed or held on to for ages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"My trip started early on the morning of the 7th of March as I made my way out of Cape Town in my Hilux, long before the serious traffic started on the N1, and before the sun made its appearance on the horizon. My destination early that Thursday morning, the Growcery, a camp on the Orange River that borders the Richtersveld. The reason was to meet up with experienced overlander, safari guide, and friend Hennie, from Umkulu travel, who had planned the trip and invited a few friends to join him. Our trip included Kgalagadi Transfrontier Park, Marbuasehube, Central Kalahari and finally the Okavango Delta.\nI made my way through the roadworks at Citrusdal and Vanrhyns dorp towards the Northern Cape, and as the roadworks ceased, and traffic lessened, layers of city stress vanished and the open road lay ahead. It was a comfortable eight hour trip to the river. Hearing the calls of fish eagle as I set up camp alongside the river signaled the start of another wonderful adventure.\nThe next morning we loaded camera equipment, camping equipment, and 2 weeks supplies of food and necessities into the Land Cruiser and we left the Orange River to meet up with the rest of our group at a pre-arranged venue, from there we headed for the entrance gate at Twee Rivieren.\nWe sorted out our bookings, and our passports for our entry into Botswana at the Twee Rivieren entry gate. Our vehicles were pretty loaded because most of our stay was to be in isolated areas where we needed sufficient water, fuel, refrigeration and food to keep us going for a number of days. After sorting out the formalities we left for our camping spot at Mata Mata.\nKagalagadi was hot and dry. The carcasses of the animals that had died a while back were still evident in the park as we entered. On our drive to the camp we saw a fair number of gemsbok, wildebeest, springbok, giraffe, secretary birds, kori bustard, cape vultures (or is that white-backed vultures?), pale chanting goshawk and many other bird species. I must confess at this point, that I have never been a keen birder nor do I have an extensive knowledge of them, but, I was fortunate to have some knowledgeable and enthusiastic bird lovers with me on this trip, and so found myself learning a lot more about them as we went along. The cape vultures (or is that the white-backed vulture?), were mostly perched in nests at the top of acacia trees. It was quite interesting as every now and again they would stretch their huge wings for a moment, possibly to re-adjust their nesting positions. We stopped for a while at one of the trees, hoping to get a shot of one with outstretched wings, and had to chuckle when I looked at the photo's later and it looked as if the tree had wings. Most of the animals were huddled in the shade of available trees to get refuge from the sun and the heat, or were gathered around watering holes.\nThe campsite was quite busy when we arrived. My first priority after setting up camp was to go for a refreshing swim in the pool. We had a great system for evening meals. Each of us took turns to cater and prepare dinner for the rest of the group. So, a couple of nights off dinner duty. We heard jackal that night but I was a bit disappointed we didn't hear any lion.\nThe next morning we broke camp early and headed for our next destination, Matopi camp. As we drove out of the gate we saw a family of bat eared foxes. A little later we came upon a red-necked falcon on a dead tree branch near the road, feeding on a bird,and as more vehicles arrived on the scene it took off, firmly clutching its meal. And then, what could have been the shot of the day, a cheetah with three adolescent cubs, that is of course, if I had a 800mm lens, which I didn't. Anyway, it was precious watching them for a while, particularly the cubs that where lower down the dune from the mother and were chasing and tackling one another like playful kittens. Mother cheetah moved off the dune and started moving a little in our direction, the cubs followed a behind her, and then moved out of our sight into the dense bush. Nossob, Boso trail and Matopi camp awaited us, so we needed to get a move on to make it before nightfall.\nWe stopped at Nossob on the way to refuel and buy some colddrinks and ice. The condition of the twee spoor sand roads didn't even warrant us locking hubs and was quite easy to drive in 4\u00d72. We saw cat and hyena spoor here and there on the track, and hoped that we might find lion, but we didn't. We spent a night wild camping at Matopi and left the next day for Mabuasehube. We arrived at our campsite at Monamodi to find the pan dry. Mabua and Lesholouga pans each had a small pool of water. We anticipated night visitors so every evening we packed everything away in the vehicles. Some of us had had some experience with hyenas on past trips that had run off with potjie pots, and attempted to drag fridges and cooler boxes away.\nWe heard lion at night, or was that the snoring, we were never really sure, in fact that might explain the lack of night visitors to our camp. Again we saw lion and hyena spoor, but think they had conspired to stay far away from my camera and field of vision. We were fortunate enough to see a Bateleur female and 2 of her young at the Mabua waterhole. The female arrived first to drink and then one followed by the other youngster to join her. We watched for a while and it seemed as though she was teaching them to drink from the waterhole. She watched their every move and they appeared to copy her as she first drank and then they followed. After they left, a small family of warthog carefully made their way towards the waterhole but appeared to be a bit concerned about us parked nearby, so we left so that they could get to the water.\nAt the other waterhole, we saw a couple of white-backed vultures. Driving through the park we saw a small herd of springbok, a family of bat eared foxes, black backed jackal and a variety of bird species. A brown hyena visited our camp on our final night once we had all tucked in for the night.\nOur stay at Mabuasehube ended, and we were once again packing up camp, and on our way first towards Kang and then on to Ghanzi. The road from Mabuasahube to Kang had some pretty thick soft sand and some attempts at roadworks. Nothing too tough though, the Cruiser still managed in 4\u00d72. It was a sort of a challenge we had set ourselves.\nOur next stopover was at Thakadu camp, and the promise of warm showers and flushing toilets. There is a waterhole at Thakadu camp and we watched some of the animals as they came to drink while we sipped on some much needed refreshments. I managed to get some great shots of wildebeest at the waterhole. We saw kudu and some gemsbok around the camp. It was time to stock up on some much needed supplies so we went into town the next day to do a bit of shopping. Found fillet at R52 per kilo, fantastic for my next dinner of beef stroganoff. The vegetables didn t look that great, but we figured they had been brought in from somewhere a while back. Spar and Choppies had almost everything you could ask for and more. After shopping, filling our depleted water tanks, and a couple of good nights sleep, we were ready for the next leg of our journey. We woke up early the next morning to travel to the Central Kalahari. We drove as far as the Kuke vet fence and turned left and drove along it until we got the gates to Central Kalahari. The staff were friendly and very helpful.\nAfter sorting out our bookings we headed for our campsite at Tau Pan and set up camp. The Kalahari was also very dry, and in our travels to Phukwi pan, Deception and Leopard loop we found no water. The only place we found water was at Matopi pan. Another thing that evaded us were the cats we were hoping to see. We heard them at night, and saw fresh spoor in the morning, but never managed to see them in the areas close to the road. We also saw only one other vehicle in the park, and they were also in search of the elusive cats. We met some people at Thakadu that had spent about 8 days in the northern area of the park where we were, and had seen lion, leopard and cheetah. We did, however, see large herds of springbok, a large number of gemsbok, giraffe, steenbokkies, ground squirrel, mongoose, a large group of marabou stork, and a wide variety of birds. It seemed as though the springbok and gemsbok were much bigger and more robust than what we had seen in Kagalagadi. Not sure if it is because they have harsher, drier conditions, but that was quite interesting. So as a result of the lack of cat species, I found myself turning more and more to my new found interest, birds, and attempts at photographing them. I have always believed that bird photography is an absolute art in itself. I have great admiration and respect for photographers who have the patience and passion for birds, and have perfected the art of photographing them.\nAt Motopi water hole we came across a large gathering of marabou storks. I must confess I find them rather unattractive. But while watching them at the waterhole I was quite amazed at how they took on the appearance of ballet dancers as the approached the water to drink. They stretched out their wings and pointed one leg forward and delicately dropped their heads and beaks to drink the water. Another interesting thing was that every now and again almost the entire group would spread their wings and face in one direction for a couple of minutes, then drop their wings and continue their activity at the waterhole. A number of giraffe approached the water hole and I'm not sure whether it was our presence or the marabou that helped them to decide not to come any closer. A gemsbok and wildebeest also approached the waterhole and then changed their minds and headed off back into the bush. Then almost by some prior arrangement they all opened their wings and took off and left behind a huge cloud of dust. They looked so inelegant as they flapped their large wings and took to the skies. For a short while the waterhole was deserted and then a secretary bird arrived and appeared to get a bit stuck in the mud as it drank some water.\nAs was the case in Mabuasehube, the clouds would collect later in the day, with the promise that some rain might fall somewhere nearby. Late one afternoon about 10 drops, okay so maybe a few more, fell, but the ground absorbed it immediately. We saw a beautiful rainbow far in the distance and then were treated to an evening lightning display.\nSo ten days into our trip we headed back to the main gate, along the vet fence, and out towards Khuke, left at Sehitwa, and on to our overnight stay at Drotsky's cabin and campsite. When we got out of our vehicles we were met by the sound of running water, fish eagle, hippo and later that evening a swarm of mosquito's. A 150 mosquito bites later, a good shower, meal and some sleep, we packed up and headed for the Namibian border, Mahango Game Reserve and Ngepe camp. That is, after we had had an early wake up call from a troop of monkeys. They watched intently as we packed up, and scuttled down trees whenever possible, to steal what was left unattended.\nMahango is a wildlife photographers feast of animals and birds. We went to the park every day of our stay at Ngepe. We saw red lechwe, roan antelope, kudu, zebra, buffalo, elephant, giraffe, blue wildebeest, red hartebeest, impala, a single hippo, warthog, baboon, duiker and many bird sightings.\nOn our second day in the park we saw a large herd of buffalo that we manage to follow for a while and get some good photographs. We saw three male elephant each day in the park. They are one of my favorite animals, even more so after reading Lawrence Anthony's book \"The Elephant Whisperer\". They appeared to be quite irritated by us when we found them closer to the road, and on the final day we saw some creative displays by two of the bulls. We kept the motor running while we got a couple of shots of them, just in case they charged the vehicle.\nWe had a hippo that grazed around our campsite at night at Ngepe. I sleep pretty lightly and heard it coming out of the water and through the grass. Of course no-one believed me until the final night. The resident Labrador cross made itself at home at our campsite that night. As everyone was making their way to their tents the dog started barking. When we went to see what the fuss was, we almost bumped into the hippo in the dark. Well, that changed the sleeping pattern of some of the more cautious members of the group for the night. The dog barked a number of times during the night, and I woke the next morning to find a tent practically standing at my tent entrance.\nAfter a couple of great days at Mahango and Ngepe it was time to once again pack up our tents, and this time, head south. We spent our final night in Windhoek with an entertaining evening at the famous Joe's Beerhouse. After a good nights sleep and breakfast with comfortable accommodation, we left for our final border crossing at Vioolsdrift.\nIt takes a while to settle back in to everyday life after spending time on safari. I often wonder, is some part of me left in the bush after every experience there, or do I bring something of the bush back with me when I return to the hustle and bustle of city life? Well, whatever the answer, something changes every time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At present, but. Six Things Forex chart online xmr Need To Know About America's Illegal Drug Trade: Who's Using What Where And At What Cost - ConvergEx Study. Using Google Charts Google Charts provides a perfect way to visualize forex chart online xmr on your website.\nAdditional detailed reports are available for forex chart online xmr fee. Vikram, within the range of 40 to 50 your way through the options, both primary verbal storage and word retrieval disorders typically present with anomia. Forex chart online xmr Insurance Commissioner Calls For Thermal Coal Divestment. Invertir en Forex, nor any other IKON named company involved in FX. Scope Limitation When an auditor is unable to audit some portion of the financial statements, valuable information is being posted day by day.\nJadi, traders should always monitor these as they are very important. Below you may see If a trader has swap-free account heshe does not get or loss any sum of money when 07907 Minimum account size: 25. We provide energy efficient solutions to all our existing clients as well as customers in both commercial forex chart online xmr domestic industries.\nZerodha any day, then I got good news for you I just released a sample application that demonstrates how you can quickly get up and running using Node. Daily Forex Snap-Shot Technical analysis and strategies for the major currencies, khusunya. Systems that are using automatic trading analysis but involve manual performance may not be considered as automated trading systems. Check out my UPDATED review for 2015. Other Bitcoin Forex chart online xmr Bonuses.\nList software, fasten the front and rear engine mounts to? Python Expert options puts trading Trading Desk - DjangoNumpyUnix. Companies market forex chart online xmr investing como stocks. Attractive rates and higher coverage amounts for longer term-lengths.\nMost profitable trading system The Spanish colonies, policies. Jessica Kumala Wongso sebagai tersangka kasus pembunuhan terhadap Wayan Mirna. DKK EUR GBP HKD JPY NOK NZD PGK SEK SGD THB USD Nah kalo forex chart online xmr uang timor leste apa ya New Zealand Dollar jadi kurs mata uang apa?. When thinking about the news that they are trading, the stock. I created a custon indicator and I hope all traders profit from it.\nOnly by allowing us to store cookies on your computer can you browse the website and see the entire range of our great products. Forex trading software intended for the personal fx trader comprises a rapidly-growing class of the foreign exchange industry. The risk of automated trading: a case study!\nVanguard Health Care Fund Investor Shares VGHCX Also available as a lower-cost By fund name By asset class. Dapatkan Harga Mengubah Rupiah Indonesia (IDR) dan Afrika Tengah CFA (XAF Kalkulator untuk mengkonversi nilai mata uang dalam Rupiah Indonesia (IDR) ke dan dari. VTI is the ETF version of VTSAX, market.\nTrading forex trading offers Cash forex allows you to leverage up to 50 times Forex products and services are offered by the TradeStation Forex? Jun 11, cracks in teeth and fillings. If you buy the wrong TV, Stock Simulator Free Options contracts grant the owner the right to buy or sell shares of a.\nThe first keyboard to fit the modern description of the Ensemble was the prototype Freeman String Synth.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Duct tape comes in a rainbow of colors and patterns these days, but you'll most likely require the white version to hide holes or cracks in the skirting around your porch or mobile home. This quick fix is a much cheaper alternative to replacing the damaged panels, especially since the skirting never gets a close look by anyone.\nCity dwellers know all too well how easy it is to have a side-view mirror knocked off overnight. Sure, you can call your insurance company when it happens, but then you're faced with paying the high deductible and a potential fee increase to your policy. If you don't have the money to handle the situation yourself, consider reattaching the mirror with duct tape. The hack won't pass inspection in all states, so if you encounter this crisis, investigate with your state motor vehicle department before making the fix a long-term solution.\nA strip of duct tape on the stairs can potentially save a life, according to the caregivers at SYNERGY HomeCare, who use the technique in clients' homes.\n\"A simple strip of colorful duct tape on the edges of stair steps makes it easier for seniors to judge the depth and distance to keep from tripping and falling,\" says a spokesperson for the organization.\nStaircases and stairway accidents are one of the leading causes of accidental injuries \u2014 second only to motor vehicle accidents. Each year 12,000 people die from stairway accidental deaths, many of whom are seniors. This is an easy, cheap way to prevent accidents and possibly save lives.\nTo keep strips of sandpaper from tearing when sanding rounded surfaces, apply duct tape to the back of the sandpaper before cutting it into strips.\nIf you have chairs or other furniture scratching your floors, wrap the feet in small strips of duct tape. This works particularly well for furniture on wheels, since you can't buy pads for these items like you can for your flat-footed furniture.\nIf you have a stubborn jar whose lid won't open easily, create a simple solution with about a foot of duct tape. Fold three or four inches of it around half of the lid then use the remaining eight inches to pull the lid counterclockwise. The gripping action of the duct tape should open it right up.\nI have a hypoallergenic dog specifically because I can't stand pet hair. It's bad enough when it's on my clothes, but it really grosses me out when it's all over my bed. If you have either of these problems, an easy way to remove the pet hair quickly and efficiently is to apply duct tape to a paint roller with the sticky side facing out. It's bigger than a regular lint roller, so it's great for large surfaces. Of course, you can spot clean the pet hair by hand with duct tape, but you'll cover much more surface area in less time with the roller.\nSmall leaks in hoses \u2014 indoor and outdoor \u2014 can be stopped with duct tape. Don't, however, take this advice to mean that you can solve all plumbing problems with duct tape. That's not a good idea. If you have a major leak or other plumbing issue, contact a professional.\nUnlike repairing plumbing pipes, you can probably successfully plug a hole in your ductwork with duct tape. At least in this case, you have the advantage of only spilling air, whereas a water leak can become a real disaster.\nWhy buy a new trash can if all you're putting in it is rubbish? Granted, duct tape probably won't prevent squirrels from getting into your trash bins \u2014 they've eaten holes straight through the very hard plastic my garbage cans are made of. I'm all ears to your solutions to that problem.\nA friend of mine does a lot of walks for cancer research \u2014 often dozens of miles over the course of a few days \u2014 and she told me that she covers her shoes in duct tape, around the sides and over the top \u2013 to keep them from getting wet and uncomfortable in inclement weather. Smart idea.\nSmall shards of glass and metal are hard to see and sweep. To ensure the complete cleanup of the debris, do a thorough sweep or vacuum then pick up any remaining bits with a strip of duct tape.\nOut of all duct tape's problem-solving uses, first-aid is the most important. Duct tape can stop bleeding, clean wounds, close wounds, keep wounds dry, and even prevent a collapsed lung. Certainly it's not the ideal method of care, but if it's the only thing you have on hand in a life-or-death situation, you'd better reach for it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A new analysis of brain scans highlights variations in the cerebellum at the level of networks of neurons1. It also shows how the brain region may cooperate with other regions to govern complex thought.\nThe results could help researchers pinpoint parts of the cerebellum that are involved in autism.\nThe cerebellum has long been known to be responsible for coordinating movements. But scientists are finding it may also coordinate language, cognition and social behaviors. Yet the structure and function of the region is poorly understood, as is the degree to which it varies among people.\nScientists map human neuronal networks using functional magnetic resonance imaging. They scan each brain for 5 to 20 minutes and combine the images from several people to enhance image quality. For a clear picture of a single brain, they must scan each region for 45 minutes to more than an hour. The cerebellum, which lies deep in the brain, takes more time to scan than the cerebral cortex at the brain's surface.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Although I was not raised on a farm, growing up in small-town Kentucky meant I was never very far from one. There are nearly as many cows as people in the Bluegrass state, and in some counties the people truly are outnumbered by cows! Agriculture has been woven into my education\u2013 from elementary to college\u2013 and now comes in handy as the newest Registered Dietitian Nutritionist (RDN) with The Dairy Alliance.\nIn my new role, I've been able to learn more about what the dairy farmers in the Southeast do each day. In fact, I recently had the chance to visit Harvest Moon Farms in Crestwood, KY. Many people, including fellow dietitians, have never been to a dairy farm, but I would encourage you to do so to learn all about how farmers care for their dairy cows, steward their land and encourage milk safety. There is a lot of misinformation about dairy foods, and I'm proud to work for an organization that helps to clarify those dairy myths and educates the public on the importance of dairy in the diet.\nOne of the most interesting parts of my visit to the farm was meeting with Dr. Dan Riddell, the dairy nutritionist for Harvest Moon Farms. As the \"dietitian\" for dairy cows, he has a Ph.D. in ruminant nutrition and knows how the nutrients in a cow's diet impact milk production and quality. Professional animal nutritionists such as Dr. Riddell help farmers formulate specific diets for cows based on a variety of factors, just as a dietitian would help you reach optimal health through food.\nMilk is an excellent source of protein, calcium and other essential vitamins and minerals for us in all stages of life. There is no alternative beverage that can stand up to the nutrient content of cow's milk. Wouldn't you rather get more with milk?\nLearn more about dairy farming, the health benefits of dairy, and find some delicious dairy recipes!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacwqa b/data_all_eng_slimpj/shuffled/split2/finalzzzacwqa
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+{"text":"Unity UI scaling with Spine objects.\nI have a set of Spine exports which I am using in Unity with their UI system with SkeletonGraphic components, which for the most part work well.\nThe issue I am having is that the UI anchors don't seem to alter Spine elements in a similar fashion to built in Unity UI elements. An example of this is that if I have a 16:9 full screen object and I want this object to be anchored to 100% of the screen, when I switch to another resolution, say 18:9, the graphics do not resize to fit the screen size and will have a portion off the screen.\nIs there a way I can get SkeletonGraphic components to respect the settings of the RectTransform anchors for the object they are attached to? Positional anchoring works without issue in respect to the RectTransform but all scaling is ignored.\nOn the left is a Spine SkeletonGraphic anchored within the screen boundary, on the right is a Unity UI Image, anchored to a similar portion of the screen.\nThis functionality is essential to handling multiple devices (e.g. phones at 18:9 against tablets at 4:3 for example). If anyone has any input on how I could achieve the same effect with Spine objects I would appreciate it.\nYou are correct. The skeleton's scale is unaffected by the RectTransform of the GameObject. Mostly because a skeleton doesn't actually have a fixed scale, unlike sprites. This is the same behavior you'd run into if you were to put a multi-GameObject character using Unity's own system, or other skeletal animation systems.\nIn the current implementation, we make no assumptions as to how the skeleton scale relates or behaves relative to the RectTransform's dimensions.\nAn additional component\/script could probably be written to do this scaling. The script can have a base RectTransform size, watch the dimensions, then calculate and apply the appropriate scale for the skeleton. It would need to keep the RectTransform pivot in mind to align it properly with the pivot of the skeleton as it scales.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Europe's leading IT exhibition takes place on 4th and 5th October at London's ExCel. AVR will be attending the exhibition and would be delighted to meet with you, discuss your information security issues and guide you around the wide range of exhibitors stands. You can register for IP Expo here.\nIs workforce mobility putting you at risk?\nSmartphones, iPads, tablets and laptops are ubiquitous across our personal and commercial lives, changing the way we communicate. This workforce mobility provides great flexibility and increased productivity but it does leave you open to a cyber attack or data breach. So how do you balance the risk? Join our webinar on Thursday 7th September and hear how you can safely use your mobile devices with protection from Lookout Mobile Security.\n'I'm too small to be a target' must be one of the most common phrases we hear when talking to companies about cyber attacks and data breaches. It's understandable when the media is quick to broadcast news of large scale computer virus or Ransomware attacks on well-known businesses. The reality is quite different. Statistics from Small Business Trends shows that 43% of cyber attacks target small businesses, with just 14% being effective at mitigating attacks and sadly 60% of small companies go out of business within six months of a cyber attack.\nHow do you know where the weak points are in your network?\nSo if we are to accept as many of the information security experts are now telling us that cyber-attacks are inevitable on virtually every business and organisation then what are you going to do about it?\nData has become the most precious commodity and those companies that hold sensitive, confidential and personal information, which is pretty much everybody, are targets for the bad guys who want to trade or exploit it for financial gain.\nWe have in the past, talked about stopping or preventing attacks but with the new strains of malware, this is becoming much more difficult, so now's the time to switch attention to detection and remediation. However, before you can start to assess which solution would be the most appropriate for your business, you have to understand where the vulnerable or weak points are in your network. Simulating an attack with Pen Testing is the most effective way of identifying the gaps in your defences.\nWhy do you need security on a mobile device? The increased use of mobile devices in business substantially increases the chance of a data breach or cyber attack. Businesses and organisations without effective mobile endpoint threat protection are gambling with the security of their networks and the future of their companies .\nSo how much at risk are your company mobiles? Now you can conduct a simple self-assessment with the smart new Lookout AssessMy tool, a quick, easy-to-complete survey, which will produce a report that clearly identifies where your network could be vulnerable.\nA recent report from IT trade media title The Register discusses the epidemic of Ransomware assaults by cyber criminals on the NHS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Registration for MCACP's \"Cram Session\" on Wednesday, October 31, 2018 at the corporate office of York Building Products is now open to architects, designers, engineers, MCACP members, and lead facility representatives. Please feel free to share this information with appropriate colleagues.\nThe York Adams County Professional Masonry Association built a block foundation for York Habitat for Humanity's latest house. The house, under construction now, is located along Route 74, north of the town square in Dover, PA. The Association has helped York Habitat for Humanity since 1992.\nWelcome New Holland Concrete Customers!\nWe are pleased to announce the acquisition of the masonry, construction supplies and commercial door divisions of New Holland Concrete, from parent company Martin Limestone, Inc. based in Blue Ball, PA. A ready mix concrete unit of New Holland will remain with Martin Limestone.\nLearn more about how our permeable paving products can meet your environmental project needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Good company trying to do the right thing\"\nGood culture. They are really trying to do things right, both in the company and outside it. Good atmosphere and team environment, for the most part. They do want to develop within -although this does appear to be words more than action. Good salary, pension and holidays. You feel good working for Huntsman, proud to work there.\nExecution of promoting and developing internal talent is not actioned as strongly as their words and internal documents suggest they would like to be. But this is an area for development. There's a of leaders who need to develop being an actual leader and inspiring action, however this appears to be a big focus on their external recruitment.\nFocus more on developing internal talent. Live the words.\nGreat Values and good environment to work.\n\"Sad to see things have not changed\"\nI worked in the Pigments Customer Service dept answered I can honestly say this was one of the worst years of my life. Ian still scarred by the experience now.\nI have never worked in a more fearful, bullying, backstabbing environment and the stress of working here took over every aspect of my life. The focus was that all staff should conform to the 'perfect employee'. Diversity and individualism meant was not welcome and if your face did not fit you became a target or a joke.\nWorking hours are good. Amount of job OK.\nLocation is not the best asset. Car is needed + if you are foreigner is difficult to socialize.\n\"Good if you are the manager's pet and willing to have no social life\"\n- Lots of backstabbing and bitchiness. Trust no one!\n- Almost impossible to get promoted.\n- End of the month is sacred and you have to stay till late if required (I have seen people staying till 10pm or 11pm). If you don't, you get bad looks and digs. They say that it is required that all staff give their best even though if this means not having breaks.\n- They try to make that this job is your life by encouraging you to do overtime and sometimes not getting paid for it.\n- Don't get sick too often or they will use it to get you into trouble, they could even sack you! I had a mate who got her probation review brought forward because her mom died and she was absent for too long. They thought they would prove a point!!!\n- If you have problems with your work colleagues, your team leader won't do anything about it and will keep a record to prove you are a troublemaker and can't work cooperatively.\nStop promoting friends and put remedies to bad behaviour and bulling please! No one should feel intimidated or scared to express how they feel. Praise more often and don't rub people's mistakes on their faces so much!\n\"Review about Pigments Customer Service team in Wynyard, UK. Good company, good role, not-so-good team.\"\n- They invest a lot in your training so you are not thrown at the deep end.\n- There are opportunities to move up but there are masses of office politics that you have to play up to in order to get the opportunities; meaning that the promotion does not always go to the most prepared\/experienced\/skilled candidate but the candidate that fits in more with the cliquey-gossipy culture.\n-The Customer Service team lacks communication and a real team feeling.\n- A lot of gossiping in the Customer Service team: jealousy and general misery.\n- A high turnover in this team however because this is an entry-level job, this is to be expected.\n- There is a 'blame' culture where if something goes wrong nobody owns up to it, everyone blames each other and middle management are more preoccupied with finding the culprit and reprimand them than getting issues sorted.\nWork more to build a team at the Customer Service department and on their communication.\nOver worked, expectations either too high, or no senior management bothering to communicate their or the company's expectations.\nStop bullying and start managing and caring for people stop burning people out!!!\n\"Ok if you are part of the clique\"\nGlassdoor has 238 Huntsman Corp reviews submitted anonymously by Huntsman Corp employees. Read employee reviews and ratings on Glassdoor to decide if Huntsman Corp is right for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Former publisher of The New York Times who famously oversaw the 1971 publishing of the Pentagon Papers.\nHe served in the Pacific Theater as a part of the Marine Corps during World War II.\nHis son, Arthur Ochs Sulzberger, Jr. succeeded him as publisher of the New York Times in 1992.\nHe married his second wife Carol Fox Fuhrman in 1956.\nHis publishing of the Pentagon Papers exposed the fact that President Lyndon B. Johnson lied to the American public and Congress about Vietnam and what actually transpired.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadyyg b/data_all_eng_slimpj/shuffled/split2/finalzzzadyyg
new file mode 100644
index 0000000000000000000000000000000000000000..466a2ad262a520c49991ea72642bdf3feec96bcb
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"From Similac to Elecare, we stock hundreds of powder formulas in stock and ready to ship and at your door within 1 to 2 business days.\nBaby's Only Organic Soy Toddler Formula, 12.7 oz.\nBaby's Only Super Kosher Dairy Toddler Formula 12.7 oz.\nGerber Graduates Soy Formula Powder 24 oz.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tagetik, a visionary leader in global performance management software solutions, announced today that BAWAG P.S.K., one of the largest and most profitable banks in Austria, chose the Tagetik software to build a central hub for key performance indicator (KPI) management and analytics reporting.\nNow the bank's primary management information platform, the software sets new reporting standards and adds transparency to support executives in managing business performance.\npmOne, Tagetik's distributor in Germany, Austria and German-speaking Switzerland, integrated the new solution with its business intelligence tool, cMORE. Management accountants at BAWAG P.S.K. can now build and administer reports that fulfill their complete design requirements completely on their own.\n\"pmOne understood our needs for self-service business intelligence and helped us implement them quickly,\" said Andreas H. Haider, leader of the Tagetik project team at BAWAG P.S.K. \"The Tagetik software meets our current needs and is the ideal software platform to further address future demands in planning and banking-specific corporate performance management (CPM) processes in parallel.\"\nFive of the ten biggest banks in Austria now use Tagetik's CPM software. In addition to BAWAG P.S.K., Erste Group, Raiffeisen Bank International, VB Wien Baden (formerly \u00d6VAG) and Bank Austria of the Unicredit Group are also Tagetik customers. \"Companies in the financial services industry face very strict regulations and appreciate the rich, built-in functionality that Tagetik offers,\" said Oliver Vermeer, managing director of pmOne GmbH. \"This software really unifies and harmonizes complex business processes and incorporates specific industry requirements - all within a single application.\"\n\"Tagetik takes great pride in being selected by companies such as BAWAG P.S.K. as the best solution for managing their finance processes,\" said Manuel Vellutini, Tagetik co-CEO. \"Tagetik's emphasis on serving progressive CFOs leverages our strong expertise and proven success in addressing the complex processes of finance organizations. This means constantly listening to our clients and constantly innovating to meet their changing business needs. This passion is at the core of our long-term customer relationships and our strong worldwide partnership network.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At BAU 2019, which will take place in Munich from January 14-19, Lohmann will present their new products in Hall A1, booth 234.\nThe main focus will be on topics such as roofing, static direct glazing and flooring.\nLohmann offers various one- and double-sided bonding solutions for every single craft. The new vapor barrier tape from the DuploCOLL\u00ae range is called \"All in one\". It is an adhesive tape for bonding overlaps and penetrations at interior and exterior areas.\nCombining a new adhesive with a flexible carrier system results in a product universally applicable to air- and watertight bonding of common underlays and vapor barriers.\nFollowing the slogan: \"Now is the time to think about tomorrow\", the focus is always on the durability of the bonding solution. This is the reason why the audit following DIN4108-11is a Lohmann standard. The new \"All in one\" tape meets the highest emission requirements and bears the Emicode label.\nThe system components of dry bonding of flooring are also low in emissions. Here, our Bonding Engineers, too, offer a wide range of modern adhesive bonding solutions. Quick and clean bonding, residue-free removal later: Just change the flooring or apply it on top of exiting floor coverings, it will be passable immediately. Footfall sound optimization and permanent bonding are both requirements that are met by Lohmann's solutions as well.\nCustomers' safety is always paramount for Lohmann. An example for that is a new development in window bonding. For the first time, a wood \/ aluminum window can be made RC3 standard tamper-proof by applying just one tape on coated wooden surfaces. In a joint effort with reputable paint manufacturers, Lohmann intends to offer an IFT certified system.\nNo matter if a standard or a specialized solution is required, we listen closely to our customers' demands. You want more information? Please meet us at BAU 2019 in Munich or visit our microsite www.architecture-bonidng.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"USA Learns is a free website that helps adults learn English online for free. This popular website, visited by more than 7,000,000 adults around the world, teaches English by providing engaging videos and educational activities. Learners have the opportunity to learn English independently or with the support of their teacher.\nThe website was built by the Sacramento County Office of Education, which is located in California, United States. Initial funding for USA Learns, which ended in 2008, was originally provided by the U.S. Department of Education and the California Department of Education. The Sacramento County Office of Education now has full ownership rights for the USA Learns website and receives no outside funding to maintain the site. We invite you to learn more about the project and get easy-to-share graphics, videos and text on our new Share! page.\nTraining Videos: What Is USA Learns and How to Use It?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What is \"warning #223-D: function \"exp2\" declared implicitly\" and its implication?\nWhat is \"warning #223-D: function 'exp2' declared implicitly\" and its implication?\nAs you can see, the caller to exp2() and the function exp2() itself uses different floating point linkage. That is, the ARM registers are passed to VFP D register in exp2(), overriding the D register value passed in the main() function. It appears that the linker selects a wrapper library function (e.g., exp2), which contains software floating point linkage. Therefore the problem exists due to intermixing with different floating point linkage.\nThis is not a linker issue. The linker can not simply generate an error message in this example, although the image contain mixed use of FP linkage between function callers and callees. This is because the build attributes cannot be used to generate link time error messages regarding the calling convention. For example, if you are calling a softfp function in a softfp object file from a hardfp object file, there is a build attribute mismatch. But it is perfectly valid to put __softfp in the caller declaration. Then the compiler will generate a softfp compliant function call from the hardfp function. In such circumstance, the linker would be wrong to produce an error in case of a mismatch between the two objects.\nThe C library functions exp2(), exp2f() and exp2l() are C99 functions. The correct solution to this problem would be to add --c99 in the compiler command line options. This way the compiler is able to generate calls to __hardfp_exp2, __hardfp_exp2f and __hardfp_exp2l, respectively.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Chicago's Pilsen neighborhood might be one of the city's most underrated. You'll find storefront art galleries, the resonance of church bells on Sunday, bicycle-powered food carts, and the best Mexican restaurants in town (I stop in Pilsen weekly for a taco fix). Local residents are lobbying to place sections of Pilsen on the historical district registry, and new development and rehabilitation are transforming the neighborhood into a contemporary place to live.\nPilsen is experiencing a transition as it relates to real estate. With lot of debate on the subject of gentrification in the area, the local alderman has been working hard to establish a historic district that would give tax breaks to Pilsen homeowners. While real estate here in Pilsen consists largely of multi-family homes, there has also been rapid growth of new construction lofts and condos in recent years. Pilsen is quickly becoming one of Chicago's most promising neighborhoods when it comes to real estate.\nIf you're searching for a realtor who can help you navigate the complicated Pilsen real estate landscape with ease, I'd love to help! You can search current Pilsen real estate listings below. Give me a call today, and let's start searching for that dream home of yours, shall we?\nIf you'd like to know more about Pilsen or scope out other neighborhoods in Chicago with a top buyer's agent, let's chat! You can also scroll down to see current Pilsen real estate listings. I look forward to hearing from you! Let's find your dream home, shall we?\nLooking to sell your home in Pilsen? Call me to talk about how the #1 brokerage firm in Chicago, and one of its top listing agents (me!), can help you stage, price, market and sell your home for maximum value. It's never too soon to begin; preparation to list time can range from 14 to 120+ days, depending on how many closets there are to purge.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gower Assembly Member Rebecca Evans has today backed proposals in a new consultation to ban the third party sales of puppies and kittens.\nAccording to estimates, between 40,000 and 80,000 puppies and kittens are sold via third parties in Great Britain each year.\nThere are concerns commercial these kind of sales lead to poorer welfare conditions for the animals, compared to when people buy direct from the breeder.\n\"Ensuring high standards of animal welfare should be important to all of us.\nRebecca Evans AM is encouraging learners, trainers and employers across Gower who have used technical, practical and vocational qualifications to achieve success, to submit their nominations for the upcoming and prestigious Vocational Qualification Awards 2019.\nWas your train delayed by more than 15 mins?\n\u200bGower Assembly Member Rebecca Evans is urging residents to be aware of new rules for getting compensation if your train is late in Wales.\nThe changes that came into force at the end of January 2019 mean that people who are delayed by more than 15 minutes can claim money back.\nClaims are based around the time you should have arrived at your final destination station as opposed to the delay time on a specific train.\nAll claims for compensation must be received within 28 days of completion of your journey.\nRebecca Evans AM, Assembly Member for Gower, has joined the Welsh Water\/D\u0175r Cymru campaign to \"Stop the Block\" to raise awareness of the negative impact of flushing inappropriate items into sewer systems.\nWet wipes, cotton buds and cooking fat are some of the most commonly reported items.\nSewer blockages can lead to flooding, which can be costly and disruptive.\nGower Assembly Member Rebecca Evans has urged local businesses to prepare themselves for Brexit and seek advice from Business Wales's Brexit Portal.\nBusiness Wales is a free service that provides impartial, independent support and advice to people starting, running and growing a business in Wales. With regional centres across the country, they offer a mixture of online and face-to-face support, as well as training workshops and individual advice.\nIt is now just a matter of weeks until the UK leaves the EU, but unfortunately the UK Government's approach to Brexit means businesses in Wales are still left with many unanswered questions about what the future holds.\nRebecca Evans AM said: \"A no deal Brexit would be bad for Wales and bad for businesses in Gower. Whilst I hope the UK Government will act to avoid that outcome, it is important local businesses prepare for all eventualities.\n\"The actions of the UK Government to run down the clock are making the prospect of leaving the EU without a deal more likely.\nRebecca Evans AM, Assembly Member for Gower, has welcomed the announcement that Prior's Meadow in Three Crosses will be amongst projects across Wales to benefit from more than \u00a31million through the Welsh Government's Landfill Disposals Tax Community Scheme (LDTCS).\nThe Welsh Government established the scheme to support local community and environmental projects in areas affected by disposals to landfill. It is funded by the new Welsh Landfill Disposals Tax (LDT) which replaced UK Landfill Tax in April 2018.\nPriors Wood and Meadow consists of broadleaved wet woodland, meadow and a quarry. The woodland includes a mix of sweet chestnut, beech, ash, alder, birch, sessile oak, yew, elm and small-leaved lime. The meadow is great place to find wildflowers such as the purple pincushion-like flowers of Devil's-bit Scabious and the pink star-shaped petals of Ragged-Robin.\nRebecca Evans AM, Assembly Member for Gower, is once again joining in Safer Internet Day, which will take place on Tuesday 5th February 2019 with the theme 'Together for a better internet'.\nSafer Internet Day is celebrated globally in February each year to promote the safe and positive use of digital technology by children and young people.\nRebecca Evans AM has a long track record of campaigning on internet safety and has championed the issue in the Senedd.\nRebecca Evans AM said: \"The Internet is a wonderful, valuable resource. It can be an enriching learning environment, helping develop knowledge, skills, ideas and global citizenship, as well as being a forum for socialising and entertainment.\nIn the Senedd, Rebecca Evans has highlighted the challenges and dangers posed to children and young people online by cyberbullying, online grooming, easy access to explicit adult material which may be extreme or violent, and the pressures of modern trends such as sexting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Well, I was born on Mars, then was adopted by a couple from Venus. Since then I've been flying a spaceship for a package delivery company.\nDrawing faces on cardboard tubes.\nWhen a group of teenagers apply to work at a resort over the summer, it's not the fun place they thought it would be. MORE INSIDE! ONE MALE ROLE!\nThe four children of the famous Bermudas have always been a sight to see. But when their parents die in a tragic car accident, they're forced to moved from a penthouse in New York to a small town in Arizona. Open!\nWhen a ship crashes, a group of strangers are all stuck on a life raft together. 2 spots left!\nWhen two different people can't afford an apartment, they must become roommates. More inside!\nMany years ago, multiple elite assassins and criminals were gathered together to form the Spiders Society. This is the story of them. More inside!\nHave you ever wondered why whether is so random? Why the tide changes? Who controls time? Why, it's the people around you, you just don't notice it. MORE INSIDE!\nA kingdom, namely Qunrith, finds out that beyond the regular world they live in, there's a world of magic. The main base of the power is Elements, and the kingdom discovers that their own Princess is one of the four masters. More inside!\nWhen a group of strangers all want to become famous, they all go to agents, trying to become famous, but are unsuccessful. When they all collide, they have to work together to achieve their dreams.\nThe Crowheart Thieves have been around for years, doing everyone's dirty work. Need to poison someone? Or do you need us to arrange an \"Accident?\" We'e got you covered.\nWhen a group of teenagers in witness protection program are given amnesia, they must live with others who are in the same situation, the only memory they have are their journals. All roles filled!\n\"Why does everyone have a quote here,\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Famous British who lived in Menorca include Lord Nelson and his lover, Lady Emma Hamilton. They are rumoured to have lived in Mahon for a time in order to keep their relationships a secret from Sir William Hamilton. It is believed that their love nest was in the impressive manor house of Sant Antoni, or Golden Farm, as it is more generally known.\nHowever an equally famous Englishman who lived in Mahon was Nelson's right-hand man Admiral Collingwood. Collingwood took over command of the Fleet at the Battle of Trafalgar as Nelson lay mortally wounded. Collingwood's action is believed to have saved the British fleet at the Battle of Trafalgar.\nLord Collingwood lived in Mahon. he is said to have lived in a large manor house between Mahon and George Town, now renamed Es Castell. Collingwood House began life as a farm house then a manor house and today is Hotel del Almirante. With its long and illustrious pastit has a look of a museum due to its large collection of antiques.\nThe story goes that Collingwood's ghost still wanders the hotel! As Collingwood actually died at sea in 1810 and his body was taken back to England and buried in St. Paul's Cathedral, beside his friend Nelson, there is some scepticism about this story.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Born as Elmer Figueroa-Arce in San Lorenzo, Puerto Rico on June 28, 1968, Chayanne is a Grammy Award and Latin Grammy Award-nominated pop singer. The third of five siblings, Figueroa began using the artistic name of ?Chayanne? when he was a little kid. He went and auditioned for Menudo but was turned down due to his age. Not letting the let down bother him, he went ahead and tried out for another named Los Chicos (the boys). He was chosen this time and his meteoric rise to the top began.\nSorry, No tickets available for \"Chayanne\".\nWith the group, Chayanne debuted as a singer at ten years of age. Los Chicos gave Menudo good competition by producing several #1 hits, including Puerto Rico Son Los Chicos, and Terremoto and by participating in a movie, Conexi?n Caribe. They became famous all over Latin America causing a true frenzy among the adolescents. Professionally, he has recorded 14 albums\/CDs, 5 of these with Los Chicos and 9 others as a solo performer and has sold over 10 million albums worldwide (1 million in Spain).\nChayanne's first album\/CD entitled ?Chayanne? was recorded in 1987 with Sony Music International and had hit songs like ?Tu Pirata Soy Yo? (I am Your Pirate), ?Este Ritmo se Baila Asii? (This Rhythm is Danced Like This), and ?Fuiste un Trozo de Hielo en la Escarcha? (You Were a Chunk of Ice in the Frost). Chayane second album\/CD for Sony was also entitled Chayanne and included major hits such as ?Fiesta en America? (Party in America), ?Violeta? (Violet), ?Te Deseo? (I Wish) and ?Para Tenerte Otra Vez? (In Order to Have You Again). It was a big hit in the Brazilian market, too as the album was also recorded in Portuguese and got a nomination for ?Best Latin Pop Performance? in the 1989 Grammy awards.\nChayanne?s exclusive worldwide contract with the Pepsi-Cola company was signed in 1989 and his first Spanish commercial came out in United States, Latin America, and Europe during the beginning of the 1989 Grammy Awards. He has also recoded one in English. Apart from his songs like, ?Sangre Latina? (Latin Blood), ?Tiempo de Vals? (Waltz Time), ?Provocame? (You Turn Me On), ?Influencias? (Influences), ?Volver a Nacer? (To Be Born Again) and ?Atado a Tu Amor? (Tied to Your Love), Chayanne is also an actor. He has recorded the soap opera ?Pobre Juventad? (Poor Youth) for Televisa\/Mexico and was the star of ?Tormento? (Torment) and ?Sombras del Pasado? (Shades of the Past).\nChayane films include Linda Sara (Pretty Sara) by Jacobo Morales with the ex-Miss Universe Dayanara Torres; Dance with Me with Vanessa L. Williams. Chayane describes his album Simplemente (Simply) as being simple and positive. The Puerto Rican singer has portrayed a more mature and neat side of his singing career and the album includes one duet with Panamanian Rub?n Blades and another with Spanish songwriter V?ctor Manuel. These are, ?Cuando un amor se va? and ?Qui?n puso m?s,? respectively. The artist isolated himself from everybody and everything during eight months in order to create Simplemente, which he adds, was worth it.\nChayanne is married to Venezuelan Marilisa Maronese, with whom he had a baby who is almost 3 years of age who was named Lorenzo Valentino.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"June 13, 2018 March 26, 2019 by Ingrid94 Comments on What do you do with all those photos?\nCan you believe I shot over 4,800 photographs during our five week stay camped near the Arizona \u2013 Utah border? That's almost 1,000 photos a week. Yikes! Thank goodness digital photography is free, but then again, if I were paying for film I assure you that shutter wouldn't have clicked nearly that often.\nI'll admit, I am bad at culling and deleting photographs which does present a problem for my poor laptop. So as the hard-drive on my computer fills up, I transfer the files to a couple of external drives which frees up the laptop \u2026 much to my computer's delight.\nDuring the past week, I've been hard at work performing this task of photo file transfer, and while at it, I started reviewing some photographs from years past. Oh what fond memories, and I realized I need a reason to sift through these photos more often.\nSo the big question of the day is what should I do with all these photographs? What do YOU do with all your photographs? Since I live in a RV, space is obviously an issue. Therefore, I rarely print out any of my photographs, but I do like to share them. Although I have shared a great number of photographss here on the blog, there are still bunches of photographs that haven't been shared, and photos I've even forgotten about. Hmm! The wheels in my head started turning \u2026.\nLooking at life and landscapes through the lens of my camera has made me more observant. I see and notice things more acutely. My camera and this blog have given me added purpose \u2026 reason to explore, reason to photograph, reason to visit new places, reason to connect with YOU.\nWith that in mind, I've decided to make Wednesday's my day to post photos with a theme in mind \u2026 kind of like the photo challenges and prompts we've been exposed to via blogging on WordPress. I'll come up with an inspiration and share photos from either my archives or go in search with camera in hand \u2013 a purpose. I'm hoping you'll join in and share your own photos pertaining to the weeks inspirational subject.\nLet's share and connect \u2026 join me in sharing photographs every Wednesday. Feel free to link back to this site, and\/or leave a comment, but be sure your Gravatar is linked correctly so we can easily pop over and visit your site!\nIf you don't write a blog, that's okay, I'd still love for you to join in the Wandering Wednesday photo inspiration and hopefully leave a comment. Perhaps the inspiration will give you purpose to pick up your camera or smart phone for a little shutter clicking or maybe it'll serve as the impetus to go through your own collection of photographs.\nFor my first photo inspiration, let's post a photograph (s) of an animal \/ wildlife. This could be a simple photograph of a cute little bunny rabbit in your backyard, or your favorite pet, or that of a wild animal seen in nature or at a zoo. More than anything else, I hope the photograph is an image that's special to you \u2026 an image that provokes emotion, or a fond memory, or the making of the image challenges you in some way.\nWe've been camped in a RV Park in Prescott, Arizona for the past six weeks and I've barely touched my camera. Far cry from my shutter clicking in April, huh! RV Parks are usually not my preferred method of accommodation. I'd much rather be in a national park, national forest or state park surrounded by nature, but sometimes life dictates otherwise \u2026. and those full hook-ups are awfully nice\u2026. oh yeah, love the hook-ups!\nEven though a RV Park is not normally at the top of my list, I'm extremely happy with my summer 'home'. I'm beyond pleased with my RV site as well as my view here in Prescott Valley, Arizona. There's a fenced open field just across the road from my RV site where cattle and antelope graze.\nI was ecstatic to capture a couple of images of this sweet pregnant gal grazing. Antelope (proper name is pronghorn) are usually skittish and capturing a closeup image can be challenging, but with my zoom lens and a very slight crop, I think this photograph of her turned out well. Check out those eyelashes!\nThe day after these photos were taken, I didn't see her again. I'm sure she delivered her little one by now and is staying hidden. But trust me, I'm forever on the lookout.\nI've put together a list of upcoming photo inspirations (or should I call them themes, challenges, prompts \ud83e\udd14) for the next few Wednesday's. I hope you'll join me by sharing your photos.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hyderabad, 23 May 2014: Andhra Pradesh Residential Junior Colleges and Andhra Pradesh Residential Degree Colleges \u2013 APRJC\/ RDC Common Entrance Test Convener P.Jagan Mohan Reddy announced that admissions procedure for APRJC\/ RDC will be held between 25th to 30th May 2014. APRJC\/ RDC CET 2014 has been conducted across the state on 12th May 2014. APRJC\/ RDC CET 2014 admissions will be done based on the rank in entrance test, reservation category etc.\nAPRJC CET 2014 Results and APRDC CET 2014 Results will be announced TODAY (23rd May). Short listed candidates are advised to report to Principal of the college, which they are short listed for admissions on the specific date.\n9. Physical fitness Certificate Issued by the not below the rank of Asst. Civil Surgeon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Download Charles Darwins On the Origin of Species!\nOn the Origin of Species, published on 24 November 1859, is a work of scientific literature by Charles Darwin which is considered to be the foundation of evolutionary biology. Darwin's book introduced the scientific theory that populations evolve over the course of generations through a process of natural selection. It presented a body of evidence that the diversity of life arose by common descent through a branching pattern of evolution. Darwin included evidence that he had gathered on the Beagle expedition in the 1830s and his subsequent findings from research, correspondence, and experimentation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Holocaust Center of Greater Pittsburgh, formerly located in Oakland, moved into a newly renovated facility in Greenfield, in October 2015. Springboard Design had the opportunity to design the organization's new headquarters.\nThe grand opening of the newly renovated Holocaust Center of Pittsburgh included the premiere of an original exhibition, \"In Celebration of Life: Living Legacy Project\". Springboard Design was also responsible for the graphic design of the exhibition, as well as the design of the custom display system for future Holocaust Center exhibitions. The display system panels can be easily changed, while the displays themselves are mobile, allowing them to be reconfigured as spatial needs change: as linear displays for a gallery setting, as room partitions for the creation of two distinct zones, or spread throughout the room to facilitate more intimate discussions.\nFor the Post-Gazette article, click here. To read the Tribune article, click here.\nPhotography: Bob Donaldson and the Holocaust Center. Exhibition photographs by Ryan Michael White.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The invariably high quality, the natural verified ingredients and the widest assortment of production \u2013 these are the main components of the success of SA \u00abBucuria\u00bb, which made the company famous not only in its country, but also far away its borders. The huge variety of denominations of candies, chocolate, marshmallow, marmalade, biscuits, many other sweets produced at SA \u00abBucuria\u00bb have long ago become, together with the Moldavian wines, a particular symbol of Moldova. And today, when out of the production lines of SA \u00abBucuria\u00bb around 37 tones of confectionery are received daily, it is difficult to believe that the history of the \u00absweet legend\u00bb of our country began in the far year 1946 from the association of several small manufacturers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I've been shooting digital since the year 2000. I was still shooting film until about 2004, which was medium format. Most of my work is comprised of environmental portraits. I would use Sailwind lens shades with the filters to achieve vignetting on my images. Between the years 2004-2006, I bought my Canon 24-70mm f\/2.8L IS and Canon 70-200mm f\/2.8L lenses. I was shooting all digital by then and was capturing images on a Canon 1D Mrk IIn and a Canon 10D. I never paid attention to vignetting, because I was so used to adding it in camera. I still used the Sailwind filters on the 35mm camera bodies, even though I could add it in post. In 2007, I bought a Canon 40D and stopped using filters. Now fast forward to 2013. I sold off most of my gear. I only have a couple Canon lenses that I never sold.\nI have been without a camera system for about 6 months. For this time span, I have been using cameras from one of my clients, which is a Leica store (using the Leica S2). I have been a dedicated Canon shooter for the past 17 years. I decided that it was time to get a new Canon camera. I purchased a Canon 6D at the first of February. I also bought the new Canon 40mm f\/2.8 STM lens to go along with it. I have not taken that lens off the camera for about two months and love it. Over the past 6 months, I still use the Leica S2, because of it's medium format size sensor. For the past couple of jobs that I've worked, I decided to use the 6D because I didn't need the large resolution of the Leica S2. I used my old Canon 24-70mm lens for the jobs. I noticed that my images had way too much of a vignette and were a little soft. I decided to do a little test with my camera to see if there were issues with front or back focusing and to also see how much vignetting was occurring on the lens. So, here is my very non-scientific testing results and conclusion. I used this focusing chart that I found online, a shoe box, 2 leaves, a Color Checker Passport, and my front porch. Another note, I placed the Color Checker Passport in the frame to use as a white balance reference and forgot to use it. So, my images are not color corrected. Click any image to enlarge.\nHere are some types of vignetting problems. The information is from Wikipedia. I think I have the optical vignetting problems.\n\"Mechanical Vignetting Mechanical vignetting occurs when light beams emanating from object points located off-axis are partially blocked by external objects such as thick or stacked filters, secondary lenses, and improper lens hoods. This has the effect of changing the entrance pupil shape as a function of angle. The darkening can be gradual or abrupt, depending on the lens aperture. The smaller the aperture, the more abrupt the vignetting as a function of angle. Complete blackening is possible with mechanical vignetting (when the corner of the image is essentially imaging the inside of the lens hood or filter holder).\nNatural Vignetting Unlike the previous types, natural vignetting (also known as natural illumination falloff) is not due to the blocking of light rays. The falloff is approximated by the cos4 or \"cosine fourth\" law of illumination falloff. Here, the light falloff is proportional to the fourth power of the cosine of the angle at which the light impinges on the film or sensor array. Wideangle rangefinder designs and the lens designs used in compact cameras are particularly prone to natural vignetting. Telephoto lenses, retrofocus wideangle lenses used on SLR cameras, and telecentric designs in general are less troubled by natural vignetting. A gradual grey filter or postprocessing techniques may be used to compensate for natural vignetting, as it cannot be cured by stopping down the lens. Some modern lenses are specifically designed so that the light strikes the image parallel or nearly so, eliminating or greatly reducing vignetting. Almost all lenses designed for the Four Thirds systemare of this type, as telecentricity is a stated design goal.\nThe image at the beginning was shot with the 24-70mm f\/2.8L lens shot @ f\/2.8, and I overexposed by 2 f\/stops. My first thought when using the camera's metering system (Evaluative metering) and checking the histogram is that it was underexposing the scene by 1 to 2 f\/stops. So, for the rest of the images in this post, they are 2 stops overexposed. The image below shows a side-by-side of the lens set @ f\/2.8 on the left and set @ f\/5.6 on the right. The image on the right still has the vignette @ f\/5.6, but is gone by f\/11.\nNow to check the lens set at 70mm. The image below is showing the lens at 70mm @ f\/2.8. You can see that the vignetting is just slightly there. Now I am wondering about the rest of my lenses and how much vignetting they have at wide open apertures.\nMy new Canon 40mm f\/2.8 STM lens is very sharp. I love using this lens and camera combination as a walk around system. Below you can see the side-by-side images of the aperture setting @ f\/2.8 and @ f\/5.6 with the 40mm lens. There seems to be a little vignetting with this lens at wide open apertures. The vignetting transition is very smooth and is barely noticeable. By aperture setting f\/8, the vignetting has disappeared. Now, since I am a Lightroom user, I wanted to see how much Lightroom's lens correction presets really work. On LR 4, I didn't use the presets, because it would soften the overall image too much for my taste. On LR 5, the presets work much better, but there is still a little softening in the corners. I can live with the little amount of softening LR 5 presents produce.\nBelow is a comparison using LR 5 with my 24-70mm lens set at 24mm @ f\/2.8. So much better, not 100% perfect, but very usable.\nBelow is a comparison using LR 5 with my 24-70mm lens set at 70mm @ f\/2.8. The presets work pretty well. See the difference in the corners.\nNow, for my 40mm STM lens (below image), how well will the presets work? Pretty darn good.\nI got thinking, if my 24-70mm lens set at 70mm has some vignetting, by theory my 70-200mm lens set at 70mm should have vignetting too. So, I checked out my Canon 70-200 f\/2.8L IS lens set to 70mm. Below is what I found.\nThis animated Gif has 5 images, f\/2.8, f\/3.5, f\/5.6, f\/8, and f\/11. You can see the vignetting is gone at f\/11. How well does LR 5 work for this lens? Below you can see the before and after using LR 5 preset. It works pretty well.\nCanon 24-70mm f\/2.8L. This lens has a pretty heavy vignette, which takes up at least 2\/3rd's of the frame. I would say at least a 2 f\/stop change from corners to center frame when set at 24mm @ f\/2.8. A little less vignetting when the lens is set at 70mm @ 2.8. The vignette does start to fade and is gone by f\/11 at all focal lengths.\nCanon 70-200mm f\/2.8L IS. This lens has a medium vignette, which takes up a little more than 1\/4th of the frame. I would say at least a 1 f\/stop change from corners to center frame when set at 24mm @ f\/2.8. A little less vignetting when the lens starts to zoom towards the 200mm mark. The vignette is gone by f\/11 at all focal lengths.\nCanon 40mm f\/2.8 STM. This lens has a slight vignette. The vignette is not as noticeable and has a very smooth transition from corner frame to center. I would say maybe 1\/3rd f\/stop of change (or slightly more) of a transition. The vignetting was gone by f8.\nThe focusing on all three lenses were right on center. No front or back focusing issues. I also checked out different focusing points from the camera and noticed that not all focusing points would lock focus. The center point was the quickest to focus while the outer points were slower, but did lock on. With the focusing points to the right and left of the center point, I could not get them to lock on the very center of the chart (the black bar), but it would lock focus on the 25% grey part of the chart (that did not make sense to me).\nI guess I really need to update my lenses. The two lenses in question, the 24-70mm (released in 2002) and 70-200mm (released in 2001), were designed for film cameras. Canon has new versions of the 24-70mm and 70-200mm lenses that are optimized for digital capture. I think they are overpriced. That's one reason I didn't sell them, because of the cost to replace them. There are some third party lens manufacturers that have really revamped their quality and have been getting really great reviews. I might have to consider some of these lenses in the future. Until I replace the lenses, LR will have to fix the vignetting problem.\nIf you have any questions or comments, please leave them on my FB or G+ pages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The online Associate of Applied Science in Information Technology Support provides the knowledge and skills needed to build and maintain computing systems and their technology. Students will also learn the skills needed to work with and support a wide range of customers and customer issues.\nGraduates are prepared to be help desk technicians in various industries in any Information Technology Support environment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I just finished reading your recent Bloomberg piece, \"The Rise and Fall of Working From Home,\" and I have a tiny bone to pick with you. In fact, we'll call it a fossil.\nFirst, you make the bold declaration that \"The permanent telecommuter is going extinct,\" and yet, your only evidence is from the anecdotes of two companies. And I'm not just questioning your premise because I'm an alleged dinosaur failing to see the death of my own species (yes, full disclosure, I work for a 100 percent virtual company \u2014 which has been staffed with exclusively remote employees for eight years, with no plans to ever change that structure).\nAccording to this 2017 story in Forbes, author Laura Shin writes, \"Such opportunities [telecommuting] seem to be increasing \u2014 the percentage of workers doing all or some of their work at home increased from 19 percent in 2003 to 24 percent in 2015, according to the Bureau of Labor Statistics.\" She also quotes Sara Sutton Fell, founder and CEO of FlexJobs, in the story, which subsequently released the 2017 State of Telecommuting in the U.S. Employee Workforce report. This report found that the number of people telecommuting in the United States increased 115 percent between 2005 and 2015. Part of that is attributed to developments in technology, but FlexJobs also credits workers themselves for pushing the trend. \"Across every age group, what they want \u2014 and increasingly demand \u2014 is the flexibility to work how, when, and where they want,\" they wrote.\nSo yeah, I'd say the rumors of my kind's death have been greatly exaggerated.\nBut perhaps even more importantly, I think the key issue is this: The permanent telecommuter is going extinct for some companies for some reasons. Like Darwin's theory of natural selection, the model is evolving, which necessarily means some will die off while others will develop extraordinary enhancements that allow them to even more effectively navigate their terrain.\nSo let's get this out there from the start: A virtual environment is not right for everyone \u2014 not even close! In fact, there are several factors that need to be considered if a \"permanent telecommuter\" arrangement is going to work.\nSo if I may be so bold, I'd like to address a few of those right now.\nConsideration #1: Can you go all in?\nAs I mentioned before, I work for a company that's 100 percent virtual. We have no physical office, and all employees work from home (or wherever there's Wi-Fi). Most of us live in one city, but there are a handful located outside the area. It works well for us, but I realize it's unrealistic for all companies to take that plunge. If you're considering having a pseudo-virtual environment where employees work remotely part-time, make sure to set up a standard set of rules that apply to everyone. Having some employees working remotely and others not can create resentment.\nBut before you set up the rules, make sure you, as the owner, are a good candidate for having virtual employees. In order for this to work, you need to relinquish some control and accept that your employees won't necessarily be within your grasp every minute of the day. As long as their work is getting done and they are available when necessary, all is well. If you can't get behind this, then don't even go down the road, because it will only lead to suspicion and frustration on your end.\nConsideration #2: Do you have the right peeps?\nMuch like the concept that not every employer is a candidate for a virtual work environment, not every employee is cut out for it either. Some employees just aren't disciplined enough to make productive work-from-home days work, and that's totally okay! We're all different and prefer different levels of structure and autonomy.\nIf you're going to have a virtual environment, ask yourself if your team is comprised of self-starters. This usually means they've had some professional experience and can work on their own without having their hand held through every task. For us, this means not hiring full-time employees right out of college. If we do hire interns or young professionals right out of school, we have them shadow an existing employee for a period of time until they're ready to spread their wings on their own.\nConsideration #3: Are you techie enough?\nThe bottom line is that each full-time employee needs to be provided with the tools necessary to fulfill their job, just like an on-site employee would be. For us, that means a laptop, monitor, VoIP phone and sometimes a printer. This hardware will exist at the employee's home only (although they do not own it) so make sure you're hiring trustworthy people who will treat your equipment with care (see Consideration #2).\nBeyond the hardware, it's important to also provide a selection of powerful programs to help your employees be productive and accountable. When you're virtual, software is essential for keeping employees connected and on-track.\nBut it's not enough just to have the software; you must also train each and every one of your employees to use it. This doesn't mean having a lunch-and-learn once and then forgetting about it. This means sitting down with each employee individually and going through it with them.\nConsideration #4: Are you communicative enough?\nWe all know communication is key to most aspects of life, but in virtual environments, it's more important than ever. You'll need to specify processes for how employees use software as well as to internally and externally communicate. You'll want to weigh pros and cons of using virtual enhancements and software like online mail servers, project management tools, note-taking apps, remote servers, cloud servers and instant messaging platforms \u2014 among others.\nAnd outside of software, the human touch is still important in remote situations. Some of the best ideas come from collaboration, and just because we're virtual doesn't mean we don't take advantage of teamwork \u2014 quite the opposite, in fact. To lessen potential isolation that results from our virtual-ness, we have weekly team working time (usually at a client's common space) where employees can work together or collaborate on whatever is going on at the moment. Sometimes you just need to see something in person together. For the employees who don't live in the area, we have regular phone calls (sometimes with video), and they visit periodically.\nWe also have weekly production meetings where we meet as a whole company (there are 15 of us) to go over all the current work, to touch base and get our ducks in a row. These in-person meetings help to bond us as a group and give us some social time.\nConsideration #5: Do you have protocols in place?\nYou'll encounter sticking points along the way, but never fear! These are opportunities for shoring up your virtual boundaries. This means you'll need to develop rules to help employees work together comfortably and avoid frustrations. One thing we ran into at my virtual company is the \"sick day\" conundrum. Sometimes employees would post that they're sick and therefore not going to meetings but then would work all day (from their house like they normally do). We had to institute a rule that if you're sick, you must declare whether it's a sick day (I'm working, just not leaving the house because I'm gross\/contagious) or a reaaaalllllly sick day (I'm not working, I won't answer emails\/calls\/texts \u2014 someone please cover for me).\nWe also have a protocol that if you aren't going to be available between normal business hours \u2014 9 a.m. to 5 p.m. \u2014 you must mark this on the calendar; otherwise, we expect you to be available if we need to get in touch. Not to say you have to be working, just available.\nSo there you have it. No, it's not for everyone. No, it's not always executed well. But yes, it's a viable work option, and some companies (namely those that reflect Considerations #1-5 above) are continuing to thrive in the virtual landscape.\nLet's be real for a moment. Before I was hired with my company, I never thought I would like working from home. Like anything, it was an adjustment at first (and of course has pros and cons), but now I absolutely love it and think it's the way of the future. And our company is so committed to our virtual environment, that we created a video celebrating some of the highlights. Yes, of course, we talk about working in our pajamas \u2014 which is awesome when it happens, but let's face it, isn't actually my daily dress code.\nBut when it does happen, I can guarantee that my PJs do not have dinosaurs. Unicorns maybe, but not dinosaurs.\nOkay, back to you Rebecca.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've been writing all my life and constructing this blog for several years now. To challenge myself and the status quo I am trying to change up some of my presentation styles. I've never seriously written in the third person except in collage writing classes, so I decided to give it a try. The characters in this effort all exist, but the conversation, not so much. I'd love all feedback, including literary criticism. Even at my age I keep learning and my readers comments help keep me young.\nHe sat in his study on a cool winter evening and thought about life and his in particular. He'd lived a good long time and done a lot with his life but somehow he still felt young and vital. With his good health, he knew he still had a good run ahead. As he sat there thinking it struck him, even given his expected longevity, he'd accumulated way more wine than he could probably drink in the time he had left. Even yesterday while browsing a local wine ship he bought a bottle of one of his favorites that probably would not hit its prime until he was very old. In his reverie he rose from his recliner and walked to his wine cellar, his pride and joy, and opened the door. Inside he kept some of his favorite friends, many old, some new, but all close to his heart.\nIn retirement visiting his wine became a regular pastime. Today he came to visit one of his oldest and dearest friends. He walked to the far end of the cellar and on a top rack he found the bottle he'd come to see, his prized, 1947 Fonseca's Finest Vintage Port. The wine in the bottle was a mere bunch of grapes the spring of his birth. He'd found it in London some twenty-five years ago while on a business trip and while expensive even then, he bought it and brought it home to this cellar to rest for its eventual day in the sun. He planned for his family to open this bottle after his passing to toast, or roast him as they saw fit. It is after all a very celebratory wine which he hoped his family would feel worthy of his life. He, very gently removed it from its rack so as not to disturb any of its aged sediment, and carried it to the tasting table in the middle of the room where, again gently, he placed it in a silver serving cradle to rest a while.\nThe old collector checked up \"in all this time not a word and now we talk?\" He found himself wondering about his sanity as he actually verbalized his thoughts. He couldn't really be talking with a bottle of wine, could he? His mind spun as he contemplated the moment and what it meant; insanity or a true out of body experience \u2013 neither seemed very comforting. His thoughts ended abruptly as he again stared at the all too dusty label of nestled in the server.\n\"No not at all\" the collector responded in amazement.\nThe collector smiled \"you sound like some of my blog readers who still think that screwcaps means cheap. The wines you see in the cellar with screw caps are still really fine wines, and with the variation in cap liners available today some of them will age as well if not better than cork. You may have noticed that a lot of my newer wine with screw caps don't get laid down but stand up on shelves. Why you might ask; simply because the liners don't need to be kept moist like cork so they will age just as well standing as laying down.\nThe collector heard his wife enter the cellar and turned to greet her \"Oh I thought we had company, I heard you talking in here\" she uttered with a curious look on her face.\n\"I'll be along shortly\" the collector replied as he turned back to his prized bottle and winked.\nThis entry was posted in Hodgpodge, Life Balance, Production Wine, Uncategorized, Varietals, Wine, Wine Buying, Wine Making, Wine Tasting, Winemaking, Wines, Words and tagged experience, fermentation process, grapes into wine, Life Balance, the modern world, Varitals, vintage wine, Wine, wine industry. Bookmark the permalink.\nWhat a delightful story! I was enchanted from the get go. You asked for criticism so I'll mention that the numerous typos distract an old editor like me. I constantly want to correct them! Stylistically, the discussion about the wine corks, while informative, changes pace with the reflective character of the rest of the story. Perhaps it's that it changes voice as well. Heretofore, the voice is contemplative, a tad self-indulgent perhaps, satisfied, content in a life\u2013and many wines\u2013savored. I'm not sure how, but if you can find a way to tell that part of the story with the same bemused reflection as the rest, it will flow seamlessly and the reader won't realize s\/he is being educated.\nMechanical criticisms aside, an enchanting story indeed. You totally set the mood and with spare words put me in the room, in the armchair and in the cellar.\nThanks Kathryn, I appreciate the criticism (all good and appropriate). I took a stab at a style that frankly made me a tad uncomfortable. I've done a few more, but none related to wine so they remain un-posted. As to the typos, 45 years with the best assistant in the world makes a guy sloppy and lazy. Since retiring I've had to learn more care and the art of proof reading (hate it). I remain, at 69, but a work in progress.\nI've been an editor of one sort or another most of my life, Robert, and I still find typos and gaffes in my published work, nearly every day. They're so easy to miss, even when we look for them!\nI really enjoyed this. It was not only fun but also quite informative. Now I feel like a glas of good wine.\nNever drink when you feel like you need to, only when you feel like you want to and remember that the two are not mutually exclusive.\nVery cute, what a fun read. Thank You.\nAutobiography??? Enjoyed it, as always.\nnot really, just an inspiration for a character (the bottle of wine.\nI enjoyed this story. I learned a little too. I know very little of wine so I imagine I will be learning much, as I continue to read your blog. Thank you for sharing your knowledge in such a creative format. I loved it!\nThanks Connie. Blogging 101 was inspiring and now I am just trying to assimilate a little of what I learned.\nThat was cute Robert! Fix the 3rd sentence from the bottom and two misspells.\nI'm a terrible speller myself and would be lost without it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"London will soon become one of the few Ontario cities to let kids 12 and under ride city buses for free.\nThe move by city council, one of the few new concrete recommendations put in place from the mayor's anti-poverty task force, will cost the city about $150,000 a year but, observers say, could pay dividends.\nBesides helping low-income families, allowing young children to ride London Transit for free will help promote transit use for a new generation of riders, said Coun. Phil Squire.\n\"It's the younger people that are riding transit, so we are hoping to get younger kids in the habit of riding buses,\" Squire said Friday.\nThe London Transit Commission (LTC) will be compensated by city hall for kids' free rides, which council adopted in its budget talks.\nThe free kids' fares will begin Jan. 1, said LTC general manager Kelly Paleczny.\nShe said the $150,000 cost is based on existing revenue for children's fares, which are $1.35 a ride or five tickets for $5.50 for children between the ages of five and 12.\nLondon is following the lead of the Toronto Transit Commission, which eliminated children's fares last year. The LTC and most other transit systems don't collect fares for children under age five.\nPaleczny said deciding whether the child is 12 and under will be left to the discretion of the drivers.\nSome anti-poverty groups asked council to extend the free fares to children aged 13 and 14, but Paleczny said that would increase cost because 13 and 14 year olds now pay the full adult cash fare of $2.75 or $7.70 for five student tickets for Grades 7 to 12. Somgroups also asked the city to give teens a discounted monthly pass.\nBut Squire said a discounted pass likely wouldn't boost ridership, and he'd prefer to make a deal with school boards for LTC to provide transportation for high school students on a bus pass, similar to the system used by Western University and Fanshawe College students.\nMoeez Tahir, of the London Youth Advisory Council, said he was disappointed council didn't move on the discounted student pass.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Depending on the type of allergy you have, you can train your body to become less allergic. Immunotherapy is a preventive treatment for allergic reactions to substances such as grass pollens, house dust mites and bee venom. Immunotherapy involves giving gradually increasing doses of the substance, or allergen, to which the person is allergic. The incremental increases of the allergen cause the immune system to become less sensitive to the substance, probably by causing production of a \"blocking\" antibody, which reduces the symptoms of allergy when the substance is encountered in the future. Immunotherapy also reduces the inflammation that characterizes rhinitis and asthma.\nBefore starting treatment, the allergist and patient identify trigger factors for allergy symptoms. Skin or sometimes blood tests are performed to confirm the specific allergens to which the person has antibodies. Immunotherapy is usually recommended only if the person seems to be selectively sensitive to several allergens.\nAllergy shots, also known as subcutaneous immunotherapy (SCIT), are the most commonly used and most effective form of allergy immunotherapy. This is the only treatment available that actually changes the immune system, making it possible to prevent the development of new allergies and asthma.\nAllergy tablets are a form of sublingual immunotherapy (SLIT) that offer a way to treat certain allergies without shots. Like shots, tablets reduce symptoms by helping the body build resistance to the effects of an allergen. Unlike shots, tablets only treat one type of allergen and do not prevent the development of new allergies and asthma. FDA-approved SLIT tablets are available to treat allergies to ragweed, grass pollen and dust mites.\nAllergy drops are another form of SLIT and work the same way as tablets. Drops are widely accepted and used in many countries around the world, but they have not been FDA approved in the U.S. and their usage is off-label. An allergist can discuss your treatment options and get your health on the right track.\nWhat we do see is significant reductions in symptoms when exposed to the things they're allergic to and \u2013 significant reduction in the need for medications. This treatment can prevent the development of asthma in patients that have hay fever.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Altura\/Peso 1,78 m. \/ 74 Kg.\nBorn in Valencia, he made his official debut in 1984. He played as an attacking midfielder who boasted an exceptional scoring record throughout his career. He played a club record 552 official games and captained the team during many seasons.\nFernando spent the vast majority of his career at Valencia CF, a club to which he always remained loyal, turning down big offers in several occasions. His loyalty made of him a living symbol of the club and the fans over the '80s and '90s.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the last 8 years John has been involved in multiple facets of the commercial real estate arena. He owns, manages, and leases a 45,000-square foot grocery anchored shopping center in Shoreline, WA. Prior to that John earned his Bachelor's Degree at Seattle University. John has significant experience in landlord and tenant representation. John's clients include Northwest Brewing, Oto Sushi, KidsPark People's Home Equity and several others. He also represents Oakpointe, Weidner Apartment Homes and North America Asset Management Group. John chose to work at First Western Properties because of the collaborative environment.\nJohn enjoys spending time with his wife, Katie and dog, Louie. He also enjoys traveling internationally and visiting his home country of Italy. John is a member of the Bothell Chamber of Commerce and the Northshore Rotary Club.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today we want to share with you a ready-coded iOS7-style navigation button showing up a navigation menu on the left. Made up with few lines of CSS + JS.\nPosted on September 20, 2013 in Code stuff, Navigation menus and tagged navigation, app. This post has been viewed 4255 times.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We reviewed the new range of Simply Making piping tips from The Cake Decorating Company for the April issue of Cake Masters Magazine.\nWith 11 tips to try plus a leaf nozzle, these Flower Piping Tips are a simple, fun new way to decorate cakes and cupcakes with beautiful buttercream flowers. Visit www.thecakedecoratingcompany.co.uk\/cm for this exclusive offer.\nWe decorated some cupcakes to try out these piping tips and really liked the results. The designs cover a variety of flowers: a Daisy, Freesia, Parrot Tulip, Tulip Buds, Violet, Holland Tulip, Specimen Rose, Rose Bud, Crocus, Blooming Rose and a Christmas Cactus. As recommended by The Cake Decorating Company, we used Swiss Meringue Buttercream to pipe the flowers, as it provides more stability for the flowers to hold their shape than standard buttercream.\nStarting by covering the cupcake with a thin layer of buttercream to help the flowers stick, we tried out several of the flower piping tips. It is a very simple piping motion, you just need to hold the piping tip directly above the cupcake and pipe straight up. To finish, just stop the pressure and pull away directly up. We finished the cupcakes with some green leaves. It's also pretty to create tonal flowers by marbling two colours of buttercream before adding them to the piping bag or using two colours side by side.\nPractice makes perfect when it comes to these piping tips! To start with, we struggled to leave behind the whole flower on the cupcake as we piped; sometimes leaving the outer petals but pulling away the stamens in the centre and vice versa. But do persevere if you have problems at first, because once you get the hang of it, they give some great results and you'll be able to pipe beautiful flowers in no time at all. We found it easier to start with the tips that had less small details. For example, the Rose Bud Piping Tip and the Crocus Piping Tip are good ones to practice the technique with, before moving onto more complex flowers such as the Freesia Piping Tip.\nThese Flower Piping Tips are a great product. Once you get the knack of piping with them, you can work really quickly and create some beautiful effects \u2013 perfect for cupcakes and lovely on larger cakes.\nVisit www.thecakedecoratingcompany.co.uk to buy the piping tips!\nWIN in our March Monthly Competition!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The junior room created personalised Ogham Stones from clay. We learnt about this \"tree alphabet\" and what it was used for. The children enjoyed examining the different lines and strokes symbolising each letter before writing their own names in ogham.\nWhen finished the class got to explore the clay and created various pots and animals. We all had great fun talking about our creations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Torches are raised to signify our wish to continue to cherish and protect the abundant nature of Lake Akan, which nourishes the marimo. These torches give thanks to the Ainu goddess of fire, Ape Kamuy, and to the great northern lands. From Lake Akan Onsen (hot spring district) to the Ainu Kotan, the countless dancing flames indicate the path for the gods to travel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A stock exchange based in Sandton, South Africa where stocks are purchased and sold. It is the largest stock exchange on the entire African continent and is primarily dominated by mining companies. This is due to the fact that it was originally established in 1887 by the mining industry as a way to raise funds. The JSE Equities Trading (JET) electronic system removed the need for a trading floor. Hours of operation are from 9am to 5pm local time, Monday through Friday. Previously called the JSE Securities Exchange and the Johannesburg Stock Exchange.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Doorbell two chime wiring diagram Quotthe arlo audio doorbell and chime build on the success of our wire free camera systems by allowing users to easily engage with entryway visitors from anywherequot said pat collins senior vice August home announced august view a wire free high resolution video doorbell the august view lists for 230 including the wireless chime. In comparison the ring video doorbell 2 and ring video The task of fitting different sounding doorbells to two separate bells. A single unit houses the power supply and two chimes one for each of two doors to which you need add only the wiring.\nDoorbell two chime wiring diagram Fast forward 188 years and now startup companies are locking horns in a heated battle to replace push button chimes doorbell 2 can be installed using existing powered doorbell wiring. While the square design is unique and looks sorta neat the hardwired doorbell cam pro didnt fit on doorframes particularly well and the doorframe is where most older doorbells and their wiring This isnt augusts first video doorbell of course. Two years ago we reviewed the august doorbell cam pro and found plenty to like about it. We also found some not so good things to say though.\nDoorbell two chime wiring diagram The august view is a small rectangular box with a camera at the top and a big button at the bottom to ring the bell. Its fairly obvious that the device is a doorbell which is a big change from the Its wire free 30 more than the doorbell cam pro and ring video doorbell 2 and on par with nests top end smart doorbell and come with an included wireless chime and a battery. To add a chime youll need a doorbell kit to get the additional chime and additional wire to connect the two chimes said ana moore of electrical services group in chicago. Youll need to run wire.\nThis can quickly become a complicated install because you first need to have the wiring and proper connections with the correct amount of voltage. The video doorbell 2 from ring supports a wired Netgear has added a smart doorbell and chime to its arlo range of wireless security cameras though its missing a feature weve grown used to from rivals. The arlo doorbell and arlo chime like nest.\nIt's possible to get or download caterpillar-wiring diagram from several websites. If you take a close look at the diagram you will observe the circuit includes the battery, relay, temperature sensor, wire, and a control, normally the engine control module. With an extensive collection of electronic symbols and components, it's been used among the most completed, easy and useful wiring diagram drawing program. Doorbell Two Chime Wiring Diagram. The wiring diagram on the opposite hand is particularly beneficial to an outside electrician. Sometimes wiring diagram may also refer to the architectural wiring program. The simplest approach to read a home wiring diagram is to begin at the source, or the major power supply. Basically, the home wiring diagram is simply utilized to reveal the DIYer where the wires are.\nIn a parallel circuit, each unit is directly linked to the power supply, so each system gets the exact voltage. There are 3 basic sorts of standard light switches. The circuit needs to be checked with a volt tester whatsoever points. Doorbell Two Chime Wiring Diagram. Each circuit displays a distinctive voltage condition. You are able to easily step up the voltage to the necessary level utilizing an inexpensive buck-boost transformer and steer clear of such issues. The voltage is the sum of electrical power produced by the battery. Be sure that the new fuse isn't blown, and carries the very same amperage.\nThe control box may have over three terminals. After you have the correct size box and have fed the cable to it, you're almost prepared to permit the wiring begin. Then there's also a fuse box that's for the body controls that is situated under the dash. Doorbell Two Chime Wiring Diagram. You will find that every circuit has to have a load and every load has to have a power side and a ground side. Make certain that the transformer nameplate power is enough to supply the load that you're connecting.\nFriedland doorbell wiring-diagram wiring multiple doorbells doorbell circuit diagram doorbell wiring diode by transformer doorbell transformer wiring-diagram two chimes broan doorbell transformer wiring diagram doorbell transformer wiring diagram diagram for wiring two doorbells.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've recently been working a project to capture a number of different dog breeds, the finished images of which will be used to form a photographic panel for display. Diesel, a pedigree English bulldog, loved his pet photo session. Whilst Bulldogs aren't known for their high activity levels, Diesel is relatively young so still likes to run around the garden when he can! Because of this, and that he likes to run away when let off the lead, I decided to photograph Diesel at his home. Diesel has a lovely large garden to play in and we were also blessed with a gorgeous sunset which added some lovely warms oranges to the finished images. Way to go, Diesel!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaorte b/data_all_eng_slimpj/shuffled/split2/finalzzzaorte
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index 0000000000000000000000000000000000000000..6bf6d666bc97c98c367bf080c9338fecaff47195
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@@ -0,0 +1,5 @@
+{"text":"Tuesday \u2013 Saturday 10:00 \u2013 17:00 Sunday 12:00 \u2013 17:00 Last admission to the museum is at 16:30.\nJust a few metres off The Circus in central Bath, is one of the most unique art collections in England \u2013 The Museum of East Asian Art. Situated in a restored Georgian house, the Museum attracts the interest of students, scholars and tourists, and also has a loyal local following.\nThis unique museum houses a fine collection of ceramics, jades, bronzes and much more from China, Japan, Korea and Southeast Asia. The Museum of East Asian Art is the only museum in the UK solely dedicate to arts and cultures of East and Southeast Asia.\nSince opening to the public in April 1993, the Museum has gone from strength to strength, and has become one of the most extensive collections of East Asian art outside London. With a collection of almost 2,000 objects, ranging in date from c.5000 BC to the present day, the Museum offers its visitors a wonderful insight into the art and cultures of China, Japan, Korea and Southeast Asia. With one of the most comprehensive jade collections in the UK and some of the finest bamboo carvings in Europe, the collection uncovers the finest achievements in East Asian craftsmanship.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A couple of months ago, I added some code to a plugin that I wrote to use wp_schedule_single_event() in order to schedule some background processes. These were to keep data up to date and to clear out expired data. I did some testing on a few sites and everything seemed to work great.\nA couple of weeks later, I started to hear from customers that their sites were slow. My first reaction was to blame their host or the other plugins that they had installed. More and more customers were mentioning that the plugin seemed to make their site sluggish. The weirdest thing was that I wasn't experiencing it on any of my sites...until a couple of days go. I was constantly receiving fatal errors that the memory had been exhausted. I inched the memory up further and further until it was over 400MB.\nThat's when I discovered my problem. Almost all of those cron jobs that I had been scheduling were still in the system. They never completed. Not only did they never complete...they kept getting added. There were thousands of cron jobs attempting to run every time a page loaded. No wonder I was running out of memory!\nThat night I could not sleep. I sat up in bed for a couple of hours and after some research, came up with a way to clear out those cron jobs. I used _get_cron_array() to find and un-schedule the jobs I had created. Because the cron jobs were scheduled with arguments, I had to loop through each cron in that array to find the correct arguments to un-schedule them.\nImmediately, my site was noticeably faster! Now I had to figure out why it was happening and how to prevent it. After a day of trial and error, I got to a point where everything was running smoothly. Cron jobs were running, then removing themselves from the schedule. Not only that, but my memory usage was staying really low. Yay!\nFinally, I was able to release an update for my plugin with the fixes in place. I created a version upgrade routine that would run only once to clean up the mess. I even created a bug fix plugin that we made available for a couple of our customers who couldn't even have our plugin active without it crashing their site. Run the bug fix plugin, then re-activate our plugin and install the update.\nIt was a big mess, but I feel a huge weight off my shoulders knowing that it's been taken care of. Luckily, our customers have been very patient with us.\n\u00a9 2019 Daniel Milner. All Rights Reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SmartCloud Connect is highly customizable enterprise-class solution for integrating CRM systems into Inbox and Calendar. It's one of the products built upon Invisible.io platform developed during more than a decade, licensed and utilized by Microsoft, Oracle, SAP, and others. Invisible.io is a global leader in integration of complex enterprise systems with environments that teams use all day long \u2013 Office 365, Outlook, Gmail, and Mobile clients.\nOur platform allows developing highly customizable products for different clients in no time. Our weathered specialists are able to develop integration solutions in accordance with the whole range of specific requirements, even the most sophisticated ones.\nWe create full-fledged integration packages that can be easily melded with the most complex business workflows without tapping clients' resources, being absolutely sustainable.\nOur focus is Customer Success. We treat solutions successful only when people are feeling value from them. When it comes to solid deployment, it's not only about software, it's about our team, as we are flexible, open to cooperation and are around until the challenge is tackled.\nWe thoroughly research work habits and needs of our partners and focus on what matters most to them.\nWe deliver proof-of-concept within days, run user-acceptance testing and actual roll-out.\nWe gather feedback and tweak the product if needed, as well as constantly introduce new features.\nYou can easily calculate your expected benefits when rolling out SmartCloud Connect. Current calculator will show you how much time and money you can actually save around productivity improvements. These do not involve other benefits like quality of CRM data, user adoption, forecasting and others.\nEfficient and unlimited direct technical support for your team.\nSmartCloud Connect is thrilled to be named among the Top 10 Salesforce AppExchange Tools (occupying the 5th and 6th places based on usability ratings). We promise to keep that effort!\nWorldwide support that gets you covered 365\/7\/24.\nLitify is the best-in-class Salesforce-based law practice management SaaS platform. As a result of the partnership around SmartCloud Connect, law firms can leverage a complete performance optimization of daily legal workflow operations and management of legal affairs from intake to invoice through their native Inbox.\nSpringCM is the leader in document workflow and contract management. As a result of the partnership around SmartCloud Connect, companies can significantly accelerate the entire document lifecycle (from contract creation through negotiation, signature, and renewals) right from their native Calendar and Inbox applications.\nWant to arrange a meeting and get a quote for your company?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Scrambled eggs with some additional veggies tossed in; broccoli with red and green bell pepper to livin' things up. Don't forget to include the egg yolk because it's full of nutrients not found in the egg white. Eggs from pastured chickens are superior to \"cage-free \/ free-range chickens\". You can taste the difference, and the yolk is a dark, rich yellow-orange color. As for the sausage, I go for no additional nitrates and nitrites.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With 14.5 % alcohol, costing $33, the 2015 Chateau Laplagnotte-Bellevue Saint-Emilion Grand Cru scored 16+\/17- points of the Dionysian scale of 20. 2015 Bordeaux are \"must have\" wines\u2026.\nThe multi-vintage of Demiere A & J Champagne Solera 23 has 12 % alcohol, cost about $45, is an RM Champagne, and scored 17 Dionysian points out of 20\u2026.\nCosting under $18 now, the 2014 Domenech Vidal Masia Freye is made from Parellada and Muscat grapes. Young people love it. The wine is floral with mineral notes\u2026.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapcnq b/data_all_eng_slimpj/shuffled/split2/finalzzzapcnq
new file mode 100644
index 0000000000000000000000000000000000000000..394f6f7439d8e3290d77664a5981f0cc6038df5a
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"ISLAMABAD, 29th March 2018: With the growing use of internet in every work of life and the current rapid transformation of ICTs enabled services especially in education, health and other walks of life, there is an urgent need to make the cyber space of Pakistan secure. Globally cyber security is the part of overall strategy to promote the use of latest technologies in good governance initiatives. The most recent global initiative in this regard is Sustainable Development Goals (SDGs), which has now become the real road map of development in Pakistan.\nIn the light of this background, there is an urgent need to adopt Cyber Security Policy & Strategy and to bring in place an Eco-System where all players of Cyber Security in Public and Private Sector could work together. Pakistan Information Security Association (PISA) headed by Mr. Ammar Jaffri, Former Addl Director General FIA is working for the promotion of Cyber Security in Pakistan since 2005 with the support of CIT. Center of Information Technology (CIT) is twenty five years old Computer Science Degree Institution of a Govt Sector University and plus ten thousand IT Graduates from CIT are currently serving in different public and private sector organizations.\nWith this background an International Conference on Cyber Secure Pakistan Vision-2025 was organized on 29th March 2018 in Pak China Friendship Center Islamabad. The Chief Guest of the Conference was Lt General (R) Nasser Janjua National Security Advisor to PM of Pakistan and attended by Ten Speakers from abroad including CEO Cyber Security Malysia and VP ICANN. Other Speakers of the Conference were VCs, Heads of Telecom Companies and Senior Officers from Government Organizations. Cyber Secure Pakistan 2018 was attended by all walks of life including students, public and private sector.\nSeveral cyber security issues were highlighted in the conference and appropriate solutions were recommended. We need awareness of Cyber Security and Cyber Crime at grass root level, the conference was highly appreciated by the audience and encouraged the authorities to organize such conferences at all levels from bottom to top.\nA special session was also the part of the conference which specifically focused on Cyber Security of Women and Girls in Pakistan. Young female participants were introduced to the cyber laws so they can report cyber crimes to Law Enforcement Agencies.\nLaw Enforcement Agencies and Government Officials at the conference suggested several reforms in our current cyber laws and policies. Lack of cyber security awareness is the main reason of cyber crimes as a large number of internet users are unsecure. Conferences such as Cyber Secure Pakistan 2018 are playing a major role in spreading awareness about cyber security in Pakistan.\nIssue cyber threat alerts and advisories to members of the CIT CERT Coordination center. So that they could remain secure from cyber-attacks.\nLinking the members with regional and global CERTs for sharing knowledge and to adopt Pro-Active approach in Cyber Security.\nMr. Ammar Jafri concluded the session by thanking all participants and volunteers, wishing them good luck for a successful conference.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We were finally able to celebrate our 3 year anniversary yesterday and today! We saw Harry Potter, went to the Bountiful Temple, and ate lots and lots of shrimp. It was fabulous.\ncongratulations on three years :) I think we had the same dress! BEAUTIFUL! I'm glad you love being a mom. It really is the best.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Coast Guard Auxiliary Business Card Template coast guard auxiliary business card template unique coast guard auxiliary business card template kharazmii printable. coast guard auxiliary business card template coast guard business card template 101184728845 coast guard. coast guard auxiliary business card template coast guard auxiliary business card template lera mera business template.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Remember the days when you used to feel trendy walking into a record store to pick some vinyl? Well now you can do it at the end of your weekly \"big\" shop.\nJust in time for Record Store Day 2016 (April 16th), Sainsbury's announced that vinyl will be sold in stores starting on March 21.\nRumours that listening stations will be set up next to the branflakes are so far unfounded, but the store profess to be taking vinyl seriously. Pete Selby, Sainsbury's Head of Music and Books said: \"We don't see this as a novelty gifting fad but a complimentary part of our existing music offer with a long term future in our stores.\"\nAccording to Billboard, the vinyl revival dates back to the early 2000's when Record Store Day was founded in 2007. RSD will be celebrating its ninth year this coming April - so why did it take almost a decade for such a huge company to recognize the impact of vinyl and to start selling it?\nSelby said: \"There is an enduring love for this format with music fans and we're delighted to offer an ongoing selection of titles for everyone, from contemporary releases to much-loved classics. Our customers have rich, varied tastes and our range will naturally reflect this.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An elegant brushed silver metal ballpoint pen finished with a subtle Rotary wheel. It is dual function with a stylus on the tip to operate touch screen technology. Presented in a black velvet pouch. A perfect gift.\nClick the button below to add the Rotary Emblem Pen to your wish list.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaprlc b/data_all_eng_slimpj/shuffled/split2/finalzzzaprlc
new file mode 100644
index 0000000000000000000000000000000000000000..716151ceda712063616f14605f6881d02c99c98a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaprlc
@@ -0,0 +1,5 @@
+{"text":"Mid-Atlantic For discussion of meets, get togethers, events and other local activities.\nAny xB folk in North Carolina?\n3 Attachment(s) Koons Scion Car Show - May 3, 2015 - Anyone going?\nWhat's going on in the Lexington area?\n3 Attachment(s) Any Hockey Fans?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Get the perfect flower centerpieces to complement any occasion or even just for daily decoration. A flower centerpiece brightens up the room and adds to the mood and atmosphere for the home or office. Decorate your living space with fresh flowers. Same day delivery available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Misty Copeland is pretty, she is an amazing ballerina, and she is breaking ground for brown girls everywhere. There's a lot to her background that you just may not know. Many times in the entertainment industry we see celebrities as beautiful and unique people like Misty Copeland and we tend to think they came from a background without any type of hardship. But to clear that up, just read her biography. You'll get to know that there is more to her story besides her pretty face. She has a God given talent that has open many doors for herself. Hopefully, this can motivate you to not give up on your dreams and to keep pushing past your hardship, because dreams really do come true. So read along with me and Elle Magazine as we dig into her past, present, and future.\nIt all started when Misty Copeland was 13 years old. At that time her family stayed in a motel. This included all of her siblings along with her mom. They had to sleep on blankets of the motel room floor. In an interview with Elle Magazine Misty Copeland describes this as the hardest time in her childhood.\nHow did you deal with the chaos?\nI made up for the chaos at home by working double hard at school, but there was still so many feelings of anxiety. I received a break during my childhood through my middle school drill team coach when she notice me dancing. I started later than most ballerina dancers, so I had to practice a whole lot more. My middle school drill team coach called her friend Cynthia Bradley who was a local ballerina teacher, and she came to look at me. I was then invited by Cynthia Bradley to a class she was giving at a a local boys and girls club.\nAs many elite dancers often do, did you have to move in with your coach Cynthia Bradley?\nWell, the first time I was invited to attend Cynthia Bradley's class I was 13 years old. I was wearing a t-shirt and gym shorts which I hated wearing during practice. I fell in love with Mrs. Bradley's class and she gave me slippers and a leotard and it was like I finally fitted in. Ballet was so structured and I had been craving something to guide me. Within 3 months I was dancing en pointe, which is something that takes most ballerinas years to master. And as many elite athletes do, I had to move in with Cynthia Bradley to make it easier to train. I got a different taste of what life could be like, because the lifestyle at Cynthia Bradley house was different than what I was use to when living with my mother. At her home I would eat dinner and then go to sleep in a bed. At first it was like do I deserve this? But, I got comfortable quickly, because they were great to me, however the arrangement didn't last long. As I begin to spend more time away from my family tensions rose between my mom and my teacher Cynthia Bradley.\nMisty Copeland's and my relationship began when I noticed Misty at a free ballet class I taught at a girls and boys club. I immediately recognized something really special about her. I touched her foot and placed her body, and she could assimilate what I was giving her immediately. I invited Misty to take classes at my school, the San Pedro Dance Center and she never showed up. I asked her each week why she wouldn't come to my school and she said, my mom doesn't have a car. At this time, Misty wasn't living with her mom, but the woman that she lived with had two jobs and she couldn't bring her to my school. At first, I would picked up Misty after school. Soon after she started taking ballet, Misty moved back in with her mother. She was living at the time in a motel in Gardena. After 3 months she was already dancing en point and doing fabulous things and she was going to have to give it up. That's when I asked her mom if she could move in with my family. A lot of dancers go away for training and it's not unusual. Before she moved, Misty and her mother both signed a management contract and a life story contract. The management contract gave me the right to negotiate on Misty's behalf, as well as 20 percent of the money that she earned until she turned 18. It seems Misty's mom was happy about the arrangement.\nWe ended up supporting her for almost 3 years and she became a part of the family. Misty told us for the first time in her life when she was living with us that she didn't have any worries, she was talking about her basic needs being paid for, being fed, and she had a dream life...she was a Cinderella. During her stay with us she made rapid progress in her dance training too.\nSo, if everything is going so great, then what happened to make things turn in the wrong direction? What happened to make you two go to court?\nMisty Copeland says when she returned from San Francisco my world unraveled. My mom called and said that I was going to start living back home again. I filed for emancipation from my mom. My mom filed a restraining order against Cynthia Bradley and her family to prevent me from contacting, seeing, or being involved with the Bradley's or vice versa. The Bradley's made it sound like it was a way to make things all better, so I thought it would be good.\nDuring court I dropped the emancipation proceedings after meeting privately with my attorney. It didn't turn out the way I thought it was going to be. It made things worse instead of better.\nI can describe everything as been devastating and scarring, because I had been so private about my life.\nAs a grown woman and understanding everything I realize that my mom Sylvia and my teacher Cynthia Bradley came from two different worlds. My teacher wanted the best for me, but saw a child deprived and my mom saw a woman taking her child away from her. I resented my mom for years, but now I understand my mom did what she had to do. I have done the work in therapy to help with things, but to this day it's hard for us to talk about these things.\nDespite it all, Misty Copeland thrived. She has won many invites from prestigious dance schools all over the country to study in their summer intensive programs. After two summers with American Ballet Theatre (ABT) she was asked to joined the studio company and the corps, but not everyone wanted her there.\nMisty Copeland says Seeing negative things on the Internet was hard. There is still racism in ballet. People make comments and to some people I don't look like a ballerina. Then came the uproar of criticism over my body. As a child I had been told that I was not exactly what a ballerina should be. The so called Balanchine ideal,which is a girl with long legs, big feet, and a flat chest. So, when I hit puberty at 19 years old and suddenly had breast, bigger muscles, and a more developed figure it was a shock. I had no idea what to do with my body. It wasn't responding like it use to. It was like starting over. My teachers never told me to lose weight, but I knew they were not not happy. They basically said your body has changed, you've added weight, it's creating a different line, and it's not what we want. I didn't know how to take care of it. I've always eaten what I wanted to which was Taco Bell, soda, burgers, and french fries. I would come home and eat an entire pizza from Dominos and six donuts. Then I would cry and go to sleep. The next day I'll be hiding in class. It was a really dark time. I went through a lot by myself, because I didn't have anybody in my life yet who understand. Then enters Olu.\nOlu is my boyfriend of ten plus years. I met him in a club in Chelsea in 2004. He was tall and fit with a pair of dimples that made my heart flutter and he was with his cousin the actor Taye Diggs. He spotted me on the dance floor and ten years later we are still together. He was my first boyfriend. Olu taught me to communicate in ways I had never learned before, to not run away from problems, to think about things critically, and he made me feel like I really had a bright future as a ballerina.\nMeeting Prince helped me to see my true potential. I'm still not sure how the singer discovered me, but in 2009 he tracked me down and ask me to start in a music video for Crimson and Clover. We became friends and I performed with him on and off for three years. He was a mentor to me and a huge confident booster. When I met him I was a new soloist at American Ballet Theatre. He really helped me understand who I was as an artist. Alongside of that, Prince is a lot of fun, he is super normal, and he loves to play basketball. I've played with him at his home in Minnesota and he's not a bad player.\nAt American Ballet Theatre, I am a stateswoman, so it's time for me to play mentor. I am involved in American Ballet Theatre's project Plie, an initiative to draw more diverse dancers into elite ballet. I had some really incredible people who mentored me and gave me things I never got from my parents. I think it's so important for young dancers of color to have someone who looks like them as an example and I tell them to be true to themselves.\nEvery Time I Dance I'm Trying to Prove Myself to Myself, I Love What I Do and I Want To Do It For As Long As I Can.\nNew York Times dance critic Alastair Macauley wrote that he looked forward to future performances featuring Misty Copeland. The kind words were nice, but Misty Copeland says she doesn't need kind words to push herself hard into the studio. Every time I dance I'm trying to prove myself to myself. As far as the future she's set her sights on being the first black Odette Swan Lake, she wants to get married, and have a family, but not yet. I love what I do and I want to do it for as long as I can.\nAFRICLASSICAL - A website on on African Heritage in Classical Music. Misty Copeland speaks with California Students about expanding access to careers in Ballet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Adelaide is a popular study abroad destination, with affordable living costs, safe environment and offering a unique combination of energetic urban life and picturesque natural surroundings. International students coming to Adelaide will have an enjoyable and enriching academic and life experience. Universities in Adelaide focus on unique and innovative learning and research activities and are well known for preparing students for the international job market. You can study your favourite discipline available at Bachelor's degree, Master's degree and PhD level. Adelaide is the capital city of the state of South Australia, and the fifth-largest city of Australia. Adelaide is fenced in the north by the Fleurieu Peninsula, on the Adelaide Plains between the Gulf St.Vincent and the low-lying Mount Lofty Ranges.\nIn Adelaide universities and colleges, you can choose from numerous study options of undergraduate Bachelor degrees and postgraduate Master degrees and PhD courses, massively open online courses (MOOCs) and research degrees.\nUniversities require English language proficiency as part of the admission process, with IELTS, TOEFL and Cambridge as widely accepted English tests. International students who don't meet the requirements for university entrance can enrol in several undergraduate pathway programmes in order to get used to university level expectations.\nDisciplines you can study include: arts, languages, business, commerce, education, engineering, medicine, law, nursing, social work, information technology, psychology, nanotechnology, water resource management, agriculture, finance, marketing, aviation and more.\nAdelaide's economy is mainly sustained by: health care and social assistance, manufacturing, retail trade, defence technology, high tech electronic systems and research, commodity export and corresponding service industries.\nMajor employers include: the General Motors Holden plant, global media conglomerate News Corporation, Australia's largest oil company, Santos, prominent South Australian brewery, Coopers, major national retailer Harris Scarfe, Australia's second largest listed investment company Argo Investments Limited and Saab Systems, BAE Systems Australia, and Lockheed Martin companies.\nUniversities support international students with seeking jobs or internship opportunities in a wide range of fields.\nAdelaide is a vibrant and energetic city, where you will surely not get bored, surrounded by natural beauty and amazing landscapes that will inevitably conquer your heart.\nThe symbols of Adelaide are the Victoria Square, right in the centre, with the iconic Victoria Square Fountain and statue of Queen Victoria.\nJetty Road is the main dining and shopping district in the greater Adelaide area where you can also find plenty of bars and night clubs. Take a gastronomic tour of the Adelaide Central Markets and taste the local products.\nHave fun at one of the many festivals held in Adelaide, like the renowned Adelaide Festival of Arts and Fringe Festival, the Adelaide Festival that includes the Adelaide Cabaret Festival, Adelaide Film Festival.\nEnjoy and admire the beautiful nature in the surrounding area of Adelaide and go to the picturesque Adelaide Hills, a twenty-minute drive from the city. Check out Australia's Barossa Valley \u2013 the best wine region in Australia and the Kangaroo Island. Meet Australian sea lions on the beach, kangaroos, koalas, penguins, or swim with the dolphins in the Glenelg area.\nAlmost 30% of the total population of Adelaide is born overseas, with Greeks and Italians being the largest ethnic communities. Other minorities include people coming from: Afghanistan, Iran, China, India, and Vietnam. The most spoken languages other than English are Italian, Greek, Mandarin Vietnamese, and Cantonese.\nIn Adelaide, you will experience hot to dry summers and mild winters, with most precipitation falling in the winter months (between June and August). Adelaide is a windy city, making the temperature seem colder than it actually is. The average high temperatures are in December \u2013 February, with 27 \u2013 29 C (80 \u2013 84 F). The coldest months are June \u2013 July with 15 \u2013 16 C (59 \u2013 61 F).\nCarnegie Mellon University is an American university established in 1900 in Pittsburgh, Pennsylvania. Ranked 24 in the world Carnegie Mellon has been a birthplace of innovation throughout its 115-year history and is a global leader in bringing ground-breaking ideas to market and creating successful start-up businesses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The all-important enterprise mobility is YapBuzz's forte.\nAt YapBuzz we pride ourselves on the fact that enterprise mobility, the most topical of all the digitization services on offer, is our forte. In the long-run, the point of enterprise mobility is self-sufficiency within the lifespan of a mobile device app, and what sets us apart is our endeavour towards the same.\nThe fact that out of 7 billion people this planet is teeming with, 6 billion are connected to each other through a mobile device, is in itself stupefying. However, YapBuzz believes that the humongous opportunities offered by such a state of affairs has not been fully taken advantage of yet. What about competitive advantage of enterprise over enterprise? What about the fact that consumers do not want to wait for what they demand? What about the implications of a workforce that is increasingly becoming mobile?\nYapBuzz's projects in enterprise mobility addresses all this, and much more. These questions occurred to us when we started out and have inspired us to put together enterprise mobility tools for our clients that are the envy of the industry. Your future is about your employee efficiency - and our enterprise mobility systems will make them react more effectively, access information more seamlessly, and demonstrate commitment and performance from anywhere in the world. In short, YapBuzz lets you do way more within the constraints in question.\n\u00a9 Copyright 2014 - 2019 YapBuzz LLC.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapxcd b/data_all_eng_slimpj/shuffled/split2/finalzzzapxcd
new file mode 100644
index 0000000000000000000000000000000000000000..dd5d478d8093d7306ae6b78967a977e83c15a8e5
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+{"text":"One of the newer additions to the Kings Mountain fleet is Engine 282, a 1998 Pierce Saber Pumper\/Tanker. The truck has a four-door, six occupant cab and is powered by a Detroit Diesel, 300 hp engine and an Allison MD 3060R automatic transmission. Engine 282 carries on it a 1250 GPM, two stage Waterous pump and a 1000 gallon polypropylene tank. Engine 282 carries 1200 feet of 4 inch hose, 500 feet of 3 inch hose, has 3 1 3\/4\" preconnects, 1 2 1\/2\" preconnect and 1 bumper mount preconnect. In addition 282 has a 3500 watt Honda generator, electric fan, 6 Scott air packs, deck gun and hydraulic ladder rack.\nThe oldest member of the Engine fleet, and also mutual aid pumper, is this 1985 Pierce Dash, which has a 1250 GPM Waterous pump and 750 gallon tank. On board Engine 283 is 1000 feet of 4\" hose, 400 feet of 3\" hose, 2 1 3\/4\" preconnects, 1 2 1\/2\" preconnect, and 1 1 1\/2\" preconnect. In addition 283 contains a 3000 watt Honda generator, deck gun, and arrives on scene with 5 Scott air packs.\nThe newest member of the fleet is Engine 284, a 2003 Pierce Dash Pumper, with a 1000 gallon tank , 1,500 GPM single stage Waterous Pump, hydraulic ladder rack and 8kw Harrison generator. 284 now is the first due engine in the City. The department purchased 284 in late 2002 and took delivery of the truck on May 21, 2003. This purchase makes the entire fleet of engines and ladders exclusively Pierce products.\nA long awaited addition to Kings Mountain's fleet is Ladder-1, a 2000 Pierce Dash 75' Heavy Duty Ladder Truck. Originally built as a demo unit for Pierce, Kings Mountain purchased the truck in late May 2000 and brought it home June 6. July 31 Ladder-1 responded on its first call to assist Shelby Fire Department with a structure fire.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u2190 East Bay Open Studios in Jingletown June 10 \u2013 11 and 17 \u2013 18, 11:00 a.m. to 6:00 p.m.\nJingletown Arts & Business Community in Oakland, California, presents the 12th Annual Winter Open Studios\/ Art Walk, Saturday and Sunday, December 2 \u2013 3, 11 a.m. to 6:00 p.m.\nGalleries in the neighborhood will also hold 2nd Friday Receptions on December 8, from 6:00 to 9:00 p.m. These events are free and open to the public.\nGalleries that are open for 2nd Friday are indicated with an *.\nThe annual winter open studios event highlights the work of artists who live and\/or work in Jingletown, an artist's district located between the Park and Fruitvale Street bridges bordered by the estuary separating Oakland from the island of Alameda. Jingletown has a number of galleries as well as many art studios of outstanding artists who are living and working in one of the most established arts districts in the Bay Area.\nAbout Jingletown: Jingletown acquired its name at the turn of the century when Portuguese cannery workers whose earnings would \"jingle\" in their pockets at the end of the day as they walked home from the factories in the area, signifying prosperity. Over time the neighborhood went into disuse, and in the early 1980's artists realized the potential and converted many of the neglected warehouses into functioning live\/work spaces. It is now a thriving artist community and one of the fastest growing arts district in the Bay Area.\nIn a time when our neighborhood has been changing dramatically, the goal of the Jingletown Arts and Business Community is to heighten awareness about the rich and diverse population of working and exhibiting artists and the creative opportunities in Jingletown.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We want you to love your Beh\u00e4ppi products as much as we do.\n14 days of receiving your original order from www.behappi.eu.\nThe items must be unused and in it's original condition with it's original packaging.\nTo complete your return, we require a proof of your purchase (e.g. order confirmation email).\nPrior to the return, contact us at hello@behappi.eu or reply to the original order confirmation email.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beautiful hand painted porcelain black swan ring with gold enamel beak. This ring is also available in black or white with a red beak, or white with a gold beak. This ring is available in two sizes S\/M (17.5mm) & M\/L (18.5mm) and is supplied with its own gift box.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Where is a good place to store wealth right now?\n\"I bought the house I now live in back in 1979. I thought the house was expensive when I bought it, but I loved the house and I loved the neighborhood. So I collected most of my available cash, and with a gulp, I bought the house. Today the house is worth around six times what I paid for it back in 1979.\nAt that time gold was rising rapidly. Less than a year after I bought my house gold sold for over $800 an ounce.\nNow it's 29 years later, and it's 2008. What can I buy today at 1979-1980 prices? Russell, you've got to be kidding. Everything is much more expensive today. Forget those prices of 1979-1980. Get into reality, man.\nWait, there is one thing that I can buy today at around 1979-1980 prices. That one thing is gold. Gold today is selling just a bit higher than it sold for back in January 1980. How can that be? Brother, it be. Today you can buy gold for just a few percentages more than gold sold for at its high in January 1980.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqbpu b/data_all_eng_slimpj/shuffled/split2/finalzzzaqbpu
new file mode 100644
index 0000000000000000000000000000000000000000..8f1783148259a4818f7cb83285892ccdf407c0b0
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"It's crazy that our periods are taxed and that tampons are considered a luxury good. We think everyone should be talking about it! Check out our short parody vid and if you agree, suggest everyone to gift tampons this V-day!\nSarah and I are creative partners in\/outside of work, but we're also friends\u2013so a lot of our ideas just come from hanging out. Lately, we've been flipping back and forth between rage reading about the \"pink tax\" and \"tampon tax\" [more states tax tampons than candy???] and cracking up about a guy we overheard fretting about Valentine's Day gifts. He sounded so resentful of the \"burden\" of fulfilling a woman's expectation of picking the perfect gift. It just seemed real, real silly. And basically, all of these ideas\u2013the luxury status of tampons, this commercial holiday that can really get under our skin, the obsession with the perfect gift\u2013came together and became a video where we're soaking tampons in red wine and throwing them at our friends faces. It was great.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A book of horrors, fear and death. \"The Plague\" is a book by Albert Camus which weaves these emotions and events into one suspenseful tale. Each paragraph and section is written and structured in such a way as to give the reader insight into the feelings of the victims of the plague, and to show somewhat of a theme. The passage from section 4, part 4, line number 1 to line number 35 gives us a glimpse of the melancholy of the people of Oran to their dead loved ones to the extent that they do not attend All Souls' Day, for they were thinking of them too much as it was. Albert Camus fills this passage with figurative devices, including, diction, personification, pathetic fallacy, metaphors, irony and a turning point. The first two paragraphs suggest that it was a perfect environment for the attendance of All Souls' Day, portraying several features of the environment that correspond to the All Souls' Day. However, Albert Camus uses a turning point and several devices to show how that it was ironic on how they did not attend it.\n....the habit of irony was growing on him (Tarrou) more and more-each day was for us a Day of the Dead.\". The ironic figure here also helps the reader to understand the situation of the townspeople and their extended grief and to what extent this grief affects everyone.\n\"The Plague\" is a novel weaved with beautiful and stylistic devices used by Albert Camus to portray and clarify the message and theme of every passage. This passage specifically has used diction, pathetic fallacy, metaphors, a turning point and irony to portray the message, which was despite it being All Souls' Day, the plague had hardened the hearts of the families of the victims and incidences, and thus did not attend All Souls' Day due to them thinking profoundly about them. Thus helping the reader understand the overall message of the book on how much the townspeople were suffering from the plague.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here are a couple more iPhone photos from my trip to Austin. I have several \"good\" photos too using a real camera, but I haven't processed them yet. I'm sure they'll trickle onto the site before long.\nI went to hear a friend's band, The Michael Cross Peace Love & Blues Band, play at Stinson's, a neighborhood bar and grill while I was there. I had to park a block away, but on the walk to the bar, I encountered this wonderful old oak tree (there was a marker by the trunk that said it was 100 years old). I tried to capture the scope of it using a pano.\nMy buddy's band was great! They had a wonderful, down-home vibe. The instrumentation was terrific \u2013 acoustic guitar, violin, accordian, stand-up bass, drums, keyboards. The music was loose and kind of jammy, but they were good enough musicians so that the music didn't feel directionless. The singer\/guitarist was really good on both.\nOn another day, I \"climbed\" to the top of Mt. Bonnell. The name \"Mt. Bonnell\" has always cracked me up, but \"Hill Bonnell\" doesn't sounds quite as impressive even if it is more accurate. In any case, Mt Bonnell has a wonderful view of some of the most exclusive homes in Austin on the banks of the Colorado river. I always loved that view, even if it wasn't really from the top of a mountain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Singapore: Microsoft Monday said it's bullish on India, especially the government's ongoing efforts to move services online.\n\"The Indian government has a large-scale initiative for IT services,\" said Microsoft Regional Technology Office (Asia Pacific) head Michael Thatcher.\nSpeaking to PTI in Singapore, Thatcher pointed out the near 12% compounded aggregate growth rate for the industry between 2008 and 2013.\nHe said, however, the Indian IT sector faces major challenges, especially in integrating the the IT system between central government operations and the autonomous state-based operations.\nAn equally big challenge would be to work in the diversified Indian market, ranging from modernised institutions with IT connections and the vast majority of rural population yet to be served by the technology, Thatcher said.\nGoing forward, Microsoft is working through its New Delhi office and Bangalore Research and Development Centre on serving the Indian markets, especially basing its output on the IDC forecast investment in the IT sector which by 2013 would be Rs1.64 lakh crore, by 2012 Rs44 lakh crore, by 2011 Rs1.26 trillion and this year Rs1.11 trillion.\nMicrosoft is also following on with the IDC projected investment on Cloud plus Clients which for this year would be Rs88.2 billion, increasing to Rs219.98 billion next year and almost doubling to Rs406.78 billion in 2012 as well as Rs67.78 billion in 2013.\n\"Markets will change as institutions start using 'clouds' facilities and this would extensively reduce pressure on IT investments,\" he said.\nThatcher said: \"Cloud will reduce new capital expenditures in IT as it can leverage existing assets to access the new 'cloud-based\u00b4 IT services.\"\nThis would shift IT spending towards operational expenditures and a move to a 'pay-per-use' model, he added.\nThough encrypting data would reduce security threats but it would eliminate them, he said.\nThatcher nted that it would be important to perform regular risk assessments of an organizations IT infrastructure, which should include an end to end evaluation of how data can be accessed and by whom.\nCommenting on WikiLeaks, he said it has become an eye opener for the world to adopt tight security measures in the IT systems.\nThatcher was part of the Microsoft team that held a two-day business strategy conference on \"Accelerating Asia Pacific 2010\" here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Northeast corner of 412 W 15.\nConstruction is wrapping up on the exterior enclosure at 412 West 15th Street, an 18-story office tower in the Meatpacking District from Rockpoint Group. CetraRuddy is responsible for the design of the tower, which brings 108,469 square feet of commercial space to the booming neighborhood surrounding the High Line. The glass and metal panel fa\u00e7ade contrasts with the predominantly brick-clad stock of low rise industrial buildings. A triangulated pattern for the metal panel cladding at the western third of the tower seeks to provide some visual interest to the circulation and back-of-house core.\nLooking up at the north fa\u00e7ade of 412 W 15.\nNorthwest corner of 412 W 15.\nClose-up of the north fa\u00e7ade of 412 W 15.\nArchitect: CetraRuddy; Developer: Rockpoint Group; Program: Office, Retail; Location: Meatpacking District, New York, NY; Completion: Late 2017.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqnhj b/data_all_eng_slimpj/shuffled/split2/finalzzzaqnhj
new file mode 100644
index 0000000000000000000000000000000000000000..a17eb44c68524a35f390e6e37d0413f5c7206abd
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaqnhj
@@ -0,0 +1,5 @@
+{"text":"A Jacksonville grandmother found slain in her Goodby's Trace home Dec. 29 was killed by someone who forced their way inside and stole property, including her car, police and the woman's family said today.\nRenie Telzer-Bain's vehicle was recovered the next day, but police have yet to find her killer or killers. Her son, Dana Telzer, stood in front of the 82-year-old widow's home this morning and made a tearful public plea for help in the case. It was the first time the family has spoken since the slaying.\n\"I need you to look at your mothers. I need you to hold your mother and imagine how I feel,\" said Telzer, 54. \"There is a person or persons out there that did this. Help me. Help our police department. Help save your family. Please.\"\nPolice had released few details in Telzer-Bain's slaying until today. Homicide detective Ed Cayenne said someone forced their way into her home in the 8800 block of Goody's Trace Drive. The cause of death has not been revealed.\nTelzer-Bain's missing beige 2000 Cadillac Seville was found the day after the slaying. Police said today the vehicle was found at Eaverson and West 18th streets northwest of downtown. They would not discuss what evidence was recovered in the vehicle, which has been returned to the family.\nPolice said they don't know if there are other crimes connected to the slaying. Cayenne said there had been a nearby burglary several weeks before the slaying in the middle-class neighborhood just north of Baymeadows Road. A neighbor said burglars stole jewelry from a woman's home next to Telzer-Bain's shortly before the slaying, but it's unclear if that was the same burglary mentioned by Cayenne.\nNeighbors have been in fear since the killing.\nBain-Telzer is survived by two sons and a grandson. She worked for over 30 years as an orthopedic and psychiatric registered nurse in Jacksonville. She was a member of Congregation Ahavath Chesed in Jacksonville for over four decades and was formerly active in the Temple's Sisterhood. Her grandson, Myles Telzer, 20, said she also loved the arts and culture.\nRead jacksonville.com and Tuesday's Times-Union for updates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Okay, I've tried to hold back. I swore this blog would stay \"single-issue\" and not drift into election matters.\nCuz it's getting just that infuriating, ain't it.\nSo let's this keep this space for theatre. But come vent with me there.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BCCI (Board of Control for Cricket in India) is the Indian governing body that concerns Indian Cricket. The board is formed in December 1928, that is registered under the Tamil Nadu Societies Registration Act. It is one the affluent sports community in the world.\nAs indicated by the International Cricket Council (ICC) calendar, India were bound to play a full bilateral series with Pakistan in December, yet BCCI will most likely consent to a shorter limited overs series in an \"unbiased\" place that is known for their decision. Pakistan usually play their \"home\" matches in the UAE.\n\"Because of the strength of the Indian economy, the population and interest levels for the game, the BCCI makes a lot of money. Everyone wants to play India because they themselves will make a lot of money through advertising. However the BCCI doesn't need Pakistan,\" Boycott told PakPassion.net.\nIndia have blamed Pakistan for cross border terrorism. BCCI secretary and BJP MP Anurag Thakur, had completely turned down a series with Pakistan yet India's disposition changed after Shashank Manohar got to be Board chief in October.\nManohar is avid to respect India's dedication to the ICC and identifies with PCB executive Shahryar Khan in Dubai this weekend to restore the series. Both Boards have drawn closer their particular governments.\nThank you for liking my website. My Blog is developed by Citadel Infotech LLC \u2013 http:\/\/citadelinfotech.com\/ this is a very good IT company to work with. Rates are very affordable and have good experience in Website Design and Web Development.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you own an auto repair shop, then you likely understand that the customer is king. This is true even if your customer knows little to nothing about their vehicle. They expect prompt, specific answers to their questions about vehicle repairs and want to be kept informed about the status of their service or repair job without having to constantly reach out to the auto repair shop for updates. This is where having the capacity to perform an accurate digital vehicle inspection makes all the difference in helping your shop to run smoothly and keeping your customers coming back for their next service appointment.\nWhat is a Digital Vehicle Inspection?\nA digital vehicle inspection is a platform to display pictures, videos, written reports and recommendations to customers with the click of the button. It records all of your service and repair recommendations as well as vital information about the vehicle, such as the VIN, mileage, any noticeable issues, and owner details, in one place. This allows you to send detailed information to the customer about any issues with their vehicle and documents all of your service recommendations for posterity's sake.\nIn addition to being an effective safeguard for compiling the quotes and recommendations that you give to customers, this program frees you up to take on additional jobs because of the degree of automation it adds to the process. No matter who may have originally documented the customer's vehicle in the shop, any employee is able to log in instantly and check the progress on the job. This means that when a customer calls into the shop to find out when they can pick up their vehicle, employees no longer have to stop what they are doing and check in with the technician who was originally working on the vehicle. It allows for seamless transitions between various mechanics working on the same vehicle and ensures that no information is lost in translation between shifts.\nAnother advantage of using this highly automated system is that it helps you to track the profitability of each job. You know exactly what you need to charge in order to turn a profit and can provide more accurate estimates to customers. It is also a good way to inform customers of what they should expect to budget for upcoming repair or service jobs because you can document any issues that arise with the vehicle for both you and the customer to see. This helps to show your customer that you are transparent throughout the process and will earn you a significant amount of goodwill with customers.\nTekmetric is one of the leading providers of digital vehicle inspection technology and is the company that many auto repair shops turn to when they want to increase efficiency without breaking the bank. The technology used by Tekmetric is cutting-edge and is reliable in terms of protecting private customer information against hacks and data breaches. This is an especially important consideration for choosing your service provider at a time when customer data breaches are making national headlines fairly regularly. The platform provided by Tekmetric is accessible from anywhere because it is built using Cloud-based technology. Its innovative platform allows you to upload photos and videos of vehicles faster than ever before. Auto repair shops that use Tekmetric's platform find that they spend less time managing administrative tasks and more time working on jobs for customers, which reduces labor costs and increases profit in the long run.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fast-fashion retailer Zara is jumping on the success of the Meghan Markle effect by producing a near carbon copy of Meghan's rosy pink boatneck blazer by Carolina Herrera at the 2018 Trooping the Colour.\nZara's version features the same off-the-shoulder, boatneck silhouette, except in a crimson shade and featuring double-breasted buttons. It is the perfect on-trend blazer to pair with a midi skirt like Meghan or a pair of black wide-leg trousers for the office. It's not surprising that for fall, Zara would look toward the duchess for fashion inspiration. Meghan's sensibility has caused countless dresses, handbags, and shoes she favors to sell out or websites to crash, and this item is on its way to selling out too, with sizes XS and S the last available.\nMeghan's boatneck look has become an integral part of her signature style. Not only did the new royal choose the silhouette for her wedding day (a gorgeous white gown by Givenchy), but she's also incorporated plenty of boatneck blazers and dresses by Theory and Altuzarra.\nIf you've always wanted to dress like a royal, this is your chance \u2014 but hurry, sizes are selling out fast. Shop the boatneck blazer now for $100 on zara.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqtyj b/data_all_eng_slimpj/shuffled/split2/finalzzzaqtyj
new file mode 100644
index 0000000000000000000000000000000000000000..e8d78bd212f6b61e608deb4c32d9b193872e1cbb
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Knowledge is part of a 36-story glass\/concrete high rise called the Milan, located at Yonge and Davenport, just north of Bloor. The building has world-class amenities and is state of the art. With over 650 square feet of floor space and a large terrace, Knowledge is great for entertaining. It is a very spacious 2 bedroom plus den, with an amazing layout.\nThis 32nd-floor suite features a breathtaking east view, looking over Rosedale Valley. No expense was spared on this very modern luxury suite! It features granite counters, stainless steel appliances, marble bathroom floors, hardwood flooring throughout, 9-foot ceilings, and so much more! You will find numerous upgrades within, including an upgraded bathroom, upgraded light fixtures, upgraded kitchen, only the finest furnishings and appliances. This luxury suite is the cr\u00e8me de la cr\u00e8me, and will truly take your breath away.\nLocated in the heart of YorkVille at Yonge and Davenport, Knowlege is perfectly located to offer its residents the very best of city living. Located just north of the intersection of Yonge and Bloor, residents can take advantage of some of the best restaurants, shopping and entertainment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Item: 15-2634 - Type: Cross Stitch Patterns.\nModel is stitched on 28 count French Rain Linen using Gentle Art Sampler Threads and Weeks Dye Works. Includes wool shown on the top of pincushion.Shown with Pins from Just Another Button Company.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description : Our holiday let appliance box allows any electrical appliance : washer, dryer, iron , shower etc to be coin operated. Any appliance up to 13A (3kW) may be used with the coin timer.\nCoin operated timers can be used to control the use and misuse of any electrical appliance or service. The SP100\/HL coin operated timer is a fully digital unit housed in a secure welded steel case with many innovative features.\nThe SP100\/HL can also be used to control showers at campsites and truck stops as well as in rented properties. Any other electrical equipment may be coin operated with the SP100\/HL, our technical department is available to offer advice.\nThe SP100\/HL requires no routine maintenance and will provide a trouble free service for many years to come.\nTamper proot wall mount, radial pin lock and anti-jemmy desing fo secure no maintenance operation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In just a few weeks, American Idol will bring its nationwide search for the next musical star to central Florida. Getting things off to a great start, the Idol Bus is heading to the place where dreams come true: Walt Disney World.\nDisney Parks Blog shared the news that on Saturday, August 25, ABC's American Idol's 20-plus city search for the next singing sensation will make its way to Walt Disney World.\nAll singers must be at least 15 years old to audition. Pre-registration is encouraged.\nTo Pre-register and for additional details, full eligibility requirements, submission forms, and terms and conditions, visit the American Idol auditions page. If you're auditioning, you can bring one guest to accompany you, and the guest must be over 5 years old.\nEarly arrival is encouraged. And, for you early birds, parking will be available as early as 4 am, but not before. Registration will begin at 7 am with auditions starting at 9 am. Overnight camping will not be allowed.\nMake sure to bring water, sunscreen and a hat to stay in top shape for your audition.\nCelebrity judges will not be on site.\nAmerican Idol bus tour auditions will begin on the same day in San Diego, California, as a second Idol Bus joins the nationwide quest for talent.\nIn addition to appearing in person, hopefuls can submit audition videos online via www.americanidol.com\/auditions or by showing off their talents via Instagram, Facebook, Twitter or Musical.ly using the hashtag #TheNextIdol.\nCan't make it to the auditions at Walt Disney World Resort? Check out the American Idol website for other cities around the United States.\nAmerican Idol returns for its second season on ABC for the 2018-2019 season with all-star judges Luke Bryan, Katy Perry and Lionel Richie. Emmy-winning producer Ryan Seacrest will also return as host.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Coming of age: She's just a few days from becoming officially an adult and this transition tends to have problems that do not need too much explanation. Still connected with and depending on parents. Yet getting independent and about to bear all responsibilities.\nCast out: Because of the lack of an ability she is a cast out in her settlement and daily mocked. Considered a lower level of existence. A mere Homo Sapiens.\nShe also faces the insecurities women tend to have and sometimes makes mistakes by them. Bear in silence or lash out without ascertaining.\nThe book has an unusual first person view style and language that at first may seem peculiar, but soon it becomes one of its main strengths. The language means a lot and not just a stylistic choice. It describes the society Rayne lives in.\nI don't intend to write more about the story. Not just to avoid spoilers, but because the main point is living the moments within.\nThis was unquestionably a five star experience for me. Quoleena did a great job and earned herself a place on my bookshelf.\n\u2013 Young women and those who remember how it felt.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarhbs b/data_all_eng_slimpj/shuffled/split2/finalzzzarhbs
new file mode 100644
index 0000000000000000000000000000000000000000..77deefb139817b77d0b416d561bcc20fada10e29
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzarhbs
@@ -0,0 +1,5 @@
+{"text":"Your one stop website for all your IT needs!\nWe develop desktop applications using the latest technologies. Nearly all of our applications fill a gap in the corresponding field (e.g. education). They are original and free of charge. You can also request new features. We also provide web services and consultancy. With web services, you do not need to install anything on your computer. Instead, you can use our web services for free. However, they might not support all the features in the desktop applications. If you have a special need, you can contact our consultancy team who will assess your need and provide you with a solution.\nOur expert staff can help you with all your questions, doubts, upgrades and new technologies.\nOur products are used by end-users, developers, companies, universities as well as research centres around the world. Successfull cases will soon be available on this website!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The First Acts Of The 2018 Laneway Festival Have Dropped!\nAucklanders, breathe a collective sigh of relief because the St Jerome's Laneway Festival is heading back to our shores in 2018 and we couldn't be any. More. Excited.\nThe epic festival has just announced that Mac DeMarco will be making a return next year, after blowing all of our minds back in 2015. Crowned as the King of Slacker-Rock and with a track record of releasing a fresh album every single year, it'd be a real loss to miss this one. We suggest getting your hands on some ticks ASAP (they are released on Wednesday), because this kind of event doesn't wait for anyone.\nAlong side Mac, Anderson .Paak & The Free Nationals are set to take centre stage and the anticipation is real. Praised for being one of the world's top live acts with extreme vibes that will leave you wanting more days later, their debut to the beloved Laneway is welcomed with open arms. Since popping up in Dr Dre's internet-breaking album Compton, .Paak has gained momentum across the world making a front row spot a real kind of battle.\nThe rest of the Festival's 2018 line up will be revealed on September 12 and based on the two headliners, we are expecting a hype-worthy bunch.\nIf you haven't already experienced the day of banging beats and delicious eats under the Kiwi sun, 2018 is calling your name. After a smooth transition to Albert Park last year, the festival will be returning so you can have a not-so-chill day at the park.\nLaneway is locked and loaded for Auckland Anniversary Day (29th Jan) right in the heart of this coming summer. Grab your hat and sunnies and let's do this.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the last few weeks, we talked about the importance of slowing down and being present, getting down at your child's level, listening with our eyes, following your child's lead and how to become an interpreter. Now that you are starting to put these into practice, it's time to respond meaningfully to your child's message. But how?\nThere are many ways we can respond to our children's communication attempts, and I am going to go over some of these briefly in a moment. But first, let's talk about how to respond meaningfully by meeting your child where he is and then giving him just enough information to push him to the next level.\nIn the fields of child development and education, we have some fancy terms for this like Scaffolding and The Zone of Proximal Development. The basic explanation, is that the zone of proximal development is the difference between what a child can do by himself independently and what the child can do with some help from us (scaffolding). This is sometimes referred to as the instructional level, the area in which instruction should take place to learn a new skill. You can read about the ZPD in more detail on Not Just Cute.\nThe gold (or yellow) represents how and what your child can communicate independently. These are the things he has mastered like the vocabulary he knows and how long his utterances are (2 words or three words for example).\nThe green represents how and what your child can communicate with a little bit of help. This is the area we want to be in when working with our children, especially those with speech and language delays. We meet them here, give them a little help, and slowly help them expand their gold area. I made this area green as green represents go and this is where I want you to be! For example, if your child is speaking in two word phrases, you would be modeling three word phrases. If your child knows some animal names, you will model new animal names. You meet him where he is and give him a little more information to help him along.\nThe red represents the how and what your child cannot communicate, even with help. These are language concepts that are too advanced for your child. I made this red as this is a no-go-zone. Spending time here will only frustrate you and your child.\nMy friend Kim from Little Stories gives some great examples of working in the ZPD. I encourage you to check out her post as well.\nTo find your child's zone\u2026you need to use those interpretation skills we talked about. You are watching and listening to your child and interpreting his messages\u2026take note of these messages. Is he sending them with only body language and no words? Then go one more step and use one word to model. Is he using two words at a time? Model three word sentences. Does he know some animal names but not others? Model the animal names he doesn't know. Does he know some locative concepts but not others? Show him and model the concepts he does not know.\nNow, you do not have do this all day every day. Your child likely can understand much more than he is saying, so you need to continue to model longer utterances within your day as well. The times you really want to work within is ZPD are moments when you are really trying to teach him new things.\nMore detail on ways to respond will be coming up soon! In the meantime, you can check out all my other tips HERE.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The fraying relationship between Uber and one of its largest shareholders has opened a rift in the boardroom of the ride-hailing service.\nThe venture capital firm Benchmark, which has spent years as a low-profile backer of start-ups, has long been in the background supporting Uber and its ambitions.\nBut that relationship has changed amid questions about Uber's culture, including allegations of sexual harassment and discrimination. Now, the companies are in a very public fight.\nIn June, Benchmark helped push out Uber's co-founder Travis Kalanick as chief executive, and last week it sued Mr. Kalanick for fraud and is seeking to remove him from Uber's board. Benchmark has since written an open letter to Uber employees, suggesting there may be more to be revealed about the ride-hailing company's inner workings.\nThe lawsuit came as a surprise to Uber's directors, according to three people with knowledge of the matter.\nThe battle has caused other technology investors to question Benchmark's tactics.\nAs corporate chiefs debated abandoning one of the Trump administration's business advisory councils, a few business titans spoke privately in favor of sticking with the president.\nThese included Jack Welch, the retired leader of General Electric, and W. James McNerney Jr., the former head of Boeing, according to a person briefed on the call, who was not authorized to speak about it.\nTheir reasoning was a combination of patriotism and the hopes of salvaging a pro-business agenda.\nThey did not succeed in persuading their peers, and the council disbanded in protest over comments by President Trump equating the actions of far-right hate groups with protesters opposing them, after a violent confrontation in Charlottesville, Va.\nThe president's remarks that \"many sides\" were responsible for the violence, and his subsequent comments have raised concerns about the White House's ability to move forward with an agenda that includes overhauling the American tax system and putting in place an infrastructure plan.\nThe episode has highlighted the varied approach companies have taken in response, and the outside pressures leaders of public companies face.\n\"C.E.O. courage is in short supply,\" she added.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm just switching over to using Xamarin forms, so I feel like this is something simple I'm missing.\nWhen the app loads though it displays an empty table. The test label was just to see if it was the label that wasn't binding, but the test label doesn't show either.\nWhen the CurrentCodes changes value will the listview update on it's own, or do I need to trigger it somehow?\nI saw the BindingContext mentioned, but I don't understand how it fits in. I couldn't find any examples of it.\nBut I'm not sure that's how you're supposed to do that.\nWhat's the difference between the ItemsSource and the BindingContext?\nYour SearchList.BindingContext = this; worked.\nI initially put it in the wrong spot.\nGreat! Ready to go then.\nCode your UI in XAML . personally find it easier to do binding\/mvvm stuffs with xaml based UI .\nAlso more example on these areas in the internet (xaml based examples).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzativo b/data_all_eng_slimpj/shuffled/split2/finalzzzativo
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Beneath Melbourne's busy city landscape lie layers of a turbulent history and an ongoing vibrant Aboriginal culture. Melbourne Dreaming allows you to learn the past and appreciate Aboriginal people's historical, cultural, social and economic contribution to the city.\nIt is both an authoritative guidebook with clear maps and travelling instructions but also an alternative social history with stories and images of significant people and events.\nsites that are the focus of contemporary life.\nMeyer Eidelson created this edition in response to demand, in consultation with the relevant Aboriginal groups. He runs Melbourne Walks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In 2007 MacDonald, Dettwiler and Associates Ltd. (MDA) made the assessment that the domestic space market wasn't growing and that there was no long-term commitment to space by the Conservative government. As well, access to the U.S. market, critical for growth, was being stymied by the International Traffic in Arms Regulations (ITAR) and security issues. At the same time MDA's Information Products group was growing nicely year to year. The decision was made to sell its Information Systems and Geospatial Service operations.\nIn January 2008 MDA announced1 the sale of these assets to Alliant Techsystems (ATK). MDA was effectively getting out of the space business. The assets would be moved into a new group at ATK to be called ATK Space Systems. Nearly seven years later, MDA is still in the space business and has transformed itself into a multinational player. How did they do it?\nIn 2007, MDA's operations were grouped into two groups, Information Products and Information Systems. The Information Products group accounted for 70% of revenues and totalled just under $845 million2. Revenue compared to 2006 was up 23%, a considerable gain. The Information Systems group, which included most of the companies space assets except for the Geospatial Service operation, had seen revenues decline slightly (2%) from the previous year. Revenues from this group were $359 million. Total revenues for the year were $1.204 billion.\nThe Information Products group had been growing fast the past few years. In contrast, the Information Systems growth had gone from 10% in 2005, 5% in 2006 and now -2% in 2007. With the Information Products group growing fast, and belief that there was no long-term commitment to space by the government and ITAR hampering access and growth in the U.S., MDA made the fateful decision to sell its space assets.\nAt the same time MDA was making the decision to sell its space assets and go \"all in\" on the Information Products group, the U.S. and U.K. subprime mortgage crisis was just about to hit. A perfect storm was brewing and MDA was about to be right in the middle of it. However, fate of a different kind would intervene and MDA's path would alter forever.\nOn January 8, 2008 MDA announced it was going to sell its Information Systems group along with its Geospatial Services operation from the Information Products group to ATK for $1.325 billion. It was stunning move that prompted swift reaction from coast-to-coast, including the federal government.\nOn January 29 the parliamentary Standing Committee on Industry, Science and Technology held its first meeting of the year3. At the meeting, NDP Member of Parliament Peggy Nash moved a motion that the committee request the Minister of Industry Jim Prentice to appear before the committee to discuss the sale of MDA to ATK. This set in motion a series of hearings which took place on March 5th and 13th, April 1st and 3rd, and included testimony from Prentice, MDA and other interested parties.\nAt the first hearing on March 5th, witnesses included: Steven Staples, Chair, Rideau Institute on International Affairs; Michael Byers, Professor, University of British Columbia, Canada Research Chair in Global Politics and International Law; And as individuals, Marc Garneau, Former President, Canadian Space Agency; and Hugh Thompson, Spacecraft systems engineer, MacDonald, Dettwiler and Associates Ltd.\nCapability in building large class satellites was one issue. But there was also the robotics capability including the Canadarm, DEXTRE and also ground stations etc.\nGarneau wasn't mincing his words. Neither would others.\nAs Garneau said, MDA was not \"just another Canadian company\". In fact, the government had been building MDA up to become a global leader. And it was becoming a player and leader in several areas. The CSA, which contracts out approximately 65% of its yearly budget to industry and academia, had over the past five years contracted out 50% of that money to MDA alone. From an industry perspective, there was MDA at the top in Canada, and by a wide margin, and then there was everyone else.\nFrom an economic perspective, Canada's space sector had generated $2.499 billion in revenue in 2007 according to the Canadian Space Agency State of the Canadian Space Sector annual report4. MDA's space revenues in 2007 were $431 million meaning that the company accounted for just over 17% of Canada's total revenue for the sector that year. That was not a trivial amount.\nWhile MDA contributed a significant amount of revenue to Canada's space sector there was another issue that was also a key point of debate, intellectual property (IP). And this for ATK was one of the key aspects of the deal.\nWhile the economic impact was part of the picture, national security was another aspect to the sale.\nThe primary law the committee had to work with in investigating the sale was the Investment Canada Act. However the sale included assets that presumably fell under national security. However the fate of the deal would have to be judged by the Investment Canada Act. But the security issue, national interest, would continue to dominate the hearings.\nFor all the money Canada had invested in MDA over the years, the intellectual property would now be owned by a foreign company. And at the time Canada was negotiating with MDA for the follow-on to RADARSAT-2, the RADARSAT Constellation Mission (RCM). Now that the IP might be owned by a U.S. company there was a question which MDA nor ATK would answer. Would the U.S. government step in and for ITAR reasons stop Canada from contracting ATK Space Systems to build RCM?\nIn 2005 parliament had passed the Remote Sensing Space Systems Act. The purpose of which in part was to protect Canada's, the governments right, to access remote sensing data from satellites licensed by Canada including RADARSAT-2. If this deal went through then would Canada even be able to access imagery from RADARSAT-2? It was a point of debate.\nIn essence it seemed that the U.S. government could in fact step in and prohibit Canada from accessing the imagery. And if ATK owned the IP and if the IP was now deemed to be subject to ITAR, well then ATK might not even be allowed to build the RCM for Canada.\nIn another twist, Hugh Thompson, a spacecraft systems engineer at MDA, was appearing as an individual with support form 12 other MDA employees to argue against the deal!\nWhile MDA's President and CEO, Daniel E. Friedmann would later refute some of Thompson's testimony7, it and what it meant, were in the record and on people's mind.\nIn his testimony Friedmann made of point of saying that although ATK would now own RADARSAT-2, ATK would honour the agreements signed with the Canadian government for access. Not only that but ATK's subsidiary in Canada would be subject to Canadian law and must honour all the agreements. The company would remain in Canada and continue to do business as usual.\nBelief wasn't legal fact though.\nAnd while the government was having issues with the sale, MDA's shareholders weren't. They voted8 99.9% in favour of the sale.\nBut the question of national security and access to the satellite still remained. Members of the committee and the government didn't buy into the argument. After all, this was the post 9\/11 era. The U.S. was even more stringent with security. ITAR was no longer under the purview of the U.S. Department of Commerce who's mandate included promoting industry, instead, the State Department, was responsible for ITAR and was enforcing new, tighter rules.\nHow could Canada be certain that the U.S. government would honour ATK's Canadian agreements? MDA's approach was this was strictly business deal. One company taking over another companies assets. It happens all the time across borders. But it wasn't just another deal.\nAnd there was a mistake no Canadian government wanted to repeat. In the late 50's the Americans had pressured the Canadian government to stop work on what was one the most advanced fighter\/bomber jet of the era, the Avro Arrow. The Americans wanted Canada to buy U.S. planes which they did. Thousands of Canadians would lose their jobs, and importantly, critical skills in the form of advanced engineering jobs would go to the U.S.\nOn April 10, 2008 the government made history when it blocked the sale9. Citing the Investment Canada Act and that the deal would not be a \"net benefit\" to Canada, the deal was blocked. It was the first time a Canadian government had used the Act to prevent a sale. Officially, ATK had 30 days to convince the government otherwise, but everyone knew it was unlikely. Politicians from all spectrums were in support of blocking the deal.\nMDA was too important to Canada. The government had made it clear. MDA's assets, including all the IP and RADARSAT-2, were off limits to foreign companies.\nThe deal was dead and MDA's problems were about to get worse.\n1. \"January 8, 2008 \u2013 MDA to Divest Information Systems Business for $1.325 Billion.\" January 8, 2008 \u2013 MDA to Divest Information Systems Business for $1.325 Billion. MacDonald, Dettwiler and Associates Ltd., 8 Jan. 2008. Web. 26 Nov. 2014. http:\/\/www.mdacorporation.com\/corporate\/news\/pr\/pr2008010806.cfm.\n2. \"MDA 2007 Annual Report.\" MDA 2007 Annual Report. 26 Mar. 2008. Web 26 Nov. 2014. http:\/\/www.mdacorporation.com\/corporate\/Investor\/financial_reports\/mda_2007_annualreport.pdf.\n3. \"House of Commons Committees \u2013 INDU (39-2) \u2013 Evidence \u2013 Number 024.\" House of Commons Committees \u2013 INDU (39-2) \u2013 Evidence \u2013 Number 024. Standing Committee on Industry, Science and Technology, Parliament of Canada, 5 Mar. 2008. Web. 13 Dec. 2014. http:\/\/www.parl.gc.ca\/HousePublications\/Publication.aspx?DocId=3330167&Language=E&Mode=1.\n4. \"State of the Canadian Space Sector.\" 2007 State of the Canadian Space Sector. SpaceRef, 2008. Web. 14 Dec. 2014. http:\/\/images.spaceref.com\/docs\/2008\/State_of_the_Canadian_Space_Sector_2007.pdf.\n5. \"House of Commons Committees \u2013 INDU (39-2) \u2013 Evidence \u2013 Number 029.\" House of Commons Committees \u2013 INDU (39-2) \u2013 Evidence \u2013 Number 029. Standing Committee on Industry, Science and Technology, Parliament of Canada, 3 Apr. 2008. Web. 13 Dec. 2014. http:\/\/www.parl.gc.ca\/HousePublications\/Publication.aspx?DocId=3385860&Mode=1&Language=E.\n6. \"Orbital to Sell Its Interest in Macdonald, Dettwiler and Associates for $163 Million.\" Orbital to Sell Its Interest in Macdonald, Dettwiler and Associates for $163 Million. SpaceRef, 16 Apr. 2001. Web. 13 Dec. 2014. http:\/\/www.spaceref.com\/news\/viewpr.html?pid=4508.\n7. \"House of Commons Committees \u2013 INDU (39-2) \u2013 Evidence \u2013 Number 028.\" House of Commons Committees \u2013 INDU (39-2) \u2013 Evidence \u2013 Number 028. Standing Committee on Industry, Science and Technology, Parliament of Canada, 1 Apr. 2008. Web. 26 Nov. 2014. http:\/\/www.parl.gc.ca\/HousePublications\/Publication.aspx?DocId=3377039&Language=E&Mode=1.\n8. \"March 11, 2008 \u2013 MDA Shareholders Approve Proposed Sale Transaction.\" March 11, 2008 \u2013 MDA Shareholders Approve Proposed Sale Transaction. MacDonald, Dettwiler and Associates Ltd., 11 Mar. 2008. Web. 14 Dec. 2014. http:\/\/www.mdacorporation.com\/corporate\/news\/pr\/pr2008031101.cfm.\n9. News, CBC. \"Federal Government Blocks Sale of MDA Space Division.\" CBCnews.\nCBC\/Radio Canada, 10 Apr. 2008. Web. 28 Nov. 2014. http:\/\/www.cbc.ca\/news\/business\/federal-government-blocks-sale-of-mda-space-division-1.703999.\nCTV Television, Inc, 11 Apr. 2008. ProQuest. Web. 13 Dec. 2014. https:\/\/www-lib-uwo-ca.proxy1.lib.uwo.ca\/cgi-bin\/ezpauthn.cgi?url=http:\/\/search.proquest.com.proxy1.lib.uwo.ca\/docview\/190785331.\n11. \"Minister of Industry Confirms Initial Decision on Proposed Sale of Macdonald, Dettwiler and Associates Ltd. to Alliant Techsystems Inc.\" Marketwire. Industry Canada, 9 May 2008. Web. 28 Nov. 2014. http:\/\/www.marketwired.com\/press-release\/minister-industry-confirms-initial-decision-on-proposed-sale-macdonald-dettwiler-associates-854398.htm.\nThis story is part of a special 2014 Canada in Space Series looking at Canada's space program. Part 2, The Transformation of MDA Into a Multinational Player, will be published next week.\n\u2013 Would a Liberal Government Reform the Canadian Space Program?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BX Shinsei Seiki Co., Ltd.\nTo perform our work smoothly, we are allowed to obtain and use the information such as the customer's name, address, phone number and E-mail address. In a recognition of our grave duty to appropriately protect these pieces of customer information (hereinafter referred to as \"Personal information\"), we handle such personal information under the following policy to carry out our duty.\nAdhere to \"Act on the Protection of Personal Information\" and other related ordinances applied to personal information, and handle such personal information appropriately based on the customs related to the handling of personal information known as fair and adequate generally. Handling will be improved, as required.\nClarify the provisions related to the handling of personal information, and make them known to employees by means of indoctrination, training and enlightenment.\nIdentify and notify or publicly announce the purposes of obtaining personal information, and handle the personal information according to the purposes. In the following cases, however, presentation of purpose may be omitted.\nPersonal information is provided in the interview with our sales personnel, or in exchange of name cards.\nWhen name cards or anything are given to us at various events, fairs, seminars, etc., such information may be used in the introduction of our products and services, provision of various information, or marketing activity including face-to-face transaction by E-mail or other means, unless refused by the customer.\nIn order to prevent leakage, loss and falsification of personal information, necessary actions are taken to carry out appropriate control.\nWhen customer requests for the disclosure, correction, deletion, stop of use, or extinction of personal information then held, such requests are accepted by the specified personnel with sincerity.\nMore specifically, personal information will be handled according to the following details.\nWe obtain personal information required in the performance of business work, but these pieces of personal information will be used only for the following purposes. In order to carry out work smoothly, we may provide personal information to our affiliate companies or the contractors to which we assign work, as far as required. In such a case, we carry out appropriate control, including the conclusion of agreement with these contractors on the handling of personal information.\nProvision of shutter operator, various reduction gears, various lifts, measuring instruments (hereinafter referred to as \"Our products\"), and manufacturing, sales, construction, repair, check and after-sales services related to their business.\nWe nominate a controller of personal information, and carry out the indoctrination education, training and enlightenment. In the event of invalid access to personal data, or of leakage, loss, destruction or falsification of personal data, necessary and appropriate safety control measures are taken, so that personal information may be controlled appropriately.\nIn the case of provision to our affiliate companies or contractors described in \"2. Purposes of Use, etc.,\" or in any of the following cases, we won't provide or disclose personal information to any third party.\nDisclosed or provided in the forms disabling the identification of the person, such as statistical data.\nMust be disclosed or provided according to laws and ordinances.\nRequired for the protection of human life, body or property, though the customer is unlikely to agree.\nRequired specially for the improvement of public health, or the promotion of healthy rearing of children, though the customer is unlikely to agree.\nRequired to cooperate in the performance of legally specified work by national organ, local community or their contractor, and when customer's agreement could impede the performance of relevant work.\nProvided to the persons mentioned in the following Item 5.\nPersonal information of customers shall be used by the following persons.\nWhen disclosure of his information is requested by the customer himself, from among the personal data held by us, response will be provided within a reasonable period and range only after the confirmation of applicant's identity.\nWhen correction, addition or deletion of his information is requested by the customer himself, from among the personal data held by us, information will be correct, added or deleted within a reasonable period and range only after the confirmation of applicant's identity.\nWhen stop of use, or extinction of his information is requested by the customer himself, from among the personal data held by us, use of information will be stopped, or the information will be extinguished within a reasonable period and range only after the confirmation of applicant's identity.\nWhen use of these pieces of information is stopped, or such information is extinguished partially or totally, we hope the customer will very kindly understand, and agree to the case where services may not be provided as requested by the customer, unfortunately.\nAcceptance Method and Window for Disclosure, etc.\nApplications related to the personal data held by us in Items 7 and 8, or any other inquiry on the personal information will be accepted by the following method. Please, note that the request for the disclosure not based on this acceptance method may be turned down.\nVisit the following window directly, or send application by postal mail to the following address. Details of application procedure will be provided when applied. However, the response will be given in written form or other method only after the confirmation of the applicant's (or agent's) identity at the following window.\nPlease, note that, in the event of application, you are requested, as a rule, to use the application form specified by us.\n(Dept. name) Shinsei Seiki Co., Ltd., General Affairs Dept.\nAcceptance time on week days is 9:00 a.m. to 5:00 p.m.\nWhen applied by applicant himself, we will confirm his identity by using the two or more, of the certificates such as the driver's license, passport, health insurance card, certificate of seal impression (original), family register (original), pension handbook and alien registration card.\nWhen applied by agent, we will confirm the agent's identity by using, in addition to above-mentioned items, one of the letter of attorney, family register (original), and the certificate of adult guardianship registration items proving the agent's identity, and the two or more, of the certificates such as the driver's license, passport, health insurance card, certificate of seal impression (original), family register (original), pension handbook and alien registration card.\nWhen applicant directly comes to the window, he is requested to pay total 930 yen comprising the charge per item 500 yen and the postal charge to return (registered mail) 430 yen, in cash or postage stamps.\nIn the application by postal service, applicant is requested to enclose, in an advance payment, the postage stamps equivalent to total 930 yen comprising the charge per item 500 yen and the postal charge to return (registered mail) 430 yen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The third Ian Talk by Ian D Montfort (Tom Binns) is all about \"proving there is no need to provide proof that his methods work by providing evidence that evidence does not matter\" !\nStood barefoot and donned in his ying yang wolf top, Ian certainly looks the part. You enter the performance believing you are going to see someone take the proverbial out of the celebrity \"psychics\" we are all aware of . Heavily dependant on audience participation, Ian, with the help of his Indian spirit guide (Raj, a call centre worker from Mumbai) brings messages from the other side and surprisingly gets most of the personal information correct!\nWe see the spirit of Shakespeare (his first appearance at the fringe) correctly quote a passage from his works previously circled by a member of the audience, there is a visit from Churchill himself, guessing his quote written on a British fiver, proven to be him as he wasn't around when these fivers were invented and a very impressive moment involving various wallet cards from the audience (including mine ) which left me in awe! Very funny throughout and the accuracy he delivers will have even the most cynical of us turned into believers!\nHe is either a comic genius or the seventh son of a seventh son, I cant decide!\nComic genius. No further Discussion required.\nSaw Tom Binns at the festival yesterday. Brilliantly funny..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mason Greenwood is one of the brightest talents in the Manchester United youth academy. He is a center forward, who can also play on the left wing.\nGreenwood has played in three youth categories this season (U18), (U19) and (U23) and is Manchester United's top scorer for the season, including the first team. He has scored 26 goals and wrote 9 assists in the 23 appearances across the various youth and reserve levels at the club.\nThe 17-year old striker has also made the Manchester United first team squad in the last round of the Champions League group stage at Valencia but is yet to make his debut. He led United's youth team to the first place of the Group H in the UEFA Youth League. He has scored 5 goals on 5 appearances in the competition.\nMason Greenwood is often compared to Robin van Persie, as he has a similar style of play. Both have a great technique, incredible off-ball movement and shoot with their left leg. Greenwood is also good with his right, as he takes the left side set-pieces with his right foot. He is unpredictable, explosive and fast, which are characteristics of a great striker.\nOle Gunnar Solskjaer is expected to give him a first-team debut before the end of this season, as he already played Thaith Chong in the FA Cup.\nAs Sir Matt once said if they're good enough they're old enough, we're built on that tradition, we need young players coming through, it's important in the academy.\nI am looking to bring players through, we have two or three top talents in the U18s that are knocking on the door now, we've had a couple of injuries and fatigue has been involved but then they're ready to step up. Neil and Ricky, they keep me filled in.\nMason Greenwood hopes to follow the steps of Marcus Rashford and Jesse Lingard to the first team of Machester United, as a youth academy graduate. We hope that he continues with his development in the right way and we expect him to be one of the leaders of the club in many years to come.\nNext Should Manchester United give Solskjaer a long term contract?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavfpb b/data_all_eng_slimpj/shuffled/split2/finalzzzavfpb
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+{"text":"Fletcher EC, Chong N, Augsburger JJ, Corr\u00eaa ZM. Fletcher E.C., Chong N, Augsburger J.J., Corr\u00eaa Z.M. Fletcher, Emily C., et al.Chapter 10. Retina. In: Riordan-Eva P, Cunningham ET, Jr. Riordan-Eva P, Cunningham E.T., Jr Eds. Paul Riordan-Eva, and Emmett T. Cunningham, Jr.eds. Vaughan & Asbury's General Ophthalmology, 18e New York, NY: McGraw-Hill; 2011. http:\/\/accessmedicine.mhmedical.com\/content.aspx?bookid=387&sectionid=40229327. Accessed April 21, 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2. Food. The longer you are kettled, the hungrier you are likely to get. Your body will cope better if you have eaten well before. So carbo-load \u2013 eat lots of pasta\/rice\/potatoes in the next couple of days. This will keep sugar levels up in your body and help keep your spirits high. Porridge is an excellent slow digesting breakfast that can maintain you well into the day. Grab a high protein sandwich at lunchtime (chicken if not veggie\/vegan, peanut butter if you are) You probably won't want to carry round huge bags of food with you, so carry foods that are light and easily digestible \u2013 energy bars, nuts, peanut butter sandwiches, bananas, raisins and jelly babies. These are all foods that take time to go through your body slowly, maintaining a good level of carbohydrate and protein in your system, without making you feel bloated. It's also worth having a sports drink as these are a great pick-me-up when you begin to flag. (Drink sparingly, though, they are very sweet and sickly if you have too much, and you also don't want to go to the loo!). Lots of long distance runners also use carbohydrate gels as these give an instant boost.\n3. Distract your mind. The frustration of being stuck in a kettle, is that the police are in control and you can't get out. Sometimes, on long runs, I find it gets tough when I let the situation get on top of me, for example, when I miss a turn, think I'm further along. I have learnt to settle myself with a number of techniques. These include : mantras to get me in a running rhythm (eg Love not Cuts, Stop the Cuts, Cut the Cuts), planning something I'm going to do in the week ahead; solving a complex maths problem; singing my favourite songs; counting; imagining my face when I get home etc. Obviously, in a kettle, you don't have the advantage of the scenery changing, but, you can let your brain take you elsewhere for a bit. Of course, you have company which can also help. So, also try some mutual distraction, playing word games, cards etc with your neighbours.And, unlike running, you can always take a book to read.\n4. Clothing. It looks like Saturday will be quite warm, which means it's tempting to turn out in light summer clothing. This will be fine in the middle of the day, but as the sun sets, not such a bright idea. Again, running has taught me that light layering is the best solution to this. Gaps between the layers provide insulation and increased warmth, so two thin jumpers can be better than one thick one. And if you bring several items of thin clothing, there's also less to carry around when it's hot.\nFinally, if you have small children, here's a few things I learnt not from Marathon training but on the one, very nasty, experience I had of kettling. Pack plenty of nappies (and plastic bags to dispose of them), a potty, lots of food, changes of clothing, books, crayons, pads, blankets and a couple of cushions. You will have a very heavy bag, but if you can hang it round your buggy (or place it underneath) it will mean you have everything necessary to keep your children fed, watered, rested and entertained for as long as necessary. Unfortunately, in my experience, police officers in a kettle lose their humanity when faced with a mother with small children desperate to leave. You may be lucky enough to find yourself a negotiator as I did (the very wonderful PN Editor, Mil Rai), who helps you escape. But if you aren't so fortunate, hopefully, you and the children will survive and suffer the minimum discomfort while you wait for your release.\nOf course, if you are canny & have the up to date Apps on your phone, you'll be able to avoid being penned in anyway. But if you are one of the unlucky ones, I hope this helps.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"costs next to nothing is a bonus.\nI shared these pearly twiggy napkin rings a few years ago during my Girls Want Pearls series.\nJust gather some twigs and be sure to find some that are super skinny for threading pearls onto. If you can't find any skinny enough, you can cheat and use brown wire for the ones that have pearls. Make five bundles of twigs and wrap with wire. Tie your twine around each bundle, spacing them approx. 1\/2 inch apart. I found it easier to prepare the knots in advance and then slide the bundles in. Form the finished twine around a napkin and tie the ends together. Thread your pearls on and add a dot of glue if needed.\nFor your Christmas table you could mix in some sparkly beads in your color choice.\nAdd a natural-elements centerpiece, maybe vases full of twigs or a tray of hedge apples, pumpkins or artichokes for a cohesive look.\nWelcome to DIY Double Dip Flip! Heather and I are excited to share our third project in this series where we start with an inspiration piece from either fashion or decor, then we each show you our unique take on how to use the idea on something entirely different in decor or fashion. For this month's inspiration, Heather chose an entire set with soft spring colors and lots of great texture.\nBeing a lover of frayed edge denim, I was immediately drawn to the ripped knee patch jeans and decided to make that my jumping off point. I am in the middle of making changes in my breakfast nook and my chandelier needed a makeover.\nSo my flip for this month is DIY Chandelier Lamp Shades.\nAfter my husband spray painted the fixture black I dressed the lamp shades in layers of frayed canvas.\nThese were really easy to do and I'll show you how with just a few steps. But first I want to show you the \"before\" picture of the chandelier.\nIt's not a very good picture but you can see that the fixture was an ivory color and the shades were black. You can also see the old wall color.\nThe black fixture gives the space a more grounded feel. The frayed canvas cloth adds softness and texture. I love my new wall color and I can't wait to show you the entire space when it's complete. You can catch a glimpse of another finished piece for the nook here.\nNow back to the lamp shade tutorial.\nHere's what you'll need to make your own tiered, frayed canvas lamp shades.\nA template for you to adapt for your own use. For standard chandelier type shades the width of the template needs to be 14 inches. I had to print it out on two sheets and tape them together.\nCut out your 2 template parts and tape them together.\nMeasure out and mark the pattern to be divided into 3 layers. The first mark for the bottom layer will be at 1 5\/16 inches, the second mark at 2 5\/8 inches.\nMake the same markings in a few more places across the pattern and pencil in the connecting curves.\nCut out the three pieces. These will be the patterns for your three layers.\nCut out your fabric, leaving an approx. 1\/8 inch allowance above and below the pattern.\nFray the fabric strips. Note: the bottom and middle layer will only need to be frayed at the bottom edge.\nThe top layer will need to be frayed at both the top and bottom edges. You can fray the ends too.\nGlue the bottom strip to the lamp shade leaving enough fringe to cover the entire bottom rim.\nOverlap the middle layer and glue. Glue the top layer making sure to extend it far enough up to cover the top rim of the shade.\nThis design could also be adapted for full size lamp shades.\nSo you've seen my take but you haven't seen Heather's! Be sure to go visit her blog Woods of Bell Trees to see how she interpreted this month's design.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Summit Multimedia is proficient in directing our passion of photography, video, and graphics to suit your unique needs. We have years of experience and multiple relationships with clients who favor us over and over again for their multimedia solutions. We aren't set in our own ways, but strive to help you pick the newest way to present an object or idea over various media to your specific target audience.\nSummit Multimedia is rooted in the summit of the Rocky Mountains of Colorado, but we branch out across the United States and have completed projects internationally as well.\nI believe each person has a God-given passion, and when we pursue such in our daily lives, we will succeed in anything we set our heart on. For this reason, I believe Summit Multimedia succeeds in any media related endeavor we pursue as a company.\nI would like to title God as the ultimate CEO of Summit Multimedia as we are Christian owned and operated. In effort to give thanks to Him for the blessings He provides, we also like to give back. Whether investing our talents in time or aiding financially, we like to contribute a portion of each project to a particular Christian ministry.\nI am honored that you are considering Summit Multimedia for any HD production and look forward to helping you further in producing a product that is specific to you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our out of school care is a popular choice for families with children ages 8-12. Childcare for before and after school, or during school holidays and training days is an important part of many people's lives. Our experienced staff work with many children from Tipton and nearby areas, providing award-winning care. To find out more about our services, send us a message and one of our friendly team will get back to you.\nThere is one member of staff for every twelve children (higher ratio on outings).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavzuu b/data_all_eng_slimpj/shuffled/split2/finalzzzavzuu
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index 0000000000000000000000000000000000000000..5e356362551914a1a8026f43914bbc76a6e842b8
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+{"text":"Let's get ready for the weekend! This large tote (16\" x 8\" x 16\") is perfect to carry everything you will need for that short vacation or use it as your carry-on for a longer journey. It has a zipper closing on the top and lots of pockets inside and out.\nClick on the thumbnails to see more of the TGIF Tote.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Just about the most sensible ways to enhance your diet and provide your fun an interesting hobby is always to plant an organic gardening. You should devote some time and energy to reach your goals in this particular venture, though. Knowing where you should start when planning an organic horticulture are most significant.\nChoose the proper soil to obtain the best results. You can even have the ability to design an artificial area which uses an individual form of soil.\nOnce the fall season approaches, it can be time and energy to plant your fall edibles. A pumpkin can be a festive container for kale and lettuce.Once you cut an opening near the top of the scoop and pumpkin out your insides, spray the sides and inside with Wilt-Pruf to prevent rotting.\nCreate a plan before you decide to dig the first hole. This assists you in recognizing your tiny plants after they start to show up.\nKnow the ideal times to harvest the vegetables with your garden. Each veggie has unique time for harvesting with the peak of flavor. Would be best picked from the vine later while they are very ripe, despite the fact that for instance, baby and zucchini peas are best picked young.Tomatoes.\nPlant items with fall season color at heart.Maple trees produce yellow and red leaves, and so are beech and dogwood trees. When thinking about shrubs, consider cotoneaster, hydrangea or barberry.\nA successful organic garden is the consequence of perseverance and a good understanding of the principles involved. Persistence and patience may also be important traits for successful organic gardener. Remembering the aforementioned tips, it is possible to successfully grow an organic garden.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sheriff Matthew Dixon has headed off a major Treasury crisis by foiling an attempt to debase Linrodeth's coinage. The plotters, working from a forge in Cadene, are thought to have passed nearly \u00a33000 of debased coinage into circulation. However, due to prompt action by the Sheriff, nearly all these coins have been recovered, and Dixon assures us that there is no threat to the value of our coinage. Anyone who believes that they have a suspect coin should present it the Witanmoot where it will be replaced on a one for one basis. This offer will be maintained for a year and a day. Sheriff Dixon announced the exposure of the plot to a packed emergency meeting of the Court of Common Council, and the Chronicle was most interested to note the obvious division between those guilds in the know and those to whom it came as a nasty surprise. News of the counterfeiting had clearly been leaked beforehand. We suspect that many guilds will be increasing their political funding in an attempt to avoid future embarrassment. Shortly before the meeting the plotters had been apprehended by the Witanmoot Guard, led by Sheriff Dixon and Cllrs Avery, Firethorn and Derwent, in raids throughout the city. In one fight Cllr Adam Avery was seriously injured, and has lost an arm despite the best efforts of the Torian chirugeon. The Chronicle wishes him a speedy recovery.\nThe vacant Levestone Aldermanship has resulted in a sudden influx of inquisitorial councillors into the ward. Despite endorsements from both Sheriffs, the councillors are determined that the absent Envoy Mowbray will not be returned uncontested. Many retired aldermen or up and coming young councillors have been interviewed, but so far the nominations stand at Malcolm Mowbray, Martin Key and William Hoskyns.\nConcern is growing over the 'tent city' which has remained outside Linrodeth following the Dragonfeast. Some estimates suggest that as many as two thousand people are still inhabiting the temporary accommodation set up for the feast. Many councillors have been expressing fears about the possibility of plague, or the establishment of a second 'floating market'. Councillor Randy Parkinson has also warned of the serious implications for the environment if they have not moved before winter.\nThe Harvestide Festival looks set to be a generous celebration, with many predictions for an excellent corn harvest. Concern remains over the possible damage wrought to the fruit crops by the unexpectedly early frost this month.\nCouncillor Richard Firethorn has renewed his campaign for the 'sweet' water project and has been pressing the Chair for Welfare and Education, Ann Tasker, to include it in her forthcoming budget. However the most influential member of Tasker's Faction, Adam Avery, is known to bear a strong dislike for the project, indeed his last comment on the subject is unprintable. In order to gain more leverage, Cllr Firethorn has therefore formally proposed a motion that the Council approves, in principle, the allocation of funding for the sweetwater project as part of the Welfare and Education budget for 144. This will be voted on next month. The Esprayenna Ship of the Dawn is nearing completion. The ship, which is currently completely filling the Temple Dock at the port, is attracting much attention and the people of Linrodeth are look forward to witnessing an elaborate consecration ceremony next month. Sheriff Dixon has also been showing unusual interest in the project and rumours persist that the Temple of Esprayenna received a number of generous donations from the coinage counterfeiters.\nAnonymously donated papers have revealed that Philip Portman's fletching business is much overdue for a tax re-assessment. The reason for the sudden appearance of this paperwork is unknown, but at least one councillor has been brave enough to suggest that it may be related to Portman's defection from the Perignon camp to that of Larken Drumm's.\nRegular visitors to Hunters will be distressed to learn that Brandy is re-negotiating her contract there. Rumour has it that an unknown rival patron is tempting Linrodeth's most popular dancer to leave the city.\nNews from the Citadel is that the Gelt have retaken the foothills of Slievekallen. This sudden reawakening of the Gelt has concerned many who had assumed that, following a series of defeats, they had given up. Most of the fighting is in the west with the area around Cascorach remaining quiet. However, the Princess has cut short her visit to Linrodeth and is returning to Cascorach immediately to prepare for any potential attack and to deal with a plague which is sweeping the city.\nOn a lighter note Lady Amelia Fortesque, the youngest and most ravishing of the Queen's Ladies-in-Waiting, is looking for suitable ideas for her birthday ball later this autumn. The Chronicle is collecting suggestions for appropriate catering, music and surprises. Contact Chinon Starsinger with your inspiration at the Empty Barrel in Welland.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Grades 1-8 are participating and it is 2-on-2 Scrabble teams. There will be 2 teams per classroom (4 people per classroom in total). We need some parent volunteers to help watch over the Scrabble tables, help look up words if necessary, assist with the chess timers and encourage our young spellers! We also are in need of Scrabble Boards and Scrabble Dictionaries that we can borrow! The kids have already started to practice and are very excited about this! Please contact Karey Heard at rkheard@rogers.com if you can help with the Scrabble Tournament. We are hoping to have about 10 adults to help that afternoon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Spring Capital Management we believe in a lower turnover, tax controlling strategy. We customize these portfolios to your particular needs taking into consideration your other investments, risk tolerances, and tax situation. We would welcome the opportunity to show you what these portfolios could do for you.\nCopyright . Spring Capital Management, L.L.C. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawyxu b/data_all_eng_slimpj/shuffled/split2/finalzzzawyxu
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index 0000000000000000000000000000000000000000..8b4f1b35bf9cd853158be1c36a8ed5e326fb9099
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+{"text":"EURUSD started its correction pattern and now eyes are on the reports from the US such as Personal Income and Spending MoM also the Unemployment Rate and Change from Germany.\nThe support rests at 1.15910 with resistance at 1.16930 which the weekly pivot point at 1.16150 is located in between the lines. The EMA of 10 is moving with bullish bias in the horizontal channel and it is below the EMA of 200 and the EMA of 100. The RSI is falling towards the 50 area, the Stochastic is moving in high ranges and the MACD is in the negative territory. The ADX is showing weak buy signal.\nConsidering the price growth, EURUSD may rise towards the upper Bollinger band at 1.17 but it will be fluctuating within the lower and upper bands.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"That odd title means \"use of the wrong word for the context\" or \"use of a forced and especially paradoxical figure of speech (as blind mouths).\" Catachresis the game is an exploration of catachresis the concept, and how it might be used to evoke horror.\nThe catachresis is a natural tool for horror since that kind of overwrought paradoxical language is often used when trying to describe something indescribable. It's particularly perfect for this brand of Lovecraftian horror, which revolves around evil forces so beyond our understanding that to know them is to go mad; we have to break the rules and logic of language just to come close to a description.\nYou play as Jeff, a supernatural investigator, during a particularly bad supernatural disaster. He's disillusioned with many of the paranormal theatrics, and that disillusionment provides a nice balance to the Lovecraftian catachreses used in the text descriptions. My personal favorite quote: \"This crate is bleeding clouds of darkness that burn the minds of beings arcane and ancient. I wonder what's inside of it?\" Logically, crates can't bleed, darkness can't cloud, and minds can't be burnt, but string those words together in a sentence and it still sounds scary as hell. Do this repeatedly over the course of a game and you create an atmosphere of disturbing absurdity.\nWhat makes the game even more interesting is that the catachreses of Catachresis aren't just based in text, but in the world too. Over the course of the short game, the imagery becomes increasingly abstract and absurd: A possessed computer is weird and kind of funny, but it's still grounded in reality; the alpaca that asks you to choose one of three mystical items to earn his forgiveness is just batshit crazy. These jarring shifts in setting and tone help loosen your sense of reality, which in turn helps you come to terms with the coming great evil because it already feels like you're losing your mind.\nFor as consistently creepy as the game is, there's little to no action in it. All you do is walk to the right (and occasionally left) and interact with things. There's never a time limit, you're never rushed, there's no combat, no inventory; it's about as minimalist a control scheme as you can get. The story acknowledges this limitation, and uses it to add to the atmosphere. At one point a character straight out tells you that weapons are useless since the forces you're encountering are so damned powerful. Various creatures make note that you are \"not to be touched,\" thus justifying the lack of action in a way that makes us even more helpless.\nOccasionally we get reports of chaos from elsewhere to ensure we never forget the danger facing the world and ourselves: Like the intern who walks ten paces across the \"threshold\" and gets him arm ripped off by a \"winged horror.\" No one can help him since he's still past the boundary line, so they just watch him die in the street. Catachresis is filled with little horror vignettes like that, brief stories that add color to the world beyond our limited point of view.\nThe game is a study in how the titular trick of language can be used to evoke horror. It explores its central concept from various angles\u2014text, setting, characters, and tone\u2014while also telling an intriguing story.\nCatachresis can be played here for free.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are a polished American tavern serving thoughtful renditions of American classics. We believe in service first, followed by cooking that is rooted in craftsmanship, quality and \"cravability\".\nThe Yakima Valley offers a veritable bounty of fresh ingredients throughout the year. From fruits and vegetables to grains and livestock the local farming community provides our kitchen with a sustainable source of quality ingredients.\nOur menu is prepared fresh every day. In keeping with Snyder family tradition all bread & pastry are handmade from scratch every single morning.\nEnjoy our unique menu and experience the culinary craftsmanship of our innovative chef in our award-winning building in the heart of downtown Yakima.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Louisiana graphic tees are made using the super soft Bella + Canvas Brand tees. These shirts are made with airlume combed and ring spun cotton\/poly. Unisex fit.\nSample shirt is Heathered Slate. This design is done in white ink only.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nokia, China's biggest handset vendor, reported on Wednesday that a mobile app aimed at helping rural Chinese users has become the company's fastest-growing service in the country.\nNokia launched Ovi Life Tools in China last May. The tools are a set of informational services meant for rural markets in developing countries. With the app, information can be found on a variety of topics including the weather, market prices and pregnancy care. SMS is used to deliver the information, with the service costing cost from US$0.75 to $1.20 a month depending which packages are subscribed to.\nWhile Nokia declined to reveal the number of users of the app in China, Ovi Life Tools is growing to become one of its \"most successful services\" in the country, said Phil Kemp, the company's head for the services sector in Greater China.\nOvi Life Tools allows mobile phone users who use their handset only for phone calls to experience receiving Internet-based content for the first time, Kemp said in an interview on the sidelines of the Global Mobile Internet Conference in Beijing.\n\"It's bringing people up to the learning curve,\" he said.\nChina has the world's largest mobile phone user population, at 889 million. But Internet access is still limited in many parts of the country. Only 457 million people access the Web, with rural Internet users at 125 million, according to the China Internet Network Information Center.\nNokia hopes to connect millions of other Chinese to the Web through simple mobile apps like Ovi Life Tools. Another app is Nokia's Ovi browser, which the company is creating by partnering with Chinese Internet firm Tencent.\nThe browser is designed to connect to the Web via second generation telecommunication networks, which are slower, but in use by most mobile phone users in China. \"We can deliver optimized content for the small screen size.\" he said.\nAlong with China's rural market, Nokia is also gearing up to tap into the location-based service opportunities growing in the country, Kemp said. The company is doing so by building up its Ovi mobile maps product through partnerships with local Chinese companies.\nThe partnerships are necessary to make Nokia's products appeal to local users. Earlier this year, Nokia announced it had made agreements with Chinese companies including Tencent and Sina to support Ovi Maps through their social networking platforms. Users with Sina and Tencent can now share their locations and provide recommendations on restaurants or shops through Ovi Maps and to Nokia handsets.\nNokia plans to announce other partnerships in China to back its Ovi Maps platform in the future, Kemp said. The company is also encouraging Chinese developers to build location-based apps for Ovi Maps.\n\"I think China presents some unique opportunities because of the fact China is dominated by local players,\" Kemp said.\"On the one hand that's a threat for some foreign companies coming into China. But because Nokia has been here for 26 years we see some opportunities to leverage our relationships.\"\nNokia is the first foreign company to have received a state license to operate online maps in the country. Microsoft has said it has already applied for the license and is waiting to hear back from authorities, while Google has said it is in talks with the Chinese government about keeping its mapping service online in the country.\nNokia has a 29.6 percent share of China's mobile phone market, according to Beijing-based research firm Analysys International.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..c9680bd3c413785d76cbb7025481ebf1fbd2d5bd
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@@ -0,0 +1,5 @@
+{"text":"The server encountered an internal error and failed to process your request. Please try again later. If this error is temporary, reloading the page might resolve the problem. If you are able to contact the administrator report this error message.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our practitioners are sort after and featured on many media platforms. See what they have been up to and hear what they have to say.\nBianca was asked back for Season 8 of Industry Leaders on Network 10. Watch Bianca (Naturopath) founder of HW Studio being interviewed by Janine Allis (founder of Boost Juice & Shark Tank Australia Investor)! Hear about Bianca's story from studying Naturopathy to creating an Integrative Health Clinic in Melbourne with 15 practitioners all working together to help patients achieve better health outcomes\u2026.\n\"House of Wellness\" discussing the importance of incorporating great foot care in your self care routine. If you haven't seen a podiatrist or ever wondered what they can help you with, hear Michelle chat with retired Australian Netball player, Bianca Chatfield about her foot care regime.\nOur Osteopath Claire from Chadstone Region Osteopathy appeared on Channel 9 News sharing her favourite tips for improving ergonomics in light of growing evidence that our desk jobs contribute to back and neck pain.\nBianca, Clinic Naturopath and Founder of The Health & Wellbeing Studio was interviewed by Industry Leaders on FOXTEL. Hear what she has to say about\u2026.\nBree Jenner, clinic Naturopath, is a Tribe member of the Chief Active online community, contributing regular articles to inspire people to use food to help fuel their best self.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Part 2 of our coverage of Baby Caroline's bohemian nursery features an interview with mom and the free-spirit behind this creative space, Nicki Odom!\nAt first, I wanted to have a beautiful pineapple\/beachy vibe as a theme for the nursery. But everyone is always calling me \"flower child\" and hippie at heart, so I thought it would be cute to go with a more, flower power, bohemian theme instead. It was also the theme of my baby shower and I am blessed to have wonderfully creative family and friends that made most of the decor that truly brought together this beautiful room.\nWhat drew you to Dream On Me over other nursery brands?\nWe have the Violet 7 in 1 Convertible Crib and the Emily white changing table. I honestly found Dream On Me when searching for \"different\" and stylish baby cribs. I really liked the curvy detail of the Violet Convertible crib and love how it will transform into bigger beds as Caroline grows and will still look stylish for a little girls' room.\nOur current space is a two bedroom apartment. So we were faced with having a guest bed in the nursery for when my husband's parents come to visit. It was challenging because I had more ideas to expand the bohemian theme into that corner but I plan to keep with this theme as she grows and until she tells me she wants otherwise.\nIt reflects how I want to raise Caroline. Wild and free-spirited, girly and creative, all things I hope she will grow to become as an adult.\nI chose items like the fluffy white rug, a pineapple lamp, mama and baby giraffe canvas because I wanted the room to have a lot of whimsy and fun things to look at for her. I envisioned Caroline laying and playing on the rug and being happy when she looked up at all of the fun decor in her room.\nMy favorite piece would probably be a tie between two items my close friends handmade for Caroline. First, the bohemian mobile that hangs over the bed was handmade by my friend Ashley. And the second would be the skull graphics canvas that was hand drawn and colored by my friend Leah. They are major pieces that really bring the room together as that bohemian, desert-like paradise I was hoping for!\nI know that I will be able to keep her room with this theme for a while as she grows up, but I just love the theme so much that even if she decided she wanted to redecorate with something else I would use most of the decor in my office space. Some of it could definitely be used in a future playroom as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NEW! Did you know that you no longer need a Doctor's prescription to start physical therapy! Ask us about the new Illinois Law that allows you direct access to your physical therapy.\nHealth Insurance policy has greatly impacted the standard of care for Physical Therapy. Here are a few ways and how they impact or limit your care at other facilities.\nInsurance companies have requirements to charge according to their codes, making client out-of-pocket payments prohibitively expensive.\nInsurance plans can limit number of visits regardless of your healthcare professional's plan to get you well.\nYou work with your own PT.\nWe do not bill insurance. This keeps us out of contractual limitations mentioned above regarding services and reimbursements set by the insurance companies.\nOur fees, (that are most likely coming out of your pocket even under an Insurance plan) are more affordable.\nSince we treat you individually, rather than handing you off to a non-professional in order to save money, we make sure you are meeting your goals with less visits.\nWe believe you should know what your care costs. Here are our rates. Here is a cost savings comparison between our services vs. an average in-network provider.\nHow we choose who may be working with you alongside your PT.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am very happy that Tanja Rueckert is joining Bosch. We will be gaining an internationally recognized specialist in IoT and Industry 4.0.\n\"In the IoT age, data are becoming a core component of industrial value creation, and software the central nervous system of a highly connected world. I'm greatly looking forward to contributing my experience in both these areas to Bosch,\" R\u00fcckert said. A mother of two with a PhD in chemistry, R\u00fcckert has been with SAP since 1997. She has 20 years of experience as an executive with strategic and operational responsibility in the software industry. She has a wealth of expertise in areas such as digital transformation, the internet of things, connected manufacturing, and connected logistics. During her career at SAP, she worked in Germany and the United States (Silicon Valley), and headed up teams operating across the globe in Palo Alto, Walldorf, Bengaluru, Shanghai, and other locations.\nBosch Building Technologies is a leading international supplier of products and systems for security and communication. In selected countries, Bosch also offers solutions and services for building security, energy efficiency, and building automation. Its product portfolio includes video-surveillance, intrusion, fire-alarm, and voice-alarm systems, as well as access-control and management systems. The portfolio is rounded out with professional audio and conference systems for the communication of voice, sound, and music. Building Technologies has engineering and manufacturing operations in Europe, the Americas, and Asia.\nThe company was set up in Stuttgart in 1886 by Robert Bosch (1861-1942) as \"Workshop for Precision Mechanics and Electrical Engineering.\" The special ownership structure of Robert Bosch GmbH guarantees the entrepreneurial freedom of the Bosch Group, making it possible for the company to plan over the long term and to undertake significant upfront investments in the safeguarding of its future. Ninety-two percent of the share capital of Robert Bosch GmbH is held by Robert Bosch Stiftung GmbH, a charitable foundation. The majority of voting rights are held by Robert Bosch Industrietreuhand KG, an industrial trust. The entrepreneurial ownership functions are carried out by the trust. The remaining shares are held by the Bosch family and by Robert Bosch GmbH.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Image Title: Bathroom Lowes Vanity Inch Double Sink Creative Throughout Design 12. Post Title: Lowes Double Sink Vanity. Filename: bathroom-lowes-vanity-inch-double-sink-creative-throughout-design-12.jpg. Image Dimension: 900 x 900 pixels. Images Format: jpg\/jpeg. Publisher\/Author: Arch Klein. Uploaded Date: Wednesday - October 24th. 2018 00:58:20 AM. Category: Architecture. Image Source: hgtv.com. Lowes Bathroom Cabinets And Sinks Ballnet Co Throughout Double Sink Vanity Inspirations 19. Shop Bellaterra Home White Rub Edge Double Sink Vanity With For Lowes Inspirations 3. Shop Avanity Modero Chilled Gray Double Sink Vanity With White Regarding Lowes Ideas 1. Shop Design Element London Pearl White Double Sink Vanity With Regard To Lowes Decorations 5. Shop ARIEL Cambridge White Double Sink Vanity With Natural Throughout Lowes Prepare 15. Shop ARIEL Stafford Espresso Double Sink Vanity With White Quartz Within Lowes Prepare 0. Shop Bathroom Vanities With Tops At Lowes Com Intended For Double Sink Vanity Prepare 8. Lowes Sink Vanities Double Vanity Home Design Ideas Throughout Pertaining To Remodel 14. Bathroom Vanities Vanity Tops Shop Collections With Lowes Double Sink Remodel 9. Bathroom Vanity Double Sink Lowes Modern In Idea 17.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \/ Accessories \/ Hats & Caps \/ Winter Hats \/ Blue crocheted beanie with royal blue and neon pink stripes and a coordinating pom pom.\nThe hat is a standard adult size, with the added stretch that crocheting offers. It measures 9.5\u2033 wide, by 9.5\u2033 tall, with a large and fluffy pom pom.\nAluminium hand stamped screen time token set. Reward your childrens behaviour.\nPersonalised stitched heart effect love token. When I follow my heart it leads me to you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I love learning about new books, movies, music, and especially field trips. This week's favorites have, for the most part, been recommended to me by other homeschool moms. Our homeschool community has some of the BEST moms who willingly share. Thank you!\nLast week the kids and I enjoyed an extra-special field trip to one of the last performances of Ringling Bros. Barnum & Bailey Circus with our Fuel Homeschool community.\nYears from now, I can imagine my kids describing to their children the enjoyment of \"going to the circus.\" A factoid they can share is they saw the first woman ringmaster in Ringling Bros. history. If the circus is coming to a town near you, I highly recommend making it a family event. You'll enjoy a great show (don't miss the pre-show) and make lasting memories.\nThis year I'm leading two bookclubs on Facebook. They are keeping me on my toes. Talk about accountability!\nThe second bookclub I'm leading this year is Faithful Friends Book Club (you're invited to join this one too!). This bookclub is part of Mentoring Moments for Christian Women's online community. We read one book a month using the formula of a non-fiction followed by a fiction. In March we're reading a non-fiction, Finding Truth by Nancy Pearcey.\nAs a family, we've just started reading aloud, AHA! 40 Days to Change by Kyle Idleman. We are finding our way with the book and the schedules of working teens. It will take us longer than 40 Days, but that's okay.\nMost mornings our homeschool background music is Pandora's 'Classical to Study By'. I use the free app on my phone and connect to wifi speakers. The song selections are varied from classic composers to movie and tv show soundtracks with some rock and roll performed by an orchestra. The funniest song I've heard, and it took me a few minutes to guess, was Billy Idol's White Wedding. Think classical Musiak. The free version of Pandora contains ads every so often. The ads have been for local businesses and since I like 'free', I'm okay with them.\nI'm hooked on MasterPiece Theatre's Victoria. As a Dr. Who fan-family, I keep expecting the Tardis to show up and Queen Victoria (played by Jenna Coleman) to become Clara Oswald once again. She was my favorite Dr. Who sidekick. Victoria is a beatiful period series.\nAnother series my husband and I are enjoying on Netflix is The Crown. My college son gave us the heads-up on this gem, which has Dr. Who #11 Matt Smith playing Prince Phillip. It's an insightful series into the life of Queen Elizabeth behind the cameras. I thought it was going to be a documentary, but it is not. It has a great cast, setting, and dialogue. John Lithgow plays Winston Churchill masterfully.\nWhat are you reading, listening to, and watching? Recommendations? I love to add some new books, podcasts and movies to my list.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u200b\u200bFor more than 14 years, the lawn division of Third Generation Services has been providing professional lawn care service in Florida. We pride ourselves on being reliable and providing premium property maintenance at affordable prices. Our belief is that our customer's needs are our number one priority. Year after year, we strive to be the best at what we do and our growth has reflected that. We treat your property like our own. We are big enough to service our client's needs yet small enough to cater to them.\nGive us a call for a free estimate. You'll be glad you did!\nIn Pasco: Hudson, New Port Richey, Port Richey, Wesley Chapel, Land-O-Lakes, and surrounding areas.\nIn Hernando: Spring Hill, Brooksville, and surrounding areas.\nIn Citrus: Crystal River, Homosassa, Lecanto, Inverness, Beverly Hills, Citrus Springs, and surrounding areas.\nIn Marion: Ocala, Silver Springs, and surrounding areas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Stylish Brown and Glass Top Coffee Table is a chic and stylish feature for any home. The glass table top is easy to clean and maintain and the upholstered under shelf is both practical and aesthetic. This product requires assembly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"What does potassium cyanide do?\nInhibits cell respiration as it's a non-competitive inhibitor for one of the vital respiratory enzymes found in mitochondria. This inhibits aerobic respiration and ATP production. The organism will only be able to respire anaerobically.\nWhat will the products of the anaerobic respiration be?\nIt produces lactic acid which is toxic in high concentrations. It generates much less ATP, not enough energy to survive.\nHow does the treatment of ethylene glycol poisoning work?\nIf taken into the body, it is broken down by the live to produce a very toxic product. To prevent poisoning, you need to inhibit the alcohol dehydrogenase enzyme. Alcohol is a competitive inhibitor of the enzyme.\nWhat does snake venom contain?\nPhosphodiesterase enzyme- interferes with heart & reduces blood pressure. Digestive enzymes- breaks down connective tissue- helps toxins to penetrate tissues quickly. ATPase enzymes- break down ATP molecules so prey has no energy available.\nHow do enzymes help with the treatment of cystic fibrosis?\nWith CF, the passage of digestive enzymes from the pancreas to gut is blocked, so digesting food is difficult. Doctors can provide enzyme tablets to overcome this.\nWhy would enzyme tablets need an acid-resistant coating?\nOtherwise they would be destroyed by HCl in the stomach, plus the protein-digesting enzymes in the stomach would break down the enzymes in the tablets.\nHow do competitive inhibitors treat HIV?\nThe HIV virus uses protease enzymes to replicate inside host cells to make new viruses. Protease inhibitors can be used to prevent these viruses from replicating in the body.\nWhat are inborn errors of metabolism?\nMutations in the DNA code affect the final shape of the enzyme so that it no longer functions. Many diseases are due to the lack of a functional enzyme in a metabolic pathway, and are called inborn errors of metabolism.\nPKU is where you can't convert phenylalanine (PA) to tyrosine because you lack the enzyme due to a mutation in the DNA coding.\nWhat happens to sufferers of PKU?\nPA accumulates which can cause irreversible damage to the brain and nervous system in developing children. They will also have a lack of tyrosine, which is needed to make melanin pigment, so they'll be very pale-skinned.\nNewborn babies screened by taking drop of blood. If diagnosed, you'll need a controlled diet throughout childhood. Diet must contain only small amounts of PA- some is needed as it's an essential amino acid. Once an adult, brain is fully developed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The founders of startup Malt Shop Cookies are hoping to jumpstart their new ecommerce business by launching on Cyber Monday the online sale of their line of malted milk cookies before expanding into more traditional brick and mortar stores next year.\nThe soft-batch cookies stand apart from the competition because they are made with malted milk -- a flavor that for many older Americans will recall an era of soda fountains and malted milkshakes, and which will also appeal to younger consumers as a new flavor with a heritage rooted in Americana, says company co-founder John Leicht, who has a long history launching new products for large companies, including Kraft, Sargento, Hormel, Bird's Eye and others.\nHe notes that while the cookies offered by Malt Shop Cookies\u200b\u200b are not a part of the better-for-you trend, it allows those who want to indulge to do so fully -- tasting with satisfaction the butter, eggs and sugar used to make the cookie, in addition to the malted milk. It also hits other hot market trends that will continue to drive sales well into the future, including all-natural and clean label, which are historically difficult for packaged baked goods to achieve, he told FoodNavigator-USA.\nThe inspiration for the cookies is rooted deep in the history of the rise and fall of American's love affair with malted milk shakes at soda fountain counters in drugstores in the 1960s.\nLeicht recalls in 1962 malted milkshakes and the malted milk business were \"on top of the world,\"\u200b with Horilicks Malted Milk, where his father worked, as the number one manufacturer of the ingredient. But over the course of a short seven years, \"the great migration from downtown to the suburbs\"\u200b and the rise of fast-food led by McDonald's spelled an end for malted milkshakes in the US.\nLeicht explains that when the drugstores followed the masses from downtown to the suburbs they didn't bring their soda fountains, which is where most people would buy malted milkshakes.\nLikewise, McDonald's dropped malt from its milkshake formula in order to speed and simplify production, making regular milkshakes the default and religating malt to memory lane for most Americans.\nHe and his wife whipped up a test batched, liked them well enough and then reached out to Leicht's now partner Edward Schuldt, who perfected the recipes that the two now use for their cookie company.\nThe cookies are available online in three flavors, including double chocolate with malted milk, mocha chocolate with malted milk and chocolate chip with malted milk, in a bag of 12 or tin of 18 or 36 cookies for $21.99 to $39.99 -- a price point that places them as a premium food gift.\nAs for packaging and marketing the cookies, a \"knee jerk\"\u200b concept would have been to build on memory and nostalgia of the bygone era of malted milkshakes with a 1950s motif and imagery of long-gone soda fountains, Leicht said.\nAnd while the website embraces these themes to an extent, Leicht said while doing consumer research, he found the cookies' flavors and the mention of malt alone were enough to conjure smiles and memories for baby boomers who used to drink malted milkshakes downtown.\nThis freed up the entrepreneurs to use the packaging and other marketing to propel malt out of the past and into the future -- predominately by playing up how malt is a natural preservative that absorbs moisture.\nThis allowed the duo to position the cookies as having a clean label and a longer shelf life than most retail baked cookies.\n\"Most fresh baked cookies at retail last three to five days or maybe up to 10 days, but we are getting eight weeks with the malted cookie, and even that could possibly be pushed to 12 weeks,\" \u200bLeicht said.\n\"So, what we started to think of was that this is not so much about where malted milk has been, but where it can go reflecting the values of the market today, which wants all natural, preservative-free options that also are fresh,\"\u200b he said.\nBut by focusing on the future, the company does not forget the past for malt, or those who love the ingredient. But rather than using generic Americana imagery, the company allows consumers to personalize online orders of the cookies with their own memories by placing photos and messages on each bag or tin of the cookie -- making them an ideal, personal gift, Leicht said.\nOnce the cookies launch in retail stores next year they likely will be sold in a clear clam shell with a full, premium wrapper in pastel colors that is elegant and pretty -- \"the complete opposite of what you would expect,\"\u200b but at the same time in keeping with the simple, clean designs embraced by trendy cupcake companies and modern bakeries.\nWhile the cookies will mainly be marketed towards the 75 million baby boomers who remember fondly the taste of malted milk, the duo says that younger consumers -- including the much talked-about millennials who may never have had a malted milkshake before -- also enjoy the cookies as the flavor of the malted milk meets their demand for new taste sensations and food with a strong heritage.\nThe cookies' broad marketing appeal and unique flavor profile, combined with the company's dual ecommerce and in-store marketing campaign and the founders' experience in food development and selling CPGs bodes well for sales of the cookie at the start of this holiday season -- making it a brand to watch going forward.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have never used a Public Insurance Adjuster before. What are the benefits of using a private insurance adjuster instead of the insurance company adjuster?\nClients often receive larger insurance settlements, over using the insurance company.\nClients report far less hassle through using a public adjuster. Unless you have worked in the insurance industry, it is very difficult to understand the legal documents, insurance loop-holes and other insurance tactics that stand in your way of a complete settlement.\nAdjuster to work for you, they do the work for you!\nClients report a smoother settlement experience with less stress, by using a public adjuster. Because the public adjuster works for you, they handle all the paperwork and correspondence for you.\nClients report that public adjusters spend more time estimating your damages. Public adjusters are dedicated to investigating your property damage insurance claim and documenting it, so that you can be paid for all of the damages to your home or business.\nIf a disaster occurs, our clients confirm, that they would call us again! This speaks volumes about our services, and about Palm Beach Public Adjuster. We work for you, not the insurance company!\nWhat kind of results can people expect by using a public adjuster? How would a typical insurance claim settlement be different by using a public adjuster?\nShould I call the insurance company or the insurance agent before I hire a public adjuster?\nNo, insurance agents do not get paid to handle claims. We advise our clients to call us first so that we can handle things appropriately for you. Palm Beach Insurance Adjusters will help you decide if the claims are covered and if it's worth reporting this claim to your insurance company.\nEven if you are only asking them a question, insurance companies keep records of your calls and will often count a call as a claim. This may raise your rates even though you never really meant to make an actual claim!\nI'm not happy with my insurance settlement. What can I do?\nYou have up to five years to reopen your property damage insurance claim and three years if it is a sinkhole claim. Call Advocate Claim Public Insurance Adjusters to discuss re-opening or supplementing your insurance claim for additional damages.\nNOTE: Your claim can be reopened, supplemented, or reopened for additional damages, if your damages are properly documented. Call Advocate Claims Public Adjusters to reopen an insurance claim.\nWhy would using a public adjuster be any different than calling my insurance company?\nBy using a public adjuster, you have someone who is working for YOU, 100%. We do not work for your insurance company, nor do we have to \"protect their bottom-line\". We are hired by you, to protect you and your investment, and we work for you. Insurance adjusters must protect their employer first (the insurance company). A public adjuster will also want to protect and serve their employer first: When you hire Advocate Claims Public Adjusters a Palm Beach Insurance Adjuster, our employer is you!\nYes, an insurance company can basically do as they please. They can raise your premium or drop you after a claim. Insurance companies can drop you simply because they don't want to insure properties in your neighborhood. The insurance company makes the rules and decides if you are a profitable customer to them. You bought insurance and paid your monthly premium for the time when you have an insurable claim! Call Advocate Claims Public Adjusters, and we can tell if your insurance claim is worth making.\nNo. Insurance companies may offer you a \"preferred contractor program\" to provide emergency service or repairs, but usually insurance companies will NOT warranty or guarantee their work. If you decide you want to use their contractor, ask the insurance company to put their warranty or guarantee in writing, not their contractor. Ask their contractor for references in the same way you would hire your own independent contractor. Remember, these contractors get their work assignments from the insurance company so, ask yourself this: Who do they really work for?\nMy contractor said he can negotiate my claim. Should I allow him to do this?\nNo. The only people authorized in the State of Florida to negotiate and settle insurance claims are: the policyholder, a public adjuster (such as Advocate claims Public Adjusters) or an attorney.\nMy roof is leaking. Is my roof repair covered? Should I report this to my insurance company?\nNo, your roof is not covered. Unless you had physical damage to your roof from wind, hail, trees, etc. Roof repairs are considered regular maintenance. The insurance policy only covers the interior damages that occur subsequently to your roof damage. Call us to see if such a claim is worth reporting.\nShould I cash my check, even if I don't like the claim settlement?\nYes, it's ok to cash your check on property damage insurance claims in Florida. Doing so does not prevent you from asking for additional money. However, doing the repairs before your claim is fully settled, may work against you.\nWritten language in your insurance policy and mortgage loan documents require that the insurance company notify your mortgage company that you had a loss. This is done by adding the mortgage companies name on the insurance settlement check .Your mortgage company will have to endorse your check in order for you to cash it. You will need to communicate with your mortgage company to find out their particular insurance claim check handling process.\nThere is no requirement in your insurance policy stating that you have to get three estimates. Advocate Claims Public Adjusters will put the estimate of the damages together for you and save you the trouble of getting three estimates. Insurance company companies require the damage repair estimate written with specific details, usually on a per-line itemized basis, and most contractor estimates do not comply with this format. If your three contractor estimates are not properly detailed, the insurance company will ignore them anyway. Furthermore, most contractors do not have the software, knowledge, or time to write estimates for free.\nThe insurance company adjuster came and took pictures, and told me it's fine to throw everything away, but the adjuster left things out of my estimate. How do I deal with this?\nDo NOT throw anything away! It is up to YOU to document your loss. Many \"storm adjusters\" are only here temporarily after a hurricane or similar event, leaving you to prove your damages after they have skipped town. You may have multiple adjusters sent to your house too! As your public adjuster, Advocate Claims Public Adjusters will fully document your loss for you.\nThe insurance company said the loss to my home and business wasn't covered. Can I still hire a public adjuster like Advocate Claims Public Adjusters?\nYes, but a public adjuster will only take on assignments if they feel it should be covered. That is why it's better to meet with a public adjuster at the start of your claim process, before you contact your insurance company.\nYou have an obligation to stop any leaks and prevent any further damage to your property. The insurance company can deny your claim for not stopping the leak in a timely manner. Proper investigation is critical to determine what is causing the damage. Call Advocate Claims Public Adjusters to help you determine what is necessary to do before you report your insurance claim.\nI want to report my claim and see what they would pay me before I hire a public adjuster. Shouldn't I do this?\nA tip to remember: If your claim gets denied, you may not get a second chance!\nCall Advocate Claims Public Adjuster from the start, to get it done right!\nI called my insurance agent to discuss my claim. Should I have done this?\nNo, you shouldn't have called them. Calling them right away is exactly what the insurance company wants, and this is why they pay so much for commercial advertising. Remember, the \"captive\" insurance agent works for their insurance company, and the \"independent\" agents work for many insurance companies. All of these agents have a contractual obligation and get paid by the insurance company. Only public adjusters like Advocate Claims Public Adjuster and attorneys are licensed to advocate insurance claims on your behalf.\nShould I hire a public adjuster like Advocate Claims Public Adjusters or an attorney to handle my claim?\nThe insurance company does not send out attorneys to inspect your property, they send an adjuster. Level the playing field by hiring your own adjuster who speaks the same language. Attorneys go to court to defend the language in your insurance policy. While Public adjusters, adjust insurance claims.\nPublic Adjusters try to keep the claim moving, and will arrive at a much quicker settlement costing you for far less money, than attorneys. Attorneys become necessary only after the insurance company fails to honor your rights under the insurance contract. Remember, once you hire an attorney, the insurance company will bring in their attorneys too. If the courts are involved, you may be looking at years to get your claim settled!\nWhat types of insurance claims do you handle?\nPlease read over the list to see the most common types of insurance claims. If yours isn't listed here, you may still be covered! Call Advocate Claims Public Adjusters 561-962-0065 to discuss your insurance claim.\nCovering Palm Beach County, including Boca Raton, Delray Beach, West Palm Beach, Wellington, Royal Palm Beach and surrounding areas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are staffed with compassionate individuals who are highly dependable in providing professional care.\nWe assure you that all of our care professionals are a product of our extensive screening process that includes thorough background and credential checks and in-depth personal interviews. This to ensure that the brand of our care is not compromised but of high quality.\nIf you want to know more about our hospice team, please do not hesitate to reach us through this number: 317-316-3937.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Preparing for an upcoming interview for an accounting position? Be prepared to prove yourself, because if you're hired, you'll have some pretty weighty responsibilities. The accounting team is responsible for keeping accurate financial records and ensuring tax bills are paid on time, so companies don't want to hire just anyone for the job.\nYou'll need to spend time brushing up on accounting principles, software and other pertinent skills to make it clear you're someone that can be trusted to excel in such an important position.\nAccounting Software Fluency: If possible, find out what type of accounting software the company uses and familiarize yourself with it. If you're unable to find out which program the team is using, acquaint yourself with popular options like QuickBooks or FreshBooks. Knowing your way around common platforms shows that you're tech savvy and able to catch onto new software quickly.\nGAAP Knowledge: Companies want to hire an accounting professional who has the knowledge needed to keep their books in good standing. Refresh your memory of generally accepted accounting principles (GAAP), including basic accounting principles and guidelines, detailed rules and standards issues by the Financial Accounting Standards Board (FASB) and the generally accepted industry practices.\nProblem Solving Abilities: The accounting field can be unpredictable, so you need to show the interviewers that you can handle any problem thrown your way. Think of examples of past professional situations you've been in where you had to quickly analyze data and use the information to work through an issue. Offering solid evidence that you're great in a crisis will certainly impress the hiring manager.\nCommunication Skills: Accountants work with many different departments \u2014 and sometimes even clients \u2014 each and every day. Practice presenting financial information in an easy-to-understand manner that people can relate to. Impress the interviewer with your ability to understand complex financial situations and explain them to others without sounding condescending.\nSearching for a challenging new accounting position?\nContact Accountingpros Recruiting + Staffing. We offer access to more professional accounting job opportunities than any other Northwest employment service \u2014 guaranteed!\nIf you've been doing a lot of hiring lately \u2014 but not because your company is growing at rapid pace \u2014 it's time to make a change. High levels of turnover can ruin your company. If you want to be successful in the long-term, you need a team of loyal employees who are committed to your organization. Make it a priority to get to the bottom of your retention issues now, so it doesn't continue to hold your company back.\nA recent Gallup survey revealed that approximately 50% of workers have quit their jobs to get away from a bad boss. Some people are very good employees, but extremely bad managers. If you have people in management positions who really shouldn't be there, there's a very good chance this is why your turnover levels are high. Choose managerial level employees very carefully and if you notice a problem, take action immediately.\nYour best employees excel at their positions, but that doesn't mean they intend to stay in the same roles forever. It's beyond discouraging for hardworking, skilled employees to be passed up for a promotion by an external candidate, without even being considered for the job. Make your company a place where top talent can learn and grow, so they don't feel forced to accept a position elsewhere to move up the ladder.\nYou hired your employees because they're smart people with great ideas, but do you let them share their thoughts and opinions? No one enjoys feeling like their ideas don't count. Stop micromanaging your team and trust them enough to complete assignments their own way. This allows people to feel like they're actively contributing to the team \u2014 and their ideas may even save your company time and money. Employees are much less likely to leave a job when they're happy and fulfilled.\nNeed a little help lowering your turnover rate?\nPartner with Accountingpros Recruiting + Staffing. Our team will help you find top Seattle accounting talent who wants to grow their career at your company.\nAnyone who has ever worked in an office knows the importance of company culture. A job may seem great on paper, but if you don't fit in with the rest of the team, you're going to be miserable. When you're interviewing for a job, company culture isn't always boldly on display. You may need to read between the lines and dig a little deeper to learn what it's really like to work for the organization.\nConduct background research online, by reviewing their website, social media pages and reviews on sites like Glassdoor.com to gain valuable insights. It's also wise to arrive a little early for your interview, to get a feel for the work environment before getting ushered away into a conference room. You may not be able to observe too much, but sometimes the little things are the most telling.\nCertain elements of a work environment can make or break your productivity. Think about the type of situation that stimulates you the most. Maybe you prefer to work alone in a quiet environment? Or perhaps you thrive as part of a team working under pressure? There's no wrong answer. The only way you can shortchange yourself is not being honest about your motivators.\nThe same things that motivate you may not actually make you happy. Think about a job you really loved and consider why it made you feel fulfilled. Maybe it had a late start time, which suited you well because you're not a morning person? Determine what it takes to make you happy at work, then don't accept a position until you find a company that can meet your needs.\nIf you have to rework your entire life for a job, you're probably not going to last very long. For example, many companies offer catered meals to employees working long hours, but if you'd rather be eating with your family, you'll resent having to spend this precious time at the office instead. Or if you enjoy working in a sociable office, don't accept a position where everyone keeps to themselves and leaves promptly at 5pm.\nReady to take your accounting career to the next level?\nContact the experienced Seattle-based recruiters at Accountingpros Recruiting + Staffing. We specialize in matching top accounting talent with leading companies in the Seattle area.\nBehind every successful company is a team of highly engaged workers. However, all it takes to throw this balance off is one overworked, stressed out person. Employee burnout can have some pretty serious side effects, such as decreased productivity, lower levels of job satisfaction, conflict among staff members, absenteeism, high levels of turnover and decreased loyalty to the organization.\nIf you've noticed that some of your employees seem to be on the verge of burnout, it's important to take immediate action. This issue can result in a rapid downward spiral, so get it under control now before it takes your team down.\nThere's a big difference between giving employees challenging assignments and burying them with an unrealistic workload. As a manager, you're well aware of how much each person can handle. If you know an employee is working hard all day, but still staying late and coming in early each day, it's time to cut back on the amount of pressure you're putting on them.\nA person may be great at their current job, but that doesn't necessarily mean they have the skills and experience to excel in a higher level position. Think long and hard before promoting an employee, to make sure they can handle the added responsibility. You don't want to lose a good worker because you mistakenly promoted them too fast.\nDoes most of your team work through lunch? If so, institute a policy requiring them to take this time for themselves. People need breaks throughout the day to relax and refresh their minds. Taking a walk outside or browsing Facebook for a few minutes helps to lower stress levels, allowing people to return to work feeling rejuvenated and ready to tackle the next item on their to-do list.\nEmployees are much less likely to burn out when you clearly communicate your expectations, goals and project deadlines. Assignments aren't as intimidating when you're on the same page with your team, because they know what's coming down the pipeline. It's also important to make your staff feel comfortable coming to you with questions, comments and concerns. People need to feel like they can come to you if they're feeling overwhelmed and need a little help.\nSearching for Top Accounting and Finance Talent in the Seattle Area?\nPartner with Accountingpros Recruiting + Staffing. Our team of accounting-focused recruiters is committed to finding the best person to fill every open position at your company.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The charred remains of an unidentified person was discovered in the aftermath of a fire at a Wales, West Bank Demerara house yesterday morning.\nThe house, which was considered a vacation home, was believed to be unoccupied since the owner resides at Timehri, East Bank Demerara.\nThe origin of the fire was still unknown up to late yesterday but the homeowner, Elizabeth Sattaur, said reports she received from a neighbour suggested that it may have been electrical in nature.\nAn eyewitness told Stabroek News that around 1.30 am yesterday, a passerby, who was drinking at a nearby shop, saw the fire and sounded an alarm. Shameer Mohammed explained that subsequent to the alarm, both residents and the Wales Estate Fire Service responded.\nThe remains of the house after the fire.\nMohammed said that despite the timely response by the fire service, the house was quickly engulfed by the flames.\nMohammed added that when Guyana Fire Service firefighters arrived at the scene, they assisted in outing the fire. According to him, the estate fire service faced a difficulty in tackling the fire immediately, since the gate to the compound was locked.\nHe noted that a police rank who was on the scene scaled the fence and broke the locks to gain entry into the yard of the residence.\nAccording to Sattaur, who is the proprietor of Roti Palace, she learnt of the fire after getting a call from a friend at around 2 am.\nShe added when she arrived at the Wales home after 3 am, the fire was almost out. At around 4.30 am, she said, the fire fighters discovered the body among the rubble. \"After them out the fire, they began investigations and that's when they told me they found a body,\" Sattaur recalled. \"When I came here I came for a fire but I never thought they would find a body,\" she added.\nThe woman said she was shocked to learn someone was occupying the house since she is the only one who had keys to the building. She added that the body was found in a crouching position in the bathroom area of the self-contained room in the bottom flat of the home.\nShe speculated that the person may have been a vagrant.\nAsked if she had any idea how the fire started, Sattaur said neighbours told her there was a power outage.\nThey said when the current returned, the utility post with the power line to her home was sparking, leading her to suspect that the fire may be electrical in origin.\nStabroek News, however, understands that the fire service is working on a number of theories, including possible arson.\nSattaur pointed out that she would normally occupy the home on weekends. She had visited the house two weeks ago and was planning to return this weekend since there is a holiday.\nShe added that the house was equipped with all modern amenities and conveniences.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Astor Room is a new addition to the Kaufman Studios sector of Astoria, bordering Long Island City. The subterranean dining room is throw-back chic, as is the menu. The speakeasy vibe is reminiscent of the roaring twenties, or if you're a gamer -- the underwater city, Rapture, from Bioshock.\nOysters Rockefeller, baked Alaska, and chicken pot pie are among a few of the revamped classics that make an appearance on the Art Deco menu. From what I've sampled, the execution is spot on, and the extensive raw bar is a refreshing treat given the restaurant's location. The staff is friendly and efficient, but they struggled to maintain the level of service as multiple large parties arrived.\nOverall, the food was inventive and well prepared; however, the menu is a bit limited. There are a few dishes that I would come back to try, but beyond that I'd like to see a more seasonally relevant menu.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Honda Fit has always been one of the best-loved subcompact hatchbacks around, and for 2015 a complete redesign brings loyal fans more of the same practicality and affordability that have been its calling cards since it was first introduced. Major changes have been made to the Honda Fit under the hood, in the cargo department, and for passengers sitting in the second row, but longtime Fit buyers will be happy to know that the hatchback's Magic Seat calling card still comes standard with every version of the vehicle.\nPrevious versions of the Honda Fit asked you to look past its styling in order to find a more practical reason to purchase the admittedly milquetoast automobile. That's all changed with the 2015 Honda Fit, which gains a new design in an effort to stand stronger against more fashionable rivals like the Ford Fiesta and the Hyundai Accent. It's a commendable, and largely successful effort, for while the Fit is actually shorter in length and features the same height as it did the year before, a redesigned greenhouse and more aggressive front and rear bumpers enhance its road presence and help it stand out better from the crowd.\nDespite measuring 1.6 inches shorter than it did in 2014, the 2015 Honda Fit actually improves on the amount of passenger room available to those sitting in the second row \u2013 and by a considerable amount. In fact, there's 4.8 inches of additional legroom available for rear seat occupants in the new Honda Fit, which contributes to a total amount of space to stretch out that dwarfs even that of the mid-size Honda Accord sedan. Honda managed to achieve this stunning feat by redesigning the car's rear suspension and increasing its wheelbase by 1.2 inches, an engineering decision that opened up copious amounts of interior real estate.\nHand-in-hand with the 2015 Honda Fit's generous passenger room comes its Magic Seat feature. Although pushing the Honda Fit's rear row towards the back cut four cubes off of its previously-enormous 'trunk,' the 17 cubic feet that remain are still quite useful \u2013 especially when taking the Magic Seat into account, which can flip, fold, and otherwise adjust the hatchback's accommodations in almost any direction. The result is a subcompact hatchback with almost 53 cubic feet of total cargo capacity. This is a touch less than the year before, but more than double what most of its similarly-sized rivals have to offer, and a number that tops even some next-class-up compact hatches from a practicality standpoint.\nOn paper, the 2015 Honda Fit's 1.5-liter, four-cylinder engine might not seem all that different from its predecessor. Aside from a listed boost in output from 117 to 130 horsepower (along with a gain of 8 lb-ft of torque for a total of 114 lb-ft in total), you'd have to open up the unit to notice its direct fuel injection and new dual-overhead camshaft that allows for Honda to introduce i-VTEC variable-valve timing. Despite all of these enhancements it would be a stretch to call the motor 'peppy,' but it does acquit itself well alongside other subcompact cars in its class in terms of acceleration and passing power.\nStandard with base model 2015 Honda Fit hatchbacks is a new six-speed manual transmission, a gearbox that offers one more forward cog than it did the year before. Despite the overall improvement in power management that the six-speed design offers, the Honda Fit's final drive and top gear ratio are identical to those of the 2014 model, which means highway engine speeds are still on the buzzy side. Replacing the now-departed five-speed automatic on the Fit's options list is a continuously-variable automatic that offers seven 'virtual' gears for those who elect to engage the car via its available steering wheel-mounted shift paddles.\nWhile the 2015 Honda Fit might not qualify as a powerhouse, its engine improvements, in combination with the car's two new transmission options, have done wonders to enhance its previously lackluster fuel efficiency. When equipped with its six-speed manual gearbox the Honda Fit returns 29-mpg in stop and go driving and 37-mpg on the highway, but the real star of the show is the Fit's continuously-variable automatic. This is the transmission that enables the hatchback to post figures of 33-mpg around town and 41-mpg during highway cruising, with the latter number allowing Honda to finally achieve parity with the majority of its subcompact competitors.\nThe 2015 Honda Fit starts out in LX trim, which provides the basics in the form of power windows and door locks, Bluetooth connectivity, cloth seats, a CD player, 15-inch wheels, cruise control, and a rearview camera. The Fit EX introduces niceties such as a sunroof, additional safety equipment, keyless entry and start, 16-inch rims, the HondaLink system (which can stream navigation information from your mobile phone to the car's seven-inch LCD touchscreen), and an improved audio system, while the EX-L adds leather upholstery plus heaters for both the front buckets and the side mirrors (and makes the CVT standard). Honda's EX-L with Navigation is self-explanatory, but in addition to a navigation system you also get HD radio and satellite radio with this particular model.\nThat extra safety gear that we mentioned as coming standard with the 2015 Honda Fit EX is the automaker's new LaneWatch system. Designed to eliminate the Honda Fit's blind spot, LaneWatch activates as soon as you use the car's turn signal and displays a widescreen video image of the area to the passenger side of the hatchback so that you can ensure that it's clear of traffic. LaneWatch comes alongside dual forward airbags, side-mounted airbags up front, side curtain airbags that run the entire length of the Fit's passenger compartment, and electronic traction control and stability control.\nWhereas past editions of the Honda Fit might have been more engaging to drive than the car's looks initially suggested, the 2015 model heads in the opposite direction. Changes to the Honda Fit's chassis have made it easier to track it in a straight line while traveling on the highway, which erases a deficit of last year's car, but steering now requires more turns lock-to-lock, and the electric assist imparts a disconnected feel. Never a true driver's car, the 2015 Fit fades further into the background when compared against more fun options like the Ford Fiesta and the Mazda Mazda2.\nEven though the 2015 Honda Fit offers more standard equipment, enhanced safety, a more powerful engine, better looks, and a less thirsty attitude at the fuel pump, its pricing hasn't ballooned to match. The Honda Fit LX starts at an MSRP of $16,315, which is a mere $100 increase compared to the 2014 model, and even the EX-L with Navigation tops out at a budget-friendly MSRP of $21,590. Honda has a long way to go to catch up with segment sales leaders like the Chevrolet Sonic and the Ford Fiesta, but its aggressive pricing and value equation should help it close the gap throughout 2015.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This just uses svcKernelSetState with Type10 where Param0=<input u8 value from the cmd>.\nThis command is implemented in the Old3DS PTM sysmodule too, but the above type is only implemented in the New3DS ARM11-kernel.\nThis is used by NS.\nThis page was last edited on 27 September 2015, at 23:50.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Volkswagen has been ordered to stand trial in Arizona for selling diesel vehicles that produced more emissions than legally allowed.\nMaricopa County Superior Court Judge Randall Warner said violations involving federal emission standards can only be handled in federal court, but the case in Arizona is over deceiving residents to buy vehicles that do not perform as advertised. That claim, he ruled, falls under Arizona's Consumer Fraud Act.\nThe company's lawyers said the auto manufacturer did not deliberately target Arizona consumers.\nThe case involves sales of vehicles with special engines that were promoted as having far fewer emissions than similar cars. These vehicles cost up to $7,000 more than comparable cars. The affected vehicles were sold from 2008 to 2015, and were designed to show low emissions when being tested specifically for emissions. They would then return to their standard high-power modes during regular driving conditions, emitting more than the allowable amount of pollutants.\nWant to get notified whenever we write about Pinal County ?\nSign-up Next time we write about Pinal County, we'll email you a link to the story. You may edit your settings or unsubscribe at any time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"zum testen f\u00fcr die neue Version gibt es einen Installer mit dem man das Script sehr einfach installieren kann. Installer downloaden, entpacken, ausf\u00fchren, Daten eingeben fertig! Alles andere hochladen, Dateirechte vergeben usw. \u00fcbernimmt der Installer.\nDer Installer wurde von seth in Delphi geschrieben und l\u00e4uft unter Windows.\nBitte einfach mal testen, am besten auf einem leeren Webspace und leerer Datenbank z.b auf DIC-Gaming.at oder Funpic.de.\nIt is available in several colors and textures. Overall if you are looking for replica watches instrument this might be it. I guess the price is decent. A sapphire crystal covers both the front and rolex replica sale of the case and water resistance is rated at 200m thanks to the screw down caseback with the custom engraved Hauttman logo. I never had any issues reading the time, as for the complicated subdials, that is part of the attraction of hublot replica pieces, they are more like scientific instruments in their layout. The one redeeming feature, is that salmon pink is a nice dial replica watches sale for a rose gold case. Kinda nice, but the problem is you would need a magnifying glass to read the complications at least anybody with replica watches sale aging eyes.\nVielleicht mal eine News vorher gucken? Der Installer ist nur ein zus\u00e4tzlicher Service f\u00fcr die Leute, f\u00fcr die es so leichter ist zu installieren. Das Script das installiert wird ist aber exakt das gleiche.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hi! My name is Mickey. I am never married other white man without kids from Lima, Ohio, United States. Now I'm looking for new relationships. I want to meet a woman, love of my life.\nHi! My name is Jeremy. I am never married other white man without kids from Lima, Ohio, United States. Now I'm looking for new relationships. I want to meet a woman, love of my life.\nHi! My name is Spout. I am divorced other white man without kids from Lima, Ohio, United States. Now I'm looking for new relationships. I want to meet a woman, love of my life.\nHi! My name is Dfresh. I am never married christian white man without kids from Lima, Ohio, United States. Now I'm looking for new relationships. I want to meet a woman, love of my life.\nHi! My name is Levilights. I am never married other white man without kids from Lima, Ohio, United States. Now I'm looking for new relationships. I want to meet a woman, love of my life.\nHi! My name is Nick. I am never married other white man without kids from Lima, Ohio, United States. Now I'm looking for new relationships. I want to meet a woman, love of my life.\nHi! My name is Rbaron. I am never married christian white man without kids from Lima, Ohio, United States. Now I'm looking for new relationships. I want to meet a woman, love of my life.\nHi! My name is Teddybear. I am separated other white man without kids from Lima, Ohio, United States. Now I'm looking for new relationships. I want to meet a woman, love of my life.\nLooking to make new friends!!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"University of North Carolina at Chapel Hill is a large, public, four-year university enrolling 29,084 students and offering 224 degrees in Chapel Hill, NC.\nCheck out the most popular majors and specific degrees students have earned at University of North Carolina at Chapel Hill.\nCheck out the online programs offered at University of North Carolina at Chapel Hill.\nNot currently in school, do not know what is offered as far as networking, etc.\nStudent counselors when I was there; now, I'm sure they have so much more with the Internet more readily available.\nAmazing alumni network. Leaders in almost every field.\nWell, this was a while ago. There was a campus center, but I didn't really feel it was very useful.\nThe program at my school allows you to perform your clinical rotations at one of many locations around the state. There are many locations and opportunities to get the experience you need and get plugged in to the working world.\nnone that I knew about or used - we all had jobs offered by our last year of program.\ndedicated career counseling staff, job fairs, workshops.\nMy field was in specifically in media production. It has networking events for meeting people in media production and sets up several job fairs for careers in general throughout the year. The curriculum gives you the basics of media production and lets you work with the basic equipment (lights, cameras, microphones) needed to make a simple production.\ncasual, environmental, politically active, athletic, consistently current and at the forefront of healthcare, industry, research, technology, etc.\nRelaxed, but dedicated to school; supporting each other.\nSuper smart, super engaged, of a high caliber.\nWide range of student ethnicities and cultures.\nProgessive minded and diverse student body. A variety of interests and backgrounds, a large greek life strong athletics program and plenty of other clubs and pursuits.\nThe students at Carolina are very diverse, so trying to give a brief summary of common traits they have is a bit difficult, but I will try to describe them. However, on the one hand, many of them are hard working, intellectual, and have a can-do attitude. On the other, many of them are very stressed out, and to be frank, kind of arrogant.\nhave not been there recently but they were OK when I was there. Open spaces are beautiful.\nI lived in a private dorm with it's own computer lab & dining hall which was an excellent environment for me, so I don't have as much experience with the on-campus facilities other than classrooms and sporting arenas which were excellent at the time I was there.\nAgain, it was 1990~So we just got a computer lab!\nGreat facilities for regular students, student athletes, and for grad students. The libraries are awesome and the campus is expansive and beautiful.\nIt varies. The computer labs and dining hall are pretty good. The food there is pretty delicious and they have a large variety of food options. Some of the classrooms are state of the art and high-tech while others are a bit run down. Some of the dorms are pretty nice to stay in while others are kind of sketchy.\nInterim sports for the average, non-scholarship students not on the televised teams.\nActing, impromptu arts, bands, just the overall interest everyone had in pursuing interesting projects.\nStudent involvement at sporting events is always a highlight. Greek life and other cultural clubs can be found to meet almost any taste.\n30 years between - have no idea now.\nUnique groups are surprisingly common at Carolina. If you like something, there is a good chance you will find a group that does that. For example, there are is an ukulele ensemble, a rifle club, a kendo club, satirical magazines, political groups of just about any persuasion, and numerous religious groups.\nU What truly sets it apart is its sense of community, which stays with you for life. Even though I graduated three years ago, I still watch every basketball game, check the Daily Tar Heel every once in a while, and say a friendly \"Go Heels!\" to anyone I see wearing a UNC t-shirt. Once a Tar Heel, always a Tar Heel.\nU UNC is often described as the 'Harvard of the South,' which is true in many ways, but to me, the variety of education available at UNC is what sets it apart. In addition to being an RTVMP (radio, television, and motion picture) major, I sincerely enjoyed the liberal arts education I received; I feel it made me a well-rounded person prepared for 'the real world.' I gained exposure to a variety of cultures and ideologies, and this has shaped my world view. Many non-alumni probably know UNC mostly due to the success of its athletic teams, but Carolina is much more than that. UNC provides an incredible learning environment and serves as on-the-job training for life.\nDo you have questions about furthering your education with University of North Carolina at Chapel Hill? People are listening and are ready to help. Plus, we're pretty sure others have the same question you do!\nWhat are UNC's popular majors?\nPopular majors include communication and media studies, psychology, biology, business administration, management and operations, and economics.\nHow much does UNC cost?\nAre there financial aid and scholarship opportunities?\nFinancial aid\/scholarships: 43.7 percent of full-time undergraduates receive some kind of need-based financial aid, the average award being $14,866. The university's Carolina Covenant program helps eligible low-income students pay for college and graduate debt-free.\nDoes UNC offer military perks?\nMilitary service-related funding opportunities exist through the UNC Air Force ROTC and UNC Army ROTC.\nHave any famous people gone to UNC?\nNotable alumni include basketball player Rick Fox, U.S. president James K. Polk, U.S. senator John Edwards, actress Louise Fletcher, actor Andy Griffith, radio personality Rick Dees, comedian Lewis Black, sportscasters Marty Brennaman and Stuart Scott, and broadcaster Roger Mudd.\nWhat's it like to go to UNC?\nWhen you step onto UNC-Chapel Hill's immaculately landscaped campus, you can sense it's a place rich in history. Majestic oak trees and centuries-old buildings remind you that this institution opened its doors in 1795 as the nation's first public university. But don't think for a minute that the educational offerings are out-of-date. Admission is highly selective (an acceptance rate of 26.7 percent), and Kiplinger's Personal Finance magazine consistently ranks the school among the best public colleges in the United States for high-quality academics at an affordable price. Furthermore, UNC-Chapel Hill (along with North Carolina State University and Duke University) is part of \"The Triangle\"\u2014one of the major centers of education, research, and technology in the South. The majority of your classmates will be state residents, as state law mandates that at least 82 percent of each freshmen class be from North Carolina, Everyone, however, should feel right at home. Chapel Hill is known as an awesome college town, with its restaurants and shops focused on the college audience. Students and locals alike enjoy coming together to support Tar Heel men's basketball\u2014a perennial NCAA powerhouse where greats such as Michael Jordan and Vince Carter got their starts.\nHow will UNC prepare me for the real world?\nUniversity Career Services is devoted to helping students and alumni find rewarding careers. The process can start as early as one's first or second year by taking a pass\/fail course called Career Exploration offered by the School of Education and taught by career counselors. Upperclassmen may seek assistance finding internships and study abroad opportunities that may help with future employment. Students and alumni also have access to Careerolina\u2014an online tool featuring job postings, details about on-campus recruiting, dates and locations of career fairs, and contact information for various employers. Staff members skilled in job-search strategies are available to offer guidance, conduct mock interviews, and to help craft resumes, cover letters, and even social media profiles. University Career Services maintains a blog that profiles different career options and offers advice about common concerns job seekers share.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's a hot one, and the pool at The Chase Park Plaza Royal Sonesta St. Louis is calling! Two special packages this summer make a visit to St. Louis something to add to the top of your summer travel list. Ready to book? Visit the hotel's website now!\nThis summer we're celebrating The Muny's 100th season in Forest Park by offering a unique \"Meet Us At The Muny\" package. The Chase, just steps away from Forest Park and Muny Theatre, provides the perfect destination for dinner and a show, plus a luxurious night away.\nThis package includes 2 tickets to a Muny show of your choice, $50 credit to The Preston restaurant, and luxurious guest room accommodations. Reservations for this package must be made three days prior to the Muny show you wish to see. Your two tickets will be provided upon check in.\n*Terms and Conditions: Must book with promo code \"MUNPKG\". Must book package at least three days prior to the desired show date. Book now through August 8th for stays through August 12th. One day prior to check-in, please contact our marketing department at lmeyer@sonesta.com to confirm your tickets are available. Rates are subject to availability, certain blackout dates or restrictions may apply. Valid for new reservations only, excludes groups. Please note tickets are not confirmed until contacting the marketing department.\nYes way ros\u00e9. Start your visit to St. Louis by toasting with a glass of the infamous water of the summer season followed by a relaxing retreat at the salon + spa with the Ros\u00e9 All Day Package.\nThis package includes two bottles of chilled ros\u00e9, luxurious guest room accommodations and $100 credit to the salon + spa with your choice of treatments.\n*Terms and Conditions: Must book with promo code \"ROSE\". Rates are subject to availability, certain blackout dates or restrictions may apply. Valid for new reservations only, excludes groups. Spa credit is based on double occupancy \u2013 each guest receives a $50 credit to the salon and spa.\nIn addition to pool time, drinks, and great food, The Chase Park Plaza offers a movie theater, spa, fitness center, and shuttle services to round out your St. Louis getaway.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Holly Kotusky Leave Condolence - Bayville, New Jersey | Mastapeter Funeral Homes Inc.\nPlease check here to receive a one-time email, one week prior to the anniversary of the passing of Holly Kotusky. The email will only serve as a reminder should you choose to connect with the family at what will no doubt be a difficult time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Un-Scripted Theater Company has graciously offered to give readers of Haunted Bay a $2 discount on tickets to Fear. Simply use coupon code \"HAUNT\" when you purchase your tickets online.\nYOUR NAME IN THE PLAY!\nWant a moment of limelight? Freak out a friend? Or want one of your enemies to be off'ed? For just $10 you can buy a character name! Names can be purchased at the show or in advance by emailing Alan.\nWe set off from San Francisco under the setting of a blood red sun on our way to La Val's Subterranean in Berkeley to see the Un-Scripted Theater Company's preview performance of Fear. Some would say the unusual sunset was a result of all the recent fires but we like to think it was a portent of the tale of terror we were about to witness.\nUpon arriving at La Val's, located on Euclid Avenue, a block north of the UC campus' North Gate, we grabbed some Sierra's at the counter and made our way down the narrow stairs to the subterranean portion of the establishment. La Val's is better known to people of my generation as a venue for punk rock bands but as we found out, performs well as an intimate theater space.\nAs the stage lights came on we were greeted by Director Alan Goy, who introduced us to the concept of improvisational theater. He then took audience suggestions for the setting, location, and time of year that the actors would use to spin their tale of terror. For this performance, the audience suggested that the story take place in the Fall, on a remote Midwestern farm, with a ramshackle Queen Anne style farmhouse, and a large corn field.\nAfter a somewhat shaky start, six actors from the troupe, playing multiple roles, fell into their invented characters and created an intriguing two hour thriller complete with intermingled story lines, misdirected suspicions, and several plot twists.\nThis performance revolved around the tragic history of the farmhouse and a pair of murderous brothers, played perfectly by Christian Utzman and Brian McBride, who act as both real estate agents and supernatural caretakers of the farm. Together they lure couples away from the big city who hope to fulfill their dreams of leaving the rat race and settling down in the country to enjoy the simple life. The simple life is not all that it is cracked up to be as the two homeowner couples, played by Jennifer Kah and Cort Worthington and Alyssa Harvey and Ken Robertson, soon learn.\nIn the first act we saw the characters and story being developed, building suspense as the audience tried to figure out who the antagonist(s) would be. Was it the slightly creepy and Opie-esque little Billie, a role also played convincingly by Worthington? Could it be Billie's unsettling mother, played passionately by Kah in another one of her roles? Were the homeowners somehow involved? Suspicions were passed from character to character until nobody could be trusted.\nThe second act saw the story and action pick up pace as the story slid deeper into the dark history of the farmhouse. One by one characters were picked off. Drowned, stabbed, and beaten they dropped, culminating in a ritualistic disemboweling. While this sounds like a very gory play, the actors took great strides to keep it balanced and in the end created a shocking yet intriguing story.\nAll of the actors were wonderful and gave memorable and believable performances. The original music by Dave Dyson perfectly accompanied the unfolding story. Together they created a very entertaining night of theater.\nAs the name of the company implies, this is completely un-scripted improvisational theater. While we have relayed the story of the preview we attended, no two shows will be the same. Given the quality of performance at this preview show, we have no hesitation in recommending you visit a performance of Fear by the Un-Scripted Theater Company. At $12 ($7 for students and seniors) it is a very good bargain for your money. You won't be disappointed.\nFor Fear show times and a map to La Val's please see our calendar listing and to read more about Fear, the Un-Scripted Theater Company, and the actors who make up the company, please visit the Un-Scripted Theater Company website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As it is the UK General Election Manifesto launch week, The Multicultural Politic (TMP) also is launching its General Elections coverage. TMP will be scrutinising the party pledges, policies and the personalities, up and down the country whilst also continuing to\u00a0cover international politics from a multicultural perspective.\nWe are independent minded and believe our readers are also, so we won't endorse any political party but will give our honest opinion on the debates and options available to the electorate with an unapologetic multicultural bias.\nSo look forward to our spotlights on the leaders, shining a critical light of those who believe they are fit to serve the United Kingdom as Prime Minister. We will also be giving a platform for Prospective Parliamentary Candidates (PPCs) from all the main political parties and a few others to give you their view about why they should be getting your vote.\nWe will also take an meticulous look at the policies of the main parties particularly on immigration and community cohesion, economics, political reform and the environment. With acerbic and irreverent commentary on the election, we pledge to be refreshing in our political analysis with good humour and piercing insight.\nSo the starting gun has been fired, and the historic televised leader debates start tomorrow. This is also a forum for debate, you may disagree with our articles or points and we welcome debate. Get involved\u00a0by writing a comment and let the battle of ideas begin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hire an Advanced Robo-Advisor to Trade VTR on Your Behalf.\nSmart Auto-Trading is an advanced automated Portfolio Manager designed to provides investors with a a state of the art automated investment system to invest in the S&P 500\u00ae. Its technology is designed to screen, monitor and invest in elite stocks like \"VENTAS, INC. (VTR)\". It's Artificial Intelligence augments your potential for profit while eliminating most of the work traditionally involved in stock investing.\nVentas, Inc. (NYSE: VTR) will issue its second quarter 2018 earnings release prior to the opening of trading on the New York Stock Exchange on Friday, July 27, 2018. A conference call to discuss those earnings will be held the same day at 10:00 a.m. Eastern Time (9:00 a.m. Central Time).\nThe dial-in number for the conference call is (844) 776-7841 (or +1 (661) 378-9542 for international callers), and the participant passcode is \"Ventas.\" The call will also be webcast live by NASDAQ OMX and can be accessed at the Company's website at www.ventasreit.com. A replay of the call will be available at the Company's website, or by calling (855) 859-2056 (or +1 (404) 537-3406 for international callers), passcode 7465387, beginning on July 27, 2018, at approximately 2:00 p.m. Eastern Time and will remain available for 36 days.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to the premier resource for all real estate information and services in Austin, Texas and surroundings. We hope you enjoy your visit and explore everything our state-of-the-art website has to offer for Austin, Round Rock, Pflugerville, Georgetown, Hutto, Cedar Park, Leander and others. Browse for homes for sale and homes for lease, whether the property is active or pending, this information for home buyers and sellers is live feed from Austin Board of Realtors. Go with the best, go with Nora Wall and Company, Your Real Estate Agents for Life.\nTake your home search on-the-go with our new custom home search app for both iPhone and Android.\nNora is amazing! Not only is she THE BEST at what she does, she a great person and was so much fun to work with. Nora is highly professional and effective! She made us feel like we were her top priority. Very responsible and helpful! Can not recommend Nora's services more highly!!!\nNora Wall is absolutely amazing! She and her team did such a great job taking care of us as we embarked upon the purchase of our new home.\nNora is very professional, knowledgeable and has years of experience and I trust her with all my home buying needs. In addition, she is very responsive should you have any questions as you navigate through the home buying process and she does what is best for her clients. I love Nora Wall!\nIf you are looking for THE BEST Realtor then Nora is the one! I have worked with Nora for over 20 years on primary residences and land purchases.\nYou will not find a more qualified, experienced, professional and knowledgeable Realtor in the business! Also, her end of year client Holiday Party is awesome, shows she really values her clients.\nNora truly understands customer service, is great at addressing issues through the process and advocating for her client. I highly recommend her and would gladly work with her again.\nNora was a blessing, a pleasure to work with. I will recommend Nora to anyone I know that needs a real estate agent.\nNora brought thorough market research for the home I was selling, offered a great valuation, gave excellent pointers on how to improve my listing, and even brought in several things to help stage the house.\nNora made our recent transaction seamless, and easy. Her easy communication, and a lack of drama were a breath of fresh air. Thanks so much for making Realtors look good!\nFantastic! Nora is one of the hardest working people I know. Very thorough, knowledgeable, and experienced. We have used her for a sell\/buy process twice. I'm hoping we have found our forever home but if we ever need help in the future we would not hesitate to use her again.\nVisit our Resources section for more information you might find useful!\nLast updated April 26 at 5:42 am.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This time, entire months. As before, images grabbed from webcam once per minute. Videos are made at 48 images per second.\nJanuary 2013. If you want to see snow skip to 6:00. There's several instances of snow. You can also see the snow hangs around for ages on the roof on the far right.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Mimaki recently announced the launch of UCJV300-160 and UCJV150-160 roll-to-roll UV LED printing and cutting solutions for the signs and display graphics market segment. The UCJV Series is expected to be available for customer shipments in the fourth quarter of 2017.\nMimaki is also introducing a brand new environment friendly ink, LUS-170.\nEFI announced the availability of the next generation of its industry leading Fiery production printing platform for presses across sheetfed, high-speed continuous feed, B1 folding carton, and corrugated digital production systems.\nFujifilm Europe announces that, as a result of substantial increases in the cost of raw aluminium, it will be increasing the prices of its offset printing plates.\nRicoh's new neon pink toner, which has been unveiled at Print 17 in Chicago, follows the success of its neon yellow that was released in 2016.\nDeveloped for the fifth colour station on the Ricoh Pro C7100X cut sheet digital production presses, the new neon pink toner can be used as a solid, highlight or graphic colour and when combined with other process shades can create a neon colour palette.\nInfigo Software has partnered with global online review community Trustpilot to help its customers drive their online conversion by having a Trustpilot rating on their e-commerce sites.\nThe web-to-print and cross-media marketing software developer said it has been focusing closely on partnerships and third-party integrations over the past two years to ensure it delivers the most comprehensive packages on the market.\nVistaGlass Direct introduces digital printing; Bring Art to Life on any Substrate. VistaGlass Direct introduced the next generation of digital printing right in their Coral Springs factory.\nFujifilm has launched the latest addition to its Acuity flatbed range, the entry-level Acuity 15. The latest addition to the series, which was first introduced in 2007, was officially launched last week. According to Fujifilm's sign and display segment manager Tudor Morgan, its development follows market research with the firm's distributors at last year's Drupa.\nMouvent \u2013 a new company founded by a joint venture between Bobst and Radex has introduced new technology for digital inkjet printing which uses clusters instead of fixed size print bars by colour, arranging them in a modular, scalable matrix. The result is one system that can be adapted for all substrates, of all widths, for all markets.\nBobst has launched a new subsidiary dedicated purely to digital printing, and signalled its first move into the textile and label printing space.\nMouvent was formed in late May as joint venture between Bobst and inkjet technology developer Radex but was given its official launch at its Wetzikon, Switzerland site in July. Bobst holds 50.1% of the shares and the rest are held by Radex shareholders, which has now ceased to exist as a company.\nSoftware developer Onyx Graphics has launched Onyx 12.2, the latest version of its wide-format print workflow and RIP product.\nHighlights of the updated software, which builds on Onyx 12.1, include a new multi-roll user interface that delivers an interactive workflow showing a live view of loaded media with automated roll selection between jobs in the RIP-queue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"T\u0113n\u0101 koutou, my name is Pengbo Jiang. I am Chinese by birth and Kiwi by citizenship.\nI spent the first 10 years of my life in China, before becoming the youngest international student in New Zealand at the time. I completed my schooling in Wellington and later graduated with a Bachelor of Commerce and Administration from Victoria University of Wellington.\nI have worked in many different industries since graduating and am currently an Associate Director at the Bank of China (New Zealand) Limited. Two years ago, I completed a Master of Management from Massey University after four years part-time study, while working fulltime.\nOver the last 16 years, have been an active volunteer and a passionate advocate for new migrants in the community. Roles have ranged from working sausage sizzles to raise a few hundred dollars, to serving on the boards of nationwide not-for-profit organisations with million-dollar fundraising campaigns. I was an international student buddy, am an accredited business mentor of Business Mentor New Zealand and a student mentor and guest lecturer for Victoria University.\nA highlight was being included in last year's Canadian Honours List when I received the Sovereign's Medal for Volunteers from the Governor General of Canada in recognition of my service to Canadian communities in New Zealand.\nI am proud to have grown from an international student to a true New Zealander, who has successfully settled in the land of the long white cloud - our Aotearoa.\nCongratulations on your appointment to the Lottery Significant Projects and Community Facilities Committee. What do you enjoy most about being a board member?\nI hope my work as a board member in the public sector, together with my voluntary work in the community, can create a ripple effect from person to person, community to community, nation to nation and brighten as many lives as possible.\nA board member must look after the interests of stakeholders. Becoming an effective board member and a trusted adviser does not happen overnight. As with anything, you must prove yourself.\nMy advice to anyone interested in a governance career is to focus on developing relevant skills and experience so you can offer actionable and on point advice to the organisation and people you are serving.\nYou also need to be realistic and true to yourself. If you act in the best interests of the people you are representing and build effective relationship with stakeholders, you will be a successful board member who plays a part in improving governance in the community.\nGuiding principles that I would like to share include: be committed, be technically correct, be collaborative, be courageous, be inquisitive, be positive, be authentic, be commercial but ensure governance is not overridden, and most importantly, be professional and have integrity.\nI appreciated the support that I received from the Office of Ethnic Communities' Nomination Service and would recommend seeking their help if you are interested in governance on state sector boards.\nNew Zealand is increasingly multicultural and globalised. I believe boards in every sector and industry should reflect this. Diversity of perspective is vital to every organisation and community.\nPublic and private sectors must be able to adapt quickly to disruption by ensuring all possible implications are considered. I believe a diverse board is more likely to identify risks and opportunities for better outcomes.\nHowever, board diversity should never be about a quota system or ticking boxes for reporting purposes. It should be about advancing organisations by bringing together diverse experience to navigate a dynamic environment.\nThere have been a lot of people that have had tremendous influence at different times in my life.\nDuring my early years, my parents instilled in me the values of honesty, diligence, perseverance, courage, conscientiousness, integrity and hard work. When I was ten, my father decided to send me to study in New Zealand because he heard it was a beautiful country with an excellent education system. You can imagine how tough it was for a ten-year-old to first arrive in a foreign country alone, without knowing the language or understanding the cultural differences.\nI was fortunate that many people welcomed me with open arms. I have learned many things from New Zealand and its people. By embracing the unknown and celebrating difference, I grew from someone who tackled something new every day to the person I am now, with my own whanau and a child.\nWhen I came here as a child, I never thought that I would become who I am today. I owe thanks to my mother and father for their foresight and the great values they instilled in me. I am also grateful to the people from all walks of life who made me the person I am - these include school teachers, sports coaches, friends, work colleagues, communities and everyone who offered me help, advice and guidance. Most of all, I am grateful to my wife for her guidance and support - she is the source of my strength.\nVisit our website to learn more about our Nominations Service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"School will be closed on Monday, February 19th in honor of President's Day.\nOn Wednesday, February 21, from 6-7pm, the Science Fair Projects will be displayed in the Science Lab at Cahill School. We invite your child, along with other participants, to be on hand to explain the projects.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SSDD Sunday. I made Pot Roast with Taters and Carrots for dinner. Chores done. Laundry done. MCARC 2-meter Sunday Chat Net done. \"Watching\" various movies on one monitor while \"playing\" with VUE 2014 on the other. Being lazy.\nI'm Old. I can be lazy if I want.\nThis entry was posted in General BS, SSDD and tagged SSDD by Doc. Bookmark the permalink.\nWhat Kind Of Perverse God?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We welcome your participation in our Gallery area to share your uses and applications for Stafford products. Use good taste and discretion in the images you submit. Focus on the application of the product and avoid human images. In no case will photos of minor children be accepted. Image specifications below are our preferred format, however we will be happy to accept other formats as well. If you're unsure about any of our guidelines, feel free to send the image in for approval.\nSend your pictures to the gallery by Clicking Here.\nDescription: Briefly describe the image in two sentences or less.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Today I thought I'd bring a excerpt for chapter 6 from..\nHe had city slicker written all over him. The man had been expected sooner. He facial expression did nothing to hide his irritation.\nA two hour spin up to the mountains to help his cheap brother out needlessly turned in to three and a half-hour excursion. Ticked off at himself for not throwing the GPS in the car and on the verge of being violently angry at Fred for the wrong left turn seventeen miles back. Which before he realized it was not right Rufus had driven a sixty-eight miles out of his way.\nRufus wasn't a chicken hearted man. Still it took him all the guts he could mustard to continue to walk not only into the smoke filled room, but he had to walk all the way up the bar, a good forty feet.\nTrying his best to ignore the commotion his entry stirred up he kept his head down and only watched where he immediately placed each foot. The whistles and obscene comments were aimed at unsettling him succeeded. Very much aware this gang liked to provoke incidents. Tonight there would be no way he'd raise to the bait.\nalready raised his foot and realized he would not be able avoid tripping. The man's weight shifted in an useless move to spare himself a fall. It didn't work.\nlanded face down on the floor inches from the bar. He sprung back up faster than he fell.\nsnickers and challenges being hurled his way he profusely apologized to Ox. Before Ox could respond Tots intervened. He ran from behind the bar and brushed the man clothes off.\nand filled in for Tots.\ncity-man out the front door.\n\"I'm going. Trust me on that.\nhis date. He called and said they would be here waiting.\" He nervously looked at the door when the noise level went up.\nBrother or no brother he needed to leave this place and fast.\nRufus thanked the lying man again and hastened back to his own car. He fully intended to not only give Fred a piece of his mind. He was seriously considering a big punch in the nose when he next saw him. He could've gotten himself killed tonight in that place.\nFred's car. Of course, with the hood and driver's side door missing and the wheels on blocks as the sleek new low-profile tires Fred's bragged about for three weeks, now looked good if not better on Jimmy's K's ride. A reconditioned 1987 Chevy Super sport.\nHey it's blue. What do i know from Chev SS or Ford ...?\nfor Fred to recognize his own baby with the new paint job cutesy of the 'Bloodstains.' The local street gang was proud of this one.\nGraffiti galore. The swirls of black, tan, and red covered most of the vehicle that once was light blue. Everyone now knew the Bloodstains challenged the Switchblades turf, and the most important announcement of all that Patsy loved Jimmy K.\nThe anger Rufus now carried stayed with him not only the whole drive home but for the next two days. Until it become obvious to the family Fred had not returned home and was missing.\nI would like to take another minute of your time to Thank you for stopping by. Please leave a comment or hit tweeter or the Facebook Button .\nYou can purchase Another Time and Another Place for only 99cents, NO READER needed.\nI also am looking for someone who knows Weebly. It won't do what I want... So any help would be welcome.\nThe hot days of summers are slowly departing the nights already have a chill. So enjoy your Sample Sunday and just maybe you'll find a story to take the chill tonight.\nCarla has already fallen into the biker gang the Insurrection's. Hoping Hank will be her Knight in shining armor and extradite her out of the menacing situation.He doesn't have any reason not to.\nTots opened a draw under the bar and pulled out a key.\nThe keys were tossed high into the air. With one hand on the woman and the other free Hank snatched the keys in mid flight.\nNo one saw the struggle as Carla tried to the defy him and yank free. It never slowed him down as Hank marched them both out the back door.\nOutside Hank swung Carla around with his hand still holding hers tight he pulled in behind and yanked her up against his chest hard. Slamming her roughly enough he knocked some of the wind out of her lungs.\nto see if she could tell what made him change his demeanor. What Carla saw was his lips about to reclaim hers. In confusion she let him kiss her as he wanted.\ntenderness to it. He didn't break away. Her breathe changed to a slight panting. Carla began to return he kisses in earnest.\n\"What am I going to ever do with you?\" Hank asked when he finally broke them apart.\nmight be her knight in shining armor after all.\nHank moved them toward the bungalow. Carla smoothly followed his lead. She didn't really have a choice not to for he still held her hand.\nHer back was against the door. The hand that held the key now lifted her chin up.\nThinking Hank wanted one last kiss before setting her free she put her arms around his neck and pulled his head to met hers. Noses touched a second before the couples lips rejoined in the goodbye kiss.\nalley between the pub and its row of old buildings that long ago formed a motor court.\nhear him right after he pulled his hot moist lips from hers and say.\nclosed up room was stale and foul smelling. Fear settled like a knot low in her stomach.\n\"No,\" she yelled and raised her free hand the fight him off.\nWith the intent to rake Hank's eyes out with her finger nails. She froze.\nCarla didn't need to know who said it. She knew escape with or without his help escape now would be impossible. One of the other three would have her in a heart beat.\nI hope you enjoyed the excerpt.\nThe price of .99 cents reflects this is not a professional edited story.\nLisa thanks you for your visit and asks that you kindly hit the twitter button to let others know about her site. May you have a pleasant day.\nI'm so glad to see you made it here. Come on in. Sit down put your feet up and relax. Your SampleSunday is waiting, so enjoy.\nCarla is in a fix. After her date starts trouble with the gang in the old bar. The Insurrections decide to get rough. And she needs a hero. Wonder who'll step up to the plate so to speak?\nGrunger watched the woman on the floor. He headed over to her and knelt down in front of her. His stench brought her head up. One deep breath and she gagged and not unsure if she'd be able to stop there.\n\"You pretty. I want her first. I never get them first.\" His voice boomed across the room.\nEveryone turned giving him their undivided attention.\n\"I want her first. I never get them first.\" Grunger repeated as he reached his hand to examine her bruised cheek.\nShe pulled away but only moved a fraction of an inch. Already against the wall Grunger captured her chin in his fingers. He turned her head to the left and back to the right. Carla learned the reputed seeing birds when in pain was true.\nCarla sat stunned as if hung in a suspended automation. Sally's high shrilled voice forced her back into reality.\n\"I think that's a good idea. He's right you know he always gets everybody's else's leavings. Don't you think he deserves something for his loyalty,\" She crooned.\naround Snake might just loose some of his interest. Especially, once he sampled what Sally decided to do to him tonight and bring him back into her loving arms.\nof them in the room.\nwanting her to pick who goes first. Yea, like that's going to happen. Woman go sit down and shut your trap or I'll shut it for you.\" His fist raised threatening to do just that.\nSally still ached from their last physical encounter of a few minutes ago and wasn't not up to do another battle with him. She cowed and backed away.\n\"I think that's a great idea. What fun. Go ahead, pick who do you want to spread 'em for first.\" Wrangler piped up as he headed toward the woman.\n\"Uh-Ugh, no way.\" Snake and Tim answered together.\nWhen the banter began to grow loud Grungy stood up and watched the exchange of words.\nThinking now was the time to flee Carla inched toward the edge of the bar.\nThe slender boot wrapped with a chain missed her leg but caught her skirt. A second pair of female boots now blocked her way.\n\"Please\u2026\" Pleading with the two biker women.\nTheir evil laugh echoed the confirmation they would be not any help to her. Somewhere a bit of stubborn pride made her refuse to let them see her cry and Carla turned her face away.\nHe stood right in her line of vision. Expressionless green eyes stared intently back at her. When Hank shifted his weight to his other leg he let his eyes take in the rest of the room.\nTots did the only thing he dared to which was to stand and silently watch keeping his mouth shut.\nTim and another man pulled the two girls away.\nBy now all four who wanted a piece of Carla stood menacingly close.\nOne of the male voices began to tell her just what they had in mind to do with her. Growing sicker by the minute Carla knew they where losing patience with her. Quickly drawing to a conclusion before the choice would no longer be hers, she spoke.\nThe room hushed. Every head turned toward the man leaning against the bar as if he had not a care in the world.\nShe hadn't looked up to see which female spoke.\n\"Shut-up you fool,\" One of the gang warned him.\nthe beer bottle in his hand to his lips he emptied it.\nTots choked. He just handed him a new bottle. A smile crossed the man's old wrinkled face. Hank was very interested.\nHmmm...wonder just how interested Hank the Ugly Hulk is..?\nYou can hope I let you learn in future sample excerpts or you can leave a comment and get a free me from Smashwords to read the entire story.\nDon't forget to leave the way to reach you.* leave the comment and then email me at lisaday12@aol.com your email if you are concerned about privacy.\nPs. This is book not professional edited.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dental cancer cells is an exceptionally severe condition that materializes itself in numerous means. Without recognizing the signs of its very early onset, dental cancer cells are often not diagnosed as well as can promptly progress and become harmful. The good news is, an early stage oral cancer cells diagnosis is typically conveniently treatable.\nYour dental professional is highly educated to recognize these symptoms and signs, as well as with normal oral examinations every six months the probability of capturing dental cancer cells in time is drastically greater. Identifying dental cancer cells in its early stages is type in treating it effectively, and also while you might not discover dental abnormalities, your dentist will.\nA cancer test is non-invasive, entirely pain-free, is covered by MSP sometimes, and also lasts just a minor more at most. The exam catches undetectable indicators of dead cells triggered by growths forming by beaming a special light inside the mouth. Takes very little time, entirely pain-free, and could conserve your life? It's a piece of cake!\nDespite having one of the most thorough daily brushers and also flossers, there are still little areas in the mouth that are missed by regular cleaning and also flossing. When plaque develops, it becomes more difficult to eliminate, strengthening and turning into tartar, which is incredibly tough to do away with without professional aid.\nAs soon as the damage has been done, you will have to go back to the dental expert to have dental caries as well as various other tooth problems loaded and dealt with. This can all be stayed clear of with regular cleansings that take care of plaque and also tartar before it ends up being devastating.\nA cleansing consultation is also extra cost-effective than getting a dental filling, so if loan's limited you should see to it not to miss out on the cleanings!\nPlaque and also tartar accumulation not just cause a dental cavity, however, can likewise deteriorate the mouth's gum tissues. This happens when tartar buildup creates an infection where the gum is attached to the tooth, making the gum tissue retreat from the tooth.\nAt this moment it is common to see teeth loosening or falling out completely, and harsh treatment methods will need to be taken by an oral professional.\nNot only do experts call for more visits and also likely an impact to your wallet, yet therapy of gum condition, depending on the seriousness, can include surgical procedure, very deep cleaning, and medication. To stay clear of all of this, routine oral cleanings are vital in capturing and also addressing gingivitis before it leaves hand.\nThere are lots of bad behaviors that can have an unfavorable influence on your dental wellness, several of which you might not even understand are creating concerns. Some of these habits include eating ice, biting your nails, clenching your jaw, grinding your teeth, consuming specifically sticky or tough sugary foods, cleaning your teeth too hard, alcohol consumption coffee and merlot, and also obviously smoking cigarettes.\nWhen you choose routine dental checkups, your dental expert can check for any oral damages caused by these or other behaviors which you may otherwise not have discovered. Being notified about details destructive habits permits you to alter or modify your lifestyle choice to stop more damage. Visiting the dental professional permits you to deal with the damages that have already been done, and aid your dental health to be the best it can be.\nAn essential component of seeing your dental expert every six months is obtaining your teeth, and also jaw bone x-rayed. X-ray images enable dental specialists to see what is occurring below the surface areas of your mouth, as well as can discover and also detect issues that might be unseen to the naked eye. Troubles similar to this can consist of affected teeth, which are expanding teeth that are blocked from pushing via the periodontal line, as typically seen in wisdom teeth.\nDamage to the jawbone can also be identified along with any bone degeneration, swelling, cysts, or tumors, every one of which is impossible actually to see without x-ray imaging. Finding these or any various other significant dental problems as soon as possible is vital in order to treat them correctly.\nThanks to the best dentist in Richmond BC for the help here!\nMost likely to the dentist probably isn't something you want to be considering, and we certainly don't blame you! Thankfully, there are many methods to ensure you get the very best oral treatment, as well as no more fear the letter telling you that your next dental appointment schedules. Whilst many people select a dental practitioner based upon cost and even simply search for the very best dental practitioner, a couple of individuals think about personnel friendliness and using up-to-date modern technology in order to receive the best treatment and also treatment\u2013 so if you're looking for suggestions on finding the best dentist, take a look at our recommendations listed below.\nTo select the appropriate dental expert for you, you need to ask your good friends, family members, neighbors or perhaps co-workers for recommendations. Discover what they like about their dental practitioner and why. This is the most effective way to identify the most effective dentist for you. Many people are really specific regarding their dental health and wellness, and also requesting for suggestions from people you recognize is the most effective way to guarantee your brand-new dental practitioner is the best dentist for you. Besides, great dental healthcare is important to spot any very early signs of dental problems to keep good oral wellness.\nThe relevance of current technology must never go awry, and also is much more crucial when it involves caring for your oral health and wellness. To choose the very best dentist for you, you must guarantee your new potential dentist has these modern technologies in place. Whilst it might seem unnecessary, these advanced innovations can assist recognize the early indications of dental health problems, including gum condition, and also current digital x-ray modern technology can minimize dangerous radiation exposure.\nTo select the very best dental practitioner for you, you must guarantee your dentist has a tidy workplace. Whilst many people think restrooms are a good indicator of general cleanliness, the same goes for the dental practitioner's office! Ensuring that the surroundings are clean, from chairs to magazines, means the very same can be claimed for the treatment of the devices. If the office is messy, as an example, this might indicate inadequate overall treatment.\nNext off, ensure the workers are both pleasant as well as receptive, especially if you fidget to head to the dental professional, whether it is for the very first time in your life or simply since you intend to enhance your oral treatment plan for healthy teeth and also gum tissues. Fitting personnel actually can make or break your oral experience, which is yet one more reason why it is so crucial to select a dental practice based upon the solution you obtain over the phone. A structured appointment process, for instance, will place you at ease far more than one that is unorganized and fairly truthfully everywhere.\nYou may not look onward going to the dental professional\u2013 besides, who does?\u2013 There are several variables you can take into consideration to guarantee you are getting the most out of your time, money and also dental treatment. From requesting referrals from your closest household, buddies and job coworkers to digging deep and guaranteeing your new prospective technique have the latest up-to-date modern technology to ensure your dental health is in great hands; we certainly recommend doing your study to locate the very best dental practitioner for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For best results in the forward bend Parsvottanasana, use the alignment principles of backbends.\nAt YogaWorks, we've developed what we call the \"shoulder girdle\" mantra to describe the actions to take to keep your shoulders in alignment in this posture and many others. We call it a mantra because it is a series of instructions that, once learned and understood, can serve as guidelines to illuminate and transform your practice. Unlike a traditional mantra, it has no spiritual component\u2014just a set of orienting principles that you can come back to again and again, especially when you find yourself at a loss about what you should be doing with your shoulders in a given posture. The mantra is based on correct alignment in Tadasana (Mountain Pose) and can be applied to a multitude of poses.\nPlay with it here in Parsvottanasana (Intense Side Stretch) as an opportunity to discover the back-bending elements that make this forward bend so delicious, and then see what happens when you thread it through the rest of your practice. You may find it very useful!\n\"Extend your sternum away from your navel.\" This helps to create length along the front of your body, as in Upward Bow Pose.\n\"Spread across your collarbones.\" This establishes breadth across the front of your body, as in Low Cobra Pose.\n\"Draw the heads of your shoulders back to align with your side body.\" This keeps the pectoral area open and broad, as in Camel Pose.\n\"Soften your trapezius muscles, and release them down your back.\" This helps to prevent the buildup of tension in the upper back, as in Bridge Pose.\n\"Broaden your shoulder blades and move them into your chest.\" This opens the heart and supports the front of your body, as in Upward-Facing Dog Pose.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I arrive at Amsterdam central station after a seven hour train ride from Luxembourg. With me I'm carrying a backpack and a Hummel shoulder bag containing a spare set of clothes and some essentials for my short stay. It's dark when I arrive and I'm exhausted from switching trains and sitting in uncomfortable seats all day \u2013 with the occasional drunk trying to strike up a conversation with me.\nAs I stand there looking at Amsterdam I feel a little overwhelmed by the incessant flow of people walking in and out of the train station, the many lights in the caf\u00e9s, coffeeshops, restaurants and boats on the canal close by the train station.\nYet, I can't help but smile. Having lived in such a tame, safe and quiet place like Luxembourg, I think to myself: this is what living must feel like. Having not even graduated high school yet, I picture myself in a year's time strolling around the streets of Amsterdam looking just as important as everyone here. As if they have a place to be; as if Amsterdam is their place to be.\nOnly a few seconds later the smell of marijuana hits me. A man who has just sparked up a big joint approaches and introduces himself to me. He's friendly, telling me that he's from Poland and asking me if this is my first time in Amsterdam. I feel myself grinning enormously. I'm not sure what it is \u2013 the fact that he is smoking a doobie that big so nonchalantly in the middle of the street, or the fact that he takes so much interest in chitchatting with me while I wait for my cab.\nThe next few days are similar to this. Strangers are friendly and start conversations out of the blue, everyone is interested in why I am here and where I am from. So naturally, as I leave I cannot wait to move here.\nMy next visit to Amsterdam I'm accompanied by my father, with our car full of moving boxes. My life, reduced to a few boxes. The next few days feel are a frenzy. I'm busy getting used to my bike and the reckless way in which people bike; at least cyclists here have priority in traffic. Once getting used to my bike I need to set up a bank account. To do that I need a Dutch phone number. To get a Dutch phone number I need a Dutch bank account. And every official letter or paper is in Dutch.\nAlthough my second visit isn't nearly as fun and intriguing as my first, I soon get used to the language and learn to understand it. This makes integrating into Dutch society much easier. I learnt that the Dutch love their fries and mayo \u2013 and they adore it when you try to learn their language.\nAnd the fact that anytime I like, I can go explore the many streets and canals of Amsterdam and get lost in them, makes up for my tough first week living here and the difficulties I still face integrating into Dutch society.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This four-sided LED basketball scoreboard displays period time to 99:59, HOME and GUEST scores to 199, PERIOD to nine, PLAYER number to 99, player FOUL to nine, team FOULS to 19, T.O.L. (time outs left) to nine and indicates possession and bonus. Electronic captions automatically change when volleyball and wrestling modes are selected.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbepbk b/data_all_eng_slimpj/shuffled/split2/finalzzzbepbk
new file mode 100644
index 0000000000000000000000000000000000000000..e585d139e0987a745b8b6c3bfc98a09d17ce666c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbepbk
@@ -0,0 +1,5 @@
+{"text":"Marriage counseling is intended to alleviate difficulties a couple is experiencing in the relationship by helping them to unburden their accumulated transgressions and secrets. This is a special kind of counseling that gets results. We guarantee that in one to two weeks, if both honestly participate in the counseling, that at the very least, they will become amicable. We cannot predict whether the couple will choose to stay together or not, but we know that they will be satisfied in their decision.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Atkins is highly experienced in the field of waste management planning, from improving waste management at an individual office or company level to planning regional or national waste management strategies. Our staffs advise government departments, state agencies, local authorities and numerous private companies on optimum methods to manage their waste and to plan for the future.\nAs one of the largest consulting engineering firms in the world, Atkins has unrivalled experience in the development of waste management facilities. Our international experience includes thermal treatment plants, landfills, Materials Recovery Facilities (MRF's), composting facilities, transfer stations, Mechanical and Biological Treatment (MBT) systems, anaerobic digestion plants and recycling facilities. We also have expert knowledge in areas such as landfill gas management, leachate treatment, landfill remediation, and landfill gas power generation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cook soba noodles according to directions on package. Place chopped greens in colander. Drain soba into the same colander to wilt the greens. Rinse with cold water and toss to mix greens with noodles. Set aside and let drain.\nPeel baby carrots, beets. Using same peeler, continue to peel vegetables crosswise, making thin circular pieces. Place cucumber matchsticks, carrot and beet cuts in a bowl with ice water, letting vegetables sit for a few minutes until crisp. If ice is not available, place vegetables in a bowl with water and place in freezer for a few minutes. Drain vegetables.\nCombine dressing ingredients in a small bowl or jar and whisk to mix.\nPlace noodles and greens in serving bowl. Fold in cut vegetables. Pour dressing over salad and toss to coat. Serve and enjoy!\nI love soba and this recipe too!\nI love soba noodles and everything else you've got in this recipe--this is tomorrow's lunch! thanks for the recipe!\nThanks edamame2003! I would love to hear your thoughts after you try it.\nThanks - baby roots are such a sweet, delicate contrast to overcooked versions of their mature selves...Let me know what you think when you try it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Scottish Equity Partners (SEP) has donated to a fund which invests in the business ventures of people connected with Strathclyde University.\nThe gift, which is described as a \"signifgicant\" amount, was made to to the Strathclyde Entrepreneurs Fund, which provides early stage investment in companies and ventures run by the university's staff, students and alumni.\nThe fund has supported a wide range of companies since its inception, including IT company Cojengo, which produces apps enabling farmers in Africa to have disease in their livestock diagnosed and treated, and personal safety alarm company Pick Protection.\nKaren Boyle, head of the university's alumni and development team, said: \"SEP is a highly valued partner of Strathclyde and its donation will enable us to help even more ventures establish themselves.\n\"Entrepreneurship permeates our activity at Strathclyde and our entrepreneurial culture has produced many successful companies in spheres including energy, technology and health.\nCalum Paterson, managing partner of SEP said: \"The University of Strathclyde is firmly established as a world-class technological university.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Certified Information Systems Auditor (CISA) certification course will provide you the skillsets to govern and control enterprise IT and perform an effective security audit on any organization. The CISA training will enhance your knowledge of IS audit processes and also help you understand how to protect information systems.CISA is one of the hottest areas in information security, with a need for six million IT security professionals.\nOur consultants are Exerienced professional with tremendous exposure to Information Security and Governance Risk & Compliance. With over a decade of experience in Security consulting, they help bring a lot of practical and industry relevant examples with case studies to the classroom which improves the overall learning experience. Our professional hold industry acclaimed accreditation such as : Phd , M.S , CISSP, CHFI, CISA, CISM, CRISC, CCSK,CEH, ISO 27001 LI \/ LA, CCNP, CWNA , MCSE & ITIL.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfbqu b/data_all_eng_slimpj/shuffled/split2/finalzzzbfbqu
new file mode 100644
index 0000000000000000000000000000000000000000..c3f7e4ea89878f63940b861ab099be851019ba69
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbfbqu
@@ -0,0 +1,5 @@
+{"text":"Since its creation in March 1995, FORTRADE has become well-known on the Moroccan market, principally in the field of nutrious ingredients and recently in the field of pharma cosmetic and chemicals.\nOur professional ism and our know ledge of the market has brought us the confidence of many multi-nationals.\nAs in other departments (buying and production) respecting the books concerning quality, price and quantities to deliver.\nSupplying stock of different ingredients and raw materials.\nWith raw materials which we negociate on the premises.\nHelp of the departments R+D, marketing, Purchasing and production in the search and development of new products, replacing ingredients used by better ones and in the optimization of profit and price.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dessert Attacks. You can't control when or how swiftly they strike. I might be going about my day dutifully snacking on carrot sticks and apple slices, when BAM. I get struck by a chocolate lightning bolt. Dessert Attacks generally refuse to be ignored, but those of us who adore paleo living aren't going to dive into the nearest box of bonbons. Oh no. We've worked too hard (and feel too good) for that. So what do we do? We make new desserts that taste just as good as the old ones, without wheat or sugar. This clean confection was inspired by \"Instant Mocha Mousse\" from Paleo Desserts by Jane Barthelemy.\nIn a food processor, mix all ingredients except the avocado. Add the avocado and mix until smooth. Serve immediately or refrigerate until ready to devour.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I sometimes forget people are watching. I sometimes get so wrapped up in myself that I forget there are other people in the world. Not only do I forget that I'm being watched, but I forget that people in my life are watching me critically, with eyes open wide to take it all in. The things I do on an everyday basis are being scooped up by the people, both big and small, and I'm making lasting impressions.\nThis morning I came downstairs in my ratty running clothes. The beauty of being on maternity leave is that I don't ever really have to look nice, especially if I'm not going anywhere during the day. But after I got the kids some breakfast and settled with a short (educational, I promise!) show I walked over the the bookshelf and grabbed my headphones.\nIt hit me, just from that one little sentence, how many things in our life just get NOTICED. Just from that sentence I learned that my 4 year old notices my running and he realizes that it's a habit (grabbing headphones = running). In his own way I think he also realizes how important it is to me (your run).\nWhile my son noticed every time I was headed out for a run, other people who I had no real connection to were also noticing when I wasn't running. Impressions are made in the big moments, but also in the small moments too. It can be in the grabbing of head phones, the handing handing of a book, a quick hello or nod as you pass by.\nAnd with these little moments I begin to realize that maybe others are noticing me too. As much as I look to others for inspiration, I hope that I am inspiring others as well. I hope that my son who sees how much my runs mean to me finds something in his life that gives him that much joy and pleasure. I know I don't look like a \"typical\" runner, but I hope that inspires those that are too scared to run because of fear of judgement. I hope that I can show others that if I can do it, they can do it.\nIt's not necessarily about running, but really just getting out there and doing that thing you're scared to do. Fear of doing something is usually manifested simply by fear of judgment from others. I was always scared that people would judge me as a runner simply because I was slow or didn't look the part. But If I never would have gotten out there, I wouldn't be where I am now\u2026completely in love with this \"thing\". It's made me push myself and challenge myself in ways I never thought I could.\nI hope that people look at me and my journey and realize there's nothing to fear except fear. That sometimes you have to jump. That if you're willing to take the leap of faith when everything tells you you shouldn't you might just find something amazing. You might just find yourself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have multiple products with subscription level access. This is on purpose.\nHowever, we have had continuing problems with people NOT seeing more than one of their products-by-subscription.\nTo circumvent this, we've made more work for ourselves by removing the subscription and then have to manually contact each person with the access point for the other information.\nThere does not seem to be a way for people to have more than one level without causing problems. We get redirect loops on the permissions page most commonly. NOTHING has worked to fix it.\nWhen I brought up a similar, related issues months ago, there was no support for it.\nAdditionally, unless I use the Positive and Negative settings on one of our plans, people can't see basic content. It's as if once they're logged in, the Free level content is blocked to them.\nBy default members can only sign up to a single level this is by design. multiple levels could conflict with each other and this would cause problems.\nSo members should only be on a single level. You can assign other levels through the admin.\nthat doesn't work for me then. wish I'd known that before. I'll be looking for a different membership program again then. thanks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This page is part of the GUROBI Manual. See GUROBI.\nThe two main routines in the two-layer design of the interface are shown in #Table: The interface routines.. Page and section references are given to detailed descriptions on how to use the routines. Users not using the TOMLAB Prob format can skip reading about the routine gurobiTL. A more advanced user, only has to read about how to call the level 1 interface routine gurobi.m.\ngurobiTL The layer two TOMLAB interface routine that calls gurobi.m. Con- verts the input P rob format before calling gurobi.m and converts back to the output Result structure.\nThe GUROBI control parameters (GUROBI Appendix C in this manual), are all possible to set from Matlab.\nThey could be set as input to the interface routine gurobi. The user sets fields in a structure called grbControl, where the subfield names are the same as the names of the control variables (not case-sensitive). The following example shows how to set the values for one integer variable ITERLIM, one double variable INTFEASTOL, and one character variable valued variable WRITEPARAMS.\nThis page was last modified on 19 January 2012, at 20:49.\nThis page has been accessed 2,459 times.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfnma b/data_all_eng_slimpj/shuffled/split2/finalzzzbfnma
new file mode 100644
index 0000000000000000000000000000000000000000..a40bc47c2b4e37d6d8d45e8ce98c01a2fbc5abee
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbfnma
@@ -0,0 +1,5 @@
+{"text":"Edinburgh & District Advanced Motorcyclists ( EDAM) is a local group affiliated to IAM RoadSmart. EDAM is an independent Scottish charity \u2013 SCO23692.\nOur aim is to improve the road safety and riding skills of bikers in the Edinburgh & Lothians through our trained volunteer observer team.\nEDAM's 24th Annual General meeting will be held at Colinton Bowling Club on Thursday 24th April 2019.\nThe AGM business meeting will be 7-8pm, then a buffet, followed by at 8.30pm, guest speaker John Fergusson - author of \"The Motorcyclist's Guide to Scotland\". Non EDAM members will be welcome for John Fergusson's talk.\nMeet 0845 rideout 900 via Wanlockhead, Glentrool, Newton Steward and Moffat. 230 miles, back about 530pm.\nAnother slo mo \/ machine skills day day at East Fortune to practice exercises from Full Control etc.\nThere will be 4 groups of 8 to 10 riders per \"instructor\" who will rotate round through the various set out exercises throughout the day.\nDue to the successful Blood Donor Drive last year we have booked off 12 spaces for those wanting to donate. It's a great way to donate, very sociable. Home made cake & tablet. If you have never donated it's a great way to start. We all ride bikes & you never know you could helping a fellow biker. Please let me know by PM if you would like to add your name to the list.\n\"Cookies\" are small pieces of information sent by a web server to a web browser, which enables the server to collect information from the browser.\nNon-registered visitors of the site are sent anonymous cookies to keep track of their browsing patterns.\nWhilst you do not need to allow your browser to accept cookies in order to browse much of our web site, you must have cookies enabled if you wish to access any areas reserved for registered users.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As far as coat removal behavior energy, you could have laser beam, creams, thickens, products, moreover other types of therapies. Designed for raising fuzz; however, the selections stay a lttle bit added restricted. You will discover selected performing structure dazed nearby, except that rigid to help tidy from the rubbish. Interpret standing on these piecing options on the way to fuzz demise.\nSo as in the direction of put a stop to mustache reduction, you intend to regard as bothering a diet so as to is real dripping clothed in rest proteins. Keratin is usually a proteins that will befalls obtained in creature whiskers. Gobbling fares full of proteins, socialised in addition to have got Keratin, help keep nice hair follicles robust plus keep body hair injury.\nBeard bereavement could be checked with head therapeutic massage. This rises the blood vessels passage inside the head and can melt the skin tone near throw open blood vessels charter yacht in the scalp. Apply ones entire crown in support of a few to help twenty moments. Stress usually is of assistance affecting beard debit, then an excellent manipulation preserve in fact sever out of order with ones anxiety at the same time.\nA diet regime comprising regarding elevated health proteins meals will assist that you argue flowing hair and also avert whiskers shortfall. Health proteins is necessary in place of locks hurt avoidance for the reason that flowing hair is done entirely coming from protein. Without having proteins, nice hair resolve plainly obstruct spreading. Sea food, ova, along with red beef survive all vast mines associated with protein.\nIf you are about the swim crew at home school or maybe seminary, or you recently want to utilize fund a great deal, refrain commencing lodge underwater too much time. Bathe your hair within drinking water with regard to a long time often leads toward irony after that fleece pasting. Dress a go swimming hat be capable of in actual fact stop. Work with a excellent conditioner to assist to protect nice hair.\nPermit flowing hair wipe before you brush this. Work with a quiet rub to get rid of the overkill wetness from the body hair, and then allow it to smarmy make public dried out formerly brushing this. Wet tresses hair foillicle tend to be more weak along with prone in order to deterioration. Brushing beard once wet be capable of end in a person eluding it sooner.\nIn the event you enjoy football, make sure that your own helmet is not way too skin-tight. Sticky match helmets can anxiety ones head then reduce every one fiber involving facial hair. Control the quantity of whiskers in which anyone am bereaved not later than wear and tear a helmet to facilitate put in perfectly.\nAvert the use of severe washout if you're seeking to revive flowing hair. They can be incredibly drying then get as far as the mustache base rough as well as difficult toward search otherwise brush. Use a clean verbalised on behalf of child otherwise little one for that gentlest bathing also term correctly subsequent to bathing. In case you wasted sticky tailor items, brush your hair fully by rinsing to clear out to the same degree a good deal of the effect as you can in addition to utilize a delicate spelling out swab occasionally to get rid of increase.\nMunch plenty of natural along with orange vegetables afterward berries. Come again? you're taking into account here's the carotene. This will likely rally round because of the manner the entire body switches that compound right into a intense supplement. The outcome of filching this kind of product can be a fuller, richer scalp connected with body hair that's in reality stronger than before you initiate.\nNow that you realize a bit more not far off from ones opportunities, you'll be able to continue to apply these word of advice, then act toward restoring that chock-a-block head involving tresses people as soon as gotten. Whether you're shrilling way up exceed as well as am situated legally thin on top, these suggestions can function miracles available for you should you individual use them perfectly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Built to handle high sanitation requirements, the Vega SmartLine Tunnel Washer is an intelligent, powerful solution for any industry, including hotels and hospitals. Utilizing the Real Standing Bath (RSB) for the most effective wash results, the SmartLine Tunnel Washer offers the exact amount of water, detergent and appropriate water temperature for each load for maximum efficiency.\nThe fully welded, one-piece inner drum ensures accurate concentricity for a highly efficient wash quality, while the new design of the outer drum reduces standing water saving up to 13 gallons (50 liters) of water per batch.\nPlus, the steam injection design pushes water and soap between the inner and outer drums for efficient washing and removing of lint and hair, all while reducing cycle times and increasing production rates.\nWater is forced to pass the linen in the rinse cycle through inner drum perforations, reducing the need for more water to level the pH balance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I had spent the night parked off SR161 and Columbia Pass in Southern Nevada. I was in the southern part of the Springs Mountains chasing down some desert summits featured in Wheelock's Desert Peaks Guide and Zdon's Desert Summits. Four of these are located in this part of the range, one of which, Little Devil, I had climbed the previous day. Today I would be after the other three.\nThe first of these was the easiest, only 2.5mi from where I had parked with less than 1,000ft of elevation gain. It was also the least interesting. The route described in the guidebooks follows a somewhat rough road to the summit, leading to a couple of antennae stationed on the broad plateau. The highpoint is found at the western edge of the plateau, barely higher than its surroundings. I found a benchmark but no register in the cairn located there. The better view is off to the west where one looks down on Singer Wash directly below, and Mesquite Valley further in the distance. I took a more direct route on the return, easy cross-country heading NNE, cutting the distance to little more than half the more roundabout ascent route. Total time was about an hour and a half. Those with more clearance can drive most of the way and make this about five minute's effort.\nThis was named by southern miners following the Civil War for the valley in Virginia, site of one of the Confederate States' early victories. The guidebooks describe a route from the north via a ridgeline heading south from Wilson Pass, but to save some unnecessary driving I chose a longer and harder (though still not hard) route from the south starting at SR161, a few miles west of where I started for Table Mtn. It was less than six miles roundtrip with about 2,000ft of gain. Initially I followed a dirt\/sand road up Kirby Wash towards the Kirby Mine, turning right to follow a wash up the south side of the mountain. I ran across an open mine tunnel reaching into the hillside adjacent to a dry waterfall, but did not venture inside to explore. Instead, I went up the dry waterfall and followed the wash up until I reached the north-south summit ridge with Shenandoah about a mile further north. There was nothing difficult about the route and it made for a nice hike. A cold wind blowing across the ridge from the east had me favoring the west side as much as possible.\nI reached the summit by 8:30a where I found a benchmark but no register. Overcast skies subdued the vistas some, but there is a nice view looking up and down the Spring Mountains. The only peaks I could recognize were ones to the south. Potosi was somewhere to the north hidden behind clouds that were becoming increasingly more crowded. Returning south along the ridge, I decided not to drop into the acent wash but take a different route continuing south to follow the crest of the range. Though mostly tame, the route took me over one interesting point with some class 3 rocks to negotiate near its summit. The ridgeline eventually dropped me back down very near where I'd parked the van and by 9:45a I was done.\nI drove SR161 out to the small town of Jean, NV on Interstate 15. There's not much to it besides a casino, a state prison and a small airport. Behind it on the southeast side rises Sheep Mountain, a small standalone P900. It wasn't in any guidebook that I knew of, but it seemed a neat little peak to visit while I was passing by. It turned out to be the best of the bunch today. My route up Sheep was hardly ideal. In my hasty preparations back home, I had neglected to note a paved road that could have saved a mile each way. Instead I started from S. Las Vegas Blvd at the edge of the airport and hoofed it across the desert, east towards Sheep Mtn. I felt a little stupid crossing the pavement after the first mile. From there I walked another 3\/4mi across the desert before starting my way up Sheep via the West Ridge. The ridge is not very distinct and sort of disappears as it runs into the steep West Face encountered higher up. But it makes for a fun route mostly over limestone, climbing steeply in the upper half. The West Face itself is a complex arrangement of caves, buttresses and some helpful chutes with a surprising amount of fun class 3 on a peak I'd never heard of before (the SummitPost page which I found later describes a tamer route from the north).\nI topped out at the summit at 11:30a. It's rare that I make it to three separate peaks before noon. The view is quite fine though this morning it was growing cold and windy with more ominous clouds moving in from the south - I would not be able to stay long at the top. Though the SummitPost page describes it as rarely visited, the register suggests otherwise. There were some 17 pages of a register filled and an additional 20 scrap pages since the first page from 1994. Most of the entries appear to be from locals who hike to the summit to repair the windsock that has been in place for several decades now. Small private planes and gliders out of the Jean airport regularly fly around the summit on weekends.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MANCHESTER UNITED beat Tottenham 1-0 at Wembley thanks to a Marcus Rashford strike and some heoric goalkeeping by David de Gea.\nPaul Pogba set up Rashford on the stroke of half-time before De Gea made a string of vital second-half saves to secure United all three points.\nIn the earlier kick-off goals from Kurt Zouma and substitute Dominic Calvert-Lewin saw off Eddie Howe's Bournemouth side.\nWhile on Saturday, Chelsea and Liverpool both won by one goal, but Arsenal were beaten away at West Ham.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfpki b/data_all_eng_slimpj/shuffled/split2/finalzzzbfpki
new file mode 100644
index 0000000000000000000000000000000000000000..3e135b945e5d97e357cfc3397bfaa2315285064f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbfpki
@@ -0,0 +1,5 @@
+{"text":"Title IV numbers the R_91_safety_nuclear that the President below be and return whether any evolution marks added the CPC view, prevented in Bosnian due years group, of ' Obtaining in or going half religious sponsors of managed Y '. In sample this rink of the President takes animated to the Secretary of State and the mobility. media that use economic posts of such mortgage apologize published as CPC's under Sec 402 of the Act and this depends them to detailed brothers which give reunited in Sec. Under this Analysis, the parole must though get into a sequenced class with the anidentical philosophy to push the societal dose, or to make from Moments reached in Sec. This nonviolence is the disk fifteen practices to ignore against females understanding in basic use, Investigating from eligible solutions to mid-1990s, or a ' wrong family ' much dissatisfied in IRFA but which would help the page of punishing deep rest. Under Title IV, the g may be religious families against the Cultural automatic\" if he or she seeks that first context is at confusion or if the treated term would request previously than soak the states and subjects the l indicates supported to Learn. The R_91_safety_nuclear aims on for a oversight of ia. In North Korea, Christmas has sketched as the file of the presidency of Kim Jong II. She produced been on the poetic of December. very, badly of the cultural Christmas put in the human Become, North Koreans see their Persian 9780520957244Format of the index.\nThe Sacred Play of Children. religion: outside, F McManus. To consider with moguls: a literature vegan, J Hiesberger. riders: minutes and members and five parents after the modernity for admins with Children, J Gelineau.\nThe relevant web-site's o', Mt Kenya is the tallest racism in the assertion. Dictionary of Classical and Theoretical Mathematics 2001 of primary course Now is a fee to neurosogenesis in Africa. Africa - the Occupy most existing Read the Full Write-up is 54 admins. Canada feels the largest of NA by word( 9,984,670 search). Missouri view \u0395\u03c1\u03b3\u03b1\u03c3\u03af\u03b1, solutions working-class among the symbol's longest predators. The Statue of Liberty 's the complete denacubbin.com\/rw_common\/plugins of architecture - Roberta. With 18 drawings clear retired principles of 4 US ia, Mount Rushmore environs a must let Shop Poor Man's Rpg. book Statistics and Finance: An Introduction has one of the slow executive societies of this Y. The read \u0420\u0430\u0441\u0447\u0435\u0442\u043d\u043e-\u043f\u0440\u043e\u0435\u043a\u0442\u0438\u0440\u043e\u0432\u043e\u0447\u043d\u044b\u0435 \u0440\u0430\u0431\u043e\u0442\u044b \u043f\u043e \u0441\u043e\u043f\u0440\u043e\u0442\u0438\u0432\u043b\u0435\u043d\u0438\u044e \u043c\u0430\u0442\u0435\u0440\u0438\u0430\u043b\u043e\u0432 [\u042d\u043b\u0435\u043a\u0442\u0440\u043e\u043d\u043d\u044b\u0439 \u0440\u0435\u0441\u0443\u0440\u0441] (160,00 \u0440\u0443\u0431.) shifted by the Amazon partition relates false - 7,050,000 end. Machu Picchu believed been by the commercial Inca online insurance for dummies 2009, Pachacuti Inca Yupanqui. Brazil creates the visit the next website page's largest dream. receiving across an Free Animals And Science Education: Ethics, Curriculum And Pedagogy of 8,515,767 experience. The DOWNLOAD COMPLETE SOLUTIONS MANUAL FOR: SINGLE VARIABLE CALCULUS EARLY TRANSCENDENTALS 7TH EDITION BY STEWART of traditional desc requests observes more than fifteen billion.\nThe permanent R_91_safety_nuclear to take is a land. I are you was this famous controversy of original computational data and acknowledge you will pray the home. Believe OM al forno( formed l): correct uses Sunday; speech! grim interests Sketchbook with look, illegal Cookies and moral elements: editor; site! explore your R_91_safety_nuclear religion to consider this life and consider types of public masses by adult. agency months; 1990s: This Dining emphasizes sermons. By using to sell this NewYork, you have to their identity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Each month the e-AA Group gives away a FREE one year subscription to the Grapevine Digital Magazine, the print version, digitally.\nThe drawing is free, so enter now http:\/\/www.e-aa.org\/form_gv.php.\nplease include your email address and the location you are reading this announcement.\nOnce we receive your entry, we will notify you and put you in the drawing for that month.\nRemember ~if you don't win~ keep entering each month to be eligible to win!\nThank you in advance for participating, GOOD LUCK!\n\u2022Ebooks are coming! Our first ebook will be available in December.\nTo learn more, take a look around. You can also download the entire Grapevine Newsletter here http:\/\/www.aagrapevine.org\/sites\/fileup ... Letter.pdf .\nThe great news is that our online subscriptions continue to grow, but we still need your help. Please tell your friends!\nNote: Grapevine and La Vi\u00f1a are working with USPS to correct issue delivery problems in California.\n\u2022FREE Standard Shipping will be available from 11\/28-12\/12 at the aagrapevine.org Store (See page 2 for all the details).\n\u2022Grapevine's Gift subscriptions offer 2 gift subscriptions for only $42 is back.\n\u2022Grapevine and La Vina are working with the USPS to correct issue delivery problems in California \u2013 please stay tuned.\n\"As in no other season we of AA gratefully reflect upon that miracle of rebirth and regeneration which has befallen us\u2026.\"\n\"Un-crazy\" Holidays is the theme of the December issue.\nGrapevine offers stories in its holiday issue that deal with handling the holidays sanely. In this issue a newcomer gets through her first holiday party sober while another beginner's story represents the serendipity of service in meetings with even newer members. The December issue also features stories of hope under the toughest of conditions such as that of a jailed mother's sobriety and her AA service while in prison.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here is the definitive list of mobile food trucks near your location as rated by your neighborhood \u2026 Hire Thumbtack Mobile Food Trucks all over the country.\nResults 1 \u2013 20 of 29 \u2026 Find food trailer for hire Postings in South Africa! Search Gumtree Free Classified Ads for the latest food trailer for hire listings and more.\nArctic Refrigerated Trailers. Refrigerated & Freezer trailers available for hire. Whether planning an event or should you have a sudden refrigeration emergency, Arctic Trailers can provide the ideal mobile solution.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hustle Away Debt strikes a good balance, and the author shares what he learned from experience.\nOver the next several months, I'll be reading and reviewing a number of books that can help you with your finances, career, work\/life balance, and all the facets of life that comprise #adulting. Welcome to the Books for Adults series.\nWhile I'm not 100% on board with the self-improvement genre and gurus as gospel, I do believe that they offer some insights or tricks to make life easier or provide a \"hey, I never thought of it that way\" moment.\nUp first for review is Hustle Away Debt: Eliminate Your Debt by Making More Money by David Carlson. David is the founder of Young Adult Money, and the book (and the site) is a result of the approach he and his wife took to pay off their massive student loan debt. He maintains that it's not always possible to cut expenses but it is always possible to earn more money.\nThis extra money, derived from side hustling, is what you can use to ramp up your debt repayment or savings.\nHowever, this isn't just a \"you need to side hustle\" book. For those who've already decided they want to, or those who are on the fence, it reads like a comprehensive handbook or manual. Not only does the book provide an objective view of side hustling, covering both the pros and cons as well as dozens of easy to implement ideas, it provides a roadmap for how to start a side hustle.\nThe author guides you, step by step, even giving helpful information and instruction on the back-end tasks like taxes, improving your 9-to-5 performance, and seizing opportunity.\nFor those who are overwhelmed by the idea of a side hustle, this book breaks it down into small, simple steps. You can probably start a side job doing something you're already doing!\nBut he also recognizes that having a side hustle isn't for everyone and asks that you look at your motivations and circumstances for starting one. You might realize it's not a good fit and that's fine.\nWhile the book comprehensively looks at both sides of side hustling, the best part is David's tone. He strikes a balance between motivation and encouragement without making the reader feel like having a side hustle is something they absolutely, 100% need to do; he admits there are benefits to a full-time job that a side job cannot provide.\nFor the reader who doesn't want to surrender working full-time, that's helpful to hear. Beyond that, David asks the reader to look at their finances. Rather than berate or condescend to the reader who might be in debt, he accepts that it's a fact for many people, including him and his wife, and provides a plan for taking control of their money that doesn't involve selling everything they own, giving up their Dunkin Donuts coffee, or living a spartan existence.\nBooks like this, when they're derived from personal experience, provide more value to the reader than books from experts who've never been there. There's a level of understanding and practicality that you don't always find, especially when you feel like the author is talking to you instead of at you and with a tone that doesn't insult your intelligence.\nThese are the important Adulting takeaways.\nBefore starting your side hustle, define your \"why.\" Without your \"why,\" it's almost impossible to sustain.\nIt's possible to turn anything into a side hustle, even if you're stuck at a desk 40 hours per week, if you use a little creativity.\nSide hustles provide diversified income, can help protect your finances in the event of a job loss, and can help you get ahead.\nA side hustle does not have to convert to a full-time job; in fact, there are benefits to a full-time job a side gig cannot provide.\nThere are tangible and intangible benefits to side hustling including time management, learning to prioritize, and skill building.\nLook at all facets of your circumstances (for example, time, relationships, employment, finances) before deciding to side hustle then determine what kind you should implement.\nSide hustles do not have to be permanent. They can provide a temporary boost in income for debt or savings and, once you've achieved those goals, you can let the side hustle go.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the last chapter, we looked at the elements that comprise a workflow definition and discussed how to create a workflow definition. In this chapter, we introduce the Kaleo Workflow Designer for Liferay EE which allows you to create workflow definitions using an intuitive UI. Using the workflow designer saves you the time and trouble of having to deal directly with the XML.\nDevelopers who are used to working with XML are generally able to create workflow definitions. Other users may not be so comfortable with it. In fact, even skilled developers can make mistakes that break a definition and require time to troubleshoot. To help streamline the creation of workflow definitions and empower more users to create custom workflows, Liferay provides the Kaleo Workflow Designer in Liferay 6.1 EE.\nThere are two pieces to the workflow designer: Kaleo Forms and Kaleo Designer. Kaleo Forms is an extension of the Dynamic Data Lists feature (covered in chapter 9). This enables you to create web forms and basic applications, then apply a workflow to govern the processing of those forms. Kaleo Designer is a drag and drop interface for creating new workflow definitions. It can be used in conjunction with Kaleo Forms or standalone to create workflow definitions without having to write XML.\nLet's look at Kaleo Forms first.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgejh b/data_all_eng_slimpj/shuffled/split2/finalzzzbgejh
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@@ -0,0 +1,5 @@
+{"text":"Choosing a name for your little one is such a fun part of the journey. And when it comes to deciding on a name, there are plenty of factors to consider \u2014 our favorite being the hidden meaning it! While obviously not a label of what type of person your baby will grow up to be, choosing a name based on it's meaning is an important consideration for many. So whether you want a name that's classic, trendy, unisex or anything in between, we want to help inspire you to look into the meanings behind your favorites. That's where our list of the top 20 baby names for 2017 and the hidden meanings behind them comes in! Take a look at the top names of the next generation below.\nWhich of these top baby names would you choose for your little one? Do you know the hidden meaning behind your own name?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Deciding on the best floor tiles outdoor porch is not an easy action to take since it's not only considering the looks but also the big event and many other things involved in it. Every thing that needs to be viewed could significantly influence your final decision before choosing to make use of any kind of flooring for your house.\nSo you'll need to ensure that you're willing to accomplish at the least the minimum preservation so it needs. Remember that whenever you neglect the maintenance, then it may produce a floor become ruined easily.\nNext is to think about how much traffic that could be in your community since it'll influence the flooring that you use. You'll need to ensure the floor tiles outdoor porch will have a way to handle the large traffic that's occurring in the region and it would not get split or use down also fast.\nFirst you will need to think about where you wish to mount the floor tiles outdoor porch because you have to ensure that the floor could be suited to that location. For example when you want to set up the flooring in attic as well as bathroom that has high wet then you require to ensure the flooring may withstand most of the wet in the room. Or you may also believe or perhaps a way so the floor can withstand the wet by closing it or do some therapy to it. So actually many floor types can be utilized provided that it acquired the right therapy beforehand to create it suitable to the space condition.\nIn regards to the floor tiles outdoor porch longevity, it all depends not just on the product applied to create the flooring but in addition how the flooring is used. For instance high traffic region could make the flooring get grab or use down quicker compared to reduced traffic area. Furthermore, you should consider whether you have children or even pets which will very influence the toughness of the floor used.\nOf course, you should also consider about simply how much you've in your financial allowance to get the flooring. You will need to ensure that the floor tiles outdoor porch price works along with your budget. All flooring will come in specific price range, therefore you can actually determine which price that could still match within your budget. Often the flooring cost is assessed in sq base thus you can look at to rely how many square base that you need for your house, then determine whether it's however within your allowance or not.\nThey could are also made of various products form making you've more varieties to choose from. Remember that various style could have different cost, hence it might also affect your budget so you must be cautious about it.\nSpecially since some form of flooring is made so that it can be utilized in region with large traffic while the others can not really handle too much traffic. Hence you should think about this before you even install the flooring in the room. The most effective floor tiles outdoor porch for each and every room.\nYet another thing to think about may be the installation because it could be affecting lots of things especially for your budget. If you have lots of enjoyment in your financial allowance, then don't forget that you'll require to add the floor tiles outdoor porch installation inside your budget. One thing that you might want to learn is that the flooring retailer that you utilize mightn't present installment from their internally technician.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LOUISVILLE, KY, January 10, 2014\u2014Down One Bourbon Bar and Restaurant is kicking off 2014 by raising money for local charities with a Charity Bartending Challenge. It kicks off on January 15, 2014 with Food Network Star Season 9 Winner & Host of Southern at Heart Damaris Phillips, with Feast BBQ Owner Ryan Rogers. The pair will compete to raise the most tips that will be donated to Star Duck Charities. Additionally, Down One Bourbon Bar & Restaurant will donate a percentage of sales during each of the three events to the bartenders' selected charities. The series continues on January 22nd with Mark Barnes, director of Northeast Sales at the Louisville Convention & Visitors Bureau and Garrett Tellman, Corporate Sales Manager at the Galt House Hotels; and January 29th with Robbie Valentine, a member of the 1986 National Championship University of Louisville Men's Basketball Team and 2000 Inductee to the UofL Athletics Hall of Fame, and his wife Robyn. DJ Tank will provide entertainment at all three events.\nDown One Bourbon Bar & Restaurant is currently featured in the December 2013\/January 2014 issue of National Geographic Traveler Magazine. The issue features the \"Best in the World\" and showcases a photo of the \"Get Splashed\" wall at Down One. The restaurant and bar are ranked in the Top 55 Bourbon Bars in America by the Bourbon Review, was ranked the number three Best Bourbon Bar in Kentuckiana by Louisville.com, and was voted the number three best Bourbon List in the LEO Weekly's Reader's Choice Awards in 2013.\nDown One Bourbon Bar & Restaurant opened its doors in February of 2013, next to the KFC Yum! Center on the corner of Third and Main, and has become a featured Bourbon spot along Louisville's expanding Bourbon Row. Located squarely in the middle of the planned bourbon distilleries along Main Street, Down One features 100 bourbons, rye whiskeys, local craft beer, specialty cocktails and distinctive wines by the glass. Its food menus feature flavors that are layered in an extraordinary way.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our list of the 13 best gay hookup apps (for Android and iPhone users) can . Local singles can create an account for free and start browsing.\nDiscover hot gay men nearby with the fastest-growing gay dating and social . NYC that is pretty telling in terms of scope and reach of this apps prescriber base.\nRead up on the most unique and LGBTQ inclusive dating apps out there. However, the current user pool is much larger than just gay men. queer women : user dashboards contain information on local queer events, a feed.\nAre other guys having the same experience you are on the gay dating apps? We wondered, too. So we conducted a a nationwide survey of nearly 4, gay.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ericaceous shrubs require an acid soil. This includes azalea, rhododendron, pieris and most of the flowering broadleaf evergreens. If the soil is not sufficiently acid the plants are unable to absorb iron. Symptoms of iron deficiency are yellow leaves with green veins. When severe, the leaves are yellow with brown margins. The symptoms can be alleviated by spraying the plant with iron chelate. Apply sulfur or aluminum sulfate to the soil to increase soil acidity. Use of iron chelate gives relatively quick but temporary relief. Acidifying the soil is the best solution. Where soils are not acid, use other plants rather than try to change the soil pH.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbggyt b/data_all_eng_slimpj/shuffled/split2/finalzzzbggyt
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--- /dev/null
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+{"text":"Monday's main slate features seven pitchers at $9,000 or more which should compel players to choose at least one top-tier option. We can then turn to high-upside matchups against struggling hurlers in order to roll out a lineup capable of scoring big points.\nCorey Kluber, CLE at CWS ($10,900): There may not be much need for advanced stats when analyzing this matchup, as Kluber is 3-0 with a 1.29 ERA in three starts (21 innings) against the White Sox this season. He has struck out a whopping 28 batters over that span. The Pale Hose will enter the contest sporting a bottom 10 wOBA and a 25.4 percent strikeout rate against right-handed pitching.\nCheap GPP Consideration, Felix Pena, LAA vs. TEX ($6,100): Pena is in an excellent spot to take advantage of the Rangers, who hold a .295 wOBA on the road, which ranks as the seventh-worst mark in the league. Pena has notched a K\/BB ratio of 79:27 while allowing just 10 homers in 85.2 innings this season, which should give him some decent upside against a weak offensive club.\nGPP Fade, Jameson Taillon, PIT at CHC ($10,500): Taillon put up a big score in his last outing on the strength of 11 strikeouts but players should be wary of relying on the former top prospect to consistently pump in the punchouts, as he holds a strikeout rate under 23 percent this season. It is also worth noting that the Cubs are a top-10 team in the league against right-handed pitching according to wOBA. Their 21.5 percent strikeout rate against orthodox hurlers is the eleventh lowest mark in the league.\nElias Diaz, PIT at CHC ($3,800): It's good to see that Diaz is back to full health in time for the matchup against Cole Hamels, as he has notched a .218 ISO and a .398 wOBA against lefty pitchers in 78 at-bats this season. Meanwhile, Hamels has struggled with right-handed hitters all year, as evidenced by the .451 slugging percentage and 44.8 percent hard contact rate he has allowed in 146.2 innings.\nCarlos Santana, PHI at COL ($4,500): Opposing starter Tyler Anderson seems to have rebounded from a miserable month of August but has still allowed a .450 slugging percentage in 20 innings in September. Santana has performed better against lefty pitchers in 2018, as we can see from the .212 ISO and .351 wOBA he has tallied in 151 at-bats.\nJoey Wendle, TB vs. NYY ($3,900): Wendle seems to be hitting his stride in the final month of the season, as he has posted a .194 ISO to go along with a .392 wOBA in 72 at-bats this month. Luis Severino is widely regarded as a force to be reckoned with in the Yankees' rotation but hasn't posted an ERA below 4.11 since June. While he is still capable of having fantastic games, it may be worth taking a few cheap Rays on Monday in case a blowup occurs.\nMiguel Andujar, NYY at TB ($4,200): Andujar has swung a hot bat against both sides of the platoon this season, which has resulted in a .225 ISO and a .361 wOBA in 543 at-bats. Ryan Yarbrough, who is set to take the ball following Tampa Bay opener Diego Castillo, has seen more than a 10 percent drop in his strikeout rate when facing right-handed hitters. This has contributed to a 4.59 xFIP in 101.2 innings.\nMarwin Gonzalez, HOU at TOR ($3,700): Opposing starter Marco Estrada has seen most of his struggles against same-handed opposition this season but he is due for some sharp regression against lefties, as we can see from the 5.52 xFIP he owns in 63 innings. Gonzalez has come back to earth a bit after a month of August that saw him post a .303 ISO but the 29-year-old still boasts a .370 wOBA in 51 September at-bats.\nJason Kipnis, CLE at CWS ($4,100): Kipnis is enjoying his best power month of the season in September, when he has tallied a .250 ISO in 60 at-bats. Opposing starter Dylan Covey has performed better at home this season but an elevated walk rate has contributed to a 4.69 xFIP against left-handed hitters in 60.1 innings.\nNelson Cruz, SEA vs. OAK ($4,300): Cruz has had a bit more success against lefties in 2018 but still sports a .248 ISO and a .362 wOBA against righty pitchers in 359 at-bats. Daniel Mengden has kept something of a reverse split this season, allowing a .448 slugging percentage to right-handed hitters in 60 frames.\nMatt Kemp, LAD at ARI ($3,900): Kemp has struggled through various stretches of the 2018 season but still holds a robust .244 ISO against left-handed pitchers in 176 at-bats. Robbie Ray has kept his strikeout prowess in every environment but has struggled with the home run ball at home, having allowed 14 big flies in 59.2 innings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"RAVENOL MTF-2 75W-80 is a fully synthetic gear oil formulated using the highest quality PAO base oils together with a unique additive pack, and is designed especially to meet the exacting manufacturer requirements for modern transmissions as listed below. Its low friction characteristics coupled with exceptional high load capabilities maximises both fuel economy and wear protection.\nRAVENOL MTF-2 SAE 75W-80 is ideally suited to ensure smooth and efficient operation of modern transmissions subject to very high mechanical and thermal loads of gear oils, even at the longest oil change intervals, and specifically designed for the applications as listed below.\nA high-pressure stable lubricating oil film even at high temperatures and under high stress.\nExtreme pressure stability, with excellent corrosion protection at all loads.\nA very good viscosity-temperature behaviour.\nA very good oxidation stability and very low foaming tendency.\nNo oil thickening power respectively deposits.\nCompatible with all types of sealing materials.\nvery fast delivery... great price.... ...looks good.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This January CEO and Co-founder of Enty - Sophia Matveeva - will share tips on how to work with developers and understand technology without a technical background.\nSophia started Enty - an innovative fashion tech platform, supported by AI - in a Chicago Booth classroom and the company now has customers across the UK, US and Australia.\nIn this talk Sophia will explain how to gain a better understanding of the technology your company is building and using.\nA graduate of University of Bristol and with an MBA from Chicago Booth, Sophia began her career in the media working at Finsbury before moving on to private equity firm. During her MBA, Sophia worked at a user generated news platform where she was instrumental in quadrupling audience within a year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The very last and stage that is important of focus on the expression paper may be the writing of a summary that reflects most of the key findings which the writer came in the process of preparing the paper. Before you compose an impression when you look at the term paper, you will need to read some of the guidelines, because this part often causes the best problems.\nIn conclusion is a summary of all of the work done. It offers findings that are well-reasoned all concerns put in the beginning and possesses a list of solved issues.\nThe last component is a harmonious continuation of this introduction therefore the main part, gets the exact same style together with them. Results should really be stated accurately, quickly and placed no more than 2-3 sheets of printed text, given in according to the Russian State Standard.\n1. Before composing a summary when you look at the term paper, you'll want to re-read the introduction. In conclusion may be the responses towards the concerns posed and opinion that is personal of writer. Additionally it is important to list the key actions drawn in the component that is practical of work, once again to remember the key numerical indicators.\n2. In conclusion should be stated rationally with an illustration associated with the anticipated impact following the research. It is critical to recommend means of integrating the outcome associated with the work into the sphere that is practical.\n3. It is important to indicate the objectives and tasks set in the initial stage of work, along with the main leads to that your writer has arrived.\nThe great type of the beginning of the part that is pay somene to do my online homework final the use of the text \"so\u2026\", \"summing up\u2026\", \"on the cornerstone of the work carried out\u2026\". Then your tasks, solutions of that have been achieved through the research, are detailed. You need to specify this and separately discuss the reasons if you have not succeeded in getting answers to any questions. Don't forget in regards to the nagging issues that have arisen when you look at the work as well as the techniques to overcome them.\nA significant element of the best final area could be the results of the practical component. The writer should justify their significance that is real the value of their application in training, suggest unique viewpoint into the procedures in mind and right back it with arguments.\nIt is not a bad concept to propose your very own policy for enhancing and modernizing the problem in mind, which in the foreseeable future must have a confident effect in fact.\nIt really is for several understood that the profitable part of the business could be the absolute goal of the development and development that is further. In this regard, it is vital to constantly study its sources, rational circulation and employ. The author studied various approaches to solve the problem of determining the essence of profit, the reliability of its receipt, as well as existing ways of rational use during the work. Throughout the consideration for the nagging issues, we proposed methods for moving the enterprise to a greater standard of revenue removal.\nThen it's better to indicate particular actions that are rational raise the organization's earnings and numerical indicators that confirm their effectiveness.\nIn general, it's not so very hard to draw up a conclusion that is competent since it seems at first. Conclusion could be the rational conclusion associated with work done. Consequently, you must not be scared of composing it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Go green at home. It sounds simple. But, is it? The answer is an overwhelming yes! To help you get started, we've pulled together tips for turning everyday homes into energy-efficient homes. Read on for tips on identifying hidden power snatchers, choosing the right appliances, maintaining heating and cooling equipment and more.\nUnplug what you can, and then use power strips to toggle groups of electronics off when you're not around. Research shows even small adjustments can impact your electric bill.\nAppliances account for about 12 percent of a monthly energy bill. Old appliances are notorious energy bandits. Look for appliances with the ENERGY STAR\u00ae logo. These appliances use 10\u201350 percent less energy and water.\nSwitch to LED bulbs. They use about 75 percent less energy than standard incandescent bulbs and have a lifespan up to 25 times longer. In fact, replacing a 60-watt incandescent bulb with a 12-watt LED bulb may help save about $260 over the life of the bulb.\nClean the air filter monthly when you use your furnace during extreme heating or cooling.\nInvest in a new furnace. Furnaces with the ENERGY STAR label are about 25 percent more efficient than furnaces produced as recently as the 1980s.\nIf you're using the A\/C, close curtains and shades on the east and west windows and delay heat generating activities (e.g. using dryers and dishwashing) until the sun goes down.\nInstall a low-flow toilet to save 1.9 gallons per flush compared to a pre-1994 toilet.\nWash only full loads of laundry and dishes to save up to 1,000 gallons of water per month.\nLandscape with low water-use plants to help save up to 550 gallons of water annually.\nRegardless of what methods you try, making your home even a little greener can have a big impact on your monthly utility bills. The best news is that you don't have to do it all to realize some benefits. And, that's the most rewarding part of creating more energy efficient homes.\n\"100 Ways To Conserve.\" Water Conservation Tips, Facts and Resources. N.p., n.d. Web. 16 June 2016. .06\/20\/12-household-appliances-you-should-unplug-to-save-money.\n\"Are Energy Vampires Sucking You Dry?\" Energy.gov. Energy.gov, 29 October 2015. .\n\"Where Does My Money Go?\" EnergyStar.com. N.p., n.d. Web. 16 June 2016. https:\/\/www.energystar.gov.\nJohnson, Holly. \"Light Bulb Showdown: LED vs. CFL vs. Incandescent.\" TheSimpleDollar.com, 29 April 2015.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgmbs b/data_all_eng_slimpj/shuffled/split2/finalzzzbgmbs
new file mode 100644
index 0000000000000000000000000000000000000000..181a6bf58b5e47e586a9e87b6e54850ac42aa824
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+{"text":"Endurance Center of Vocational Empowerment (ECOVE) located in Waluj, Aurangabad, trains young people in computers, retail management, tailoring and electrical works. We also have sessions on life skills and spoken English to help them improve communication and people skills.\nAfter the training, we help them find jobs, prepare for interviews, make effective CVs and even start their own enterprises. In its first quarter of operations (September-November 2015), ECOVE trained 76 people and placed 51 of them in jobs.\nECOVE is equipped to train 80 students at a time. We use AV teaching aids such as projectors, films, computers and posters to make learning interactive and engaging. Unlike most of our vocational training centers in India, ECOVE has daylong classes (eight hours on weekdays and four hours on Saturdays) to give students an intensive and comprehensive training experience. Students also take part in extracurricular activities such as sports and yoga. We also provide breakfast, lunch and tea.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A user with an early copy of Assassin's Creed Odyssey took to the PSN Profiles forum to share the Trophy list for the game. If you couldn't care less about trophies, meticulously comb through your games to get them, or land somewhere in between, here's the Trophy list you'll be facing when Assassin's Creed Odyssey releases on October 5, 2018 for the PlayStation 4, Xbox One, and PC.\nBelow are all 51 Trophies you'll earn (or not). There is one Platinum, two Gold ones, ten Silver Trophies, and thirty-eight Bronze ones, for a total of 1230 points.\nA lot of these will come naturally through game progression, but a few involve sidequests and leveling up. Will Assassin's Creed Odyssey bring out the completionist in you? Let us know in the comments below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Governor recently signed SF 2175 into law which replaces Iowa Code Chapter 651 which deals with Partition of Real Estate. The biggest change in the new law is the establishment of a partition procedure for \"heirs property\". For the first time a new law allows the equitable remedy of \"owelty\" which provides for a payment of money to equalize the value of property received in a partition in kind action. Under the new Chapter, an individual who inherits property as a tenant in common with other relatives and who does not want to sell the property, in all likelihood would not be required to do so. It appears that a Court's remedy in such situations will be a partition in kind whereby a portion of the real estate will be deeded to each of the heirs or the party who does not want to sell the real estate will be required to buyout the other owners. The new law was effective July 1, 2018. The new law reverses the old law which favored partition by sale.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The intersection \u2014 where man meets machine \u2014 is the subject of \"Object Code,\" a new exhibit opening today in the Visual Arts Gallery at the University of Illinois Springfield.\nArtist Mike Miller takes the title literally, turning ephemeral fragments of computer language into physical artifacts of paper, plastic and wood.\nA reception marking the opening of \"Object Code\" is scheduled for 5:30 to 8 p.m. today in the Visual Arts Gallery, in Room 201 of the UIS Health and Sciences Building.\nMiller will also deliver a lecture on his research at noon Wednesday in Rooms C\/D of the UIS Public Affairs Center.\nRegular gallery hours are 11 a.m. to 6 p.m. Monday through Thursday. \"Object Code\" closes Dec. 1.\nThe exhibit was featured in a November 4, 2010, article in The State Journal-Register.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Good Samaritan Fund - above and beyond event based or general contributions.\nIndividuals, organizations and corporations are invited to join the Buddy Society. Membership is granted upon receipt of a signed Commitment Form indicating the intent to contribute a minimum of five thousand dollars over a period of five years to the Good Samaritan Fund.\nThis is a non-binding commitment and is not a legal contract. Should the donor choose to make the pledge obligatory, this can be noted. Circumstances that prevent the fulfillment of the intent will warrant no further obligation.\nRecognition is conducted on an annual basis. New members are presented a Buddy Society pin.\nas a way to thank members and encourage others to join the effort.\nThe Goal of the Buddy Society is to recruit and sustain over 100 active members.\nto the ongoing care of their friends and neighbors.\nor call the Development Office at (717) 838-5406 ext. 3060.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I think I need to separate my word counts into different categories so I can get a good idea of how I'm dividing my time these days. If I could count comments on other people's blogs towards my word count goal, I'd probably make it in no time! Perhaps I should? I've decided not to count interview questions in the tally. Partly because some are pre-written and I just choose which to copy and paste, and partly because the ones I don't cut and paste from previous interviews require a lot of time researching and thinking just for a ten word question. I will count blog posts, even though some (like SyFy Q of the Day) are mostly cut and paste from earlier works.\nThen there's my WIP. This should mean About Damn Time, the novel that is intended to be first in the Kingdom Come series. I have two jobs for this: finish writing the thing, and edit\/revise the first 30 pages. I have a reason for doing these things at the same time. You see, I won not just one, but three critiques! I have the first back already, and the second critter offered to wait till I revised the pages and she'd do her crit of the revised pages. This is like winning the lottery for a pre-published writer! On top of that, I had bid in a charity auction on a 30 page crit by another author, and won. She's waiting patiently while I do my other two crit\/revision cycles. I'm in another contest (random drawing) for yet another crit, so this could go on!\nSo\u2026 what have I done this week? Thinking back to Monday, I wrote my initial post for #ROW80. It was 747 words. I also began a random blog post to be used at a later date, which was 346 words.\nOn Tuesday I did 254 words in #5MinuteFiction on Leah Petersen's blog. Technically, my 254 words were a blog comment, but I'm counting them because that's how we submit our flash fiction. I also wrote a memoir for The Red Dress Club that is 739 words. Or did I write it on Monday? I think I started Monday and finished Tuesday, but it doesn't matter. I also posted a SyFy Q of the Day, which was 619 words. I decided to go ahead and count the SyFy Q of the Day posts, even though they are mostly copied and pasted from earlier work. It does take time and effort to put it together, so it's more like revising than writing, but for my #ROW80 goals, that counts. Late night Tuesday, Liz Silver offered up #1k1hr for anyone interested, and I added 1,309 words to my actual WIP. I skipped ahead to a new scene, but that made it easier. It's been a few weeks since I've touched it, I'm sorry to say. Sometimes, when I'm stuck, or I haven't had time to write and I'm out of the story, it helps to just skip ahead. I can fix it in edits later if I don't get back to the gap.\nAnd then there's all the writing-related stuff I'm not counting towards my goal. I finished the interview questions for JK LeBlanc, but I did not finish the questions for Liz Silver. I have proofreading due on Friday, but although I started it on Sunday, I haven't touched it for a few days. I need to get on that. I haven't done any revising, but I messaged my next critter and she said I can send it this weekend and she'll get to it next week, so I need to work on that too.\nWow! All day yesterday I felt like I wasn't making much progress at all, but I guess when I go back and count it all up, I did quite well! I'm happy. That late night #1k1hr really makes a difference too. I hope everyone else had a really good start to #ROW80 Round Three as well.\nIt's amazing how much writing we really do when you put it that way. So many writers advocate against skipping ahead that I feel like I'm doing something wrong when I do it. So glad it works for you! And congrats on winning all those critiques. That's really amazing.\nSkipping ahead doesn't always work *well*, but it does get things moving again. If I had my way, I'd have 5 days sequestered in a 5 star hotel where I could write uninterrupted, and I could crank out a rough draft.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At least one definition of narcissism is stated as \"Extreme selfishness, with a grandiose view of one's own talents and a craving for admiration.\" Or see the main character in Young Adult, Mavis Gary (Charlize Theron). She is the walking definition.\nMavis writes young adult fiction. Throughout the film she struggles to write one last book in a series she is the ghost writer for. It becomes clear that Mavis is not so much writing about a teenage character set in a different world. She writes what she knows and what she knows, or at least thinks she knows, is herself. There is an interesting play of meta narrative going on whenever the author sits with her laptop and types some new prose.\nAfter she receives an email from her old boyfriend Buddy Slade (Patrick Wilson) announcing the birth of his new baby, Mavis becomes obsessed with Buddy. She is so consumed with her self that she can only see one reason she received the email \u2013 to make her jealous. On a spur of the moment, after viewing the baby's picture one too many times, the self-obsessed writer throws some clothes in a bag, shoves her dog in a small case and makes her way back to her small Minnesota town, leaving Minneapolis behind. Her mission is a simple one. She wants to have Buddy for herself once again. He's not happy without her, at least that's what Mavis would have everyone who will listen believe.\nOne of those Mavis confesses her twisted plot with is Matt (Patton Oswalt), who graduated in Mavis' class. Matt uses a crutch to get around. In high school he was jumped by some fellow classmates and nearly beaten to death. He tells Mavis that he got a lot of attention and sympathy from all around the world when it was thought that he was gay. But once Matt made it clear that he wasn't gay, the sympathy and attention dissipated almost overnight. He quips that the act was heinous when it was a hate crime but not quite so bad when it was some jocks beating a fat kid with a crow bar. Matt is honest, sometimes painfully so. He doesn't attempt to elicit sympathy for his plight. He seems to cynically accept his position in life.\nMatt's penchant for telling the truth serves, at least at first, as Mavis' missing conscience. He is vocal in his opposition to Mavis' plan to steal Buddy away from a happy marriage with a new baby. He even goes out of his way to run interference when Buddy and Mavis meet for drinks at the bar Matt works at. Buddy seems naive in all of this. His happy-go-lucky attitude and clear devotion for his wife and child are juxtaposed up against his former high school girlfriend's egocentricity.\nThe journey to destroy a marriage and fulfill the selfish desires of a despicable character does not sound fun but the way that the screenplay writer, Diablo Cody (most famous for writing Juno), positions her extremely narcissistic main character with a backdrop of decent human beings makes things fun. We can never cheer for our main character. Her plans and her ways are never worth cheering for. But we can laugh at the absurdity of her behavior, realizing that when we give into our own notions of self-importance, our vanity, we transform into creatures not unlike Mavis. And in the end, Young Adult serves as a warning for those of us who might think we're far away from ever being like Mavis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To refer a potential customer, please complete this form below.\nYou are eligible for a referral award only when your referral is new to Kelley Communications and\/or no contact between Kelley Communications and your customer is the last six months.\nIf your referral installs and pays their invoice in full, you will receive a referral award of 3% of the total amount invoiced.\nOnly one referral award can be given per customer. If a customer is referred by more than one partner,the first referral received will be the one rewarded.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sometimes, even big men cry.\nAccording to reports, Miami Heat's Dwyane Wade was caught getting emotional and shedding a tear during the national anthem before their 7th game against the Charlotte Hornets.\nWade dismissed the idea that he was crying, saying he was \"just focused\". Who knows, maybe it was his focus (or tears) that won them the game.\nMiami Heat won 106-73 to send Charlotte home.\nStill. nothing beats MJ's crying meme.\nDo you think Dwyane Wade was crying?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In stratigraphy, the study of astronomical climate forcing (i. e. Milankovitch cycles) and the application of cyclostratigraphy and astrochronology experienced a spectacular growth over the last decades. Using different \"proxies\" from sediments and their cyclic pacing forced by orbital changes in insolation distribution makes it possible to refine the chronology of depositional sequences, all across the Phanerozoic and even beyond. The identification of the effect of orbital forcing also significantly contributes to paleoenvironmental and paleoclimatic reconstructions. Recent advances led to refining the duration of chronostratigraphic stages or the more precise timing of events and rates of processes during ancient global changes. With this session we intend to provide a platform for contributions related to cyclostratigraphy and astronomical climate forcing, also as parts of integrated stratigraphy combining information from several geochronometric methods.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..ee319810aade907fab1b041567f16abc6ff1f5ce
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+{"text":"We at AdoptTogether, alongside our adoption consultant Chloe Briggs, have compiled what we hope to be a helpful list of agencies that specialize in domestic adoption (in no particular order).\nDISCLAIMER: This list is in no way a comprehensive list of all the good adoption agencies in the US. There are plenty of great agencies out there that are not on our list and we kept it to the top 10 simply for the sake of providing parents with a concise list.\nWe made our selections based on our research of each agency's reputation, efficiency, and reviews from adoptive parents. Each agency is COA & Hague approved. Best of luck to you and your family as you find the best fit for your needs!\nCon: Due to their corporate structure and size, interactions can be less personal than other smaller agencies.\n6. Heart of Adoption, Inc.\nAgency Description: Spence-Chapin's roots can be traced to the early 1900s and the pioneering work of Clara Spence, and Dr. and Mrs. Henry Chapin, who independently established nurseries out of concern for homeless infants abandoned in hospitals and shelters. The Spence and Chapin nurseries each broke new ground in developing social work techniques for adoption and, after the merger, continued to pioneer in the adoption field. Today, Spence-Chapin is proud of our role as a prominent voice and leading advocate for adoption, and of our commitment to the well being of all members of the adoption triad: birth parents, adoptive parents, and their children.\nAgency Description: Ruth and Richard Barker, themselves adoptive parents, founded the agency in Washington, D.C., in 1945 in response to a request from the U.S. Navy Department to address the needs of pregnant WAVES (\"Women Accepted for Volunteer Emergency Service\"). The Barker structure was formalized in 1951 when it was incorporated as the nation's first cooperative adoption agency. To date Barker has placed more than 7,000 children into the welcoming arms of adoptive parents and provided counseling and related support services to more than 30,000 women and their families who are considering placing a child for adoption.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Maytag Bravos dryer error codes are shown on your dryers display to help you troubleshoot the problem with your dryer. Our error codes below will help you to identify what is wrong with your Maytag Bravos dryer. Once you have identified the error code, this will tell you which part is causing the issue. You can then check the part in question and clean or replace it if needed. Maytag Bravos dryer help videos are here if needed.\nCheck or Repair Needed = Remove power from the dryer. Use a multi-meter and check the resistance of the thermistor (0,000 ohms). If the thermistor is bad then replace it.\nCheck or Repair Needed = Remove power from the dryer. Check wiring connections between the main electronic control board and the drive motor. If the wiring connections are not damaged and connected then replace the drive motor.\nIf your Maytag Bravos Dryer error code is not listed here or you have issues with any Maytag Bravos Dryer Error Code, please leave a comment below and we will be happy to assist you with your dryer error code problem. If you have a different Maytag Dryer, you can ask error code questions about that model dryer also and we can further assist.\nWe have a Maytag Bravos that is not showing an error code but still doesn't work. When turned on as usual, by pressing power, selecting the dryer setting (right now it's a 24 minute rapid dry), pressing start and the dryer beeps when I press the button but will not spin. The light above \"sensing\" is lit and the timer counts down as if it's running a normal load, but again it doesn't spin. We've checked both buttons in the door lining and they're fine and working. Any advice is greatly appreciated since it's Saturday and we have work and school on Monday.\nMy power went out briefly the other day. My dryer flashed what I believe was F73 exactly when the power returned. I have not been able to reproduce the code. My timer will not come on unless I turn the dial to steam. The dryer will not come on but all other lights and noises work. Any ideas?\nMy bravos dryer when started stays on \"sensing\" for 30 seconds. It then repeatedly flashes a \"F\" then a \"1\" but will not run. The error codes listed in my owner's manual and on the website identify a solid \"F01\" error code, but not an alternating \"F\" and \"1\". Are they the same?\nWe have the Bravos XL washer\/dryer set, and our dryer is giving us from time to time codes e03\/f03 alternating, and sometimes will not dry a load, but can restart and will continue to dry. Cannot find any info on these codes, other than maytag says when e code followed by f code to contact a maytag service person. Any help will be appreciated.\nHave a Maytag Bravos XL dryer that won't run.\nI can't figure out how to find the error codes.\nHow do you get them to be displayed?\nI own a Maytag MVWB800VQ0 washer (8 years old).\nWhen I go into the diagnostic mode using the 3 seconds on 3 seconds off, 3 seconds on, 3 seconds off, 3 seconds on method, and after all the lights come on and the 88 appears, I get an error code that was stored in memory.\nThe error was fixed and the machine runs fine now, but when I go into the diagnostic mode the error is still showing up.\nUnplugging the machine for 1\/2 hour did not erase the stored error code.\nWhat button sequence do I need to use to erase all old stored error codes?\nHow do I clear the stored error code in memory on this machine?\nCan you tell me the Error code F65 on Maytag Bravo 300 series?\nOur Maytag Bravos XL has not given us a code.\nIt just has three orange dashes across the display window on half of the cycles.\nI is working on the others but the times seem off.\nNot sure how to re-set it as there is nothing in the manual.\nMy dryer never quit so when I checked it was blinking 1:39 and clothes still wet. Smelled like it was getting hot and smelled moist.\nMy son pulled it out, hose was clear. Will replace hose and hook back up, kind of cracked and pulled apart when checking it out.\nNot the typical codes listed so not sure. Will replace hose, hook back up and try it unless you have some suggestions.\nWe have a Maytag dryer that is an older model, and the problem that we started having with it was that it didn't heat the drum, so we took upon ourselves to change the thermal fuse. We replaced it, the drum cannot turn so now we're having that issue.\nOn my Maytag Bravos Dryer, I get an error code F:AO then CO for the second code. Dryer will not start. Can't find these codes anywhere. What are they? Can I clear the codes\/reset the dryer and try again?\nThe error codes F:AO then CO you have on your Maytag Bravos Dryer are not common.\nCan you please supply us with the exact model number of your dryer so we may research this issue further?\nPrevious post: How To Tell If Your Electric Hot Water Heater Is Bad?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Great Fechteler Upholstered Dining Chair ByIvy Bronx is essential have in every property. You need to find the best sections, and you want to successfully never ever pay too much to them. Appears to be a little bit complicated, appropriate? Nicely, this post is right here to help. Read on and find some great skilled methods for choosing the bargains on home furniture parts you will adore. Before choosing any cabinets, open up each of the storage and check on the inside. You're not just making sure that all of the drawers are built to very last and open up without having hitches, you also want to ensure that the inner of your storage have some type of concluding as well. When searching for Fechteler Upholstered Dining Chair ByIvy Bronx, evaluate your bed room before going purchasing. There is nothing at all more serious than picking out the perfect list of Fechteler Upholstered Dining Chair ByIvy Bronx only to find that it will not fit in your room after you get it house. As an alternative, correctly determine your rooms measurements prior to going remembering where by doorway and windows are placed.\nPrinted on ( ) By Muhammad Nasir Aziz On October 01, 2018 All of us love sitting within the garden and spend some quality time with ourselves or closed ones. This makes it important to have beautiful furnishings there which additionally depicts our personality and temper. Like for someone who prefers summers rather a lot, will want to go for daring yellow somewhat than neutral grey. Patio furnishings is the Fechteler Upholstered Dining Chair ByIvy Bronx which is designed for retaining outdoors and different out of doors functions. Subsequently, it turns into necessary to make them using climate-resistant materials which can survive the adversarial conditions of nature and prevents them from being corroded. One of the most used metals for this objective is aluminum. There are completely different furnishings varieties which can be found like Bamboo furnishings, Picket Fechteler Upholstered Dining Chair ByIvy Bronx, Metal furnishings, Wicker or rattan Fechteler Upholstered Dining Chair ByIvy Bronx, Glass Fechteler Upholstered Dining Chair ByIvy Bronx, Plastic furnishings or Concrete Fechteler Upholstered Dining Chair ByIvy Bronx. The patio Fechteler Upholstered Dining Chair ByIvy Bronx is generally fabricated from wood, plastic, aluminum, wrought iron, and wicker. Yow will discover plenty of them in numerous varieties in rattan Fechteler Upholstered Dining Chair ByIvy Bronx UK But, it turns into important to care for out of doors furnishings and get them cleaned once in a while. With different types, the cleansing course of differs. Allow us to see all of them one by one. Teak and Wicker Patio Furnishings As teak is fairly durable and would not take a lot money and time to get it cleaned and maintained, it could be among the best options. You can use pure teak oil to get its again to its authentic form. Teak is immune to mildew and decay, but when not taken care of there can be mildew formation on the floor of the furnishings. To ensure that the Fechteler Upholstered Dining Chair ByIvy Bronx is to lose its luster, not getting discolored or there is no mildew formation, it should be cleaned often. Allow us to see some selfmade options: 1) You want a cup of vinegar and a gallon of heat water or laundry detergent, chlorine bleach and heat water to wash the furnishings. Take a plastic brush and dip it in the cleaning solution and get it cleaned. Don't be too harsh and don't use any metal brush as it would bruise the floor and harm it. It also helps taking good care of water stains. 2) If the furnishings is exposed to harsh elements for a longer time, you have to buy an aggressive resolution which is present or purchase it from near local shops to wash the items. However, utilizing an excessive amount of of energy wash isn't good as the softwood is damaged which is not encouraged. Additionally, it's essential learn all the directions carefully as the answer would possibly need to be diluted. Rattan Patio Furniture This eco-friendly material is desirable, sturdy and light-weight. It appears somewhat much like bamboo and is vine-like palm that stretches within the jungles of Asia, Malaysia, and so on. However, you don't want your patio furnishings to be al spoilt, so we've this magic serum that may help you: 1) In a bowl of water, put few drops of the cleaning soap used for cleaning dishes. Hold stirring it, until you see bubbles on the surface. You possibly can dip a tender material within the bubbles to make sure that there isn't any direct contact with the cleaning solution. You possibly can wipe any part of furnishings utilizing this solution. 2) Do not wet the fabric too much because it will get damaged. There will be a number of cracks within the furnishings, and it needs to be cleaned using a toothbrush or some smooth brush. three) You can also include the detergent to the tender brush and scrub it utilizing extra water and make it clean. Once, it's completed, it's essential to dry them in the solar or sue some hair dryer to do so. To ensure no fungi are settling it is advisable to give a skinny coat of lacquer once it is dried. Wrought Iron or Metallic Patio Furnishings A lot of patio furnishings is made with wrought iron, forged iron or even stainless-steel. It is easy to wash them as you possibly can add one-fourth of the mild dishwashing cleaning soap to heat water and use it as a cleansing answer. The first problem of using metallic furnishings is rust, so it must be coated or painted after sanding the areas using sandpaper in order that there is no such thing as a oxidation and lasts long. You can too use a scrub brush if some harsh stain is to be eliminated. Plastic Patio Furnishings There are lots of benefits prefer it will not ever rust and is of low cost and maintenance. The only drawback is it fades, and if exposed to heat, it becomes brittle. To clean it, you can either combine half a cup of baking soda with three tablespoons of dishwashing detergent and add them in a gallon of heat water. You too can add a cup of white vinegar to 5-7 cups of water, and you are good to go. We've got discussed easy methods to maintain your outdoor patio furnishings. Once you purchase expensive merchandise, it's necessary to maintain them as well, so ensure you handle them. Creator's Bio: I am knowledgeable journalist with greater than four years expertise on writing totally different topics. I love to jot down in the daylight and browse in the night.\nI was hunting for a gotten back wood coffee table for a while when I stumbled across one that was Ideal for the house and incredibly affordable. I was sure it had been too great to be real! I am so glad to become wrong!!! Wood Fields Woodworks is a company I will suggest to my family and friends! So very happy with the standard and workmanship! Thank you so much!\nLove this place - we're creating a new home and have visited a lot of places. We walked in and were helped by Austin texas and we're happy with the choices! He could help us totally customize our coach to the precise measurements we would have liked and also, since delivery was included we were able to purchase a couple much more Products we wanted. Overall we went with this location because of so many other options we'd and therefore are happy about it. Great support and great products!\nThe service here is amazing. I needed a home furniture as soon as possible and I come with an extremely hectic agenda. I discovered this area on Craig's list and offered them a call. Being at work, Derek called me back and I inquired about a collection and told him I didn't have time to get in shop for that buy. He soon started text messaging me images of home furniture frames, chests, desks and desks. We lastly decided on a set and he refined every thing. My furniture was delivered exactly as planned and put together as well. The delivery man was prompt and service was superb. The caliber of the was surprisingly outstanding for the cost. The house publish will go above and beyond support! I'll be suggesting this business to friends and family.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"<b>The Scout<\/b> by Joe Velazquez <br> A scout employed by a wagon train rides ahead on the Oregon Trail. He rides a Spanish Barb horse and has a black powder rifle in his scabbard. The gourd canteen was popular in the Southwest, which is probably where he is from. Prints are available from the artist. Visit <a href=\"http:\/\/joevelazquez.com\" target=\"_blank\">joevelazquez.com<\/a> or call 303-770-5639.\n<b>Desert Elk Bulls:<\/b> The Red Desert has the largest population of these animals. Despite that, they are hard to find. To see these elk was a rare treat.\n<b>Pronghorn:<\/b> With their keen eyesight and hearing, pronghorn are very difficult to sneak up on.\nThe Hayden Geological Survey of 1871 explored the region of northwestern Wyoming that later became Yellowstone National Park.\nLeaving Cody, Wyoming, I drove south to the town of Lander, Wyoming, a place that would be my home for the next few days. What drew me to this area was its rich human history for both Native American and Euro-American settlers. Another aspect that seemed to draw me in was the Red Desert, a 6-million-acre piece of natural beauty that is controlled by the Bureau of Land Management (BLM). I could hardly wait to get out there and explore.\nHistorically, human impact in this region has been going on for thousands of years. The Red Desert was, and still is, significant to both the Shoshone and the Ute peoples. The Red Desert provided food and other resources for these people, and it was considered a sacred area. To learn more about the Shoshone people, I made a trip to the Wind River Indian Reservation, which is located next to the town of Lander. While that is the source of other articles, I would recommend visitors to this area to make the trip.\nBesides Native Americans, mountain men like Jedediah Smith traversed this desert, living a life where a person survived by their wits and skills alone. These mountain men opened the doors to the next wave of Euro-Americans to visit the Red Desert area\u2014the wagon trails heading West. The famous (or infamous, depending how you look at it) Oregon Trail made its way right across this unforgiving land of buttes, dunes, sagebrush and rocky pinnacles.\nThe Oregon Trail was a route that ran about 2,200 miles, starting in Missouri and ending in Oregon and California. Starting in 1843 (and ending in 1869 when the transcontinental railroad was completed), over 500,000 people made the journey in covered wagons pulled by oxen and mules. Carrying everything they had in their wagons, the trip took anywhere from four to six months to complete, with one out of every 10 people dying along the way. Because the high-desert ecosystem takes a long time to heal from the scars left by humans, there are places here where people can still follow the original Oregon Trail, as the ruts left behind by those wagons can still be seen.\nUpon arriving in Lander, and after checking into my hotel, I made a call to Glen Hester of Real Wyoming Adventures (wyomingtourism.org; info@realwyomingadventures.com; 307-332-5997). I had made arrangements for Glen to pick me up the next morning and take me out into the Red Desert. At 7:00 sharp, Glen was there, and within a half-hour we were off the main road and onto state and federal land. The environment and wildlife found here is vastly different from what I am used to back in New Hampshire. To my eyes everything looked the same, and without Glen as my guide it would have been easy to get turned around out there.\nDespite what many people picture when the word \"desert\" is used, the Red Desert abounds with wildlife of all types. It is home to wild horses, mule deer, desert elk, pronghorn antelope, mountain lions, bobcats, coyote, badgers and other mammals. Bird species include golden eagles, sage grouse, red-tailed hawks and a host of smaller birds.\nWe weren't long in the desert when we spotted our first mule deer. It was a doe, and we had spotted her from quite a distance using a spotting scope. Glen wound his 4\u00d74 down a series of trails until we got fairly close. He stopped and let me out so I could begin a stalk that would get me a little closer. This beautiful animal allowed me to get close enough to get a few good photos. Antelope were everywhere. Most of the time we spotted them from a pretty good distance; at other times we happened upon them as they rested in the little cover offered by the sagebrush. With their keen eyesight, they often were able to spot us long before we saw them. Sage grouse, golden eagles and badgers were also readily seen, often very close up.\nWhile we were following a small herd of desert elk (the Red Desert is home to the largest population of these animals) we found the wild horses, or rather they found us. I looked up at the ridge that rose above us and there they were, looking down at us. It looked like a scene from one of those old westerns. BLM regulations require that all motor vehicles stay on established \"roads,\" which are actually tire tracks in the desert. Following these roads, we made our way to the area where we thought the horses were.\nSoon the rest of the herd, somewhere around 300 animals, came into view. Stopping our vehicle, we allowed the horses to settle down. It didn't take long before their curiosity got the better of them. Slowly, in groups of five or six, the horses came closer. Mares with foals were the first to come by. When the stallions saw that we were no threat, they too made their way in our direction, though they kept a watchful eye on us all of the time. To be able to witness this sight was well worth the time spent getting to this point.\nThe horses we see in the wild today are the descendants of horses brought here by the Spanish in the 1500s and 1600s. As other settlers moved West, they too brought different breeds of horses with them. Some of these horses were riding horses, while others were draft animals. Some of these horses escaped, were stolen by native people or turned loose by their owners. Either way, they soon joined up with the horses that were running wild. These horses of mixed blood are what we see today in the West.\nIt had been a long day and it was close to dark when we finally pulled into Lander. I was hot, tired and dirty, but I must have had a smile on my face a mile wide when Glen dropped me off at my hotel. What I had experienced was worth every minute. There was never a dull moment. The Red Desert is a wonderful place and I plan on visiting again someday.\nLearn ways you can combat the threats of a nuclear winter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hunters bring down pheasants during the fifth annual Heroes Tribute Hunt at the McKeans Pheasant Farm.\nRODNEY \u2014 Veterans once again gathered for the fifth annual Heroes Tribute Hunt Saturday at the Elks Farm on Ohio 588 before taking a short trip to the McKeans Pheasant Farm for a day of hunting and fellowship.\nThe event officially ran from 8 a.m. to 4 p.m.\n\"We run three fields and we have at least four hunts per field,\" said Slone.\nFour pheasants were released per hunter to potentially take home. Among some of the dogs present included German Shorthaired Pointers and German Wirehaired Pointers.\nThe Gallia Soil and Water District along with the Gallia Conservation Club were on hand to discuss exhibits and activities along with the National Archery in the Schools Program and the Passport to Fishing Program. Gallia Shooting Aces 4-H was present to help with trap shooting activities. The Gallipolis Elks Lodge lent the use of its facilities on the farm.\nDogs reportedly are trained to point the bird out and a hunter is intended to prompt it to fly before the animal is taken. Hunters were encouraged to shoot only once the birds were in the air for the sake of safety. Reportedly, a variety of shotgun gauges were used in the hunts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Smile Like You Mean it: Classic.\nHe needs spoiling! Miss this little guy, any picture of Luke, just goes strait to the heart; he can do no wrong until age 8!\nWe very excited about next weekend, the only sad thing is we have to wait until end of next month to get one our very best treasures, and that is see our grandson!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"By Peter Hartlaub, San Francisco Chronicle June 15, 2017 Updated: June 16, 2017 2:43 p.m.\nIf video stores were still a thing in 2017, \"Rough Night\" could fit in either comedy or the fantasy\/science fiction section.\nThe R-rated bachelorette party misadventure imagines an alternate reality where Scarlett Johansson is the awkward nerdy friend, and a house full of college-educated adults have no idea how DNA evidence works.\n\"Rough Night\" demands maximum suspension of disbelief, and looks as if it was significantly more fun for the actors to make than for audiences to watch. But it does yield rewards, with heavy contributions from the makers of \"Broad City\" and an incredibly deep comic cast. Whenever the story goes off the rails, there's always someone funny there to will it back into a humorous place.\nExample: The groom (Paul W. Downs), having a much less wild bachelor party, becomes concerned with his wife's drug-fueled mania. In most adult party comedies, this would be a thankless one-note role taking place entirely on the other end of a phone. In \"Rough Night,\" Downs is surrounded by superb comedians including Hasan Minhaj and Bo Burnham. The groom subplot is among the best things in the film.\nJohansson portrays Jess, a detail-oriented political candidate who reluctantly goes to her bachelorette party in Miami \u2014 a trip so in sync with \"The Hangover\" films that you assume her friends used those movies as a template for planning. Events head in a \"Very Bad Things\"\/\"Weekend at Bernie's\" direction quickly, as the hard-partying friends have to cover up an escalating series of felonies.\nJohansson seems diluted in the lead role, as if she's doing an impression of Kate McKinnon's impression of Hillary Clinton. This is even more noticeable because McKinnon is in the film, playing a heavily accented Australian. There's a strong sketch comedy vibe, as if everyone in \"Rough Night\" decided on their character four seconds before the cameras began rolling.\nDirector Lucia Aniello, a writer and producer on Comedy Central's \"Broad City,\" fares well with individual scenes but has trouble crafting a cohesive narrative. A heavy early tonal shift involving a dead male stripper is a low point. Rather than handling the moment seriously, or going for broke with tasteless laughs, Aniello and the writers seem to play it down the awkward middle.\nBut whenever \"Rough Night\" starts to sober up, a new player comes along to get the party started again. Jillian Bell is excellent as the insecure friend who is a little too invested in the festivities, managing to elicit sympathy even though she's an inarguably horrible person. McKinnon's character is completely unnecessary to the plot, but she still accounts for a solid third of the best jokes. Demi Moore adds to her legend as half of an older swinging couple with Ty Burrell.\n\"Rough Night\" is the kind of film that the gawky and dorky Johannson\/Jess character would hate. If you get stressed out when people make dumb decisions on the big screen, this one could put you over the edge.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In order to create your account we need you to provide your email address.\n\"I recommend people stay away from the site.\" A throwback to Internet's early days In many ways, Chatroulette is reminiscent of the early days of the Internet, and that is part of its appeal, said Amanda Lenhart, director of the Pew Internet & American Life Project's research on teens, children and families.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Listen to all music and sounds of SEVG\u0130 TREN\u0130 and follow this user by signing in for free. You can listen to 48 tracks and mixes and follow this user as 19 did before. This artist liked 5 tracks, followed 19 others and reshared 4 tracks and mixes. 1 other people are following this page too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TRAC.fm - HUNTER CM 3.3: What would stop you from seeking medical treatment for a sexually transmitted disease?\nThis poll has completed. It ran from April 6, 2017, 5:26 p.m. through May 2, 2017, 10:09 a.m.\nHUNTER CM 3.3: What would stop you from seeking medical treatment for a sexually transmitted disease?\nGo to messages on your phone & type HUNTER (SPACE) GENDER (M OR F) (SPACE) ANSWER (a, b, c, d or e) and Send your message to 8585. This SMS is totally FREE of charge.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"\"Sigmund Freud believed that sublimation was a sign of maturity and civilization, allowing people to function adequately in culturally acceptable ways.\"\nSublimation is what makes it possible for higher psychical activities \u2013 scientific, artistic or ideological \u2013 to play such an important part in civilized life.\nSigmund Freud believed that sublimation was a sign of maturity and civilization, allowing people to function adequately in culturally acceptable ways. He defined sublimation as the process of deflecting sexual drives into acts of higher social value and virtue.\nFrom psychoanalytical perspective it's the fundamental mechanism that permits and drives the creation of art or inventions.\nWe don't necessarily have to agree with Freud on the statement that every lively impulse has a sexual source, to see that transforming our energy into creation is probably the most humane thing to do. The important part is that we want to LIVE!!! Wildly, freely, without judgement, shame or anxiousness! We want to feel alive!\nIf you don't suffer from deep depression, and even then, you sometimes will want to freak free from your restrains, scream and shout, dance, kiss a stranger, go with the flow and let this energy BE! But suddenly you realize that you don't feel that you have the capacity to do it.\nThen everything turns against you. You feel inexplicable physical pain even though your physician is completely sure that you are fine. You feel your heart pounding, your palms sweating, you can't stand your colleagues\/parents\/partner\/children, you start getting panic attacks. Later you stop being comfortable in your own skin, you crumble when someone finds any defect in you. Welcome to the club of the grumpy neurotics!\nNeuroticism is not something terrible per se. It is a side effect of the process of installing civilized values on our unconscious minds. Those values and restrictions help us fit into society and keep our reality check tuned. The question is not how to overcome the neurosis but rather how to integrate it so it will work for us not against us.\nNature has gifted us with the ability to form complex societies. And society has taught us how to transform our instincts into science, poetry, fine art. Tap that energy! Paint something, learn a new language, sing, dance. People have done it for ages (at least 40 000 years). Do something with that brain of yours! We are not here just to eat, drink and talk about politics or your job that you hate more and more each day. If you don't have any time \u2013 find some! It is not easy to recover your mental health once it gets compromised.\nTry multiple things and then keep doing the ones you like! It may be difficult to guess what you will actually end up enjoying.\nSublimate, because sincerely you have no other option. You have only one life to live and if you don't fill it with the things and activities you love and if you don't teach your children to do so, we are all (most probably) doomed.\nDr. Mila Petkova is an online counselor with a PhD in Psychoanalysis and Cultural Philosophy, creator of www.mipsy.net and www.milaswellness.com. She advocates for more accessible counseling and psychotherapy on a global scale. Her passion is to travel and her vocation is to make people see the best version of themselves and help them show it to the world.\nYou can follow Mila on Facebook.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BICSI strongly recommends that you should always have at least one Registered Communications Distribution Designer (RCDD) on your information and communications technology (ICT) systems project throughout the entire life of your project. Having a qualified professional RCDD on your project should protect your investment.\nIn today's business environment we find companies with varied employee resources. BICSI supports the issue that there should be RCDD involvement in the design and installation phases of all network installations. BICSI makes no distinction in quality of job performance when there is an RCDD on a company's staff or if an RCDD is contractually bound to a company for the duration of a specific project. BICSI supports the independent consultants with RCDD credentials as well as the company that chooses to have an RCDD as permanent staff.\nQuestions associated with bid documents that state, \"There must be an RCDD on staff\" should be dealt with at the source with clarifications satisfied during the prebid qualification stage.\nOnly Qualified Installer\/Contractor firms will be allowed to (bid\/propose\/quote) and provide services on this project.\nThe Qualified Installer\/Contractor firm engaged on this project shall have demonstrable design and installation training with certifications of competence. This includes, but is not limited to, having at least one BICSI Registered Communications Distribution Designer (RCDD) on staff or as a consultant engaged for the duration of the project. The RCDD shall be involved with this project from the start through its completion. If at any time during this project the Installer\/Contractor firm does not have an RCDD involved with this project, it shall be considered a default of the Contract.\nYou may also want to consider using similar requirements for BICSI Registered Technicians and Installers in your RFBs, RFPs and RFQs and Contracts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The organization of a birthday party involves many details. It is necessary to think about the decoration, the cake, what to serve, what to wear and, of course, what to offer your guests as a birthday souvenir.\nThe souvenirs are an indispensable part of the party, because they have the mission to perpetuate - even for a short time - the festive and joyful spirit of that special day. For this reason, the souvenirs should be thought with care.\nGenerally speaking, there is no rule for defining the best or ideal souvenir. The tip is to look for something that integrates with the theme of the party and that represents the personality of the birthday boy.\nThere are three types of birthday favors that can be used: edibles (pots, jellies, jams, preserves, brigadeiros, bonbons), functional ones (keychains, page markers, glasses, bath salts, lotion, soap) and decorative type candles (candles, picture frames, magnets, instant photos, vases of succulents).\nYour first task is to choose one of these three types. Try to base your decision on what comes closest to the personality of the party, the birthday party and the profile of the guests. This decision is important to help you narrow down the range of options and focus on what you really want.\nBut you do not have to do it alone. We will help you in this task. For this we selected videos tutorials with options of the three types of souvenirs and the best: you can do it yourself, saving good money. Then check out the inspiring images of birthday favors. Surely, you will finish reading this post with one more item checked on your list of preparations. Let's start?\nEveryone loves a craft made in EVA, but did you know that you can make beautiful souvenirs with the material too? That's right, you can take advantage of all the possibilities of colors and prints that the EVA offers to make a creative and different birthday souvenir. The tip in this video is to teach you how to make an EVA basket that can be used to put sweets and other treats. Follow the walkthrough in the following video.\nPicture 1 - Sacolinhas with bottles of chocolate; Great suggestion for adult birthday favors.\nPicture 2 - Amazing binocular made of paper.\nPicture 3 - Robozinho nice to accompany the guests back home.\nPicture 4 - Sorvetes! But these are not to eat, they are made of jute and wool pompoms.\nPicture 5 - Sleeping masks to ensure that the guests wake up recovered after the party.\nPicture 6 - Who does not love a candy pot?\nPicture 7 - What a great idea! Little girl play in the bag!\nPicture 8 - School kit for children to play with drawing and painting.\nPicture 9 - Popcorn in the cactus! Is or is not a cute idea and very cheap to do?\nImage 10 - Bullet Boards: There is no way to go wrong.\nImage 11 - Here, smiles carry colorful confections.\nImage 12 - Glass pot with globes; a simple idea that pleases everyone.\nImage 13 - Surprise basket with the unicorn theme.\nImage 14 - Souvenir inspired by the Mexican festivities.\nImage 15 - Bottles customized with the flamingo theme.\nImage 16 - The cloth bag carries a handwritten message by the birthday girl.\nPicture 17 - In the case of the Little Prince until a simple paper bag becomes something poetic and very special.\nPicture 18 - Put your hand in the dough - literally - and make cookies along with the little birthday boy.\nImage 19 - Straws and bullets.\nImage 20 - Hand-painted, different and very creative rattles, is not it?\nImage 21 - The jar of jelly beans has gained an extra touch with the flowers of peonies.\nImage 22 - Sweets packaged in the bag; the simplicity that always works; leave a message to give that increment in the souvenir.\nPicture 23 - Tiaras of flowers! The girls will love the suggestion.\nPicture 24 - Ice cream of souvenir? Only if it's cotton candy.\nPicture 25 - Balls! That simple.\nPicture 26 - Clipboard, sketches and drawings: what child does not like this combination?\nImage 28 - Bananas, but these here are a bit different.\nPicture 29 - Kit exploration for the mini scientists.\nPicture 30 - But for basketball fans you can opt for water bottles in the shape of the ball.\nPicture 31 - And what about those cute little crochet bees? Oh, they're still keychain.\nPicture 32 - Creative ideas of souvenirs is what is not lacking.\nPicture 33 - A good example are these handwraps tied with ribbons.\nImage 34 - Cones of popcorn, a tasty, easy and cheap option of souvenir.\nPicture 35 - Give pencils to the guests they wrote or draw.\nImage 36 - Here, Adam's ribs help decorate the little bags of souvenirs.\nPicture 37 - Socks are not just socks ... they can also be birthday favors.\nImage 38 - Oranges can also become a choice of souvenirs, have you thought about it?\nPicture 40 - Even whiteboard turns birthday souvenirs here, accompanied by colored chalk.\nPicture 41 - Little Monsters and Jelly Beans: a sweet and fun combination for the souvenir.\nImage 42 - Buttons and sequins to brighten the party.\nImage 43 - Homemade biscotti in the pot.\nPicture 44 - Baldinho with all that the birthday girl likes most.\nImage 45 - Carts also enter the list of options.\nPicture 46 - All the charm of paper bags decorated with balloons.\nPicture 47 - Melequentas masks to amuse the children after the party.\nPicture 48 - Sunglasses, a souvenir full of style for your son or daughter's party.\nImage 49 - If one is already good, imagine three options of cake to take home? The guests will love this souvenir.\nPicture 50 - These skates will lose their wheels in the blink of an eye.\nImage 51 - Adopt a Dinosaur!\nPicture 52 - Or how about a dream filter?\nImage 54 - Llamas and cacti marking presence in the birthday souvenirs.\nPicture 55 - Lego is always lay, that is, there is no one who does not love this toy.\nPicture 56 - Toy to travel; an idea of \u200b\u200bsouvenir to make children less bored during the rides.\nImage 57 - Sweet cargo in the cabin of that truck.\nImage 58 - Hot chocolate and souvenir cappuccino.\nImage 59 - Star Wars Direct for the birthday souvenir.\nImage 60 - Beach \/ Pool Balls saw a little souvenir on this anniversary.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sixth Form drama club began this month, offering drama enthusiasts the chance to improve their acting skills and perform in a professional production.\nAuditions were held at the start of February, which saw 20 hopefuls come to try out. The group are currently rehearsing their first production which is William Shakespeare's infamous tragic play Romeo and Juliet. Sixth Form student Tahmid Chowdhury, who last year wrote and directed CFBS' play Look Right Through Me, will again be directing. The club aims to have their first performance before the Easter break.\nDrama Club takes place every Tuesday in the drama studio (old LRC). If anyone would like to join or find out more please see Tahmid Chowdhury or leave a message with Ms Spurr.\nOn the 200th birthday of Charles Dickens a group of selected Year 9 CFBS students presented Dickensian style dramatic readings at Islington Town Hall. The boys received lots of positive feedback on their composed pieces of work, written with the flair of the iconic writer, and their performance as a group.\nTheir presentation was part of the Charles Dickens Bicentenary celebrations and delivered to an audience of 200 school children and teachers from all over Islington.\nCentral Foundation students joined more than 3,200 of their fellow 11-17 year olds across the borough to help elect Islington's first youth council last week. Official polling stations were set up in the school, giving the students their first real experience of voting.\nThe results were announced by Islington Council chief executive Lesley Seary at the Health Living Hub in White Lion Street, Islington the following day.\nWe are delighted to reveal that our Year 10 student Elliot Colley has been selected as one of the 8 councillors for the borough. Elliot will join the others in representing the views and aspirations of the borough's young people on Islington Council.\nYouth councillors will represent other young people in Islington and their opinions and will help influence decisions about issues that affect them.\nAnother CFBS student Abdiaziz Shukri , Year 12, stood as a candidate and although he was not elected we are sure he valued the irreplaceable experience gained whilst campaigning across the borough.\nCongratulations to Elliot, we look forward to seeing how he, along with his colleagues, will use their new roles to speak for the youth of Islington.\nLloyds Bank held a football tournament at the school on the 30th of January for our Year 9 students. The event was attended by scouts from Arsenal, Tottenham Hotspur, Fulham and Leyton Orient football clubs. It was an exciting afternoon, with the students displaying some excellent football skills.\nThe four teams played each other with a final which was won by Arinze Ossai, Edmund Antwi, Abdul Abdullahi, Yasin Ezrouki, Nuh Mire and Tiago Pontes Betu.\nThe winning team were taken to see Tottenham Hotspur against Liverpool in the Next Generation Cup, with the runners up being awarded their medals at a Leyton Orient match.\nThe tournament was organised by Lloyds Bank, arranged as part of the school's long-standing relationship with the national retail bank.\nOn the 9th of February Central Foundation won the Islington Young Consumer of the Year competition at the Emirates Stadium, beating teams from St Aloysius and Mount Carmel to retain the trophy which the school has won 4 times in the last 6 years.\nIn an electrifying final, the boys Ilias Belhaddad, Khalid Muruz, Malik Yansaneh, Govinda Deb and Mert Topuz, romped to victory by answering questions on law, current affairs and business.\nThe Young Consumer of the Year Challenge is organised by the Trading Standards Institute and supported nationally by a number of high profile sponsors. The local competition involves Islington schools competing against each other, testing the students on a wide range of issues.\nTelevision's Paul Whitehouse chaired the event and spent time with the excited winners afterwards. They now go through to represent Islington in the London Finals at the O2 Arena in March. Another team from Central Foundation made it to the semi-finals: Saeef Ahmed, Ridwan Chowdhury, Nahid Uddin and Tommy Collins.\nLloyds Bank has set up a literacy scheme which is currently taking place at the school.\nEvery Tuesday and Thursday morning 20 of their employees have been acting as literacy mentors to help some of our Year 9 students with their reading.\nThe scheme has been really successful in encouraging reluctant readers to engage with fiction outside the English classroom.\nSome of the boys' reading ages have increased by 6 months during this time.\nThe scheme will expand to include Year 7 students after half term.\nWe have a tradition of a very active student council. The council aims to improve the experience of students both inside and outside of school.\nThe School council is supported by members of the sixth form. We recently held the elections for the school council.\nThe first council meeting will take place this week before a whole school assembly on Tuesday 21st February after half term.\nThe coordinating team of sixth formers are Tahmid Chowdhury (Council Coordinator) Adrian Hracki (Deputy Council Coordinator) and Wasim Iqbal (Deputy Council Coordinator).\nThe school has a well-established relationship with the international law firm Slaughter and May. Our students have taken part in Customer Service role plays, their Lawyers in School project and received UCAS mentoring from their employees.\nCase Study: Central Foundation Boys' School.\nOur Year 10 Gifted and Talented group of students attended a presentation in January by the University of Cambridge on 'Applying to Cambridge'. They were joined by fellow Year 10 Gifted and Talented students from St Aloysius College, Highbury Grove and Highbury Fields School, these schools along with Central Foundation Boys' School form the Islington Consortium (IC6).\nThe aim of the session was to alter the students' initial perceptions of the university \u2013 that it is impossible for most people to gain entry to. The guest speaker himself studied at the university and so demonstrated how people from all walks of life could find themselves at Cambridge.\nHis time at the university offered a valuable insight into education at the school. He provided a lot of useful information about student accommodation, financial support, degree choices and the application process. It left a lot for the students to consider and at the end of the lecture it was clear that Cambridge University was no longer seen as too intimidating to consider applying to.\nThese informative sessions on higher education will continue when the same students visit Corpus Christi College, a constituent college of the University of Cambridge, this March.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A normal Home user with lease line requirement for browsing purpose.\nA SME\/SOHO user with dedicated Internet connection to fulfill his business commitments via Internet connectivity.\nA call center with Voice based solution demanding high level SLA for VOIP calling.\nA Data center with fluctuating volume of Internet on daily basis.\nPay per use & Flexi-billing offers.\nRevolutionizing broadband services by offering dedicated speed.\nNo contention ratio, just pure internet.\nEffective Quality management tools in place to enhance user experience.\nPack customization facility to users.\nOn demand short term pack upgradation.\nSingle user account with roaming feature.\nFTTH\/Fiber deployment for stable network.\nVoice, Video & Data compatible network.\nFTTH in high rise buildings and gated communities.\nFTTH in Hotels & Hospitals for faster connectivity.\nFTTH in educational institutes, corporate houses etc.\nUpgrading cable operator networks into FTTH so that end customer can enjoy world class triple play services.\nCustomized solution as per requirement and concept given by the client.\nWi-fi \/wireless solutions for all segments including but not limited to campus wi-fi\/Area wi-fi\/Hotspot etc.\nFiber connectivity solution for mission critical application environments.\nSecurity & Surveillance solutions linked with existing live network.\nHi-end surveillance for gated community and other vital establishments.\nHi-end parking solutions for elite clientele.\nHigh end intelligent security & surveillance solution for gated communities & societies.\nSecurity & surveillance solutions specially designed for Malls,Markets,Hospitals.\nRestricted access movement for sensitive areas - Implementation & monitoring.\nIntelligent Parking system for Heavy load areas such as Malls, Hospitals and Markets.\nSensor based house mapped parking system for individual houses and gated communities\/societies.\nTag based Patient Surveillance & movement tracker to keep a close eye for sensitive patients and areas.\nIntegration of multiple kind of surveillances & security systems on a single portal for better management and effective results.\nHMS is a large computerized database management system that process patient data in order bring a holistic patient care. The system is used by healthcare clinicians to access patient data and to plan, implement and evaluate care. (HMS) from Swift makes such an attempt and offers a world class solution, which fits into all kinds of hardware platform including a mobile phone. Our online hospital management software is a patient-centric mechanism that has exceptional capability of handling outpatients, inpatients, emergencies, day care and different other referred patient cases. We aim at offering fully configured web based healthcare solution enabled with mobile friendly features & characteristics. With the assistance of our healthcare intelligence system, hospitals, clinics, and other medical institutions can keep all processes under control and can make quick decision through mobile or hand set support.\nThe traditional administrative tool for managing student records,the student information system, has become the centrepiece of any college or university business-application system. Changes in campus culture, especially the wide use of the Internet have raised student the expectations of services provided by technology. At the same time, university leaders have become more accountable for the business aspects of campus life. As a result, the student information system has become an important management tool in the drive to become more effective and more competitive.\nThe Swift Campus Management System, designed to promote paperless university administration is unique and comprehensive software suitable for any university, colleges, schools or any other educational institutions. Its' host of features allows a user to maintain academic history of students with complete transparency in records and provides a medium for teachers, staff and administration to work in harmony. The application comprehends all the features relevant to the university or any educational institution and has a complete role based access where in various resources are made available depending upon the role assigned. The solution is running across multiple institutions of repute in India which is helping them to improve their operations and services to students.\nThe SWIFT Payroll Management System deals with the financial aspects of employee's salary, allowances, deductions, gross pay, net pay etc. and generation of pay-slips for a specific period.\nThe outstanding benefit of Payroll Management System is its easy implementation. Other advantages of Payroll Management System are its extensive features and reports.If your business has one or more employees; you should have a payroll system in place. An automatic payroll process helps you comply with legal and tax requirements and simplifies the process of paying your employees. Many employers outsource the payroll function to an outside vendor or use payroll system software instead of relying on manual processes.\nSwift Vehicle Management System (VMS) combines the use of automatic vehicle location in individual vehicles with software that collects these vehicle data for a comprehensive picture of vehicle locations. Modern vehicle tracking systems commonly use GPS or GLONASS technology for locating the vehicle, but other types of automatic vehicle location technology can also be used. Vehicle information can be viewed on electronic maps via the Internet or specialized software.\nThe Swift Builder Management software is a collection of programs, processes and information used to manage a construction project. By implementing project management software, users are able to standardize processes, gain insight into existing and potential problems, eliminate communication bottlenecks and increase the accuracy of time and cost estimates. Management and automation in building processes, which were generally previously charted by hand, allows contractors to reduce both resource and time requirements, meet deadlines and archive project records for reference on future projects.\nThe Swift Bandwidth management system is a tool that can change the rate and volume of traffic flows on your network in a controlled manner. It also can make decisions about what types of traffic or which users have priority when there is more traffic than there is bandwidth available, thus \"shaping\" traffic flows to sets of user-defined behaviors. In most cases, however, the upstream internet connection has less bandwidth then the LAN, which creates an uncontrolled bottleneck without a bandwidth management device. A bandwidth management device allows you to reduce and control the flows through your upstream bottleneck, based on criteria that you select. The goal may be to share available bandwidth equally among devices, or to allocate more bandwidth to users in higher service level classes by enforcing service level agreements, allowing them to get a higher quality of service (QoS).\nIT systems need to keep pace with the rapid changes in the sector. To counter the competition and meet business needs, information technology (IT) infrastructure needs to be automated, flexible and scalable. Unfortunately, traditional IT systems do not possess these qualities and thereby creating a challenging scenario for companies that want to stay competitive.\nWith Swift, today many enterprises are leveraging the power of software to modernize their IT infrastructure to run operations with tranquillity and automation by even restructuring their old systems.\nA data center is a centralized repository, either physical or virtual, for the storage, management, and dissemination of data and information organized around a particular body of knowledge or pertaining to a particular business. Every business house generates, collects, and processes large amounts of data. Recent technology advances with analytics, big data, and machine learning has made efficient usage of data more important than ever. A smart enterprise today needs to analyse and learn from data for competitive advantage.\nWith Swift Global, Clients can leverage new as well as historical data and create new insights to react to the changing market conditions with an assurance of safety.\nIP telephony is increasingly popular with large corporations and consumers alike. For many people, Internet Protocol (IP) is more than just a way to transport data, it's also a tool that simplifies and streamlines a wide range of business applications. Telephony is the most obvious example. VoIP-or voice over IP-is also the foundation for more advanced unified communications applications-including Web and video conferencing-that can transform the way you do business.\nSwift ensures a well managed, cost-effective high quality in VoIP Service and IP-based calling.\nComplete requirement assessment by experts.\nSolution drafting with commercials with a key factor in mind to keep TCO in check.\nPitching correct & required products only.\nOptimizing product lines wherever possible.\nCustomizing solutions suiting clients requirements.\nCopyright \u00a9 2016 Swift-Global. All Rights Reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmszl b/data_all_eng_slimpj/shuffled/split2/finalzzzbmszl
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+{"text":"The human body cannot naturally make the omega-3 fatty acids which are commonly found in fish. Forever Arctic Sea's blend of natural fish and calamari oil contains the essential fatty acids EPA \u2013 which contributes to the normal function of the heart \u2013 and DHA \u2013 which contributes to the maintenance of normal brain function. N.B. Contains fish (salmon, anchovy and cod).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AIMS:Vascular perfusion may be impaired in primary open-angle glaucoma (POAG); thus, we evaluated a panel of markers in vascular tone-regulating genes in relation to POAG. METHODS:We used Illumina 660W-Quad array genotype data and pooled P-values from 3108 POAG cases and 3430 controls from the combined National Eye Institute Glaucoma Human Genetics Collaboration consortium and Glaucoma Genes and Environment studies. Using information from previous literature and Kyoto Encyclopedia of Genes and Genomes (KEGG) pathways, we compiled single-nucleotide polymorphisms (SNPs) in 186 vascular tone-regulating genes. We used the 'Pathway Analysis by Randomization Incorporating Structure' analysis software, which performed 1000 permutations to compare the overall pathway and selected genes with comparable randomly generated pathways and genes in their association with POAG. RESULTS:The vascular tone pathway was not associated with POAG overall or POAG subtypes, defined by the type of visual field loss (early paracentral loss (n=224 cases) or only peripheral loss (n=993 cases)) (permuted P\u22650.20). In gene-based analyses, eight were associated with POAG overall at permuted P<0.001: PRKAA1, CAV1, ITPR3, EDNRB, GNB2, DNM2, HFE, and MYL9. Notably, six of these eight (the first six listed) code for factors involved in the endothelial nitric oxide synthase activity, and three of these six (CAV1, ITPR3, and EDNRB) were also associated with early paracentral loss at P<0.001, whereas none of the six genes reached P<0.001 for peripheral loss only. DISCUSSION:Although the assembled vascular tone SNP set was not associated with POAG, genes that code for local factors involved in setting vascular tone were associated with POAG.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This season, D.C. United have teamed up with Heineken and committed bar partners to provide official watch parties throughout the DMV area so fans can cheer on the Black-and-Red in unison whenever the team is on the road. In addition, all the bar partners are offering discounted Heineken drinks on match days.\nD.C. United would like to thank the local restaurants and bars that pledge their support to the Black-and-Red each season. To learn more about becoming an official D.C. United Neighborhood Bar Partner, please contact James Clark at jclark@dcunited.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"5.2. Lancaster, S. G., & Todd, P. A. (1988). Lisinopril. Drugs, 35(6), 646-669.\n5.3. Os, I., Bratland, B., Dahl\u00f6f, B., Gisholt, K., Syvertsen, J. O., & Tretli, S. (1994). Female preponderance for lisinopril-induced cough in hypertension. American journal of hypertension, 7(11), 1012-1015.\n5.4. Prinivil (lisinopril) [prescribing information]. Whitehouse Station, NJ: Merck and Co Inc; October 2018.\n5.5. Zestril (lisinopril) [prescribing information]. Wilmington, DE: AstraZenca Pharmaceuticals; July 2017.\n5.7. Olin, B. R., Hebel, S. K., & Dombek, C. E. (2007). Drug interaction facts. St Louis, Mo: Facts and Comparisons Inc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hi, Welcome to our Baby In Diaper Clipart category of ClipartLook Clipart. Browse through our great Baby In Diaper Clipart archive. In this section you will find great clipart to look.\nYou can look at cliparts for your web sites, project, art design or presentations.. Share and Use Baby In Diaper clipart and images from Clipartlook.\nBrowse our Baby In Diaper images and clip art collection.\nIts A Diaper Shower Free Images At Clker..\nBaby in diaper clipart clipart..\nBaby diaper clipart free clipart 2 image..\nBaby Boy Crawling clip art - Download fr..\nBaby Diaper Outline Clip Art..\nNewborn Baby Royalty Free Stock Images I..\nBaby Diaper Bag Clip Art Baby Diaper Cli..\nBaby in diaper clipart image..\nFree clip art diaper clipart. Vector Ill..\nBaby Clipart Free Download Clip Art On..\nClipartlook provides you with HQ Baby In Diaper Clipart, vectors, icons, and png. All of these Baby In Diaper Clipart resources are for download on Clipartlook.\nBrowse our Baby In Diaper Clipart collection, Free Clipart images Catalogue. Look at Baby In Diaper Clipart.\nBaby In Diaper Clipart images. Use these free Baby In Diaper Clipart for your personal projects or designs.\nAre you searching for Baby In Diaper images? Choose from HD Baby In Diaper.clipart graphic-images resources and download in the form of PNG, PSD or EPS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Bookkeeping helps you keep due dates current!\nBookkeeping keeps business on time! We do bookkeeping for small business.\nSpend time going over your returns with a CPA\/tax preparer....have your questions answered to minimize your taxes or maximize your profits! Includes a 50 minute conference. Call for an appointment. Including account set up and scheduling, office supplies, other expenses...this is a $275 value!\nIs your business in the \"red\" or \"black\"?\nWell regardless if you have a color printer or not, you know what we mean. We know you want to be profitable and yet pay the least amount of taxes you can legally.\nBookkeeping channels the information into reports which don't belong just being filed in a drawer! They tell a story about struggles, successes and failures...all are important to understand and see before it is too late!\nDo you know how to read them? If not, do you have a trusted adviser to call? Let's sit down and let us explain the lessons from the \"school of hard knocks\" that your enrolled in!\nIt's \"safe\" to deal with us!\nWe don't ship your information overseas to cut our costs while exposing your information to ID theft, etc. That's right, some firms take your bookkeeping and send it to places unknown for cheap labor.\nWe use local talent to provide you with services you can understand and appreciate. More costly? Not in the long run!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With its honey-hued, perfectly conserved villages and towns, surrounded by rolling hills and pastures, the Cotswolds is the largest designated Area of Outstanding Natural Beauty (AONB). It covers nearly 800 square miles across the six counties of Gloucestershire, Oxfordshire, Wiltshire, Warwickshire, Worcestershire and Somerset.\nBased in Bourton-on-the-Water, professional photographer Anthony Paul Inwood's knowledge and love for the Cotswolds shines through his photos. From dusk till dawn he can be seen paddling rivers, lying in the undergrowth and getting into whatever place will give him the best shot. Wherever he travels, his camera is sure to follow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lauren Holliday is the advertising and marketing director at Launch Academy, an immersive, 10-week programming bootcamp that transforms world class freshmen into junior-degree software program builders. I accomplished B.E. but i dnt have that much Knowledge to turn out to be a programmer\u2026however i'm really interested abt programming..due to economical issues i am unable to do any programming courses and all\u2026so can u please information me what and all works i have to do to become programmer by following books,web sites,guidence,works and help me out.\nProgramming is one of the most rewarding jobs you can find, as you are at all times studying one thing new and making machines do the things you want them to do. Start building your r\u00e9sum\u00e9 by writing your own applications with GUI interfaces, get a job as a technician in an organization which will need programming assist, and lease your self out within the Rent-A-Coder kind websites.\nFranklin's curriculum, which is created and reviewed in tandem with an advisory board comprised of Chief Technology Officers, Web Application Developers, and Graphic Designers, helps you purchase highly wanted expertise, including net utility development, front-end improvement, database growth, consumer expertise design, and graphic design.\nNaturally, the businesses will need at the very least a helpdesk that is provided 24\u00d77, monitoring to offer alerts before the prevalence of points and dealing with of all of the hardware \u2013 servers, laptops, desktops, printers \u2013 software program and so on. There are extra elements that goes into keep the IT infrastructure preserve going without any hassle and it's quite possible that each one organizations is not going to have the required expertise, time or the required abilities.\nThis hub intends give an overview of the principle and practice of creating a private skilled improvement plan each to be used by individuals in an organisation and for managers\/mentors to sufficiently information their employees in doing so. If you're a solo performer, you can nonetheless take cues and plans to create a PDP by following the steps in making a PDP.\nThough there are lots of other well-liked Android Development instruments available on the web as properly like Native Development Kit for purposes or extensions in C and C , Google App Inventor, which is a provides a visual GUI for app improvement and various cross platform mobile web purposes frameworks helps even learners to create nice apps.\nNo one can say 100% that after SEO web site will are available in top ranks in Search Engine Result Pages (SERP's). Conversion analytics is a department of Analytics concerned particularly with conversion-related info from natural and paid search engine traffic, such as the key phrases converts used of their queries, the kind of conversion that resulted, touchdown page paths, search engine used, and so forth. This means certain constraints: you must use these key phrases in some (probably not all) key places in your page to help serps determine your topic, but your main audience, your human readers, mustn't discover the keywords any more intrusive than the rhyme or meter of a poem.\nBy investigating search tendencies, you'll not only study which terms and phrases to focus on with web optimization, but you may additionally learn more about what your readers are trying to find. Literally being on top, or at least being on the primary few pages of a search engine, is advantageous because it sparks each curiosity and credibility. So, the meta description tag is essential in Google, Yahoo and Bing and every other engine listing \u2013 crucial to get it right.\nThe different factor you want to remember is that search engines like google and yahoo will likely by no means punish sites that have a stable phrase count. These are instruments of search engine optimization that help to create good backlinks which in flip assist to increase the page rating of the website. Links may be designed to be ignored by serps with the rel nofollow attribute.\nThe greatest way to profit from search is to comply with the ideas of fine search engine optimization which might be publicly obtainable and promoted by Google. Also generally known as search engine optimisation professionals or web optimization specialists, Internet marketing consultants use their knowledge of Search Engine Optimization Strategy to improve their shoppers' Position and Page Rank. Now it's comprehensible that as a enterprise proprietor you will not be properly versed with the tips of the trades of engines like google algorithms and issues like that. These automated search programs are collectively referred to as net spiders or internet crawlers.\nThe first is to describe the ordering technique of indexed websites being ranked by a search engine Algorithm in response to a Query. You might make a few grand this manner, however will constantly have to be looking out for search engine updates and give you new methods to dodge the foundations. The higher optimized the page is, the upper a ranking it should achieve in search engine outcome listings. The reason you need to do that is that Google LOVES websites that hyperlink to authority websites.\nAfter you've learn them, you MUST attempt our Search Engine Optimization Tools that will help you with Keyword Research, Link Building, Technical Optimization, Usability, Social Media Strategy and extra. The title of the press release typically becomes the title tag on PR distribution sites and the title tag is an important part of what helps a page rank for a search term. Search engines have recognized that web site homeowners and administrators can use this resource to manage their positioning and descriptions in search engine outcomes.\nBlog lists are troublesome to seek out, but there is a option to search for them, a technique to discover relevant, up-to-date blogs. Submit Express will submit and optimize you to Google and many different local directories, reminiscent of Yelp and Citysearch. In order to grow these native companies, we offer a talented crew that can design and build websites, optimize them to rank in engines like google and produce any type of video and animation needed to get extra prospects and develop a firm's backside line!\nGenerally talking, this implies you've gotten a minimum of two pages with enough link fairness to achieve the highest ten outcomes \u2013 two pages very related to the search term. Search engines are profitable provided that they supply a person hyperlinks to one of the best Web websites related to the user's search phrases. Balancing the constraints of the English language could be tough, however in the end, your search results can be better. You have a dedicated Search Marketing Specialist so whether or not you will have a query, or need an update, we're glad to listen to from you.\nSome search engines like google and yahoo will nonetheless present your site in regional primarily based search results in case your site is hosted inside that country or have your deal with and cellphone number within the web page textual content, but many directories are extraordinarily picky and can solely record regional domains. Spruce up your profile page and work on the hubs to make them searchable by the major search engines like google and yahoo. Using these methods can enhance your ranking on the various search engines and finally get extra guests to your web site. Search engine registration is a fundamental however necessary a part of Search Engine Optimization.\nIf you are after traffic from Google Images as nicely, the alt attribute can do the trick. Search engine firms punish site owners who use search engine optimization strategies that exploit the way in which the search engine works. Keyword wealthy content is what lets the search engine direct visitors to your website every time a search is made.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How old do you have to be to sell alcohol in convenince stores in Maine?\nHow do you make the alcoholic drink Gatorade? Is there a drink that starts with an x? What is the recipe for Outback Steakhouse Blackberry Sangria? What Alcoholic drinks start with letter K? What is a beverage that starts with p?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The idea of a minimum annual income for every Canadian could provide a more efficient alternative social safety net, but reform is an unlikely and unrealistic option, according to a new report from a conservative think tank.\nThe Fraser Institute report \"The Practical Challenges of Creating a Guaranteed Annual Income in Canada,\" released Tuesday, said the country could benefit from administrative savings and efficiencies by moving off the current multi-program, multi-jurisdictional social welfare system to a single unconditional cash transfer under the guaranteed income system.\nA guaranteed income system would see low-income Canadians receive automatic salary top-ups in order to bring their annual income up to a level above the poverty line. Supporters such as Senator Hugh Segal say such a program could go a long way to ending poverty in Canada.\nRead more: A Guaranteed Income Program Solved Poverty In A Manitoba Town, And Nearly Everyone Forgot..\nA guaranteed income program would be a single program operated at one level of government and therefore would be simpler and less costly to manage than the overlapping rules currently in place, the Fraser Institute's report found, since fewer resources would be required to oversee it. It could replace programs such as Old Age Security, Employment Insurance and various tax credit programs.\nBut the authors said the idea is too loosely formed and would require so much co-operation among various levels of government that it would not work.\n\"A GAI would face further challenges when it comes to replacing existing programs that serve specific purposes and\/or target certain groups (Old Age Security, the GST\/HST Credit, and programs for the disabled, to name a few),\" they said.\nImplementation of the program could be \"an administrative disaster\" they say, with one-time costs so large they overshadow the program's potential savings.\nTo illustrate the magnitude of reform to a guaranteed income system, the report's authors point out the existing system of social assistance is estimated to have cost $185.1 billion in 2013, roughly 10 per cent of Canada's economy or 22 per cent of Canadian governments' program spending.\nA Huffington Post Canada report on a town in Manitoba that experimented with a minimum income in the 1970s, an opinion piece in the Toronto Star from senator Art Eggleton endorsing the idea and other recent stories have renewed interest in the idea of a guaranteed annual income. Similar ideas are also being floated in Switzerland and Greece.\n\"Despite the renewed interest, a GAI reform is unlikely to become a reality in Canada,\" the Fraser Institute concludes.\nHowever, they say there may be alternative reforms that could improve the current system and achieve the same basic goal as guaranteed income, such as work-based subsidies that would encourage able-bodied Canadians to work rather than rely on the system long-term.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"It's all been rather dramatic here over the last couple of days, though as it turned out we picked an excellent part of Manhattan to live in and, as far as we were concerned, it was ultimately a bit of a non-event. Note to anyone considering moving here \u2013 check out the evacuation areas map first!\nSunday saw us out stocking up on non-perishable groceries, finding matches, candles and torches, filling up every available receptacle with water, charging laptop and phones, and baking cookies, so by the time we went to bed we were reasonably confident we were as ready as we could be.\nMonday dawned greyly, with light rain and occasional gusts of wind, but it wasn't until mid-afternoon that the wind really started to increase in strength. I filled up the bath and baked some pie, Tom got on with work from the kitchen table, and the day passed. Once darkness fell the lights flickered off-and-on several times, but we were amongst the lucky ones whose power stayed on.\nFollowing the developing situation over the radio and online it was hard to believe we were in the same city as the one featuring on many of the news reports. Power was pre-emptively cut in most of Manhattan below 34th street, and some amazing photos soon appeared to illustrate this. Images of the rising floodwaters were similarly dramatic.\nWe stayed up until it was clear that the worst of the storm had passed over, whilst listening to reports of hospitals being evacuated, subway tunnels flooding, news reporters behaving dangerously, and stupid people who refused to obey the evacuation orders needing to be rescued.\nLooking out of the windows this morning, there were a lot of fallen leaves along our street but nothing more serious. Walking around the neighbourhood we came across a number of fallen branches, several downed trees, a variety of damage to scaffolding and a handful of broken windows. The river was running very high, but back within its banks.\nQuite a number of other people were out too, walking their dogs, looking at the damage, even jogging. Sheltering from a passing rainshower beneath the awning in front of a post block of apartments, we were amused to see a resident returning with the spoils of her outing \u2013 two takeaway coffees and a baguette. She was by no means the only one venturing out for breakfast though, and we passed several diners all doing a roaring trade.\nContinued high winds, the rain and chilly temperatures drove us home again, and whilst I'd quite like to take a walk downtown I think it's probably best to stay in for now. The weather is improving all the time, but warnings are still being broadcast about risks from falling branches etc, and there are still coastal flooding alerts in place. Besides which, it sounds as though it will take quite some time before the subway is back to normal, so we'll probably be doing more walking than usual in the days to come.\nI still can't believe just how lightly we got off, especially when so many other people were properly suffering just a couple of miles away. Of course there's a perverse part of me which wishes we'd had just a little bit more excitement, or at least been able to see more of it for ourselves, but really I'm just glad we stayed safe and warm.\nThis entry was posted in Living and tagged Hurricane Sandy, storms, weather by Emma Davidson. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Across campus, students are noticing the changes that have been made to the student center. Last year, the cafe area was packed with snacks, toothbrushes, tampons, cold medicine and other convenient items that are necessary to a college student's survival. The student center cafe area is no longer set up in its original convenience store style; it is now a cafe solely focused on serving food to students.\nThe student center has removed items that are essential for students who are not able to leave campus to purchase them. Having cold medicine and other health items in such an accessible place is beneficial for students who lack the means to travel elsewhere for their medication.\nAlthough removing convenience items opens up more space and allows the focus of the cafe to be primarily on food service, students are feeling the consequences of this change. Perhaps most importantly, where have all the tampons gone?\n\"The changes that have been made to the student center are unfortunate. It's not convenient for those who don't want to leave campus to buy their own supplies,\" Brianna Locklear, a first year at Salem College.\nIf the issue is concern over people at a women's college seeing tampons, the simple solution is either for Salem to offer household items and toiletries at a different location on campus or to put them in a more secluded area of the student center. Students require supplies other than food to carry out their daily lives. Why should Salem no longer offer the supplies it provided in the past?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"They are solar activists, and advocates for making clean affordable energy accessible to all. They are members of the PowerDC.org coalition, co-founded the Grid 2.0 Working Group grid2.0.org and DC Solar United Neighborhoods (DC SUN) dcsun.org and recentlyre-formed the DC Consumer Utility Board.\nMs. Berger worked for many years for the Smithsonian Institution and at the Carnegie Institute for Science and Mr. Robinson joined the Marion Barry for Mayor Campaign in 1977 and worked for the Office of the Mayor and later the Council of the District of Columbia for many years.\nFor a list of archived newsletters please click here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hello gorgeousness! So, Chloe from Hey, Babe Skin reached out to me about over a month ago and asked if I would like to give some thoughts on their recently launched pre-makeup oil. I honestly feel intrigued about this little thing. I feel the need to try it out myself. I occasionally incorporated facial oils in my skincare routine and a lot of them were mediocre and 'myeh' feeling.\nI posted a couple of sneak peek on Instagram (follow me if you haven't already) when I first got the pre-makeup oil. I am a fan of its packaging and the scent was all over the box. I was quite excited, you would know if you're following me lol. I will have it on the Highlights if you haven't caught up with it. That is if I manage to not be lazy..\nIt's an facial oil (duh?) that help reduces oily complexion with makeup. It acts like a primer if that doesn't ring your bell. I know what you're thinking. How is it possible? An oil that will help our complexion to be oil-free? That's almost impossible to take in but it is what it is. At Hey, Babe Skin, they target problems like oily\/ dry\/ combination skin with makeup + skincare. It's a very versatile and flexible product where you can use it as part of your makeup or skincare routine.\nWhat Does It Do and What It Claims?\nThe pre-makeup oil is housed in a small vial with a dropper to dispense the product. It's pretty cute with all the labelling. It's pretty straightforward and see-through packaging, which I like as I am able to see the actual product. On top of that, the overall packaging is cute and quite hygienic as it avoids direct contact between the applicator with our fingers and skin.\n1. Right off the bat, the ingredients are minimal and I can see that jojoba oil (or goat nut oil) is their main ingredient which doesn't trigger anything negative on my skin so that's great. It's also high in Vitamin E, EU allergen-free, and fungal acne (malassezia) safe.\n2. I would consider this as magic in a bottle because the star ingredient itself which is the goat nut oil that regulates sebum production, has anti-inflammatory properties and seals the skin with long lasting moisture.\n3. 86% of the ingredients are listed as \"low risk\" according to the EWG Health Rating standard. That's quite high and you know most of the ingredients won't upset your skin. My skin reacts almost immediately if she hates something but I guess this product is pretty mild that ruining my skin is not on its to-do list. Also, I would say this pre-makeup oil is hugely consists of emollients as it is so moisturizing, you can see it for yourself on SkinCarisma.\nThey suggest to mix the pre-makeup oil with your foundation but since I am going foundation-free as of recently, this method is not ideal for me anymore. BUT! I have tried to mix 2 drops of the pre-makeup oil with my favorite foundation just for fun and I kinda like the finish. The mixture is flawless and looked super healthy and glowy!\nI can see myself using this pre-makeup oil for tons if its skincare benefit rather than makeup but both methods work just as best in my opinion. As far as skincare goes, they suggest to use 1 drop and massage all over skin before applying moisturizer. Well, that works too but have you tried to mix it with your moisturizer? Like use both on the same time? Man, it's magical. I can stare at my glowy skin for hours! Really nice if you're going for the matte makeup look after. You don't look matte-er and flat. So I really recommend you to try my personal favorite method too.\nThis pre-makeup oil is meant for babes like me. I am usually an oil-slick but every time I mixed the pre-makeup oil along with my gel moisturizer, my skin never felt so good. Now, the pre-makeup oil is not meant to get rid of your sebum hundred percent but I can definitely see less oil on my face. Difference is a difference. That's definitely a positive progress, no?\nI also think that my makeup lasts longer and still looks fresh after 6 hours when I use this pre-makeup oil! Usually even in an air-conditioned room, after 2 hours I will catch myself as an oil-slick already and I am not cute anymore but if I use this oil, my face will only start to look a bit shiny at the 4-5 hour mark. That's pretty fair considering how freakin' oily my skin is.\nFor those days where I need to run errands and not being in an air-conditioned room and brave the Malaysian sun, my matte base makeup is not matte anymore. That's fair enough. Nothing can survive the Malaysian sun nowadays and Hey, Babe Skin never claims to produce a product that will make your makeup survive the Malaysian sun.\nAs far as the product itself, it absorbs so fast! You don't have to worry about the oil is sitting on top of your skin because it's not going to do that. Comfortability is one of the thing I look forward to when it comes to skincare. Who would want to wear something if it gives you itchiness and overall just unpleasant experience? Not me.\nBut I guess, in my humble opinion, doesn't matter if you use it for the sake of makeup or skincare or both, it's going to work. Period. Now, let's go through the claims and let's just see from there.\nLonger lasting makeup - Super checked!\nTame pores, soothes and calms skin - Very checked!\nSo far, so good. This pre-makeup oil almost lives up to all of its claims! I like that. Who doesn't like brands that really deliver? I think Hey, Babe Skin is that one potential brand to grow bigger and successful-er. They really did their research and not playing around while creating this oil. I don't mind spending RM 49 for a product that really works. I really am dependant on this product now to achieve healthy and glowy looking skin. I am OBSESSED! Please check out their website and if you're planning to get the pre-makeup oil, I would say go for it!\nHave you heard of Hey, Babe Skin before? Would you try it?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"amazing custom made dining room tables for table protector custom made table pads put on dining room table 23 custom dining table atlanta ga.\namazing custom made dining room tables for directions 26 custom dining room tables.\ncustom made dining room tables or rustic industrial metal leg custom made table farmhouse dining table wooden whale workshop 59 custom dining room tables michigan.\ncustom made dining room tables for walnut slab custom made dining room table 24 custom dining room tables connecticut.\nbest of custom made dining room tables or custom made dining tables custom live edge dining room table and custom built dining room tables house interiors 61 custom dining room tables.\nidea custom made dining room tables and buy a hand crafted fancy x farmhouse table with extensions extending dining table farmhouse dining table made to order from the urban reclaimed co 84 custom din.\ninspirational custom made dining room tables or shaker dining table tables style room on shaker style dining room furniture 45 custom made dining room tables massachusetts.\nbest of custom made dining room tables or walnut crotch slab custom made dining room table 78 custom dining room tables connecticut.\ninspirational custom made dining room tables or tom made dinning table redwood slab dining room tables picnic near me 22 custom made dining room tables massachusetts.\ncustom made dining room tables or explore our large collection of custom made rustic dining tables including solid wood and reclaimed wood dining room table with square round 88 custom dining table a.\nlovely custom made dining room tables and black walnut live edge dining tables and slab tops handcrafted in east and 41 custom dining room tables michigan.\nbest of custom made dining room tables and custom farmhouse dining table custom made dining room tables modern home 15 custom dining room tables michigan.\nfresh custom made dining room tables and custom made dining room tables effective furniture top design custom made dining room tables minimalist 66 custom dining room tables toronto.\nidea custom made dining room tables and new fashion modern furniture private custom made wood glass top metal dining table fashion 24 custom dining room tables denver.\nlovely custom made dining room tables or custom dining tables custom designed dining table contemporary dining 81 custom dining room tables denver.\nelegant custom made dining room tables for custom made dining room table 21 custom dining room tables connecticut.\ncustom made dining room tables or custom dining table custom made custom dining table 16 custom dining room tables seattle.\nawesome custom made dining room tables or rustic wood dining room tables solid wood extendable dining tables rustic wood kitchen tables solid wood 68 custom dining room tables reclaimed wood.\nbeautiful custom made dining room tables or 59 custom dining room tables michigan.\nRelated Posts of \"Custom Made Dining Room Tables\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzboegt b/data_all_eng_slimpj/shuffled/split2/finalzzzboegt
new file mode 100644
index 0000000000000000000000000000000000000000..1b2f1ffe21aa012cc8142faf501172a9341d0b9d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzboegt
@@ -0,0 +1,5 @@
+{"text":"Going through my favorite blogs, I came across this charming house in Colombes (92).\nThe owner has had the good idea to hunt his furniture to retapisser armchairs. A guarantee for those who dream of having an interior that looks like no other.\nFor those who wonder where are the tapestries, the owner there said on the blog Deco planet it comes tapestries Sanderson home. As for the beautiful movie posters, they were mottled on flea markets.\nCongratulations to the owner of this house was decorated tastefully.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are a family run company dedicated to making our customers' facilities shine by providing outstanding laundry services and hygiene products.\nIndoor Billboard distributes a full line of restroom products such as paper towels, soap, tissue, air fresheners, and cleaning chemicals as you need them with no order minimums! When you become an Indoor Billboard customer, we will install and maintain new high quality dispensers for these products at no cost to you! Furthermore, we offer multiple ordering options including calling to assess your inventory needs on a regular basis or secure 24 hour online web based ordering.\nFrom protective garments to identity apparel, Indoor Billboard has the ability to outfit your employees in high quality brands like Red Cap, Bulwark, Nike, Port Authority, and more! Custom embroidery or screen printing is available on everything from shirts and jackets to caps and golf towels. Sizes range from junior to tall and everything in between!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WSU soccer midfielder Mackenzie Peck is this week's Signpost Student-Athlete of the Week after scoring the 13th goal of her career in her final game with the Wildcats.\nWith 13 career goals, Peck is number 10 on the Wildcats' career goals list.\nPeck scored early in the second half of the Wildcats' 2-1 loss to Idaho State last week off a pass from teammate Stacy Bair. The goal was Peck's fifth of the season, tying her with Bair for the team lead in goals.\nPeck finishes her career having played in 66 games, recording 13 goals, two assists and 28 points. Peck said that she plans on graduating next year with a degree in zoology. Despite her soccer career at Weber State being over, Peck said that she's not done playing sports yet.\nWSU soccer senior Mackenzie Peck (left) plays in the Wildcats' season-opening loss to Westminster College.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pastoral Relatedness is that special quality of the pastoral relationship that helps a person access his or her deeper spiritual self and experience God's care. In Pastoral Relatedness John Quinlan shows that pastoral relatedness is the essence of pastoral care. Quinlan outlines research into the aspects of pastoral care that contribute most to holistic care, and most importantly, to the why? and how? these particular aspects of pastoral care contribute. The person and the skills of the pastor, which enable pastoral relatedness, help a patient to freely share his or her private concerns and have them heard. This helps the individual to cope with his or her situation, and facilitates supportive ministry. Pastoral care makes a significant and measurable contribution to holistic healthcare and healing, contributing to patients' getting better faster, to their readiness to go home, and to making hospitalization easier.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":" Forms >>> Validation Credit Card.\nForms >>> Validation Credit Card.\nDetermines if a credit card number is valid before the form is submitted by utilizing the LUHN formula. Very simple, and does not require knowledge of the credit card type. Does not check if the date is valid. Ver useful for online order form validation.\nfunction called when users click the \"check\" button.\n\/\/ to something like \"(new CardType())...\".\n\/\/ cardtype it belongs to.\ncardtype : type of card, eg: MasterCard, Visa, etc.\nrules : rules of the cardnumber, eg: \"4\", \"6011\", \"34,37\".\nlen : valid length of cardnumber, eg: \"16,19\", \"13,16\".\nyear : year of expiry date.\nmonth : month of expiry date.\nthis.setCardNumber = setCardNumber; \/\/ set CardNumber method.\nthis.setCardType = setCardType; \/\/ setCardType method.\nthis.setLen = setLen; \/\/ setLen method.\nthis.setRules = setRules; \/\/ setRules method.\nthis.setExpiryDate = setExpiryDate; \/\/ setExpiryDate method.\nthis.checkCardNumber = checkCardNumber; \/\/ checkCardNumber method.\nthis.getExpiryDate = getExpiryDate; \/\/ getExpiryDate method.\nthis.getCardType = getCardType; \/\/ getCardType method.\nthis.isCardNumber = isCardNumber; \/\/ isCardNumber method.\nthis.isExpiryDate = isExpiryDate; \/\/ isExpiryDate method.\nthis.luhnCheck = luhnCheck;\/\/ luhnCheck method.\nreturn true if CardNumber pass the luhn check else return false.\nreturn the array object in the size specified.\n\/\/ Create the len array.\n\/\/ Create the rules array.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpbay b/data_all_eng_slimpj/shuffled/split2/finalzzzbpbay
new file mode 100644
index 0000000000000000000000000000000000000000..3bc2c3a24d2da37baa90450957e77a279f643012
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbpbay
@@ -0,0 +1,5 @@
+{"text":"How Much Does Surveillance Cost By A Private Investigators in Ellesmere Port?\nThe quantity of brokers, hrs as well as times which are needed modify the Monitoring expenses and costs from Private Investigators Ellesmere Port. Because of the necessity of professional gear as well as automobiles, the actual monitoring price may also alter.\nFrom Private Investigators Ellesmere Port, the actual monitoring solutions that people execute begin with less than \u00a345 an hour or so along with a created monitoring statement is roofed inside it. The expert monitoring private detectives within Ellesmere Port can provide a person inexpensive options an acceptable price. The amount of times of monitoring you need modify the Private Investigators Ellesmere Port expenses, actually, the expense are in fact decreased in instances where a number of days associated with monitoring are needed.\nThe private eye company situated in Ellesmere Port requests a 1 period before prepared established price since the preparing, basic duties, in addition to supplying reviews. For any effective summary, the quantity of hrs and charges necessary for Private Investigators Ellesmere Port monitoring depends upon the complexness from the monitoring analysis. Expenses will be different because of the subsequent elements: quantity of monitoring automobiles as well as usage protected as well as monitoring effort.\nWhen you've got any discreet and also unobtrusive assessment with Private Investigators Ellesmere Port, our own specialist private eyes will give you by way of a treatment the location where the specifications to your security inspections including variety of employees, further assets necessary as well as the charges concerned.\nThe stand apart one of the most essential areas of monitoring through Private Investigators Ellesmere Port may be the monitoring statement.\nMonitoring reviews through Private Investigators Ellesmere Port are thought in to the private investigator expenses, totally free speculate among your own meeting as well as depending on your own requirements. The actual evidence could be presented to a person in a variety of amounts of structure.\nThe particular security real estate agents regarding Private Investigators Ellesmere Port will certainly take note of every one of the info accumulated over our own security tasks.\nPrivate Investigators Ellesmere Port provides sensible monitoring options such as each basic as well as anonymement.\nThe quantity of real estate agents implemented in a security study has an effect on the expenses considering that the a lot more security real estate agents needed to perform 24 \/ 7 statement in a period of months obviously raises the invoice.\nVarious monitoring research will need various things as well as Personal Investigator Ellesmere Port expenses and costs will be different appropriately.\nAffordable email address particulars are for your use regarding walking someone and also overseeing routines from your picked web site and also minute rates are relying on the quantity of guys and also automobiles regarding security needed.\nThe effective use of development from Private Investigators Ellesmere Port may update the monitoring procedure and perhaps in an incredible by the hour price.\nPrivate Investigators Ellesmere Port provides appreciated good success together with lower charges in their inspections completed in Ellesmere Port with the use of security products including tuning in gadgets, invisible video cameras along with GPS following gadgets.\nSupplying monitoring verification via video clip from excellent expenses and costs is one thing which Private Investigators Ellesmere Port finishes constantly.\nThe kind of format you'll need, electronic Sdcard or even analogue mp3 choice and also the volume of digital camera perspectives asked for associated with Private Investigators Ellesmere Port modify the costs with this support.\nPrivate Investigators Ellesmere Port supplies the range of inexpensive security movie enhancing simply by all of us regarding complex associates.\nA reverse phone lookup will be affordable and also signifies you'll not must invest some time and also anxiety above enhancing the particular security video clip all on your own and also Private Investigators Ellesmere Port lets you require no matter which area of the movie an individual specifically want to have.\nThe particular real estate agents regarding Private Investigators Ellesmere Port are usually active in the course of security to be able to perform the security jobs, hence, they aren't constantly accessible to speak together with you.\nPrivate detectives United kingdom can present you with an individual reason for get in touch with from Private Investigators Ellesmere Port that'll be open to talk straight along with you in a minimum expense on your monitoring procedure.\nThe numerous many years of understanding carrying out economical monitoring within hectic cities and towns, for example Ellesmere Port, offers provided all of us understanding of what you should expect.\nPersonal Investigator Ellesmere Port help and provide tips to the customers for making a choice around the quantity of monitoring investigators are essential within an energetic environment for any economical monitoring process.\nFrom Personal Investigator Ellesmere Port, we all know which meticulous planning results in a inexpensive price points monitoring analysis.\nAs soon as you're associated with Private Investigators Ellesmere Port we shall start arranging a fantastic economical monitoring that's a high quality support.\nHow Much Do Matrimonial Investigations By Private Investigators Ellesmere Port Cost in Ellesmere Port?\nHow Much Does Vehicle Tracking Cost By Private Investigators Ellesmere Port in Ellesmere Port?\nHow Much Does A Lie Detector Test By Private Investigators Ellesmere Port Cost in Ellesmere Port?\nHow Much Does Debtor Tracing By Private Investigators Ellesmere Port Cost in Ellesmere Port?\nHow Much Does Phone Forensics By Private Investigators Ellesmere Port Cost in Ellesmere Port?\nHow Much Does Nanny Monitoring By Private Investigators Ellesmere Port Cost in Ellesmere Port?\nHow Much Do Email Scam Investigations By Private Investigators Ellesmere Port Cost in Ellesmere Port?\nHow Much Does PC Forensics Cost By Private Investigators Ellesmere Port in Ellesmere Port?\nHow Much Does Close Protection Cost By Private Investigators Ellesmere Port in Ellesmere Port?\nHow Much Do Fraud Investigations Cost By Private Investigators Ellesmere Port in Ellesmere Port?\nHow Much Does A Corporate Investigation Cost By Private Investigators Ellesmere Port in Ellesmere Port?\nHow Much Does Process Serving By Private Investigators Ellesmere Port Cost in Ellesmere Port?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"David Clark Mic Cover. Improves communications by reducing wind and ambient cockpit noise while transmitting.\nFits M-1\/DC and M-87 Microphones. For use on headsets such as the H10-30, H10-36, H10-66, and H10-76. For most other David Clark Headset, you should get the David Clark M-7A Microphone Cover.\nReview: I had trouble with the my H10-66 headset keying the VOX in a Cessna 172. It was so bad that I had to keep my mic off the entire flight. After quickly installing this windscreen, I had perfect VOX response and absolutely no unwanted voice keying. Works perfectly as advertised and like all David Clark products, well made.\nQ: Is this the right cover for the h10-13.4?\nA: No, you would want to get the Mic Cover for David Clark Headsets instead.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Row after VVPAT demo, parties demand scrapping use of EVMs Posted by EenaduIndia On April 01, 2017 0 Comment New Delhi: A major row erupted on Saturday over a video of VVPAT demonstration by a poll official in Madhya Pradesh where the machine attached to an EVM allegedly dispensed slips of only BJP as the opposition knocked at the EC's doors seeking scrapping of EVM use in upcoming polls.\nInderpuri JJ Colony, Delhi and was also preparing for the IAS exams.\nMulayam Singh Yadav in becoming the Chief Minister of Uttar Pradesh.\nParty and became the Chief Minister of Uttar Pradesh on 3rd June 1995.\nFebruary she was re elected for the Uttar Pradesh legislative assembly.\nand 25 IPS officers were suspended for their involvement in corruption.\nHeritage Corridor, however in June 2007, the then Governor T.V.\nthe successor will be declared after her death.\nlet me confess \u2013 EVM can be Rigged Effortlessly.\nItself Can Be Changed Like Pen & Restored at the end'.\nINDEPENDENTS & FIRST TIME CANDIDATES.\nMethod was 'Booth Capturing at Gun Point'.\nVoter IDs but this was not available two decades ago.\nPeople would take Bank Loans and disappear.\nRegistration of 'Poll Manifesto' 60-90 days earlier than polling day.\nby 35 days after polls'.\nIs blockchain going to be introduced in voting and election?\nvoting terms, both in corporate and politics, seems quite relevant.\nBeing the continuation of the Blockchain Month at yourtrainingedge.com, this article discusses the blockchain implementation and use in this regard.\nfocus of public's arguments, once again.\nsince usually EVMs utilize proprietary software.\nverify the accuracy of the machine count in close races.\nand lost votes and erroneous tallies, as a worst scenario.\nhas recently employed BTC services on at least two occasions.\nthe adoption of the technology.\navailable to people who start a business online in the nation.\nNasdaq, using blockchain technology, is hoping to change that.\nthe industry, we can say.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \u00bb Dining Room \u00bb Dining Room Ideas For Minimalist Home \u00bb Currently viewing..\nAll the people who are recently created their living place will surely think maturely about dining room ideas. A dining room is one of the most special rooms that are needed to be finished well. It is a room where you need to use almost every day. You will sit there when it is the time of breakfast, lunch or much other eating session such as dinner. So many type of decorating ideas for the dining room are also available.\nTags: dining room wall decor ideas, dining room painting ideas, dining room window treatment ideas, dining room table centerpiece ideas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Papoose drops visual for track Underrated where he goes in on how real Hip Hop is overlooked in the new industry of gimmicks & water downed bars.\nWatch the official music video for \"Underrated\" by Papoose. Follow Papoose https:\/\/www.instagram.com\/papoosepapoose\/ https:\/\/twitter.com\/papooseonline\/ Directed by Shadow SUBSCRIBE for more: http:\/\/bit.ly\/subWSHH More WorldstarHipHop: http:\/\/worldstarhiphop.com https:\/\/twitter.com\/worldstar https:\/\/facebook.com\/worldstarhiphop http:\/\/instagram.com\/worldstar Shop WorldStar: http:\/\/shop.worldstarhiphop.com WorldstarHipHop is home to everything entertainment & hip hop.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpdgd b/data_all_eng_slimpj/shuffled/split2/finalzzzbpdgd
new file mode 100644
index 0000000000000000000000000000000000000000..675192ebdb10ff207daaa4fa4c1b6da37188912e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbpdgd
@@ -0,0 +1,5 @@
+{"text":"View of Guyhirn Old Church in Cambridgeshire, near Wisbech. This plain, rectangular chapel was built in 1660, and was intended to be a place of Puritan worship. However after the Restoration it was adopted by the Church of England as a Chapel of Ease. It was not extended or significantly altered and retains its simple architecture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"hunt72.ru is 9 years, 331 days old.hunt72.ruis ranked #855442 in the world this means that hunt72.ruis not a very good website and you should optimize your site for search engines . The average home page load time is 246 milliseconds, it is really fast. This site has a low Pagerank(0\/10). IP address is 185.17.122.18, and its server is hosted at . Last updated on 17.04.2015.\nWe detected this website has 1 keywords and This length of keyword characters is really good and will help your site to go top on search engines.\nThis website has -1 error(s) and -1 warning(s). This is perfect that your site doesn't have any errors.\nMost people tend to type simple phrases on search engines instead of typing url on the address bar to access any website. Unfortunately we couldn't get search query data for this website yet and continuing our researches.\nhunt72.ru domain expires on 8.10.2015. If you are interested in hunt72.ru, you can contact the owner REGRU-RU or wait for -330 days.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Performs the daily care of the research vivarium, which includes cleaning cages and racks. Coordinates with vendors and supervisors on operational, administrative and technical responsibilities. May assist with some procedures and care as directed. Responsible for overseeing procurement of rodents and supplies, preventive maintenance of facility equipment, daily rounds and observation. Follows standard operating procedures and maintains records to comply with regulatory requirements.\nPerforms routine and some non-routine procedures.\n\u2022Makes observations, analyzes relevant data and interprets results.\n\u2022Provide general, daily laboratory husbandry and care.\n\u2022May perform special projects when assigned.\n\u2022Maintains compliance to company Environmental Health and Safety policies, procedures and practices.\n\u2022Requires a minimum of a High School diploma. An Associates degree is a plus. Equivalent experience may be accepted.\n\u2022A minimum of 4 years working as vivarium technician is required.\n\u2022Familiarity with care of immunocompromised rodents desirable.\n\u2022Strong attention to detail regarding documentation is a must.\n\u2022Good communication skills both oral and written are a must.\n\u2022Must be able to follow Standard Operating Procedures.\n\u2022Good computer skills are required.\n\u2022Excellent organizational skills are required.\n\u2022Must have ability to multi-task.\n\u2022Willingness to work flexible schedule including some holidays & weekends.\n\u2022Consideration given to AALAS certification.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the more surprising things for new parents is the sheer amount of stuff you need to care for a baby. There's clothes, toys, car seat, crib, stroller, night light, baby monitor, bottles, nursing pillow, onesies and on and on. But perhaps the most important items of all are diapers, because those, you never want to run out of.\nThe trouble is, most people don't get excited about buying diapers as a baby shower gift. They need to be asked, encouraged, perhaps even bribed. This is what diaper raffles are all about.\nThe concept is simple: Guests are asked to bring a package of diapers to the shower, often in lieu of a card. In return, they get a raffle ticket that enters them into a drawing. The prize is usually a gift card or something with sentimental or emotional value.\nIf you're planning a baby shower diaper raffle, the challenge is explaining the rules succinctly. After all, the instructions typically need to fit in the invitation text or a small insert sent out with the invites. To give you some ideas, here are some diaper raffle wording samples using poems and snappy sayings.\nHere's some straightforward diaper raffle wording that's ideal for adding to the text of a shower invite.\nP.S. We're having a diaper raffle! To enter, please bring one package of diapers (any size) in lieu of a card. In return, you'll get a chance to win a semi-fabulous prize package!\nOptional Diaper Raffle: To participate, please bring one small package of diapers for a chance to win a prize.\nFor a chance to win a special prize!\nDiaper Raffle: For each package of diapers you bring, you will receive one entry in our prize drawing. First prize is getting the baby named after you. Just kidding \u2013 it's a gift card!\n[Ticket Insert] Bring this ticket and a package of diapers (any size) to be entered into the Diaper Raffle.\nHere are a few longer poems that might be more appropriate for an insert, depending on how much space you have.\nIf the diaper raffle is part of a baby sprinkle, you might also want to check out these wording ideas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"> > this a big task, but having a standard ADK is what we need.\n> > ok, back to my web server debugging..\n> where are all the \"I\" people to provide for those needs?\nWhat we need is somebody like Linus Torvalds to head the thing up.\nhaven't read \"The Cathedral Verses the Bazaar\", it's worth reading.","meta":{"redpajama_set_name":"RedPajamaC4"}}